Instrumentation & Control

Integral (Reset) Windup, Jacketing Logic and the Velocity PI Form

2011-03-22T13:40:00.001-07:00

A valve cannot open more than all the way. A pump cannot go slower than stopped. Yet an improperly programmed control algorithm can issue such commands.

Herein lies the problem of integral windup (also referred to as reset windup or integral saturation). It is a problem that has been around for decades and was solved long ago. We discuss why it occurs and how to prevent it to help those who choose to write their own control algorithm.

The PI Algorithm
To increase our comfort level with the idea that different vendors cast the same PI algorithm in different forms, we choose the independent, continuous, position PI form for this discussion:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = proportional gain, a tuning parameter
Ki = integral gain, a tuning parameter

Note that Kc is the same parameter in both the dependent and independent forms, though it is more typically called controller gain in the dependent form.

Every procedure and observation we have previously discussed about PI controllers applies to both forms. Both even use the same tuning correlations. To tune Ki, we compute Kc and Ti for the dependent form and then divide (Ki = Kc/Ti).

Integral (Reset) Windup
Our previous discussion of integral action noted that integration is a continual summing. As shown below, integration of error means that we continually sum controller error up to the present time.

The integral sum starts accumulating when the controller is first put in automatic and continues to change as long as controller error exists.
If an error is large enough and/or persists long enough, it is mathematically possible for the integral term to grow very large (either positive or negative):

This large integral, when combined with the other terms in the equation, can produce a CO value that causes the final control element (FCE) to saturate. That is, the CO drives the FCE (e.g. valve, pump, compressor) to its physical limit of fully open/on/maximum or fully closed/off/minimum.

And if this extreme value is still not sufficient to eliminate the error, the simple mathematics of the controller algorithm, if not jacketed with protective logic, permits the integral term to continue growing.

If the integral term grows unchecked, the equation above can command the valve, pump or compressor to move to 110%, then 120% and more. Clearly, however, when an an FCE reaches its full 100% value, these last commands have no physical meaning and consequently, no impact on the process.

Control is Lost
Once we cross over to a "no physical meaning" computation, the controller has lost the ability to regulate the process.

When the computed CO exceeds the physical capabilities of the FCE because the integral term has reached a large positive or negative value, the controller is suffering from windup. Because windup is associated with the integral term, it is often referred to as integral windup or reset windup.

To prevent windup from occurring, modern controllers are protected by either:
▪ Employing extra "jacketing logic" in the software to halt integration when the CO reaches a maximum or minimum value.
▪ Recasting the controller into a discrete velocity form that, by its very formulation, naturally avoids windup.
Both alternatives offer benefits but possess some fairly subtle drawbacks that we discuss below.

Visualizing Windup
To better visualize the problem of windup and the benefit of anti-windup protection, consider the plot from our heat exchanger process below (click for a large view).
To the left is the performance of a PI controller with no windup protection. To the right is the performance of the same controller protected by an anti-windup strategy.
For both controllers, the set point is stepped from 200 °C up to 215 °C and back again. As shown in the lower trace on the plot, the controller moves the CO to 0%, closing the valve completely, yet this is not sufficient to move the PV up to the new set point.

To the left in the plot, the impact of windup is a degraded controller performance. When the set point is stepped back to its original value of 200 °C , the windup condition causes a delay in the CO action. This in turn causes a delay in the PV response.

To the right in the plot, anti-windup protection permits the CO, and thus PV, to respond promptly to the command to return to the original SP value of 200 °C.

More Details on Windup
The plot below (click for large view) offers more detail. As labeled on the plot:

1) To the left for the Controller with Wind-up case, the SP is stepped up to 215 °C. The valve closes completely but is not able to move the PV all the way to the high set point value. Integration is a summing of controller error, and since error persists, the integration term grows very large.
The sustained error permits the controller to windup (saturate). While it is not obvious from the plot, the PI algorithm is computing values for CO that ask the valve to be open -5%, -8% and more. The control algorithm is just simple math with no ability to recognize that a valve cannot be open to a negative value.
Note that the chart shows the CO signal bottoming out at 0% while the controller algorithm is computing negative CO values. This misleading information is one reason why windup can be difficult to diagnose as the root cause of a problem from visual inspection of process data trend plots.

2) When the SP is stepped back to 200 °C, it seems as if the CO does not move at first. In reality, the control algorithm started moving the CO when the SP changed, but the values remain in the physically meaningless range of negative numbers.
So while the valve remains fully closed at 0%, the integral sum is accumulating controller errors of opposite sign. As time passes, the integral term shrinks or "unwinds" as the running sum of errors balance out.
3) When the integral sum of errors shrinks enough, it no longer dominates the CO computation. The CO signal returns from the physically meaningless world of negative values. The valve can finally move in response.
4) To the right in the plot above, the controller is protected from windup. As a result, when the set point is stepped back to 200 °C, the CO immediately reacts with a change that is proportional to the size of the SP change. The PV moves quickly in response to the CO actions as it tracks the SP back to 200 °C.

◊ Solution 1: Jacketing Logic on the Position Algorithm
The PI controller at the top of this article is called the position form because the computed CO is a specific intermediate value between full on/open/maximum and closed/off/minimum. The continuous PI algorithm is specifying the actual position (e.g., 27% open, 64% of maximum) that the final control element (FCE) should assume.

● Simple Logic Creates Additional Problems
It is not enough to have logic that simply limits or clips the CO if it reaches a maximum (COmax) or minimum (COmin) value because this does nothing to check the growth of the integral sum of errors term.
I
n fact, such simple logic was used in the "control with windup" plots just discussed. The CO seems stuck at 0% and we are unaware that the algorithm is actually computing negative valve positions as described in item 1 above.

● Anti-Windup Logic Outline
When we switch from manual mode to automatic, we assume that we have initialized the controller using a bumpless transfer procedure. That is, at switchover, the integral sum of error is set to zero, the SP is set equal to the current PV, and the controller bias is set equal to the current CO (implying that COmin < CObias < COmax).

Thus, there is nothing to cause CO to immediately change and "bump" our process at switchover.

One approach to creating anti-windup jacketing logic is to artificially manipulate the integral sum of error itself. With our controller properly initialized, the approach is to flip the algorithm around and back-calculate a value for the integral sum of error that will provide a desired controller output value (COdesired), or:

Note that COdesired can be different in different situations. For example,
▪ We do not want tuning parameter adjustments to cause sudden CO movements that bump our process. So if tuning values have changed, COdesired is the value of CO from the previous loop calculation cycle.
▪ If the PI controller computes CO values that are above COmax or below COmin, then we must be concerned about windup and COdesired is set equal to the limiting COmax or COmin value.

The anti-windup logic followed at every loop sample time, T, is thus:

1) If tuning parameters have changed since the last loop calculation cycle, then COdesired = current CO. Back calculate the integral sum of error so CO remains unchanged from the previous sample time. This prevents sudden CO bumps due to tuning changes.

2) Update SP and PV for this loop calculation cycle.
3) compute:

4) If CO > COmax or if CO < COmin, then the anti-windup (integral desaturation) logic of step 5 is required. Otherwise, proceed to step 6.

5) If CO > COmax, then CO = COdesired = COmax. if CO < COmin, then CO = COdesired = COmin. Back calculate the integral sum of error using our selected COdesired and save it for use in the next control loop calculation cycle.

6) Implement CO

◊ Solution 2 - Use the Velocity (Discrete) Controller Form
Rather than computing a CO signal indicating a specific position for our final control element, an alternative is to compute a signal that specifies a change, ∆CO, from current position for the FCE. As explained below, this is called the velocity or discrete controller form.

We employ the dependent algorithm for this presentation, but the derivation that follows can be applied in an analogous fashion to the independent PI form. To derive the discrete velocity form, we must first write the continuous, position form of the PI controller to include the independent variable on the controller output, showing it properly as CO(t) to reflect that it changes with time:

Please note that this controller is identical to all dependent PI forms as presented in other articles. The only difference is we are being more mathematically precise in our expression of CO(t).

● Deriving the Discrete Velocity Form
The first step in deriving the discrete form is to take the time derivative of the continuous form. In physics, the time derivative (rate of change) of a position is a velocity. This is why the final form of the PI controller we derive is often called the velocity form.

Taking the derivative of the continuous PI controller above with respect to time yields:

Since CObias is a constant and the derivative of a constant is zero, then:

Removing this term from the equation results in:

If we assume discrete or finite difference approximations for the continuous derivatives, then the controller becomes:

where ei is the current controller error, ei-1 is the controller error at the previous sample time, T, and ∆ei = ei - ei-1.
Recognizing that loop sample time is T = ∆t, then the PI controller becomes:

Rearranging, we arrive at the discrete velocity form of the PI controller:

● Reason for Anti-Windup Protection
Discrete velocity algorithms compute a ∆CO that signals the FCE to move a specific distance and direction from its current position. As we can see from the PI controller form above, the computation does not keep track of the current FCE position, nor does it mathematically accumulate any integral sums.

In a sense, the accumulation of integration is stored in the final control element itself. If a long series of ∆CO moves are all positive, for example, the valve, pump or compressor will move toward its maximum value. And once the FCE reaches its maximum limit, any ∆CO commands to move further will have no impact because, as stated in the first sentences of this article, a valve cannot open more than all the way and a pump cannot go slower than stopped. It is the physical nature of the FCE itself that provides protection from over-accumulation (i.e., windup).

As long as the CO never reaches COmax or COmin, the continuous position and discrete velocity forms of the PI controller provide identical performance. A properly jacketed continuous position PI controller will also provide windup protection equal to the discrete velocity form. Implicit in these statements is that sample time,T, is reasonably fast and that T and the tuning values (Kc and Ti) are the same when comparing implementations.

● Concerns with Discrete Velocity PID
Unfortunately, the usefulness of the discrete velocity form is limited because the method suffers problems when derivative action is included. We find that we must take the derivative of a derivative, yielding a second derivative. A second derivative applied to data that contains even modest noise can produce nonsense results.

Some vendors implement this form anyway and include a signal filter and additional logic sequences to address the problem. Thus, even with the anti-windup benefits of a discrete velocity algorithm, we find the need to jacket the algorithm with protective logic.

PI Disturbance Rejection in the Jacketed Stirred Reactor

2011-03-22T13:39:00.001-07:00

PI Disturbance Rejection in the Jacketed Stirred Reactor

The control objective for the jacketed reactor is to minimize the impact on reactor operation when the temperature of the liquid entering the cooling jacket changes (detailed discussion here). As a base case study, we establish here the performance capabilities of a PI controller in achieving this objective.

The important variables for this study are labeled in the graphic (click for large view):
CO = signal to valve that adjusts cooling jacket liquid flow rate (controller output, %)
PV = reactor exit stream temperature (measured process variable, oC)
SP = desired reactor exit stream temperature (set point, oC)
D = temperature of cooling liquid entering the jacket (major disturbance, oC)
We follow our industry proven recipe to design and tune our PI controller:

Step 1: Design Level of Operation (DLO)
The details of expected process operation and how this leads to our DLO are presented in this article and are summarized:
▪ Design PV and SP = 90 oC with approval for brief dynamic (bump) testing of ±2 oC.
▪ Design D = 43 oC with occasional spikes up to 50 oC.

Step 2: Collect Process Data around the DLO
When CO, PV and D are steady near the design level of operation, we bump the process as detailed here to generate CO-to-PV cause and effect response data.

Step 3: Fit a FOPDT Model to the Dynamic Process Data
We approximate the dynamic behavior of the process by fitting a first order plus dead time (FOPDT) dynamic model to the test data from step 2. The results of the modeling study are presented in detail here and are summarized:
▪ Process gain (direction and how far), Kp = - 0.5 oC/%
▪ Time constant (how fast), Tp = 2.2 min
▪ Dead time (how much delay), Өp = 0.8 min

Step 4: Use the FOPDT Parameters to Complete the Design
As in the heat exchanger PI control study, we explore what is often called the dependent, ideal form of the PI control algorithm:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value; set by bumpless transfer
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = controller gain, a tuning parameter
Ti = reset time, a tuning parameter

Aside: our observations using the dependent ideal PI algorithm directly apply to the other popular PI controller forms. For example, the integral gain for the independent algorithm form, written as:

can be computed as: Ki = Kc/Ti. The Kc is the same for both forms, though it is more commonly called the proportional gain for the independent algorithm.

• Sample Time, T
Best practice is to set the loop sample time, T, at one-tenth the time constant or faster (i.e., T ≤ 0.1Tp). Faster sampling may provide modestly improved performance, while slower sampling can lead to significantly degraded performance.
In this study, T ≤ 0.1(2.2 min), so T should be 13 seconds or less. We meet this with the sample time option available from most commercial vendors:
◊ sample time, T = 1 sec

• Control Action (Direct/Reverse)
The jacketed stirred reactor process has a negative Kp. That is, when CO increases, PV decreases in response. Since a controller must provide negative feedback, if the process is reverse acting, the controller must be direct acting. That is, when in automatic mode (closed loop), if the PV is too high, the controller must increase the CO to correct the error. Since the controller must move in the same direction as the problem, we specify:
◊ controller is direct acting

• Dead Time Issues
If dead time is greater than the process time constant (Өp > Tp), control becomes increasingly problematic and a Smith predictor can offer benefit. For this process, the dead time is smaller than the time constant, so:
◊ dead time is small and not a concern

• Computing Controller Error, e(t)
Set point, SP, is manually entered into a controller. The measured PV comes from the sensor (our wire in). Since SP and PV are known values, then at every loop sample time, T, controller error can be directly computed as:
◊ error, e(t) = SP - PV

• Determining Bias Value, CObias
CObias is the value of CO that, in manual mode, causes the PV to steady at the DLO when the major disturbances are quiet and at their normal or expected values. When integral action is enabled, commercial controllers determine the bias value with a bumpless transfer procedure.
That is, when switching to automatic, the controller initializes the SP to the current value of PV, and CObias to the current value of CO. By choosing our current operation as our design state (at least temporarily at switchover), there is no corrective action needed by the controller that will bump the process. Thus,
◊ controller bias, CObias = current CO for a bumpless transfer

• Controller Gain, Kc, and Reset Time, Ti
We use our FOPDT model parameters in the industry-proven Internal Model Control (IMC) tuning correlations to compute PI tuning values.

The first step in using the IMC correlations is to compute Tc, the closed loop time constant. Tc describes how active our controller should be in responding to a set point change or in rejecting a disturbance.

The performance implications of choosing Tc have been explored previously for PI control of the heat exchanger and the gravity drained tanks case studies.

In short, the closed loop time constant, Tc, is computed based on whether we seek:
▪ aggressive action and can tolerate some overshoot and oscillation in the PV response,
▪ moderate action where the PV will move reasonably fast but show little overshoot,
▪ conservative action where the PV will move in the proper direction, but quite slowly.

Once this decision is made, we compute Tc with these rules:
▪ Aggressive Response: Tc is the larger of 0.1·Tp or 0.8·Өp
▪ Moderate Response: Tc is the larger of 1·Tp or 8·Өp
▪ Conservative Response: Tc is the larger of 10·Tp or 80·Өp

With Tc computed, the PI controller gain, Kc, and reset time, Ti, are computed as:

Notice that reset time, Ti, is always equal to the process time constant, Tp, regardless of desired controller activity.

a) Moderate Response Tuning:
For a controller that will move the PV reasonably fast while producing little to no overshoot, choose:
Moderate Tc = the larger of 1·Tp or 8·Өp
= larger of 1(2.2 min) or 8(0.8 min)
= 6.4 min
Using this Tc and our model parameters in the tuning correlations above, we arrive at the moderate tuning values:

b) Aggressive Response Tuning:
For an active or quickly responding controller where we can tolerate some overshoot and oscillation as the PV settles out, specify:
Aggressive Tc = the larger of 0.1·Tp or 0.8·Өp
= larger of 0.1(2.2 min) or 0.8(0.8 min)
= 0.64 min
and the aggressive tuning values are:

Practitioner's Note: The FOPDT model parameters used in the tuning correlations above have engineering units, so the Kc values we compute also have engineering units. In commercial control systems, controller gain (or proportional band) is normally entered as a dimensionless (%/%) value.
For commercial implementations, we could:
▪ Scale the process data before fitting our FOPDT dynamic model so we directly compute a dimensionless Kc.
▪ Convert the model Kp to dimensionless %/% after fitting the model but before using the FOPDT parameters in the tuning correlations.
▪ Convert Kc from engineering units into dimensionless %/% after using the tuning correlations.

CO is already scaled from 0 - 100% in the above example. Thus, we convert Kc from engineering units into dimensionless %/% using the formula:

For the jacketed stirred reactor, PVmax = 250 oC and PVmin = 0 oC. The dimensionless Kc values are thus computed:
▪ moderate Kc = (- 0.6 %/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)] = - 1.5 %/%
▪ aggressive Kc = (- 3.1%/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)] = - 7.8 %/%

We use Kc with engineering units in the remainder of this article and are careful that our PI controller is formulated to accept such values. We would be mindful if we were using a commercial control system, however, to ensure our tuning parameters are cast in the form appropriate for our equipment.

Implement and Test
The ability of the PI controller to reject changes in the cooling jacket inlet temperature, D, is pictured below for the moderate and aggressive tuning values computed above. Note that the set point remains constant at 90 oC throughout the study.

As expected, the aggressive controller shows a more energetic CO action, and thus, a more active PV response.
While not our design objective, presented below is the set point tracking ability of the PI controller (click for a large view) when the disturbance temperature is held constant.

The plot shows that set point tracking performance matches the descriptions used above for choosing Tc:
▪ Use aggressive action if we seek a fast response and can tolerate some overshoot and oscillation in the PV response.
▪ Use moderate action if we seek a reasonably fast response but seek little to no overshoot in the PV response.

Important => Ti Always Equals Tp
As stated above, the rules provide a constant reset time, Ti, regardless of our desired performance. So if we believe we have collected a good process data set, and the FOPDT model fit looks like a reasonable approximation of this data, then we have a good estimate of the process time constant and Ti = Tp regardless of desired performance.

If we are going to tweak the tuning, Kc should be the only value we adjust. For example, if we seek a performance between moderate and aggressive, we average the Kc values while Ti remains constant.

Interacting Tuning Parameters

2011-03-22T13:38:00.001-07:00

Many process control practitioners tune by "intuition," fiddling their way to final tuning by a combination of experience and trial-and-error.

Some are quite good at approaching process control as art. Since they are the ones who define "best" performance based on the goals of production, the capabilities of the process, the impact on down stream units, and the desires of management, it can be difficult to challenge any claims of success.

To explore the pitfalls of a trial and error approach and reinforce that there is science to controller tuning, we consider the common dependent, ideal form of the PI controller:

Where:
CO = controller output
e(t) = controller error = set point - process variable = SP - PV
Kc = controller gain, a tuning parameter
Ti = reset time, a tuning parameter

For this form, controller activity or aggressiveness increases as Kc increases and as Ti decreases (Ti is in the denominator, so smaller values increase the weighting on the integral action term, thus increasing controller activity).

Since Kc and Ti individually can make a controller more or less aggressive in its response, the two tuning parameters interact with each other. If current controller performance is not what we desire, it is not always clear which value to raise or lower, or by how much.

Example of Interaction Confusion
To illustrate, consider a case where we seek to balance a fairly rapid response to a set point change (a short rise time) against a small overshoot. While every process application is different, we choose to call the response plot below our desired or base case performance.

Now consider the two response plots below. These were made using the identical process and controller to that above. The only difference between the base case response above and plot A and plot B below is that different Kc and Ti tuning values were used in each one.

And now the question: what tuning adjustments are required to restore the desired base case performance above starting from each plot below? Or alternatively: how has the tuning been changed from base case performance to produce these different behaviors?

There are no tricks in this question. The "process" is a simple linear second order system with modest dead time. Controller output is not hitting any limits. The scales on the plot are identical. Everything is as it seems, except PI controller tuning is different in each case.

Study the plots for a moment before reading ahead and see if you can figure it out. Each plot has a very different answer.

Some Hints
Before we reveal the answer, here is a hint. One plot has been made more active or aggressive in its response by doubling Kc while keeping Ti constant at the original base case value.

The other cuts Ti in half (remember, decreasing Ti makes this PI form more active) while keeping Kc at the base case value:

So we have:
• Base case = Kc and Ti
• Plot A or B = 2Kc and Ti
• Other Plot B or A = Kc and Ti/2

Still not sure? Here is a final hint: remember from our previous discussions that proportional action is largely responsible for the first movements in a response. We also discussed that integral action tends to increase the oscillatory or cycling behavior in the PV.

It is not easy to know the answer, even with these huge hints, and that is the point of this article.

The Answer
Below is a complete tuning map with the base case performance from our challenge problem in the center. The plot shows how performance changes as Kc and Ti are doubled and halved from the base case for the dependent, ideal PI controller form.

Starting from the center and moving up on the map from the base case performance brings us to plot B. As indicated on the tuning map axis, this direction increases (doubles) controller gain, Kc, thus making the controller more active or aggressive. Moving down on the map from the base case decreases (halves) Kc, making the controller more sluggish in its response.

Moving left on the map from the base case brings us to plot A. As indicated on the tuning map axis, this direction decreases reset time (cuts it in half), again making the controller more active or aggressive. Moving right on the map from the base case increases (doubles) reset time, making the controller more sluggish in its response.

It is clear from the tuning map that the controller is more active or aggressive in its response when Kc increases and Ti decreases, and more sluggish or conservative when Kc decreases and Ti increases.

Building on this observation, it is not surprising that the upper left most plot (2Kc and Ti/2) shows the most active controller response, and the lower right most plot (Kc/2 and 2Ti) is the most conservative or sluggish response.

Back to the question. With what we now know, the answer:
• Base case = Kc and Ti
• Plot B = 2Kc and Ti
• Plot A = Kc and Ti/2

Interacting Parameters Makes Tuning Problematic

The PI controller has only two tuning parameters, yet it produces very similar looking performance plots located in different places on a tuning map.

f our instincts lead us to believe that we are at plot A when we really are at plot B, then the corrective action we make based on this instinct will compound our problem rather than solve it. This is strong evidence that trial and error is not an efficient or appropriate approach to tuning.

When we consider a PID controller with three tuning parameters, the number of similar looking plots in what would be a three dimensional tuning map increases dramatically. Trial and error tuning becomes almost futile.

We have been exploring a step by step tuning recipe approach that produces desired results without the wasted time and off-spec product that results from trial and error tuning.

If we follow this industry-proven methodology, we will improve the safety and profitability of our operation.

Interesting Observation

Before leaving this subject, we make one more very useful observation from the tuning map. This will help build our intuition and may help one day when we are out in the plant.

The right most plot in the center row (Kc, 2Ti) of the tuning map above is reproduced below.

Notice how the PV shows a dip or brief oscillation on its way up to the set point? This is a classic indication that the proportional term is reasonable but the integral term is not getting enough weight in the calculation. For the PI form used in this article, that would mean that the reset time, Ti, is too large since it is in the denominator.

If we cover the right half of the "not enough integral action" plot, the response looks like it is going to settle out with some offset, as would be expected with a P-Only controller. When we consider the plot as a whole, we see that as enough time passes, the response completes. This is because the weak integral action finally accumulates enough weight in the calculation to move the PV up to set point.

This "oscillates on the way" pattern is a useful marker for diagnosing a lack of sufficient integral action.

2011-03-22T13:37:00.001-07:00

PI Disturbance Rejection Of The Gravity Drained Tanks

When exploring the capabilities of the P-Only controller in rejecting disturbances for the gravity drained tanks process, we confirmed the observations we had made during the the P-Only set point tracking study for the heat exchanger.

In particular, the P-Only algorithm is easy to tune and maintain, but whenever the set point or a major disturbance moves the process from the design level of operation, a sustained error between the process variable (PV) and set point (SP), called offset, results.

Further, we saw in both case studies that as controller gain, Kc, increases (or as proportional band, PB, decreases):
▪ the activity of the controller output, CO, increases
▪ the oscillatory nature of the response increases
▪ the offset (sustained error) decreases

In this article, we explore the benefits of integral action and the capabilities of the PI controller for rejecting disturbances in the gravity drained tanks process. We have previously presented the fundamentals behind PI control and its application to set point tracking in the heat exchanger.

As with all controller implementations, best practice is to follow our proven four-step design and tuning recipe. One benefit of the recipe is that steps 1-3, summarized below from our P-Only study, remain the same regardless of the control algorithm being employed. After summarizing steps 1-3, we complete the PI controller design and tuning in step 4.

Step 1: Determine the Design Level of Operation (DLO)
The control objective is to reject disturbances as we control liquid level in the lower tank. Our design level of operation (DLO), detailed here for this study is:
▪ design PV and SP = 2.2 m with range of 2.0 to 2.4 m
▪ design D = 2 L/min with occasional spikes up to 5 L/min

Step 2: Collect Process Data around the DLO
When CO, PV and D are steady near the design level of operation, we bump the CO as detailed here and force a clear response in the PV that dominates the noise.

Step 3: Fit a FOPDT Model to the Dynamic Process Data
We then describe the process behavior by fitting an approximating first order plus dead time (FOPDT) dynamic model to the test data from step 2. We define the model parameters and present details of the model fit of step test data here. A model fit of doublet test data using commercial software confirms these values:
▪ process gain (how far), Kp = 0.09 m/%
▪ time constant (how fast), Tp = 1.4 min
▪ dead time (how much delay), Өp = 0.5 min

Step 4: Use the FOPDT Parameters to Complete the Design
Following the heat exchanger PI control study, we explore what is often called the dependent, ideal form of the PI control algorithm:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value; set by bumpless transfer
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = controller gain, a tuning parameter
Ti = reset time, a tuning parameter

Aside: our observations using the dependent ideal PI algorithm directly apply to the other popular PI controller forms. For example, the integral gain, Ki, in the independent algorithm form:

can be computed directly from controller gain and reset time as: Ki = Kc/Ti.

In the P-Only study, we established that for the gravity drained tanks process:
▪ sample time, T = 1 sec
▪ the controller is reverse acting
▪ dead time is small compared to Tp and thus not a concern in the design

• Controller Gain, Kc, and Reset Time, Ti
We use our FOPDT model parameters in the industry-proven Internal Model Control (IMC) tuning correlations to compute PI tuning values.

The first step in using the IMC correlations is to compute Tc, the closed loop time constant. All time constants describe the speed or quickness of a response. Tc describes the desired speed or quickness of a controller in responding to a set point change or rejecting a disturbance.

If we want an active or quickly responding controller and can tolerate some overshoot and oscillation as the PV settles out, we want a small Tc (a short response time) and should choose aggressive tuning:
▪ Aggressive Response: Tc is the larger of 0.1·Tp or 0.8·Өp

If we seek a sluggish controller that will move things in the proper direction, but quite slowly, we choose conservative tuning (a big or long Tc).
▪ Conservative Response: Tc is the larger of 10·Tp or 80·Өp

Moderate tuning is for a controller that will move the PV reasonably fast while producing little to no overshoot.
▪ Moderate Response: Tc is the larger of 1·Tp or 8·Өp

With Tc computed, the PI controller gain, Kc, and reset time, Ti, are computed as:

Notice that reset time, Ti, is always equal to the process time constant, Tp, regardless of desired controller activity.

a) Moderate Response Tuning:
For a controller that will move the PV reasonably fast while producing little to no overshoot, choose:
Moderate Tc = the larger of 1·Tp or 8·Өp
= larger of 1(1.4 min) or 8(0.5 min)
= 4 min
Using this Tc and our model parameters in the tuning correlations above, we arrive at the moderate tuning values:

b) Aggressive Response Tuning:
For an active or quickly responding controller where we can tolerate some overshoot and oscillation as the PV settles out, specify:
Aggressive Tc = the larger of 0.1·Tp or 0.8·Өp
= larger of 0.1(1.4 min) or 0.8(0.5 min)
= 0.4 min
and the aggressive tuning values are:

Practitioner's Note: The FOPDT model parameters used in the tuning correlations above have engineering units, so the Kc values we compute also have engineering units. In commercial control systems, controller gain (or proportional band) is normally entered as a dimensionless (%/%) value.
To address this, we could:
▪ Scale the process data before fitting our FOPDT dynamic model so we directly compute a dimensionless Kc.
▪ Convert the model Kp to dimensionless %/% after fitting the model but before using the FOPDT parameters in the tuning correlations.
▪ Convert Kc from engineering units into dimensionless %/% after using the tuning correlations.

Since we already have Kc in engineering units, we employ the third option. CO is already scaled from 0 - 100% in the above example. Thus, we convert Kc from engineering units into dimensionless %/% using the formula:

For the gravity drained tanks, PVmax = 10 m and PVmin = 0 m. The dimensionless Kc values are thus computed:
▪ moderate Kc = (3.5 %/m)∙[(10 - 0 m) ÷ (100 - 0%)] = 0.35 %/%
▪ aggressive Kc = (17 %/m)∙[(10 - 0 m) ÷ (100 - 0%)] = 1.7 %/%

We use the Kc with engineering units in the remainder of this article and are careful that our PI controller is formulated to accept such values. If we were using these results in a commercial control system, we would be careful to ensure our tuning parameters are cast in the form appropriate for our equipment.

• Controller Action
The process gain, Kp, is positive for the gravity drained tanks, indicating that when CO increases, the PV increases in response. This behavior is characteristic of a direct acting process. Given this CO to PV relationship, when in automatic mode (closed loop), if the PV starts drifting above set point, the controller must decrease CO to correct the error. Such negative feedback is an essential component of stable controller design.
A process that is naturally direct acting requires a controller that is reverse acting to remain stable. In spite of the opposite labels (direct acting process and reverse acting controller), the details presented above show that both Kp and Kc are positive values.

In most commercial controllers, only positive Kc values can be entered. The sign (or action) of the controller is then assigned by specifying that the controller is either reverse acting or direct acting to indicate a positive or negative Kc, respectively.

If the wrong control action is entered, the controller will quickly drive the final control element (FCE) to full on/open or full off/closed and remain there until a proper control action entry is made.

Implement and Test
The ability of the PI controller to reject changes in the pumped flow disturbance, D, is pictured below for the moderate and aggressive tuning values computed above. Note that the set point remains constant at 2.2 m throughout the study.

The aggressive controller shows a more energetic CO action, and thus, a more active PV response. As shown above, however, the penalty for this increased activity is some overshoot and oscillation in the process response.

Please be aware that the terms "moderate" and "aggressive" hold no magic. If we desire a control performance between the two, we need only average the Kc values from the tuning rules above. Note, however, that these rules provide a constant reset time, Ti, regardless of our desired performance. So if we believe we have collected a good process data set, and the FOPDT model fit looks like a reasonable approximation of this data, then Ti = Tp always.

While not our design objective, presented below is the set point tracking ability of the PI controller when the disturbance flow is held constant:

Again, the aggressive tuning values provide for a more active response.
Aside: it may appear that the random noise in the PV measurement signal is different in the two plots above, but it is indeed the same. Note that the span of the PV axis in each plot differs by a factor of four. The narrow span of the set point tracking plot greatly magnifies the signal traces, making the noise more visible.
Comparison With P-Only Control
The performance of a P-Only controller in addressing the same disturbance rejection and set point tracking challenge is shown here. A comparison of that study with the results presented here reveals that PI controllers:
▪ can eliminate the offset associated with P-Only control,
▪ have integral action that increases the tendency for the PV to roll (or oscillate),
▪ have two tuning parameters that interact, increasing the challenge to correct tuning when performance is not acceptable.
Derivative Action
The addition of the derivative term to complete the PID algorithm provides modest benefit yet significant challenges.

Again, the aggressive tuning values provide for a more active response.

Aside: it may appear that the random noise in the PV measurement signal is different in the two plots above, but it is indeed the same. Note that the span of the PV axis in each plot differs by a factor of four. The narrow span of the set point tracking plot greatly magnifies the signal traces, making the noise more visible.

Comparison With P-Only Control
The performance of a P-Only controller in addressing the same disturbance rejection and set point tracking challenge is shown here. A comparison of that study with the results presented here reveals that PI controllers:
▪ can eliminate the offset associated with P-Only control,
▪ have integral action that increases the tendency for the PV to roll (or oscillate),
▪ have two tuning parameters that interact, increasing the challenge to correct tuning when performance is not acceptable.

Derivative Action
The addition of the derivative term to complete the PID algorithm provides modest benefit yet significant challenges.

PI Control of the Heat Exchanger

2011-03-22T13:36:00.000-07:00

PI Control of the Heat Exchanger

We investigated P-Only control of the heat exchanger process and learned that while P-Only is an algorithm that is easy to tune and maintain, it has a severe limitation. Specifically, its simple form permits steady state error, called offset, in most processes during normal operation.

Then we moved on to integral action and PI control. We focused in that article on the structure of the algorithm and explored the mathematics of how the proportional and integral terms worked together to eliminate offset.

Here we test the capabilities of the PI controller on the heat exchanger process. Our focus is on design, implementation and basic performance issues. Along the way we will highlight some strengths and weaknesses of this popular algorithm.

As with all controller implementations, best practice is to follow our proven four-step design and tuning recipe as we proceed with this case study.

Step 1: Design Level of Operation (DLO)

Real processes display a nonlinear behavior. That is, their process gain, time constant and/or dead time changes as operating level changes and as major disturbances change. Since controller design and tuning is based on these process Kp, Tp and Өp values, controllers should be designed and tuned for a specific level of operation.
Thus, the first step in our controller design recipe is to specify our design level of operation (DLO). This includes stating:
▪ Where we expect the set point, SP, and measured process variable, PV, to be during normal operation.
▪ The range of values the SP and PV might assume so we can explore the nature of the process dynamics across that range.
We will track along with the same design conditions used in the P-Only control study to permit a direct comparison of performance and capability. As in that study, we specify:
▪ Design PV and SP = 138 °C with range of 138 to 140 °C
We also should know normal or typical values for our major disturbances and be reasonably confident that they are quiet so we may proceed with a bump test. The heat exchanger process has only one major disturbance variable, and consistent with the previous study:
▪ Expected warm liquid flow disturbance = 10 L/min

Step 2: Collect Data at the DLO
The next step in the design recipe is to collect dynamic process data as near as practical to our design level of operation. We have previously collected and documented heat exchanger step test data that matches our design conditions.

Step 3: Fit an FOPDT Model to the Design Data
Here we document a first order plus dead time (FOPDT) model approximation of the step test data from step 2:
▪ Process gain (how far), Kp = -0.53 °C/%
▪ Time constant (how fast), Tp = 1.3 min
▪ Dead time (how much delay), Өp = 0.8 min

Step 4: Use the Parameters to Complete the Design
One common form of the PI controller computes a controller output (CO) action every loop sample time T as:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value; set by bumpless transfer as explained below
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = controller gain, a tuning parameter
Ti = reset time, a tuning parameter

• Loop Sample Time, T
Best practice is to specify loop sample time, T, at 10 times per time constant or faster (T ≤ 0.1Tp). For this study, T ≤ 0.13 min = 8 sec. Faster sampling may provide modestly improved performance, while slower sampling can lead to significantly degraded performance. Most commercial controllers offer an option of T = 1.0 sec, and since this meets our design rule, we use that here.

• Computing controller error, e(t)
Set point, SP, is something we enter into the controller. The PV measurement comes from our sensor (our wire in). With SP and PV known, controller error, e(t) = SP - PV, can be directly computed at every loop sample time T.

• Determining Bias Value
Strictly speaking, CObias is the value of the CO that, in manual mode, causes the PV to steady at the DLO while the major disturbances are quiet and at their normal or expected values.

• Bumpless Transfer
A desirable feature of the PI algorithm is that it is able to eliminate the offset that can occur under P-Only control. The integral term of the PI controller provides this capability by providing updated information that, when combined with the controller bias, keeps the process centered as conditions change.

Since integral action acts to update (or reset) our bias value over time, CObias can be initialized in a straightforward fashion to a value that produces no abrupt control actions when we switch to automatic. Most commercial controllers do this with a simple "bumpless transfer" feature. When switching to automatic, they initialize:
▪ SP equal to the current PV
▪ CObias equal to the current CO

With the set point equal to the measured process variable, there is no error to drive a change in our controller output. And with the controller bias set to our current controller output, we are prepared by default to maintain current operation.

We will use a controller that employs these bumpless transfer rules when we switch to automatic. Hence, we need not specify any value for CObias as part of our design.

• Computing Controller Gain and Reset Time
Here we use the industry-proven Internal Model Control (IMC) tuning correlations. The first step in using the IMC correlations is to compute Tc, the closed loop time constant. All time constants describe the speed or quickness of a response. The closed loop time constant describes the desired speed or quickness of a controller in responding to a set point change or rejecting a disturbance.

If we want an active or quickly responding controller and can tolerate some overshoot and oscillation as the PV settles out, we want a small Tc (a short response time) and should choose aggressive tuning:
▪ aggressive: Tc is the larger of 0.1·Tp or 0.8·Өp
Moderate tuning is for a controller that will move the PV reasonably fast while producing little to no overshoot.
▪ moderate: Tc is the larger of 1·Tp or 8·Өp
If we seek a more sluggish controller that will move things in the proper direction, but quite slowly, we choose conservative tuning (a big or long Tc).
▪ conservative: Tc is the larger of 10·Tp or 80·Өp

Once we have decided on our desired performance and computed the closed loop time constant, Tc, with the above rules, then the PI correlations for controller gain, Kc, and reset time, Ti, are:

Notice that reset time, Ti, is always set equal to the time constant of the process, regardless of desired controller activity.

a) Moderate Response Tuning:
For a controller that will move the PV reasonably fast while producing little to no overshoot, choose:
Moderate Tc = the larger of 1·Tp or 8·Өp
= larger of 1(1.3 min) or 8(0.8 min)
= 6.4 min
Using this Tc and our model parameters in the tuning correlations above, we arrive at the moderate tuning values:

b) Aggressive Response Tuning:

For an active or quickly responding controller where we can tolerate some overshoot and oscillation as the PV settles out, specify
Aggressive Tc = the larger of 0.1·Tp or 0.8·Өp
= larger of 0.1(1.3 min) or 0.8(0.8 min)
= 0.64 min
and the aggressive tuning values are:

Practitioner's Note: The FOPDT model parameters used in the tuning correlations above have engineering units, so the Kc values we compute also have engineering units. In commercial control systems, controller gain (or proportional band) is normally entered as a dimensionless (%/%) value.

For commercial implementations, we could:
▪ Scale the process data before fitting our FOPDT dynamic model so we directly compute a dimensionless Kc.
▪ Convert the model Kp to dimensionless %/% after fitting the model but before using the FOPDT parameters in the tuning correlations.
▪ Convert Kc from engineering units into dimensionless %/% after using the tuning correlations.

CO is already scaled from 0 - 100% in the above example. Thus, we convert Kc from engineering units into dimensionless %/% using the formula:

For the heat exchanger, PVmax = 250 oC and PVmin = 0 oC. The dimensionless Kc values are thus computed:

▪ moderate Kc = (- 0.34 %/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)] = - 0.85 %/%
▪ aggressive Kc = (- 1.7%/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)] = - 4.2 %/%

We use Kc with engineering units in the remainder of this article and are careful that our PI controller is formulated to accept such values. We would be mindful if we were using a commercial control system, however, to ensure our tuning parameters are cast in the form appropriate for our equipment.

• Controller Action
The process gain, Kp, is negative for the heat exchanger, indicating that when CO increases, the PV decreases in response. This behavior is characteristic of a reverse acting process. Given this CO to PV relationship, when in automatic mode (closed loop), if the PV starts drifting above set point, the controller must increase CO to correct the error. Such negative feedback is an essential component of stable controller design.
A process that is naturally reverse acting requires a controller that is direct acting to remain stable. In spite of the opposite labels (reverse acting process and direct acting controller), the details presented above show that both Kp and Kc are negative values.

In most commercial controllers, only positive Kc values can be entered. The sign (or action) of the controller is then assigned by specifying that the controller is either reverse acting or direct acting to indicate a positive or negative Kc, respectively.

If the wrong control action is entered, the controller will quickly drive the final control element (FCE) to full on/open or full off/closed and remain there until a proper control action entry is made.

Implement and Test

Below we test our two PI controllers on the heat exchanger process simulation. Shown are two set points step pairs from 138 °C up to 140 °C and back again.
The first set point steps to the left show the PI controller performance using the moderate tuning values computed above. The second set point steps to the right show the controller performance using the aggressive tuning values. Note that the warm liquid disturbance flow, though not shown, remains constant at 10 L/min throughout the study.
(For comparison, the performance of the P-Only controller in tracking these set point changes is pictured here).

The asymmetrical behavior of the PV for the set point steps up compared to the steps down is due to the very nonlinear character of the heat exchanger.

If we seek tuning between moderate and aggressive performance, we would average the Kc values from the tuning rules above.

But if we believe we had collected good bump test data (we saw a clear response in the PV when we stepped the CO and the major disturbances were quiet during the test), and the FOPDT model fit appears to be visually descriptive of the data, then we have a good value for Tp and that means a good value for Ti.

If we are going to fiddle with the tuning, we can tweak Kc and we should leave the reset time alone.

Tuning Recipe Saves Time and Money
The exciting result is that we achieved our desired controller performance based on one bump test and following a controller design recipe. No trial and error was involved. Little off-spec product was produced. No time was wasted.

Soon we will see how software tools help us achieve such results with even less disruption to the process.

The method of approximating complex behavior with a FOPDT model and then following a recipe for controller design and tuning has been used successfully on a broad spectrum of processes with streams composed of gases, liquids, powders, slurries and melts. It is a reliable approach that has been proven time and again at diverse plants from a wide range of companies.

Integral Action and PI Control

2011-03-22T13:35:00.000-07:00

Like the P-Only controller, the Proportional-Integral (PI) algorithm computes and transmits a controller output (CO) signal every sample time, T, to the final control element (e.g., valve, variable speed pump). The computed CO from the PI algorithm is influenced by the controller tuning parameters and the controller error, e(t).
PI controllers have two tuning parameters to adjust. While this makes them more challenging to tune than a P-Only controller, they are not as complex as the three parameter PID controller.
Integral action enables PI controllers to eliminate offset, a major weakness of a P-only controller. Thus, PI controllers provide a balance of complexity and capability that makes them by far the most widely used algorithm in process control applications.

The PI Algorithm

While different vendors cast what is essentially the same algorithm in different forms, here we explore what is variously described as the dependent, ideal, continuous, position form:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value; set by bumpless transfer as explained below
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = controller gain, a tuning parameter
Ti = reset time, a tuning parameter

The first two terms to the right of the equal sign are identical to the P-Only controller referenced at the top of this article.
The integral mode of the controller is the last term of the equation. Its function is to integrate or continually sum the controller error, e(t), over time.

Some things we should know about the reset time tuning parameter, Ti:
▪ It provides a separate weight to the integral term so the influence of integral action can be independently adjusted.
▪ It is in the denominator so smaller values provide a larger weight to (i.e. increase the influence of) the integral term.
▪ It has units of time so it is always positive.

Function of the Proportional Term

As with the P-Only controller, the proportional term of the PI controller, Kc·e(t), adds or subtracts from CObias based on the size of controller error e(t) at each time t.

As e(t) grows or shrinks, the amount added to CObias grows or shrinks immediately and proportionately. The past history and current trajectory of the controller error have no influence on the proportional term computation.

The plot below illustrates this idea for a set point response. The error used in the proportional calculation is shown on the plot:
▪ At time t = 25 min, e(25) = 60-56 = 4
▪ At time t = 40 min, e(40) = 60-62 = -2

Recalling that controller error e(t) = SP - PV, rather than viewing PV and SP as separate traces as we do above, we can compute and plot e(t) at each point in time t.

Below is the identical data to that above only it is recast as a plot of e(t) itself. Notice that in the plot above, PV = SP = 50 for the first 10 min, while in the error plot below, e(t) = 0 for the same time period.

This plot is useful as it helps us visualize how controller error continually changes size and sign as time passes.

Function of the Integral Term

While the proportional term considers the current size of e(t) only at the time of the controller calculation, the integral term considers the history of the error, or how long and how far the measured process variable has been from the set point over time.

Integration is a continual summing. Integration of error over time means that we sum up the complete controller error history up to the present time, starting from when the controller was first switched to automatic.

Controller error is e(t) = SP - PV. In the plot below, the integral sum of error is computed as the shaded areas between the SP and PV traces.

Each box in the plot has an integral sum of 20 (2 high by 10 wide). If we count the number of boxes (including fractions of boxes) contained in the shaded areas, we can compute the integral sum of error.

So when the PV first crosses the set point at around t = 32, the integral sum has grown to about 135. We write the integral term of the PI controller as:

Since it is controller error that drives the calculation, we get a direct view the situation from a controller error plot as shown below:

Note that the integral of each shaded portion has the same sign as the error. Since the integral sum starts accumulating when the controller is first put in automatic, the total integral sum grows as long as e(t) is positive and shrinks when it is negative.

At time t = 60 min on the plots, the integral sum is 135 - 34 = 101. The response is largely settled out at t = 90 min, and the integral sum is then 135 - 34 7 = 108.

Integral Action Eliminates Offset
The previous sentence makes a subtle yet very important observation. The response is largely complete at time t = 90 min, yet the integral sum of all error is not zero.

In this example, the integral sum has a final or residual value of 108. It is this residual value that enables integral action of the PI controller to eliminate offset.

As discussed in a previous article, most processes under P-only control experience offset during normal operation. Offset is a sustained value for controller error (i.e., PV does not equal SP at steady state).

We recognize from the P-Only controller:

that CO will always equal CObias unless we add or subtract something from it.

The only way we have something to add or subtract from CObias in the P-Only equation above is if e(t) is not zero. It e(t) is not steady at zero, then PV does not equal SP and we have offset.

However, with the PI controller:

we now know that the integral sum of error can have a final or residual value after a response is complete. This is important because it means that e(t) can be zero, yet we can still have something to add or subtract from CObias to form the final controller output, CO.

So as long as there is any error (as long as e(t) is not zero), the integral term will grow or shrink in size to impact CO. The changes in CO will only cease when PV equals SP (when e(t) = 0) for a sustained period of time.

At that point, the integral term can have a residual value as just discussed. This residual value from integration, when added to CObias, essentially creates a new overall bias value that corresponds to the new level of operation.

In effect, integral action continually resets the bias value to eliminate offset as operating level changes.

Challenges of PI Control

There are challenges in employing the PI algorithm:
▪ The two tuning parameters interact with each other and their influence must be balanced by the designer.
▪ The integral term tends to increase the oscillatory or rolling behavior of the process response.

Because the two tuning parameters interact with each other, it can be challenging to arrive at "best" tuning values. The value and importance of our design and tuning recipe increases as the controller becomes more complex.

Initializing the Controller for Bumpless Transfer

When we switch any controller from manual mode to automatic (from open loop to closed loop), we want the result to be uneventful. That is, we do not want the switchover to cause abrupt control actions that impact or disrupt our process

We achieve this desired outcome at switchover by initializing the controller integral sum of error to zero. Also, the set point and controller bias value are initialized by setting:
▪ SP equal to the current PV
▪ CObias equal to the current CO

With the integral sum of error set to zero, there is nothing to add or subtract from CObias that would cause a sudden change in the current controller output. With the set point equal to the measured process variable, there is no error to drive a change in our CO. And with the controller bias set to our current CO value, we are prepared by default to maintain current operation.

Thus, when we switch from manual mode to automatic, we have "bumpless transfer" with no surprises. This is a result everyone appreciates.

Reset Time Versus Reset Rate

Different vendors cast their control algorithms in slightly different forms. Some use proportional band rather than controller gain. Also, some use reset rate, Tr, instead of reset time. These are simply the inverse of each other:
Tr = 1/Ti

No matter how the tuning parameters are expressed, the PI algorithms are all equally capable.

But it is critical to know your manufacturer before you start tuning your controller because parameter values must be matched to your particular algorithm form. Commercial software for controller design and tuning will automatically address this problem for you.

Implementing a PI controller

We explore PI controller design, tuning and implementation on the heat exchanger in this article and the gravity drained tanks in this article.

Controller Gain Is Dimensionless in Commercial Systems

2011-03-22T13:32:00.000-07:00

Today I am going to dicuss a very importnat issue.

In fact two of my colleagues have experienced the thisissue & almost got embarrased in front of the client.

Whle implementing PID youn should be very careful to implement SCALLING & UNITS to tyhe controller. My colleague was there to implement as PH control loop. After setting all the parameters they were not getting the correct reading out of PH transmitter & in turn they can't use these values to implement the control loop for the industry. After a long struggle at site they have called e. My first question was about scaling & guess What! They forgot to implement scaling. This simple mistake cost them & cluient more than two hours.

So remember , Controller Gain Is Dimensionless in Commercial Systems.

IIn modern plants, process variable (PV) measurement signals are typically scaled to engineering units before they are displayed on the control room HMI computer screen or archived for storage by a process data historian. This is done for good reasons.

When operations staff walk through the plant, the assorted field gauges display the local measurements in engineering units to show that a vessel is operating, for example, at a pressure of 25 psig (1.7 barg) and a temperature of 140 oC (284 oF).

It makes sense, then, that the computer screens in the control room display the set point (SP) and PV values in these same familiar engineering units because:
• It helps the operations staff translate their knowledge and intuition from their field experience over to the abstract world of crowded HMI computer displays.
• Familiar units will facilitate the instinctive reactions and rapid decision making that prevents an unusual occurrence from escalating into a crisis situation.
• The process was originally designed in engineering units, so this is how the plant documentation will list the operating specifications.

Knowledge Base Articles Compute Kc With Units
Like a control room display, the Control Station Knowledge Base presents PV values in engineering units. In most articles, these PVs are used directly in tuning correlations to compute controller gains, Kc. As a result, the Kc values also carry engineering units.

The benefit of this approach is that controller gain maintains the intuitive familiarity that engineering units provide. The difficulty is that commercial controllers are normally configured to use a dimensionless Kc (or dimensionless proportional band, PB).

To address this issue, we explore below how to convert a Kc with engineering units into the standard dimensionless (%/%) form.

The conversion formula presented at the end of this article is reasonably straightforward to use. But it is derived from several subtle concepts that might benefit from explanation. Thus, we begin with a background discussion on units and scaling, and work our way toward our Kc conversion formula goal.

From Analog Sensor to Digital Signal
There are many ways to measure a process variable and move the signal into the digital world for use in a computer based control system. Below is a simplified sketch of one approach.

Other operations in the pathway from sensor to control system not shown in the simplified sketch might include a transducer, an amplifier, a transmitter, a scaling element, a linearizing element, a signal filter, a multiplexer, and more.

The central issue for this discussion is that the PV signal arrives at the computers and controllers in a raw digital form. The continuous analog PV measurement has been quantized (broken into) a range of discrete increments or digital integer "counts" by an A/D (analog to digital) converter.

More counts dividing the span of a measurement signal increases the resolution of the measurement when expressed as a digital value. The ranges offered by most vendors result from the computer binary 2n form where n is the number of bits of resolution used by the A/D converter.

Example: a 12 bit A/D converter digitizes an analog signal into 212 = 4096 discrete increments normally expressed to range from 0 to 4095 counts.
A 13 bit A/D converter digitizes an analog signal into 213 = 8192 discrete increments normally expressed to range from 0 to 8191 counts.
A 14 bit A/D converter digitizes an analog signal into 214 = 16384 discrete increments normally expressed to range from 0 to 16383 counts.

Example: if a 4 to 20 mA (milliamp) analog signal range is digitized by a 12 bit A/D converter into 0 to 4095 counts, then the resolution is:
(20 - 4 mA) ÷ 4095 counts = 0.00391 mA/count
A signal of 7 mA from an analog range of 4 to 20 mA changes to digital counts from the 12 bit A/D converter as:
(7 - 4 mA) ÷ 0.00391 mA/count = 767 counts
A signal of 1250 counts from a 12 bit A/D converter corresponds to an input signal of 8.89 mA from an analog range of 4 to 20 mA as:
4 mA (1250 counts)∙(0.00391 mA/count) = 8.89 mA

Scaling the Digital PV Signal to Engineering Units for Display
During the configuration phase of a control project, the minimum and maximum (or zero and span) of the PV measurement must be entered. These values are used to scale the digital PV signal to engineering units for display and storage.
Example: if a temperature range of 100 oC to 500 oC is digitized into 0 to 8191 counts by a 13 bit A/D converter, the signal is scaled for display and storage by setting the minimum digital value of 0 counts = 100 oC, and maximum digital value of 8191 counts = 500 oC
Each digital count from the 13 bit A/D converter gives a resolution of:
(500 - 100 oC) ÷ 8191 counts = 0.0488 oC/count
A signal of 175 oC from an analog range of 100 oC to 500 oC changes to digital counts from the 13 bit A/D converter as:
(175 - 100 oC) ÷ 0.0488 oC/count = 1537 counts
A signal of 1250 counts from the 13 bit A/D converter corresponds to an input signal of 161 oC from an analog range of 100 oC to 500 oC as:
100 oC (1250 counts)∙(0.0488 oC/count) = 161 oC

As discussed at the top of this article, the intuition and field knowledge of the operations staff is maintained by using engineering units in control room displays and when storing data to a historian.

For this same reason, modern control software uses engineering units when passing variables between the function blocks used for calculations and decision-making. Calculation and decision functions are easier to understand, document and debug when the logic is written using floating point values in common engineering units.

Scaling the Digital PV Signal for Use by the PID Controller
Most commercial PID controllers use a controller gain, Kc (or proportional band, PB) that is expressed as a standard dimensionless %/%.
Note: Controller gain in commercial controllers is often said to be unitless or dimensionless, but Kc actually has units of (% of CO signal)/(% of PV signal). In a precise mathematical world, these units do not cancel, though there is little harm in speaking as though they do.

Prior to executing the PID controller calculation, the PV signal must be scaled to a standard 0% to 100% to match the "dimensionless" Kc. This happens every loop sample time, T, regardless of whether we are measuring temperature, pressure, flow, or any other process variable.

To perform this scaling, the minimum and maximum PV values in engineering units corresponding to the 0% to 100% standard PV range must be entered during setup and loop configuration.

Example: if a temperature range of 100 oC to 500 oC is digitized into 0 to 8191 counts by a 13 bit A/D converter, the signal is scaled for the PID control calculation by setting the minimum digital value of 0 counts = 0%, and the maximum digital value of 8191 counts = 100%.
Each digital count from the 13 bit A/D converter gives a resolution of:
(100 - 0%) ÷ 8191 counts = 0.0122%/count
A signal of 1537 counts (175 oC) from a 13 bit A/D converter would translate to a signal of 18.75% as:
0% (1537)∙(0.0122%/value) = 18.75%
A signal of 1250 counts (161 oC) from a 13 bit A/D converter would translate to a signal of 15.25% as:
0% (1250)∙(0.0122%/value) = 15.25%

Control Output is 0% to 100%
The controller output (CO) from commercial controllers normally default to a 0% to 100% digital signal as well. Digital to analog (D/A) converters begin the transition of moving the digital CO values into the appropriate electrical current and voltage required by the valve, pump or other final control element (FCE) in the loop.

Note: While CO commonly defaults to a 0% to 100% signal, this may not be appropriate when implementing the outer primary controller in a cascade. The outer primary CO1 becomes the set point of the inner secondary controller, and signal scaling must match. For example, if SP2 is in engineering units, the CO1 signal must be scaled accordingly.

Care Required When Using Engineering Units For Controller Tuning
It is quite common to analyze and design controllers using data retrieved from our process historian or captured from our computer display. Just as with the articles in this e-book, this means the computed Kc values will likely be scaled in engineering units.

The sketch below highlights (click for a large view) that scaling from engineering units to a standard 0% to 100% range used in commercial controllers requires careful attention to detail.

The conversion of PV in engineering units to a standard 0% to 100% range requires knowledge of the maximum and minimum PV values in engineering units. These are the same values that are entered into our PID controller software function block during setup and loop configuration. The general conversion formula is:

where:
PVmax = maximum PV value in engineering units
PVmin = minimum PV value in engineering units
PV = current PV value in engineering units

Example: a temperature signal ranges from 100 oC to 500 oC and we seek to scale it to a range of 0% to 100% for use in a PID controller. We set:
PVmin = 100 oC and PVmax = 500 oC
A temperature of 175 oC converts to a standard 0% to 100% range as:
[(175 - 100 oC) ÷ (500 - 100 oC)]∙(100 - 0%) = 18.75%
A temperature of 161 oC converts to a standard 0% to 100% range as:
[(161 - 100 oC) ÷ (500 - 100 oC)]∙(100 - 0%) = 15.25%

Applying Conversion to Controller Gain, Kc
The discussion to this point provides the basis for the formula used to convert Kc from engineering units into dimensionless (%/%):

Example: the moderate Kc value in our P-Only control of the heat exchanger study is Kc = - 0.7 %/ oC. For this process, PVmax = 250 oC and PVmin = 0 oC
Kc = (- 0.7 %/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)]
= - 1.75 %/%

Example: the moderate value for Kc in our P-Only control of the gravity drained tanks study is Kc = 8 %/ oC For this process, PVmax = 10 m and PVmin = 0 m
Kc = (8 %/ m)∙[(10 - 0 m) ÷ (100 - 0%)]
= 0.8 %/%

Final Thoughts
Textbooks are full of rule-of-thumb guidelines for estimating initial Kc values for a controller depending on whether, for example, it is a flow loop, a temperature loop or a liquid level loop. While we have great reservations with such a "guess and test" approach to tuning, it is important to recognize that such rules are based on a Kc that is expressed in a dimensionless (%/%) form.

P-Only Disturbance Rejection of the Gravity Drained Tanks

2011-03-22T13:26:00.000-07:00

In a previous article, we looked at the structure of the P-Only algorithm and we considered some design issues associated with implementation. We also studied the set point tracking (or servo) performance of this simple controller for the heat exchanger process.

Here we investigate the capabilities of the P-Only controller for liquid level control of the gravity drained tanks process. Our objective in this study is disturbance rejection (or regulatory control) performance.

Gravity Drained Tanks Process
A graphic of the gravity drained tanks process is shown below :

The measured process variable (PV) is liquid level in the lower tank. The controller output (CO) adjusts the flow into the upper tank to maintain the PV at set point (SP).

The disturbance (D) is a pumped flow out of the lower tank. It's draw rate is adjusted by a different process and is thus beyond our control. Because it runs through a pump, D is not affected by liquid level, though the pumped flow rate drops to zero if the tank empties.

We begin by summarizing the previously discussed results of steps 1 through 3 of our design and tuning recipe as we proceed with our P-Only control investigation:

Step 1: Determine the Design Level of Operation (DLO)
Our primary objective is to reject disturbances as we control liquid level in the lower tank. As discussed here, our design level of operation (DLO) for this study is:

design PV and SP = 2.2 m with range of 2.0 to 2.4 m
design D = 2 L/min with occasional spikes up to 5 L/min

Step 2: Collect Process Data around the DLO
When CO, PV and D are steady near the design level of operation, we bump the CO far enough and fast enough to force a clear dynamic response in the PV that dominates the signal and process noise. As detailed here, we performed two different open loop (manual mode) dynamic tests, a step test and a doublet test.

Step 3: Fit a FOPDT Model to the Dynamic Process Data
The third step of the recipe is to describe the overall dynamic behavior of the process with an approximating first order plus dead time (FOPDT) dynamic model. We define the model parameters and present details of the model fit of step test data here. A model fit of doublet test data using commercial software confirms these values:

process gain (how far), Kp = 0.09 m/%
time constant (how fast), Tp = 1.4 min
dead time (how much delay), Өp = 0.5 min

Step 4: Use the FOPDT Parameters to Complete the Design
Following the heat exchanger P-Only study, the P-Only control algorithm computes a CO action every loop sample time T as:

CO = CObias Kc∙e(t)

Where:
CObias = controller bias or null value
Kc = controller gain, a tuning parameter
e(t) = controller error, defined as SP - PV

· Sample Time, T
Best practice is to set the loop sample time, T, at one-tenth the time constant or faster (i.e., T ≤ 0.1Tp). Faster sampling may provide modestly improved performance. Slower sampling can lead to significantly degraded performance.

In this study, T ≤ (0.1)(1.4 min), so T should be 8 seconds or less. We meet this specification with the common vendor sample time option:
▪ sample time, T = 1 sec

· Control Action (Direct/Reverse)
The gravity drained tanks has a positive Kp. That is, when CO increases, PV increases in response. When in automatic mode (closed loop), if the PV is too high, the controller must decrease the CO to correct the error (read more here). Since the controller must move in the direction opposite of the problem, we specify:
▪ controller is reverse acting

· Dead Time Issues
If dead time is greater than the process time constant (Өp > Tp), control becomes increasingly problematic and a Smith predictor can offer benefit. For this process, the dead time is smaller than the time constant, so:
▪ dead time is small and thus not a concern

· Computing Controller Error, e(t)
Set point, SP, is manually entered into a controller. The measured PV comes from the sensor (our wire in). Since SP and PV are known values, then at every loop sample time, T, controller error can be directly computed as:
▪ error, e(t) = SP - PV

· Determining Bias Value, CObias
CObias is the value of CO that, in manual mode, causes the PV to remain steady at the DLO when the major disturbances are quiet and at their normal or expected values. Our doublet plots establish that when CO is at 53%, the PV is steady at the design value of 2.2 m, thus:
▪ controller bias, CObias = 53%

· Controller Gain, Kc
For the simple P-Only controller form, we use the integral of time-weighted absolute error (ITAE) tuning correlation:

Moderate P-Only:

Aside: Regardless of the values computed in the FOPDT fit, best practice is to set Өp no smaller than sample time, T (or Өp ≥ T) in the control rules and correlations (more discussion here). In this gravity drained tanks study, our FOPDT fit produced a Өp much larger than T, so the "dead time greater than sample time" rule is met.

Using our FOPDT model values from step 3, we compute:

And our moderate P-Only controller becomes:

▪ P-Only controller: CO = 53% 8∙e(t)

Implement and Test
To explore how controller gain impacts P-Only performance, we test the controller with the above Kc = 8 %/m. Since the correlation tends to produce moderate performance values, we also explore increasingly aggressive or active P-Only tuning by doubling Kc (2Kc = 16 %/m) and then doubling it again (4Kc = 32 %/m).

The ability of the P-Only controller to reject step changes in the pumped flow disturbance, D, is pictured below (click for a large view) for the ITAE value of Kc and its multiples. Note that the set point remains constant at 2.2 m throughout the study.

As shown in the figure above, whenever the pumped flow disturbance, D, is at the design level of 2 L/min (e.g., when time is less than 30 min) then PV equals SP.

The three times that D is stepped away from the DLO, however, the PV shifts away from the set point. The simple P-Only controller is not able to eliminate this "offset," or sustained error between the PV and SP. This behavior reinforces that both set point and disturbances contribute to defining the design level of operation for a process.

The figure shows that as Kc increases across the plot:

the activity of the controller output, CO, increases,
the offset (difference between SP and final PV) decreases, and
the oscillatory nature of the response increases.

Offset, or the sustained error between SP and PV when the process moves away from the DLO, is a big disadvantage of P-Only control. Yet there are appropriate uses for this simple controller (more discussion here).

While not our design objective, presented below is the set point tracking ability of the controller (click for a large view) when the disturbance flow is held constant:

The figure shows that as Kc increases across the plot, the same performance observations made above apply here: the activity of CO increases, the offset decreases, and the oscillatory nature of the response increases.

Aside: it may appear that the random noise in the PV measurement signal is different in the two plots above, but it is indeed the same. Note that the span of the PV axis in the two plots differs by a factor of four. The narrow span of the set point tracking plot greatly magnifies the signal traces, making the noise more visible.

Proportional Band
Different manufacturers use different forms for the same tuning parameter. The popular alternative to controller gain found in the marketplace is proportional band, PB.

If the CO and PV have units of percent and both can range from 0 to 100%, then the conversion between controller gain and proportional band is:

PB = 100/Kc

Thus, as Kc increases, PB decreases. This reverse thinking can challenge our intuition when switching among manufacturers.

Many examples on this site assign engineering units to the measured PV because plant software has made the task of unit conversions straightforward. If this is true in your plant, take care when using this formula.

Integral Action
Integral action has the benefit of eliminating offset but presents greater design challenges.

2011-03-22T13:25:00.001-07:00

The P-Only Control Algorithm

The simplest algorithm in the PID family is a proportional or P-Only controller. Like all automatic controllers, it repeats a measurement-computation-action procedure at every loop sample time, T, following the logic flow shown in the block diagram below (click for large view):

Starting at the far right of the control loop block diagram above:

A sensor measures and transmits the current value of the process variable, PV, back to the controller (the 'controller wire in')
Controller error at current time t is computed as set point minus measured process variable, or e(t) = SP - PV
The controller uses this e(t) in a control algorithm to compute a new controller output signal, CO
The CO signal is sent to the final control element (e.g. valve, pump, heater, fan) causing it to change (the 'controller wire out')
The change in the final control element (FCE) causes a change in a manipulated variable
The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV

The goal of the controller is to make e(t) = 0 in spite of unplanned and unmeasured disturbances. Since e(t) = SP - PV, this is the same as saying a controller seeks to make PV = SP.

The P-Only Algorithm
The P-Only controller computes a CO action every loop sample time T as:

CO = CObias Kc∙e(t)

Where:
CObias = controller bias or null value
Kc = controller gain, a tuning parameter
e(t) = controller error = SP - PV
SP = set point
PV = measured process variable

Design Level of Operation
Real processes display a nonlinear behavior, which means their apparent process gain, time constant and/or dead time changes as operating level changes and as major disturbances change. Since controller design and tuning is based on these Kp, Tp and Өp values, controllers should be designed and tuned for a pre-defined level of operation.

When designing a cruise control system for a car, for example, would it make sense for us to perform bump tests to generate dynamic data when the car is traveling twice the normal speed limit while going down hill on a windy day? Of course not.

Bump test data should be collected as close as practical to the design PV when the disturbances are quiet and near their typical values. Thus, the design level of operation for a cruise control system is when the car is traveling at highway speed on flat ground on a calm day.

Definition: the design level of operation (DLO) is where we expect the SP and PV will be during normal operation while the important disturbances are quiet and at their expected or typical values.

Understanding Controller Bias
Let's suppose the P-Only control algorithm shown above is used for cruise control in an automobile and CO is the throttle signal adjusting the flow of fuel to the engine.

Let's also suppose that the speed SP is 70 and the measured PV is also 70 (units can be mph or kph depending on where you live in the world). Since PV = SP, then e(t) = 0 and the algorithm reduces to:

CO = CObias Kc∙(0) = CObias

If CObias is zero, then when set point equals measurement, the above equation says that the throttle signal, CO, is also zero. This makes no sense. Clearly if the car is traveling 70 kph, then some baseline flow of fuel is going to the engine.

This baseline value of the CO is called the bias or null value. In this example, CObias is the flow of fuel that, in manual mode, causes the car to travel the design speed of 70 kph when on flat ground on a calm day.

Definition: CObias is the value of the CO that, in manual mode, causes the PV to steady at the DLO while the major disturbances are quiet and at their normal or expected values.

A P-Only controller bias (sometimes called null value) is assigned a value as part of the controller design and remains fixed once the controller is put in automatic.

Controller Gain, Kc
The P-Only controller has the advantage of having only one adjustable or tuning parameter, Kc, that defines how active or aggressive the CO will move in response to changes in controller error, e(t).

For a given value of e(t) in the P-Only algorithm above, if Kc is small, then the amount added to CObias is small and the controller response will be slow or sluggish. If Kc is large, then the amount added to CObias is large and the controller response will be fast or aggressive.

Thus, Kc can be adjusted or tuned for each process to make the controller more or less active in its actions when measurement does not equal set point.

P-Only Controller Design
All controllers from the family of PID algorithms (P-Only, PI, PID) should be designed and tuned using our proven recipe:

Establish the design level of operation (the normal or expected values for set point and major disturbances).
Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level.
Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model.
Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.

The Internal Model Control (IMC) tuning correlations that work so well for PI and PID controllers cannot be derived for the simple P-Only controller form. The next best choice is to use the widely-published integral of time-weighted absolute error (ITAE) tuning correlation:

Moderate P-Only:

This correlation is useful in that it reliably yields a moderate Kc value. In fact, some practitioners find that the ITAE Kc value provides a response performance so predictably modest that they automatically start with an aggressive P-Only tuning, defined here as two and a half times the ITAE value:

Aggressive P-Only: Kc = 2.5 (Moderate Kc)

Reverse Acting, Direct Acting and Control Action
Time constant, Tp, and dead time, Өp, cannot affect the sign of Kc because they mark the passage of time and must always be positive. The above tuning correlation thus implies that Kc must always have the same sign as the process gain, Kp.

When CO increases on a process that has a positive Kp, the PV will increase in response. The process is direct acting. Given this CO to PV relationship, when in automatic mode (closed loop), if the PV starts drifting too high above set point, the controller must decrease CO to correct the error.

This "opposite to the problem" reaction is called negative feedback and forms the basis of stable control.

A process with a positive Kp is direct acting. With negative feedback, the controller must be reverse acting for stable control. Conversely, when Kp is negative (a reverse acting process), the controller must be direct acting for stable control.

Since Kp and Kc always have the same sign for a particular process and stable control requires negative feedback, then:

direct acting process (Kp and Kc positive) −› use a reverse acting controller
reverse acting process (Kp and Kc negative) −› use a direct acting controller

In most commercial controllers, a positive value of the Kc is always entered. The sign (or action) of the controller is then assigned by specifying that the controller is either reverse or direct acting to indicate a positive or negative Kc respectively.

If the wrong control action is entered, the controller will quickly drive the final control element (e.g., valve, pump, compressor) to full on/open or full off/closed and remain there until the proper control action entry is made.

Proportional Band
Some manufacturers use different forms for the same tuning parameter. The popular alternative to Kc found in the marketplace is proportional band, PB.

In many industry applications, both the CO and PV are expressed in units of percent. Given that a controller output signal ranges from a minimum (COmin) to maximum (COmax) value, then:

PB = (COmax - COmin)/Kc

When CO and PV have units of percent and both range from 0% to 100%, the much published conversion between controller gain and proportional band results:

PB = 100/Kc

Many case studies on this site assign engineering units to the measured PV because plant software has made the task of unit conversions straightforward. If this is true in your plant, take care when using these conversion formula.

Implementation Issues
Implementation of a P-Only controller is reasonably straightforward, but this simple algorithm exhibits a phenomenon called "offset." In most industrial applications, offset is considered an unacceptable weakness. We explore P-Only control, offset and other issues for the heat exchanger and the gravity drained tanks processes.

Proportional Control - The Simplest PID Controller

2011-03-22T13:23:00.001-07:00

he P-Only Control Algorithm
The simplest algorithm in the PID family is a proportional or P-Only controller. Like all automatic controllers, it repeats a measurement-computation-action procedure at every loop sample time, T, following the logic flow shown in the block diagram below (click for large view):

Starting at the far right of the control loop block diagram above:

A sensor measures and transmits the current value of the process variable, PV, back to the controller (the 'controller wire in')
Controller error at current time t is computed as set point minus measured process variable, or e(t) = SP - PV
The controller uses this e(t) in a control algorithm to compute a new controller output signal, CO
The CO signal is sent to the final control element (e.g. valve, pump, heater, fan) causing it to change (the 'controller wire out')
The change in the final control element (FCE) causes a change in a manipulated variable
The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV

Establish the design level of operation (the normal or expected values for set point and major disturbances).
Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level.
Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model.
Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.

direct acting process (Kp and Kc positive) −› use a reverse acting controller
reverse acting process (Kp and Kc negative) −› use a direct acting controller

Controller process architecture

2011-03-22T13:22:00.000-07:00

A controller seeks to maintain the measured process variable (PV) at set point (SP) in spite of unplanned and unmeasured disturbances. Since e(t) = SP - PV, this is equivalent to saying that a controller seeks to maintain controller error, e(t), equal to zero.

A controller repeats a measurement-computation-action procedure at every loop sample time, T. Starting at the far right of the control loop block diagram above:

A sensor measures a temperature, pressure, concentration or other property of interest from our process.
The sensor signal is transmitted to the controller. The pathway from sensor to controller might include: a transducer, an amplifier, a scaling element, quantization, a signal filter, a multiplexer, and other operations that can add delay and change the size, sign, and/or units of the measurement.
After all electronic and digital operations, the result terminates at our controller as the "wire in" measured process variable (PV) signal.
This "wire in" process variable is subtracted from set point in the controller to compute error, e(t) = SP - PV, which is then used in an algorithm (examples here and here) to compute a controller output (CO) signal.
The computed CO signal is transmitted on the "wire out" from the controller on a path to the final control element (FCE).
Similar to the measurement path, the signal from the controller to FCE might include filtering, scaling, linearization, amplification, multiplexing, transducing and other operations that can add delay and change the size, sign, and/or units of our original CO signal.
After any electronic and digital operations, the signal reaches the valve, pump, compressor or other FCE, causing a change in the manipulated variable (a liquid or gas stream flow rate, for example).
The change in the manipulated variable causes a change in our temperature, pressure, concentration or other process property of interest, all with the goal of making e(t) = 0.

Design Based on CO to PV Dynamics
The steps of the controller design and tuning recipe include: bumping the CO signal to generate CO to PV dynamic process data, approximating this test data with a first order plus dead time (FOPDT) model, and then using the model parameters in rules and correlations to complete the controller design and tuning.

The recipe provides a proven basis for controller design and tuning that avoids wasteful and expensive trial-and-error experiments. But for success, controller design and tuning must be based on process data as the controller sees it.

The controller only knows about the state of the process from the PV signal arriving on the "wire in" after all operations in the signal path from the sensor. It can only impact the state of the process with the CO signal it sends on the "wire out" before any such operations are made in the path to the final control element.

As indicated in the diagram at the top of this article, the proper signals that describe our complete "process" from the controller's view is the "wire out" CO and the "wire in" PV.

Complete the Circuit
Sometimes we find ourselves unable to proceed with an orderly controller design and tuning. Perhaps our controller interface does not make it convenient to directly record process data. Maybe we find a vendor's documentation to be so poorly written as to be all but worthless. There are a host of complications that can hinder progress.

Being resourceful, we may be tempted to move the project forward by using portable instrumentation. It seems reasonable to collect, say, temperature in a vessel during a bump test by inserting a spare thermocouple into the liquid. Or maybe we feel we can be more precise by standing right at the valve and using a portable signal generator to bump the process rather than doing so from a remote control panel.

As shown below, such an approach cuts out or short circuits the complete control loop pathway. External or portable instrumentation will not be recording the actual CO or PV as the controller sees it, and the data will not be appropriate for controller design or tuning.

Every Item Counts
The illustration above is extreme in that it shows many items that are not included in the control loop. But please recognize that it can be problematic to leave out even a single step in the complete signal pathway.

A simple scaling element that multiplies the signal by a constant value, for example, may seem reasonably unimportant to the overall loop dynamics. But this alone can change the size and even the sign of Kp, thus having dramatic impact on best tuning and final controller performance.

From a controller's view, the complete loop goes from "wire out" to "wire in" as shown below.

Every item in the loop counts. Always use the complete CO to PV data for process control analysis, design and tuning.

Pay Attention to Units
As detailed in this related article, signals can appear in a control loop in electronic units (e.g., volts, mA), in engineering units (e.g. oC, Lb/hr), as percent of scale (e.g., 0% to 100%), or as discrete or digital counts (e.g. 0 to 4095 counts).

It is critical that we remain aware of the units of a signal when working with a particular instrument or device. All values entered and computations performed must be consistent with the form of the data at that point in the loop.

Beyond the theory and methods discussed in this e-book, such "accounting confusion" can be one of the biggest challenges for the process control practitioner.

Nonlinear Process Behavior & DEEISGN OF CONTROL.

2011-03-22T13:20:00.000-07:00

Processes with streams comprised of gases, liquids, powders, slurries and melts tend to exhibit variations in behavior as operating level changes. This, in fact, is the very nature of a nonlinear process. For this reason, our recipe for controller design and tuning begins by specifying our design level of operation.

Controller Design and Tuning Recipe:

Establish the design level of operation (DLO), which is the normal or expected values for set point and major disturbances.
Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level.
Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model.
Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.

Nonlinear Behavior of the Gravity Drained Tanks
The dynamic behavior of the gravity drained tanks process is reasonably intuitive. Increase or decrease the inlet flow rate into the upper tank and the liquid level in the lower tank rises or falls in response.

One challenge this process presents is that its dynamic behavior is nonlinear. That is, the process gain, Kp; time constant, Tp; and/or dead time, Өp; changes as operating level changes. This is evident in the open loop response plot below.

As shown above, the CO is stepped in equal increments, yet the response behavior of the PV changes as the level in the tank rises. The consequence of nonlinear behavior is that a controller designed to give desirable performance at one operating level may not give desirable performance at another level.

Nonlinear Behavior of the Heat Exchanger
Nonlinear process behavior has important implications for controller design and tuning. Consider, for example, our heat exchanger process under PI control.

When tuned for a moderate response as shown in the first set point step from 140 °C to 155 °C in the plot below, the process variable (PV) responds in a manner consistent with our design goals. That is, the PV moves to the new set point (SP) reasonably quickly but does not overshoot the set point.

The consequence of a nonlinear process character is apparent as the set point steps continue to higher temperatures. In the third set point step from 170 °C to 185 °C, the same controller that had given a desired moderate performance now produces a PV response with a clear overshoot and some oscillation.

Such a change in performance with operating level may be tolerable in some applications and unacceptable in others. As we discuss in this article, "best" performance is something we judge for ourselves based on the goals of production, capabilities of the process, impact on down stream units and the desires of management

Nonlinear behavior should not catch us by surprise. It is something we can know about our process in advance. And this is why we should choose a design level of operation as a first step in our controller design and tuning procedure.

Step 1: Establish the Design Level of Operation (DLO)
Because, as shown in the examples above, processes have process gain, Kp; time constant, Tp; and/or dead time, Өp values that change as operating level changes, and these FOPDT model parameter values are used to complete the controller design and tuning procedure, it is important that dynamic process test data be collected at a pre-determined level of operation.

Defining this design level of operation (DLO) includes specifying where we expect the set point (SP) and measured process variable (PV) to be during normal operation, and the range of values the SP and PV might typically assume. This way we know where to explore the dynamic process behavior during controller design and tuning.

The DLO also considers our major disturbances (D). We should know the normal or typical values for our major disturbances. And we should be reasonably confident that the disturbances are quiet so we may proceed with a bump test to generate and record dynamic process data.

Step 2. Collect Dynamic Process Data Around the DLO
The next step in our recipe is to collect dynamic process data as near as practical to our design level of operation. We do this with a bump test, where we step or pulse the CO and collect data as the PV responds.

It is important to wait until the CO, PV and D have settled out and are as near to constant values as is possible for our particular operation before we start a bump test. The point of bumping a process is to learn about the cause and effect relationship between the CO and PV.

With the process at steady state, we are starting with a clean slate. As the PV responds to the CO bumps, the dynamic cause and effect behavior is isolated and evident in the data. On a practical note, be sure the data capture routine is enabled before the initial bump is implemented so all relevant data is collected.

Two popular open loop (manual mode) methods are the step test and the doublet test.

For either method, the CO must be moved far enough and fast enough to force a response in the PV that dominates the measurement noise.

Also, our bump should move the PV both above and below the DLO during testing. With data from each side of the DLO, the model (step 3) will be able to average out the nonlinear effects as discussed above.

Step Test

To collect data that will "average out" to our design level of operation, we start the test at steady state with the PV on one side of (either above or below) the DLO. Then, as shown in the plot below, we step the CO so that the measured PV moves across to settle on the other side of the DLO.

We can either start high and step the CO down (as shown above), or start low and step the CO up. Both methods produce dynamic data of equal value for our design and tuning recipe.

Doublet Test

A doublet test, as shown below, is two CO pulses performed in rapid succession and in opposite direction. The second pulse is implemented as soon as the process has shown a clear response to the first pulse that dominates the noise in the PV. It is not necessary to wait for the process to respond to steady state for either pulse.

The doublet test offers attractive benefits, including that it starts from and quickly returns to the DLO, it produces data both above and below the design level to "average out" the nonlinear effects, and the PV always stays close to the DLO, thus minimizing off-spec production. Such data does require commercial PID loop tuning software for model fitting, however.

Step 3: Fit a FOPDT dynamic model to Process Data
In fitting a first order plus dead time (FOPDT) model, we approximate those essential features of the dynamic process behavior that are fundamental to control. We need not understand differential equations to appreciate the articles on on this site, but for completeness, the first order plus dead time (FOPDT) dynamic model has the form:

Where:
PV(t) = measured process variable as a function of time
CO(t - Өp) = controller output signal as a function of time and shifted by Өp
Өp = process dead time
t = time
When the FOPDT dynamic model is fit to process data, the results describe how PV will respond to a change in CO via the model parameters. In particular:

Process gain, Kp, describes the direction and how far PV will travel,
Time constant, Tp, states how fast PV moves after it begins its response,
Dead time, Өp, is the delay from when CO changes until when PV begins to respond.

An example study that compares dynamic process data from the heat exchanger with a FOPDT model prediction can be found here. Comparisons between data and model for the gravity drained tanks can be found here and here.

Step 4: Use the model parameters to complete the design and tuning
In step 4, the three FOPDT model parameters are used in correlations to compute controller tuning values. For example, the chart below lists internal model control (IMC) tuning correlations for the PI controller and dependent ideal PID controller, and dependent ideal PID with CO filter forms:

The closed loop time constant, Tc, in the IMC correlations is used to specify the desired speed or quickness of our controller in responding to a set point change or rejecting a disturbance. The closed loop time constant is computed:

aggressive performance: Tc is the larger of 0.1·Tp or 0.8·Өp
moderate performance: Tc is the larger of 1·Tp or 8·Өp
conservative performance: Tc is the larger of 10·Tp or 80·Өp

Use the Recipe - It is Best Practice
The FOPDT dynamic model of step 3 also provides us the information we need to decide other controller design issues, including:

Controller Action

Before implementing our controller, we must input the proper direction our controller should move to correct for growing errors. Some vendors use the term "reverse acting" and "direct acting." Others use terms like "up-up" and "up-down" (as CO goes up, then PV goes up or down). This specification is determined solely by the sign of the process gain, Kp.

Loop Sample Time, T

Process time constant, Tp, is the clock of a process. The size of Tp indicates the maximum desirable loop sample time. Best practice is to set loop sample time, T, at 10 times per time constant or faster (T ≤ 0.1Tp). Faster may provide modestly improved performance. Slower than five times per time constant leads to significantly degraded performance.

Dead Time Problems

As dead time grows greater than the process time constant (when Өp > Tp), controller performance can benefit from a model based dead time compensator such as the Smith predictor.

Model Based Control

If we choose to employ a Smith predictor, or perhaps a dynamic feed forward element, a multivariable decoupler, or any other model based controller, we need a dynamic model of the process to enter into the control computer. The FOPDT model from step 3 of the recipe is usually appropriate for this task.

Sample Time Impacts Controller Performance

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There are two sample times, T, used in process controller design and tuning.

One is the control loop sample time that specifies how often the controller samples the measured process variable (PV) and then computes and transmits a new controller output (CO) signal.

The other is the rate at which CO and PV data are sampled and recorded during a bump test of our process. Bump test data is used to design and tune our controller prior to implementation.

In both cases, sampling too slow will have a negative impact on controller performance. Sampling faster will not necessarily provide better performance, but it is a safer direction to move if we have any doubts.

Fast and slow are relative terms defined by the process time constant, Tp. Best practice for both control loop sample time and bump test data collection are the same:

Best Practice:
Sample time should be 10 times per process time constant or faster (T ≤ 0.1Tp).

We explore this "best practice" rule in a detailed study here. This study employs some fairly advanced concepts, so it is placed further down in the Table of Contents.

Yet perhaps we can gain an appreciation for how sample time impacts controller design and tuning with this thought experiment:

Suppose you see me standing on your left. You close your eyes for a time, open them, and now I am standing on your right. Do you know how long I have been at my new spot? Did I just arrive or have I been there for a while? What path did I take to get there? Did I move around in front or in back of you? Maybe I even jumped over you?

Now suppose your challenge is to keep your hands at your side until I pass by, and just as I do, you are to reach out and touch me. What are your chances with your eyes closed (and loud music is playing so you cannot hear me)?

Now lets say you are permitted to blink open your eyes briefly once per minute. Do you think you will have a better chance of touching me? How about blinking once every ten seconds? Clearly, as you start blinking say, two or three times a second, the task of touching me becomes easy. That's because you are sampling fast enough to see my "process" behavior fully and completely.

Based on this thought experiment, sampling too slow is problematic and sampling faster is generally better.

Keep in mind the "T ≤ 0.1Tp" rule as we study PID control. This applies both to sampling during data collection, and the "measure and act" loop sample time when we implement our controller.

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Process Data, Dynamic Modeling and a Recipe for Profitable Control

It is best practice to follow a formal procedure or "recipe" when designing and tuning a PID (proportional-integral-derivative) controller. A recipe-based approach is the fastest method for moving a controller into operation. And perhaps most important, the performance of the controller will be superior to a controller tuned using a guess-and-test or trial-and-error method.

Additionally, a recipe-based approach overcomes many of the concerns that make control projects challenging in a commercial operating environment. Specifically, the recipe-based method causes less disruption to the production schedule, wastes less raw material and utilities, requires less personnel time, and generates less off-spec product.

The recipe for success is short:

Establish the design level of operation (DLO), defined as the expected values for set point and major disturbances during normal operation
Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level
Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model
Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.

We explore each step of this recipe in detail in other articles on this site. For now, we introduce some initial thoughts about steps 2 and 4.

Step 2: Bumping Our Process and Collecting CO to PV Data
From a controller's view, a complete control loop goes from wire out to wire in as shown below. Whenever we mention controller output (CO) or process variable (PV) data anywhere on this site, we are specifically referring to the data signals exiting and entering our controller at the wire termination interface.

To generate CO to PV data, we bump our process. That is, we step or pulse the CO (or the set point if in automatic mode as discussed here) and record PV data as the process responds. Here are three basic rules we follow in all of our examples:

Start with the process at steady state and record everything

The point of bumping the CO is to learn about the cause and effect relationship between it and the PV. With the plant initially at steady state, we are starting with a clean slate. The dynamic behavior of the process is then clearly isolated as the PV responds. It is important that we start capturing data before we make the initial CO bump and then sample and record quickly as the PV responds.

Make sure the PV response dominates the process noise

When performing a bump test, it is important that the CO moves far enough and fast enough to force a response that clearly dominates any noise or random error in the measured PV signal. If the CO to PV cause and effect response is clear enough to see by eye on a data plot, we can be confident that modern software can model it.

The disturbances should be quiet during the bump test

We desire that the dynamic test data contain PV response data that has been clearly, and in the ideal world exclusively, forced by changes in the CO.

Data that has been corrupted by unmeasured disturbances is of little value for controller design and tuning. The model (see below) will then incorrectly describe the CO to PV cause and effect relationship. And as a result, the controller will not perform correctly. If we are concerned that a disturbance event has corrupted test data, it is conservative to rerun the test.

Step 4: Using Model Parameters For Design and Tuning
The final step of the recipe states that once we have obtained model parameters that approximate the dynamic behavior of our process, we can complete the design and tuning of our PID controller.

We look ahead at this last step because this is where the payoff of the recipe-based approach is clear. To establish the merit, we assume for now that we have determined the design level of operation for our process (step 1), we have collected a proper data set rich in dynamic process information around this design level (step 2), and we have approximated the behavior revealed in the process data with a first order plus dead time (FOPDT) dynamic model (step 3).

Thankfully, we do not need to know what a FOPDT model is or even what it looks like. But we do need to know about the three model parameters that result when we fit this approximating model to process data.

The FOPDT (first order plus dead time) model parameters, listed below, tell us important information about the measured process variable (PV) behavior whenever there is a change in the controller output (CO) signal:

process gain, Kp (tells the direction and how far PV will travel)
process time constant, Tp (tells how fast PV moves after it begins its response)
process dead time, Өp (tells how much delay before PV first begins to respond)

Aside: we do not need to understand differential equations to appreciate the articles on this site. But for those interested, we note that the first order plus dead time (FOPDT) dynamic model has the form:

Where:

PV(t) = measured process variable as a function of time
CO(t - Өp) = controller output signal as a function of time and shifted by Өp Өp = process dead time t = time

The other variables are as listed above this box. It is a first order differential equation because it has one derivative with one time constant, Tp. It is called a first order plus dead time equation because it also directly accounts for a delay or dead time, Өp, in the CO(t) to PV(t) behavior.
We study what these three model parameters are and how to compute them in other articles, but here is why process gain, Kp, process time constant, Tp, and process dead time, Өp, are all important:

Tuning

These three model parameters can be plugged into proven correlations to directly compute P-Only, PI, PID, and PID with CO Filter tuning values. No more trial and error. No more tweaking our way to acceptable control. Great performance can be readily achieved with the step by step recipe listed above.

Controller Action

Loop Sample Time, T

Process time constant, Tp, is the clock of a process. The size of Tp indicates the maximum desirable loop sample time. Best practice is to set loop sample time, T, at 10 times per time constant or faster (T ≤ 0.1Tp). Sampling faster will not necessarily provide better performance, but it is a safer direction to move if we have any doubts. Sampling too slowly will have a negative impact on controller performance. Sampling slower than five times per time constant will lead to degraded performance.

Dead Time Problems

As dead time grows larger than the process time constant (Өp > Tp), the control loop can benefit greatly from a model based dead time compensator such as a Smith predictor. The only way we know if Өp > Tp is if we have followed the recipe and computed the parameters of a FOPDT model.

Model Based Control

If we choose to employ a Smith predictor, a dynamic feed forward element, a multivariable decoupler, or any other model based controller, we need a dynamic model of the process to enter into the control computer. The FOPDT model from step 2 of the recipe is often appropriate for this task.

Fundamental to Success
With tuning values, loop specifications, performance diagnostics and advanced control all dependent on knowledge of a dynamic model, we begin to see that process gain, Kp; process time constant, Tp; and process dead time, Өp; are parameters of fundamental importance to success in process control.

Control Loop Basics.

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Components of a Control Loop

A controller seeks to maintain the measured process variable (PV) at set point (SP) in spite of unmeasured disturbances (D). The major components of a control system include a sensor, a controller and a final control element. To design and implement a controller, we must:

have identified a process variable we seek to regulate, be able to measure it (or something directly related to it) with a sensor, and be able to transmit that measurement as an electrical signal back to our controller, and
have a final control element (FCE) that can receive the controller output (CO) signal, react in some fashion to impact the process (e.g., a valve moves), and as a result cause the process variable to respond in a consistent and predictable fashion.

Home Temperature Control
As shown below, the home heating control system can be organized as a traditional control loop block diagram. Block diagrams help us visualize the components of a loop and see how the pieces are connected.

A home heating system is simple on/off control with many of the components contained in a small box mounted on our wall. Nevertheless, we introduce the idea of control loop diagrams by presenting a home heating system in the same way we would a more sophisticated commercial control application.

Starting from the far right in the diagram above, our process variable of interest is house temperature. A sensor, such as a thermistor in a modern digital thermostat, measures temperature and transmits a signal to the controller.

The measured temperature PV signal is subtracted from set point to compute controller error, e(t) = SP - PV. The action of the controller is based on this error, e(t).
In our home heating system, the controller output (CO) signal is limited to open/close for the fuel flow solenoid valve (our FCE). So in this example, if e(t) = SP - PV > 0, the controller signals to open the valve. If e(t) = SP - PV < 0, it signals to close the valve. As an aside, note that there also must be a safety interlock to ensure that the furnace burner switches on and off as the fuel flow valve opens and closes.

As the energy output of the furnace rises or falls, the temperature of our house increases or decreases and a feedback loop is complete. The important elements of a home heating control system can be organized like any commercial application:

Control Objective: maintain house temperature at SP in spite of disturbances
Process Variable: house temperature
Measurement Sensor: thermistor; or bimetallic strip coil on analog models
Measured Process Variable (PV) Signal: signal transmitted from the thermistor
Set Point (SP): desired house temperature
Controller Output (CO): signal to fuel valve actuator and furnace burner
Final Control Element (FCE): solenoid valve for fuel flow to furnace
Manipulated Variable: fuel flow rate to furnace
Disturbances (D): heat loss from doors, walls and windows; changing outdoor temperature; sunrise and sunset; rain...

A General Control Loop and Intermediate Value Control
The home heating control loop above can be generalized into a block diagram pertinent to all feedback control loops as shown below:

Both diagrams above show a closed loop system based on negative feedback. That is, the controller takes actions that counteract or oppose any drift in the measured PV signal from set point.

While the home heating system is on/off, our focus going forward shifts to intermediate value control loops. An intermediate value controller can generate a full range of CO signals anywhere between full on/off or open/closed. The PI algorithm and PID algorithm are examples of popular intermediate value controllers.

To implement intermediate value control, we require a sensor that can measure a full range of our process variable, and a final control element that can receive and assume a full range of intermediate positions between full on/off or open/closed. This might include, for example, a process valve, variable speed pump or compressor, or heating or cooling element.

Note from the loop diagram that the process variable becomes our official PV only after it has been measured by a sensor and transmitted as an electrical signal to the controller. In industrial applications. these are most often implemented as 4-20 milliamps signals, though commercial instruments are available that have been calibrated in a host of amperage and voltage units.

With the loop closed as shown in the diagrams, we are said to be in automatic mode and the controller is making all adjustments to the FCE. If we were to open the loop and switch to manual mode, then we would be able to issue CO commands through buttons or a keyboard directly to the FCE. Hence:

open loop = manual mode
closed loop = automatic mode

Cruise Control and Measuring Our PV
Cruise control in a car is a reasonably common intermediate value control system. For those who are unfamiliar with cruise control, here is how it works.

We first enable the control system with a button on the car instrument panel. Once on the open road and at our desired cruising speed, we press a second button that switches the controller from manual mode (where car speed is adjusted by our foot) to automatic mode (where car speed is adjusted by the controller).

The speed of the car at the moment we close the loop and switch from manual to automatic becomes the set point. The controller then continually computes and transmits corrective actions to the gas pedal (throttle) to maintain measured speed at set point.

It is often cheaper and easier to measure and control a variable directly related to the process variable of interest. This idea is central to control system design and maintenance. And this is why the loop diagrams above distinguish between our "process variable" and our "measured PV signal."

Cruise control serves to illustrate this idea. Actual car speed is challenging to measure. But transmission rotational speed can be measured reliably and inexpensively. The transmission connects the engine to the wheels, so as it spins faster or slower, the car speed directly increases or decreases.

Thus, we attach a small magnet to the rotating output shaft of the car transmission and a magnetic field detector (loops of wire and a simple circuit) to the body of the car above the magnet. With each rotation, the magnet passes by the detector and the event is registered by the circuitry as a "click." As the drive shaft spins faster or slower, the click rate and car speed increase or decrease proportionally.

So a cruise control system really adjusts fuel flow rate to maintain click rate at the set point value. With this knowledge, we can organize cruise control into the essential design elements:

Control Objective: maintain car speed at SP in spite of disturbances
Process Variable: car speed
Measurement Sensor: magnet and coil to clock drive shaft rotation
Measured Process Variable (PV) Signal: "click rate" signal from the magnet and coil
Set Point (SP): desired car speed, recast in the controller as a desired click rate
Controller Output (CO): signal to actuator that adjusts gas pedal (throttle)
Final Control Element (FCE): gas pedal position
Manipulated Variable: fuel flow rate
Disturbances (D): hills, wind, curves, passing trucks…

The traditional block diagram for cruise control is thus:

Instruments Should be Fast, Cheap and Easy
The above magnet and coil "click rate = car speed" example introduces the idea that when purchasing an instrument for process control, there are wider considerations that can make a loop faster, easier and cheaper to implement and maintain. Here is a "best practice" checklist to use when considering an instrument purchase:

Low cost
Easy to install and wire
Compatible with existing instrument interface
Low maintenance
Rugged and robust
Reliable and long lasting
Sufficiently accurate and precise
Fast to respond (small time constant and dead time)
Consistent with similar instrumentation already in the plant

Motivation and Terminology of Automatic Process Control

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Automatic control systems enable us to operate our processes in a safe and profitable manner. Consider, as on this site, processes with streams comprised of gases, liquids, powders, slurries and melts. Control systems achieve this "safe and profitable" objective by continually measuring process variables such as temperature, pressure, level, flow and concentration - and taking actions such as opening valves, slowing down pumps and turning up heaters - all so that the measured process variables are maintained at operator specified set point values.

Safety First
The overriding motivation for automatic control is safety, which encompasses the safety of people, the environment and equipment.

The safety of plant personnel and people in the community are the highest priority in any plant operation. The design of a process and associated control system must always make human safety the primary objective.

The tradeoff between safety of the environment and safety of equipment is considered on a case by case basis. At the extremes, the control system of a multi-billion dollar nuclear power facility will permit the entire plant to become ruined rather than allow significant radiation to be leaked to the environment.

On the other hand, the control system of a coal-fired power plant may permit a large cloud of smoke to be released to the environment rather than allowing damage to occur to, say, a single pump or compressor worth a few thousand dollars.

The Profit Motive
When people, the environment and plant equipment are properly protected, our control objectives can focus on the profit motive. Automatic control systems offer strong benefits in this regard.

Plant-level control objectives motivated by profit include:

meeting final product specifications
minimizing waste production
minimizing environmental impact
minimizing energy use
maximizing overall production rate

It can be most profitable to operate as close as possible to these minimum or maximum objectives. For example, our customers often set our product specifications, and it is essential that we meet them if failing to do so means losing a sale.

Suppose we are making a film or sheet product. It takes more raw material to make a product thicker than the minimum our customers will accept on delivery. Consequently, the closer we can operate to the minimum permitted thickness constraint without going under, the less material we use and the greater our profit.

Or perhaps we sell a product that tends to be contaminated with an impurity and our customers have set a maximum acceptable value for this contaminant. It takes more processing effort (more money) to remove impurities, so the closer we can operate to the maximum permitted impurity constraint without going over, the greater the profit.

Whether it is a product specification, energy usage, production rate, or other objective, approaching these targets ultimately translates into operating the individual process units within the plant as close as possible to predetermined set point values for temperature, pressure, level, flow, concentration and the other measured process variables.

Controllers Reduce Variability
As shown in the plot below, a poorly controlled process can exhibit large variability in a measured process variable (e.g., temperature, pressure, level, flow, concentration) over time.

Suppose, as in this example, the measured process variable (PV) must not exceed a maximum value. And as is often the case, the closer we can run to this operating constraint, the greater our profit (note the vertical axis label on the plot).

To ensure our operating constraint limit is not exceeded, the operator-specified set point (SP), that is, the point where we want the control system to maintain our PV, must be set far from the constraint to ensure it is never violated. Note in the plot that SP is set at 50% when our PV is poorly controlled.

Below we see the same process with improved control. There is significantly less variability in the measured PV, and as a result, the SP can be moved closer to the operating constraint.

With the SP in the plot below moved to 55%, the average PV is maintained closer to the specification limit while still remaining below the maximum allowed value. The result is increased profitability of our operation.

Terminology of Control
We establish the jargon for this site by discussing a home heating control system as illustrated below.

This is a simplistic example because a home furnace is either on or off. Most control challenges have a final control element (FCE), such as a valve, pump or compressor, that can receive and respond to a complete range of controller output (CO) signals between full on and full off. This would include, for example, a valve that can be open 37% or a pump that can be running at 73%.

For our home heating process, the control objective is to keep the measured process variable (PV) at the set point value (SP) in spite of unmeasured disturbances (D).

For our home heating system:

PV = process variable is house temperature
CO = controller output signal from thermostat to furnace valve
SP = set point is the desired temperature set on the thermostat by the home owner
D = heat loss disturbances from doors, walls and windows; changing outdoor temperature; sunrise and sunset; rain...

To achieve this control objective, the measured process variable is compared to the thermostat set point. The difference between the two is the controller error, which is used in a control algorithm such as a PID (proportional-integral-derivative) controller to compute a CO signal to the final control element (FCE).

The change in the controller output (CO) signal causes a response in the final control element (fuel flow valve), which subsequently causes a change in the manipulated process variable (flow of fuel to the furnace). If the manipulated process variable is moved in the right direction and by the right amount, the measured process variable will be maintained at set point, thus satisfying the control objective.

This example, like all in process control, involves a measurement, computation and action:

is the measured temp colder than set point (SP - PV > 0)? Then open the valve.
is the measured temp hotter than set point (SP - PV < 0)? Then close the valve.

Note that computing the necessary controller action is based on controller error, or the difference between the set point and the measured process variable, i.e.

e(t) = SP - PV (error = set point - measured process variable)

In a home heating process, control is an on/off or open/close decision. And as outlined above, it is a straightforward decision to make. The price of such simplicity, however, is that the capability to tightly regulate our measured PV is rather limited.

One situation not addressed above is the action to take when PV = SP (i.e., e(t) = 0). And in industrial practice, we are concerned with variable position final control elements, so the challenge elevates to computing:

the direction to move the valve, pump, compressor, heating element...
how far to move it at this moment
how long to wait before moving it again
whether there should be a delay between measurement and action

AC/DC motor details.

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Basic Motor Theory

Introduction

It has been said that if the Ancient Romans, with their advanced civilization and knowledge of the sciences, had been able to develop a steam motor, the course of history would have been much different. The development of the electric motor in modern times has indicated the truth in this theory. The development of the electric motor has given us the most efficient and effective means to do work known to man. Because of the electric motor we have been able to greatly reduce the painstaking toil of man's survival and have been able to build a civilization which is now reaching to the stars. The electric motor is a simple device in principle. It converts electric energy into mechanical energy. Over the years, electric motors have changed substantially in design, however the basic principles have remained the same. In this section of the Action Guide we will discuss these basic motor principles. We will discuss the phenomena of magnetism, AC current and basic motor operation.

Magnetism

Now, before we discuss basic motor operation a short review of magnetism might be helpful to many of us. We all know that a permanent magnet will attract and hold metal objects when the object is near or in contact with the magnet. The permanent magnet is able to do this because of its inherent magnetic force which is referred to as a "magnetic field". In Figure 1 , the magnetic field of two permanent magnets are represented by "lines of flux". These lines of flux help us to visualize the magnetic field of any magnet even though they only represent an invisible phenomena. The number of lines of flux vary from one magnetic field to another. The stronger the magnetic field, the greater the number of lines of flux which are drawn to represent the magnetic field. The lines of flux are drawn with a direction indicated since we should visualize these lines and the magnetic field they represent as having a distinct movement from a N-pole to a S-pole as shown in Figure 1. Another but similar type of magnetic field is produced around an electrical conductor when an electric current is passed through the conductor as shown in Figure 2-a. These lines of flux define the magnetic field and are in the form of concentric circles around the wire. Some of you may remember the old "Left Hand Rule" as shown in Figure 2-b. The rule states that if you point the thumb of your left hand in the direction of the current, your fingers will point in the direction of the magnetic field.

Figure 1 - The lines of flux of a magnetic field travel from the N-pole to the S-pole.

Figure 2 - The flow of electrical current in a conductor sets up concentric lines of magnetic flux around the conductor.

Figure 3 - The magnetic lines around a current carrying conductor leave from the N-pole and re-enter at the S-pole.

When the wire is shaped into a coil as shown in Figure 3, all the individual flux lines produced by each section of wire join together to form one large magnetic field around the total coil. As with the permanent magnet, these flux lines leave the north of the coil and re-enter the coil at its south pole. The magnetic field of a wire coil is much greater and more localized than the magnetic field around the plain conductor before being formed into a coil. This magnetic field around the coil can be strengthened even more by placing a core of iron or similar metal in the center of the core. The metal core presents less resistance to the lines of flux than the air, thereby causing the field strength to increase. (This is exactly how a stator coil is made; a coil of wire with a steel core.) The advantage of a magnetic field which is produced by a current carrying coil of wire is that when the current is reversed in direction the poles of the magnetic field will switch positions since the lines of flux have changed direction. This phenomenon is illustrated in Figure 4. Without this magnetic phenomenon existing, the AC motor as we know it today would not exist.

Figure 4 - The poles of an electro-magnetic coil change when the direction of current flow changes.

Magnetic Propulsion Within A Motor

The basic principle of all motors can easily be shown using two electromagnets and a permanent magnet. Current is passed through coil no. 1 in such a direction that a north pole is established and through coil no. 2 in such a direction that a south pole is established. A permanent magnet with a north and south pole is the moving part of this simple motor. In Figure 5-a the north pole of the permanent magnet is opposite the north pole of the electromagnet. Similarly, the south poles are opposite each other. Like magnetic poles repel each other, causing the movable permanent magnet to begin to turn. After it turns part way around, the force of attraction between the unlike poles becomes strong enough to keep the permanent magnet rotating. The rotating magnet continues to turn until the unlike poles are lined up. At this point the rotor would normally stop because of the attraction between the unlike poles. (Figure 5-b)

Figure 5 If, however, the direction of currents in the electromagnetic coils was suddenly reversed, thereby reversing the polarity of the two coils, then the poles would again be opposites and repel each other. (Figure 5-c). The movable permanent magnet would then continue to rotate. If the current direction in the electromagnetic coils was changed every time the magnet turned 180 degrees or halfway around,then the magnet would continue to rotate. This simple device is a motor in its simplest form. An actual motor is more complex than the simple device shown above, but the principle is the same.

AC Current

How is the current reversed in the coil so as to change the coils polarity, you ask. Well, as you probably know, the difference between DC and AC is that with DC the current flows in only one direction while with AC the direction of current flow changes periodically. In the case of common AC that is used throughout most of the United States, the current flow changes direction 120 times every second. This current is referred to as "60 cycle AC" or "60 Hertz AC" in honor of Mr. Hertz who first conceived the AC current concept. Another characteristic of current flow is that it can vary in quantity. We can have a 5 amp, 10 amp or 100 amp flow for instance. With pure DC, this means that the current flow is actually 5,10, or 100 amps on a continuous basis. We can visualize this on a simple time-current graph by a straight line as shown in Figure 6.

Figure 6 - Visualization of DC
But with AC it is different. As you can well imagine, it would be rather difficult for the current to be flowing at say 100 amps in a positive direction one moment and then at the next moment be flowing at an equal intensity in the negative direction. Instead, as the current is getting ready to change directions, it first tapers off until it reaches zero flow and then gradually builds up in the other direction. See Figure 7. Note that the maximum current flow (the peaks of the line) in each direction is more than the specified value (100 amps in this case). Therefore, the specified value is given as an average. It is actually called a "root mean square" value, but don't worry about remembering this because it is of no importance to us at this time. What is important in our study of motors, is to realize that the strength of the magnetic field produced by an AC electro-magnetic coil increases and decreases with the increase and decrease of this alternating current flow.

Basic AC Motor Operation

An AC motor has two basic electrical parts: a "stator" and a "rotor" as shown in Figure 8. The stator is in the stationary electrical component. It consists of a group of individual electro-magnets arranged in such a way that they form a hollow cylinder, with one pole of each magnet facing toward the center of the group. The term, "stator" is derived from the word stationary. The stator then is the stationary part of the motor. The rotor is the rotating electrical component. It also consists of a group of electro-magnets arranged around a cylinder, with the poles facing toward the stator poles. The rotor, obviously, is located inside the stator and is mounted on the motor's shaft. The term "rotor" is derived from the word rotating. The rotor then is the rotating part of the motor. The objective of these motor components is to make the rotor rotate which in turn will rotate the motor shaft. This rotation will occur because of the previously discussed magnetic phenomenon that unlike magnetic poles attract each other and like poles repel. If we progressively change the polarity of the stator poles in such a way that their combined magnetic field rotates, then the rotor will follow and rotate with the magnetic field of the stator.

Figure 8 - Basic electrical components of an AC motor.

This "rotating magnetic fields of the stator can be better understood by examining Figure 9. As shown, the stator has six magnetic poles and the rotor has two poles. At time 1, stator poles A-1 and C-2 are north poles and the opposite poles, A-2 and C-1, are south poles. The S-pole of the rotor is attracted by the two N-poles of the stator and the N-pole of the rotor is attracted by the two south poles of the stator. At time 2, the polarity of the stator poles is changed so that now C-2 and B-1 and N-poles and C-1 and B-2 are S-poles. The rotor then is forced to rotate 60 degrees to line up with the stator poles as shown. At time 3, B-1 and A-2 are N. At time 4, A-2 and C-1 are N. As each change is made, the poles of the rotor are attracted by the opposite poles on the stator. Thus, as the magnetic field of the stator rotates, the rotor is forced to rotate with it.

Figure 9 - The rotating magnetic field of an AC motor. One way to produce a rotating magnetic field in the stator of an AC motor is to use a three-phase power supply for the stator coils. What, you may ask, is three-phase power? The answer to that question can be better understood if we first examine single-phase power. Figure 7 is the visualization of single-phase power. The associated AC generator is producing just one flow of electrical current whose direction and intensity varies as indicated by the single solid line on the graph. From time 0 to time 3, current is flowing in the conductor in the positive direction. From time 3 to time 6, current is flowing in the negative. At any one time, the current is only flowing in one direction. But some generators produce three separate current flows (phases) all superimposed on the same circuit. This is referred to as three-phase power. At any one instant, however, the direction and intensity of each separate current flow are not the same as the other phases. This is illustrated in Figure 10. The three separate phases (current flows) are labeled A, B and C. At time 1, phase A is at zero amps, phase B is near its maximum amperage and flowing in the positive direction, and phase C is near to its maximum amperage but flowing in the negative direction. At time 2, the amperage of phase A is increasing and flow is positive, the amperage of phase B is decreasing and its flow is still negative, and phase C has dropped to zero amps. A complete cycle (from zero to maximum in one direction, to zero and to maximum in the other direction, and back to zero) takes one complete revolution of the generator. Therefore, a complete cycle, is said to have 360 electrical degrees. In examining Figure 10, we see that each phase is displaced 120 degrees from the other two phases. Therefore, we say they are 120 degrees out of phase.

Figure 10 - The pattern of the separate phases of three-phase power. To produce a rotating magnetic field in the stator of a three-phase AC motor, all that needs to be done is wind the stator coils properly and connect the power supply leads correctly. The connection for a 6 pole stator is shown in Figure 11. Each phase of the three-phase power supply is connected to opposite poles and the associated coils are wound in the same direction. As you will recall from Figure 4, the polarity of the poles of an electro-magnet are determined by the direction of the current flow through the coil. Therefore, if two opposite stator electro-magnets are wound in the same direction, the polarity of the facing poles must be opposite. Therefore, when pole A1 is N, pole A2 is S. When pole B1 is N, B2 is S and so forth.

Figure 11 - Method of connecting three-phase power to a six-pole stator.
Figure 12 shows how the rotating magnetic field is produced. At time1, the current flow in the phase "A" poles is positive and pole A-1 is N. The current flow in the phase "C" poles is negative, making C-2 a N-pole and C-1 is S. There is no current flow in phase "B", so these poles are not magnetized. At time 2, the phases have shifted 60 degrees, making poles C-2 and B-1 both N and C-1 and B-2 both S. Thus, as the phases shift their current flow, the resultant N and S poles move clockwise around the stator, producing a rotating magnetic field. The rotor acts like a bar magnet, being pulled along by the rotating magnetic field.

igure 12 - How three-phase power produces a rotating magnetic field.
Up to this point not much has been said about the rotor. In the previous examples, it has been assumed the rotor poles were wound with coils, just as the stator poles, and supplied with DC to create fixed polarity poles. This, by the way, is exactly how a synchronous AC motor works. However, most AC motors being used today are not synchronous motors. Instead, so-called "induction" motors are the workhorses of industry. So how is an induction motor different? The big difference is the manner in which current is supplied to the rotor. This is no external power supply. As you might imagine from the motor's name, an induction technique is used instead. Induction is another characteristic of magnetism. It is a natural phenomena which occurs when a conductor (aluminum bars in the case of a rotor, see Figure 13) is moved through an existing magnetic field or when a magnetic field is moved past a conductor. In either case, the relative motion of the two causes an electric current to flow in the conductor. This is referred to as "induced" current flow. In other words, in an induction motor the current flow in the rotor is not caused by any direct connection of the conductors to a voltage source, but rather by the influence of the rotor conductors cutting across the lines of flux produced by the stator magnetic fields. The induced current which is produced in the rotor results in a magnetic field around the rotor conductors as shown in Figure 14. This magnetic field around each rotor conductor will cause each rotor conductor to act like the permanent magnet in the Figure 9 example. As the magnetic field of the stator rotates, due to the effect of the three-phase AC power supply, the induced magnetic field of the rotor will be attracted and will follow the rotation. The rotor is connected to the motor shaft, so the shaft will rotate and drive the connection load. That's how a motor works! Simple, was it not?

Figure 13 - Construction of an AC induction motor's rotor.

Figure 14 - How voltage is induced in the rotor, resulting in current flow in the rotor conductors.

DC Motor Theory

Preface

The intent of this paper is to provide one with an understanding of DC Motors in order that they can be applied with confidence. This paper contains basic information and specific information that applies to Reliance Medium HP and Large HP DC Motors. Due to the nature of Baldor Systems business, emphasis has been placed on the Large DC motor product line.

Section 1: Dynamo Development

The first generators and motors were called dynamos or dynamoelertric machines. Dynamo is from the Greek word dynamis which means power. Webster defines dynamoelectric as "relating to the conversion of mechanical energy into electrical energy or vice versa". The word motor is from the Latin word motus which means one that imparts motion or prime mover. The dynamo was the result of the efforts of several people, in different countries, in the mid-nineteenth century, to make electricity work for them.

Definitions

Dynamo:	From the Greek word dynamis, which means power
Dynamoelectric:	Relating to the conversion by induction of mechanical energy into electrical energy or vice versa
Dynamoelectric machine:	A dynamo or generator
Motor:	From the Latin word motus, one that imparts motion, prime mover. A device that changes electrical energy into mechanical energy.
Generator:	A device that changes mechanical energy into electrical energy. Although the terms AC and DC generator are in common usage, a generator is normally considered to be a device that provides DC current.
Alternator:	A device that changes mechanical energy into an alternating current electrical energy, an AC generator.

Landmarks Of Electric Motor Development

1820 The discovery of electromagnetism Hans Christian Oersted, Danish

1827 The statement of the law of electric conduction, Ohm's law George S. Ohm, German

1830 The discovery of electromagnetic induction Joseph Henry, American

1831 The discovery of electromagnetic induction Michael Faraday, English

The first practical dynamo, about 1867

Section 2: Electric Motor And Generator Basics

Electrodynamic Principles

Faraday's Law
In order that current can be obtained from an electric circuit, an electromotive force (voltage) must be established and maintained between the two ends of the circuit. This electromotive force may be established in several ways, one of which is by means of an electromagnetic generator.
Michael Faraday discovered that an electric potential can be established between the ends of a conductor in the following three ways:

By a conductor moving or cutting across a stationary magnetic field. (DC Generator)
By a moving magnetic field cutting across a stationary conductor. (AC Generator)
By a change in the number of magnetic lines enclosed by a stationary loop or coil. (Transformer)

Faraday's law states that, "the EMF (electromotive force) induced between the ends of a loop or coil is proportional to the rate of change of magnetic flux enclosed by the coil; or the EMF induced between the ends of a bar conductor is proportional to the time rate at which magnetic flux is cut by the conductor."
This law emphasizes rate of change or rate or flux cutting rather than density or extent of magnetic field.

Lenz's Law

Lenz's Law states that, "A change in the magnetic flux passing through or linking with, a loop or coil causes EMF to be induced in a direction to oppose any change in circuit conditions, this opposition being produced magnetically when current flows in response to the induced EMF."

Whenever there is a change in current in a magnetizing coil, which works to change the flux in the coil, a voltage is induced which tends to prevent the change. Thus, if we attempt to diminish the current flowing in a magnetizing coil, a voltage will be developed that will tend to keep the current unchanged. Likewise, if we attempt to establish a current in a magnetizing coil, a voltage will be developed that will tend to keep the current from increasing.

Generator Basic Principles

Energy Conversion
To produce voltage, it is necessary to move a conductor through a magnetic field as stated above. Mechanical energy is required to provide motion to this conductor. With the field energy remaining constant, the conductor is changing mechanical energy into electrical energy.
Voltage Generation
There is a definite relationship between the direction of the magnetic flux, the direction of motion of the conductor and the direction of the induced EMF. Figure 1 shows the motion of the conductor perpendicular to the magnetic field. The voltage and current output are perpendicular to both the motion of the conductor and the magnetic field.

Figure 1.
Voltage Generation

To illustrate this with Fleming's right hand rule, the thumb and first two fingers of the right hand are extended at right angles to one another, the thumb will indicate the direction of motion of the conductor, the forefinger will indicate the direction of the magnetic field, and the middle finger will indicate the direction of voltage or current.

Applying this rule, one can see that the current will reverse if the motion of the conductor changes from down to up. This is true even though the magnetic field does not change position. Therefore, the rotating coil in Figure 2 will produce a voltage which is continually changing direction.

Figure 2.
Revolving Coil in a Magnetic Field

Voltage Induced in Conductor Moving Through a Magnetic Field
Revolving Coil in a Magnetic Field

The coil in position AB, in figure 2, encloses the maximum amount of flux. The flux decreases as the coil moves toward position CD and becomes zero at CD, since the plane of the coil is parallel to the magnetic field. Then the flux increases in the opposite direction, reaching a negative maximum at BA and diminishing again to zero at DC. The flux reverses and increases again in the original direction to reach a maximum at AB.
Although the flux is maximum at positions AB and BA and zero at positions CD and DC, the induced EMF will be maximum at positions CD and DC and zero at positions AB and BA. This is true because the EMF depends upon the rate of change of flux or rate of cutting flux lines and not upon the quantity enclosed.
If the coil in Figure 2 were rotated at a constant speed in a uniform magnetic field, a sine wave of voltage would be obtained. This is shown in Figure 3 where both the amount of flux enclosed and the EMF induced are plotted against time.

Figure 3.
Voltage Sine Wave Produced by rotation of a coil at constant speed in a uniform magnetic field.

Value of Generated Voltage
The EMF at any instant of time is proportional to the number of turns in the coil times rate of change of flux. The C.G.S. (centimeter gram second) unit of EMF known as the abvolt is defined as that value induced, in a coil of one turn, when the flux linking with the coil is changing at the rate of one line or Maxwell per second; or as that value induced when magnetic flux is being cut by the conductor at the rate of one line per second. A volt is equal to 108 abvolts or an abvolt is equal to 10^-8 volts. Therefore, the instantaneous value of voltage is expressed as:

e = N x (d / dt) x 10^-8

where:

	e = voltage
	N = the number of turns
	d / dt = the rate of change of flux

This equation can be further developed to obtain the voltage for movement of a conductor at constant velocity through a uniform magnetic field:

E = N B v sin x 10^-8

where:

	E = voltage
	N = number of turns
	B = flux density in lines per square inch
	= length of the conductor in inches
	v = velocity in inches per second
	= the angle between the conductor and flux field

If the conductor moves directly across the field at right angles to it, then = 90° and sin = 1. The equation then becomes:

E = N B v x 10^-8

It should be noted that this equation is a special form of the original equation and is not applicable in all cases.

MOTOR BASIC PRINCIPLES

Energy Conversion
As stated above, mechanical energy is changed into electrical energy by movement of conductor through a magnetic field. The converse of this is also true. If electrical energy is supplied to a conductor lying normal to a magnetic field, resulting in current flow in the conductor, a mechanical force and thus mechanical energy will be produced.
Producing Mechanical Force
As in the generator, the motor has a definite relationship between the direction of the magnetic flux, the direction of motion of the conductor or force, and the direction of the applied voltage or current.
Since the motor is the reverse of the generator, Fleming's left hand rule can be used. If the thumb and first two fingers of the left hand are extended at right angles to one another, the thumb will indicate the direction of motion, the forefinger will indicate the direction of the magnetic field, and the middle finger will indicate the direction of current. In either the motor or generator, if the directions of any two factors are known, the third can be easily determined.
Value of Mechanical Force
The force exerted upon a current carrying conductor is dependent upon the density of the magnetic field, the length of conductor, and the value of current flowing in the conductor. Assuming that the conductor is located at right angles to the magnetic field, the force developed can be expressed as follows:

F = (B I) / 10

where:

	F = force in dynes
	B = flux density in lines per square centimeter
	= length of the conductor in centimeters
	I = current in amperes.

At the same time torque is being produced, the conductors are moving in a magnetic field and generating a voltage. This voltage is in opposition to the voltage that causes current flow through the conductor and is referred to as a countervoltage or back EMF. The value of current flowing through the armature is dependent upon the difference between the applied voltage and the countervoltage.

Sample Calculations

Generator
Given:

	N = 60 turns
	B = 40,000 lines per square inch
	= 3.0 inches
	v = 600 inches per second

Find:

	E = voltage
		E = 60 x 40,000 x 3 x 600 x 10-8 = 43.2 volts

Motor
Given:

	B = 6,000 lines per square centimeter
	= 10 Centimeters
	I = 50 amps

Find:

	F = force
	F = (6,000 x 10 x 50) / 10 = 300,000 dynes
	Newtons = Pounds x 4.44823
	Dynes = Newtons x 100,000

DC Machines, Principles Of Operation

Generator

In a generator, moving a conductor through a stationary magnetic field generates voltage. If a coil is rotated through a magnetic field as shown in Figure 4, an alternating voltage will be produced. To make this voltage available to a stationary external circuit, two slip rings and brushes must be provided. For the external circuit to produce DC voltage, it is necessary to reverse the polarity of the external leads at the same time the voltage in the coil is reversed. This is accomplished by segmenting a slip ring to form what is called a commutator. An elementary two segment commutator is illustrated in Figure 5. This single coil, two piece commutator will yield an unidirectional but pulsating voltage as shown in Figure 6. However, when a large number of commutator segments or bars is used, the resulting voltage will be more uniform as shown in Figure 7.

Figure 4.
Brushes and slip rings provide AC voltage

Figure 5.
Brushes and Commutator provides DC voltage

Figure 6.
Unidirectional, Pulsating Voltage

Figure 7.
Uniform DC Voltage

As stated above, the generated voltage in a single conductor is:

E = N B v x 10^-8

where:

	B = flux density in lines per square inch
	= length of the conductor in inches
	v = velocity in inches per second

This equation can be developed to the following equation for DC machines:

E = (Z / paths) x x poles x (rpm / 60) x 10^-8

where:

	Z = total number of conductors
	= flux per pole in lines

This equation represents the average voltage. For a given machine, it can be reduced to:

E = K₁ S

where:

	= flux per pole
	S = speed in rpm
	K₁ = all other factors

Motor

As stated previously, if current is supplied to a conductor in a magnetic field, a force will be produced. The force developed in a single conductor is:

F = (B I) / 10

where:

	F = force in dynes
	B = flux density in lines per square centimeter
	= length of the conductor in centimeters
	I = current in amperes

This equation can be developed to the following for DC motors:

T = 11.73 x (Z / paths) x x poles x I_A x 10^-10

where:

	T = torque in ft-lb
	Z = total number of conductors
	= flux per pole in lines
	I = current in amperes

For a given machine, this can be reduced to:

T = K₂ IA

where:

	= flux per pole in lines
	I_A = current in amperes
	K₂ = all other factors

K₂ is not the same as the K₁ for voltage. The above torque is not the output torque of the shaft, but rather the total torque developed by the armature. Part of this total torque is needed to overcome the inertia of the armature itself.
The horsepower output of any motor can be expressed as:

HP = T x N / C

where:

	T = output torque in ft-lb
	N = speed in rpm
	C = the constant 5252

DC General Construction

A typical DC generator or motor usually consists of: An armature core, an air gap, poles, and a yoke which form the magnetic circuit; an armature winding, a field winding, brushes and a commutator which form the electric circuit; and a frame, end bells, bearings, brush supports and a shaft which provide the mechanical support. See figure 8.

Figure 8.

Four Pole DC Motor

Armature Core or Stack
The armature stack is made up thin magnetic steel laminations stamped from sheet steel with a blanking die. Slots are punched in the lamination with a slot die. Sometimes these two operations are done as one. The laminations are welded, riveted, bolted or bonded together.

Armature Winding
The armature winding is the winding, which fits in the armature slots and is eventually connected to the commutator. It either generates or receives the voltage depending on whether the unit is a generator or motor. The armature winding usually consists of copper wire, either round or rectangular and is insulated from the armature stack.

Field Poles
The pole cores can be made from solid steel castings or from laminations. At the air gap, the pole usually fans out into what is known as a pole head or pole shoe. This is done to reduce the reluctance of the air gap. Normally the field coils are formed and placed on the pole cores and then the whole assembly is mounted to the yoke.

Field Coils
The field coils are those windings, which are located on the poles and set up the magnetic fields in the machine. They also usually consist of copper wire are insulated from the poles. The field coils may be either shunt windings (in parallel with the armature winding) or series windings (in series with the armature winding) or a combination of both.

Yoke
The yoke is a circular steel ring, which supports the field, poles mechanically and provides the necessary magnetic path between the pole. The yoke can be solid or laminated. In many DC machines, the yoke also serves as the frame.

Commutator
The commutator is the mechanical rectifier, which changes the AC voltage of the rotating conductors to DC voltage. It consists of a number of segments normally equal to the number of slots. The segments or commutator bars are made of silver bearing copper and are separated from each other by mica insulation.

Brushes and Brush Holders
Brushes conduct the current from the commutator to the external circuit. There are many types of brushes. A brush holder is usually a metal box that is rectangular in shape. The brush holder has a spring that holds the brush in contact with the commutator. Each brush usually has a flexible copper shunt or pigtail, which extends to the lead wires. Often, the entire brush assembly is insulated from the frame and is made movable as a unit about the commutator to allow for adjustment.

Interpoles
Interpoles are similar to the main field poles and located on the yoke between the main field poles. They have windings in series with the armature winding. Interpoles have the function of reducing the armature reaction effect in the commutating zone. They eliminate the need to shift the brush assembly.

Frame, End Bells, Shaft, and Bearings
The frame and end bells are usually steel, aluminum or magnesium castings used to enclose and support the basic machine parts. The armature is mounted on a steel shaft, which is supported between two bearings. The bearings are either sleeve, ball or roller type. They are normally lubricated by grease or oil.

Back End, Front End
The load end of the motor is the Back End. The opposite load end, most often the commutator end, is the Front End of the motor.

Armature Windings

Gramme Ring Winding

The old Gramme Ring type winding, now obsolete, is shown in Figure 9 and its equivalent circuit in Figure 10. It can be seen that there are an equal number of voltage-generating conductors on each side of the armature and the conductor voltages are additive from bottom to top on each side. There are two paths between the positive and negative brushes and the voltage per path is the generated voltage of the machine. Each path provides half of the current output.

Figure 9.
Two Pole Gramme Ring Winding

Figure 10.
Equivalent Circuit, Two Pole Gramme Ring Winding

Drum Winding
The Drum type winding is made of coils, one of which is illustrated in Figure 11. The straight portions of the coil are the parts rotating through the magnetic field in which the voltage is induced. Therefore, each single coil has two conductors. This has the advantage over the Gramme Ring winding where only one side of each coil is used as an active conductor. There are two classes of drum windings depending upon how the coils are connected to the commutator.

Figure 11.
Drum Type Winding Coil

Lap Winding
When the end connections of the coils are brought to adjacent bars as shown in Figure 12, a lap or parallel winding is formed. In this type winding, there are as many paths through the armature as there are poles on the machine. Therefore, to obtain full use of this type winding, there must be as many brushes as there are poles, alternate brushes being positive and negative. Any winding can be illustrated in one of two forms, the circular form or the development form. A simplex lap winding is shown in Figure 13 (circular form) and Figure 14 (development form.) In this particular circular form, the flux cutting portions of the conductors are shown as straight lines radiating from the center and are numbered for convenience in connecting them to the commutator which is in the center of the diagram. The outermost connecting lines represent the end connections on the back of the armature and the inner connecting lines represent the connections on the front or commutator end of the armature. The development form of winding represents the armature as if it were split open and rolled out flat. It is somewhat simpler to understand but the continuity of the winding is broken. The lap winding is best suited for low voltage, high current ratings because of the number of parallel paths.

Figure 12.
Lap Winding connected to commutator bars

Figure 13.
Simplex Lap Winding, Circular Form

Figure 14.
Simplex Lap Winding, Development Form

Wave Winding
When the end connections of the coils are spread apart as shown in Figure 15 a wave or series winding is formed. In a wave winding there are only two paths regardless of the number of poles. Therefore, this type winding requires only two brushes but can use as many brushes as poles. The simplex wave winding in Figure 16 (circular) and Figure 17 (development) shows that the connections to the armature do not lap back toward the coil but progress forward. The coil voltages are cumulative but it is necessary to travel several times around the armature and to traverse half the total winding in order to trace the path between the positive and negative brush. The wave winding is best suited for high voltage low current ratings since it has only two paths.

Figure 15.
Wave Winding connected to commutator bars

Figure 16.
Simplex Wave Winding, Circular Form

Figure 17.
Simplex Wave Winding, Development Form
Slots and Coils
The number and size of slots depend upon the generator or motor requirements. The slot has to be large enough to hold the correct number of conductors but at the same time, the tooth has to be large enough to pass the necessary magnetic flux. Normally, in a simple winding, there are as many coils as there are slots. This means that each slot contains two coil sides, one side of each coil being at the top of a slot and the other at the bottom of a slot. Each coil may consist of one or more turns depending on the applied or generated voltage of the unit. A typical arrangement of coil sides and slots is shown in Figure 18. Solid lines represent the front end connections to the commutator and dotted lines represent the back end connections.

Slot Pitch
Slot pitch refers to the number of slots spanned by each coil. For example, in Figure 18, the top of coil in slot 1 has its bottom in slot 4, therefore, the slot pitch is 1-4 or 3. Since the top of the coil is directly under the north pole and the bottom is directly under the south pole, the winding is known as a full pitch winding. In many cases, for various reasons, the pitch is reduced to less than full pitch. For example, if the coils in Figure 6 spanned 2 slots instead of three, the winding would become a two-thirds pitch winding.

Figure 18.
Coil Sides in Armature Slots

Field Windings

The field windings provide the excitation necessary to set up the magnetic fields in the machine. There are various types of field windings that can be used in the generator or motor circuit. In addition to the following field winding types, permanent magnet fields are used on some smaller DC products. See Figure 19 for winding types.

Shunt Wound - DC Operation
Typical Speed - Torque Curve

Shunt wound motors, with the armature shunted across the field, offer relatively flat speed-torque characteristics. Combined with inherently controlled no-load speed, this provides good speed regulation over wide load ranges. While the starting torque is comparatively lower than the other DC winding types, shunt wound motors offer simplified control for reversing service.

Compound Wound - DC Operation
Typical Speed - Torque Curve

Compound wound (stabilized shunt) motors utilize a field winding in series with the armature in addition to the shunt field to obtain a compromise in performance between a series and shunt type motor. This type offers a combination of good starting torque and speed stability. Standard compounding is about 12%. Heavier compounding of up to 40 to 50% can be supplied for special high starting torque applications, such as hoists and cranes.

Series Wound - DC Operation
Typical Speed - Torque Curve

Series wound motors have the armature connected in series with the field. While it offers very high starting torque and good torque output per ampere, the series motor has poor speed regulation. Speed of DC series motors is generally limited to 5000 rpm and below. Series motors should be avoided in applications where they are likely to lose there load because of their tendency to "run away" under no-load conditions. These are generally used on crane and hoist applications.

Permanent Magnet - DC Operation
Typical Speed - Torque Curve

Permanent magnet motors have no wound field and a conventional wound armature with commutator and brushes. This motor has excellent starting torques, with speed regulation not as good as compound motors. However, the speed regulation can be improved with various designs, with corresponding lower rated torques for a given frame. Because of permanent field, motor losses are less with better operating efficiencies. These motors can be dynamically braked and reversed at some low armature voltage (10%) but should not be plug reversed with full armature voltage. Reversing current can be no higher than the locked armature current.

Figure 19.
Field Windings

Separately Excited Winding
When the field is connected to an external power source, it is a separately excited field.

Straight Shunt Winding
This winding is connected in parallel with the armature. Shunt windings usually consists of a large number of turns of small size wire. This is a good winding for reversing applications since it provides the same amount of torque in both directions. The torque/ current curve is non-linear above full load. Shunt wound motors often have a rising speed characteristic with increased load.

Series Winding
This winding is connected in series with the armature. A series winding usually consists of a small number of turns of large size wire. With this winding, the motor can produce high starting and overload torque. This design is not used for applications with light loads or no load conditions.

Compound Winding
This winding consists of a shunt winding and a series winding. This is also known as compound excitation. The series winding can be designed as a starting series only or as a start and run series.

Stabilized Shunt Winding
Like the compound winding, this winding consists of a shunt winding and a series winding. The series or stabilizing winding has a fewer number of turns than the series winding in a compound wound machine. A stabilizing winding is used to assures a speed droop with overload. It also adds to the torque in one direction of operation and subtracts from torque in the reverse direction of operation and in regeneration.

Shunt Compensated Winding
Shunt compensated motors have a shunt winding and a pole face series winding made up of large conductors placed in slots in the face of the main field poles. The direction of current in the compensating windings is the opposite of the current in the armature conductors passing under the poles. The flux produced by the compensating windings neutralizes the flux of the armature conductors passing under the poles so that distortion of air gap flux is minimized. Shunt compensated motors maintain constant or set speed well at all loads, no load through overload. Unlike the stabilized shunt winding, the pole face winding adds to torque in both the forward and reverse direction of rotation. Shunt compensated windings, due to cost and difficulty of construction, are provided on large motors only, usually 840 frames and larger.

Commutation

The maximum voltage from an armature winding can be obtained when the brushes are in contact with those conductors, which are midway between the poles. This will result in the greatest possible number of conductors cutting the magnetic lines in one direction between a positive and negative brush. This brush position is known as the no load neutral position of the brushes. The current in a given armature coil reverses in direction as the coil sides move from one pole to another of opposite polarity, whereas the function of the commutator is to keep the current unidirectional. This reversal of current is known as commutation. The commutator acts as a switch to keep the current flowing in one direction. However, the fast rate of change in direction of the current in any given coil induces an appreciable voltage in that coil which tends to keep the current flowing in the original direction. Therefore, the current reversal is delayed causing an accelerated rate of change near the end of the commutation period. This results in an arc if the reversal is not completed before the brush breaks contact with the coil involved. Any arcing is detrimental to the operation of the machine and must be counteracted.

Armature Reaction

Since the armature conductors carry current they set up a magnetic field which distorts or opposes the main field. This is called armature reaction and is a function of the amount of load present. Figure 21 shows the MMF and flux wave shapes due to the armature reaction only; and Figure 22 shows the combined effect of both. It can be seen that armature reaction causes the flux to shift, thus tending to saturate one pole tip. If this effect is appreciable, it can be detrimental to the satisfactory performance of the machine. If severe enough, it may result in a flashover, which is the progressive arcing over successive bars until the arc extends from positive to negative brush, thus short circuiting the machine terminals.

Figure 20.
MMF and Flux Wave Shape due to Main Field only

Figure 21.
MMF and Flux Wave Shape due to Armature Reaction only

Figure 22.
Flux Wave Shape, combined effect

Brush Shifting
One method of reducing the arcing due to non-linear commutation is to shift the brushes away from the geometrical neutral position. Then commutation will occur when the applicable coil is under the influence of a weak magnetic field that will generate a voltage in the coil, which opposes the induced voltage due to current change. Therefore, this new voltage will assist rather than hinder the current reversal. In a generator, it is necessary to shift the brushes forward in the direction of rotation for good commutation. This is true because the current flow through the conductors is in the same direction as the voltage and, it commutation is delayed until the coil sides are under the next pole, it will be assisted by the current reversing voltage. In a motor, it is necessary to shift the brushes against the direction of rotation because current flow is in opposition to the induced voltage. The amount of shift necessary depends on the load so a given shift will not be satisfactory for all loads. One effect of shifting brushes is that a demagnetization component of armature reaction is introduced. In other words, when the brushes are shifted, the armature reaction will not only distort the main field flux but it will also directly oppose the main field. This will result in a reduction of the field flux. Another effect is that if the brushes are shifted far enough, it is possible to reduce the number of effective turns because there will be voltages in opposition to each other between two brushes.
In generators the demagnetization component of armature reaction would be detrimental because there will be a decrease in generated voltage with increase in load. However, in a motor, the effect would be beneficial because the speed would tend to remain constant.

Interpoles
Another method to combat the induced voltage caused by current reversal is the use of interpoles. The interpoles are located at the geometric neutral points midway between the main poles and provide reversing magnetic field of proper strength and polarity. They eliminate the need for brush shifting and, because of this, the demagnetization effect of armature reaction is eliminated. The interpole must have sufficient strength to overcome the armature reaction and provide a reversing field, therefore, it is connected in series with the armature winding. When the armature current is increased in the same proportion. In a generator, the interpole must have the same polarity as the next pole in the direction of rotation while in a motor the interpole must have the same polarity as the last pole.

Generator Characteristics

No Load Saturation Curve
A typical no load saturation curve is shown in Figure 23. This is similar to the magnetization curve mentioned previously except that it represents the entire magnetic circuit of a machine rather than one particular magnetic material. Also, it has generator output voltage plotted against field current rather than flux density against magnetizing force. This can be done since generator voltage is directly proportional to the field flux and the number of turns is fixed. There is a different saturation curve for each speed. The lower straight line portion of the curve represents the air gap because the magnetic parts are not saturated. When the magnetic parts start to saturate, the curve bends over until complete saturation is reached. Then the curve becomes a straight line again.

Figure 23.

No Load Saturation Curve

Figure 23.1
No Load Saturation Curve

Figure 23.2
No Load Saturation Curve

Generator Build Up

Generator build up usually refers to the gradual rise in voltage at the armature terminals when the machine is self-excited and operated at normal speed. This is illustrated in Figure 25 by referring to the field resistance line which shows how the field current varies as field voltage is varied. The slope of this line is the field resistance at a constant temperature. The voltage rise starts with the residual magnetism of the field iron. This provides a small voltage output E₁ that is fed back to the field as ₁. ₁ increases the flux providing a slightly larger voltage, E₂ . E₂ causes ₂ to flow. This process continues until the machine starts to saturate and stops at the point where the field resistance line intersects the saturation curve. If the speed of the machine is reduced so that the saturation curve becomes tangent to field resistance curve, the voltage will not build up. This is known as the critical speed. Also, at any given speed, if the field resistance is increased by addition of external resistance, a critical resistance can be reached.
Figure 25.

DC Motor Curves
Voltage Output The voltage equation has been expressed as:

E = K₁ S.

However, this is the generated voltage and part of it must be used to overcome the IR drops in the machine, which are caused by the resistance's of the armature, series field, interpoles, brushes, etc. If these resistance's are combined together and called armature resistance, then the voltage output at the generator terminals can be expressed as:

V = E - I_a R_a - K₁ S - I_a R_a

where:

	E = generated voltage
	I_a = armature current
	R_a = armature circuit resistance
	K₁ = machine constants
	= flux per pole
	S = speed.

External Characteristics
The curve showing the relationship between output voltage and output current is known as the external characteristic. Shown in Figure 24 are the external characteristic curves for generators with various types of excitation. If a generator, which is separately excited, is driven at constant speed and has a fixed field current, the output voltage will decrease with increased load current as shown. This decrease is due to the armature resistance and armature reaction effects. If the field flux remained constant, the generated voltage would tend to remain constant and the output voltage would be equal to the generated voltage minus the IR drop of the armature circuit. However, the demagnetizing component of armature reactions tends to decrease the flux, thus adding an additional factor, which decreases the output voltage.

Figure 24.

DC Generator Curves
In a shunt excited generator, it can be seen that the output voltage decreases faster than with separate excitation. This is due to the fact that, since the output voltage is reduced because of the armature reaction effect and armature IR drop, the field voltage is also reduced which further reduces the flux. It can also be seen that beyond a certain critical value, the shunt generator shows a reversal in trend of current values with decreasing voltages. This point of maximum current output is known as the breakdown point. At the short circuit condition, the only flux available to produce current is the residual magnetism of the armature.
To build up the voltage on a series generator, the external circuit must be connected and its resistance reduced to a comparatively low value. Since the armature is in series with the field, load current must be flowing to obtain flux in the field. As the voltage and current rise the load resistance may be increased to its normal value. As the external characteristic curve shows, the voltage output starts at zero, reaches a peak, and then falls back to zero.
The combination of a shunt field and a series field gives the best external characteristic as illustrated in Figure 24. The voltage drop, which occurs in the shunt machine, is compensated for by the voltage rise, which occurs in the series machine. The addition of a sufficient number of series turns offsets the armature IR drop and armature reaction effect, resulting in a flat-compound generator, which has a nearly constant voltage. If more series turns are added, the voltage may rise with load and the machine is known as an over-compound generator.

Voltage Regulation
Voltage Regulation is the change in terminal voltage with the change in load current at constant speed. A generator has good regulation if the change in voltage between no load and full load is small. If the change is large, the regulation is poor. Expressed in equation form:
Percent Voltage Regulation = (E_NL - E_FL ) / E_FL x 100 or for some compound machines, Percent Voltage Regulation = (E_FL - E_NL ) / E_FL x 100
Figure 24 shows that the regulation of a separately excited machine is better than that of a shunt machine. However, the best regulation is obtained with a compound machine. The series machine has practically no regulation at all and, therefore, has little practical application.

Motor Characteristics

Motor Operation
As previously stated, a conductor moving through a magnetic field due to the motor action also generates a voltage which is in opposition to the applied voltage. This is the back EMF. Then for motor action the voltage equation is:

V = E + I_A R_A = K₁ S + I_A R_A

where:

	V = applied or terminal voltage
	E = back EMF
	I_A = armature current
	R_A = armature circuit resistance's
	K₁ = machine constants
	= flux per pole
	S = speed

When comparing this equation with the voltage equation of a generator, it can be seen that in a generator the generated voltage is higher than the terminal voltage while in a motor the opposite is true. Therefore, as long as the generated voltage is less than the terminal voltage, a machine operates as a motor and takes power from the electrical side, but when the generated voltage becomes greater than the terminal voltage, the machine becomes a generator, supplies electric power, and requires mechanical energy to keep operating.
The back or counter EMF acts as a control for the amount of current needed for each mechanical load. When the mechanical load is increased, the first effect is a reduction in speed. But a reduction in speed also causes a reduction in back EMF, thus making available an increased voltage for current flow in the armature. Therefore, the current increases which in turn increases the torque. Because of this action, a very slight decrease in speed is sufficient to meet the increased torque demand. Also, the input power is regulated to the amount required for supplying the motor losses and output.

Speed Torque Curves
Speed torque curves for the three forms of excitation are shown in Figure 25. In a shunt excited motor, the change in speed is slight and, therefore, it is considered a constant speed motor. Also, the field flux is nearly constant in a shunt motor and the torque varies almost directly with armature current.
In a series motor the drop in speed with increased torque is much greater. This is due to the fact that the field flux increases with increased current, thus tending to prevent the reduction in back EMF that is being caused by the reduction in speed. The field flux varies in a series motor and the torque varies as the square of the armature current until saturation is reached. Upon reaching saturation, the curve tends to approach the straight line trend of the shunt motor. The no load speed of a series motor is usually too high for safety and, therefore, it should never be operated without sufficient load.
A compound motor has a speed torque characteristic which lies between a shunt and series motor.

Speed Regulation
Speed regulation is the change in speed with the change in load torque, other conditions being constant. A motor has good regulation if the change between the no load speed and full load speed is small.
Percent Speed Regulation = (S_NL - S_FL) / S_FL x 100 A shunt motor has good speed regulation while a series motor has poor speed regulation. For some applications such as cranes or hoists, the series motor has an advantage since it results in the more deliberate movement of heavier loads. Also, the slowing down of the series motor is better for heavy starting loads. However, for many applications the shunt motor is preferred.

Motor Starting
When the armature is not rotating, the back EMF is zero and the total applied voltage is available for sending current through the armature. Since the armature resistance is low, an enormous current would flow if voltage were applied under this condition. Therefore, it is necessary to insert an additional resistance in series with the armature until a satisfactory speed is reached where the back EMF will take over to limit the current input.

Losses And Efficiency

Friction and Windage
These losses include bearing friction, brush friction, and windage. They are also known as mechanical losses. They are constant at a given speed but vary with changes in speed. Power losses due to friction increase as the square of the speed and those due to windage increase as the cube of the speed.
Armature Copper Losses
These are the I² R losses of the armature circuit, which includes the armature winding, commutator, and brushes. They vary directly with the resistance and as the square of the currents.
Field Copper Losses
These are the I² R losses of the field circuit which can include the shunt field winding, series field winding, interpole windings and any shunts used in connection with these windings. They vary directly with the resistance and as the square of the currents.
Core Losses
These are the hysteresis and eddy current losses in the armature. With the continual change of direction of flux in the armature iron, an expenditure of energy is required to carry the iron through a complete hysteresis loop. This is the hysteresis loss. Also since the iron is a conductor and revolving in a magnetic field, a voltage will be generated. This, in turn, will result in small circulating currents known as eddy currents. If a solid core were used for the armature, the eddy current losses would be high. They are reduced by using thin laminations, which are insulated from each other. Hysteresis and eddy current losses vary with flux density and speed.
Efficiency
For generations or motors, the efficiency is equal to the output divided by the input. However, in a generator, the input is mechanical while the output is electrical. In a motor the opposite is true, therefore:

	Motor Efficiency = (Input - Losses) / Input
	Generator Efficiency = Output / (Output + Losses)

Section 3: Horsepower Basics

In 18th century England, coal was feeding the industrial revolution and Thomas Newcomen invented a steam driven engine that was used to pump water from coal mines. It was a Scott however, by the name of James Watt, who in 1769 improved the steam engine making it truly workable and practical. In his attempt to sell his new steam engines, the first question coal mine owners asked was "can your engine out work one of my horses?" Watt didn't know since he didn't know how much work a horse could do. To find out, Watt and his partner bought a few average size horses and measured their work. They found that the average horse worked at the rate of 22,000 foot pounds per minute. Watt decided, for some unknown reason, to add 50% to this figure and rate the average horse at 33,000 foot pounds per minute.
What's important is that there is now a system in place for measuring the rate of doing work. And there is a unit of power, horsepower.
If steam engines had been developed some place else in the world, where the horse was not the beast of burden, we might be rating motors in oxen power or camel power. Today, motors are also rated in Watts output.

	hp = lb x fpm / 33,000
	hp = ft-lb x rpm / 5,252
	kW = hp x 0.7457
	hp_Metric = hp x 1.0138

Horsepower as defined by Watt, is the same for AC and DC motors, gasoline engines, dog sleds, etc.

Horsepower and Electric Motors

	Torque = force x radius = lb x ft = T
	Speed = rpm = N
	Constant = 5252 = C
	HP = T x N / C

Torque and DC Motors

T = k I_a

At overload, torque increases at some rate less than the increase in current due to saturation

D² L and Torque

	258AT = 324 D² L
	259AT = 378 D² L

With the same frame diameter, the 259AT has 17% more D² L and thus 17% more and 17% more Torque. Motor torque increases with an increase in iron and copper, combined with current. It can then be said that it takes iron and copper to produce torque and torque makes products. Or to put it another way, what you purchase to make product is TORQUE and that is IRON and COPPER. The rate of doing work is power and HORSEPOWER is a unit of power.

Speed and DC Motors
Shunt wound DC motors
With motor load, temperature and field current held constant, speed is controlled by armature voltage.

E = ((Z / a) x x P x (N / 60) x 10^-8 ) + (I R_a + I R_ip + I R_b )

The sum of the voltage drop in the armature circuit can be represented as IR

N = (E - IR) / K

Speed example: given motor is design G6219, frame MC3212, 50 hp, 1150 rpm, 500 volt armature, 85 amps full load, 0.432 armature circuit resistance hot, 0.206 armature circuit resistance cold

E_drop = IR = 85 amp x 0.432 = 36.72 volts
	500 v arm - 36.72 v drop = 463.28 working volts
	Volts per rpm = 463.28 / 1150 rpm = 0.40285
N_{base speed} = 1150 rpm = (500 v - 36.72 v) / 0.40285
With 250 v on the armature, there is 213.28 working volts (250 - 36.72)
213.28 / 0.40285 = 529 rpm (not 1/2 speed, 575 rpm)
	N = 529 rpm = (250 v - 36.72 v) / 0.40285
N = (E - IR) / K = (E - IR) / 0.40285
K changes with changes in load and temperature
		HP_Metric = HP x 1.0138
		kW = HP x 0.7457

A very nice basic tutorial for motors.

2011-03-14T06:01:00.000-07:00

http://www.reliance.com/mtr/mtrthrmn.htm

Overview of Motor Types Tutorial

2011-03-14T05:58:00.002-07:00

http://www.oddparts.com/acsi/motortut.htm#AC_Motors

Coolmagnetman. Motor experiment

2011-03-14T05:58:00.000-07:00

http://www.coolmagnetman.com/magacmot.htm

PACONTROL TUTORIAL

2011-03-14T05:55:00.000-07:00

PACONTROL IS THE SITES BY WHICH WE CAN GET A LOT OF INFORMATION ABOUT THE MOTOR CONTROL.

Below is the lnk for your reveiw.
http://www.pacontrol.com/Motor1.html
http://www.pacontrol.com/3phasemotors1.html
http://www.pacontrol.com/3phasemotors2.html
http://www.pacontrol.com/3phasemotors3.html

Adjustable speed drive

2011-03-14T05:48:00.000-07:00

Tutorial Motor Basics Lecture 2.1MB

Adjustable Speed Drives Tutorial 300kB

Electric Motor Controls Tutorial 653kB

A very good tutorial for motor is given below.

2011-03-14T05:47:00.000-07:00

The venerable electric motor that was the muscle of the industrial revolution is becoming the smart muscle of the computer-controlled plant and commercial facility of the future. The advent of powerful, reliable electronic drives is keeping motors in the forefront of this technological evolution.

Electric motors have a tremendous impact on overall energy use. Between 30 to 40 percent of all fossil fuels burned are used to generate electricity, and two-thirds of that electricity is converted by motors into mechanical energy.

The Fundamentals of Energy Management series this month will focus on topics that will allow facility managers and engineers at commercial and industrial facilities to understand the basics of motors and drives. This information will help them to select and implement strategies with the goal of reducing motor and drive costs as well as decreasing downtime.

AC Induction Motors

AC induction motors are ideal for most industrial and commercial applications because of their simple construction and low number of parts, which reduce maintenance cost. Induction motors are frequently used for both constant-speed and adjustable speed drive (ASD) applications.

The two basic parts of an induction motor are the stationary stator located in the motor frame and the rotor that is free to rotate with the motor shaft. Today's motor design and construction are highly refined. For example, stator and rotor laminations have been designed to achieve maximum magnetic density with minimum core losses and heating. The basic simplicity of this design ensures high efficiency and makes them easily adaptable to a variety of shapes and enclosures.

A three-phase induction motor can best be understood by examining the three-phase voltage source that powers the motor. Three-phase currents flowing in the motor leads establish a rotating magnetic field in the stator coils. This magnetic field continuously pulsates across the air gap and into the rotor. As magnetic flux cuts across the rotor bars, a voltage is induced in them, much as a voltage is induced in the secondary winding of a transformer. Because the rotor bars are part of a closed circuit (including the end rings), a current begins to circulate in them. The rotor current in turn produces a magnetic field that interacts with the magnetic field of the stator. Since this field is rotating and magnetically interlocked with the rotor, the rotor is dragged around with the stator field.

When there is no mechanical load on the motor shaft (no-load condition), the rotor almost manages to keep up with the synchronous speed of the rotating magnetic field in the stator coils. Drag from bearing friction and air resistance prevents perfect synchronicity. As the load increases on the motor shaft, the actual speed of the rotor tends to fall further behind the speed of the rotating magnetic field in the stator. This difference in speed causes more magnetic lines to be cut, resulting in more torque being developed in the rotor and delivered to the shaft mechanical load. The rotor always turns at the exact speed necessary to produce the torque required to meet the load placed on the motor shaft at that moment in time. This is usually a dynamic situation, with the motor shaft speed constantly changing slightly to accommodate minor variations in load.

The rotor consists of copper or aluminum bars connected together at the ends with heavy rings. The construction is similar to that of a squirrel cage, a term often used to describe this type of ac induction motor.

The rotating magnetic field in the stator coils, in addition to inducing voltages in the rotor bars, also induces voltages in the stator and rotor cores. The voltages in these cores cause small currents, called eddy currents, to flow. The eddy currents serve no useful purpose and result in wasted power. To keep these currents to a minimum, the stator and rotor cores are made of thin steel discs called laminations. These laminations are coated with insulating varnish and then edge welded together to form a core. This type of core construction substantially reduces eddy current losses, but does not entirely eliminate them.

By varying the design of the basic squirrel-cage motor, almost any characteristic of speed, torque, and voltage can be controlled by the designer. To standardize motor features the National Electrical Manufacturers Association (NEMA) has established standards for a number of motor features.

The speed of an ac induction motor depends on the frequency of the supply voltage and the number of poles for which the motor is wound. The term poles refers to the manner in which the stator coils are connected to the three incoming power leads to create the desired rotating magnetic field. Motors are always wound with an even number of poles. The higher the input frequency, the faster the motor runs. The more poles a motor has, the slower it runs at a given input frequency. The synchronous speed of an ac induction motor is the speed at which the stator magnetic flux rotates around the stator core at the air gap. At 60 Hz the following synchronous speeds are obtained:

Number of poles                 RPM
        2                         3,600
        4                         1,800
        6                         1,200
        8                         900
        10                         720
        12                         600

Providing the motor is properly constructed, the output speed can be doubled for a given number of poles by running an ASD supplying the motor at an output frequency of 120 Hz.

The actual speed of an induction motor rotor and shaft is always somewhat less than its synchronous speed. The difference between the synchronous and actual speed is called slip. If the rotor rotated as fast as the stator magnetic field, the rotor conductor bars would appear to be standing still with respect to the rotating field. There would be no voltage induced in the rotor bars and no current would be set up to produce torque.

Induction motors are made with slip ranging from less than 5% up to 20%. A motor with a slip of 5% or less is known as a normal-slip motor. A normal-slip motor is sometimes referred to as a 'constant speed' motor because the speed changes very little from no-load to full-load conditions. A common four-pole motor with a synchronous speed of 1,800 rpm may have a no-load speed of 1,795 rpm and a full-load speed of 1,750 rpm. The rate-of-change of slip is approximately linear from 10% to 110% load, when all other factors such as temperature and voltage are held constant. Motors with slip over 5% are used for hard to start applications.

The direction of rotation of a poly-phase ac induction motor depends on the connection of the stator leads to the power lines. Interchanging any two input leads reverses rotation.

Torque and Horsepower
Torque and horsepower are two very important characteristics that determine the size of the motor for a particular application. Torque is the turning effort. For example, suppose a grinding wheel with a crank arm one-foot long takes a force of one pound to turn the wheel at steady rate. The torque required is one pound times one foot or one foot-pound. If the crank is turned twice as fast, the torque remains the same. Regardless of how fast the crank is turned, the torque is unchanged as long as the crank is turned at a steady speed.

Horsepower takes into account how fast the crank is turned. Turning the crank more rapidly takes more horsepower than turning the crank slowly. Horsepower is the rate of doing work. By definition, one horsepower equals 33,000 foot-pounds per minute. In other words, to lift a 33,000-pound load one foot in one minute would require one horsepower.

The discussion so far has only involved torque at a steady speed. More effort is required to start a load than to keep it going. An ac induction motor is built to supply the extra torque needed to overcome the inertia of starting a load. The speed-torque curve for a representative motor is shown in Figure 4 and illustrates in this example that the starting torque is 210% of rated-load torque.

Motor Losses and Loss Reduction Techniques

The only way to improve motor efficiency is to reduce motor losses. Since motor losses produce heat, reducing losses not only saves energy directly but can also reduce cooling load on a facility's air conditioning system.

Motor energy losses can be segregated into five major areas. Each area is influenced by the motor manufacturer's design and construction decisions. One design consideration, for example, is the size of the air gap between the rotor and the stator. Large air gaps tend to maximize efficiency at the expense of a lower power factor. Small air gaps slightly compromise efficiency while significantly improving power factor.

Motor losses may be grouped as fixed or variable losses. Fixed losses occur whenever the motor is energized and remain constant for any given voltage and speed. Variable losses increase with an increase in motor load. Core loss and friction windage losses are fixed. Variable losses include stator- and rotor-resistance losses and stray load losses.

Motor Economics
The principal factors in energy-saving calculations are motor efficiency, run hours (at a certain load), and the cost of electricity. When a motor runs at nearly full load for many hours at a facility with high electrical costs, the higher resulting savings will indicate the use of a 'premium efficiency' unit. In some cases, the savings may be great enough to warrant taking a perfectly serviceable older motor off-line and upgrading to a new, premium-efficiency model. For applications with less than continuous use or at lower than full loading, upgrading a working motor will usually not make sense.

Some other application considerations: The full-load speed of high-efficiency motors is usually somewhat higher than standard efficiency models. When applied to centrifugal loads such as pumps and fans, the higher speed can translate to an increased horsepower requirement, and this can offset the anticipated energy savings. The higher output speed may also increase overall output by allowing a piece of machinery to finish its job faster. Results will vary with the application. There are energy efficient motor designs that can exhibit some unusual characteristics such as lower power factor and higher locked rotor amperage; these must be considered when choosing a motor.

Adjustable Speed Drive Systems

Commercial and industrial firms today use adjustable-speed drive (ASD) systems for a variety of applications. Most common of these include standard pumps, fans, and blowers. Newer applications include hoists and cranes, conveyors, machine tools, film lines, extruders, and textile-fiber spinning machines.

Many applications have unique demands and characteristics.

Drive vendors have responded to this demand by producing a variety of drives. The combination of the many types of drives available and the abundance of applications has made the selection of the optimum drive for a given application a challenge.

New generation ASDs have evolved with advancements in solid-state electronics. ASDs can now be applied to ac motors regardless of motor horsepower or location within a facility and can be used to drive almost all types of motorized equipment, from a small fan to the largest extruder or machine tool. Commercial and industrial facilities can expect to dramatically reduce both energy consumption and operating and maintenance costs while offering improved operating conditions by using new generation electronic ASDs. The latest generation of ASDs allows ac induction motors to be just as controllable and efficient as their dc counterparts were.

Historically a variety of terms have been used to describe a system that permits a mechanical load to be driven at user-selected speeds. These terms include, but are not limited to:

Variable-Speed Drive
Variable-Frequency Drive
Adjustable-Frequency Drive
Adjustable-Speed Drive

The term variable implies a change that may or may not be under the control of the user. Adjustable is the preferred term since this refers to a change directly under control of the user. The term frequency can only be applied to drives with an ac output, while the term speed is preferred since this includes both ac and dc drives. Thus, the term most commonly accepted is Adjustable-Speed Drive (ASD).

Basic ASD Components
Most ASD units consist of three basic parts. A rectifier that converts the fixed frequency ac input voltage to dc. An inverter that switches the rectified dc voltage to an adjustable frequency ac output voltage. (The inverter may also control output current flow, if desired.) The dc link connects the rectifier to the inverter. A set of controls directs the rectifier and inverter to produce the desired ac frequency and voltage to meet the needs of the ASD system at any moment in time.

The advantages of ASDs do not stop with saving energy and improving control. ASD technology can now be applied to manufacturing equipment previously considered too expensive or uneconomical. Such applications are often unique to a particular industry and its equipment, or even to a particular facility. Cost benefits, such as those obtained from improved quality, may be desirable for each application.

Tutorial Motor Basic lecture.

Power Distribution Courses

2011-03-14T05:43:00.000-07:00

Power Distribution Courses

* Basics of Busway
* Basics of Circuit Breakers
* Basics of Load Centers
* Basics of Meter Mounting Equipment
* Basics of Panelboards
* Basics of Safety Switches
* Basics of Switchboards

Useful material for Motors

2011-03-14T05:41:00.000-07:00

Introductory Courses

* Basics of Electricity
* Basics of Electrical Products

Motors & Control Courses

* Basics of AC Motors
* Basics of Control Components
* Basics of DC Drives
* Basics of Motor Control Centers
* Basics of PLCs