Electronics Engineering

Application of the Divergence Theorem: Laplace's Equation

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The combination div grad , () or� is called the "Laplacian" differential operator,
The equation () f = 0 is called Laplace's equation. Static electric and steady state magnetic fields obey this equation where there are no charges or current.
Any solution to this equation in R has the property that its value at the center of a sphere within R is the average of its value on the sphere's surface. If g(r) obeys Laplace's equation inside a spherical surface, S, of radius a, centered at r', we have

Laplace's Equation

Thus solutions to Laplace's equation� are very smooth: they� have no bumps maxima or minima in R and essentially "interpolate" smoothly between their values on the boundaries of R. We prove this important fact as an application of the divergence theorem.
This result also implies that if we know the divergence of a vector v and its curl everywhere, these are differentiable everywhere, and v vanishes� at infinity, then v is uniquely determined. proof box( if there were two solutions v and v' with the same divergence and curl, then on applying the double cross product identity we find that every component of their difference obeys Laplace's equation everywhere. Its value anywhere is then its average value on a huge circle at infinity, which is 0 by assumption. The same conclusion holds if v and v' are required to behave at infinity in the same way, so that v - v' must approach 0 for large arguments.)
Proof
Suppose our function, f(r), obeys Laplace's equation within some� sphere S centered at r':

div grad f = () f = 0 inside S

We apply the divergence theorem to the vector fg - gf in the sphere with surface S excluding a tiny sphere of radius b with surface S'� having the same center. We obtain

the latter being obtained by substituting for g. The second integral on the second line vanishes here as can be seen by applying the divergence theorem again within S and noticing that the� Laplacian applied to f is 0.
The right hand side here is the average value of f on S. The similar integral over S' is evaluated in exactly the same way and is the average value of f on S'. We conclude that the average value of v on any sphere with center at r' is the same. Obviously as the sphere approaches radius 0 that average value becomes the value of f at r'.
The method used in this argument is a very important and general one that is used in dealing with many differential equations. In fact the use of the divergence theorem in the form used above is often called "Green's Theorem." And the function g defined above is called a "Green's function" for Laplaces's equation.
We can use this function g to find a vector field v that vanishes at infinity obeying div v = , �curl v = 0. (we assume that r is sufficently well behaved, integrable, vanishes at infinity etc...)� Suppose we write v as grad f.
We get

(This equation is called Poisson's equation and is obeyed by the potential produced by a charge distribution with charge density .)
A solution that vanishes at infinity is then given by

Proof
The same approach can be used to obtain a vector potential A and determine a vector v obeying

v = 0, v= j

�With v =A we obtain

((A)) = j

and with the double cross identity

((A)) = (A) - () A

and the "gauge" condition (which we are free to assume) ) A = 0, we find that each component of A obeys Poisson's equation with source - the corresponding component of j.
We may therefore find a formula for each coordinate of A exactly like the corresponding formula for the scalar potential V. Again v may be recovered from A by differentiating.
These results are of some use in the study of electromagnetic fields but they� don't solve all problems. Often the known charges and currents induce unknown charges and currents in conducting surfaces, and one wants to determine the fields, in circumstances in which there are some unknown charges and /or currents and you know conditions on the field at the conductor surfaces instead of the charges and currents in them.
Exercises

The Gradient in General Orthogonal Coordinate System

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Given a scalar field f, its gradient is a vector normal to the surface on which the field is constant, of length given by f's directional derivative in that direction.
We can always write

or , after trivial manipulation,

We may identify ds_j�= u_jdw_j, and obtain

Suppose we set two of the ds_i's to zero; then the directional derivative of f in the third direction must be the component of� the gradient in that direction. We therefore obtain

The Curl in General Orthogonal Coordinate System

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We may use Stokes' Theorem to deduce this by applying it to a tiny rectangle normal to the w₁ direction with sides obeying one of dw_2�= 0 and dw_3�= 0, and then cyclically permuting the indices.
The surface integral in Stokes' theorem then becomes

u₂u₃dw₂dw₃(v)₁

while the path integrals become

which gives us:

Maxwell's Equations

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Maxwell raised the question: what are the magnetic fields produced by non-steady currents? He noticed that Ampere's Law contradicts the conservation of charge and required modification when currents are not steady. These two laws, as we have noted, correspond to the equations

and

cB = j

Taking the divergence of both sides of the latter and using the former gives:

0 =B =j = -/ t

Thus, Ampere's Law implies that the current in it is steady.
Maxwell noted that the electrostatic equation,E =, provides a way to modify Ampere's Law so that it is consistent with non-steady currents. If we add a term E / t to its left hand side, we can eliminate the condition that be constant, and obtain a consistent set of equations: These equations are called Maxwell's Equations.
They are:
B +E / ct = j / c (Maxwell's modified Ampere's Law)
B = 0 (No magnetic sources or sinks)
E -B / ct = 0 (Faraday's Law)
E = (Gauss's Theorem)

Electromagnetic Induction: Faraday's Law

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Faraday observed that changing magnetic fields produced modifications to the laws of electrostatics: in particular, changing magnetic fields lead to non-conservative electric fields, in which the circulation integral of electric field around a closed path P does not always vanish. In fact his laws describe precisely what such integrals are: a constant multiple of the rate of change of magnetic field flux on any surface bounded by P.
His law is:

Stokes' Theorem for E is

Combining these two equations yields for any surface S:

from which physicists conclude:

cE - (B / t) = 0

as the differential form of Faraday's Law.

For convenience, we have chosen units for E and B such the constants c apearing in Ampere's Law and Faraday's Law are the same.

Ampere's Law

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Physicists conclude that electrostatics can be described by the two conditions:E = 0, E =.
No sources or sinks of Magnetic Field lines; and Ampere's Law.
We can immediately translate the statement that magnetic fields have no sources or sinks into the statement:B = 0, since, (again by the divergence theorem,) we know that sources and sinks correspond to non-vanishing divergence.
You are probably familiar with the Biot-Savart Law which describes the magnetic field produced by an electrical� current element.
Ampere's Law states that, for constant current flow, the flux of electrical current through a surface is proportional to the line integral of the magnetic field (counterclockwise) around its boundary:

Stokes' Theorem, applied to B, gives

Upon combining these two we obtain, for any surface S:

from which physicists conclude the differential equations of steady state current flow:

B = j / c, B = 0

Gauss's Theorem

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We will now review these laws and make the conversions. We will simplify matters by ignoring polarization produced by fields in media which require using both E and D as electric fields and B and H as magnetic fields. We will identify these pairs, setting E = D and B = H in the following discussion.
Electrostatics: Gauss's Theorem
You probably encountered electrostatics through Coulomb's Law, which gives the electric force between two point charges, and hence the electric field of a point charge. An equivalent formulation of elctrostatics is that the field is derivable from a potential (so that curl E = 0) and Gauss's Theorem holds.
This Theorem is the statement that the integral of the flux of electric field over the boundary, V, of a region V, is the amount of electrical charge within V:

Combining this statement with the Divergence Theorem applied to E:

yields for any volume V:

Nodal Analysis of Electric Circuits

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In this method, we set up and solve a system of equations in which the unknowns are the voltages at the principal nodes of the circuit. From these nodal voltages the currents in the various branches of the circuit are easily determined.

The steps in the nodal analysis method are:

Count the number of principal nodes or junctions in the circuit. Call this number n. (A principal node or junction is a point where 3 or more branches join. We will indicate them in a circuit diagram with a red dot. Note that if a branch contains no voltage sources or loads then that entire branch can be considered to be one node.)
Number the nodes N₁, N₂, . . . , N_n and draw them on the circuit diagram. Call the voltages at these nodes V₁, V₂, . . . , V_n, respectively.
Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.
For each node except the reference node write down Kirchoff's Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.)

For example, for the node to the right KCL yields the equation:
I_a + I_b + I_c = 0

Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R). Here are some examples:

The current downward out of node 1 depends on the voltage difference V1 - V3 and the resistance in the branch.

In this case the voltage difference across the resistance is V1 - V2 minus the voltage across the voltage source. Thus the downward current is as shown.

In this case the voltage difference across the resistance must be 100 volts greater than the difference V1 - V2. Thus the downward current is as shown.

The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n - 1). The equations are of this form:

where G₁₁, G₁₂, . . . , G_mm and I₁, I₂, . . . , I_m are constants.

Alternatively, the system of equations can be gotten (already in simplified form) by using the inspection method.
Solve the system of equations for the m node voltages V₁, V₂, . . . , V_m using Gaussian elimination or some other method.

Example 1: Use nodal analysis to find the voltage at each node of this circuit.
Solution:

Note that the "pair of nodes" at the bottom is actually 1 extended node. Thus the number of nodes is 3.
We will number the nodes as shown to the right.
We will choose node 2 as the reference node and assign it a voltage of zero.
Write down Kirchoff's Current Law for each node. Call V₁ the voltage at node 1, V₃ the voltage at node 3, and remember that V₂ = 0. The result is the following system of equations:
The first equation results from KCL applied at node 1 and the second equation results from KCL applied at node 3. Collecting terms this becomes:
This form for the system of equations could have been gotten immediately by using the inspection method.
Solving the system of equations using Gaussian elimination or some other method gives the following voltages:
V₁=68.2 volts and V₃=27.3 volts

Example 2: Use nodal analysis to find the voltage at each node of this circuit.

Solution: Click here for solution.

Example 3: Use nodal analysis to find the voltage at each node of this circuit.

Solution: Click here for solution.

Mesh Analysis Circuit

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Kirchoff´s Laws give us the basic method for analysing any complex electrical circuit, there are different ways of improving upon this method by using Mesh Current Analysis or Nodal Voltage Analysis that results in a lessening of the math's involved and when large networks are involved this reduction in maths can be a big advantage.

For example, consider the circuit from the previous section.

Mesh Analysis Circuit

One simple method of reducing the amount of math's involved is to analyse the circuit using Kirchoff's Current Law equations to determine the currents, I₁ and I₂ flowing in the two resistors. Then there is no need to calculate the current I₃ as its just the sum of I₁ and I₂. So Kirchoff's second voltage law simply becomes:

Equation No 1 : 10 = 50I₁ + 40I₂
Equation No 2 : 20 = 40I₁ + 60I₂

therefore, one line of math's calculation have been saved.

Mesh Current Analysis

A more easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which is also sometimes called Maxwell´s Circulating Currents method. Instead of labelling the branch currents we need to label each "closed loop" with a circulating current. As a general rule of thumb, only label inside loops in a clockwise direction with circulating currents as the aim is to cover all the elements of the circuit at least once. Any required branch current may be found from the appropriate loop or mesh currents as before using Kirchoff´s method.

For example: : i₁ = I₁ , i₂ = -I₂ and I₃ = I₁ - I₂

We now write Kirchoff's voltage law equation in the same way as before to solve them but the advantage of this method is that it ensures that the information obtained from the circuit equations is the minimum required to solve the circuit as the information is more general and can easily be put into a matrix form.

For example, consider the circuit from the previous section.

These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each element ON the principal diagonal will be "positive" and is the total impedance of each mesh. Where as, each element OFF the principal diagonal will either be "zero" or "negative" and represents the circuit element connecting all the appropriate meshes. This then gives us a matrix of:

Where:

[ V ] gives the total battery voltage for loop 1 and then loop 2.
[ I ] states the names of the loop currents which we are trying to find.
[ R ] is called the resistance matrix.

and this gives I₁ as -0.143 Amps and I₂ as -0.429 Amps

As : I₃ = I₁ - I₂

The current I₃ is therefore given as : -0.143 - (-0.429) = 0.286 Amps

which is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous tutorial.

Mesh Current Analysis Summary.

This "look-see" method of circuit analysis is probably the best of all the circuit analysis methods with the basic procedure for solving Mesh Current Analysis equations is as follows:

1. Label all the internal loops with circulating currents. (I₁, I₂, ...I_L etc)
2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each loop.
3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows;
- R₁₁ = the total resistance in the first loop.
- R_nn = the total resistance in the Nth loop.
- R_JK = the resistance which directly joins loop J to Loop K.
4. Write the matrix or vector equation [V] = [R] x [I] where [I] is the list of currents to be found.

As well as using Mesh Current Analysis, we can also use node analysis to calculate the voltages around the loops, again reducing the amount of mathematics required using just Kirchoff's laws. In the next tutorial about DC Theory we will look at Nodal Voltage Analysis to do just that.

The Bridge Rectifier

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The Bridge Rectifier
When four diodes are connected as shown in figure 4-8, the circuit is called a BRIDGE RECTIFIER. The input to the circuit is applied to the diagonally opposite corners of the network, and the output is taken from the remaining two corners.
One complete cycle of operation will be discussed to help you understand how this circuit works. We have discussed transformers in previous modules in the NEETS series and will not go into their characteristics at this time. Let us assume the transformer is working properly and there is a positive potential at point A and a negative potential at point B. The positive potential at point A will forward bias D3 and reverse bias D4. The negative potential at point B will forward bias D1 and reverse bias D2. At this time D3 and D1 are forward biased and will allow current flow to pass through them; D4 and D2 are reverse biased and will block current flow. The path for current flow is from point B through D1, up through R_L, through D3, through the secondary of the transformer back to point B. This path is indicated by the solid arrows. Waveforms (1) and (2) can be observed across D1 and D3.

DIODES

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What are Diodes?

Diodes are semiconductor devices which might be described as passing current in one direction only. The latter part of that statement applies equally to vacuum tube diodes. Diodes however are far more versatile devices than that. They are extremely versatile in fact. It might pay you to review the topic of Electron theory and atoms
Diodes can be used as voltage regulators, tuning devices in rf tuned circuits, frequency multiplying devices in rf circuits, mixing devices in rf circuits, switching applications or can be used to make logic decisions in digital circuits. There are also diodes which emit "light", of course these are known as light-emitting-diodes or LED's. As we say diodes are extremely versatile.

Schematic symbols for Diodes

A few schematic symbols for diodes are:

Figure 1 - schematic symbols for diodes

Types of Diodes

The first diode in figure 1 is a semiconductor diode which could be a small signal diode of the 1N914 type commonly used in switching applications, a rectifying diode of the 1N4004 (400V 1A) type or even one of the high power, high current stud mounting types. You will notice the straight bar end has the letter "k", this denotes the "cathode" while the "a" denotes anode. Current can only flow from anode to cathode and not in the reverse direction, hence the "arrow" appearance. This is one very important property of diodes.
The second of the diodes is a zener diode which are fairly popular for the voltage regulation of low current power supplies. Whilst it is possible to obtain high current zener diodes, most regulation today is done electronically with the use of dedicated integrated circuits and pass transistors.
The next of the diodes in the schematic is a varactor or tuning diode. Depicted here is actually two varactor diodes mounted back to back with the DC control voltage applied at the common junction of the cathodes. These cathodes have the double bar appearance of capacitors to indicate a varactor diode. When a DC control voltage is applied to the common junction of the cathodes, the capacitance exhibited by the diodes (all diodes and transistors exhibit some degree of capacitance) will vary in accordance with the applied voltage. A typical example of a varactor diode would be the Philips BB204G tuning diodes of which there are two enscapsulated in a TO-92 transistor package. At a reverse voltage Vr (cathode to anode) of 20V each diode has a capacitance of about 16 pF and at Vr of 3V this capacitance has altered to about 36 pF. Being low cost diodes, tuning diodes have virtually replaced air variable capacitors in radio applications today.
The next diode is the simplest form of vacuum tube or valve. It simply has the old cathode and anode. These terms were passed on to modern solid state devices. Vacuum tube diodes are mainly only of interest to restorers and tube enthusiasts.
The last diode depicted is of course a light emitting diode or LED. A led actually doesn't emit as much light as it first appears, a single LED has a plastic lens installed over it and this concentrates the amount of light. Seven LED's can be arranged in a bar fashion called a seven segment LED display and when decoded properly can display the numbers 0 - 9 as well as the letters A to F.

Rectifying Diodes

The principal early application of diodes was in rectifying 50 / 60 Hz AC mains to raw DC which was later smoothed by choke transformers and / or capacitors. This procedure is still carried out today and a number of rectifying schemes for diodes have evolved, half wave, full wave and bridge rectifiers.

Figure 2 - rectifying diodes

As examples in these applications the half wave rectifier passes only the positive half of successive cycles to the output filter through D1. During the negative part of the cycle D1 does not conduct and no current flows to the load. In the full wave application it essentially is two half wave rectifiers combined and because the transformer secondary is centre tapped, D1 conducts on the positive half of the cycle while D2 conducts on the negative part of the cycle. Both add together. This is more efficient. The full wave bridge rectifier operates essentially the same as the full wave rectifier but does not require a cetre tapped transformer. Further discussion may be seen on the topic power supplies.
A further application of rectifying diodes is in the conversion or detection of rf modulated signals to audio frequencies. Typical examples are am modulated signals being detected and early detection schemes for fm also used diodes for detecting modulation.

Voltage Regulating Diodes

For relatively light current loads zener diodes are a cheap solution to voltage regulation. Zener diodes work on the principle of essentially a constant voltage drop at a predetermined voltage (determined during manufacture). An example is a Philips BZX79C12 type with a regulation range between 11.4V and 12.7V but typically 12V and a total power dissipation of 500 mW in a DO-35 package. The dissipation can be extended by using a series pass transistor, see power supplies. Notice in figure 3 there is a resistor to miminmise current drawn but mainly as an aid to dropping the supply voltage and reducing the burden on the zener diodes.

Figure 3 - zener voltage regulation diodes

In the second schematic of figure 3 we have three zener diodes in series providing voltages of 5V, 10V, 12V, 22V and 27V all from a 36V supply. This configuration is not necessarily recommended especially when the current being drawn is seriously mismatched between voltages. It is presented purely out of interest.

Varactor or Tuning Diodes

These types of diodes work on the principle that all diodes exhibit some capacitance. Indeed the zener diode BZX79C12 quoted above has, according to the data book, a capacitance of 65-85 pF at 0V and measured at 1 Mhz.
For AM Radio band applications a specific diode has been devised. The Philips BB212 in a TO-92 case is one such type. Each of the diodes has a capacitance of 500 - 620 pF at a reverse bias of 0.5V and <22 pF at 8V. This diode's capacitance ratio is quoted at 22.5:1 which could not be achieved easily if at all with an air variable capacitor. This type of diode is depicted in figure 1 above.
Several obvious advantages come immediately to mind, a small transistor type package, very low cost, ease of construction on a circuit board, can be mounted away from heat generating devices, frequency determining circuitry entirely dependent upon resistor values and ratios, DC voltage control can be either from frequency synthesiser circuits or perhaps a multi-turn potentiometer. Such a potentiometer aids band spreading and fine tuning if two potentiometers are used. The only real limitation is your imagination and the calculations involved.

Diodes as frequency multipliers

Just one more example of the versatility of diodes is the frquency doubling circuit depicted in Figure 4. Now if that looks a lot like the full wave rectifier from figure 2 above you would be correct. That is why the ripple frequency for 50 / 60 Hz always comes out at 100 / 120 Hz.

Figure 4 - diodes as frequency multipliers

Here the input is a wide band transformer and the signal passes to a full wave rectifier comprising two 1N914 diodes. The DC component caused by the rectification passes to ground through RFC which of course presents a high impedance to the rf porion of the signal but essentially a short circuit for DC. The original signal should be down about 40 dB and with this type of circuit there would be a loss of somewhere around 7.5 dB so the 2 X signal would require further amplification to restore that loss.

Diodes as mixers

With some subtle re-arrangement to figure 4 we can get the circuit to function as a two diode frequency mixer. Note that there are other diode arrangements as well in this application. See mixers.

Figure 5 - diodes as frequency mixers

The diodes here act as switches and it can be mathematically shown that only the sum and difference signals will result. For example, if F1 was 5 Mhz and F2 was 3 Mhz then the sum and difference signals from the diodes would be 8 Mhz and 2 Mhz. None of the original signals appear at the output and this is a most important property of using diodes as mixers.
It should be noted that although 1N914 diodes are depicted you would normally use hot carrier diodes in any serious application and the diodes need to be well matched.

Applications of switching Diodes

Similar types of diodes have been developed specifically for band switching purposes. Although a typical 1N914 type switching diode can be used for such purposes it is preferable to use diodes which have been optimised for such purposes because the Rd on is much lower. This means the diode resistance Rd can have a serious affect on rf circuits in particular the "Q" of a tuned circuit. One example of a low Rd device is the Philips BA482 diode used for band switching in television tuners. It has a typical Rd of 0.4 ohms at a forward current of 10 mA.
In figure 5 we have one application where switching diodes operate. All diodes serve to switch in or out capacitors in the diagram which is presented here just to illustrate one single application of switching diodes, many, many more applications exist. Again the limit is your imagination.

Figure 6 - applications of switching diodes

The switching diodes in figure 4 switch in or out successively higher values of capacitors as each control select line is "grounded". The voltage from the +5V feed line proceeds through the diode at DC thus opening the diode and making it appear "transparent" for rf purposes. The capacitor with the value attached is then "switched" into circuit. Other components marked RFC and Cbp are chokes and bypass capacitors for "clean" switching. The bypass capacitors and choke values would be determined by the frequency of operation.
We could just have easily have switched inductors instead of capacitors. Note why Rd is quite important on overall circuit performance. If we were using inductors the diode resistance Rd would have a significant affect on inductor "Q" which in turn would affect filter performance, if it was in fact an LC filter application.

Switching Diodes in Logic Circuits

If you you completed the tutorial on digital basics you should be aware of binary numbers. There are a whole range of digital building blocks available and just by way of one illustration of using diodes we have presented the 74HC4040 twelve stage binary ripple counter (there are others with varying number of stages).
In the schematic of figure 7 we have this counter which divides by successive division of two for twelve stages. Initially because there is no voltage drop across the resistor a high appears on all anodes as well as on pin 4 the master reset causing the counter to reset forcing all outputs low and in turn a voltage drop across each diode and across the resistor and a low on reset.
Progressively each of the outputs change from low to high for a certain period of time and without unduly complicating matters when all outputs as selected by our diode combination (in this particular case 1 + 2 + 32 + 64 = 99) are simultaneously high the voltage drop across the resistor will cease and cause pin 11 (reset which was formerly low) to go high and reset all the internal ripple counters.

Figure 7 - applications of switching diodes in digital logic circuits

At the same time pin 4 changes state also with reset. It can been shown this happens once every 99 periods. Simply by placing diodes on the right outputs we can select to divide by any number up to 4095 using this particular counter.

Light-Emitting-Diodes or LED's

Many circuits use a led as a visual indicator of some sort even if only as an indicator of power supply being turned on. A sample calculation of the dropping resistor is included in figure 8.

Figure 8 - connecting light emitting diodes (LED's) to supply

Most leds operate at 1.7V although this is not always the case and it is wise to check. The dropping resistor is simply the net of supply voltage minus the 1.7V led voltage then divided by the led brightness current expressed as "amps" (ohms law). Note the orientation of both cathode and anode with respect to the ground end and the supply end. Usually with a led the longer lead is the anode.
UPDATE - 13th August, 2010. LED's are now being used in domestic lighting and a frien

Thevenin's Theorem

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Thevenin's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). If we're dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor components) which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.
Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it. Let's take another look at our example circuit:

Let's suppose that we decide to designate R₂ as the “load” resistor in this circuit. We already have four methods of analysis at our disposal (Branch Current, Mesh Current, Millman's Theorem, and Superposition Theorem) to use in determining voltage across R₂ and current through R₂, but each of these methods are time-consuming. Imagine repeating any of these methods over and over again to find what would happen if the load resistance changed (changing load resistance is very common in power systems, as multiple loads get switched on and off as needed. the total resistance of their parallel connections changing depending on how many are connected at a time). This could potentially involve a lot of work!
Thevenin's Theorem makes this easy by temporarily removing the load resistance from the original circuit and reducing what's left to an equivalent circuit composed of a single voltage source and series resistance. The load resistance can then be re-connected to this “Thevenin equivalent circuit” and calculations carried out as if the whole network were nothing but a simple series circuit:

. . . after Thevenin conversion . . .

The “Thevenin Equivalent Circuit” is the electrical equivalent of B₁, R₁, R₃, and B₂ as seen from the two points where our load resistor (R₂) connects.
The Thevenin equivalent circuit, if correctly derived, will behave exactly the same as the original circuit formed by B₁, R₁, R₃, and B₂. In other words, the load resistor (R₂) voltage and current should be exactly the same for the same value of load resistance in the two circuits. The load resistor R₂ cannot “tell the difference” between the original network of B₁, R₁, R₃, and B₂, and the Thevenin equivalent circuit of E_Thevenin, and R_Thevenin, provided that the values for E_Thevenin and R_Thevenin have been calculated correctly.
The advantage in performing the “Thevenin conversion” to the simpler circuit, of course, is that it makes load voltage and load current so much easier to solve than in the original network. Calculating the equivalent Thevenin source voltage and series resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit, replaced with a break (open circuit):

Next, the voltage between the two points where the load resistor used to be attached is determined. Use whatever analysis methods are at your disposal to do this. In this case, the original circuit with the load resistor removed is nothing more than a simple series circuit with opposing batteries, and so we can determine the voltage across the open load terminals by applying the rules of series circuits, Ohm's Law, and Kirchhoff's Voltage Law:

The voltage between the two load connection points can be figured from the one of the battery's voltage and one of the resistor's voltage drops, and comes out to 11.2 volts. This is our “Thevenin voltage” (E_Thevenin) in the equivalent circuit:

To find the Thevenin series resistance for our equivalent circuit, we need to take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure the resistance from one load terminal to the other:

With the removal of the two batteries, the total resistance measured at this location is equal to R₁ and R₃ in parallel: 0.8 Ω. This is our “Thevenin resistance” (R_Thevenin) for the equivalent circuit:

With the load resistor (2 Ω) attached between the connection points, we can determine voltage across it and current through it as though the whole network were nothing more than a simple series circuit:

Notice that the voltage and current figures for R₂ (8 volts, 4 amps) are identical to those found using other methods of analysis. Also notice that the voltage and current figures for the Thevenin series resistance and the Thevenin source (total) do not apply to any component in the original, complex circuit. Thevenin's Theorem is only useful for determining what happens to a single resistor in a network: the load.
The advantage, of course, is that you can quickly determine what would happen to that single resistor if it were of a value other than 2 Ω without having to go through a lot of analysis again. Just plug in that other value for the load resistor into the Thevenin equivalent circuit and a little bit of series circuit calculation will give you the result.

REVIEW:
Thevenin's Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load.
Steps to follow for Thevenin's Theorem:
(1) Find the Thevenin source voltage by removing the load resistor from the original circuit and calculating voltage across the open connection points where the load resistor used to be.
(2) Find the Thevenin resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.
(3) Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the Thevenin resistance. The load resistor re-attaches between the two open points of the equivalent circuit.
(4) Analyze voltage and current for the load resistor following the rules for series circuits.

What are Transducers?

2011-04-09T13:40:00.000+05:30

Transducers are electric or electronic devices that transform energy from one manifestation into another. Most people, when they think of transducers, think specifically of devices that perform this transformation in order to gather or transfer information, but really, anything that converts energy can be considered a transducer.
Transducers that detect or transmit information include common items such as microphones, Geiger meters, potentiometers, pressure sensors, thermometers, and antennae. A microphone, for example, converts sound waves that strike its diaphragm into an analogous electrical signal that can be transmitted over wires. A pressure sensor turns the physical force being exerted on the sensing apparatus into an analog reading that can be easily represented. While many people think of transducers as being some sort of technical device, once you start looking for them, you will find transducers everywhere in your everyday life.
Most transducers have an inverse that allows for the energy to be returned to its original form. Audio cassettes, for example, are created by using a transducer to turn the electrical signal from the microphone pick-up – which in turn went through a transducer to convert the sound waves into electrical signal – into magnetic fluctuations on the tape head. These magnetic fluctuations are then read and converted by another transducer – in this case a stereo system – to be turned back into an electrical signal, which is then fed by wire to speakers, which act as yet another transducer to turn the electrical signal back into audio waves.
Other transducers turn one type of energy into another form, not for the purpose of measuring something in the external environment or to communicate information, but rather to make use of that energy in a more productive manner. A light bulb, for example, one of the many transducers around us in our day-to-day lives, converts electrical energy into visible light. Electric motors are another common form of electromechanical transducer, converting electrical energy into kinetic energy to perform a mechanical task. The inverse of an electric motor – a generator – is also a transducer, turning kinetic energy into electrical energy that can then be used by other devices. As in all energy conversions, some energy is lost when transducers operate. The efficiency of a transducer is found by comparing the total energy put into it to the total energy coming out of the system. Some transducers are very efficient, while others are extraordinarily inefficient. A radio antenna, for example, acts as a transducer to turn radio frequency power into an electromagnetic field; when operating well, this process is upwards of 80% efficient. Most electrical motors, by contrast, are well under 50% efficient, and a common light bulb, because of the amount of energy lost as heat, is less than 10% efficient.

boolean algebra

2010-07-16T14:17:00.000+05:30

Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of conjunction x∧y, disjunction x∨y, and complement ¬x. The Boolean operations are these and all other operations that can be built from these, such as x∧(y∨z). These turn out to coincide with the set of all operations on the set {0,1} that take only finitely many arguments; there are 22n such operations when there are n arguments.

The laws of Boolean algebra can be defined axiomatically as certain equations called axioms together with their logical consequences called theorems, or semantically as those equations that are true for every possible assignment of 0 or 1 to their variables. The axiomatic approach is sound and complete in the sense that it proves respectively neither more nor fewer laws than the semantic approach.

Values

Boolean algebra is the algebra of two values. These are usually taken to be 0 and 1, as we shall do here, although F and T, false and true, etc. are also in common use. For the purpose of understanding Boolean algebra any Boolean domain of two values will do.

Regardless of nomenclature, the values are customarily thought of as essentially logical in character and are therefore referred to as truth values, in contrast to the natural numbers or the reals which are considered numerical values. On the other hand the algebra of the integers modulo 2, while ostensibly just as numeric as the integers themselves, was shown to constitute exactly Boolean algebra, originally by I.I. Zhegalkin in 1927 and rediscovered independently in the west by Marshall Stone in 1936. So in fact there is some ambiguity in the true nature of Boolean algebra: it can be viewed as either logical or numeric in character.

More generally Boolean algebra is the algebra of values from any Boolean algebra as a model of the laws of Boolean algebra. For example the bit vectors of a given length, as with say 32-bit computer words, can be combined with Boolean operations in the same way as individual bits, thereby forming a 232-element Boolean algebra under those operations. Any such combination applies the same Boolean operation to all bits simultaneously. This passage from the Boolean algebra of 0 and 1 to these more general Boolean algebras is the Boolean counterpart of the passage from the algebra of the ring of integers to the algebra of commutative rings in general. The two-element Boolean algebra is the prototypical Boolean algebra in the same sense as the ring of integers is the prototypical commutative ring. Boolean logic as the subject matter of this article is independent of the choice of Boolean algebra (the same equations hold of every nontrivial Boolean algebra); hence, there is no need here to consider any Boolean algebra other than the two-element one. The article on Boolean algebra (structure) treats Boolean algebras themselves.
operations

Basic operations

After values, the next ingredient of any algebraic system is its operations. Whereas elementary algebra is based on numeric operations multiplication xy, addition x + y, and negation −x, Boolean algebra is customarily based on logical counterparts to those operations, namely conjunction x∧y (AND), disjunction x∨y (OR), and complement or negation ¬x (NOT). In electronics, the AND is represented as a multiplication, the OR is represented as an addition, and the NOT is represented with an overbar: x ∧ y and x ∨ y, therefore, become xy and x + y.
Conjunction is the closest of these three to its numerical counterpart, in fact on 0 and 1 it is multiplication. As a logical operation the conjunction of two propositions is true when both propositions are true, and otherwise is false. The first column of Figure 1 below tabulates the values of x∧y for the four possible valuations for x and y; such a tabulation is traditionally called a truth table.
Disjunction, in the second column of the figures, works almost like addition, with one exception: the disjunction of 1 and 1 is neither 2 nor 0 but 1. Thus the disjunction of two propositions is false when both propositions are false, and otherwise is true. This is just the definition of conjunction with true and false interchanged everywhere; because of this we say that disjunction is the dual of conjunction.
Logical negation however does not work like numerical negation at all. Instead it corresponds to incrementation: ¬x = x+1 mod 2. Yet it shares in common with numerical negation the property that applying it twice returns the original value: ¬¬x = x, just as −(−x) = x. An operation with this property is called an involution. The set {0,1} has two permutations, both involutary, namely the identity, no movement, corresponding to numerical negation mod 2 (since +1 = −1 mod 2), and SWAP, corresponding to logical negation. Using negation we can formalize the notion that conjunction is dual to disjunction via De Morgan's laws, ¬(x∧y) = ¬x ∨ ¬y and ¬(x∨y) = ¬x ∧ ¬y. These can also be construed as definitions of conjunction in terms of disjunction and vice versa: x∧y = ¬(¬x ∨ ¬y) and x∨y = ¬(¬x ∧ ¬y).

Various representations of Boolean operations

Figure 2 shows the symbols used in digital electronics for conjunction and disjunction; the input ports are on the left and the signals flow through to the output port on the right. Inverters negating the input signals on the way in, or the output signals on the way out, are represented as circles on the port to be inverted.

Derived operations

Other Boolean operations are derivable from these by composition. For example implication x→y (IMP), in the third column of the figures, is a binary operation which is false when x is true and y is false, and true otherwise. It can be expressed as x→y = ¬x∨y (the OR-gate of Figure 2 with the x input inverted), or equivalently ¬(x∧¬y) (its De Morgan equivalent in Figure 3). In logic this operation is called material implication, to distinguish it from related but non-Boolean logical concepts such as entailment and relevant implication. The idea is that an implication x→y is by default true (the weaker truth value in the sense that false implies true but not vice versa) unless its premise or antecedent x holds, in which case the truth of the implication is that of its conclusion or consequent y.
Although disjunction is not the exact counterpart of numerical addition, Boolean algebra nonetheless does have an exact counterpart, called exclusive-or (XOR) or parity, x⊕y. As shown in the fourth column of the figures, the exclusive-or of two propositions is true just when exactly one of the propositions is true; equivalently when an odd number of the propositions is true, whence the name "parity". Exclusive-or is the operation of addition mod 2. The exclusive-or of any value with itself vanishes, x⊕x = 0, since the arguments have an even number of whatever value x has. Its digital electronics symbol is shown in Figure 2, being a hybrid of the disjunction symbol and the equality symbol. The latter reflects the fact that the negation (which is also the dual) of XOR, ¬(x⊕y), is logical equivalence, EQV, being true just when x and y are equal, either both true or both false. XOR and EQV are the only binary Boolean operations that are commutative and whose truth tables have equally many 0s and 1s. Exclusive-or together with conjunction constitute yet another complete basis for Boolean algebra, with the Boolean operations reformulated as the Zhegalkin polynomials.
Another example is Sheffer stroke, x|y, the NAND gate in digital electronics, which is false when both arguments are true, and true otherwise. NAND is definable by composition of negation with conjunction as x |y = ¬(x∧y). It does not have its own schematic symbol as it is easily represented as an AND gate with an inverted output. Unlike conjunction and disjunction, NAND is a binary operation that can be used to obtain negation, via the definition ¬x = x|x. With negation in hand one can then in turn define conjunction in terms of NAND via x∧y = ¬(x|y), from which all other Boolean operations of nonzero arity can then be obtained. NOR, ¬(x∨y), as the evident dual of NAND serves this purpose equally well. This universal character of NAND and NOR makes them a popular choice for gate arrays, integrated circuits with multiple general-purpose gates.
The above-mentioned duality of conjunction and disjunction is exposed further by De Morgan's laws, ¬(x∧y) = ¬x∨¬y and ¬(x∨y) = ¬x∧¬y. Figure 3 illustrates De Morgan's laws by giving for each gate its De Morgan dual, converted back to the original operation with inverters on both inputs and the outputs. In the case of implication, taking the form of an OR-gate with one inverter on disjunction, that inverter is cancelled by the second inverter that would have gone there. The De Morgan dual of XOR is just XOR with an inverter on the output (there is no separate symbol); as with implication, putting inverters on all three ports cancels the dual's output inverter. More generally, changing an odd number of inverters on an XOR gate produces the dual gate, an even number leaves the gate's functionality unchanged.
As with all the other laws in this section, De Morgan's laws may be verified case by case for each of the 2ⁿ possible valuations of the n variables occurring in the law, here two variables and hence 2² = 4 valuations. De Morgan's laws play a role in putting Boolean terms in certain normal forms, one of which we will encounter later in the section on soundness and completeness.
Figure 4 illustrates the corresponding Venn diagrams for each of the four operations presented in Figures 1-3. The interior (respectively exterior) of each circle represents the value true (respectively false) for the corresponding input, x or y. The convention followed here is to represent the true or 1 outputs as dark regions and false as light, but the reverse convention is also sometimes used.

All Boolean operations

There are infinitely many expressions that can be built from two variables using the above operations, suggesting great expressiveness. Yet a straightforward counting argument shows that only 16 distinct binary operations on two values are possible. Any given binary operation is determined by its output values for each possible combination of input values. The two arguments have 2 × 2 = 4 possible combinations of values between them, and there are 2⁴ = 16 ways of assigning an output value to each of these four input values. The choice of one of these 16 assignments then determines the operation; so all together there are only 16 distinct binary operations.
The 16 binary Boolean operations can be organized as follows:
Two constant operations, 0 and 1.
Four operations dependent on one variable, namely x, ¬x, y, and ¬y, whose truth tables amount to two juxtaposed rectangles, one containing two 1s and the other two 0s.
Two operations with a "checkerboard" truth table, namely XOR and EQV.
Four operations are obtained from disjunction with some subset of its inputs negated, namely x∨y, x→y, y→x, and x|y; their truth tables contain a single 0.
The final four come from the same treatment applied to conjunction, having a single 1 in their truth tables.
10 of the 16 operations depend on both variables; all are representable schematically as an AND-gate, an OR-gate, or an XOR-gate, with one port optionally inverted. For the AND and OR gates the location of each inverter matters, for the XOR gate it does not, only whether there is an even or odd number of inverters.
Operations of other arities are possible. For example the ternary counterpart of disjunction can be obtained as (x∨y)∨z. In general an n-ary operation, one having n inputs, has 2ⁿ possible valuations of those inputs. An operation has two possibilities for each of these, whence there exist 2^2ⁿ n-ary Boolean operations. For example, there are 2³² = 4,294,967,296 operations with 5 inputs.
Although Boolean algebra confines attention to operations that return a single bit, the concept generalizes to operations that take n bits in and return x bits instead of one bit. Digital circuit designers draw such operations as suitably shaped boxes with n wires entering on the left and m wires exiting on the right. Such multi-output operations can be understood simply as m n-ary operations. The operation count must then be raised to the m-th power, or, in the case of n inputs, (2^2ⁿ)^m = 2^m2ⁿ operations. The number of Boolean operations of this generalized kind with say 5 inputs and 5 outputs is 1.46 × 10⁴⁸. A logic gate or computer module mapping 32 bits to 32 bits could implement any of 5.47 × 10^{41,373,247,567} operations, more than is obtained by squaring a googol 28 times.

Axioms

With values and operations in hand, the next aspect of Boolean algebra is that of laws or properties. As with many kinds of algebra, the principal laws take the form of equations between terms built up from variables using the operations of the algebra. Such an equation is deemed a law or identity just when both sides have the same value for all values of the variables, equivalently when the two terms denote the same operation.
Numeric algebra has laws such as commutativity of addition and multiplication, x + y = y + x and xy = yx. Similarly, Boolean algebra has commutativity in that x ∨ y = y ∨ x for disjunction and x ∧ y = y ∧ x for conjunction. Not all binary operations are commutative; Boolean implication is not commutative, like subtraction and division.
Another equally fundamental law is associativity, which in the case of numeric multiplication is expressed as x(yz) = (xy)z, justifying abbreviating both sides to xyz and thinking of multiplication as a single ternary operation. All four of numeric addition and multiplication and logical disjunction and conjunction are associative, giving for the latter two the Boolean laws x ∨ (y ∨ z) = (x ∨ y) ∨ z and x ∧ (y ∧ z) = (x ∧ y) ∧ z.
Again numeric subtraction and logical implication serve as examples, this time of binary operations that are not associative. On the other hand exclusive-or, being just addition mod 2, is both commutative and associative.
Boolean algebra does not completely mirror numeric algebra however, as both conjunction and disjunction satisfy idempotence, expressed respectively as x ∧ x = x and x ∨ x = x. These laws are easily verified by considering the two valuations 0 and 1 for x. But since 2 + 2 = 2 × 2 = 4 in arithmetic, clearly numeric addition and multiplication are not idempotent. With arithmetic mod 2 on the other hand, multiplication is idempotent, though not addition since 1 + 1 = 0 mod 2, reflected logically in the idempotence of conjunction but not of exclusive-or.
A more subtle difference between number and logic is with x(x + y) and x + xy, neither of which equal x numerically. In Boolean algebra however, both x ∧ (x ∨ y) and x ∨ (x ∧ y) are equal to x, as can be verified for each of the four possible valuations for x and y. These two Boolean laws are called the laws of absorption. These laws (both are needed) together with the associativity, commutativity, and idempotence of conjunction and disjunction constitute the defining laws or axioms of lattice theory. (Actually idempotence can be derived from the other axioms.)
Another law common to numbers and truth values is distributivity of multiplication over addition, when paired with distributivity of conjunction over disjunction. Numerically we have x(y + z) = xy + xz, whose Boolean algebra counterpart is x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). On the other hand Boolean algebra also has distributivity of disjunction over conjunction, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), for which there is no numeric counterpart, consider 1 + 2 × 3 = 7 whereas (1 + 2) × (1 + 3) = 12. Like associativity, distributivity has three variables and so requires checking 2³ = 8 cases.
Either distributivity law for Boolean algebra entails the other. Adding either to the axioms for lattices axiomatizes the theory of distributive lattices. That theory does not need the idempotence axioms because they follow from the six absorption, distributivity, and associativity laws.
Two Boolean laws having no numeric counterpart are the laws characterizing logical negation, namely x ∧ ¬x = 0 and x ∨ ¬x = 1. These are the only laws thus far that have required constants. It then follows that x ∧ 0 = x ∧ (x ∧ ¬x) = (x ∧ x) ∧ ¬x = x ∧ ¬x = 0, showing that 0 works with conjunction in logic just as it does with multiplication of numbers. Also x ∨ 0 = x ∨ (x ∧ ¬x) = x by absorption. Dualizing this reasoning, we obtain x ∨ 1 = 1 and x ∧ 1 = x. Alternatively we can justify these laws more directly simply by checking them for each of the two valuations of x.
The six laws of lattice theory along with these first two laws for negation axiomatize the theory of complemented lattices. Including either distributivity law then axiomatizes the theory of complemented distributive lattices. For convenience we collect these nine laws in one place as follows.

		associativity
		commutativity
		absorption
		distributivity
		complements

The next two sections show that this theory is sufficient to axiomatize all the valid laws or identities of two-valued logic, that is, Boolean algebra. It follows that Boolean algebra as commonly defined in terms of these axioms coincides with the intuitive semantic notion of the valid identities of two-valued logic.

Derivations

While the Boolean laws enumerated in the previous section are certainly highlights of Boolean algebra, they by no means exhaust the laws, of which there are infinitely many, nor do they even exhaust the highlights. As it is out of the question to proceed in the ad hoc way of the preceding section for ever, the question arises as to how best to present the remaining laws.
One way of establishing an equation as being a law is to verify its truth for all valuations of its variables, sometimes called the method of truth tables. This is the method we depended on in the previous section to justify each law as we introduced it, constituting the semantic approach to establishing laws. From a practical standpoint the method lends itself to computer implementation for 20-30 variables because the enumeration of valuations is straightforward to program and boring to carry out, making it ideal work for a computer. Because there are 2ⁿ valuations to check the method starts to become impractical as 40 variables is approached. Beyond that the approach becomes of value mainly as the in-principle semantic definition of what constitutes an identically true or valid equation.
In contrast the syntactic approach is to derive new laws by symbolic manipulation from already established laws such as those listed in the previous section. (This is not to imply that derivations of a law shorter than the length of a semantic verification of that law need exist, although some thousand-variable laws impossible to verify by enumeration of valuations can have quite short derivations.) Here is an example showing the derivation of (w∨x)∨(y∨z) = (w∨y)∨(x∨z) from just the commutativity and associativity of disjunction.
(w∨x)∨(y∨z)
= ((w∨x)∨y)∨z
= (w∨(x∨y))∨z
= (w∨(y∨x))∨z
= ((w∨y)∨x)∨z
= (w∨y)∨(x∨z)
The first two and last two steps appealed to associativity while the middle step used commutativity.
The rules of derivation for forming new laws from old can be assumed to be those permissible in high school algebra. For definiteness however it is worthwhile formulating a well-defined set of rules showing exactly what is needed. These are the domain-independent rules of equational logic, as sound for logic as they are for numerical domains or any other kind.
Reflexivity: t = t. That is, any equation whose two sides are the same term t is a law. (While arguably an axiom rather than a rule since it has no premises, we classify it as a rule because like the other three rules it is domain-independent, making no mention of specific logical, numeric, or other operations.)
Symmetry: From s = t infer t = s. That is, the two sides of a law may be interchanged. Intuitively one attaches no importance to which side of an equation a term comes from.
Transitivity: A chain s = t = u of two laws yields the law s = u. (This law of "cutting out the middleman" is applied four times in the above example to eliminate the intermediate terms.)
Substitution: Given two laws and a variable, each occurrence of that variable in the first law may be replaced by one or the other side of the second law. (Distinct occurrences can be replaced by distinct sides, but every occurrence must be replaced by one or the other side.)
While the first equation in the above example might seem simply a straightforward application of the associativity law, when analyzed more carefully according to the above rules it can be seen to require something more. We can justify it in terms of the reflexivity and substitution rules. Beginning with the laws x∨(y∨z) = (x∨y)∨z and w∨x = w∨x, we use substitution to replace both occurrences of x by w∨x to arrive at the first equation. All five equations in the chain are accounted for along similar lines, with commutativity in place of associativity in the middle equation.

Soundness and completeness

It can be shown that the two approaches, semantic and syntactic, to constructing all the laws of Boolean algebra lead to the same set of laws. We say that the syntactic approach is sound when it yields a subset of the semantically obtained laws, and complete when it yields a superset thereof. We can then restate this coinciding of the semantic and syntactic approaches as the soundness and completeness of the syntactic approach with respect to (or as calibrated by) the semantic approach.
Soundness follows firstly from the fact that the initial laws or axioms we started from were all identities, that is, semantically true laws. Secondly it depends on the easily verified fact that the rules preserve identities.
Completeness can be proved by first deriving a few additional useful laws and then showing how to use the axioms and rules to prove that a term with n variables, ordered alphabetically say, is equal to its n-ary normal form, namely a unique term associated with the n-ary Boolean operation realized by that term with the variables in that order. It then follows that if two terms denote the same operation (the same thing as being semantically equal), they are both provably equal to the normal form term denoting that operation, and hence by transitivity provably equal to each other.
There is more than one suitable choice of normal form, but complete disjunctive normal form will do. A literal is either a variable or a negated variable. A disjunctive normal form (DNF) term is a disjunction of conjunctions of literals. (Associativity allows a term such as x∨(y∨z) to be viewed as the ternary disjunction x∨y∨z, likewise for longer disjunctions, and similarly for conjunction.) A DNF term is complete when every disjunct (conjunction) contains exactly one occurrence of each variable, independently of whether or not the variable is negated. Such a conjunction uniquely represents the operation it denotes by virtue of serving as a coding of those valuations at which the operation returns 1. Each conjunction codes the valuation setting the positively occurring variables to 1 and the negated ones to 0; the value of the conjunction at that valuation is 1, and hence so is the whole term. At valuations corresponding to omitted conjunctions, all conjunctions present in the term evaluate to 0 and hence so does the whole term.
In outline the general technique for converting any term to its normal form, or normalizing it, is to use De Morgan's laws to push the negations down to the variables. This yields monotone normal form, a term built from literals with conjunctions and disjunctions. For example ¬(x ∨ (¬y∧z)) becomes ¬x ∧ ¬(¬y∧z) and then ¬x ∧ (¬¬y∨¬z). Applying ¬¬x = x then yields ¬x ∧ (y∨¬z).
Next use distributivity of conjunction over disjunction to push all conjunctions down below all disjunctions, yielding a DNF term. This makes the above example (¬x∧y) ∨ (¬x∧¬z).
Then for each variable y, replace each conjunction x not containing y with the disjunction of two copies of x, with y conjoined to one copy of x and ¬y conjoined to the other, in the end yielding a complete DNF term. (This is one place where an auxiliary law helps, in this case x = x∧1 = x∧(y∨¬y) = (x∧y) ∨ (x∧¬y).) In the above example the first conjunction lacks z while the second lacks y; expanding appropriately yields the complete DNF term (¬x∧y∧z) ∨ (¬x∧y∧¬z) ∨ (¬x∧¬z∧y) ∨ (¬x∧¬z∧¬y).
Next use commutativity to put the literals in each conjunction in alphabetical order. The example becomes (¬x∧y∧z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧¬y∧¬z). This brings any repeated copies of literals next to each other; delete the redundant copies using idempotence of conjunction, not needed in our example.
Lastly order the disjuncts according to a suitable uniformly applied criterion. The criterion we use here is to read the positive and negative literals of a conjunction as respectively 1 and 0 bits, and to read the bits in a conjunction as a binary number. In our example the bits are 011, 010, 010, 000, or in decimal 3, 2, 2, 0. Ordering them numerically as 0, 2, 2, 3 yields (¬x∧¬y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧z). Note that these bits are exactly those valuations for x, y, and z that satisfy our original term ¬(x∨(¬y∧z)). Complete DNF amounts to a canonical way of representing the truth table for the original term as another term.
Repeated conjunctions can then be deleted using idempotence of disjunction, which simplifies our example to (¬x∧¬y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧z).
In this way we have proved that the term we started with is equal to the normal form term for the operation it denotes. Hence all terms denoting that operation are provably equal to the same normal form term and hence by transitivity to each other.

A Greener Apple

2010-07-16T13:47:00.002+05:30

Apple has been criticized by some environmental organizations for not being a leader in removing toxic chemicals from its new products, and for not aggressively or properly recycling its old products. Upon investigating Apple’s current practices and progress towards these goals, I was surprised to learn that in many cases Apple is ahead of, or will soon be ahead of, most of its competitors in these areas. Whatever other improvements we need to make, it is certainly clear that we have failed to communicate the things that we are doing well.

It is generally not Apple’s policy to trumpet our plans for the future; we tend to talk about the things we have just accomplished. Unfortunately this policy has left our customers, shareholders, employees and the industry in the dark about Apple’s desires and plans to become greener. Our stakeholders deserve and expect more from us, and they’re right to do so. They want us to be a leader in this area, just as we are in the other areas of our business. So today we’re changing our policy.

Now I’d like to tell you what we are doing to remove toxic chemicals from our new products, and to more aggressively recycle our old products.
Removing Toxic Chemicals

Lead

Many of the dangerous chemicals we all want to eliminate from electronic products are found in very small amounts, but there’s one toxic substance that some companies still ship by the pound, and that’s the lead contained in their cathode-ray tube (CRT) displays. A typical CRT contains approximately 3 pounds (1.36 kg) of lead. In mid-2006, Apple became the first company in the computer industry to completely eliminate CRTs. The effect has been stunning — our first CRT-based iMac contained 484 grams of lead; our current third-generation LCD-based iMac contains less than 1 gram of lead.

Apple completely eliminated the use of CRTs in mid-2006.

A note of comparison — Dell, Gateway, Hewlett Packard and Lenovo still ship CRT displays today.
Cadmium
Hexavalent Chromium
Decabromodiphenyl Ether

The European Union is generally ahead of the U.S. in restricting toxic substances in electronic products. Their latest restrictions, known as RoHS, went into effect in July 2006. All Apple products worldwide comply with RoHS. Our manufacturing policies had already restricted or banned most of the chemicals covered by RoHS, and Apple began introducing fully RoHS-compliant products a year before the European deadline.

Almost a year later, however, some electronics companies can only claim their products are RoHS compliant because of certain little-known exemptions granted by the EU. Despite the tough restrictions of RoHS, these exemptions let companies ship electronics that still contain high concentrations of two hazardous substances — hexavalent chromium, the carcinogen against which Erin Brockovich famously campaigned, and the brominated flame retardant decabromodiphenyl ether (DecaBDE), which is also feared to have adverse health effects. Apple phased out these and many other chemicals several years ago through design innovations and the use of higher quality metals and plastics.

Apple products met both the spirit and letter of the RoHS restrictions on cadmium, hexavalent chromium and brominated flame retardants years before RoHS went into effect.

A note of comparison — Some electronics companies, whose names you know, still rely on RoHS exemptions and use these toxic chemicals in their products today.
Arsenic
Mercury

Arsenic and mercury are industry standard materials used in liquid crystal displays (LCDs). Arsenic is added during the manufacturing of the high performance glass used in LCDs to prevent the formation of defects, and the fluorescent lamps used to illuminate LCDs contain minute amounts of mercury. Apple is on track to introduce our first displays using arsenic-free glass in 2007. A small number of high performance integrated circuits (ICs) will continue to contain a minute amount of arsenic as an element of the semiconductor substrate.

To eliminate mercury in our displays, we need to transition from fluorescent lamps to light-emitting diodes (LEDs) to illuminate the displays. Fortunately, all iPod displays already use LEDs for illumination, and therefore contain no mercury. We plan to introduce our first Macs with LED backlight technology in 2007. Our ability to completely eliminate fluorescent lamps in all of our displays depends on how fast the LCD industry can transition to LED backlighting for larger displays.

Apple plans to completely eliminate the use of arsenic in all of its displays by the end of 2008.

Apple plans to reduce and eventually eliminate the use of mercury by transitioning to LED backlighting for all displays when technically and economically feasible.
Polyvinyl Chloride
Brominated flame retardants

Some companies have made promises to phase out other toxic chemicals like polyvinyl chloride (PVC), a type of plastic primarily used in the construction industry but also found in computer parts and cables, and brominated flame retardants, or BFRs, which reduce the risk of fire. Apple began phasing out PVC twelve years ago and began restricting BFRs in 2001. For the past several years, we have been developing alternative materials that can replace these chemicals without compromising the safety or quality of our products. Today, we’ve successfully eliminated the largest applications of PVC and BFRs in our products, and we’re close to eliminating these chemicals altogether. For example, more than three million iPods have already shipped with a BFR-free laminate on their logic boards.

Dell and Lenovo have publicly stated that they plan to eliminate the use of PVC and BFRs in their products in 2009. Hewlett Packard has not yet publicly stated when they will eliminate the use of PVC and BFRs in their products, but has said that they will publish a plan by the end of 2007 which will state when in the future they will eliminate the use of these toxic chemicals in their products.

Apple plans to completely eliminate the use of PVC and BFRs in its products by the end of 2008.

A note of comparison — In 2007 HP stated that they will remove PVC from all their packaging. Apple did this 12 years ago. Last year, Dell began the process of phasing out large quantities of brominated flame retardants in large plastic enclosure parts. Apple’s plastic enclosure parts have been bromine-free since 2002.

In one environmental group’s recent scorecard, Dell, HP and Lenovo all scored higher than Apple because of their plans (or “plans for releasing plans” in the case of HP). In reality, Apple is ahead of all of these companies in eliminating toxic chemicals from its products.
Recycling Our Products (E-Waste)

Apple started recycling in 1994 and today we operate recycling programs in countries where more than 82% of all Macs and iPods are sold. By the end of this year, that figure will increase to 93%. How successful are these programs?

Currently, there is no industry standard way to measure the effectiveness of a company’s recycling programs. Dell has proposed a simple measure - assume a seven year product lifetime, and measure the percentage of the total weight you recycle each year compared to the total weight of what you sold seven years earlier. This makes sense to us, and has the added advantages of clarity and simplicity.

Apple recycled 13 million pounds of e-waste in 2006, which is equal to 9.5% of the weight of all products Apple sold seven years earlier. We expect this percentage to grow to 13% in 2007, and to 20% in 2008. By 2010, we forecast recycling 19 million pounds of e-waste per year — nearly 30% of the product weight we sold seven years earlier.
Weight Recycled as % of Past Sales
Chart shows an upward trend starting with 1.5% in 2002, up to an actual 9.5% in 2006, and an estimated 28% in 2010.

A note of comparison — the latest figures from HP and Dell are each around 10% per year, and neither company has yet disclosed plans to grow this percentage in the future. By 2010, Apple may be recycling significantly more than either Dell or HP as a percentage of past sales weight.

All the e-waste we collect in North America is processed in the U.S., and nothing is shipped overseas for disposal. We carefully review “environmental fate” submissions from each vendor, so we know how raw materials are handled at the end of the recycling process. We hold our recycling vendors to the highest environmental standards in the industry. In addition to annual compliance audits, we also review the performance of their downstream vendors. They must comply with all applicable health and safety laws, and we do not allow the use of prison labor at any stage of the recycling process.

Producers must also take responsibility for the design and material choices that create the product in the first place. It is these choices that fundamentally determine the weight and recycling value of material waste at the end of a product’s life. The iMac is a world-class example of material efficiency, having shed 60% of its weight since its debut in 1998. Our designs use aircraft-grade aluminum, stainless steel and high-grade plastics that are in high demand from recyclers, who recover and resell these raw materials for use in other types of products. Few of our competitors do the same.

Let me take a moment to talk specifically about iPods, even though they are included in the above data. All of Apple’s U.S. retail stores, which now number more than 150, take back unwanted iPods for environmentally friendly disposal free of charge. As an incentive, we even offer customers a 10% discount on a new iPod when they bring their old iPod to our stores for proper disposal. This summer we’re expanding it to Apple retail stores worldwide, and we’re also extending it to include free shipping from anywhere in the U.S. No product purchases are required for any of our free take back programs. In a few months, we think we’ll have ‘best of breed’ iPod recycling programs in the U.S., and we plan to continue to expand our free iPod recycling programs globally in the future.

By 2010, Apple may be recycling significantly more than either Dell or HP as a percentage of past sales weight.

All the e-waste we collect in North America is processed in the U.S., and nothing is shipped overseas for disposal.

Apple products are designed using high quality materials that are in high demand from recyclers.
The Future

Today is the first time we have openly discussed our plans to become a greener Apple. It will not be the last. We will be providing updates of our efforts and accomplishments at least annually, most likely around this time of the year. And we plan to bring other environmental issues to the table as well, such as the energy efficiency of the products in our industry. We are also beginning to explore the overall carbon “footprint” of our products, and may have some interesting data and issues to share later this year.

I hope you are as delighted as I was when I first learned how far along Apple actually is in removing toxic chemicals from its products and recycling its older products. We apologize for leaving you in the dark for this long. Apple is already a leader in innovation and engineering, and we are applying these same talents to become an environmental leader. Based on our tangible actions and results over time, hopefully our customers, employees, shareholders and professional colleagues will all feel proud of our ongoing efforts to become a greener Apple.

types of capacitor

2010-04-14T09:26:00.000+05:30

Types of Capacitors

There are a very large variety of different types of Capacitors available in the market place and each one has its own set of characteristics and applications from small delicate trimming capacitors up to large power metal can type capacitors used in high voltage power correction and smoothing circuits. Like resistors, there are also variable types of capacitors which allow us to vary their capacitance value for use in radio or "frequency tuning" type circuits. Either way, capacitors play an important part in electronic circuits so here are a few of the more "Common" types of capacitors available.
1. Dielectric

Dielectric Capacitors are usually of the variable type such as used for tuning transmitters, receivers and transistor radios. They have a set of fixed plates and a set of moving plates that mesh with the fixed plates and the position of the moving plates with respect to the fixed plates determines the overall capacitance. The capacitance is generally at maximum when the plates are fully meshed. High voltage type tuning capacitors have relatively large spacings or air-gaps between the plates with breakdown voltages reaching many thousands of volts.
Variable Capacitor Symbols
Radial Lead Film Capacitor

As well as the continuously variable types, preset types are also available called Trimmers. These are generally small devices that can be adjusted or "pre-set" to a particular capacitance with the aid of a screwdriver and are available in very small capacitances of 100pF or less and are non-polarized.
2. Film Capacitors

Film Capacitors are the most commonly available of all types of capacitors, consisting of a relatively large family of capacitors with the difference being in their dielectric properties. These include polyester (Mylar), polystyrene, polypropylene, polycarbonate, metallized paper, teflon etc. Film type capacitors are available in capacitance ranges from 5pF to 100uF depending upon the actual type of capacitor and its voltage rating. Film capacitors also come in an assortment of shapes and case styles which include:

* Wrap & Fill (Oval & Round) - where the capacitor is wrapped in a tight plastic tape and have the ends filled with epoxy to seal them.
*
* Epoxy Case (Rectangular & Round) - where the capacitor is encased in a moulded plastic shell which is then filled with epoxy.
*
* Metal Hermetically Sealed (Rectangular & Round) - where the capacitor is encased in a metal tube or can and again sealed with epoxy.

with all the above case styles available in both Axial and Radial Leads.

Examples of film capacitors are the rectangular metallized film and cylindrical film & foil types as shown below.
Radial Lead Type
Radial Lead Film Capacitor
Axial Lead Type
Metallized Film Capacitor

The film and foil types of capacitors are made from long thin strips of thin metal foil with the dielectric material sandwiched together which are wound into a tight roll and then sealed in paper or metal tubes. These film types require a much thicker dielectric film to reduce the risk of tears or punctures in the film, and is therefore more suited to lower capacitance values and larger case sizes.

Metallized foil capacitors have the conductive film metallized sprayed directly onto each side of the dielectric which gives the capacitor self-healing properties and can therefore use much thinner dielectric films. This allows for higher capacitance values and smaller case sizes for a given capacitance. Film and foil capacitors are generally used for higher power and more precise applications.
3. Ceramic Capacitors

Ceramic Capacitors or Disc Capacitors as they are generally called, are made by coating two sides of a small porcelain or ceramic disc with silver and are then stacked together to make a capacitor. For very low capacitance values a single ceramic disc of about 3-6mm is used. Ceramic capacitors have a high dielectric constant (High-K) and are available so that relatively high capacitances can be obtained in a small physical size. They exhibit large non-linear changes in capacitance against temperature and as a result are used as de-coupling or by-pass capacitors as they are also non-polarized devices. Ceramic capacitors have values ranging from a few picofarads to one or two microfarads but their voltage ratings are generally quite low.

Ceramic types of capacitors generally have a 3-digit code printed onto their body to identify their capacitance value. For example, 103 would indicate 10 x 103pF which is equivalent to 10,000 pF or 0.01μF. Likewise, 104 would indicate 10 x 104pF which is equivalent to 100,000 pF or 0.1μF and so on. Letter codes are sometimes used to indicate their tolerance value such as: J = 5%, K = 10% or M = 20% etc.
4. Electrolytic Capacitors

Electrolytic Capacitors are generally used when very large capacitance values are required. Here instead of using a very thin metallic film layer for one of the electrodes, a semi-liquid electrolyte solution in the form of a jelly or paste is used which serves as the second electrode (usually the cathode). The dielectric is a very thin layer of oxide which is grown electro-chemically in production with the thickness of the film being less than ten microns. This insulating layer is so thin that it is possible to make large value capacitors of a small size. The majority of electrolytic types of capacitors are Polarized, that is the voltage applied to the capacitor terminals must be of the correct polarity as an incorrect polarization will break down the insulating oxide layer and permanent damage may result.

Electrolytic Capacitors are generally used in DC power supply circuits to help reduce the ripple voltage or for coupling and decoupling applications. Electrolytic's generally come in two basic forms; Aluminum Electrolytic and Tantalum Electrolytic capacitors.
Electrolytic Capacitor
Electrolytic Capacitor
1. Aluminium Electrolytic Capacitors

There are basically two types of Aluminium Electrolytic Capacitor, the plain foil type and the etched foil type. The thickness of the aluminium oxide film and high breakdown voltage give these capacitors very high capacitance values for their size. The etched foil type differs from the plain foil type in that the aluminium oxide on the anode and cathode foils has been chemically etched to increase its surface area and permittivity. This gives a smaller sized capacitor than a plain foil type of equivalent value but has the disadvantage of not being able to withstand high AC currents compared to the plain type. Also their tolerance range is quite large up to 20%. Etched foil electrolytic's are best used in coupling, DC blocking and by-pass circuits while plain foil types are better suited as smoothing capacitors in power supplies. Typical values of capacitance range from 1uF to 47000uF. Aluminium Electrolytic's are "polarized" devices so reversing the applied voltage on the leads will cause the insulating layer within the capacitor to be destroyed along with the capacitor, "so be aware".
2. Tantalum Electrolytic Capacitors

Tantalum Electrolytic Capacitors or Tantalum Beads, are available in both wet (foil) and dry (solid) electrolytic types with the dry or solid tantalum being the most common. Solid tantalums use manganese dioxide as their second terminal and are physically smaller than the equivalent aluminium capacitors. The dielectric properties of tantalum oxide is also much better than those of aluminium oxide giving a lower leakage currents and better capacitance stability which makes them suitable for timing applications. Also tantalum capacitors although polarized, can tolerate being connected to a reverse voltage much more easily than the Aluminium types but are rated at much lower working voltages. Typical values of capacitance range from 47nF to 470uF.
Aluminium & Tantalum Electrolytic Capacitor
Aluminium and Tantalum Electrolytic Capacitor

2010-03-25T16:10:00.000+05:30

Voltage Regulator (regulator), usually having three legs, converts varying input voltage and produces a constant regulated output voltage. They are available in a variety of outputs.

The most common part numbers start with the numbers 78 or 79 and finish with two digits indicating the output voltage. The number 78 represents positive voltage and 79 negative one. The 78XX series of voltage regulators are designed for positive input. And the 79XX series is designed for negative input.

Examples:

· 5V DC Regulator Name: LM7805 or MC7805

· -5V DC Regulator Name: LM7905 or MC7905

· 6V DC Regulator Name: LM7806 or MC7806

· -9V DC Regulator Name: LM7909 or MC7909

The LM78XX series typically has the ability to drive current up to 1A. For application requirements up to 150mA, 78LXX can be used. As mentioned above, the component has three legs: Input leg which can hold up to 36VDC Common leg (GND) and an output leg with the regulator's voltage. For maximum voltage regulation, adding a capacitor in parallel between the common leg and the output is usually recommended. Typically a 0.1MF capacitor is used. This eliminates any high frequency AC voltage that could otherwise combine with the output voltage. See below circuit diagram which represents a typical use of a voltage regulator.

Note:

As a general rule the input voltage should be limited to 2 to 3 volts above the output voltage. The LM78XX series can handle up to 36 volts input, be advised that the power difference between the input and output appears as heat. If the input voltage is unnecessarily high, the regulator will overheat. Unless sufficient heat dissipation is provided through heat sinking, the regulator will shut down.

LM7805 3-Terminal 1A Positive Voltage Regulator

2010-03-25T16:05:00.000+05:30

General Description
The LM78XX series of three terminal positive regulators are available in the TO-220 package and with several fixed output voltages, making them useful in a wide range of applications. Each type employs internal current limiting, thermal shut down and safe operating area protection, making it essentially indestructible. If adequate heat sinking is provided, they can deliver over 1A output current. Although designed primarily as fixed voltage regulators, these devices can be used with external components to obtain adjustable voltages and currents.

Features

• Output Current up to 1A
• Output Voltages of 5, 6, 8, 9, 10, 12, 15, 18, 24
• Thermal Overload Protection
• Short Circuit Protection
• Output Transistor Safe Operating Area Protection