But the electrical resistance between these two points can depend on many factors such as the conductors length, its cross-sectional area, the temperature, as well as the actual material from which it is made. For example, let’s assume we have a piece of wire (a conductor) that has a length *L*, a cross-sectional area *A* and a resistance *R* as shown.

The electrical resistance, R of this simple conductor is a function of its length, L and the conductors area, A. Ohms law tells us that for a given resistance R, the current flowing through the conductor is proportional to the applied voltage as I = V/R. Now suppose we connect two identical conductors together in a series combination as shown.

Here by connecting the two conductors together in a series, we have effectively doubled the total length of the conductor, 2L while the cross-sectional area, A remains exactly the same. But as well as doubling the length, we have also doubled the total resistance of the conductor, giving 2R. Thus the resistance of the conductor is proportional to its length, that is: **R ∝ L**. In other words, we would expect the electrical resistance of a conductor (or wire) to be proportionally greater the longer it is.

Note also that by doubling the length and therefore the resistance of the conductor (2R), to force the same current, *i* to flow through the conductor as before, we need to double (increase) the applied voltage as now I = (2V)/(2R). Next suppose we connect the two identical conductors together in parallel combination as shown.

Here by connecting the two conductors together in a parallel combination, we have effectively doubled the total area giving 2A, while the conductors length, L remains the same as the original single conductor. But as well as doubling the area, by connecting the two conductors together in parallel we have effectively halved the total resistance of the conductor, giving 1/2R as now each half of the current flows through each conductor branch.

Thus the resistance of the conductor is inversely proportional to its area, that is: **R 1/∝ A**, or R ∝ 1/A. In other words, we would expect the electrical resistance of a conductor (or wire) to be proportionally less the greater is its cross-sectional area.

Also by doubling the area and therefore halving the total resistance of the conductor branch (1/2R), for the same current, *i* to flow through the parallel conductor branch as before we only need half (decrease) the applied voltage as now I = (1/2V)/(1/2R).

So hopefully we can see that the resistance of a conductor is directly proportional to the length (L) of the conductor, that is: R ∝ L, and inversely proportional to its area (A), R ∝ 1/A. Thus we can correctly say that resistance is:

But as well as length and conductor area, we would also expect the electrical resistance of the conductor to depend upon the actual material from which it is made, because different conductive materials, copper, silver, aluminium, etc all have different physical and electrical properties. Thus we can convert the proportionality sign (∝) of the above equation into an equals sign simply by adding a “proportional constant” into the above equation giving:

Where: R is the resistance in ohms (Ω), L is the length in meters (m), A is the area in square meters (m^{2}), and where the proportional constant ρ (the Greek letter “rho”) is known as **Resistivity**.

The electrical resistivity of a particular conductor material is a measure of how strongly the material opposes the flow of electric current through it. This resistivity factor, sometimes called its “specific electrical resistance”, enables the resistance of different types of conductors to be compared to one another at a specified temperature according to their physical properties without regards to their lengths or cross-sectional areas. Thus the higher the resistivity value of ρ the more resistance and vice versa.

For example, the resistivity of a good conductor such as copper is on the order of 1.72 x 10^{-8} ohms per meter (or 17.2 nΩ/m), whereas the resistivity of a poor conductor (insulator) such as air can be well over 1.5 x 10^{14} or 150 trillion Ω/m.

Materials such as copper and aluminium are known for their low levels of resistivity thus allowing electrical current to easily flow through them making these materials ideal for making electrical wires and cables. Silver and gold have much low resistivity values, but for obvious reasons are more expensive to turn into electrical wires.

Then the factors which affect the resistance (R) of a conductor in ohms can be listed as:

- The resistivity (ρ) of the material from which the conductor is made.
- The total length (L) of the conductor.
- The cross-sectional area (A) of the conductor.
- The temperature of the conductor.

Calculate the total DC resistance of a 100 meter roll of 2.5mm^{2} copper wire if the resistivity of copper at 20^{o}C is 1.72 x 10^{-8} Ω per meter.

Given: resistivity of copper at 20^{o}C is 1.72 x 10^{-8}, coil length L = 100m, the cross-sectional area of the conductor is 2.5mm^{2} giving an area of: A = 2.5 x 10^{-6} meters^{2}.

We said previously that resistivity is the electrical resistance per unit length and per unit of conductor cross-sectional area thus showing that resistivity, ρ has the dimensions of ohms per meter, or Ω.m as it is commonly written. Thus for a particular material at a specified temperature its electrical resistivity is given as.

While both the electrical resistance (R) and resistivity (or specific resistance) ρ, are a function of the physical nature of the material being used, and of its physical shape and size expressed by its length (L), and its sectional area (A), **Conductivity**, or specific conductance relates to the ease at which electric current con flow through a material.

Conductance (G) is the reciprocal of resistance (1/R) with the unit of conductance being the siemens (S) and is given the upside down ohms symbol mho, ℧. Thus when a conductor has a conductance of 1 siemens (1S) it has a resistance is 1 ohm (1Ω). So if its resistance is doubled, the conductance halves, and vice-versa as: siemens = 1/ohms, or ohms = 1/siemens.

While a conductors resistance gives the amount of opposition it offers to the flow of electric current, the conductance of a conductor indicates the ease by which it allows electric current to flow. So metals such as copper, aluminium or silver have very large values of conductance meaning that they are good conductors.

Conductivity, σ (Greek letter sigma), is the reciprocal of the resistivity. That is 1/ρ and is measured in siemens per meter (S/m). Since electrical conductivity σ = 1/ρ, the previous expression for electrical resistance, R can be rewritten as:

Then we can say that conductivity is the efficiency by which a conductor passes an electric current or signal without resistive loss. Therefore a material or conductor that has a high conductivity will have a low resistivity, and vice versa, since 1 siemens (S) equals 1Ω^{-1}. So copper which is a good conductor of electric current, has a conductivity of 58.14 x 10^{6} siemens per meter.

A 20 meter length of cable has a cross-sectional area of 1mm^{2} and a resistance of 5 ohms. Calculate the conductivity of the cable.

Given: DC resistance, R = 5 ohms, cable length, L = 20m, and the cross-sectional area of the conductor is 1mm^{2} giving an area of: A = 1 x 10^{-6} meters^{2}.

That is 4 mega-siemens per meter length.

We have seen in this tutorial about resistivity, that resistivity is the property of a material or conductor that indicates of well the material conducts electrical current and also that the electrical resistance (R) of a conductor depends not only on the material from which the conductor is made, copper, silver, aluminium, etc. but also on its physical dimensions.

The resistance of a conductor is directly proportional to its length (L) as R ∝ L. Thus doubling its length will double its resistance, while halving its length would halve its resistance. Also the resistance of a conductor is inversely proportional to its cross-sectional area (A) as R ∝ 1/A. Thus doubling its cross-sectional area would halve its resistance, while halving its cross-sectional area would double its resistance.

We have also learnt that the resistivity (symbol: ρ) of the conductor (or material) relates to the physical property from which it is made and varies from material to material. For example, the resistivity of copper is generally given as: 1.72 x 10^{-8} Ω.m. The resistivity of a particular material is measured in units of Ohm-Meters (Ω.m) which is also affected by temperature.

Depending upon the electrical resistivity value of a particular material, it can be classified as being either a “conductor”, an “insulator” or a “semiconductor”. Note that semiconductors are materials where its conductivity is dependent upon the impurities added to the material.

Resistivity is also important in power distribution systems as the effectiveness of the earth grounding system for an electrical power and distribution system greatly depends on the resistivity of the earth and soil material at the location of the system ground.

Conduction is the name given to the movement of free electrons in the form of an electric current. Conductivity, σ is the reciprocal of the resistivity. That is 1/ρ and has the unit of siemens per metre, S/m. Conductivity ranges from zero (for a perfect insulator) to infinity (for a perfect conductor). Thus a super conductor has infinite conductance and virtually zero ohmic resistance.

]]>A thermistor is basically a two-terminal solid state thermally sensitive transducer made from sensitive semiconductor based metal oxides with metallised or sintered connecting leads onto a ceramic disc or bead. This allows it to change its resistive value in proportion to small changes in temperature. In other words, as its temperature changes, so too does its resistance and as such its name, “Thermistor” is a combination of the words THERM-ally sensitive res-ISTOR.

While the change in resistance due to heat is generally undesirable in standard resistors, this effect can be put to good use in many temperature detection circuits. Thus being non-linear variable-resistance devices, thermistors are commonly used as temperature sensors having many applications to measure the temperature of both liquids and ambient air.

Also, being a solid state device made from highly sensitive metal oxides, they operate at the molecular level with the outermost (valence) electrons becoming more active and producing a negative temperature coefficient, or less active producing a positive temperature coefficient as the temperature of the thermistor is increased. This means that they can have very good reproducible resistance verses temperature characteristics allowing them to operate up to temperatures of about 200^{o}C.

Typical Thermistor

While the primarily used of thermistors are as resistive temperature sensors, being resistive devices belonging the the resistor family, they can also be used in series with a component or device to control the current flowing through them. In other words, they can also be used as current-limiting devices.

Thermistors are available in a range of types, materials and sizes depending on the response time and operating temperature. Also, hermetically sealed thermistors eliminate errors in resistance readings due to moisture penetration while offering high operating temperatures and a compact size. The three most common types are: Bead thermistors, Disk thermistors, and Glass encapsulated thermistors.

These heat-dependent resistors can operate in one of two ways, either increasing or decreasing their resistive value with changes in temperature. Then there are two types of thermistors available: negative temperature coefficient (NTC) of resistance and positive temperature coefficient (PTC) of resistance.

Negative temperature coefficient of resistance thermistors, or *NTC thermistors* for short, reduce or decrease their resistive value as the operating temperature around them increases. Generally, NTC thermistors are the most commonly used type of temperature sensors as they can be used in virtually any type of equipment where temperature plays a role.

NTC temperature thermistors have a negative electrical resistance versus temperature (R/T) relationship. The relatively large negative response of an NTC thermistor means that even small changes in temperature can cause significant changes in its electrical resistance. This makes them ideal for accurate temperature measurement and control.

We said previously that a thermistor is an electronic component whose resistance is highly dependent on temperature so if we send a constant current through the thermistor and then measure the voltage drop across it, we can thus determine its resistance and temperature.

NTC thermistors reduce in resistance with an increase in temperature and are available in a variety of base resistances and curves. They are usually characterised by their base resistance at room temperature, that is 25^{o}C, (77^{o}F) as this provides a convenient reference point. So for example, 2k2Ω at 25^{o}C, 10kΩ at 25^{o}C or 47kΩ at 25^{o}C, etc.

Another important characteristic is the “B” value. The B value is a material constant which is determined by the ceramic material from which it is made and describes the gradient of the resistive (R/T) curve over a particular temperature range between two temperature points. Each thermistor material will have a different material constant and therefore a different resistance versus temperature curve.

Then the B value will define the thermistors resistive value at the first temperature or base point, (which is usually 25^{o}C), called T1, and the thermistors resistive value at a second temperature point, for example 100^{o}C, called T2. Therefore the B value will define the thermistors material constant between the range of T1 and T2. That is B_{T1/T2} or B_{25/100} with typical NTC thermistor B values given anywhere between about 3000 and about 5000.

Note however, that both the temperature points of T1 and T2 are calculated in the temperature units of Kelvin where 0^{0}C = 273.15 Kelvin. Thus a value of 25^{o}C is equal to 25^{o} + 273.15 = 298.15K, and 100^{o}C is equal to 100^{o} + 273.15 = 373.15K, etc.

So by knowing the B value of a particular thermistor (obtained from manufacturers datasheet), it is possible to produce a table of temperature versus resistance to construct a suitable graph using the following normalised equation:

- Where:
- T1 is the first temperature point in Kelvin
- T2 is the second temperature point in Kelvin
- R1 is the thermistors resistance at temperature T1 in Ohms
- R2 is the thermistors resistance at temperature T2 in Ohms

A 10kΩ NTC thermistor has a B value of 3455 between the temperature range of 25 to 100^{o}C. Calculate its resistive value at 25^{o}C and at 100^{o}C.

Data given: B = 3455, R1 = 10kΩ at 25^{o}. In order to convert the temperature scale from degrees Celsius, ^{o}C to degrees Kelvin add the mathematical constant 273.15

The value of R1 is already given as its 10kΩ base resistance, thus the value of R2 at 100^{o}C is calculated as:

Giving the following two point characteristics graph of:

Note that in this simple example, only two points were found, but generally thermistors change their resistance exponentially with changes in temperature so their characteristic curve is nonlinear, therefore the more temperature points are calculated the more accurate will be the curve.

Temperature ( ^{o}C) |
10 | 20 | 25 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |

Resistance (Ω) |
18476 | 12185 | 10000 | 8260 | 5740 | 4080 | 2960 | 2188 | 1645 | 1257 | 973 | 765 | 608 |

and these points can be plotted as shown to give a more accurate characteristics curve for the 10kΩ NTC Thermistor which has a B-value of 3455.

Notice that it has a negative temperature coefficient (NTC), that is its resistance decreases with increasing temperatures.

So how can we use a thermistor to measure temperature. Hopefully by now we know that a thermistor is a resistive device and therefore according to Ohms law, if we pass a current through it, a voltage drop will be produced across it. As a thermistor is an active type of a sensor, that is, it requires an excitation signal for its operation, any changes in its resistance as a result of changes in temperature can be converted into a voltage change.

The simplest way of doing this is to use the thermistor as part of a potential divider circuit as shown. A constant voltage is applied across the resistor and thermistor series circuit with the output voltage measured across the thermistor.

If for example we use a 10kΩ thermistor with a series resistor of 10kΩ, then the output voltage at the base temperature of 25^{o}C will be half the supply voltage.

When the resistance of the thermistor changes due to changes in temperature, the fraction of the supply voltage across the thermistor also changes producing an output voltage that is proportional to the fraction of the total series resistance between the output terminals.

Thus the potential divider circuit is an example of a simple resistance to voltage converter where the resistance of the thermistor is controlled by temperature with the output voltage produced being proportional to the temperature. So the hotter the thermistor gets, the lower the voltage.

If we reversed the positions of the series resistor, R_{S} and the thermistor, R_{TH}, then the output voltage will change in the opposite direction, that is the hotter the thermistor gets, the higher the output voltage.

We can use ntc thermistors as part of a basic temperature sensing configuration using a bridge circuit as shown. The relationship between resistors R_{1} and R_{2} sets the reference voltage, V_{REF} to the value required. For example, if both R_{1} and R_{2} are of the same resistive value, the reference voltage will be equal to half of the supply voltage. That is Vs/2.

As the temperature and therefore the resistance of the thermistor changes, the voltage at V_{TH} also changes either higher or lower than that at V_{REF} producing a positive or negative output signal to the connected amplifier.

The amplifier circuit used for this basic temperature sensing bridge circuit could act as a differential amplifier for high sensitivity and amplification, or a simple Schmitt-trigger circuit for ON-OFF switching.

The problem with passing a current through a thermistor in this way, is that thermistors experience what is called self-heating effects, that is the I^{2}.R power dissipation could be high enough to create more heat than can be dissipated by the thermistor affecting its resistive value producing false results.

Thus it is possible that if the current through the thermistor is too high it would result in increased power dissipation and as the temperature increases, its resistance decreases causing more current to flow, which increases the temperature further resulting in what is known as *Thermal Runaway*. In other words, we want the thermistor to be hot due to the external temperature being measured and not by itself heating up.

Then the value for the series resistor, R_{S} above should be chosen to provide a reasonably wide response over the range of temperatures for which the thermistor is likely to be used while at the same time limiting the current to a safe value at the highest temperature.

One way of improving on this and having a more accurate conversion of resistance against temperature (R/T) is by driving the thermistor with a constant current source. The change in resistance can be measured by using a small and measured direct current, or DC, passed through the thermistor in order to measure the voltage drop produced.

We have seen that thermistors are primarily used as resistive temperature sensitive transducers, but the resistance of a thermistor can be changed either by external temperature changes or by changes in temperature caused by an electrical current flowing through them, as after all, they are resistive devices.

Ohm’s Law tells us that when an electrical current passes through a resistance R, as a result of the applied voltage, power is consumed in the form of heat due to the I^{2}R heating effect. Because of the self-heating effect of current in a thermistor, a thermistor can change its resistance with changes in current.

Inductive electrical equipment such as motors, transformers, ballast lighting, etc, suffer from excessive inrush currents when they are first turned-on. But series connected thermistors can be used to effectively limit these high initial currents to a sfe value. NTC thermistors with low values of cold resistance (at 25^{o}C) are generally used for current regulation.

Inrush current suppressors and surge limiters are types of series connected thermistor whose resistance drops to a very low value as it is heated by the load current passing through it. At the initial turn-on, the thermistors cold resistance value (its base resistance) is fairly high controlling the initial inrush current to the load.

As a result of the load current, the thermistor heats up and reduces its resistance relatively slowly to the point were the power dissipated across it is sufficient to maintain its low resistance value with most of the applied voltage developed across the load.

Due to the thermal inertia of its mass this heating effect takes a few seconds during which the load current increases gradually rather than instantaneously, so any high inrush current is restricted and the power it draws reduces accordingly. Because of this thermal action, inrush current suppression thermistors can run very hot in the low-resistance state so require a cool-down or recovery period after power is removed to allow the resistance of the NTC thermistor to increase sufficiently to provide the required inrush current suppression the next time it is needed.

Thus the speed of response of a current limiting thermistor is given by its time constant. That is, the time taken for its resistance to change by by 63% (i.e. 1 to 1/e) of the total change. For example, suppose the ambient temperature changes from 0 to 100^{o}C, then the 63% time constant would be the time taken for the thermistor to have a resistive value at 63^{o}C.

Thus NTC thermistors provide protection from undesirably high inrush currents, while their resistance remains negligibly low during continuous operation supplying power to the load. The advantage here is that they able to effectively handle much higher inrush currents than standard fixed current limiting resistors with the same power consumption.

We have seen here in this tutorial about thermistors, that a thermistor is a two terminal resistive transducer which changes its resistive value with changes in surrounding ambient temperature, hence the name thermal-resistor, or simply “thermistor”.

Thermistors are inexpensive, easily-obtainable temperature sensors constructed using semiconductor metal oxides, and are available with either a negative temperature coefficient, (NTC) of resistance or a positive temperature coefficient (PTC) of resistance. The difference being that NTC thermistors reduce their resistance as the temperature increases, while PTC thermistors increase their resistance as the temperature increases.

NTC thermistors are the most commonly used (especially the 10KΩ ntc thermistor) and along with an addition series resistor, R_{S} can be used as part of a simple potential divider circuit so that changes to its resistance due to changes in temperature, produces a temperature-related output voltage.

However, the operating current of the thermistor must be kept as low as possible to reduce any self-heating effects. If they pass operating currents which are too high, they can create more heat than can be quickly dissipated from the thermistor which may cause false results.

Thermistors are characterised by their base resistance and their B value. The base resistance, for example, 10kΩ, is the resistance of the thermistor at a given temperature, usually 25^{o}C and is defined as: R_{25}. The B value is a fixed material constant that describes the shape of the slope of the resistive curve over temperature (R/T).

We have also seen that thermistors can be used to measure an external temperature or can be used control a current as a result of the I^{2}R heating effect caused by the current flowing through it. By connecting an NTC thermistor in series with a load, it is possible to effectively limit the high inrush currents.

As part of a new wave of technology enablement, objects like door locks, wearable devices, and heavy machinery are all being hooked up to the internet and cloud services. According to ABI Research, 30 billion everyday physical objects will be part of the Internet of Things (IoT) by 2020.

Despite these promising figures, the IoT market is still considered to be in the early stages of growth. It’s a bit perplexing, what with the vast amount of chips and design solutions that have flooded the market in recent years, each promising to power up a variety of IoT applications.

A closer look at all of these options reveals that on one hand, what we have are power hungry application processors that provide ample performance, but also negate the basic energy-efficiency premise for running battery-powered IoT devices.

To read more, download the white paper at the bottom.

]]>They don’t call it the “Internet of Everything” for nothing. These days, almost every product is eligible to become part of the Internet of Things (IoT) — from the electric toothbrush to the jet engine.

The addition of internet connectivity has become a key ingredient of a new generation of “smart products,” i.e., devices that combine sensing, processing, and communications capabilities with internet-based services and applications. The world’s leading businesses are rushing to offer these smart IoT products to gain an edge over the competition.

Because of this, the total available market for IoT devices is expected to double during the next three years, rising to about 1.6 billion units in 2019, up from 800 million in 2016, according to a forecast compiled by Cypress Semiconductor Corp.

To accelerate their time-to-market, companies are turning to IoT kits, if you will, to provide the essential building blocks required for IoT development efforts.

Today, Arrow Electronics has introduced two new kit solutions, both complete with its own powerful set of tools designed to streamline the process of evaluating, prototyping, and going into production with new IoT device designs.

To read more, download the white paper at the bottom.

]]>A passive RC differentiator is nothing more than a capacitance in series with a resistance, that is a frequency dependant device which has reactance in series with a fixed resistance (the opposite to an integrator). Just like the integrator circuit, the output voltage depends on the circuits RC time constant and input frequency.

Thus at low input frequencies the reactance, Xc of the capacitor is high blocking any d.c. voltage or slowly varying input signals. While at high input frequencies the capacitors reactance is low allowing rapidly varying pulses to pass directly from the input to the output.

This is because the ratio of the capacitive reactance (Xc) to resistance (R) is different for different frequencies and the lower the frequency the less output. So for a given time constant, as the frequency of the input pulses increases, the output pulses more and more resemble the input pulses in shape.

We saw this effect in our tutorial about Passive High Pass Filters and if the input signal is a sine wave, an **rc differentiator** will simply act as a simple high pass filter (HPF) with a cut-off or corner frequency that corresponds to the RC time constant (tau, τ) of the series network.

Thus when fed with a pure sine wave an RC differentiator circuit acts as a simple passive high pass filter due to the standard capacitive reactance formula of Xc = 1/(2πƒC).

But a simple RC network can also be configured to perform differentiation of the input signal. We know from previous tutorials that the current through a capacitor is a complex exponential given by: i_{C} = C(dVc/dt). The rate at which the capacitor charges (or discharges) is directly proportional to the amount of resistance and capacitance giving the time constant of the circuit. Thus the time constant of a RC differentiator circuit is the time interval that equals the product of R and C. Consider the basic RC series circuit below.

For an RC differentiator circuit, the input signal is applied to one side of the capacitor with the output taken across the resistor, then V_{OUT} equals V_{R}. As the capacitor is a frequency dependant element, the amount of charge that is established across the plates is equal to the time domain integral of the current. That is it takes a certain amount of time for the capacitor to fully charge as the capacitor can not charge instantaneously only charge exponentially.

We saw in our tutorial about RC Integrators that when a single step voltage pulse is applied to the input of an RC integrator, the output becomes a sawtooth waveform if the RC time constant is long enough. The RC differentiator will also change the input waveform but in a different way to the integrator.

We said previously that for the RC differentiator, the output is equal to the voltage across the resistor, that is: V_{OUT} equals V_{R} and being a resistance, the output voltage can change instantaneously.

However, the voltage across the capacitor can not change instantly but depends on the value of the capacitance, C as it tries to store an electrical charge, Q across its plates. Then the current flowing into the capacitor, that is *i _{t}* depends on the rate of change of the charge across its plates. Thus the capacitor current is not proportional to the voltage but to its time variation giving: i = dQ/dt.

As the amount of charge across the capacitors plates is equal to Q = C x Vc, that is capacitance times voltage, we can derive the equation for the capacitors current as:

Therefore the capacitor current can be written as:

As V_{OUT} equals V_{R} where V_{R} according to ohms law is equal too: i_{R} x R. The current that flows through the capacitor must also flow through the resistance as they are both connected together in series. Thus:

Thus the standard equation given for an RC differentiator circuit is:

Then we can see that the output voltage, V_{OUT} is the derivative of the input voltage, V_{IN} which is weighted by the constant of RC. Where RC represents the time constant, τ of the series circuit.

When a single step voltage pulse is firstly applied to the input of an RC differentiator, the capacitor “appears” initially as a short circuit to the fast changing signal. This is because the slope dv/dt of the positive-going edge of a square wave is very large (ideally infinite), thus at the instant the signal appears, all the input voltage passes through to the output appearing across the resistor.

After the initial positive-going edge of the input signal has passed and the peak value of the input is constant, the capacitor starts to charge up in its normal way via the resistor in response to the input pulse at a rate determined by the RC time constant, τ = RC.

As the capacitor charges up, the voltage across the resistor, and thus the output decreases in an exponentially way until the capacitor becomes fully charged after a time constant of 5RC (5T), resulting in zero output across the resistor. Thus the voltage across the fully charged capacitor equals the value of the input pulse as: V_{C} = V_{IN} and this condition holds true so long as the magnitude of the input pulse does not change.

If now the input pulse changes and returns to zero, the rate of change of the negative-going edge of the pulse pass through the capacitor to the output as the capacitor can not respond to this high dv/dt change. The result is a negative going spike at the output.

After the initial negative-going edge of the input signal, the capacitor recovers and starts to discharge normally and the output voltage across the resistor, and therefore the output, starts to increases exponentially as the capacitor discharges.

Thus whenever the input signal is changing rapidly, a voltage spike is produced at the output with the polarity of this voltage spike depending on whether the input is changing in a positive or a negative direction, as a positive spike is produced with the positive-going edge of the input signal, and a negative spike produced as a result of the negative-going input signal.

Thus the RC differentiator output is effectively a graph of the rate of change of the input signal which has no resemblance to the square wave input wave, but consists of narrow positive and negative spikes as the input pulse changes value.

By varying the time period, T of the square wave input pulses with respect to the fixed RC time constant of the series combination, the shape of the output pulses will change as shown.

Then we can see that the shape of the output waveform depends on the ratio of the pulse width to the RC time constant. When RC is much larger (greater than 10RC) than the pulse width the output waveform resembles the square wave of the input signal. When RC is much smaller (less than 0.1RC) than the pulse width, the output waveform takes the form of very sharp and narrow spikes as shown above.

So by varying the time constant of the circuit from 10RC to 0.1RC we can produce a range of different wave shapes. Generally a smaller time constant is always used in RC differentiator circuits to provide good sharp pulses at the output across R. Thus the differential of a square wave pulse (high dv/dt step input) is an infinitesimally short spike resulting in an RC differentiator circuit.

Lets assume a square wave waveform has a period, T of 20mS giving a pulse width of 10mS (20mS divided by 2). For the spike to discharge down to 37% of its initial value, the pulse width must equal the RC time constant, that is RC = 10mS. If we choose a value for the capacitor, C of 1uF, then R equals 10kΩ.

For the output to resemble the input, we need RC to be ten times (10RC) the value of the pulse width, so for a capacitor value of say, 1uF, this would give a resistor value of: 100kΩ. Likewise, for the output to resemble a sharp pulse, we need RC to be one tenth (0.1RC) of the pulse width, so for the same capacitor value of 1uF, this would give a resistor value of: 1kΩ, and so on.

So by having an RC value of one tenth the pulse width (and in our example above this is 0.1 x 10mS = 1mS) or lower we can produce the required spikes at the output, and the lower the RC time constant for a given pulse width, the sharper the spikes. Thus the exact shape of the output waveform depends on the value of the RC time constant.

We have seen here in this **RC Differentiator** tutorial that the input signal is applied to one side of a capacitor and the the output is taken across the resistor. A differentiator circuit is used to produce trigger or spiked typed pulses for timing circuit applications.

When a square wave step input is applied to this RC circuit, it produces a completely different wave shape at the output. The shape of the output waveform depending on the periodic time, T (an therefore the frequency, ƒ) of the input square wave and on the circuit’s RC time constant value.

When the periodic time of the input waveform is similar too, or shorter than, (higher frequency) the circuits RC time constant, the output waveform resembles the input waveform, that is a square wave profile. When the periodic time of the input waveform is much longer than, (lower frequency) the circuits RC time constant, the output waveform resembles narrow positive and negative spikes.

The positive spike at the output is produced by the leading-edge of the input square wave, while the negative spike at the output is produced by the falling-edge of the input square wave. Then the output of an RC differentiator circuit depends on the rate of change of the input voltage as the effect is very similar to the mathematical function of differentiation.

]]>In Electronics, the basic series connected resistor-capacitor (RC) circuit has many uses and applications from basic charging/discharging circuits to high-order filter circuits. This two component passive RC circuit may look simple enough, but depending on the type and frequency of the applied input signal, the behaviour and response of this basic RC circuit can be very different.

A passive RC network is nothing more than a resistor in series with a capacitor, that is a fixed resistance in series with a capacitor that has a frequency dependant reactance which decreases as the frequency across its plates increases. Thus at low frequencies the reactance, Xc of the capacitor is high while at high frequencies its reactance is low due to the standard capacitive reactance formula of Xc = 1/(2πƒC), and we saw this effect in our tutorial about Passive Low Pass Filters.

Then if the input signal is a sine wave, an **rc integrator** will simply act as a simple low pass filter (LPF) with a cut-off or corner frequency that corresponds to the RC time constant (tau, τ) of the series network and whose output is reduced above this cut-off frequency point. Thus when fed with a pure sine wave an RC integrator acts as a passive low pass filter.

As we have seen previously, the RC time constant reflects the relationship between the resistance and the capacitance with respect to time with the amount of time, given in seconds, being directly proportional to resistance, R and capacitance, C.

Thus the rate of charging or discharging depends on the RC time constant, τ = RC. Consider the circuit below.

For an RC integrator circuit, the input signal is applied to the resistance with the output taken across the capacitor, then V_{OUT} equals V_{C}. As the capacitor is a frequency dependant element, the amount of charge that is established across the plates is equal to the time domain integral of the current. That is it takes a certain amount of time for the capacitor to fully charge as the capacitor can not charge instantaneously only charge exponentially.

Therefore the capacitor current can be written as:

This basic equation above of i_{C} = C(dVc/dt) can also be expressed as the instantaneous rate of change of charge, Q with respect to time giving us the following standard equation of: i_{C} = dQ/dt where the charge Q = C x Vc, that is capacitance times voltage.

The rate at which the capacitor charges (or discharges) is directly proportional to the amount of the resistance and capacitance giving the time constant of the circuit. Thus the time constant of a RC integrator circuit is the time interval that equals the product of R and C.

Since capacitance is equal to Q/Vc where electrical charge, Q is the flow of a current (i) over time (t), that is the product of i x t in coulombs, and from Ohms law we know that voltage (V) is equal to i x R, substituting these into the equation for the RC time constant gives:

Then we can see that as both i and R cancel out, only T remains indicating that the time constant of an RC integrator circuit has the dimension of time in seconds, being given the Greek letter tau, τ. Note that this time constant reflects the time (in seconds) required for the capacitor to charge up to 63.2% of the maximum voltage or discharge down to 36.8% of maximum voltage.

We said previously that for the RC integrator, the output is equal to the voltage across the capacitor, that is: V_{OUT} equals V_{C}. This voltage is proportional to the charge, Q being stored on the capacitor given by: Q = VxC.

The result is that the output voltage is the integral of the input voltage with the amount of integration dependent upon the values of R and C and therefore the time constant of the network.

We saw above that the capacitors current can be expressed as the rate of change of charge, Q with respect to time. Therefore, from a basic rule of differential calculus, the derivative of Q with respect to time is dQ/dt and as i = dQ/dt we get the following relationship of:

Q = ∫idt (the charge Q on the capacitor at any instant in time)

Since the input is connected to the resistor, the same current, i must pass through both the resistor and the capacitor (i_{R} = i_{C}) producing a V_{R} voltage drop across the resistor so the current, (i) flowing through this series RC network is given as:

therefore:

As i = V_{IN}/R, substituting and rearranging to solve for V_{OUT} as a function of time gives:

So in other words, the output from an RC integrator circuit, which is the voltage across the capacitor is equal to the time Integral of the input voltage, V_{IN} weighted by a constant of 1/RC. Where RC represents the time constant, τ.

Then assuming the initial charge on the capacitor is zero, that is V_{OUT} = 0, and the input voltage V_{IN} is constant, the output voltage, V_{OUT} is expressed in the time domain as:

So an RC integrator circuit is one in which the output voltage, V_{OUT} is proportional to the integral of the input voltage, and with this in mind, lets see what happens when we apply a single positive pulse in the form of a step voltage to the RC integrator circuit.

When a single step voltage pulse is applied to the input of an RC integrator, the capacitor charges up via the resistor in response to the pulse. However, the output is not instant as the voltage across the capacitor cannot change instantaneously but increases exponentially as the capacitor charges at a rate determined by the RC time constant, τ = RC.

We now know that the rate at which the capacitor either charges or discharges is determined by the RC time constant of the circuit. If an ideal step voltage pulse is applied, that is with the leading edge and trailing edge considered as being instantaneous, the voltage across the capacitor will increase for charging and decrease for discharging, exponentially over time at a rate determined by:

Capacitor Charging

Capacitor Discharging

So if we assume a capacitor voltage of one volt (1V), we can plot the percentage of charge or discharge of the capacitor for each individual R time constant as shown in the following table.

Time Constant |
Capacitor Charging |
Capacitor Discharging |

τ | % Charged | % Discharged |

0.5 | 39.4% | 60.6% |

0.7 | 50% | 50% |

1 | 63.2% | 36.7% |

2 | 86.4% | 13.5% |

3 | 95.0% | 4.9% |

4 | 98.1% | 1.8% |

5 | 99.3% | 0.67% |

Note that at 5 time constants or above, the capacitor is considered to be 100 percent fully charged or fully discharged.

So now lets assume we have an RC integrator circuit consisting of a 100kΩ resistor and a 1uF capacitor as shown.

The time constant, τ of the RC integrator circuit is therefore given as: RC = 100kΩ x 1uF = 100ms.

So if we apply a step voltage pulse to the input with a duration of say, two time constants (200mS), then from the table above we can see that the capacitor will charge to 86.4% of its fully charged value. If this pulse has an amplitude of 10 volts, then this equates to 8.64 volts before the capacitor discharges again back through the resistor to the source as the input pulse returns to zero.

If we assume that the capacitor is allowed to fully discharge in a time of 5 time constants, or 500mS before the arrival of the next input pulse, then the graph of the charging and discharging curves would look something like this:

Note that the capacitor is discharging from an initial value of 8.64 volts (2 time constants) and not from the 10 volts input.

Then we can see that as the RC time constant is fixed, any variation to the input pulse width will affect the output of the RC integrator circuit. If the pulse width is increased and is equal too or greater than 5RC, then the shape of the output pulse will be similar to that of the input as the output voltage reaches the same value as the input.

If however the pulse width is decreased below 5RC, the capacitor will only partially charge and not reach the maximum input voltage resulting in a smaller output voltage because the capacitor cannot charge as much resulting in an output voltage that is proportional to the integral of the input voltage.

So if we assume an input pulse equal to one time constant, that is 1RC, the capacitor will charge and discharge not between 0 volts and 10 volts but between 63.2% and 38.7% of the voltage across the capacitor at the time of change. Note that these values are determined by the RC time constant.

So for a continuous pulse input, the correct relationship between the periodic time of the input and the RC time constant of the circuit, integration of the input will take place producing a sort of ramp up, and then a ramp down output. But for the circuit to function correctly as an integrator, the value of the RC time constant has to be large compared to the inputs periodic time. That is RC ≫ T, usually 10 times greater.

This means that the magnitude of the output voltage (which was proportional to 1/RC) will be very small between its high and low voltages severely attenuating the output voltage. This is because the capacitor has much less time to charge and discharge between pulses but the average output DC voltage will increase towards one half magnitude of the input and in our pulse example above, this will be 5 volts (10/2).

We have seen above that an *RC integrator* circuit can perform the operation of integration by applying a pulse input resulting in a ramp-up and ramp-down triangular wave output due to the charging and discharging characteristics of the capacitor. But what would happen if we reversed the process and applied a triangular waveform to the input, would we get a pulse or square wave output?

When the input signal to an RC integrator circuit is a pulse shaped input, the output is a triangular wave. But when we apply a triangular wave, the output becomes a sine wave due to the integration over time of the ramp signal.

There are many ways to produce a sinusoidal waveform, but one simple and cheap way to electronically produce a sine waves type waveform is to use a pair of passive RC integrator circuits connected together in series as shown.

Here the first RC integrator converts the original pulse shaped input into a ramp-up and ramp-down triangular waveform which becomes the input of the second RC integrator. This second RC integrator circuit rounds off the points of the triangular waveform converting it into a sine wave as it is effectively performing a double integration on the original input signal with the RC time constant affecting the degree of integration.

As the integration of a ramp produces a sine function, (basically a round-off triangular waveform) its periodic frequency in Hertz will be equal to the period T of the original pulse. Note also that if we reverse this signal and the input signal is a sine wave, the circuit does not act as an integrator, but as a simple low pass filter (LPF) with the sine wave, being a pure waveform does not change shape, only its amplitude is affected.

We have seen here that the RC integrator is basically a series RC low-pass filter circuit which when a step voltage pulse is applied to its input produces an output that is proportional to the integral of its input. This produces a standard equation of: Vo = ∫Vi*dt* where Vi is the signal fed to the integrator and Vo is the integrated output signal.

The integration of the input step function produces an output that resembles a triangular ramp function with an amplitude smaller than that of the original pulse input with the amount of attenuation being determined by the time constant. Thus the shape of the output waveform depends on the relationship between the time constant of the circuit and the frequency (period) of the input pulse.

An RC integrators time constant is always compared to the period, T of the input, so a long RC time constant will produce a triangular wave shape with a low amplitude compared to the input signal as the capacitor has less time to fully charge or discharge. A short time constant allows the capacitor more time to charge and discharge producing a more typical rounded shape.

By connecting two RC integrator circuits together in parallel has the effect of a double integration on the input pulse. The result of this double integration is that the first integrator circuit converts the step voltage pulse into a triangular waveform and the second integrator circuit converts the triangular waveform shape by rounding off the points of the triangular waveform producing a sine wave output waveform with a greatly reduced amplitude.

]]>We have seen in this section about transformers, that a transformer is an electrical device which allows an sinusoidal input signal (such as an audio signal or voltage) to produce an output signal or voltage without the input side and output side being physically connected to each other. This coupling is achieved by having two (or more) wire coils (called windings) of insulated copper wire wound around a soft magnetic iron core.

When an AC signal is applied to the primary input winding, a corresponding AC signal appears on the output secondary winding due to the inductive coupling of the soft iron core. The turns ratio between the input and output wire coils provides either an increase or a decrease of the applied signal as it passes through the transformer.

Then audio transformers can be considered as either a step-up or step-down type, but rather than being wound to produce a specific voltage output, audio transformers are mainly designed for impedance matching. Also, a transformer with a turns ratio of 1:1, does not change the voltage or current levels but instead isolates the primary circuit from the secondary side. This type of transformer is known commonly as an Isolation Transformer.

Transformers are not intelligent devices, but can be used as bidirectional devices so that the normal primary input winding can become an output winding and the normal secondary output winding can become an input and due to this bidirectional nature, transformers can provide a signal gain when used in one direction or a signal loss when used in reverse to help match signal or voltage levels between different devices.

Note also that a single transformer can have multiple primary or secondary windings and these windings may also have multiple electrical connections or “taps” along their length. The advantage of multi-tap audio transformers is that they offer different electrical impedances as well as different gain or loss ratios making them useful for impedance matching of amplifiers and speaker loads.

As their name suggests, **audio transformers** are designed to operate within the audio band of frequencies and as such can have applications in the input stage (microphones), output stage (loudspeakers), inter-stage coupling as well as impedance matching of amplifiers. In all cases, the frequency response, primary and secondary impedances and power capabilities all need to be considered.

Audio and impedance matching transformers are similar in design to low frequency voltage and power transformer, but they operate over a much wider frequency range of frequencies. For example, 20Hz to 20kHz voice range. Audio transformers can also conduct DC in one or more of their windings for use in digital audio applications as well as transforming voltage and current levels at high frequency.

One of the main applications for *audio frequency transformers* is in impedance matching. Audio transformers are ideal for balancing amplifiers and loads together that have different input/output impedances in order to achieve maximum power transfer.

For example, a typical loudspeaker impedance ranges from 4 to 16 ohms whereas the impedance of a transistor amplifiers output stage can be several hundred ohms. A classic example of this is the LT700 Audio Transformer which can be used in the output stage of an amplifier to drive a loudspeaker.

We know that for a transformer, the ratio between the number of coil turns on the primary winding (N_{P}) to the number of coil turns on the secondary winding (N_{S}) is called the “turns ratio”. Since the same amount of voltage is induced within each single coil turn of both windings, the primary to secondary voltage ratio (V_{P}/V_{S}) will therefore be the same value as the turns ratio.

Impedance matching audio transformers always give their impedance ratio value from one winding to another by the square of the their turns ratio. That is, their impedance ratio is equal to its turns ratio squared and also its primary to secondary voltage ratio squared as shown.

Where Z_{P} is the primary winding impedance, Z_{S} is the secondary winding impedance, (N_{P}/N_{S}) is the transformers turns ratio, and (V_{P}/V_{S}) is the transformers voltage ratio.

So for instance, an impedance matching audio transformer that has a turns ratio (or voltage ratio) of say 2:1, will have an impedance ratio of 4:1.

An audio transformer with a turns ratio of 15:1 is to be used to match the output of a power amplifier to a loudspeaker. If the output impedance of the amplifier is 120Ω’s, calculate the nominal impedance of the loudspeaker required for maximum power transfer.

Then the power amplifier can efficiently drive an 8-ohm speaker.

Another very common impedance matching application is for 100 volt line transformers for the transmission of music and voice over public address tannoy systems. These types of ceiling based speaker systems use multiple loudspeakers located some distance from the power amplifier.

By using line isolating transformers, any number of low-impedance loudspeakers can be connected together in such a way that they properly load the amplifier providing impedance matching between the amplifier (source) and speakers (load) for maximum power transfer.

As power loss of signals through speaker cables is proportional to the square of current (P = I^{2}R) for a given cable resistance, the output voltage of an amplifier used for public address (PA) or tannoy systems uses a standard and constant voltage output level of 100 volts peak, (70.7 volts rms).

So for example, a 200 watt amplifier driving an 8-ohm speaker delivers a current of 5 amps, whereas a 200 watt amplifier using a 100 volt line at full power delivers only 2 amps allowing smaller gauge cables to be used. Note however that this 100 volts only exists on the line when the power amplifier driving the line is operating at full rated power otherwise there is reduced power ( sound volume) and line voltage.

So for a 100V (70.7V rms) line speaker system, the line transformer steps up the audio output signal voltage to 100 volts so that the transmission line current for a given power output is comparatively low, reducing signal losses allowing smaller diameter or gauge cables to be used.

Since the impedance of a typical loudspeaker is generally low, an impedance matching step-down transformer (usually called a line to voice-coil transformer) is used for each loudspeaker connected to the 100V line as shown.

Here the amplifier uses a step-up transformer to provide a constant 100 volts transmission line voltage at reduced current, for a given power output. The loudspeakers are connected together in parallel with each speaker having its own impedance matching step down transformer to reduce the secondary voltage and increase the current, thereby matching the 100V line to the low impedances of the loudspeakers.

The advantage of using this type of audio transmission line is that many individual speakers, tannoy’s or other such sound actuators can be connected to a single line even if they have different impedances and power handling capabilities. For example, 4 ohms at 5 watts, or 8 ohms at 20 watts.

Generally transmission line matching transformers have multiple connections called tapping points on the primary winding allowing for suitable power levels (and therefore sound volume) to be selected for each individual loudspeaker. Also, the secondary winding has similar tapping points offering different impedances to match that of the connected loudspeakers.

In this simple example, the 100V line-to-speaker transformer can drive 4, 8, or 16 Ohm speaker loads on its secondary side with amplifier power ratings of 4, 8 and 16 watts on its primary side depending on the tapping points selected. In reality, PA system line transformers can be selected for any combination of series and parallel connected speaker loads with power handling capabilities up to several kilo-watts.

But as well as constant voltage impedance matching line transformers, audio transformers can be used to connect low impedance or low signal input devices such as microphones, turntable moving coil pick-ups, line inputs, etc to an amplifier or pre-amplifier. As input audio transformers must operate over a wide range of frequencies, they are usually designed so that the internal capacitance of their windings resonates with its inductance to improve its operating frequency range allowing for a smaller transformer core size.

We have seen in this tutorial about **audio transformers**, that audio transformers are used to match impedances between different audio devices, for example, between an amplifier and speaker as a line driver, or between a microphone and amplifier for impedance matching. Unlike power transformers which operate at low frequencies such as 50 or 60Hz, audio transformers are designed to operate over the audio frequency range, that is from about 20Hz to 20kHz or much higher for radio-frequency transformers.

Due to this wide frequency band, the core of audio transformers are made from special grades of steel, such as silicon steel or from special alloys of iron which have a very low hysteresis loss. One of the main disadvantages of audio transformers is that they can be somewhat bulky and expensive, but by using special core materials allows for a smaller design because generally a transformers core size increases as the supply frequency decreases.

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