2.28.1 Power Factor

Another way to represent the amount of apparent power to reactive power within a circuit is to use what's called the power factor. The power factor of a circuit is the ratio of consumed power to apparent power:

In the example in Fig. 2.167:

Power factor is frequently expressed as a percentage, in this case 45 percent.

An equivalent definition of power factor is:

where ϕ is the phase angle (between voltage and current). In the example in Fig. 2.167, the phase angle is -63.4°, so:

as the earlier calculation confirms.

The power factor of a purely resistive circuit is 100 percent, or 1, while the power factor of a purely reactive circuit is 0.

Since the power factor is always rendered as a positive number, the value must be followed by the words "leading" or "lagging" to identify the phase of the voltage with respect to the current. Specifying the numerical power factor is not always sufficient. For example, many dc-to-ac power inverters can safely operate loads having large net reactance of one sign but only a small reactance of the opposite sign. The final calculation of the power factor in the example in Fig. 2.167 yields the value of 0.45 lagging.

In ac equipment, the ac components must handle reactive power as well as real power. For example, a transformer connected to a purely reactive load must still be capable of supplying the voltage and be able to handle the current required by the reactive load. The current in the transformer windings will heat the windings as a result of I²R losses in the winding resistances.

As a final note, there is another term to describe the percentage of power used in reactance, which is called the reactive factor. The reactive factor is defined by:

Using the example in Fig. 2.167:

First we calculate the inductive and capacitive reactances:

Since the inductor and capacitor are in series, the math is easy—simply add complex numbers in rectangular form:

In polar form, this is 60 Ω ∠ 90°. The fact that the phase is 90°, or the result is positive imaginary, means the impedance is net inductive. Don't let the imaginary part fool you—the impedance is actually felt—it provides 60 Ω of impedance, even though it's not resistive.

The current can now be found using ac Ohm's law:

Note that we used 60 Ω ∠ 90° (polar) to make the division easy. The -90° result means the source current (total current) lags the source voltage by 90°. Since this is a series circuit: I_S = I_L = I_C.

The voltage across the inductor and capacitor can be found using ac Ohm's law (or the ac voltage divider):

Notice that the voltage across the inductor is greater than the supply voltage; the capacitor supplies current to the inductor as it discharges.

To convert these snapshots into a continuous set of functions, we convert all RMS values to true value (× 1.414), add the ωt term to the phase angles, and convert to trigonometric form, then delete the imaginary part. Our results would all be in terms of cosines, but to make things pretty, we select sines. Doing this doesn't make any practical difference. See graphs and equations to left.

The apparent power due to the total impedance is:

The real (true) power consumed by the circuit is:

Only reactive power exists, and for the inductor and capacitor:

Power factor is:

A power factor of 0 means the circuit is purely reactive.

First we find the inductive and capacitive reactances:

Since the inductor and capacitor are in parallel, the math is relatively easy—use two components in parallel formula, and multiply and add complex numbers:

(Tricks used to solve: j × j = -1, 1/j = -j)

In polar form, the result is 13.33 Ω ∠ -90°. The fact that the phase is -90°, or negative imaginary, means the impedance is net capacitive. Don't let the imaginary part fool you—the impedance is actually felt—it provides 13.3 Ω of impedance, even though it's not resistive.

The total current can now be found using ac Ohm's law:

Note that we used 13.3 Ω ∠ -90° (polar form) to make the division easy. The +90° result means the total current leads the source voltage by 90°. Since this is a parallel circuit: V_S = V_L = V_C.

The current through each component can be found using ac Ohm's law (or the current divider relation):

Notice that the capacitor current is larger than the supply current; the collapsing magnetic field of the inductor supplies current to the capacitor.

To convert these snapshots into a continuous set of functions, we convert all RMS values to true value (× 1.414), add the ωt term to the phase angles, and convert to trigonometric form, then delete the imaginary part. Our results would all be in terms of cosines, but to make things pretty, we select sines. Doing this doesn't make any practical difference. See graphs and equations in Fig. 2.171.

The apparent power is:

Only reactive power exists, and for the inductor and capacitor:

Power factor is:

Again, a power factor of 0 means the circuit is purely reactive.

First let's find the reactances of the inductor and capacitor:

To find the total impedance, take the L, C, and R in series:

In polar form, the result is 2.33 Ω ∠ -64.5°. The fact that the phase is -64.5°, or the imaginary part is negative, means the impedance is net capacitive. Don't let the imaginary part fool you—the impedance is actually felt—it provides 2.33 Ω of impedance, even though it's not entirely resistive.

The total current can now be found using ac Ohm's law:

Note that we used 2.33 Ω ∠ -64.5° (polar form) to make the division easy. The +64.5° result means the total current leads the source voltage by 64.5°. Since this is a series circuit: I_S = I_L = I_C = I_R.

The voltage across each component can be found using ac Ohm's law:

Notice that the voltage across the inductor and capacitor are huge at this particular phase when compared to the supply voltage; the capacitor supplies current to the inductor as it discharges, while the inductor supplies current to the capacitor as its magnetic field collapses. To convert these snapshots into a continuous set of functions, we convert all RMS values to true value (× 1.414), add the ωt term to the phase angles, and convert to trigonometric form, then delete the imaginary part. Our results would all be in terms of cosines, but to make things pretty, we select sines. Doing this doesn't make any practical difference. See graphs and equations in Fig. 2.172.

The apparent power is:

The real (true) power, or power dissipated by resistor is:

The reactive powers for the inductor and capacitor:

Power factor is:

It should have become apparent from the last example that the VARs for the inductor and the capacitor became surprisingly large. When dealing with real-life components, the VAR values become important. Even though volt-amps do not contribute to the overall true power loss, this does not mean that the volt-amps aren't felt by the reactive components. Components like inductors and transformers (ideally reactive components) are usually given a volt-amp rating that provides the safety limit to prevent overheating the component. Again, it is the internal resistances within the inductor or transformer that must be considered.

Since the inductor and capacitor are in parallel, the math is relatively easy—use two components in general formula, and multiply and add complex numbers:

The fact that there is no reactive part to the total impedance is quite interesting and makes our life easy in terms of calculations. Before getting into the interesting matter, let's finish solving.

The total current can now be found using ac Ohm's law:

Since there is no phase angle, the source current and voltage are in phase. Since this is a parallel circuit, V_S = V_L = V_C = V_R.

The current through each component can be found using ac Ohm's law:

You can convert the voltages and current back to true sinusoidal form to get the graph shown in Fig. 2.173.

The apparent power is:

The real (true) power, or power dissipated by resistor, is:

The reactive powers for the inductor and capacitor:

Power factor is:

A power factor of 1 indicates the circuit is purely resistive. How can this be? In this case we have a special condition where a circulating current is "trapped" within the LC section. This only occurs at a special frequency called the resonant frequency. We'll cover resonant circuits in a moment.

2.30.1 Resonance in RLC Circuits

The previous LC series- and parallel-resonant circuits are ideal in nature. In reality, there is internal resistance or impedance within the components that leads to a deviation from the true resonant effects we observed. In most real LC resonant circuits, the most notable resistance is associated with losses in the inductor at high frequencies (HF range); resistive losses in a capacitor are low enough at those frequencies to be ignored. The following example shows how a series RLC circuit works.

In polar form:

The current through the total impedance is, ignoring phase:

If you plug in the L = 100 µH and C = 62.5 nF, and ω = 2πf, the current expressed as a function of frequency is:

Now, unlike the ideal LC series-resonant circuit, when we plug in the resonant frequency:

the total current doesn't go to infinity; instead it goes to 2 A, which is just V_S/R = 10 VAC/5 Ω = 2 A. The resistance thus prevents a zero impedance condition that would otherwise be present when the inductor and capacitor reactances cancel at resonance.

The unloaded Q is the reactance at resonance divided by the resistance:

As pointed out in Fig. 2.178, at resonance, the reactance of the capacitor cancels the reactance of the inductor, and thus the impedance is determined solely by the resistance. We can therefore deduce that at resonance the current and voltage must be in phase—recall that in a sinusoidal circuit with a single resistor, the current and voltage are in phase. However, as we move away from the resonant frequency (keeping component values the same), the impedance goes up due to increases in reactance of the capacitor or the inductor. As you go lower in frequency from resonance, the capacitor's reactance becomes dominant—capacitors increasingly resist current as the frequency decreases. As you go higher in frequency from resonance, the inductor's reactance becomes dominant—inductors increasingly resist current as the frequency increases. Far from resonance in either direction, you can see that resistance has an insignificant effect on current amplitude.

Now if you take a look at the graph in Fig. 2.178, notice how the current curve looks like a pointy hilltop. In electronics, describing the pointiness of the current curve is an important characteristic of concern. When the reactance of the inductor or capacitor is of the same order of magnitude as the resistance, the current decreases rather slowly as you move away from the resonant frequency in either direction. Such a curve, or "hilltop," is said to be broad. Conversely, when the reactance of the inductor or capacitor is considerably larger than the resistance, the current decreases rapidly as you move away from the resonant frequency in either direction. Such a curve, or "hilltop," is said to be sharp. A sharp resonant circuit will respond a great deal more readily to the resonant frequency than to frequencies quite close to resonance. A broad resonant circuit will respond almost equally well to a group or band of frequencies centered about the resonant frequency.

As it turns out, both sharp and narrow circuits are useful. A sharp circuit gives good selectivity. This means that it has the ability to respond strongly (in terms of current amplitude) at one desired frequency and is able to discriminate against others. A broad circuit, on the other hand, is used in situations where a similar response over a band of frequencies is desired, as opposed to a strong response at a single frequency.

Next, we'll take a look at quality factor and bandwidth—two quantities that provide a measure of the sharpness of our RLC resonant circuit.

2.30.2 Q (Quality Factor) and Bandwidth

As mentioned earlier, the ratio of reactance or stored energy to resistance or consumed energy is by definition the quality factor Q. (Q is also referred to as the figure of merit, or multiplying factor.) As it turns out, within a series RLC circuit (where R is internal resistance of the components), the internal resistive losses within the inductor dominate energy consumption at high frequencies. This means the inductor Q largely determines the resonant circuit Q. Since the value of Q is independent of any external load to which the circuit might transfer power, we modify the resonant circuit Q, by calling it the unloaded Q or Q_U of the circuit. See Fig. 2.179.

In the example RLC series-resonant circuit from Fig. 2.178, we can determine the unloaded Q of the circuit by dividing the reactance of either the inductor or the capacitor (they have the same relative impedance at resonance) by the resistance:

As you can see, if we increase the resistance, the unloaded Q decreases, giving rise to broad current response curves about resonance, as shown in the graph in Fig. 2.179. With resistances of 10, 20, and 50 Ω, the unloaded Q decreases to 4, 2, and 0.8, respectively. Conversely, if the resistance is made smaller, the unloaded Q increases, giving rise to a sharp current response curve about resonance. For example, when the resistance is lowered to 2 Ω, the unloaded Q becomes 20.

2.30.3 Bandwidth

An alternative way of expressing the sharpness of a series-resonant circuit is using what is called bandwidth. Basically, what we do is take the quality factor graph in Fig. 2.179 and convert it to the bandwidth graph in Fig. 2.179. This involves changing the current axis to a relative current axis and moving the family of curves for varying Q values up so that all have the same peak current. By assuming that the peak current of each curve is the same, the rate of change of current for various values of Q and the associated ratios of reactance to resistance are more easily compared. From the curves, you can see that lower Q circuits pass frequencies over a greater bandwidth of frequencies than circuits with a higher Q. For the purpose of comparing tuned circuits, bandwidth is often defined as the frequency spread between the two frequencies at which the current amplitude decreases to 0.707 or times the maximum value. Since the power consumed by the resistance R is proportional to the square of the current, the power at these points is half the maximum power at resonance, assuming that R is constant for the calculations. The half-power, or -3-dB, points are marked on the drawing.

For Q values of 10 or greater, the curves shown in Fig. 2.179 are approximately symmetrical. On this assumption, bandwidth (BW) can be easily calculated:

where BW and f are in units of hertz.

2.30.4 Voltage Drop Across Components in RLC Resonant Circuit

The voltage drop across a given inductor or a capacitor within an RLC resonant circuit can be determined by applying ac Ohm's law:

As we discovered earlier, these voltages may become many times larger than the source voltage, due to the magnetic and electric stored energy returned by the inductor and capacitor. This is especially true for circuits with high Q values. For example, at resonance, the RLC circuit of Fig. 2.178 has the following voltage drops across the capacitor and inductor:

The actual amplitude of the voltage when we convert from the RMS values is a factor of 1.414 higher, or 113 V. High-Q circuits such as those found in antenna couplers, which handle significant power, may experience arcing from high reactive voltages, even though the source voltage to the circuit is well within component ratings. When Qs of greater than 10 are considered, the following equation gives a good approximation of the reactive voltage within a series-resonant RLC circuit at resonance:

2.30.5 Capacitor Losses

Note that although capacitor energy losses tend to be insignificant compared to inductor losses up to about 30 MHz within a series-resonant circuit, these losses may affect circuit Q in the VHF range (30 to 300 MHz). Leakage resistance, principally in the solid dielectric between the capacitor plates, is not exactly like the internal wire resistive losses in an inductor coil. Instead of forming a series resistance, resistance associated with capacitor leakage usually forms a parallel resistance with the capacitive reactance. If the leakage resistance of a capacitor is significant enough to affect the Q of a series-resonant circuit, the parallel resistance must be converted to an equivalent series resistance before adding it to the inductor's resistance. This equivalent series resistance is given by:

where R_P is the leakage resistance and X_C is the capacitive reactance. This value is then added to the inductor's internal resistance and the sum represents the R in an RLC resonant circuit.

When calculating the impedance, current, and bandwidth of a series-resonant circuit, the series leakage resistance is added to the inductor's coil resistance. Since an inductor's resistance tends to increase with frequency due to what's called the skin effect (electrons forced to the surface of a wire), the combined losses in the capacitor and the inductor can seriously reduce circuit Q.

2.30.6 Parallel-Resonant Circuits

Although series-resonant circuits are common, the vast majority of resonant circuits are parallel-resonant circuits. Figure 2.180 shows a typical parallel-resonant circuit. As with series-resonant circuits, the inductor internal coil resistance is the main source of resistive losses, and therefore we put the series resistor in the same leg as the inductor. Unlike the series-resonant circuit whose impedance goes toward a minimum at resonance, the parallel-resonant circuit's impedance goes to a maximum. For this reason, RLC parallel-resonant circuits are often called antiresonant circuits or rejector circuits. (RLC series-resonant circuits go by the name acceptor). The following example will paint a picture of parallel resonance behavior.

In polar form:

If you plug in L = 5.0 µH and C = 50 pF, R = 10.5 Ω and ω = 2πf, the total impedance, ignoring phase, is:

The total current (line current), ignoring phase, is:

Plugging these equations into a graphing program, you get the curves shown in Fig. 2.180. Note that at a particular frequency, the impedance goes to a maximum, while the total current goes to a minimum. This, however, is not at the point where X_L = X_C—the point referred to as the resonant frequency in the case of a simple LC parallel circuit or a series RLC circuit. As it turns out, the resonant frequency of a parallel RLC circuit is a bit more complex and can be expressed three possible ways. However, for now, we make an approximation that is expressed as before:

The unloaded Q for this circuit, using the reactance of L, is:

The lower graph in Fig. 2.180 shows the quality factor and how it is influenced by the size of the parallel resistance in the inductor leg.

Unlike the ideal LC parallel resonant circuit we saw in Fig. 2.177, the addition of R changes the conditions of resonance. For example, when the inductive and capacitive reactances are the same, the impedances of the inductive and capacitive legs do not cancel—the resistance in the inductive leg screws things up. When X_L = X_C, the impedance of the inductive leg is greater than X_C and will not be 180° out of phase with X_C. The resultant current is not at a true minimum value and is not in phase with the voltage. See line (A) in Fig. 2.181.

Now we can alter the value of the inductor a bit (holding Q constant) and get a new frequency where the current reaches a true minimum—something we can accomplish with the help of a current meter. We associate the dip in current at this new frequency with what is termed the standard definition of RLC parallel resonance. The point where minimum current (or maximum impedance) is reached is called the antiresonant point and is not to be confused with the condition where X_L = X_C. Altering the inductance to achieve minimum current comes at a price; the current becomes somewhat out of phase with the voltage—see line (B) in Fig. 2.181.

It's possible to alter the circuit design of our RLC parallel-resonant circuit to draw some of the resonant points shown in Fig. 2.181 together—for example, compensating for the resistance of the inductor by altering the capacitance (retuning the capacitor). The difference among the resonant points tends to get smaller and converge to within a percent of the frequency when the circuit Q rises above 10, in which case they can be ignored for practical calculations. Tuning for minimum current will not introduce a sufficiently large phase angle between voltage and current to create circuit difficulties.

As long as we assume Qs higher than 10, we can use a single set of formulas to predict circuit performance. As it turns out, what we end up doing is removing the series inductor resistance in the leg with the inductor and substituting a parallel-equivalent resistor of the actual inductor loss series resistor, as shown in Fig. 2.182.

This parallel-equivalent resistance is often called the dynamic resistance of the parallel-resonant circuit. This resistance is the inverse of the series resistance; as the series resistance value decreases, the parallel-equivalent resistance increases. Alternatively, this means that the parallel-equivalent resistance will increase with circuit Q. We use the following formula to calculate the approximate parallel-equivalent resistance:

Since the coil Q_U remains the inductor's reactance divided by its series resistance, we get:

Multiplying Q_U by the reactance also provides the approximate parallel-equivalent resistance of the inductor's series resistance.

At resonance, assuming our parallel equivalent representation, X_L = X_C, and R_P now defines the impedance of the parallel-resonant circuit. The reactances just equal each other, leaving the voltage and current in phase with each other. In other words, at resonance, the circuit demonstrates only parallel resistance. Therefore, Eq. 2.84 can be rewritten as:

In the preceding example, the circuit impedance at resonance is 9510 Ω.

At frequencies below resonance, the reactance of the inductor is smaller than the reactance of the capacitor, and therefore the current through the inductor will be larger than that through the capacitor. This means that there is only partial cancellation of the two reactive currents, and therefore the line current is larger than the current with resistance alone. Above resonance, things are reversed; more current flows through the capacitor than through the inductor, and again the line current increases above a value larger than the current with resistance alone. At resonance, the current is determined entirely by R_P; it will be small if R_P is large and large if R_P is small.

Since the current rises off resonance, the parallel-resonant-circuit impedance must fall. It also becomes complex, resulting in an increasing phase difference between the voltage and the current. The rate at which the impedance falls is a function of Q_U. Figure 2.180 presents a family of curves showing the impedance drop from resonance for circuit Qs ranging from 10 to 100. The curve family for parallel-circuit impedance is essentially the same as the curve family for series-circuit current. As with series-tuned circuits, the higher the Q of a parallel-tuned circuit, the sharper the response peak. Likewise, the lower the Q, the wider the band of frequencies to which the circuit responds. Using the half-power (-3-dB) points as a comparative measure of circuit performance, we can apply the same equations for bandwidth for a series-resonant circuit to a parallel-resonant circuit, BW = f/Q_U. Table 2.11 summarizes performance of parallel-resonant circuits at high and low Qs, above and below resonance.

TABLE 2.11 Performance of Parallel-Resonant Circuits

2.30.7 The Q of Loaded Circuits

In many resonant circuit applications, the only practical power lost is dissipated in the resonant circuit internal resistance. At frequencies below around 30 MHz, most of the internal resistance is within the inductor coil itself. Increasing the number of turns in an inductor coil increases the reactance at a rate that is faster than the accompanying internal resistance of the coil. Inductors used in circuits where high Q is necessary have large inductances.

When a resonant circuit is used to deliver energy to a load, the energy consumed within the resonant circuit is usually insignificant compared to that consumed by the load. For example, in Fig. 2.185, a parallel load resistor R_LOAD is attached to a resonant circuit, from which it receives power.

If the power dissipated by the load is at least 10 times as great as the power lost in the inductor and capacitor, the parallel impedance of the resonant circuit itself will be so high compared with the resistance of the load that for all practical purposes the impedance of the combined circuit is equal to the load impedance. In these circumstances, the load resistance replaces the circuit impedance in calculating Q. The Q of a parallel-resonant circuit loaded by a resistance is:

where Q_LOAD is the circuit-loaded Q, R_LOAD is the parallel load resistance in ohms, and X is the reactance in ohms of either the inductor or the capacitor.

The loaded Q of a circuit increases when the reactances are decreased. When a circuit is loaded with a low resistance (a few kiloohms) it must have low-reactance elements (large capacitance and small inductance) to have reasonably high Q.

Sometimes parallel load resistors are added to parallel-resonant circuits to lower the Q and increase the circuit bandwidth, as the following example illustrates.

The desired bandwidth, 400 kHz, requires a loaded circuit Q of:

Since the desired Q is half the original value, halving the resonant impedance or parallel-resistance value of the circuit will do the trick. The present impedance of the circuit is:

The desired impedance is:

or half the present impedance.

A parallel resistor of 24,500 Ω will produce the required reduction in Q as bandwidth increases. In real design situations, things are a bit more complex—one must consider factors such as shape of the bandpass curve.

2.31.1 Alternative Decibel Representations

It is often convenient to compare a certain power level with some standard reference. For example, suppose you measured the signal coming into a receiver from an antenna and found the power to be 2 × 10^-13 mW. As this signal goes through the receiver, it increases in strength until it finally produces some sound in the receiver speaker or headphones. It is convenient to describe these signal levels in terms of decibels. A common reference power is 1 mW. The decibel value of a signal compared to 1 mW is specified as "dBm" to mean decibels compared to 1 mW. In our example, the signal strength at the receiver input is:

There are many other reference powers used, depending upon the circuits and power levels. If you use 1 W as the reference power, then you would specify dBW. Antenna power gains are often specified in relation to a dipole (dBd) or an isotropic radiator (dBi). Anytime you see another letter following dB, you will know that some reference power is being specified. For example, to describe the magnitudes of a voltage relative to a 1-V reference, you indicate the level in decibels by placing a "V" at the end of dB, giving units of dBV (again, impedances must stay the same). In acoustics, dB, SPL is used to describe the pressure of one signal in terms of a reference pressure of 20 µPa. The term decibel is also used in the context of sound (see Sec. 15.1).

2.32.1 Input Impedance

Input impedance Z_IN is the impedance "seen" looking into the input of a circuit or device (see Fig. 2.186). Input impedance gives you an idea of how much current can be drawn into the input of a device. Because a complex circuit usually contains reactive components such as inductors and capacitors, the input impedance is frequency sensitive. Therefore, the input impedance may allow only a little current to enter at one frequency, while highly impeding current enters at another frequency. At low frequencies (less than 1 kHz) reactive components may have less influence, and the term input resistance may be used—only the real resistance is dominant. The effects of capacitance and inductance are generally more significant at high frequencies.

When the input impedance is small, a relatively large current can be drawn into the input of the device when a voltage of specific frequency is applied to the input. This typically has an effect of dropping the source voltage of the driver circuit feeding the device's input. (This is especially true if the driver circuit has a large output impedance.) A device with low input impedance is an audio speaker (typically 4 or 8 Ω), which draws lots of current to drive the voice coil.

When the input impedance is large, on the other hand, a small current is drawn into the device when a voltage of specific frequency is applied to the input, and, hence, doesn't cause a substantial drop in source voltage of the driver circuit feeding the input. An op amp is a device with a very large input impedance (1 to 10 MΩ); one of its inputs (there are two) will practically draw no current (in the nA range). In terms of audio, a hi-fi preamplifier that has a 1-MΩ radio input, a 500-kΩ CD input, and a 100-kΩ tap input all have high impedance input due to the preamp being a voltage amplifier not a current amplifier—something we will discuss later.

As a general rule of thumb, in terms of transmitting a signal, the input impedance of a device should be greater than the output impedance of the circuit supplying the signal to the input. Generally, the value should be 10 times as great to ensure that the input will not overload the source of the signal and reduce the strength by a substantial amount. In terms of calculations and such, the input impedance is by definition equal to:

For example, Fig. 2.187 shows how to determine what the input impedance is for two resistor circuits. Notice that in example 2, when a load is attached to the output, the input impedance must be recalculated, taking R₂ and R_load in parallel.

In Sec. 2.33.1, which covers filter circuits, you'll see how the input impedance becomes dependent on frequency.

2.32.2 Output Impedance

Output impedance Z_OUT refers to the impedance looking back into the output of a device. The output of any circuit or device is equivalent to an output impedance Z_OUT in series with an ideal voltage source V_SOURCE. Figure 2.186 shows the equivalent circuit; it represents the combined effect of all the voltage sources and the effective total impedance (resistances, capacitance, and inductance) connected to the output side of the circuit. You can think of the equivalent circuit as a Thevenin equivalent circuit, in which case it should be clear that the V_SOURCE present in Fig. 2.186 isn't necessarily the actual supply voltage of the circuit but the Thevenin equivalent voltage. As with input impedance, output impedance can be frequency dependent. The term output resistance is used in cases where there is little reactance within the circuit, or when the frequency of operation is low (say, less than 1 kHz) and reactive effects are of little consequence; the effects of capacitance and inductance are generally most significant at high frequencies.

When the output impedance is small, a relatively large output current can be drawn from the device's output without significant drop in output voltage. A source with an output impedance much lower than the input impedance of a load to which it is attached will suffer little voltage loss driving current through its output impedance. For example, a lab dc power supply can be viewed as an ideal voltage source in series with a small internal resistance. A decent supply will have an output impedance in the milliohm (mΩ) range, meaning it can supply considerable current to a load without much drop in supply voltage. A battery typically has a higher internal resistance and tends to suffer a more substantial drop in supply voltage as the current demands increase. In general, a small output impedance (or resistance) is considered a good thing, since it means that little power is lost to resistive heating in the impedance and a larger current can be sourced. Op amps, which, we saw, have large input impedance and tend to have low output impedance.

When the output impedance is large, on the other hand, a relatively small output current can be drawn from the output of a device before the voltage at the output drops substantially. If a source with a large output impedance attempts to drive a load that has a much smaller input impedance, only a small portion appears across the load; most is lost driving the output current through the output impedance.

Again, the rule of thumb for efficient signal transfer is to have an output impedance that is at least ¹⁄₁₀ that of the load's input impedance to which it is attached.

In terms of calculations, the output impedance of a circuit is simply its Thevenin equivalent resistance R_THEV. The output impedance is sometimes called the source impedance. In terms of determining the output impedance in circuit analysis, it amounts to "killing" the source (shorting it) and finding the equivalent impedance between the output terminals—the Thevenin impedance.

For example, in Fig. 2.188, to determine the output impedance of this circuit, we effectively find the Thevinin resistance by "shorting the source" and removing the load, and determining the impedance between the output terminals. In this case, the output impedance is simply R₁ and R₂ in parallel.

In Sec. 2.3.1, when we cover filter circuits, you'll see how the output impedance becomes dependent on frequency.

2.33.1 Filters

By combining resistors, capacitors, and inductors in special ways, you can design networks that are capable of passing certain frequencies of signals while rejecting others. This section examines four basic kinds of filters: low-pass, high-pass, bandpass, and notch filters.

2.33.2 Attenuators

Often it is desirable to attenuate a sinusoidal voltage by an amount that is independent of frequency. We can do this by using a voltage divider, since its output is independent of frequency. Figure 2.198 shows a simple voltage divider attenuator network inserted between a source and a load circuit to decrease the source signal's magnitude before it reaches the load. As we go through this example, you will pick up additional insight into input and output impedances.

In (b), the attenuator network has input and output impedances of:

We can come up with a transfer function for this, taking the same current to flow through R₂:

If you rearrange terms, you can see this is simply a voltage divider.

In (c), the load has an input impedance of:

However, when we assemble the circuit, the useful input and output impedances change. Now, looking at things from the point of view of the source, as shown in (d), the source sees an input impedance of the attenuator combined with the load impedance, which is R₁ in series with the parallel combination of R₂ and R_L:

But now, from the point of view of the load, as shown in (e), the output impedance of the attenuator and source combined is R₂ in parallel with the series combination of R₁ and R_S:

(The input impedance to the load is still R_L.)

This output impedance is equivalent to the Thevinen equivalent impedance Z_THEV, as shown in (f). If we substitute R₁ = 100 Ω, R₂ = 3300 Ω, R_S = 1 Ω, and V_S = 10 VAC, we get the graph shown in (f). If we set the load R_L = 330 Ω, using the Thevenin circuit, we find:

2.34.1 Series RLC Circuit

There's another transient example, which is a bit heavy in the math but is a classic analog of many other phenomena found in science and engineering. See Fig. 2.209.

Assume the capacitor is charged to a voltage V₀, and then at t = 0, the switch is closed. Kirchhoff's voltage law for t ≥ 0:

Rewriting in the standard form gives:

This is an example of a linear second-order homogeneous differential equation. It is reasonable to guess that the solution is of the same form as for the first-order homogeneous differential equation encountered earlier:

Substituting this into the differential equation gives:

Note that a solution of the form e^αt always reduces a linear homogeneous differential equation to an algebraic equation in which first derivatives are replaced by α and second derivates by α², and so forth. A linear second-order homogeneous differential equation then becomes a quadratic algebraic equation, and so on. This particular algebraic equation has the following solutions:

Since either value of α represents a solution to the original differential equation, the most general solution is one in which the two possible solutions are multiplied by arbitrary constants and added together:

The constants I₁ and I₂ must be determined from the initial conditions. An nth-order differential equation will generally have n constants that must be determined from initial conditions. In this case the constants can be evaluated from a knowledge of I(0) and dI/dt(0). Since the current in the inductor was zero for t < 0, and since it cannot change abruptly, we know that:

The initial voltage across the inductor is the same as across the capacitor, so that:

The solution for the current in the series RLC circuit is thus:

where α₁ and α₂ are given in Eq. 2.92.

The solution to the preceding equation has a unique character, depending on whether the quantity under the square root in Eq. 2.92 is positive, zero, or negative. We consider these three cases.

2.35.1 Fourier Series

A time-dependent voltage or current is either periodic or nonperiodic. Figure 2.213 shows an example of a periodic waveform with period T.

The wave is assumed to continue indefinitely in both the +t and the -t directions. A periodic function can be displaced by one period, and the resulting function is identical to the original function:

A periodic waveform can be represented as a Fourier series of sines and cosines:

where ω₀ is called the fundamental angular frequency,

2ω₀ is called the second harmonic, and so on. The constants a_n and b_n are determined from:

The constant term a₀/2 is the average value of V(t). The superposition theorem then allows you to analyze any linear circuit having periodic sources by considering the behavior of the circuit for each of the sinusoidal components of the Fourier series. Although most of the examples that we will use have voltage or current as the dependent variable and time as the independent variable, the Fourier methods are very general and apply to any sufficiently smooth function f(t).

Using Euler's expression, e^jθ = cos θ + j sin θ, we can convert Eq. 2.98 into a general expression for a periodic waveform as the sum of complex numbers:

By allowing both positive and negative frequencies (n > 0 and n < 0), it is possible to choose the C_n in such a way that the summation is always a real number. The value of C_n can be determined by multiplying both sides of Eq. 2.102 by e^-jmω₀t, where m is an integer, and then integrating over a period. Only the term with m = n survives, and the result is:

Note that C_-n is the complex conjugate of C_n, and so the imaginary parts of Eq. 2.102 will always cancel, and the resulting V(t) is real. The n = 0 term has a particularly simple interpretation; it is simply the average value of V(t):

and corresponds to the dc component of the voltage. Whether the integrals in the preceding expression are over the interval -T/2 to T/2 or some other interval such as 0 to T is purely a matter of convenience, so long as the interval is continuous and has duration T.

The following example illustrates a Fourier series of a squarewave, as depicted in Fig. 2.214.

To create a mathematical expression in terms of a series of complex numbers, as Eq. 2.102 requests, we first determine the constants from Eq. 2.103 by breaking the integral into two parts for which V(t) is constant:

Since ω₀T = 2π, the preceding equation can be written as

With the use of equation e^jθ = cos Θ + j sin Θ, the preceding equation becomes

Note that cos nπ is +1 when n is even (0,2,4, …) and -1 when n is odd (1,3,5, …), so that all the even values of C_n are zero. Any periodic function, when displaced in time by half a period, is identical to the negative of the original function:

In this case V(t) is said to have half-wave symmetry, and its Fourier series will contain only odd harmonics. The squarewave is an example of such a function. If the wave remained at +V₀ and -V₀ for unequal times, the half-wave symmetry would be lost, and its Fourier series would then contain even as well as odd harmonics.

In addition to its half-wave symmetry, the squarewave shown in Fig. 2.214 is an odd function, because it satisfies the relation:

This property is not a fundamental property of the wave but arises purely out of the choice of where, with respect to the wave, the time origin (t = 0) is assumed. For example, if the squarewave in Fig. 2.214 were displaced by a time of T/4, the resulting squarewave would be an even function, because it would then satisfy the relation:

It's important to note that an odd function can have no dc component, since the negative parts exactly cancel the positive parts on opposite sides of the time axis. The cosine is an even function, and the sine is an odd function. Any even function can be written as a sum of cosines (b_n = 0 in Eq. 2.98), and any odd function can be written as a sum of sines (a_n = 0 in Eq. 2.98). Most periodic functions (such as the one in Fig. 2.213) are neither odd nor even.

The Fourier series calculation can often be simplified by adding or subtracting a constant to the value of the function, or by displacing the time origin so that the function is even or odd or so that it has half-wave symmetry.

The odd-numbered coefficients of the Fourier series representation of the squarewave are given by

and the Fourier series is

With the use of Euler's equation and the fact that sin θ = -sin(-θ) and cos θ = cos(-θ), the preceding equation becomes

The first three terms of the preceding series (n = 1,3,5) along with their sum are plotted in Fig. 2.215. Note that the series, even with as few as three terms, is beginning to resemble the squarewave.

For waveforms more complicated than a squarewave, the integrals are more difficult to perform, but it is still usually easier to calculate a Fourier series for a periodic voltage than to solve a differential equation in which the same time-dependent voltage appears. Furthermore, tables of Fourier series for the most frequently encountered waveforms are available and provide a convenient shortcut for analyzing many circuits. Some common waveforms and their Fourier series are listed in Fig. 2.216.

Since the source is periodic, the current I(t) is also periodic with the same period, and it can be written as a Fourier series:

Each C′_n is a phasor current representing one frequency component of the total current in the same way that each C_n represents a component of the phasor voltage in the previous section. The relationship between the two phasors is determined by dividing by the circuit impedance:

Substituting the value of C_n derived earlier for the squarewave gives:

for n odd. For n even, C′_n is zero, since C_n is zero for even n. The corresponding current is then

With the use of Euler's relation, the preceding current can be written as:

The voltage across the resistor and capacitor can be determined from the definitions of an ideal resistor and an ideal capacitor:

The sum of the first three terms (n = 1, n = 3, n = 5) of the Fourier series for V_C(t) and V_R(t) are shown in Fig. 2.217b. As n approaches infinity, the waveforms approach the real deal.

Note that circuits with squarewave sources can also be analyzed as transient circuits. In the circuit in Fig. 2.217a, during a half period (such as 0 < t < T/2) when the source voltage is constant, the voltage across the capacitor is expressed as:

The constants A and B can be determined from

The first equation results from knowing that if the source remains at +V₀ forever, the capacitor would charge to voltage V₀. The second equation is required to ensure that the function has half-wave symmetry. Hence,

The capacitor voltage is then:

for 0 < t < T/2. The waveform repeats itself for t > T/2 with each half cycle alternating in sign.

2.37.1 How SPICE Works

Here we take a look at the heart of simulators—SPICE. What are the mathematical tricks the code in SPICE uses to simulate electrical circuits described by nonlinear differential equations? At the core of the SPICE engine is a basic technique called nodal analysis. It calculates the voltage at any node, given all of the circuit resistances (or, inversely, their conductances) and the current sources. Whether the program is performing dc, ac, or transient analysis, SPICE ultimately casts its components (linear, nonlinear, and energy-storage elements) into a form where the innermost calculation is nodal analysis.

Kirchhoff discovered that the total current entering a node is equal to the total current leaving a node. Stated another way, the sum of currents in and out of a node is zero. These currents can be described by equations in terms of voltages and conductances. If you have more than one node, then you get more than one equation describing the same system (simultaneous equations). The trick then is finding the voltage at each node that satisfies all the equations simultaneously.

For example, let's consider the simple circuit in Fig. 2.220. In this circuit there are three nodes: node 0 (which is always ground), node 1, and node 2.

Using Kirchhoff's current law in the form of "the sum of current in and out of a node is zero," we can create two equations for the two nodes 1 and 2 (0 is by default ground):

Since the mission here is to calculate the node voltages, we rewrite these equations in order to separate V₁ and V₂:

The trick now becomes finding values of V₁ and V₂ that satisfy both equations. Although we could solve one variable in terms of the other from one equation and plug it into the second equation to find the other variable, things get messy (so many R variables to deal with). Instead, we'll take a cleaner approach—one that involves conductance G, where G = 1/R. This will make the bookkeeping easier, and it becomes especially important when the nodal number increases with complex circuits.

Writing resistors in terms of total conductance:

Thus, the system of equations is transformed into:

Solving the second equation for V₁:

we then stick this into the first equation and solve for V₂:

See how much cleaner the manipulations were using conductances? V₂ is described by circuit conductances and I_S alone. We still must find the numerical value of V₂ and stick it back into the V₁ equation while, in the process, calculating the numerical values of the conductances. However, we have found circuit voltages V₁ and V₂ that satisfy both system equations.

Although the last approach wasn't that hard, or messy, there are times when the circuits become big and the node and device counts so large that bookkeeping terms become a nightmare. We must go a step further in coming up with a more efficient and elegant approach. What we do is use matrices.

In matrix form, the set of nodal equations is written:

Or in terms of total conductances and source currents:

Treating each matrix as a variable, we can rewrite the preceding equation in compact form:

In matrix mathematics, you can solve for a variable (almost) as you would in any other algebraic equation. Solving for v you get:

where G^-1 is the matrix inverse of G. (1/G does not exist in the matrix world.) This equation is the central mechanism of the SPICE algorithm. Regardless of the analysis—ac, dc, or transient—all components or their effects are cast into the conductance matrix G and the node voltages are calculated by v = G^-1 · i, or some equivalent method.

Substituting the component values present in the circuit in Fig. 2.218 into the conductance matrix and current matrix (we could use an Excel spreadsheet and apply formulas to keep things clean), we get:

Hence, the voltage is:

V₁ = 9.9502 and V₂ = 4.9751

2.37.2 Limitations of SPICE and Other Simulators

Simulation of a circuit is only as accurate as the behavior models in the SPICE devices created for it. Many simulations are based on simplified models. For more complex circuits or subtle behaviors, the simulation can be misleading or incorrect. This can mean disaster if you're relying entirely on SPICE (or an advanced simulator based on SPICE) when developing circuits. SPICE can also be deceiving, since simulations are free of noise, crosstalk, interference, and so on—unless you incorporate these behaviors into the circuit. Also, SPICE is not the best predictor of component failures. You must know what dangers to look for and which behaviors are not modeled in your SPICE circuit. In short, SPICE is not a prototype substitute—the performance of the actual breadboard provides the final answer.

2.37.3 A Simple Simulation Example

As an example, we will look at simulating a simple RLC crossover network (which we will meet again in Chap. 15) with the free and easy-to-use online simulator CircuitLab (www.circuitlab.com).

The first step is to draw the schematic using the editing tool (see Fig. 2.221). The LRC network is responsible for separating an audio signal into two parts: a low-pass filter to drive the low-frequency woofer speaker and a high-pass filter to drive the tweeter.

As well as adding the appropriate components to the design and connecting them, we have added an ac voltage source to represent the input signal and attached two labels (Tweeter and Woofer), so that we can see the results of the simulation (there will be labels next to our graph plots).

When we are ready, we can run a simulation. In this case, we are going to carry out a frequency-domain simulation so that we can determine the crossover frequency of the network. To do this, we specify V₁ as being the input as well as start and end frequencies. We also specify the outputs that we wish to see plotted (see Fig. 2.222).

When we click the Simulate button, CircuitLab will run the simulation and produce an output plot, as shown in Fig. 2.223.

From Fig. 2.223, we can see that the crossover frequency where both speakers are receiving the same magnitude of signal is at around 1.6 kHz.

CircuitLab is a good starting point for understanding simulators. We strongly encourage you to try some of the sample schematics on the website and experiment with a few simulations.