Impedance

When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other’s effect. [link] shows an RLC series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an RLC circuit is the frequency dependence of $X_L$ and $X_C$, and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important “resonance” features that are the basis of many applications, such as radio tuners.

The combined effect of resistance $R$, inductive reactance $X_L$, and capacitive reactance $X_C$ is defined to be impedance, an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an RLC circuit are related by an AC version of Ohm’s law:

Here $I_0$ is the peak current, $V_0$ the peak source voltage, and $Z$ is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for $Z$ in terms of $R$, $X_L$, and $X_C$, we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled $V_R$, $V_L$, and $V_C$ in [link].

Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in $R$, $L$, and $C$ are equal and in phase. But we know from the preceding section that the voltage across the inductor $V_L$ leads the current by one-fourth of a cycle, the voltage across the capacitor $V_C$ follows the current by one-fourth of a cycle, and the voltage across the resistor $V_R$ is exactly in phase with the current. [link] shows these relationships in one graph, as well as showing the total voltage around the circuit $V=V_R+V_L+V_C$, where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit $V$ is also the voltage of the source.

You can see from [link] that while $V_R$ is in phase with the current, $V_L$ leads by $\text{90º}$, and $V_C$ follows by $\text{90º}$. Thus $V_L$ and $V_C$ are $\text{180º}$ out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage $V_0$ of the source does not equal the sum of the peak voltages across $R$, $L$, and $C$. The actual relationship is

where $V_{0R}$, $V_{0L}$, and $V_{0C}$ are the peak voltages across $R$, $L$, and $C$, respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive, we substitute $V_0=I_{0}Z$ into the above, as well as $V_{0R}=I_{0}R$, $V_{0L}=I_0{X}_L$, and $V_{0C}=I_0{X}_C$, yielding

$I_0$ cancels to yield an expression for $Z$:

which is the impedance of an RLC series AC circuit. For circuits without a resistor, take $R=\text{0}$; for those without an inductor, take $X_L=0$; and for those without a capacitor, take $X_C=0$.

Resonance in RLC Series AC Circuits

How does an RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, $I_{\text{rms}}=V_{\text{rms}}/Z$, and the expression for impedance $Z$ from $Z=\sqrt{R^2+(X_L-X_C{)}^2}$ gives

The reactances vary with frequency, with $X_L$ large at high frequencies and $X_C$ large at low frequencies, as we have seen in three previous examples. At some intermediate frequency $f_0$, the reactances will be equal and cancel, giving $Z=R$ —this is a minimum value for impedance, and a maximum value for $I_{\text{rms}}$ results. We can get an expression for $f_0$ by taking

Substituting the definitions of $X_L$ and $X_C$,

Solving this expression for $f_0$ yields

where $f_0$ is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At $f_0$, the effects of the inductor and capacitor cancel, so that $Z=R$, and $I_{\text{rms}}$ is a maximum.

Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation—in this case, forced by the voltage source—at the natural frequency of the system. The receiver in a radio is an RLC circuit that oscillates best at its $f_0$. A variable capacitor is often used to adjust $f_0$ to receive a desired frequency and to reject others. [link] is a graph of current as a function of frequency, illustrating a resonant peak in $I_{\text{rms}}$ at $f_0$. The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.

Power in RLC Series AC Circuits

If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in [link], voltage and current are out of phase in an RLC circuit. There is a phase angle $\varphi$ between the source voltage $V$ and the current $I$, which can be found from

For example, at the resonant frequency or in a purely resistive circuit $Z=R$, so that $\text{cos}\varphi =1$. This implies that $\varphi =0º$ and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because $I_{\text{rms}}$ is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is

Thus $\text{cos}\varphi$ is called the power factor, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, $\text{cos}\varphi =1$.

Power delivered to an RLC series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in [link]. The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels’ motion is a maximum if the bumps in the road are hit at the resonant frequency.

A pure LC circuit with negligible resistance oscillates at $f_0$, the same resonant frequency as an RLC circuit. It can serve as a frequency standard or clock circuit—for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time. [link] shows the analogy between an LC circuit and a mass on a spring.