RLC Series AC Circuits

In addition to the delay between current and voltage, there is also the question of how large the peaks of current and voltage become, whenever they happen. Read this text, which introduces a new concept to characterize the relative heights of the current and voltage peaks, called the reactance.

It is the generalization of the concept of resistance, in that the reactance is defined as the ratio of voltage to current – but because both are variable we must take their rms values. Recall that rms stands for root-mean-square and represents an average of a time-varying quantity. In fact, you can equivalently get the reactance by dividing the peak voltage by the peak current.

The reactance of a capacitor depends on the capacitance, and the reactance of an inductor depends on the inductance. Both also depend on the frequency of the sinusoidally-varying voltage. Because inductors act like they have inertia, they will produce less peak current for an input voltage that changes direction rapidly, making the ratio of voltage to current (i.e. their reactance) larger for high frequencies.

Capacitors behave the other way around: their reactance becomes smaller at high frequencies. This is because when the charging current changes direction rapidly, the capacitor never gets enough time to charge fully, which never allows the voltage to grow to large values. As a result, the ratio of voltage to current is small.

Even though reactances have the same units as the familiar resistance we know from Ohm's Law, there is one big difference: it is the time delay between voltage and current that we see (in different directions) for inductors and capacitors – but there is no such delay for "normal" resistors.

In the text below we consider the most complex circuit of this course: a combination of resistor, capacitor and inductor in series. The circuit diagram does not look complex, but the difficulties are hidden in the delay between current and voltage. Because that delay is affected in opposite ways by capacitors and inductors, and not at all by a resistor, the combination of all three produces a delay that depends on the parameters L, R and C.

Fortunately, when such a circuit is driven by a sinusoidally-varying voltage, it behaves a lot like a mechanical oscillator driven by a harmonically-varying force. In particular, the current and voltages will all respond by oscillating at the same frequency as the driving voltage. And just as we saw for a driven oscillator, there is a specific driving frequency at which the circuit responds particularly strongly.

To understand what is going on, you can think about the AC voltage supply as the analogue of a mechanical driving force and the peak current through the circuit as the analogue of the maximum speed of the oscillator. Also remember that the inductor acts as if it gives inertia to the current that is trying to flow.

On the other hand, the capacitor acts analogously to a spring. This is because whenever the current through the capacitor stops, it is done charging and has therefore reached its maximum voltage across the plates. This voltage is analogous to the restoring force of a spring. And if you imagine when an oscillating spring reaches its maximum restoring force, that is precisely when it turns around so that its speed is momentarily zero. Zero speed is the analogue of a stopped current.

With these mechanical analogies, it should come as no surprise that a circuit consisting of capacitor and inductor will exhibit the phenomenon of resonance. This is indeed the case: when the frequency of the AC voltage supply hits an optimal value, the circuit will show a current oscillating back and forth with especially large amplitude.

Read this text, which comes to this conclusion using another concept which generalizes the idea of resistance even further (and has the same units): impedance. It is still defined as the ratio of peak voltage to peak current, just allowing for arbitrary delays between when those peaks occur. The resonance phenomenon is identified by an impedance that becomes very small (or zero). This corresponds to very large currents in relation to the voltage.

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. Figure 23.48 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 figure describes an R LC series circuit. It shows a resistor R connected in series with an inductor L, connected to a capacitor C in series to an A C source V. The voltage of the A C source is given by V equals V zero sine two pi f t. The voltage across R is V R, across L is V L and across C is V C.

Figure 23.48 An RLC series circuit with an AC voltage source.

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:

$I_{0}=\frac{V_{0}}{Z}$ or $I_{rms}=\frac{V_{rms}}{Z}$ [Equation 23.63]

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 Figure 23.48.

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.

Figure 23.49 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 Figure 23.49 that while $V_{R}$ is in phase with the current, $V_{L}$ leads by 90º, and $V_{C}$ follows by 90º. Thus $V_{L}$ and $V_{C}$ are 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

$V_{0}=\sqrt{V_{0R}^2+(V_{0L}-V_{0C})^2}$ [Equation 23.64]

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

$I_{0}Z=\sqrt{I_{0}^{2}R^{2}+(I_{0}X_{L}-I_{0}X_{C})^{2}}=I_{0}\sqrt{R^{2}+(X_{L}-X_{C})^{2}}$ [Equation 23.65]

$I_{0}$ cancels to yield an expression for $Z$ :

$Z=\sqrt{R^{2}+(X_{L}-X_{C})^2}$ [Equation 23.66]

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

Figure 23.49 This graph shows the relationships of the voltages in an RLC circuit to the current. The voltages across the circuit elements add to equal the voltage of the source, which is seen to be out of phase with the current.

Example 23.12 Calculating Impedance and Current

An RLC series circuit has a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for $R$ and $C$ are the same as in Example 23.10 and Example 23.11. (b) If the voltage source has $V_{rms}=120\: V$ , what is $I_{rms}$ at each frequency?

Strategy

For each frequency, we use $Z=\sqrt{R^{2}+(X_{L}-X_{C})^2}$ to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.

Solution for (a)

At 60.0 Hz, the values of the reactances were found in Example 23.10 to be $X_{L}=1.13\:\Omega$ and in Example 23.11 to be $X_{C}=531\:\Omega$ . Entering these and the given 40.0 Ω for resistance into $Z=\sqrt{R^{2}+(X_{L}-X_{C})^2}$ yields

$Z=\sqrt{R^{2}+(X_{L}-X_{C})^2}$ [Equation 23.67]

$=\sqrt{(40.0\:\Omega)^{2}+(1.13\:\Omega-531\:\Omega)^2}$

$=531\:\Omega\: at\:60.0\:Hz$

Similarly, at 10.0 kHz, $X_{L}=188\:\Omega$ and $X_{C}=3.18\:\Omega$ , so that

$Z=\sqrt{(40.0\:\Omega)^{2}+(188\:\Omega-3.18\:\Omega)^2}$ [Equation 23.68]

$=190\:\Omega\: at\:10.0\:kHz$

Discussion for (a)

In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that $X_{L}$ dominates at high frequency and $X_{C}$ dominates at low frequency.

Solution for (b)

The current $I_{rms}$ can be found using the AC version of Ohm’s law in Equation $I_{rms}=V_{rms}/Z$ :

$I_{rms}=\frac{V_{rms}}{Z}=\frac{120\:V}{531\:\Omega}=0.226\:A\:at\:60.0\:Hz$

Finally, at 10.0 kHz, we find

$I_{rms}=\frac{V_{rms}}{Z}=\frac{120\:V}{190\:\Omega}=0.633\:A\:at\:10.0\:Hz$ at 10.0 kHz

Discussion for (a)

The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in Example 23.11. The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in Example 23.10. The inductor dominates at high frequency.

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, $I_{rms}=V_{rms}/Z$ , and the expression for impedance $Z$ from $Z=\sqrt{R^{2}+(X_{L}-X_{C})^2}$ gives

$I_{rms}=\frac{V_{rms}}{\sqrt{R^{2}+(X_{L}-X_{C})^2}}$ [Equation 23.69]

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 $I_{rms}$ results. We can get an expression for $f_{0}$ by taking

$X_{L}=X_{C}$ [Equation 23.70]

Substituting the definitions of $X_{L}$ and $X_{C}$ ,

$2\pi f_{0}L=\frac{1}{2\pi F_{0}C}$ [Equation 23.71]

Solving this expression for $f_{0}$ yields

$f_{0}=\frac{1}{2\pi\sqrt{LC}}$ [Equation 23.72]

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 $I_{rms}$ 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.

Figure 23.50 is a graph of current as a function of frequency, illustrating a resonant peak in $I_{rms}$ 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.

Figure 23.50 A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at $f_{0}$ , but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude $V_{0}$ .

Example 23.13 Calculating Resonant Frequency and Current

For the same RLC series circuit having a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor: (a) Find the resonant frequency. (b) Calculate $I_{rms}$ at resonance if $V_{rms}$ is 120 V.

Strategy

The resonant frequency is found by using the expression in $f_{0}=\frac{1}{2\pi\sqrt{LC}}$ . The current at that frequency is the same as if the resistor alone were in the circuit.

Solution for (a)

Entering the given values for $L$ and $C$ into the expression given for $f_{0}$ in $f_{0}=\frac{1}{2\pi\sqrt{LC}}$ yields

$f_{0}=\frac{1}{2\pi\sqrt{LC}}$ [Equation 23.73]

$=\frac{1}{2\pi\sqrt{(3.00\times^{-3}\:H)(5.00\times 10^{-6}\:F)}}=1.30\:kHz$

Discussion for (a)

We see that the resonant frequency is between 60.0 Hz and 10.0 kHz, the two frequencies chosen in earlier examples. This was to be expected, since the capacitor dominated at the low frequency and the inductor dominated at the high frequency. Their effects are the same at this intermediate frequency.

Solution for (b)

The current is given by Ohm’s law. At resonance, the two reactances are equal and cancel, so that the impedance equals the resistance alone. Thus,

$I_{rms}=\frac{V_{rms}}{Z}=\frac{120\:V}{40.0\:\Omega}=3.00\:A$ [Equation 23.74]

Discussion for (b)

At resonance, the current is greater than at the higher and lower frequencies considered for the same circuit in the preceding 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 Figure 23.49, voltage and current are out of phase in an RLC circuit. There is a phase angle $\Phi$ between the source voltage $V$ and the current $I$ , which can be found from

$cos\:\Phi=\frac{R}{Z}$ [Equation 23.75]

For example, at the resonant frequency or in a purely resistive circuit $Z=R$ , so that $cos\:\Phi =1$ . This implies that $\Phi=0^{o}$ 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 $I_{rms}$ 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

$P_{ave}=I_{rms}V_{rms}\:cos\:\Phi$ [Equation 23.76]

Thus $cos\:\Phi$ 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, $cos\:\Phi=1$ .

Example 23.14 Calculating the Power Factor and Power

For the same RLC series circuit having a 40.0 Ω resistor, a 3.00 mH inductor, a 5.00 μF capacitor, and a voltage source with a 𝑉rms of 120 V: (a) Calculate the power factor and phase angle for $f=60.0\:Hz$ . (b) What is the average power at 50.0 Hz? (c) Find the average power at the circuit’s resonant frequency.

Strategy and Solution for (a)

The power factor at 60.0 Hz is found from

$cos\:\Phi=\frac{R}{Z}$ [Equation 23.77]

We know $Z=531\:\Omega$ from Example 23.12, so that

$cos\:\Phi=\frac{40.0\:\Omega}{531\:\Omega}=0.0753\:at\:60.0\:Hz$ [Equation 23.78]

This small value indicates the voltage and current are significantly out of phase. In fact, the phase angle is

$\Phi=cos^{-1}\:0.0753=85.7^{o}\:at\:60.0\:Hz$ [Equation 23.79]

Discussion for (a)

The phase angle is close to 90º, consistent with the fact that the capacitor dominates the circuit at this low frequency (a pure RC circuit has its voltage and current 90º out of phase).

Strategy and Solution for (b)

The average power at 60.0 Hz is

$P_{ave}=I_{rms}V_{rms}\:cos\:\Phi$ [Equation 23.80]

$I_{rms}$ was found to be 0.226 A in Example 23.12. Entering the known values gives

$P_{ave}=(0.226\:A)(120\:V)(0.0753)=2.04\:W\:at\:60.0\:Hz$ [Equation 23.81]

Strategy and Solution for (c)

At the resonant frequency, we know $cos\:\Phi=1$ , and $I_{rms}$ was found to be 6.00 A in Example 23.13. Thus,

$P_{ave}=(3.00\:A)(120\:V)(1)=360\:W$ at resonance (1.30 kHz)

Discussion

Both the current and the power factor are greater at resonance, producing significantly greater power than at higher and lower frequencies.

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 Figure 23.51.

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.

The figure describes the path of motion of a wheel of a car. The front wheel of a car is shown. A shock absorber attached to the wheel is also shown. The path of motion is shown as vertically up and down.

Figure 23.51 The forced but damped motion of the wheel on the car spring is analogous to an RLC series AC circuit. The shock absorber damps the motion and dissipates energy, analogous to the resistance in an RLC circuit. The mass and spring determine 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. Figure 23.52 shows the analogy between an LC circuit and a mass on a spring.

Figure 23.52 An LC circuit is analogous to a mass oscillating on a spring with no friction and no driving force. Energy moves back and forth between the inductor and capacitor, just as it moves from kinetic to potential in the mass-spring system.

Source: Rice University, https://openstax.org/books/college-physics/pages/23-12-rlc-series-ac-circuits
This work is licensed under a Creative Commons Attribution 4.0 License.

Last modified: Tuesday, August 31, 2021, 4:43 PM