Electromagnetic Induction, AC Circuits, and Electrical Technologies

57 RLC Series AC Circuits

Learning Objectives

  • Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and current in a series RLC circuit.
  • Draw and interpret the circuit diagram of a series RLC circuit.
  • Explain the physical significance of electrical resonance and describe applications in healthcare and technology.

Impedance in RLC Circuits

In the previous section, we examined how resistors, inductors, and capacitors each respond individually to alternating current (AC). Real electrical circuits, however, often contain all three components connected together. Examples include medical imaging systems, wireless communication devices, biomedical sensors, and electronic filters used in patient-monitoring equipment.

When a resistor, inductor, and capacitor are connected in series, they do not simply oppose current independently. Instead, their effects combine in a more complex way because inductors and capacitors respond to AC in opposite manners. Inductive reactance increases with frequency, while capacitive reactance decreases with frequency. As a result, these two effects can partially—or even completely—cancel one another at certain frequencies.

Figure 57.1 shows a typical series RLC circuit connected to an AC voltage source.

Circuit diagram of a series RLC circuit consisting of a resistor, an inductor, and a capacitor connected to an alternating-current voltage source.
Figure 57.1 A series RLC circuit connected to an AC voltage source. The resistor, inductor, and capacitor each contribute differently to the circuit's response to alternating current.

The combined opposition that a series RLC circuit presents to alternating current is called its impedance. Impedance is the AC equivalent of resistance in a DC circuit. Like resistance, impedance is measured in ohms (Ω), but unlike resistance, it depends on the frequency of the applied voltage because both inductive and capacitive reactance vary with frequency.

The AC version of Ohm's law is therefore

[latex]I_0=\frac{V_0}{Z}\qquad\text{or}\qquad I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{Z}.[/latex]

where

  • [latex]V_0[/latex] and [latex]I_0[/latex] are the peak voltage and current,
  • [latex]V_{\mathrm{rms}}[/latex] and [latex]I_{\mathrm{rms}}[/latex] are the rms voltage and current, and
  • [latex]Z[/latex] is the impedance of the circuit.

As expected, increasing the impedance decreases the current flowing through the circuit.

To determine the impedance, we must examine how the voltages across the resistor, inductor, and capacitor combine. Because the components are connected in series, the same current flows through each of them at every instant. However, the voltages across the components are not synchronized because each element responds differently to alternating current.

  • The voltage across the resistor is exactly in phase with the current.
  • The voltage across the inductor leads the current by 90°.
  • The voltage across the capacitor lags behind the current by 90°.

Figure 57.2 illustrates these phase relationships.

Graphs comparing the current and the voltages across the resistor, inductor, capacitor, and source in a series RLC circuit. The resistor voltage is in phase with the current, the inductor voltage leads by 90 degrees, and the capacitor voltage lags by 90 degrees.
Figure 57.2 Instantaneous voltages and current in a series RLC circuit. The resistor voltage is in phase with the current, while the inductor and capacitor voltages are shifted by ±90°. The source voltage is the vector sum of the three component voltages.

Notice that the inductor and capacitor voltages are 180° out of phase with each other. When one reaches its positive maximum, the other reaches its negative maximum. Consequently, these voltages partially cancel each other instead of adding directly.

Because the voltages are not in phase, the peak source voltage is not equal to the arithmetic sum of the peak voltages across the three components. Instead, they combine according to vector addition, giving

[latex]V_0=\sqrt{V_{0R}^{\,2}+\left(V_{0L}-V_{0C}\right)^2}.[/latex]

Here

  • [latex]V_{0R}[/latex] is the peak voltage across the resistor,
  • [latex]V_{0L}[/latex] is the peak voltage across the inductor, and
  • [latex]V_{0C}[/latex] is the peak voltage across the capacitor.

Using the AC version of Ohm's law together with the definitions of inductive and capacitive reactance, we substitute

[latex]V_{0R}=I_0R,\qquad V_{0L}=I_0X_L,\qquad V_{0C}=I_0X_C,\qquad V_0=I_0Z.[/latex]

This gives

[latex]I_0Z = \sqrt{I_0^2R^2+\left(I_0X_L-I_0X_C\right)^2} = I_0\sqrt{R^2+\left(X_L-X_C\right)^2}.[/latex]

Since the current appears in every term, it cancels, leaving the expression for the impedance of a series RLC circuit:

[latex]Z=\sqrt{R^2+\left(X_L-X_C\right)^2}.[/latex]

This equation shows that impedance depends on three quantities:

  • the resistance [latex]R[/latex],
  • the inductive reactance [latex]X_L[/latex], and
  • the capacitive reactance [latex]X_C[/latex].

Notice that the reactances are subtracted, not added. This occurs because inductors and capacitors oppose AC in opposite ways. If the inductive and capacitive reactances happen to be equal, they cancel completely, leaving the impedance equal to the resistance alone.

Impedance in an RLC Circuit

Impedance is the total opposition that an AC circuit presents to current. It combines the effects of resistance and the frequency-dependent reactances of inductors and capacitors.

[latex]Z=\sqrt{R^2+\left(X_L-X_C\right)^2}.[/latex]

When [latex]X_L>X_C[/latex], the circuit behaves primarily as an inductive circuit. When [latex]X_C>X_L[/latex], it behaves primarily as a capacitive circuit. When [latex]X_L=X_C[/latex], the reactances cancel and the impedance reaches its minimum value.

The concept of impedance is essential in many biomedical and engineering applications. For example, impedance determines how electrical currents travel through biological tissues during electrocardiography (ECG), electrical impedance tomography, and bioimpedance analysis, where small alternating currents are used to estimate body composition or monitor lung function. Engineers also design RLC circuits to selectively transmit or block certain frequencies in MRI systems, ultrasound electronics, wireless communication devices, and physiological monitoring equipment.

Example 57.1: Calculating Impedance and Current in a Series RLC Circuit

A series RLC circuit contains a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor.

  1. Calculate the impedance of the circuit at 60.0 Hz and 10.0 kHz. (These are the same frequencies and component values used in the previous chapter when calculating the inductive and capacitive reactances.)
  2. If the source provides an rms voltage of 120 V, determine the rms current at each frequency.

Strategy

For each frequency, use the impedance equation

[latex]Z=\sqrt{R^2+\left(X_L-X_C\right)^2},[/latex]

where the reactances were previously found to be:

  • At 60.0 Hz: [latex]X_L=1.13\,\Omega[/latex] and [latex]X_C=531\,\Omega[/latex]
  • At 10.0 kHz: [latex]X_L=188\,\Omega[/latex] and [latex]X_C=3.18\,\Omega[/latex]

Once the impedance has been calculated, determine the current using the AC version of Ohm's law:

[latex]I_{\rm rms}=\frac{V_{\rm rms}}{Z}.[/latex]

Solution

Step 1: Calculate the impedance at 60.0 Hz.

[latex]\begin{aligned} Z&=\sqrt{(40.0\,\Omega)^2+\left(1.13\,\Omega-531\,\Omega\right)^2}\\ &=\sqrt{1600+529.87^2}\\ &\approx531\,\Omega. \end{aligned}[/latex]

Step 2: Calculate the impedance at 10.0 kHz.

[latex]\begin{aligned} Z&=\sqrt{(40.0\,\Omega)^2+\left(188\,\Omega-3.18\,\Omega\right)^2}\\ &=\sqrt{1600+184.82^2}\\ &\approx190\,\Omega. \end{aligned}[/latex]

Step 3: Calculate the rms current at each frequency.

At 60.0 Hz:

[latex]I_{\rm rms} =\frac{120\,\text{V}}{531\,\Omega} =0.226\,\text{A}.[/latex]

At 10.0 kHz:

[latex]I_{\rm rms} =\frac{120\,\text{V}}{190\,\Omega} =0.632\,\text{A}.[/latex]

Discussion

This example illustrates how the behavior of an RLC circuit depends strongly on the frequency of the applied voltage.

  • At 60 Hz, the capacitive reactance ([latex]X_C[/latex]) is much larger than the inductive reactance ([latex]X_L[/latex]). The circuit therefore behaves primarily like a capacitor, resulting in a large impedance and a relatively small current.
  • At 10.0 kHz, the inductive reactance becomes much larger than the capacitive reactance. The circuit now behaves primarily like an inductor, again limiting the current, although less strongly than at 60 Hz.

Notice that the impedance is not equal to the sum of the resistance and the two reactances. Instead, the inductive and capacitive reactances partially cancel each other because they are 180° out of phase. This cancellation is one of the defining features of RLC circuits and leads to the important phenomenon of electrical resonance, discussed in the next section.

Resonance in Series RLC Circuits

One of the most important properties of a series RLC circuit is that its behavior depends strongly on the frequency of the applied AC voltage. As the frequency changes, the inductive and capacitive reactances change in opposite directions. This means that the total impedance of the circuit is not constant, and therefore neither is the current.

Using the AC version of Ohm's law together with the expression for impedance, the rms current is

[latex]I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}} {\sqrt{R^2+\left(X_L-X_C\right)^2}}.[/latex]

This equation shows that the current depends on frequency because both reactances depend on frequency:

  • The inductive reactance increases as the frequency increases:
    [latex]X_L=2\pi fL.[/latex]
  • The capacitive reactance decreases as the frequency increases:
    [latex]X_C=\frac{1}{2\pi fC}.[/latex]

At low frequencies, the capacitor dominates because its reactance is large while the inductive reactance is small. At high frequencies, the opposite occurs—the inductor dominates because its reactance becomes large while the capacitor's reactance becomes small.

Between these two extremes lies a special frequency at which the inductive and capacitive reactances are exactly equal:

[latex]X_L=X_C.[/latex]

Substituting the expressions for the two reactances gives

[latex]2\pi f_0L=\frac{1}{2\pi f_0C}.[/latex]

Solving for the frequency yields

[latex]f_0=\frac{1}{2\pi\sqrt{LC}},[/latex]

where [latex]f_0[/latex] is called the resonant frequency of the circuit.

At this frequency, the inductive and capacitive reactances cancel each other exactly:

[latex]X_L=X_C.[/latex]

As a result, the impedance becomes simply

[latex]Z=R,[/latex]

which is the smallest possible impedance for the circuit. Because the impedance is minimized, the current reaches its maximum value:

[latex]I_{\mathrm{rms,max}}=\frac{V_{\mathrm{rms}}}{R}.[/latex]

Resonance in a Series RLC Circuit

A series RLC circuit is said to be in resonance when the inductive and capacitive reactances are equal.

[latex]X_L=X_C.[/latex]

At resonance:

  • The impedance is at its minimum value, [latex]Z=R[/latex].
  • The current reaches its maximum value.
  • The circuit transfers electrical energy most efficiently between the source and the resistor.

The resonant frequency is also called the circuit's natural frequency because it is the frequency at which energy naturally oscillates back and forth between the electric field of the capacitor and the magnetic field of the inductor. The voltage source simply replaces the energy that is lost due to the resistor.

An intuitive way to understand resonance is to compare it with pushing someone on a swing. If each push is applied at exactly the right moment, the swing reaches a large amplitude with relatively little effort. If the pushes are too fast or too slow, the motion is much smaller. Likewise, an RLC circuit responds most strongly when it is driven at its natural frequency.

Electrical resonance is widely used in science, engineering, and medicine. For example:

  • Radio receivers use resonant circuits to select one broadcasting frequency while rejecting all others.
  • Magnetic resonance imaging (MRI) relies on resonant interactions between radio-frequency electromagnetic fields and atomic nuclei to produce detailed medical images.
  • Wireless communication systems, including Bluetooth, Wi-Fi, and mobile phones, use resonant circuits to tune transmitters and receivers to specific frequencies.
  • Biomedical instruments often include resonant filters that isolate desired physiological signals while reducing electrical noise.

Figure 57.3 illustrates how the current changes with frequency for two different RLC circuits.

Graph of rms current versus frequency for two series RLC circuits. Both curves show a maximum current at the resonant frequency. The circuit with lower resistance has a taller, narrower resonance peak, while the circuit with higher resistance has a lower, broader peak.
Figure 57.3 The current in a series RLC circuit reaches a maximum at the resonant frequency. A circuit with lower resistance has a sharper, taller resonance peak, making it more selective for a narrow range of frequencies.

The graph shows that both circuits resonate at the same frequency because they have the same values of L and C. However, the circuit with the smaller resistance produces a much taller and narrower peak. This means it responds strongly only within a narrow frequency range, making it highly selective. Increasing the resistance broadens the resonance curve and lowers the peak current, reducing the circuit's ability to distinguish between nearby frequencies.

This property is essential in applications such as radio tuning, where the receiver must isolate one station from many others broadcasting at nearby frequencies. In medical electronics, resonance and frequency-selective circuits are similarly used to improve signal quality by enhancing desired frequencies while suppressing unwanted electrical noise.

Example 57.2: Calculating the Resonant Frequency and Current in a Series RLC Circuit

Consider the same series RLC circuit used in the previous example, containing a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor.

  1. Calculate the resonant frequency of the circuit.
  2. If the source provides an rms voltage of 120 V, determine the rms current when the circuit is operating at resonance.

Strategy

The resonant frequency is found using

[latex]f_0=\frac{1}{2\pi\sqrt{LC}}.[/latex]

Once the resonant frequency is known, remember that at resonance the inductive and capacitive reactances are equal:

[latex]X_L=X_C.[/latex]

Therefore, the impedance reduces to the resistance alone:

[latex]Z=R.[/latex]

The current can then be found directly from the AC version of Ohm's law.

Solution

Step 1: Calculate the resonant frequency.

[latex]\begin{aligned} f_0 &=\frac{1}{2\pi\sqrt{LC}}\\[4pt] &=\frac{1}{2\pi\sqrt{(3.00\times10^{-3}\,\text{H})(5.00\times10^{-6}\,\text{F})}}\\[4pt] &=1.30\times10^3\,\text{Hz}\\[4pt] &=1.30\,\text{kHz}. \end{aligned}[/latex]

Step 2: Calculate the current at resonance.

At resonance, the impedance equals the resistance:

[latex]Z=R=40.0\,\Omega.[/latex]

Using the AC version of Ohm's law,

[latex]\begin{aligned} I_{\rm rms} &=\frac{V_{\rm rms}}{Z}\\[4pt] &=\frac{120\,\text{V}}{40.0\,\Omega}\\[4pt] &=3.00\,\text{A}. \end{aligned}[/latex]

Discussion

The resonant frequency of 1.30 kHz lies between the two frequencies examined in the previous example (60 Hz and 10.0 kHz). This is expected because the circuit changes from being primarily capacitive at low frequencies to primarily inductive at high frequencies. At the resonant frequency, these two effects exactly balance one another.

Because the inductive and capacitive reactances cancel, the impedance reaches its minimum possible value, equal to the resistance alone. Consequently, the current reaches its maximum value of 3.00 A, which is much larger than the currents calculated at either 60 Hz or 10.0 kHz.

This behavior explains why resonance is so useful in practical applications. A resonant circuit naturally responds much more strongly to one particular frequency than to others. Radio receivers use this property to select a single broadcast station, while MRI systems rely on resonance between radio-frequency electromagnetic waves and atomic nuclei to produce detailed medical images. In both cases, operating near the resonant frequency greatly improves the efficiency and selectivity of the system.

Power in Series RLC AC Circuits

In a purely resistive AC circuit, the average electrical power delivered to the resistor is simply the product of the rms voltage and rms current:

[latex]P=V_{\rm rms}I_{\rm rms}.[/latex]

In a series RLC circuit, however, the situation is more complicated because the voltage and current are generally not in phase. As we saw in the previous section, the resistor, inductor, and capacitor each affect the phase of the voltage differently. The resistor dissipates electrical energy as heat, whereas the inductor and capacitor temporarily store energy in magnetic and electric fields and then return that energy to the circuit.

Because of this continual exchange of stored energy, not all of the electrical energy supplied by the source is converted into useful work or heat. The result is that the average power delivered to the circuit is less than the simple product of voltage and current.

The difference between the voltage and current is described by the phase angle, denoted by [latex]\phi[/latex]. For a series RLC circuit, the phase angle satisfies

[latex]\cos\phi=\frac{R}{Z},[/latex]

where R is the resistance and Z is the impedance of the circuit.

At resonance, the impedance equals the resistance ([latex]Z=R[/latex]), so

[latex]\cos\phi=1,[/latex]

which means that

[latex]\phi=0^\circ.[/latex]

In this special case, the voltage and current are exactly in phase, and the circuit behaves like a purely resistive circuit.

For any other frequency, the average power delivered by the AC source is

[latex]P_{\rm ave}=V_{\rm rms}I_{\rm rms}\cos\phi.[/latex]

The quantity [latex]\cos\phi[/latex] is called the power factor. Because its value ranges from 0 to 1, it tells us what fraction of the apparent electrical power is actually converted into useful power.

  • Power factor = 1: All of the supplied power is converted into useful energy, as in a purely resistive circuit or an RLC circuit operating at resonance.
  • Power factor between 0 and 1: Some of the energy is continually stored and returned by the capacitor and inductor instead of being dissipated in the resistor.
  • Power factor = 0: The circuit is purely inductive or purely capacitive. Although current flows, no average power is consumed because energy simply oscillates back and forth between the source and the reactive component.

A high power factor is desirable in nearly all electrical systems because it means that more of the supplied electrical energy is converted into useful work. Electric motors, hospital imaging equipment, industrial machinery, and electrical power distribution systems are often designed with power-factor correction circuits that increase efficiency and reduce unnecessary current in the power lines.

Power Factor in AC Circuits

Only the resistor consumes electrical energy on average. Inductors and capacitors temporarily store energy and then return it to the circuit. The power factor, [latex]\cos\phi[/latex], measures how effectively electrical power is converted into useful work.

Example 57.3: Calculating the Power Factor and Average Power

An RLC series circuit contains a 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF capacitor connected to a 120 V (rms) AC source.

Calculate:

  1. the power factor and phase angle at 60.0 Hz,
  2. the average power delivered to the circuit at 60.0 Hz, and
  3. the average power delivered at the resonant frequency.

Strategy

Use the impedance obtained in Example 57.1 together with

[latex]\cos\phi=\frac{R}{Z}[/latex]

to determine the power factor and phase angle. Then calculate the average power using

[latex]P_{\rm ave}=V_{\rm rms}I_{\rm rms}\cos\phi.[/latex]

At resonance, the inductive and capacitive reactances cancel, so the power factor becomes 1 and the impedance equals the resistance.

Solution (a): Power Factor and Phase Angle

From Example 57.1, the impedance at 60.0 Hz is

[latex]Z=531\ \Omega.[/latex]

Therefore,

[latex]\cos\phi=\frac{R}{Z} =\frac{40.0\ \Omega}{531\ \Omega} =0.0753.[/latex]

The corresponding phase angle is

[latex]\phi=\cos^{-1}(0.0753)=85.7^\circ.[/latex]

Discussion (a)

The power factor is much less than 1, indicating that the voltage and current are nearly 90° out of phase. At 60 Hz the capacitor dominates the circuit, so relatively little of the electrical energy supplied by the source is converted into useful power.

Solution (b): Average Power at 60.0 Hz

From Example 57.1, the rms current at 60.0 Hz is

[latex]I_{\rm rms}=0.226\ \text{A}.[/latex]

Substituting into the average power equation gives

[latex]P_{\rm ave} =(120\ \text{V})(0.226\ \text{A})(0.0753) =2.04\ \text{W}.[/latex]

Discussion (b)

Although the source provides 120 V, only about 2 W of average power is delivered because both the current and the power factor are small. Most of the energy is temporarily stored in the capacitor's electric field and then returned to the source rather than being dissipated in the resistor.

Solution (c): Average Power at Resonance

At resonance,

[latex]\cos\phi=1 \qquad\text{and}\qquad Z=R=40.0\ \Omega.[/latex]

From Example 57.2,

[latex]I_{\rm rms}=3.00\ \text{A}.[/latex]

Therefore,

[latex]P_{\rm ave} =(120\ \text{V})(3.00\ \text{A})(1) =360\ \text{W}.[/latex]

Discussion

At resonance, the inductive and capacitive reactances exactly cancel, so the circuit behaves like a purely resistive circuit. The current reaches its maximum value, the power factor equals 1, and the source delivers the greatest possible average power. This is why resonance is so important in applications such as radio receivers, MRI systems, wireless power transfer, and many electronic filters, where maximum energy transfer at a specific frequency is desired.

In an ideal series RLC circuit, only the resistor dissipates electrical energy. The inductor and capacitor do not consume energy continuously. Instead, they temporarily store energy and return it to the circuit. The inductor stores energy in its magnetic field, while the capacitor stores energy in its electric field.

During each AC cycle, energy moves back and forth between these two components. When the current is large, much of the circuit's stored energy is in the magnetic field of the inductor. When the capacitor is fully charged and the current is momentarily zero, the stored energy is in the capacitor's electric field. The resistor gradually converts some of this electrical energy into thermal energy, and the voltage source replaces the energy that is lost.

This description assumes that very little energy leaves the circuit as electromagnetic radiation. In some systems, such as antennas, radiation is intentional. In the RLC circuits considered here, however, radiation is assumed to be negligible.

Mechanical Analogy for an RLC Circuit

The behavior of a driven RLC circuit can be compared with the motion of a car wheel traveling over a regularly corrugated road, as shown in Figure 57.4. The evenly spaced bumps act like the alternating-voltage source, repeatedly driving the wheel upward and downward.

The parts of the mechanical system correspond to the electrical components as follows:

  • The shock absorber is analogous to the resistor because it dissipates energy and limits the amplitude of the oscillation.
  • The mass of the wheel is analogous to the inductor because energy associated with motion corresponds to energy stored in the magnetic field.
  • The spring is analogous to the capacitor because stored elastic potential energy corresponds to energy stored in the capacitor's electric field.

If the wheel encounters the bumps at the mechanical system's natural frequency, its motion reaches a maximum amplitude. Likewise, the current in a series RLC circuit reaches a maximum when the applied AC frequency equals the circuit's resonant frequency.

A car wheel attached to a spring and shock absorber moves vertically as the vehicle travels over a regularly uneven road. The driven and damped motion is analogous to oscillations in a series RLC circuit.
Figure 57.4: The forced, damped motion of a car wheel and suspension is analogous to a series RLC circuit. The road provides the periodic driving force, the shock absorber dissipates energy like a resistor, and the mass and spring determine the natural frequency of the mechanical system.

Oscillations in an LC Circuit

A pure LC circuit contains an inductor and a capacitor but has negligible resistance. If energy is initially stored in either component, the circuit can oscillate at its natural frequency,

[latex]f_0=\frac{1}{2\pi\sqrt{LC}}.[/latex]

This is the same resonant frequency as that of a series RLC circuit with the same inductance and capacitance. Because there is very little resistance, only a small input of energy is needed to maintain the oscillation.

The LC circuit is analogous to a mass attached to an ideal spring with no friction. The mechanical energy continually changes between kinetic energy and elastic potential energy. Similarly, the electrical energy in an LC circuit moves back and forth between the magnetic field of the inductor and the electric field of the capacitor.

The stages of one electrical oscillation are illustrated in Figure 57.5:

  1. The capacitor is fully charged, the current is zero, and the energy is stored in the capacitor's electric field.
  2. The capacitor begins to discharge. The current increases, and energy is transferred into the magnetic field of the inductor.
  3. The capacitor becomes fully charged with the opposite polarity, the current again becomes zero, and the energy is once more stored in the electric field.
  4. The capacitor discharges in the opposite direction, reversing the current and rebuilding the magnetic field with the opposite orientation.

The process then repeats. In a real circuit, some resistance is always present, so the oscillations gradually decrease unless an external source continually replaces the lost energy.

Four stages compare an oscillating mass on a spring with an LC circuit. When the mass is momentarily at rest at either extreme, mechanical energy is stored in the spring, corresponding to electrical energy stored in the capacitor. When the mass moves fastest through equilibrium, mechanical energy is kinetic, corresponding to electrical energy stored in the inductor's magnetic field.
Figure 57.5: An LC circuit is analogous to a frictionless mass-spring system. Energy alternates between the capacitor's electric field and the inductor's magnetic field, just as mechanical energy alternates between elastic potential energy and kinetic energy.

Oscillating LC circuits are used as frequency references and timing elements in electronic devices. They also form the basis of tuned circuits that select particular frequencies in communication systems, biomedical instruments, and radio-frequency equipment. In practice, a small energy input is usually needed to compensate for unavoidable resistive losses.

Interactive Exploration: Circuit Construction Kit (AC+DC) Virtual Lab

Alternating current (AC) and direct current (DC) circuits power nearly every electrical device used in homes, hospitals, and laboratories. In this simulation, you can build circuits containing batteries, AC voltage sources, resistors, capacitors, inductors, switches, and light bulbs, while using virtual voltmeters and ammeters to measure electrical quantities.

As you experiment, compare how resistors, capacitors, and inductors respond to both DC and AC sources. Pay particular attention to how changing the frequency affects current, voltage, and the behavior of reactive components. These observations will help you understand the concepts of reactance, impedance, phase relationships, and resonance introduced in this chapter.

Figure 57.6. PhET Circuit Construction Kit (AC+DC) Virtual Lab, which allows users to build and analyze both AC and DC circuits using realistic electrical components and measurement tools.

Guided Exploration

Use the simulation to investigate how AC circuits differ from DC circuits by answering the following questions.

  1. Build a simple DC circuit consisting of a battery and a resistor. Measure the voltage across the resistor and the current through it. Do your measurements satisfy Ohm's law?
  2. Replace the battery with an AC voltage source while keeping the resistor. How does the current change throughout one cycle? Does the resistor behave differently under AC than under DC?
  3. Replace the resistor with a capacitor. Observe how the current changes as you increase the frequency of the AC source. Does the capacitor allow low-frequency or high-frequency signals to pass more easily?
  4. Replace the capacitor with an inductor. Repeat the experiment by changing the frequency. How does the current compare with the capacitive circuit?
  5. Build an RC circuit and then an RL circuit. Measure the voltage across each component. Can you observe evidence that the voltage and current are no longer perfectly in phase?
  6. Create a series RLC circuit. Experiment by changing the AC frequency. Is there a frequency at which the current reaches a maximum? How does this relate to the concept of resonance?
  7. Based on your observations, explain how resistance, inductive reactance, and capacitive reactance combine to determine the overall impedance of an AC circuit.

After completing the activity, compare your observations with the concepts presented in this chapter. Notice that resistors behave the same at all frequencies, while capacitors and inductors respond very differently as the frequency changes. These frequency-dependent properties allow engineers to design filters, tuning circuits, medical instruments, and communication systems that operate only within selected frequency ranges.

Section Summary

  • In an AC circuit containing resistors, inductors, and capacitors, the combined opposition to current is called the impedance, [latex]Z[/latex]. It is the AC equivalent of resistance and is related to voltage and current by the AC version of Ohm's law:
    [latex]I_0=\frac{V_0}{Z} \qquad\text{or}\qquad I_{\rm rms}=\frac{V_{\rm rms}}{Z},[/latex]

    where [latex]I_0[/latex] and [latex]V_0[/latex] are the peak current and voltage, and [latex]I_{\rm rms}[/latex] and [latex]V_{\rm rms}[/latex] are their rms values.

  • The impedance of a series RLC circuit is
    [latex]Z=\sqrt{R^2+\left(X_L-X_C\right)^2},[/latex]

    where [latex]R[/latex] is the resistance, [latex]X_L[/latex] is the inductive reactance, and [latex]X_C[/latex] is the capacitive reactance.

  • The circuit reaches resonance when the inductive and capacitive reactances are equal:
    [latex]X_L=X_C.[/latex]

    At resonance, the impedance is minimized ([latex]Z=R[/latex]) and the current reaches its maximum value.

  • The resonant frequency of a series RLC circuit is
    [latex]f_0=\frac{1}{2\pi\sqrt{LC}}.[/latex]
  • In an AC circuit, the voltage and current are generally not in phase. Their phase difference is described by the phase angle [latex]\phi[/latex], which satisfies
    [latex]\cos\phi=\frac{R}{Z}.[/latex]
  • For a purely resistive circuit, or for an RLC circuit operating at resonance, the phase angle is zero ([latex]\phi=0^\circ[/latex]), so the voltage and current are exactly in phase.
  • The average power delivered to a series RLC circuit is
    [latex]P_{\rm ave}=V_{\rm rms}I_{\rm rms}\cos\phi.[/latex]

    The quantity [latex]\cos\phi[/latex] is called the power factor. It ranges from 0 to 1 and indicates how effectively electrical power supplied by the source is converted into useful work or thermal energy.

Conceptual Questions

  1. Does the resonant frequency of a series RLC circuit depend on the amplitude (peak voltage) of the AC source? Explain your reasoning.
  2. An electric motor operates with a power factor significantly less than 1. Why is improving the motor's power factor generally a better strategy for increasing its efficiency than simply increasing the applied voltage?

Problems & Exercises

  1. An RL circuit consists of a [latex]40.0\ \Omega[/latex] resistor and a 3.00 mH inductor.
    1. Find its impedance [latex]Z[/latex] at 60.0 Hz and 10.0 kHz.
    2. Compare these values of [latex]Z[/latex] with those found in Example 57.1, in which the circuit also contained a capacitor.
  2. An RC circuit consists of a [latex]40.0\ \Omega[/latex] resistor and a [latex]5.00\ \mu\text{F}[/latex] capacitor.
    1. Find its impedance at 60.0 Hz and 10.0 kHz.
    2. Compare these values of [latex]Z[/latex] with those found in Example 57.1, in which the circuit also contained an inductor.
  3. An LC circuit consists of a 3.00 mH inductor and a [latex]5.00\ \mu\text{F}[/latex] capacitor.
    1. Find its impedance at 60.0 Hz and 10.0 kHz.
    2. Compare these values of [latex]Z[/latex] with those found in Example 57.1, in which the circuit also contained a resistor.
  4. What is the resonant frequency of a 0.500 mH inductor connected to a [latex]40.0\ \mu\text{F}[/latex] capacitor?
  5. To receive AM radio signals, an RLC circuit must be able to resonate at any frequency between 500 kHz and 1650 kHz. This is accomplished using a fixed [latex]1.00\ \mu\text{H}[/latex] inductor connected to a variable capacitor. What range of capacitance is required?
  6. Suppose you have inductors ranging from 1.00 nH to 10.0 H and capacitors ranging from 1.00 pF to 0.100 F. What range of resonant frequencies can be achieved by combining one inductor with one capacitor?
  7. What capacitance is required to produce a resonant frequency of 1.00 GHz when using an 8.00 nH inductor?
  8. What inductance is required to produce a resonant frequency of 60.0 Hz when using a [latex]2.00\ \mu\text{F}[/latex] capacitor?
  9. The lowest frequency in the FM radio band is 88.0 MHz.
    1. What inductance is needed to produce this resonant frequency when connected to a 2.50 pF capacitor?
    2. The capacitor is variable, allowing the resonant frequency to be increased to 108 MHz. What must the capacitance be at this frequency if the inductance remains unchanged?
  10. A series RLC circuit contains a [latex]2.50\ \Omega[/latex] resistor, a [latex]100\ \mu\text{H}[/latex] inductor, and an [latex]80.0\ \mu\text{F}[/latex] capacitor.
    1. Find the circuit's impedance at 120 Hz.
    2. Find the circuit's impedance at 5.00 kHz.
    3. If the source has an rms voltage of [latex]V_{\text{rms}}=5.60\ \text{V}[/latex], what is [latex]I_{\text{rms}}[/latex] at each frequency?
    4. What is the resonant frequency of the circuit?
    5. What is [latex]I_{\text{rms}}[/latex] at resonance?
  11. A series RLC circuit contains a [latex]1.00\ \text{k}\Omega[/latex] resistor, a [latex]150\ \mu\text{H}[/latex] inductor, and a 25.0 nF capacitor.
    1. Find the circuit's impedance at 500 Hz.
    2. Find the circuit's impedance at 7.50 kHz.
    3. If the source has an rms voltage of [latex]V_{\text{rms}}=408\ \text{V}[/latex], what is [latex]I_{\text{rms}}[/latex] at each frequency?
    4. What is the resonant frequency of the circuit?
    5. What is [latex]I_{\text{rms}}[/latex] at resonance?
  12. A series RLC circuit contains a [latex]2.50\ \Omega[/latex] resistor, a [latex]100\ \mu\text{H}[/latex] inductor, and an [latex]80.0\ \mu\text{F}[/latex] capacitor. Assume the source has an rms voltage of [latex]5.60\ \text{V}[/latex].
    1. Find the power factor at [latex]f=120\ \text{Hz}[/latex].
    2. What is the phase angle at 120 Hz?
    3. What is the average power at 120 Hz?
    4. Find the average power at the circuit's resonant frequency.
  13. A series RLC circuit contains a [latex]1.00\ \text{k}\Omega[/latex] resistor, a [latex]150\ \mu\text{H}[/latex] inductor, and a 25.0 nF capacitor. Assume the source has an rms voltage of [latex]408\ \text{V}[/latex].
    1. Find the power factor at [latex]f=7.50\ \text{kHz}[/latex].
    2. What is the phase angle at this frequency?
    3. What is the average power at this frequency?
    4. Find the average power at the circuit's resonant frequency.
  14. A series RLC circuit contains a [latex]200\ \Omega[/latex] resistor and a 25.0 mH inductor. At 8000 Hz, the magnitude of the phase angle is [latex]45.0^\circ[/latex].
    1. What is the impedance?
    2. Find the circuit's capacitance.
    3. If [latex]V_{\text{rms}}=408\ \text{V}[/latex] is applied, what is the average power supplied?
  15. Referring to Example 57.3, find the average power at 10.0 kHz.
  16. Critical Thinking. A 4.000 m length of wire is used to detect a magnetic field. The wire is formed into a single square loop and rotated at a rate of 400 cycles per second.
    1. If the magnetic field strength is 0.02000 T, what is the magnitude of the average emf generated during the first quarter-cycle, assuming the plane of the loop is initially perpendicular to the magnetic field?
    2. Suppose the wire is instead formed into two square loops and rotated at the same rate, starting with the same orientation as in part (a). Is the magnitude of the average emf different? If so, calculate the average emf generated during the first quarter-cycle.
    3. Suppose the wire is formed into a figure-eight shape, with the wire crossing itself, and rotated at 400 cycles per second in the same magnetic field. What average emf is generated during the first quarter-cycle if the two loops begin with the same orientation relative to the magnetic field?
    4. Based on your results, does the shape and arrangement of the loop matter? Explain.

Glossary

impedance
The total opposition that an AC circuit offers to the flow of current. Impedance combines resistance, inductive reactance, and capacitive reactance, and is calculated using
[latex]Z=\sqrt{R^2+(X_L-X_C)^2}.[/latex]
resonant frequency
The frequency at which the inductive and capacitive reactances are equal, producing the minimum impedance and the maximum current in a series RLC circuit. It is given by
[latex]f_0=\frac{1}{2\pi\sqrt{LC}}.[/latex]
phase angle
The angle, denoted by [latex]\phi[/latex], that describes the difference in phase between the voltage and the current in an AC circuit.
power factor
A measure of how efficiently electrical power is delivered to an AC circuit. The power factor is equal to
[latex]\cos\phi[/latex]
and ranges from 0 to 1. A power factor of 1 indicates that the voltage and current are in phase and power transfer is maximized.
definition

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Introductory Physics for the Health and Life Sciences II Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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