Chapter 9 — The Impedance Code: AC Circuit Analysis

This chapter unifies the concepts of signals, circuits, and complex numbers to analyze circuits driven by sinusoidal sources. Phasors transform time-varying signals into fixed complex vectors, replacing calculus with algebra. Impedance extends Ohm's Law to capacitors and inductors, whose opposition to current depends on frequency. The same series, parallel, and voltage-divider techniques from DC analysis carry over directly once impedances replace resistances. The decibel scale and Bode plots provide practical tools for characterizing how circuits respond across a wide range of frequencies. Passive RC, RL, and RLC circuits then demonstrate how frequency-selective filtering arises naturally from impedance.

Learning Objectives:

• Extend DC circuit analysis techniques to AC circuits using the phasor concept.
• Explain how the frequency of the source affects component behavior in AC circuits.
• Transform time-varying sinusoidal signals into complex phasors ($A_{o}\cos(\omega t+\phi) \Rightarrow A = A_{o}e^{j\phi}$).
• Define impedance ($Z$) as the AC generalization of resistance and recognize its complex nature ($Z = R + jX$).
• Describe how capacitive reactance ($X_{C} = -1/(2\pi fC)$) and inductive reactance ($X_{L} = 2\pi fL$) vary with frequency.
• Apply the decibel scale and interpret Bode plots of magnitude and phase.
• Analyze passive RC, RL, and RLC filters using impedance, phasors, and the voltage-divider relationship.

The earlier chapters established two parallel threads. One thread covers circuit analysis: Kirchhoff's laws, resistor networks, and the transient responses of capacitors and inductors. The other covers signals: sinusoidal waveforms, the Fourier series, and the frequency spectrum. The previous chapter introduced complex numbers and phasors as the mathematical bridge between these two threads. This chapter completes the crossing.

In AC analysis, circuits are driven by harmonic signals of the form $A_{0}\cos(\omega t + \phi)$. Unlike DC circuits, where voltages and currents are constant, in AC circuits the frequency of the source determines how each component behaves. The central goal is to determine how a circuit modifies the amplitude and phase of a signal passing through it.

Phasors accomplish this by converting time-varying signals into fixed complex vectors in the complex plane:

$$A_{o}\cos(\omega t + \phi) \;\;\Longrightarrow\;\; A = A_{o}\,e^{\,j\phi}$$

This transformation replaces time-domain differential equations with algebraic equations. The voltage-current relationships for each component become:

$$v = iR \;\;\Longrightarrow\;\; V = IR$$

$$i = C\,\frac{dv}{dt} \;\;\Longrightarrow\;\; I = j\omega C\,V$$

$$v = L\,\frac{di}{dt} \;\;\Longrightarrow\;\; V = j\omega L\,I$$

The imaginary unit $j$ in the capacitor and inductor relationships encodes a 90-degree phase shift between voltage and current. For a capacitor, the $-j$ that results from rearranging the capacitor equation indicates that voltage lags current by 90 degrees: a capacitor stores charge proportional to voltage, so a rapidly changing voltage demands a large current before the voltage itself rises. For an inductor, the $+j$ indicates that voltage leads current by 90 degrees: the magnetic field storing energy must build up before current can increase, so the voltage peaks ahead of the current.

These relationships are the foundation for the rest of this chapter.

Section 1 — Impedance: Ohm's Law for AC Circuits

In the phasor domain, the voltage across every component is proportional to the current through it. This observation motivates a generalized form of Ohm's Law valid for AC circuits:

$$V = IZ$$

The quantity $Z$ is called impedance. It plays the same role in AC circuits that resistance plays in DC circuits: it quantifies how strongly a component opposes the flow of current. The key distinction is that impedance is a complex number:

$$Z = R + jX$$

The real part $R$ is the resistance, familiar from DC analysis. Resistance results from electron collisions within a material; it dissipates energy as heat and is independent of frequency. The imaginary part $X$ is the reactance. Reactance arises from energy storage in electric or magnetic fields. Unlike resistance, reactance depends on frequency and contributes no heat loss.

The impedance of the three passive components follows directly from the phasor voltage-current relationships:

$$Z_R = R, \qquad Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C}, \qquad Z_L = j\omega L$$

Figure: Impedance as a complex number in the complex plane. The real part $R$ is the resistance and the imaginary part $X$ is the reactance. The polar-form equivalents are the magnitude $|Z|$ and phase angle $\phi$.

The complex nature of impedance explains why voltages and currents in AC circuits can be out of phase with each other, a phenomenon that is impossible in purely resistive circuits. The magnitude $|Z|$ determines how much the circuit reduces the amplitude of a signal; the phase angle $\phi$ determines how much the circuit shifts it in time.

All DC analysis techniques, including Kirchhoff's laws, series and parallel combinations, voltage dividers, and node analysis, apply unchanged to AC circuits provided that impedances replace resistances and all voltages and currents are treated as complex phasors.

Note: Impedance is the AC extension of resistance. Every technique learned for DC resistive circuits carries over to AC analysis when impedances replace resistances and all quantities are expressed as complex numbers.

Section 2 — Frequency Dependence of Impedance

The impedances of capacitors and inductors depend on frequency, and their behaviors are complementary opposites. This section examines each component in turn.

Capacitive Reactance

The impedance of a capacitor is:

$$Z_C = \frac{1}{j\omega C}$$

with magnitude:

$$|Z_C| = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$

Figure: Capacitor impedance magnitude versus frequency. At low frequencies the capacitor approaches an open circuit; at high frequencies it approaches a short circuit.

At low frequencies, $|Z_C|$ is large: the capacitor strongly resists current flow and behaves like an open circuit. At the limiting case of direct current ($f = 0$), the impedance is infinite and the capacitor blocks current entirely after an initial transient. At high frequencies, $|Z_C|$ is small: the capacitor offers little opposition and behaves like a short circuit.

The capacitive reactance, the imaginary part of $Z_C$, is:

$$X_C = -\frac{1}{2\pi f C}$$

The negative sign encodes the phase relationship: the current through a capacitor leads the voltage across it by 90 degrees. The capacitor must accept charge before a voltage develops across it, so the current peaks before the voltage does.

Example: Capacitor Impedance at Two Frequencies

A capacitor with $C = 1\,\mu\text{F}$ has the following impedance magnitudes:

At $f = 100\,\text{Hz}$:
$$|Z_C| = \frac{1}{2\pi \times 100 \times 10^{-6}} \approx 1{,}592\,\Omega$$

At $f = 10\,\text{kHz}$:
$$|Z_C| = \frac{1}{2\pi \times 10{,}000 \times 10^{-6}} \approx 16\,\Omega$$

A hundredfold increase in frequency reduces the impedance by the same factor, confirming the inverse relationship.

Inductive Reactance

Inductors behave in the opposite manner to capacitors. The impedance of an inductor is:

$$Z_L = j\omega L$$

with magnitude:

$$|Z_L| = \omega L = 2\pi f L$$

Figure: Inductor impedance magnitude versus frequency. At low frequencies the inductor approaches a short circuit; at high frequencies it approaches an open circuit.

At low frequencies, $|Z_L|$ is small: the inductor offers little opposition and behaves like a short circuit. For direct current ($f = 0$), an ideal inductor is simply a wire. At high frequencies, $|Z_L|$ is large: the inductor strongly resists changes in current and behaves like an open circuit.

The inductive reactance is:

$$X_L = 2\pi f L$$

The positive sign gives the phase relationship: the current through an inductor lags the voltage across it by 90 degrees. The magnetic field must be established before the current can rise, so the voltage peaks ahead of the current.

Example: Inductor Impedance at Two Frequencies

An inductor with $L = 10\,\text{mH}$ has the following impedance magnitudes:

At $f = 100\,\text{Hz}$:
$$|Z_L| = 2\pi \times 100 \times 10 \times 10^{-3} \approx 6.3\,\Omega$$

At $f = 10\,\text{kHz}$:
$$|Z_L| = 2\pi \times 10{,}000 \times 10 \times 10^{-3} \approx 628\,\Omega$$

A hundredfold increase in frequency increases the impedance by the same factor, confirming the direct proportionality.

Caution: Real inductors and capacitors always carry some resistance. Real inductors have winding resistance; real capacitors have leakage paths. These resistive contributions affect behavior at frequency extremes and will be apparent in laboratory measurements.

Section 3 — Series and Parallel Impedances

The combination rules for resistors extend directly to impedances. The only change is that the arithmetic now involves complex numbers.

Series Connections

Impedances in series carry the same current. Their total impedance is the sum of the individual impedances:

$$Z_{\text{total}} = Z_1 + Z_2 + \cdots$$

Figure: Series connection of impedances. The same phasor current flows through every element.

Because the sum involves complex numbers, the total impedance can have both a real and an imaginary part even when the individual components are purely resistive or purely reactive. For a resistor $R$ in series with a capacitor:

$$Z_{\text{total}} = R + \frac{1}{j\omega C} = R - \frac{j}{\omega C}$$

Example: Series RC Circuit

In a series circuit with $R = 100\,\Omega$ and $C = 1\,\mu\text{F}$ at $f = 1\,\text{kHz}$:

$$Z_C \approx -j159\,\Omega$$
$$Z_{\text{total}} = 100 - j159\,\Omega$$
$$|Z_{\text{total}}| = \sqrt{100^{2} + 159^{2}} \approx 188\,\Omega$$
$$\phi = \tan^{-1}\!\left(\frac{-159}{100}\right) \approx -58^{\circ}$$

The negative phase angle confirms that voltage lags current by 58 degrees in this RC combination at this frequency.

Parallel Connections

Impedances in parallel share the same voltage. Their total impedance follows the reciprocal rule:

$$\frac{1}{Z_{\text{total}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots$$

Figure: Parallel connection of impedances. The same phasor voltage appears across each element while the current divides among the branches.

Working with admittance $Y = 1/Z$ (measured in siemens, S) often simplifies parallel calculations, because admittances add directly:

$$Y_{\text{total}} = Y_1 + Y_2 + \cdots$$

Example: Parallel RL Circuit

Consider $R = 200\,\Omega$ in parallel with $L = 50\,\text{mH}$ at $f = 1\,\text{kHz}$:

$$Z_L = j \cdot 2\pi \times 1000 \times 50 \times 10^{-3} = j314\,\Omega$$
$$Y_R = \frac{1}{200} = 5.00 \times 10^{-3}\,\text{S}$$
$$Y_L = \frac{1}{j314} = -j3.18 \times 10^{-3}\,\text{S}$$
$$Y_{\text{total}} = (5.00 - j3.18) \times 10^{-3}\,\text{S}$$

Converting back to impedance:
$$Z_{\text{total}} = \frac{1}{Y_{\text{total}}} = \frac{(5.00 + j3.18)\times 10^{-3}}{(5.00)^{2} + (3.18)^{2}} \times 10^{6} \approx 138 + j88\,\Omega$$

Note: For the special case of two impedances in parallel, the product-over-sum formula applies:
$$Z_{\text{total}} = \frac{Z_1 Z_2}{Z_1 + Z_2}$$
The arithmetic involves complex numbers, but the formula is structurally identical to its DC counterpart.

Section 4 — Impedance Voltage Divider

The voltage divider configuration carries over from DC to AC with no change in form. For a series connection of two impedances driven by a source $V_{\text{in}}$:

$$V_{\text{out}} = V_{\text{in}} \times \frac{Z_2}{Z_1 + Z_2}$$

Figure: Impedance voltage divider. The output voltage $\mathbf{V}_{out}$ is taken across $Z_2$.

The derivation is identical to the DC case: the same current $I$ flows through both $Z_1$ and $Z_2$; the voltage across $Z_2$ is $V_B = I Z_2$; and the input voltage is $V_{\text{in}} = I(Z_1 + Z_2)$. Dividing gives the voltage divider formula.

The critical difference from the DC case is that $Z_1$ and $Z_2$ are complex and frequency-dependent, so the output voltage amplitude and phase both vary with frequency. This frequency-selective behavior is the foundation of filtering.

Example: RC Voltage Divider Across Frequencies

A voltage divider has $Z_1 = R = 1\,\text{k}\Omega$ and $Z_2 = Z_C = 1/(j\omega C)$ with $C = 1\,\mu\text{F}$. With $V_{\text{in}} = 10\,\text{V}$:

At $f = 100\,\text{Hz}$ (where $Z_C \approx -j1{,}592\,\Omega$):
$$V_{\text{out}} = 10 \times \frac{-j1592}{1000 - j1592} \approx 8.45\angle{-58^{\circ}}\,\text{V}$$

At $f = 10\,\text{kHz}$ (where $Z_C \approx -j15.9\,\Omega$):
$$V_{\text{out}} \approx 0.16\angle{-89^{\circ}}\,\text{V}$$

At low frequencies, the capacitor presents a large impedance and captures most of the source voltage. At high frequencies, the capacitor presents a small impedance and the output drops to nearly zero. The same circuit therefore attenuates high frequencies while passing low ones.

Note: The impedance voltage divider formula is structurally identical to the resistive version from DC analysis. The result is now a complex number, meaning the output voltage may have a different phase than the input. A purely resistive divider cannot produce a phase shift; a reactive divider always does.

Section 5 — The Decibel Scale and Bode Plots

When examining how a circuit responds across a range of frequencies, the ratio of output to input voltage, the gain, can span many orders of magnitude. For a typical filter, the gain might vary from nearly 1 in the passband to $10^{-4}$ or less deep in the stopband. Plotting such a range on a linear scale makes it nearly impossible to see the behavior near the cutoff frequency. Two tools address this problem: the decibel scale and the Bode plot.

The Decibel

The decibel (dB) expresses a ratio on a logarithmic scale. For voltage amplitudes:

$$G_{\text{dB}} = 20\log_{10}\!\left(\frac{|V_{\text{out}}|}{|V_{\text{in}}|}\right)$$

For power ratios the corresponding formula uses a factor of 10:

$$G_{\text{dB}} = 10\log_{10}\!\left(\frac{P_{\text{out}}}{P_{\text{in}}}\right)$$

The factor of 20 in the voltage form arises because power is proportional to the square of voltage: $P \propto V^2$, so $10\log_{10}(V^2) = 20\log_{10}(V)$. The two definitions are consistent for circuits where input and output impedances are equal.

The table below lists the key reference points that appear repeatedly in filter analysis.

The $-3\,\text{dB}$ point is particularly important: it is the frequency at which signal power has dropped to half its maximum value, corresponding to a voltage ratio of $1/\sqrt{2} \approx 0.707$. This point defines the cutoff frequency of a filter and is the standard convention used in all filter specifications.

Bode Plots

A Bode plot displays the frequency response of a circuit in two panels, both plotted against a logarithmic frequency axis:

1. Magnitude plot: gain in dB versus $\log_{10}(f)$.
2. Phase plot: phase angle in degrees versus $\log_{10}(f)$.

A logarithmic frequency axis is essential because filters often operate over many decades of frequency. A linear axis from 1 Hz to 100 kHz would compress everything below 10 kHz into less than 10% of the plot width.

Figure: Bode plot sketch for a first-order low-pass filter. Above the cutoff frequency $f_c$, the magnitude decreases at $-20\,\text{dB}$ per decade and the phase approaches $-90^{\circ}$.

The slope of the magnitude response beyond cutoff is characteristic of the filter order. A first-order filter rolls off at $-20\,\text{dB}$ per decade: for every tenfold increase in $f$ beyond $f_c$, the gain drops by 20 dB. Equivalently, the gain drops $-6\,\text{dB}$ per octave (every doubling of frequency). Second-order filters roll off at $-40\,\text{dB}$ per decade.

The phase response of a first-order low-pass filter starts near $0^{\circ}$ well below cutoff, reaches exactly $-45^{\circ}$ at $f_c$, and approaches $-90^{\circ}$ well above cutoff. This $90^{\circ}$ total phase shift is a fundamental property of first-order systems.

Section 6 — Passive Filters

A filter is a circuit that selectively passes or attenuates signals based on their frequency. Passive filters use only resistors, capacitors, and inductors; they require no external power supply and introduce no energy gain. The impedance voltage divider is the fundamental building block of first-order passive filters.

Filter Types

The four standard filter types are classified by the frequencies they pass: a low-pass filter passes low frequencies and attenuates high frequencies; a high-pass filter passes high frequencies and attenuates low frequencies; a band-pass filter passes a defined frequency band while attenuating frequencies outside it; and a band-stop filter attenuates a defined band while passing frequencies on either side. This chapter focuses on low-pass and high-pass first-order filters, which are the building blocks for more complex designs.

Applications for these filter types span audio systems (bass and treble controls, crossovers), radio receivers (station selection), power supplies (ripple removal), sensor systems (noise reduction), and communication systems (channel selection).

Figure: Idealized frequency response of low-pass and high-pass filters. In practice the transition from passband to stopband is gradual, with the cutoff frequency $f_c$ defined as the $-3\,\text{dB}$ point.

Transfer Functions

The transfer function $H(j\omega)$ relates the phasor output to the phasor input across all frequencies:

$$H(j\omega) = \frac{V_{\text{out}}(j\omega)}{V_{\text{in}}(j\omega)}$$

For a first-order low-pass filter with time constant $\tau$:

$$H_{\text{LP}}(j\omega) = \frac{1}{1 + j\omega\tau}$$

For a first-order high-pass filter:

$$H_{\text{HP}}(j\omega) = \frac{j\omega\tau}{1 + j\omega\tau}$$

In both cases the time constant $\tau$ is related to the cutoff frequency by:

$$f_c = \frac{1}{2\pi\tau}$$

At $f = f_c$, the magnitude of both transfer functions equals $1/\sqrt{2}$, corresponding to $-3\,\text{dB}$.

Low-Pass RC Filter

The simplest first-order low-pass filter places a resistor in series with a capacitor and takes the output across the capacitor:

Figure: First-order low-pass RC filter. The output is taken across the capacitor.

The circuit is an impedance voltage divider with $Z_1 = R$ and $Z_2 = 1/(j\omega C)$:

$$V_{\text{out}} = V_{\text{in}} \times \frac{1/(j\omega C)}{R + 1/(j\omega C)} = V_{\text{in}} \times \frac{1}{1 + j\omega RC}$$

This matches the low-pass transfer function with $\tau = RC$, giving:

$$f_c = \frac{1}{2\pi RC}$$

At low frequencies, $|Z_C|$ is large, so most of the source voltage appears across the capacitor. At high frequencies, $|Z_C|$ is small and most of the voltage drops across $R$, leaving little at the output.

Figure: Bode plot of a first-order low-pass RC filter with $f_c = 159\,\text{Hz}$. The blue curve is the magnitude response (left axis) and the red curve is the phase response (right axis). The magnitude falls through $-3\,\text{dB}$ at $f_c$ and rolls off at $-20\,\text{dB}$/decade beyond.

Example: Designing a Low-Pass RC Filter

To design a low-pass filter with $f_c = 1\,\text{kHz}$:

1. Choose $C = 0.1\,\mu\text{F}$ (a convenient standard value).
2. Calculate $R$:
   $$R = \frac{1}{2\pi f_c C} = \frac{1}{2\pi \times 1000 \times 0.1 \times 10^{-6}} \approx 1.59\,\text{k}\Omega$$
3. Select the nearest standard resistor value: $1.6\,\text{k}\Omega$.

High-Pass RC Filter

Swapping the positions of $R$ and $C$ produces a high-pass filter:

Figure: First-order high-pass RC filter. The output is taken across the resistor.

With $Z_1 = 1/(j\omega C)$ and $Z_2 = R$:

$$V_{\text{out}} = V_{\text{in}} \times \frac{R}{1/(j\omega C) + R} = V_{\text{in}} \times \frac{j\omega RC}{1 + j\omega RC}$$

This matches the high-pass transfer function with $\tau = RC$. The cutoff frequency is the same as for the low-pass RC filter: $f_c = \frac{1}{2\pi RC}$.

At low frequencies, $|Z_C|$ dominates and blocks signal from reaching the output. At high frequencies, the capacitor approximates a short circuit and the output approaches the input. A notable consequence: because a capacitor blocks DC, a high-pass filter removes any constant (DC) offset from a signal.

Figure: Bode plot of a first-order high-pass RC filter with $f_c = 159\,\text{Hz}$. The magnitude rises through $-3\,\text{dB}$ at $f_c$ and approaches $0\,\text{dB}$ above the cutoff frequency.

RL Filters

Inductors can replace capacitors to build first-order filters. Because inductors are bulkier and more expensive than capacitors at audio frequencies, RC filters are more common in most applications. RL filters do appear in power electronics and radio-frequency circuits, where inductors are already present for other reasons.

Low-Pass RL Filter

Figure: First-order low-pass RL filter. The output is taken across the resistor.

With $Z_1 = j\omega L$ and $Z_2 = R$:

$$V_{\text{out}} = V_{\text{in}} \times \frac{R}{R + j\omega L} = V_{\text{in}} \times \frac{1}{1 + j\omega L/R}$$

The time constant is $\tau = L/R$ and the cutoff frequency is:

$$f_c = \frac{R}{2\pi L}$$

High-Pass RL Filter

Figure: First-order high-pass RL filter. The output is taken across the inductor.

With $Z_1 = R$ and $Z_2 = j\omega L$:

$$V_{\text{out}} = V_{\text{in}} \times \frac{j\omega L}{R + j\omega L} = V_{\text{in}} \times \frac{j\omega L/R}{1 + j\omega L/R}$$

The cutoff frequency is again $f_c = R/(2\pi L)$, identical to the low-pass RL filter.

Note: The cutoff frequency formulas for all first-order filters share the same form when expressed in terms of the time constant: $f_c = 1/(2\pi\tau)$, where $\tau = RC$ for RC circuits and $\tau = L/R$ for RL circuits. The same time constant that governs the transient step response also determines the cutoff frequency in AC analysis.

Section 7 — RLC Circuits: Resonance

When a resistor, inductor, and capacitor appear together in the same circuit, the opposing frequency behaviors of $Z_L$ and $Z_C$ interact to produce resonance: a frequency at which the inductive and capacitive reactances cancel exactly, leaving only resistance. This phenomenon underlies bandpass and band-stop filters, oscillators, and tuned amplifiers.

Series RLC Circuit

Figure: Series RLC circuit.

The total impedance is:

$$Z_{\text{RLC}} = R + j\omega L + \frac{1}{j\omega C} = R + j\!\left(\omega L - \frac{1}{\omega C}\right)$$

The imaginary part is the difference between inductive and capacitive reactance. At frequencies below resonance, $1/(\omega C)$ exceeds $\omega L$ and the circuit is net capacitive. At frequencies above resonance, $\omega L$ exceeds $1/(\omega C)$ and the circuit is net inductive. At resonance the two reactive terms cancel.

Figure: Impedance magnitude of a series RLC circuit versus frequency. The minimum occurs at the resonant frequency $f_0$, where the impedance equals $R$.

The Resonant Frequency

Resonance occurs when the imaginary part of $Z_{\text{RLC}}$ equals zero:

$$\omega_0 L = \frac{1}{\omega_0 C}$$

Solving for $\omega_0$ and converting to hertz:

$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$

At resonance, impedance is minimum and purely resistive ($Z = R$), current is maximum, and voltage and current are in phase.

Example: Calculating the Resonant Frequency

For $L = 10\,\text{mH}$ and $C = 100\,\text{nF}$:
$$f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{10\times10^{-3}\times100\times10^{-9}}} = \frac{1}{2\pi\times31.6\times10^{-6}} \approx 5.03\,\text{kHz}$$

Parallel RLC Circuit

Figure: Parallel RLC circuit. The same phasor voltage appears across all three components.

Working with admittance is more convenient for the parallel case:

$$Y_{\text{RLC}} = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C = \frac{1}{R} + j\!\left(\omega C - \frac{1}{\omega L}\right)$$

The behavior is the dual of the series circuit: at resonance ($\omega_0 = 1/\sqrt{LC}$) the admittances of the inductor and capacitor branches cancel, and the total admittance reduces to $1/R$, which corresponds to the maximum impedance. Away from resonance in either direction, the impedance falls.

Bandpass Filter

A parallel RLC combination used as the shunt element in a voltage divider produces bandpass behavior. Near resonance the shunt impedance is large, so most of the input voltage appears at the output. Well above or below resonance the shunt impedance is small, and the output is attenuated.

Figure: Bandpass filter using a parallel RLC network as the shunt element. The $100\,\Omega$ series resistor and the parallel RLC combination form a voltage divider whose shunt impedance peaks at the resonant frequency.

Figure: Voltage waveform of the parallel RLC circuit at resonance.

Figure: Current waveform of the parallel RLC circuit at resonance. At the resonant frequency the impedance is purely resistive, so voltage and current are in phase.

Example: Radio Tuning with an RLC Circuit

Early radio receivers used a variable capacitor in parallel with a fixed inductor to select broadcast stations. Rotating the tuning knob changed the capacitance, shifting the resonant frequency to match the carrier frequency of the desired station while attenuating all others.

For FM radio (88–108 MHz) with $L = 1\,\mu\text{H}$, the required capacitance range is:
$$C_{\min} = \frac{1}{(2\pi f_{\max})^2 L} = \frac{1}{(2\pi \times 108 \times 10^6)^2 \times 10^{-6}} \approx 2.2\,\text{pF}$$
$$C_{\max} = \frac{1}{(2\pi f_{\min})^2 L} = \frac{1}{(2\pi \times 88 \times 10^6)^2 \times 10^{-6}} \approx 3.3\,\text{pF}$$

Note: In later courses you will encounter additional characterizations of RLC circuits: bandwidth, quality factor ($Q$), and the relationship between damping and the sharpness of the resonance peak. The resonant frequency and the series/parallel impedance behaviors developed here provide the foundation for those topics.

Chapter Summary

This chapter developed impedance as the unifying tool for AC circuit analysis. Once every passive component is characterised by its impedance, all DC analysis techniques carry over directly: Kirchhoff's laws, series and parallel combinations, and the voltage divider all apply with complex arithmetic replacing real arithmetic.

The key physical insight is that reactive impedances depend on frequency. A capacitor's impedance falls as frequency rises; an inductor's rises. This complementary behavior is the basis for passive filters, which use a reactive component in an impedance voltage divider to produce a frequency-dependent output. First-order RC and RL filters roll off at $-20\,\text{dB}$ per decade beyond the cutoff frequency. When a capacitor and inductor are combined in an RLC circuit, their reactances cancel at the resonant frequency, producing the sharp frequency selectivity used in bandpass filters and tuned circuits.

Key Formulas — Chapter 9