6. AC Analysis: Sinusoids, Phasors, and Impedance

Up to now we have assumed DC: voltages and currents that do not change with time. But most real circuits (radios, audio, motors, anything that interacts with a wall outlet) operate on AC (sinusoidal signals) or at least on signals that change with time.

The good news: with the right tricks, AC analysis becomes essentially the same as DC analysis, with complex numbers replacing real numbers. Once you internalize this, the entire field of frequency-domain electronics opens up.

6.1 The sinusoid

$v(t) = V_m \cos(\omega t + \phi)$

with peak amplitude $V_m$ (volts), angular frequency $\omega = 2\pi f$ (rad/s; 60 Hz means 377 rad/s), and phase $\phi$. Period $T = 1/f$ (60 Hz: 16.67 ms).

The RMS value of a sinusoid is $V_{RMS} = V_m/\sqrt{2}$, the equivalent DC voltage that delivers the same power into a resistor. "120 V mains" is RMS; peak is $\sqrt{2}$ times that, ~170 V.

6.2 Why AC is so common

Three reasons: (1) spinning generators naturally produce sinusoids; (2) transformers work only on AC, letting us step up for transmission and down for distribution; (3) sinusoids are the eigenfunctions of linear time-invariant systems, sinusoid in, sinusoid out at the same frequency, possibly with different amplitude and phase. This last fact is what we will exploit.

6.3 Phasors: the trick that makes AC analysis tractable

Direct calculation with $\cos(\omega t + \phi)$ and integrals is painful. Apply a sinusoidal voltage to an LC network and you end up solving differential equations like $L \, di/dt = v - i R$. Each new circuit, a new ODE. There must be a better way.

Deriving the phasor

Euler's formula bridges sines and exponentials: $e^{j\theta} = \cos\theta + j\sin\theta$, so $\cos\theta = \mathrm{Re}\{e^{j\theta}\}$. Hence

$v(t) = V_m \cos(\omega t + \phi) = \mathrm{Re}\{V_m e^{j\phi} \cdot e^{j\omega t}\}$

Two factors. $V_m e^{j\phi}$ holds the amplitude and phase. $e^{j\omega t}$ is the time-rotation common to every signal at frequency $\omega$. Define the phasor

$\bar{V} = V_m e^{j\phi} = V_m \angle\phi$

stripping the time-varying part. The crucial property: differentiation in time becomes multiplication by $j\omega$ in the phasor domain. If $v(t) = \mathrm{Re}\{\bar{V} e^{j\omega t}\}$, then $dv/dt$ has phasor $j\omega \bar{V}$. Similarly, integration becomes division by $j\omega$. Differential equations become algebraic.

From $v = L \, di/dt$ to $\bar{V} = j\omega L \bar{I}$

Apply the rule to an inductor: $v = L\,di/dt$ becomes $\bar{V} = j\omega L \, \bar{I}$. The inductor's impedance (ratio $\bar{V}/\bar{I}$) is $j\omega L$. For a capacitor, $i = C\,dv/dt$ becomes $\bar{I} = j\omega C \, \bar{V}$, so $\bar{V} = (1/j\omega C)\bar{I}$ and impedance is $1/(j\omega C) = -j/(\omega C)$. Resistors are unchanged: $\bar{V} = R\bar{I}$.

Summary:

• Resistor: $Z = R$ (real).
• Capacitor: $Z = 1/(j\omega C)$ (negative imaginary).
• Inductor: $Z = j\omega L$ (positive imaginary).

Every method of this chapter, KVL, KCL, mesh, nodal, Thevenin, Norton, generalizes directly to AC by replacing $R$ with $Z$ and using complex arithmetic. That is the magic.

Why impedance can be imaginary

Resistor impedance is real because voltage and current are in phase. Capacitor and inductor impedances are imaginary because their voltage and current are 90° out of phase. The factor of $j$ encodes that 90° shift compactly: $j\omega L$ says "voltage leads current by 90°, with magnitude $\omega L$."

6.4 Impedance and reactance

Let $Z = R + jX$ with $R$ the resistance and $X$ the reactance (positive for inductors, negative for capacitors). $|Z| = \sqrt{R^2 + X^2}$ measures AC opposition; $\angle Z = \arctan(X/R)$ tells whether voltage leads or lags current.

A capacitor's voltage lags current by 90° (water flows first, pressure builds after). An inductor's voltage leads current by 90°. Inductive reactance grows with frequency: 1 mH is 6.28 Ω at 1 kHz, 6.28 kΩ at 1 MHz. Capacitive reactance shrinks with frequency: 1 µF is 159 Ω at 1 kHz, 0.159 Ω at 1 MHz. Inductors block high frequencies; caps pass them.

6.5 Series and parallel impedances

Combine impedances like resistors, just with complex arithmetic:

$Z_{series} = Z_1 + Z_2, \qquad \frac{1}{Z_{parallel}} = \frac{1}{Z_1} + \frac{1}{Z_2}$

So a series RL circuit has total impedance $R + j\omega L$. A parallel RC has impedance $R \cdot (1/j\omega C)/(R + 1/j\omega C) = R/(1 + j\omega RC)$. These show up in basic filter design.

6.6 Worked phasor example: series RLC at a single frequency

Consider a series RLC driven by a 10 V (peak) source at 1 kHz, with $R = 100$ Ω, $L = 10$ mH, $C = 10$ µF. Then $\omega = 2\pi \cdot 1000 \approx 6283$ rad/s.

• $Z_R = 100$ Ω.
• $Z_L = j\omega L = j62.83$ Ω.
• $Z_C = 1/(j\omega C) = -j15.92$ Ω.
• $Z = 100 + j46.91$ Ω, $|Z| \approx 110.5$ Ω, $\angle Z \approx 25.1°$.

Current: $\bar{I} = \bar{V}/Z = 10\angle 0° / (110.5 \angle 25.1°) = 0.0905 \angle (-25.1°)$ A. Peak 90.5 mA, lagging the voltage by 25.1°. The lag makes sense: $Z_L$ dominates $Z_C$ here, so the circuit is net inductive.

Element voltages:

• $\bar{V}_R = 9.05 \angle (-25.1°)$ V.
• $\bar{V}_L = 5.69 \angle 64.9°$ V.
• $\bar{V}_C = 1.44 \angle (-115.1°)$ V.

Note $\bar{V}_L$ and $\bar{V}_C$ are 180° apart in phase, so they partially cancel. Adding rectangular components gives $\bar{V}_R + \bar{V}_L + \bar{V}_C = 10 \angle 0°$ V (the source), as KVL demands. Voltages across individual elements that exceed the source amplitude become common at resonance.

6.7 Resonance: when reactance cancels

In a series RLC circuit, the total impedance is $Z = R + j(\omega L - 1/\omega C)$. The reactive part vanishes when $\omega L = 1/\omega C$.

Derivation of resonant frequency

Set the imaginary part of $Z$ to zero:

$\omega L = \frac{1}{\omega C} \implies \omega^2 = \frac{1}{LC} \implies \omega_0 = \frac{1}{\sqrt{LC}}$

In hertz:

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

At resonance, the circuit looks purely resistive. The inductor's and capacitor's reactances cancel exactly. The current is maximized (just $V/R$). For frequencies far from $\omega_0$, the impedance is large (the reactive part dominates) and current is small.

This frequency-selective behavior is what makes radio possible: a tank circuit (LC in parallel) tuned to one station's carrier frequency has very low impedance at that frequency only, picking that station out of the air.

Q factor: how sharp is the peak?

The Q factor ("quality factor") measures how sharp the resonance is. For series RLC:

$Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$

A high-Q circuit has a sharp peak: only frequencies very close to $\omega_0$ get through. A low-Q circuit has a broad peak: a wider range of frequencies pass.

The intuitive interpretation: $Q$ is roughly the ratio of energy stored to energy dissipated per cycle. A high-Q circuit is one where the inductor and capacitor exchange energy back and forth efficiently, with very little loss to the resistor each cycle. The energy can ring around for many cycles before dying out.

Crystal oscillators reach $Q$ in the tens of thousands, which is why your watch keeps time to a second per day. The crystal's mechanical vibration is so lossless that energy stored in one cycle persists for tens of thousands of cycles before being dissipated.

Bandwidth and the BW = $f_0/Q$ relation

The bandwidth of a resonant circuit is the frequency range over which the response is within 3 dB of the peak (i.e., half-power, or magnitude-attenuation $1/\sqrt{2}$). For a high-Q resonator:

$BW = \frac{f_0}{Q}$

So a circuit with $f_0 = 1$ MHz and $Q = 100$ has a bandwidth of 10 kHz. A high $Q$ means narrow bandwidth (good for picking out a single radio station among many at nearby frequencies). A low $Q$ means wide bandwidth (good when you want a broad pass, like an FM radio's IF stage which must handle the carrier ± 75 kHz).

Voltage magnification at resonance

Here is something striking. At resonance in a series RLC, the current is $V_s/R$. The voltage across the inductor is $\omega_0 L \cdot V_s/R = Q \cdot V_s$. So if $Q = 100$, the voltage across the inductor at resonance is one hundred times the source voltage. This is real, measurable, and useful (or dangerous). High-voltage Tesla coils exploit this for spectacular effect.

Series vs parallel resonance

In a parallel RLC circuit (where R, L, C are all in parallel and a current source drives them), the impedance is maximum at resonance instead of minimum. This is the form used in radio tuning circuits. A small antenna current is forced into the parallel LC, and the LC develops a large voltage across it only at the resonant frequency. Off-resonance, the LC's impedance is small and the voltage stays small.

Series RLC:                Parallel RLC:
                                  
 +─[R]─[L]─[C]─+           +────●─────●─────●──+
 │             │           │    │     │     │  │
 V             V           I    R     L     C  V
 │             │           │    │     │     │  │
 +─────────────+           +────●─────●─────●──+
 
 Z is min @ f0             Z is max @ f0
 I peaks @ f0              V peaks @ f0
 used as bandpass          used as tank

Phasor diagrams at resonance. In series RLC, the inductor voltage (pointing up by 90° from current) and capacitor voltage (pointing down by 90° from current) cancel:

 V_L (up)        V_R    V_R (in phase with current)
   ↑              →     V_L + V_C cancel
   |
   ●─────────●─→ I
   |
   ↓
 V_C (down)

In parallel RLC, the inductor current and capacitor current cancel, leaving only the small resistor current:

 I_C (up)        I_R     V across the parallel combo
   ↑              →      
   |
   ●─────────●─→ V
   |
   ↓
 I_L (down)

Real-world resonance

• AM/FM radio tuning. A variable capacitor (or varactor in modern radios) sweeps $f_0$ across the broadcast band.
• Crystal filters in superheterodyne receivers. Crystal $Q$ in the thousands makes for very sharp filters; SSB radios use them to pick out a 2.5 kHz sideband from a crowded ham band.
• MRI machines drive a tuned coil at the proton precession frequency (around 64 MHz for 1.5 T magnets) and detect the response.
• RFID tags. A passive tag's tank tuned to the reader's carrier (13.56 MHz) rectifies the field to a DC supply, then modulates its loading to send data back.
• Power-line filters use parallel LC traps to reject specific harmonics.

6.8 Real-world AC examples

• Wall power. 120 V (US) or 230 V (Europe) at 60 or 50 Hz. RMS-quoted; peak is $\sqrt{2}$ times higher.
• AC adapters. Transformer down, rectifier, filter cap, regulator. Clean DC out.
• AM radio. ~1 MHz carrier; tank selects the station. Without resonance, you'd hear all stations at once.
• Switch-mode power supplies. Hundreds of kHz: the inductor's $\omega L$ at that frequency stores and releases energy efficiently.
• Brushless DC motors. Internally three-phase AC even though input is DC; the ESC generates the AC waveform.