AC Circuits and Phasors

Every wall socket on Earth delivers a voltage that reverses direction a hundred or a hundred and twenty times a second. Nothing about the circuits you have mastered so far breaks under that reversal — Kirchhoff's rules still hold at every instant — but solving them the honest way now means solving a differential equation for every loop, because capacitors and inductors respond to change, and in an AC circuit everything changes all the time. This article is about the trick that tamed that problem so completely that power engineers, audio designers, and radio builders almost never write a differential equation at all. The trick is a change of arithmetic: represent the sinusoidal voltages and currents as complex numbers — phasors — and every derivative turns into a multiplication. Each element then carries a single complex number, its impedance: $R$ for a resistor, $j\omega L$ for an inductor, $1/j\omega C$ for a capacitor. Ohm's law returns as $V = IZ$ , and the entire reduction toolkit you built for resistor networks — series, parallel, dividers, equivalent sources — transfers wholesale. You arrive prepared. The LC and RLC article gave you the natural frequency and the dictionary to mechanics; RC and RL gave you each element's stubbornness in the time domain. What you gain here is the frequency domain: the point of view in which a circuit is not a thing that evolves but a filter that responds, frequency by frequency. That point of view powers the next article's resonance story, the transformer's power engineering, and every equalizer, crossover, and tuned receiver ever built.

Key takeaways

In sinusoidal steady state, every voltage and current in a linear circuit oscillates at the source's frequency $\omega$ — only amplitudes and phases differ. That is why one complex number per signal suffices.
Complex exponentials are eigenfunctions of $d/dt$ : differentiation becomes multiplication by $j\omega$ , so calculus becomes algebra.
Each element gets an impedance: $Z_R = R$ , $Z_L = j\omega L$ , $Z_C = 1/j\omega C$ . Inductor voltage leads current by 90°; capacitor voltage lags by 90° — "ELI the ICE man."
Impedances combine exactly like resistances (series add, parallel add reciprocals), so the whole DC network toolkit transfers.
A series RLC has $Z = R + j(\omega L - 1/\omega C)$ : inductive above resonance, capacitive below, purely resistive at $\omega_0$ .
Peak values and rms values differ by $\sqrt 2$ ; phasor lengths must be read in one convention consistently.
KEY IDEA: represent sinusoids as complex exponentials and every linear AC circuit becomes a resistor network with complex resistances — one algebra problem per frequency.

The core insight

Because $e^{j\omega t}$ turns differentiation into multiplication by $j\omega$ , a driven linear circuit in sinusoidal steady state obeys $V = IZ$ with $Z_R = R$ , $Z_L = j\omega L$ , $Z_C = 1/j\omega C$ — AC circuit analysis is resistor-network algebra with complex numbers carrying the phases.

Where this lives today

The purest industrial descendant of this article is impedance spectroscopy: sweep $\omega$ , measure $Z(\omega)$ , and read the inner life of a device you cannot open. Battery-management systems do it in situ — a lithium cell's impedance spectrum separates electrolyte resistance, interface chemistry, and diffusion into distinct frequency bands, so a few milliohms of drift at the right frequency flags an aging cell long before capacity visibly fades. The same measurement runs in biology (cell counters and wearables that estimate body composition are impedance meters) and in materials labs. On the power grid, the phasor became an instrument: phasor measurement units (PMUs) sample voltage and current waveforms, GPS-timestamp them to better than a microsecond, and stream live amplitude-and-phase data so that operators watch the continental grid as a single synchronized phasor diagram — phase drift between regions is the earliest warning of instability. And in every audio crossover, anti-aliasing stage, and radio front end, the humble $RC$ corner of Derivation C is still being placed, one $1/2\pi RC$ at a time.

The physical picture

Drive the same sinusoidal current through the three basic elements and watch the voltage each one develops. The resistor is the conformist: its voltage is just $iR$ , rising and falling in perfect step with the current. The inductor is the prophet: it responds to $di/dt$ , which is largest when the current crosses zero on the way up — so its voltage peaks a quarter cycle before the current does. The capacitor is the historian: its voltage is the accumulated charge, largest only after the current has been pouring in for a quarter cycle — its voltage peaks a quarter cycle after the current.

Figure

The three element personalities: same current, three different voltage timings

The mnemonic "ELI the ICE man" compresses the picture: in an inductor (L), the EMF E comes before the current I; in a capacitor (C), I comes before E. And the two limits you already own from the transient articles survive intact: at very low frequency an inductor is just a piece of wire and a capacitor is an open gap, while at very high frequency they swap roles —

element	$\omega \to 0$ (DC)	$\omega \to \infty$	phase of $V$ relative to $I$
resistor	resistor	resistor	$0°$
inductor	short circuit (wire)	open circuit	leads by $90°$
capacitor	open circuit	short circuit (wire)	lags by $90°$

This table is the frequency-domain restatement of the switch-closing shortcuts from aq-em-rc-circuits and aq-em-rl-circuits: "inductor resists change" becomes "inductor impedes high frequencies," and "capacitor holds its voltage" becomes "capacitor impedes low frequencies."

Why sinusoids get special treatment

The privilege of sinusoids rests on one mathematical fact. Take the operation every capacitor and inductor performs — differentiation in time — and ask which waveforms pass through it unchanged in shape. Try $v(t) = e^{j\omega t}$ :

\frac{d}{dt}\,e^{j\omega t} = j\omega\, e^{j\omega t}

where: $j = \sqrt{-1}$ (engineering's symbol, keeping $i$ free for current) and $\omega$ is the angular frequency. The exponential is an eigenfunction of the derivative: differentiating it returns the same function multiplied by the constant $j\omega$ . No other bounded waveform has this property. Feed a linear circuit a complex exponential and every derivative in every loop equation collapses to a factor of $j\omega$ — the differential equations become simultaneous algebraic equations. Real sinusoids ride along for free, because $\cos\omega t$ is the real part of $e^{j\omega t}$ and every circuit operation here (adding, scaling, differentiating) commutes with taking the real part. One more fact completes the license: in steady state, a linear circuit driven at frequency $\omega$ responds only at frequency $\omega$ . Transients at the circuit's own natural frequency die out exactly as aq-em-lc-rlc-oscillations showed — the $e^{-Rt/2L}$ ringdown — leaving every branch oscillating at the source's rhythm with its own amplitude and phase. Two numbers per signal; one complex number. That complex number, conventionally the amplitude-and-phase factor $\tilde V$ in $v(t) = \mathrm{Re}\,[\tilde V e^{j\omega t}]$ , is the phasor.

Derivation AThe three impedances, from the element laws

We want the voltage–current relation of each basic element for sinusoidal signals, expressed as a single complex constant $Z = \tilde V/\tilde I$ — the impedance. The strategy is direct: write the time-domain element law you already own (from the capacitance and self-inductance articles), substitute the complex exponential forms $v = \tilde V e^{j\omega t}$ , $i = \tilde I e^{j\omega t}$ , and read off the ratio. Watch two things: where the factor $j$ lands (it carries the 90° phase shifts of the hero figure), and how each impedance's magnitude scales with frequency (it carries the DC/HF limit table).

Step 1. The resistor. Its law is $v = iR$ at every instant, so $\tilde V e^{j\omega t} = R\,\tilde I e^{j\omega t}$ and

Z_R = \frac{\tilde V}{\tilde I} = R

Real and frequency-independent: voltage and current in phase at every frequency — the left panel of the hero figure.

Step 2. The inductor. Its law is $v = L\,di/dt$ (the back-EMF of aq-em-self-inductance). Substituting and using the eigenfunction property:

\tilde V e^{j\omega t} = L\,\frac{d}{dt}\!\left(\tilde I e^{j\omega t}\right) = j\omega L\,\tilde I e^{j\omega t} \quad\Longrightarrow\quad Z_L = j\omega L

Why does multiplying by $j$ mean "leads by 90°"? Because $j = e^{j\pi/2}$ : multiplying a complex number by $j$ rotates it a quarter turn counterclockwise. The voltage phasor sits 90° ahead of the current phasor — ELI. The magnitude $|Z_L| = \omega L \equiv X_L$ is the inductive reactance: it grows linearly with frequency, vanishing at DC (a wire) and diverging at high frequency (an open) — the middle column of the limit table.

Step 3. The capacitor. Its law is $i = C\,dv/dt$ (charge bookkeeping from aq-em-capacitance). The same substitution gives $\tilde I = j\omega C\,\tilde V$ , so

Z_C = \frac{1}{j\omega C} = -\,\frac{j}{\omega C}

Why the minus sign — what does dividing by $j$ do? Dividing by $j$ rotates a quarter turn clockwise: the voltage sits 90° behind the current — ICE. The magnitude $|Z_C| = 1/\omega C \equiv X_C$ is the capacitive reactance, diverging at DC (open gap) and vanishing at high frequency (effectively a wire).

Step 4. State the generalized Ohm's law. For any network of these elements, Kirchhoff's rules — linear equations — hold for the phasors, and every element obeys $\tilde V = Z\tilde I$ . Since these are formally the same equations as a DC resistor network, all of the reduction machinery of aq-em-series-parallel-networks (series addition, parallel reciprocals, dividers, the equivalent-source theorem) applies verbatim with $Z$ in place of $R$ . Verified: dimensions — $\omega L$ has units $(1/\mathrm{s})(\mathrm{V\,s/A}) = \mathrm{V/A} = \Omega$ , and $1/\omega C$ has $(\mathrm{s})(\mathrm{V/C}) = \mathrm{V\,s/(A\,s)} = \Omega$ ; at $f = 50\ \mathrm{Hz}$ , a 0.191 H coil gives $X_L = 2\pi(50)(0.191) = 60.0\ \Omega$ , the value used in Worked example 1.

The series RLC: one impedance to rule the loop

String all three elements in series with a source $\tilde V_s$ . Impedances in series add — same phasor current through each, voltages summing by Kirchhoff — so the loop presents

Z = R + j\!\left(\omega L - \frac{1}{\omega C}\right)

where: the real part is the resistance and the imaginary part, the net reactance $X = X_L - X_C$ , is a tug-of-war. The inductor's contribution grows with frequency while the capacitor's shrinks, so the sign of $X$ tells you which personality dominates: below $\omega_0 = 1/\sqrt{LC}$ the circuit is net-capacitive (current leads), above it net-inductive (current lags), and exactly at $\omega_0$ the two reactances cancel and the source sees pure resistance. That cancellation frequency is precisely the natural ring frequency of aq-em-lc-rlc-oscillations — the driven circuit is most transparent exactly where the free circuit likes to oscillate. The next article makes that coincidence the star; here we need the algebra around it.

Go deeper · Derivation B · 2 minAmplitude and phase of the driven current

Derivation BAmplitude and phase of the driven current

We want the steady-state current in the series RLC — its amplitude and its phase lag — as explicit functions of frequency, and we want to see the geometry of the phasor diagram fall out. The strategy is to write $Z$ in polar form, because dividing by a complex number in polar form divides magnitudes and subtracts angles: one line of algebra replaces the trigonometric brute force. This result is, through the dictionary of the oscillations article, exactly the driven response of the damped mechanical oscillator solved in the mechanics track (Resonance, Driven Oscillations) — we are not re-deriving that physics, just meeting it in its native electrical dress.

Step 1. Polar form of the impedance. A complex number $R + jX$ has magnitude and angle

|Z| = \sqrt{R^2 + \left(\omega L - \tfrac{1}{\omega C}\right)^2}, \qquad \tan\varphi = \frac{\omega L - 1/\omega C}{R}

so $Z = |Z|\,e^{j\varphi}$ .

Step 2. Divide. With the source phasor $\tilde V_s = V_0$ (choose its phase as the reference),

\tilde I = \frac{\tilde V_s}{Z} = \frac{V_0}{|Z|}\,e^{-j\varphi} \quad\Longrightarrow\quad i(t) = \frac{V_0}{|Z|}\cos(\omega t - \varphi)

Why does the current lag by the impedance angle? Dividing by $e^{j\varphi}$ subtracts $\varphi$ from the phase. A net-inductive loop ( $X > 0$ , $\varphi > 0$ ) makes the current arrive late; a net-capacitive loop makes it arrive early ( $\varphi < 0$ ). At resonance $\varphi = 0$ : source and current in lockstep.

Step 3. Read the voltage phasors as geometry. The element voltages are $\tilde V_R = R\tilde I$ (along the current), $\tilde V_L = j\omega L \tilde I$ (90° ahead), $\tilde V_C = \tilde I/j\omega C$ (90° behind). $\tilde V_L$ and $\tilde V_C$ point in opposite directions, so they partially cancel, and Kirchhoff's loop rule is a vector sum:

\tilde V_s = \tilde V_R + \tilde V_L + \tilde V_C

— a right triangle with legs $V_R$ and $|V_L - V_C|$ and hypotenuse $V_0$ . Squaring the triangle reproduces $|Z|$ of Step 1 term by term. Why don't the voltmeter readings add up? A meter across each element reports rms magnitudes — the lengths of the phasors — and lengths of vectors at right angles don't add arithmetically. In Worked example 1 the resistor and inductor read 128 V and 191 V across a 230 V source: $\sqrt{128^2 + 191^2} = 230$ , but $128 + 191 = 319 \neq 230$ . Nothing is wrong; the readings simply peak at different instants.

Step 4. Sanity-check the limits. As $\omega \to 0$ , $|Z| \to \infty$ through the capacitor (open at DC): no steady current, matching the fully-charged capacitor endgame of aq-em-rc-circuits. As $\omega \to \infty$ , $|Z| \to \infty$ through the inductor. In between, $|Z|$ has a minimum $R$ at $\omega_0$ — maximal current, the resonance the next article anatomizes. Verified: at $f = 250\ \mathrm{Hz}$ with $R = 25\ \Omega$ , $L = 30\ \mathrm{mH}$ , $C = 8\ \mu\mathrm{F}$ : $X_L = 47.1\ \Omega$ , $X_C = 79.6\ \Omega$ , $|Z| = \sqrt{25^2 + 32.5^2} = 41.0\ \Omega$ , $\varphi = -52.4°$ — the numbers of Worked example 2, current leading as the sign convention demands below resonance.

Figure

Phasor geometry of the series RLC: the voltage triangle and the impedance angle

The toolkit transfers: dividers and filters

Because impedances obey the resistor algebra, the divider theorems of aq-em-series-parallel-networks now answer frequency-dependent questions. Two impedances in series split the source voltage as $\tilde V_2 = \tilde V_s\, Z_2/(Z_1 + Z_2)$ — but now the ratio depends on frequency, and that single fact is the origin of every filter. The simplest and most consequential case: a resistor feeding a capacitor, with the output taken across the capacitor. That is the promised payoff of the teaser in aq-em-rc-circuits — the same two components that made a timer in the time domain make a low-pass filter in the frequency domain.

Go deeper · Derivation C · 1 minThe RC low-pass filter, end to end

Derivation CThe RC low-pass filter, end to end

We want the frequency response of the R-then-C divider: how much of the input survives to the output as a function of frequency, and with what phase. The strategy is one line of divider algebra followed by a careful reading — magnitude, phase, corner frequency, and the asymptotic rolloff that engineers quote in decibels. This is the article's method in miniature: no differential equation is ever written, yet the answer contains the complete steady-state physics.

Step 1. Apply the divider with $Z_1 = R$ and $Z_2 = 1/j\omega C$ :

H(\omega) \equiv \frac{\tilde V_{\text{out}}}{\tilde V_{\text{in}}} = \frac{1/j\omega C}{R + 1/j\omega C} = \frac{1}{1 + j\omega RC}

Step 2. Read magnitude and phase:

|H| = \frac{1}{\sqrt{1 + (\omega RC)^2}}, \qquad \text{phase} = -\arctan(\omega RC)

Why does $RC$ — the timer's time constant — reappear as the frequency dial? Because the divider compares $X_C = 1/\omega C$ with $R$ , and they are equal exactly at $\omega = 1/RC$ . A circuit that takes time $RC$ to respond cannot follow wiggles faster than $\sim 1/RC$ : the time-domain sluggishness of aq-em-rc-circuits is the frequency-domain rolloff, two views of one fact.

Step 3. Name the landmarks. At the corner frequency $f_{3\mathrm{dB}} = 1/2\pi RC$ , $|H| = 1/\sqrt 2 \approx 0.707$ (half the power, hence "−3 dB") and the phase is exactly −45°. Well below the corner, $|H| \to 1$ : the capacitor is an open and the output tracks the input. Well above, $|H| \approx 1/\omega RC$ : each doubling of frequency halves the output — a −6 dB/octave, −20 dB/decade rolloff.

Step 4. Put numbers on it. With $R = 1.0\ \mathrm{k\Omega}$ and $C = 159\ \mathrm{nF}$ : $f_{3\mathrm{dB}} = 1/(2\pi \times 10^3 \times 1.59\times10^{-7}) = 1.0\ \mathrm{kHz}$ . At 10 kHz the response is $|H| = 1/\sqrt{1 + 100} = 0.0995$ — attenuated to 10%, i.e. −20.0 dB, with the phase at −84°. Swap the two components (output across $R$ ) and every low is blocked instead: the high-pass twin, $|H| = \omega RC/\sqrt{1 + (\omega RC)^2}$ — same corner, mirrored roles. Tone knobs, DC-blocking inputs, rumble filters: all are these two circuits. Verified: $|H(10 f_{3\mathrm{dB}})| = 0.0995 = -20.04\ \mathrm{dB}$ and phase $-84.3°$ , recomputed from the complex formula; the corner value $1/\sqrt2 = -3.01\ \mathrm{dB}$ and $-45.0°$ exactly.

One convention, stated honestly

A phasor's length can carry either the peak value $V_0$ or the rms value $V_0/\sqrt2$ , and the literature uses both. Physics texts (and this article's derivations) tend to carry peaks; power engineering carries rms almost exclusively, because — as the next article shows — rms values plug directly into power formulas. Nothing physical hangs on the choice, but ratios of a mixed pair are nonsense: a 230 V rms mains supply has a 325 V peak, and quoting "230 V" while computing with peak-convention formulas silently loses a factor $\sqrt2$ . Pick one, label it, and convert at the boundary. The mains numbers to keep on hand: "230 V, 50 Hz" means $V_0 = 230\sqrt2 = 325\ \mathrm{V}$ and $\omega = 314\ \mathrm{rad/s}$ ; "120 V, 60 Hz" means $V_0 = 170\ \mathrm{V}$ , $\omega = 377\ \mathrm{rad/s}$ .

Limiting cases

case	result	meaning
$\omega \to 0$	$Z_L \to 0$ , $Z_C \to \infty$	DC intuition recovered: coils are wires, caps are gaps
$\omega \to \infty$	$Z_L \to \infty$ , $Z_C \to 0$	roles swap: coils block, caps conduct
$X_L = X_C$ (at $\omega_0 = 1/\sqrt{LC}$ )	$Z = R$ , $\varphi = 0$	series resonance — maximal current, next article's subject
$R = 0$ , off resonance	$Z$ purely imaginary, $\varphi = \pm 90°$	current flows but (next article) zero average power
$\omega RC \ll 1$ in the low-pass	$\|H\| \to 1$	signal too slow for the filter to notice
$\omega RC \gg 1$ in the low-pass	$\|H\| \approx 1/\omega RC$	−20 dB/decade rolloff
single element only	$Z = R$ , $j\omega L$ , or $1/j\omega C$	the three personalities, unmixed

Common trap“The voltmeter readings across R, L and C should add up to the source voltage”

They should not, and a bench measurement proves it within minutes: in Worked example 1 the meters across a 40 Ω resistor and a 60 Ω-reactance coil read 128 V and 191 V — "319 volts" across a 230 V source. No energy law is violated. Each meter reports an rms magnitude, and the three element voltages peak at different instants: $V_L$ a quarter cycle before the current, $V_C$ a quarter cycle after, $V_R$ in step. At any single instant the instantaneous voltages do sum to the source's instantaneous value — Kirchhoff never fails — but magnitudes of out-of-phase sinusoids add like the sides of a triangle, $V_s = \sqrt{V_R^2 + (V_L - V_C)^2}$ , not like numbers on a receipt. The same trap in reverse: near resonance, $V_L$ and $V_C$ can each exceed the source voltage while cancelling each other — the magnification the next article quantifies.

Worked examples

Example 1 — a coil on the mains (confidence builder). A solenoid with resistance $R = 40\ \Omega$ and inductance $L = 0.191\ \mathrm{H}$ is plugged into the 230 V (rms), 50 Hz mains. Find the current, its phase, and the meter readings across each part.

Step 1. Reactance: $X_L = \omega L = 2\pi(50)(0.191) = 60.0\ \Omega$ .

Step 2. Impedance: $|Z| = \sqrt{40^2 + 60^2} = 72.1\ \Omega$ , with phase $\varphi = \arctan(60/40) = 56.3°$ — the current lags the mains voltage by 56.3°, or about 3.1 ms.

Step 3. Current: $I_{\mathrm{rms}} = 230/72.1 = \mathbf{3.19\ A}$ .

Step 4. Meter readings: $V_R = (3.19)(40) = 128\ \mathrm{V}$ and $V_L = (3.19)(60) = 191\ \mathrm{V}$ . Triangle check: $\sqrt{128^2 + 191^2} = 230\ \mathrm{V}$ — the source recovered, though the arithmetic sum is 319 V.

Example 2 — a full series RLC, above and below resonance (exam level). A source of amplitude $V_0 = 48\ \mathrm{V}$ drives $R = 25\ \Omega$ , $L = 30\ \mathrm{mH}$ , $C = 8.0\ \mu\mathrm{F}$ in series at $f = 250\ \mathrm{Hz}$ . Find the current amplitude and phase; then state what happens at the resonant frequency.

Step 1. Reactances at 250 Hz: $X_L = 2\pi(250)(0.030) = 47.1\ \Omega$ ; $X_C = 1/[2\pi(250)(8.0\times10^{-6})] = 79.6\ \Omega$ . Net reactance $X = 47.1 - 79.6 = -32.5\ \Omega$ : capacitive, so expect the current to lead.

Step 2. Impedance: $|Z| = \sqrt{25^2 + 32.5^2} = 41.0\ \Omega$ ; $\varphi = \arctan(-32.5/25) = -52.4°$ .

Step 3. Current: $I_0 = 48/41.0 = \mathbf{1.17\ A}$ , leading the source by 52.4°.

Step 4. Resonance check: $f_0 = 1/2\pi\sqrt{LC} = 1/2\pi\sqrt{(0.030)(8.0\times10^{-6})} = 325\ \mathrm{Hz}$ . There $X_L = X_C = 61.2\ \Omega$ , the reactances cancel, and $I_0 = 48/25 = 1.92\ \mathrm{A}$ at zero phase — a 64% larger current for the same source, obtained by changing nothing but the frequency. Why the circuit rewards that particular frequency is the next article's opening question.

Origins · 1 minLeonhard Euler and how this was discovered

Leonhard Euler

1707–1783 · Basel, St. Petersburg, Berlin

The engine under every phasor is one line of Euler's 1748 Introductio in analysin infinitorum: $e^{j\theta} = \cos\theta + j\sin\theta$ — the formula Richard Feynman later called "the most remarkable in mathematics." Euler built it decades before anyone could use it on electricity, as pure analysis: the exponential function, fed an imaginary argument, is a rotation, which makes it the natural language for anything that turns or oscillates. That is precisely the property this article lives on — $e^{j\omega t}$ passes through $d/dt$ unchanged in shape — and it is why, when telegraph and power engineers of the 1880s–90s found themselves drowning in AC trigonometry, the rescue came from reaching back to Euler: Oliver Heaviside's operational methods, then Kennelly and Steinmetz packaging the complex-number method that turned AC design from virtuoso calculus into bookkeeping. (Euler appears once before in this track, and differently: the Electric Flux article credits his 1750s hydrodynamics for the very idea of flux through a surface. Same man, two of the track's load-bearing ideas, a century before the electricity.)

History

1748

— Euler's Introductio publishes

e^{j\theta} = \cos\theta + j\sin\theta

, the identity that will one day make AC circuits algebra.

1832

— Pixii's hand-cranked alternator produces the first artificial AC; for fifty years it is treated as a nuisance to be rectified.

1886

— Heaviside coins "impedance," making the frequency-domain resistance a named object (his operational calculus already treated

d/dt

as a symbol to divide by).

1893

— Kennelly's paper "Impedance" and Steinmetz's complex-number method, presented to the electrical congress in Chicago, give engineers the phasor toolkit essentially as taught today.

1918

— Fortescue's symmetrical components extend phasor algebra to unbalanced three-phase systems — the analysis backbone of the modern grid.

Going further

The payoff of this algebra is one article away: aq-em-ac-power-resonance asks what power actually flows through these phase angles and why the series RLC's minimum-impedance point turns a circuit into a frequency-selective instrument. The transformer then puts phasor power to continental use in aq-em-transformers. Behind this article stand aq-em-lc-rlc-oscillations (the natural frequency and the mechanics dictionary), aq-em-rc-circuits and aq-em-rl-circuits (the time-domain personalities), and aq-em-series-parallel-networks (the reduction toolkit that transferred wholesale). On the mechanics side, the driven response meeting you in Derivation B is solved and dissected in Driven Oscillations and Resonance — the same Lorentzian physics in mechanical dress. Provenance: Authored fresh for the SuperSTEM E&M track's circuits chapter (recon-§d gap heal); sources: v3 circuit pilots (rewritten to UG register) + legacy wiki circuit articles (narrative) — no AllQFT base exists for this topic.

Is j√−1 physically real? What does an imaginary voltage mean?

Nothing in the circuit is ever imaginary — the physical voltage is always the real part, $v(t) = \mathrm{Re}[\tilde V e^{j\omega t}]$ . The imaginary part is deliberately carried excess baggage: a second copy of the signal, shifted a quarter cycle, that comes along so that differentiation acts simply on the pair. Because every operation in a linear circuit (add, scale, differentiate) treats real and imaginary parts independently, you can compute with the full complex object and take the real part once, at the very end. The $j$ 's in intermediate results encode phase shifts, not mystical quantities: $jV$ means " $V$ , a quarter cycle early."

Why does an inductor's voltage lead its current? I keep flipping ELI and ICE.

Go back to the element law rather than the mnemonic. An inductor obeys $v = L\,di/dt$ : its voltage is large when the current is changing fastest, which for a sine is at the zero crossing — a quarter cycle before the current peaks. So the voltage peak arrives first: V leads I. A capacitor obeys $i = C\,dv/dt$ : now the current is the derivative, so the current peaks first and V lags. If you remember which element differentiates which variable, the mnemonics reconstruct themselves.

Reactance is measured in ohms. Does a reactance dissipate power like a resistance does?

No — and the distinction is the whole point of the next article. An ohm of reactance limits current exactly like an ohm of resistance (that is why both appear in $|Z|$ ), but the 90° phase shift means the element absorbs energy for a quarter cycle and returns it the next quarter cycle: average power zero. Only the real part $R$ of an impedance converts electrical energy irreversibly to heat. A capacitor in series with a motor limits the current at no heating cost — a trick single-phase appliance designers use constantly.

Does the phasor method work for square waves or audio signals, which aren't sinusoids?

Directly, no — a phasor represents one frequency. But linearity rescues everything: any realistic signal is a sum of sinusoids (its Fourier content), each frequency passes through the circuit independently with its own $H(\omega)$ , and the output is the reassembled sum. That is exactly what a "−20 dB/decade rolloff" means in practice: feed the low-pass filter a square wave and its high harmonics are attenuated and phase-shifted, rounding the corners. The phasor method plus Fourier decomposition is the general theory of linear circuits; only nonlinear elements (diodes, saturated cores) break it, because they mix frequencies.

Why is the corner of the RC filter at 1/RC and not somewhere else? The rolloff seems gradual, so what's special about that point?

It is the frequency at which the two impedances in the divider have equal magnitude: $X_C = 1/\omega C = R$ . Below it the capacitor's impedance dominates the series pair, so most of the input voltage appears across the capacitor (the output); above it the resistor dominates and the output starves. Nothing abrupt happens there — the response is $1/\sqrt2$ , down 3 dB, phase −45° — but it is the natural boundary between the "passes" and "blocks" regimes, and conveniently it is set by the same $RC$ product that sets the transient time constant. One number, two jobs.

How can V_L and V_C both be larger than the source voltage without breaking Kirchhoff's loop rule?

Because they point in opposite directions in the phasor diagram, their sum can be small while each is large. Near resonance $X_L \approx X_C$ can each be many times $R$ ; the common current is $\approx V_s/R$ , so each element develops $\approx (X/R)V_s$ — potentially far above the source — while their difference stays modest and the loop rule holds at every instant. This is real and measurable (and occasionally destructive: capacitors in tuned circuits must be rated for the magnified voltage, not the source voltage). The factor $X/R$ at resonance is the quality factor $Q$ , quantified in aq-em-ac-power-resonance.

My function generator says 5 V but the scope shows the wave going from +5 V to −5 V. Which number goes in the formulas?

Welcome to the convention swamp — instruments disagree. Generators usually quote amplitude ( $V_0 = 5$ V) or peak-to-peak ( $V_{pp} = 10$ V); multimeters in AC mode quote rms ( $V_0/\sqrt2 = 3.54$ V for that wave). Every formula in this article works in either pure convention: use peaks throughout and get peak currents; use rms throughout and get rms currents. The only fatal move is mixing — dividing an rms voltage by an impedance and calling the result a peak current. Label every number p, pp, or rms when you write it down and the swamp is harmless.