Phasor Representation and Complex Impedance

Every AC circuit with resistors, capacitors, and inductors requires solving differential equations — unless you convert sinusoids into phasors and treat capacitors and inductors as frequency-dependent "resistors" called impedances. This single trick is what lets you apply Ohm's law, KVL, KCL, voltage dividers, and Thevenin's theorem directly to AC circuits, without ever writing d/dt. It's the algebraic backbone of every filter, resonant circuit, and impedance-matching network you'll design.

The Problem: Sinusoids and Calculus Don't Mix Well Algebraically

A sinusoidal signal v(t) = Vm·cos(ωt + φ) is fully described by three numbers: amplitude Vm, frequency ω, and phase φ. In a linear circuit driven at a single frequency ω, every voltage and current in steady state is a sinusoid at that same frequency — only amplitude and phase differ from branch to branch.

The trouble is that capacitors and inductors relate voltage and current through derivatives and integrals:

i_C(t) = C·dv/dt
v_L(t) = L·di/dt

Writing KVL/KCL for a circuit with these elements gives you a system of differential equations. Solvable, but painful, and it obscures the algebraic structure (dividers, Thevenin equivalents) that makes resistive analysis fast.

The Phasor Transform

A phasor is a complex number that encodes the amplitude and phase of a sinusoid, with the ejωt time dependence stripped out because it's common to every signal in the (single-frequency) circuit.

v(t) = Vm·cos(ωt + φ)   ⟷   V = Vm·∠φ = Vm·e^(jφ)   (phasor, complex amplitude)

Convention: phasors here use the amplitude Vm; some textbooks use RMS instead (Vm/√2) — pick one convention per problem and stay consistent, since power calculations depend on which you used.

The transform relies on Euler's identity, e^(jθ) = cos θ + j·sin θ, and the fact that v(t) = Re{Vm·e^(j(ωt+φ))} = Re{V·e^(jωt)}. Since every signal in the circuit shares the same e^(jωt) factor, KVL and KCL — which are just sums of these signals — hold equally well for the phasors themselves (Re{} is linear, so a sum of real parts equals the real part of the sum).

Why this kills the calculus: differentiation in time becomes multiplication by jω in the phasor domain:

d/dt[Vm·e^(j(ωt+φ))] = jω·Vm·e^(j(ωt+φ))

So d/dt → ×jω and ∫dt → ÷jω. A linear ODE in time becomes a linear algebraic equation in complex numbers.

Complex Impedance

Apply the phasor transform to each element's V-I relationship:

Impedance Z = R + jX (ohms) generalizes resistance to include the 90° phase relationship reactive elements impose between V and I. Its reciprocal is admittance Y = 1/Z = G + jB (siemens). Reactance X is the imaginary part of Z: positive (inductive) for Z_L, negative (capacitive) for Z_C. Because Z has the same units and algebraic role as R, every resistive-circuit tool — series/parallel combination, voltage/current dividers, KVL/KCL, Thevenin/Norton, superposition — applies unchanged to impedances, just with complex arithmetic instead of real.

Key physical points baked into the math:

• Inductor: Z_L = jωL means current lags voltage by 90° (j = 1∠90°). At DC (ω=0), Z_L = 0 — a short. At high frequency, Z_L → ∞ — an open.
• Capacitor: Z_C = -j/(ωC) means current leads voltage by 90°. At DC, Z_C → ∞ — an open (blocks DC, as expected). At high frequency, Z_C → 0 — a short.
• Resistor: Z_R = R, purely real — voltage and current stay in phase, and it's frequency-independent.

Worked Example: Series RC at a Specific Frequency

A resistor R = 1 kΩ and capacitor C = 100 nF are in series, driven by a sinusoidal source v_s(t) = 10·cos(2π·1000·t) V (10 V amplitude, f = 1 kHz). Find the current phasor and the voltage across each element.

Step 1 — Source phasor: Vs = 10∠0° V.

Step 2 — Angular frequency: ω = 2πf = 2π(1000) = 6283 rad/s.

Step 3 — Impedances:

Z_R = 1000 Ω
Z_C = -j/(ωC) = -j/(6283 × 100×10⁻⁹) = -j/(6.283×10⁻⁴) = -j1592 Ω

Step 4 — Total series impedance:

Z_total = Z_R + Z_C = 1000 - j1592 Ω
|Z_total| = √(1000² + 1592²) = √(1,000,000 + 2,534,464) = √3,534,464 ≈ 1880 Ω
∠Z_total = atan(-1592/1000) = atan(-1.592) ≈ -57.9°

So Z_total ≈ 1880∠-57.9° Ω.

Step 5 — Current phasor (Ohm's law for phasors):

I = Vs / Z_total = 10∠0° / 1880∠-57.9° = 5.32×10⁻³ ∠57.9° A ≈ 5.32∠57.9° mA

The current leads the source voltage by 57.9° — expected, since the circuit is capacitive (net negative reactance).

Step 6 — Voltage across R:

V_R = I·Z_R = (5.32×10⁻³∠57.9°)(1000∠0°) = 5.32∠57.9° V

Step 7 — Voltage across C:

V_C = I·Z_C = (5.32×10⁻³∠57.9°)(1592∠-90°) = 8.47∠-32.1° V

(Z_C = -j1592 = 1592∠-90°.)

Check — KVL: V_R + V_C should reconstruct Vs = 10∠0°. Convert to rectangular form:

V_R = 5.32∠57.9° = 5.32(cos57.9° + jsin57.9°) = 2.82 + j4.51 V
V_C = 8.47∠-32.1° = 8.47(cos(-32.1°) + jsin(-32.1°)) = 7.18 - j4.50 V
Sum = (2.82+7.18) + j(4.51-4.50) = 10.00 + j0.01 ≈ 10∠0° V ✓

The check closes (rounding accounts for the residual 0.01 j-term), confirming the phasor arithmetic is self-consistent. Note also |V_R| and |V_C| don't simply add to 10 V (5.32 + 8.47 = 13.79 ≠ 10) — that's expected, since they're 90° apart in phase, not in-phase quantities; only the complex (phasor) sum obeys KVL.

Practical Implications

• Filter design: a transfer function like H(jω) = V_out/V_in is computed purely with impedance dividers — e.g., an RC low-pass is Z_C/(Z_R+Z_C) — no differential equations needed once you're in the phasor domain.
• Resonance: in a series RLC circuit, Z_L and Z_C cancel (jωL = j/(ωC)) at ω₀ = 1/√(LC), leaving purely resistive impedance — the basis of resonance and Q-factor analysis.
• Power factor: the phase angle of Z directly gives the power factor cos(∠Z) between voltage and current, which determines real vs. reactive power delivered to a load.
• Impedance matching: matching networks (for RF, audio, or power transfer) are designed entirely in the impedance domain, choosing reactive components to cancel unwanted reactance at the frequency of interest.
• Limits of validity: phasors and impedance strictly apply to linear circuits under sinusoidal steady-state excitation at a single frequency. Transients, non-sinusoidal waveforms, or nonlinear elements require time-domain analysis or the more general Laplace transform (Z(s) with s = jω as a special case).

Key Takeaways

• Phasors represent sinusoids as complex numbers (magnitude + phase), stripping out the shared e^(jωt) term so KVL/KCL apply directly to complex amplitudes.
• Differentiation in time becomes multiplication by jω in the phasor domain, turning differential equations into complex algebra.
• Impedance Z = R + jX generalizes resistance: Z_R = R, Z_L = jωL, Z_C = 1/(jωC) — all combine with the same series/parallel/divider rules as resistors.
• Reactive elements impose a 90° phase shift between V and I (inductor: current lags; capacitor: current leads); resistors keep V and I in phase.
• Phasor/impedance analysis is valid only for linear circuits in sinusoidal steady state at a single frequency — transients and nonlinearities need time-domain or Laplace methods.
• Always verify results with a KVL or KCL check in complex (rectangular) form — phase-shifted quantities don't add as simple magnitudes.