Impedance (Z), Admittance (Y), Reactance (X)

Every AC circuit calculation beyond a single resistor requires generalizing Ohm's Law to handle phase shifts between voltage and current. Impedance, admittance, and reactance are the complex-number machinery that makes this possible — they let you apply KCL, KVL, voltage dividers, and Thevenin/Norton equivalents to capacitors and inductors exactly as you would to resistors, provided you work in the frequency domain with phasors. Without this framework, filter design, resonance analysis, and power-factor correction would require solving differential equations by hand every time.

Impedance: The Generalized Resistance

Impedance Z is the AC (frequency-domain) generalization of resistance. It relates the phasor voltage across a component to the phasor current through it:

V(jω) = Z(jω) · I(jω)

Z is a complex number, generally written in rectangular form:

Z = R + jX

• R — resistance, the real part, dissipates energy as heat.
• X — reactance, the imaginary part, stores and returns energy (no net dissipation).

Impedance can also be written in polar form: Z = |Z|∠θ, where |Z| = √(R² + X²) and θ = atan(X/R). |Z| scales the current magnitude, and θ is the phase angle by which voltage leads current.

Units: ohms (Ω), same as resistance, because it's a ratio of volts to amps.

Reactance: The Imaginary Part

Reactance X quantifies how much a reactive element (capacitor or inductor) opposes current change without dissipating power. Its sign convention distinguishes the two:

• Inductive reactance X_L = ωL grows with frequency — an inductor looks like an open circuit at high frequency, a short at DC.
• Capacitive reactance X_C = -1/(ωC) shrinks in magnitude as frequency rises — a capacitor looks like a short at high frequency, an open at DC.

The j (or -j) factor encodes a 90° phase shift: current through an inductor lags voltage by 90°; current through a capacitor leads voltage by 90°. This is the origin of the mnemonic "ELI the ICE man" — voltage (E) leads current (I) in an inductor (L); current (I) leads voltage (E) in a capacitor (C).

In a series RLC combination, reactances add algebraically because they share the same phase reference (j):

X_total = X_L + X_C = ωL - 1/(ωC)

At the frequency where X_L = -X_C (i.e., X_total = 0), the circuit is purely resistive — this is the series resonance condition, ω₀ = 1/√(LC), covered in depth in the RLC series resonance article.

Admittance: The Reciprocal View

Admittance Y is simply the reciprocal of impedance:

Y = 1/Z = G + jB

• G — conductance, the real part (not generally equal to 1/R unless X = 0 — see the worked example below).
• B — susceptance, the imaginary part.

Units: siemens (S), same as conductance.

Admittance is the natural choice for parallel combinations, just as impedance is natural for series combinations — admittances of parallel elements add directly:

Y_total = Y₁ + Y₂ + Y₃ + ...

This mirrors how conductances add in parallel resistor networks, generalized to the complex domain.

Why G ≠ 1/R in General

A common mistake: assuming G = 1/R and B = 1/X. This is only true when Z is purely real or purely imaginary. For a general Z = R + jX, you must rationalize:

Y = 1/(R + jX) = (R - jX) / (R² + X²)

So:

G = R / (R² + X²)
B = -X / (R² + X²)

Note G depends on both R and X — it is not simply 1/R unless X = 0.

Worked Example: Series R and L, Convert to Admittance

Circuit: A resistor R = 100 Ω in series with an inductor L = 50 mH, driven at f = 1 kHz.

Step 1 — Compute ω:

ω = 2πf = 2π × 1000 ≈ 6283 rad/s

Step 2 — Compute reactance:

X_L = ωL = 6283 × 0.05 ≈ 314.2 Ω

Step 3 — Impedance:

Z = R + jX_L = 100 + j314.2 Ω

Polar form:

|Z| = √(100² + 314.2²) = √(10000 + 98741) = √108741 ≈ 329.8 Ω
θ = atan(314.2/100) = atan(3.142) ≈ 72.3°

Step 4 — Convert to admittance:

R² + X² ≈ 108741
G = R/(R²+X²) = 100/108741 ≈ 919.6 µS
B = -X/(R²+X²) = -314.2/108741 ≈ -2890 µS

So Y ≈ (919.6 - j2890) µS.

Check: magnitude of Y should equal 1/|Z|.

|Y| = √(919.6² + 2890²) µS = √(0.846 + 8.351) mS² ≈ √9.197 mS ≈ 3.033 mS
1/|Z| = 1/329.8 ≈ 3.032 mS  ✓ (matches within rounding)

Check phase: Y's angle should be -θ (reciprocal of a complex number negates its angle).

∠Y = atan(-2890/919.6) ≈ atan(-3.143) ≈ -72.3°  ✓ matches -θ

Both checks close, confirming the conversion.

Series vs. Parallel: When to Use Z or Y

This is exactly analogous to resistors and conductances in DC circuits, extended into the complex plane. In practice, engineers switch between Z and Y representations depending on which makes the topology (series or parallel) algebraically simpler — the same complex number, viewed two ways.

Practical Implications

• Filter design: Reactance's frequency dependence (X_L rising, X_C falling with ω) is the mechanism behind every RC/RL/RLC filter's frequency-selective behavior.
• Impedance matching: Maximum power transfer to an AC load requires the load impedance to be the complex conjugate of the source impedance (Z_load = Z_source*), not simply equal magnitude — reactive parts must cancel.
• Resonance: At resonance, total reactance is zero and impedance is purely resistive (series) or purely conductive/at a peak (parallel) — the basis of tuned circuits and oscillators.
• Power factor: The angle θ of Z directly sets the power factor cos θ in AC power calculations — a large reactance relative to resistance means poor power factor and higher reactive power Q.
• Two-port and network analysis: Z-parameters and Y-parameters (covered separately) extend this same impedance/admittance duality to multi-port networks like transmission lines and amplifier models.

Key Takeaways

• Impedance Z = R + jX generalizes Ohm's Law to AC circuits; R dissipates energy, X (reactance) stores and returns it.
• X_L = ωL (positive, grows with frequency); X_C = -1/(ωC) (negative, shrinks with frequency) — this sign difference sets the 90° phase relationships in inductors and capacitors.
• Admittance Y = 1/Z = G + jB is the reciprocal of impedance; use Z for series combinations and Y for parallel combinations since each adds directly in its respective topology.
• G ≠ 1/R and B ≠ 1/X in general — correct conversion requires rationalizing 1/(R+jX) = (R-jX)/(R²+X²).
• Zero total reactance defines resonance, where impedance becomes purely resistive — the foundation for tuned circuits, filters, and impedance matching.