Figure 1: In series, impedances add as complex numbers. In parallel, add the reciprocals then invert the result, or use Z = (Z₁ × Z₂)/(Z₁ + Z₂) for two impedances.
Table of Contents
What Is Complex Impedance?
Complex impedance (Z) extends the concept of resistance to AC circuits by combining resistance and reactance into a single complex number: Z = R + jX. The real part R represents energy dissipation; the imaginary part X represents energy storage (inductive if positive, capacitive if negative). Together they describe both the magnitude of opposition to current and the phase shift between voltage and current.
Working with complex impedance allows you to analyse AC circuits using the same series and parallel combination rules as DC resistance, except with complex arithmetic. The Impedance Calculator computes impedance from component values at a specific frequency; this calculator combines pre-calculated impedances in series or parallel networks.
Rectangular vs Polar Form
Polar: Z = |Z| ∠ θ — use for multiplication/division (V = IZ, parallel combination)
Convert: |Z| = √(R²+X²), θ = arctan(X/R), R = |Z|cosθ, X = |Z|sinθ
The Voltage Phasor Calculator uses the same rectangular/polar conversion for voltage phasors.
Series and Parallel Rules
Series: Impedances add directly as complex numbers. Ztotal = Z₁ + Z₂ + Z₃. Add the real parts and imaginary parts separately. This is identical to adding resistors in series, but using complex arithmetic.
Parallel: Add the reciprocals (admittances), then invert. 1/Ztotal = 1/Z₁ + 1/Z₂. For two impedances, the product-over-sum shortcut works: Ztotal = (Z₁ × Z₂) / (Z₁ + Z₂). Complex division requires converting to polar form or using the conjugate method.
Worked Example — Series RL at 1 kHz
Z₁ = 100 + j0 Ω
Z₂ = 0 + j62.83 Ω
Zseries = (100+0) + j(0+62.83) = 100 + j62.83 Ω
|Z| = √(100² + 62.83²) = 118.1 Ω ∠ 32.1°
The 32.1° phase angle means voltage leads current by that amount — characteristic of an inductive circuit. At higher frequencies, the inductive reactance grows and the phase angle approaches 90°. The Inductive Reactance Calculator shows how XL changes with frequency.
Worked Example — Parallel RC at 50 Hz
Z₁ = 1000 + j0 Ω, Z₂ = 0 − j318.3 Ω
Y₁ = 1/1000 = 0.001 S, Y₂ = j(1/318.3) = j0.00314 S
Ytotal = 0.001 + j0.00314 S
Zpar = 1/(0.001 + j0.00314) = 91.8 − j288.3 Ω = 303.5 Ω ∠ −72.3°
The negative phase angle confirms this is a capacitive network. The parallel combination always has lower impedance than either branch alone. The Capacitive Reactance Calculator can compute XC for different frequencies.
Worked Example — Mixed Series-Parallel Network
Step 1 — Parallel section: Zpar = (200)(0−j150) / (200 − j150)
= −j30000 / (200 − j150) = 72 − j96 Ω
Step 2 — Total: Z = 100 + (72 − j96) = 172 − j96 Ω = 197 Ω ∠ −29.2°
For mixed networks, calculate the parallel sections first, then add series impedances. The calculator handles one combination type at a time — do the parallel step first, then enter its result alongside the series resistor.
Admittance — The Reciprocal of Impedance
Admittance Y = 1/Z = G + jB, where G is conductance and B is susceptance. Admittance is the natural quantity for parallel combinations: Ytotal = Y₁ + Y₂. It is measured in siemens (S). Just as impedance tells you how much a circuit opposes current, admittance tells you how easily current flows.
Impedance and Phasor Analysis
Impedance connects voltage and current phasors through Ohm’s law in phasor form: V = I × Z. If you know two of the three (V, I, Z) you can find the third. Multiplying phasors uses polar form (multiply magnitudes, add angles), while adding uses rectangular form. The Phase Angle Calculator focuses specifically on the angle relationship between V and I.
Frequently Asked Questions
What is the j operator in impedance?
Can impedance be negative?
What does the phase angle tell me?
How do I handle mixed series-parallel circuits?
What is the difference between impedance and reactance?
Related AC Circuit Calculators
Browse all Electronics Calculators →