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Impedance (Z) extends resistance to AC circuits by combining resistance with reactance. It is a complex number, Z = R + jX, whose magnitude sets how much current a given voltage drives, and whose phase angle shows how far the current leads or lags the voltage.
| Quantity | Series | Parallel |
|---|---|---|
| Combine | Z = R + j(XL−XC) | 1/Z = 1/R + j(1/XL−1/XC) |
| Magnitude | |Z| = √(R²+X²) | |Z| = 1/|Y| |
| Phase angle | θ = atan(X/R) | θ = atan(Im(Y)/Re(Y)) |
A positive angle means inductive (current lags voltage); negative means capacitive (current leads voltage).
Impedance combines resistance and reactance, but not by simple addition — they are 90° apart, so they add as perpendicular vectors:
|Z| = √(R2 + X2) φ = arctan(X/R)
With R = 10 Ω and XL = 31.42 Ω, the magnitude is √(100 + 987) ≈ 32.97 Ω at a phase angle of 72.35°. Adding them arithmetically would give 41.4 Ω — wrong by over 25%.
That 72.35° is not an abstract quantity. Its cosine is the power factor: cos 72.35° ≈ 0.303. So this circuit, despite drawing substantial current, converts only about 30% of its apparent power into real work. Impedance magnitude tells you how much current flows; the phase angle tells you how much of it is doing anything useful.
In series, impedances add as complex numbers — resistances sum and reactances sum, with inductive and capacitive reactance opposing each other. In parallel, it is the admittances that add, so the reciprocals combine, and the result can behave unintuitively: a parallel LC combination near resonance presents a far higher impedance than either branch alone.
This is also why impedance cannot be measured with an ordinary ohmmeter. A multimeter applies DC, at which inductors are near short circuits and capacitors are open circuits, so it reports the resistance of the copper and nothing about the circuit's behaviour at its working frequency.
Impedance (Z) is the total opposition to AC current, combining resistance (which dissipates energy) and reactance (which stores and returns energy). It is written as a complex number Z = R + jX.
Resistance only opposes current and dissipates power; impedance also accounts for the frequency-dependent, phase-shifting effect of capacitors and inductors.
The angle by which current leads or lags voltage. A positive angle means inductive behaviour (current lags); a negative angle means capacitive behaviour (current leads).
Add the resistances and reactances separately: Z = R + j(XL−XC), then find the magnitude √(R²+X²) and phase atan(X/R).
Add their admittances (1/Z) rather than the impedances directly, since parallel impedances do not simply average like parallel resistors with reactance involved.
XL and XC cancel, so a series circuit's impedance drops to just R (minimum), while a parallel circuit's impedance rises to its maximum.
The phase angle of the impedance is the same as the angle between voltage and current, which directly sets the power factor: PF = cos(θ).
The reciprocal of impedance, Y = 1/Z, measured in siemens. It is convenient for combining parallel branches, since admittances simply add.
Yes — resistance is constant, but XL = 2πfL grows with frequency and XC = 1/(2πfC) shrinks with frequency, so total impedance changes across the spectrum.
One where the reactive part is zero (XL=XC, or no reactive components at all), so the phase angle is 0° and Z = R.
A source delivers maximum power to a load when the load impedance matches (or is the complex conjugate of) the source impedance, minimizing reflections and maximizing transfer.
The magnitude is always positive, but the reactive part X can be negative (capacitive) or positive (inductive), which shows up as a negative or positive phase angle.
|Z| = √(R² + X²), because resistance and reactance are 90° apart and combine as perpendicular vectors. The phase angle is arctan(X/R).
Because they are out of phase. Adding 10 Ω and 31.42 Ω arithmetically gives 41.4 Ω, but the correct vector result is 32.97 Ω — an error of more than 25%.
It is the angle by which current leads or lags voltage, and its cosine is the power factor. A large phase angle means most of the current is reactive and does no useful work.
No. A multimeter measures DC resistance, at which an inductor looks like a short and a capacitor like an open circuit. Measuring impedance requires an LCR meter or an impedance analyser operating at the frequency of interest.
Admittances add, so the reciprocals combine. The result is not intuitive near resonance: a parallel LC combination can present a much higher impedance than either branch on its own.
A property of a transmission line, equal to √(L/C) per unit length, describing the ratio of voltage to current for a wave travelling along it. It is a different concept from the lumped impedance of a component, though both are measured in ohms.
Reactance (XC & XL) • RLC Resonant Frequency • Power Triangle • All Calculators