Impedance in AC Circuits: Definition & Key Equations

Definition / Summary

The impedance of a load (a combination of components) in an AC circuit is a complex number $Z = R + jX$, where $R$ is resistance and $X$ is reactance.

Polar form

The same impedance can be written in polar form: $Z = |Z|e^{j\phi}$, or as the phasor $|Z|\angle\phi$.

Units and frequency dependence

Impedance is the AC equivalent of resistance and is used in the AC form of Ohm’s Law: $V_{complex} = I_{complex}Z$ (or $(V_{max}/I_{max})\angle\phi = Z$, where $\phi$ is the phase difference by which the voltage leads the current). Impedance depends on frequency except for pure resistances, and it is measured in ohms $\Omega$.

Equations

For a load across which the voltage leads the current by a phase angle $\phi$:

$Z = |Z|\cos\phi + j|Z|\sin\phi = R + jX$

(Polar form: $Z = |Z|e^{j\phi}$; Phasor notation: $Z = |Z|\angle\phi$.)

Ohm’s Law

$V_{complex} = I_{complex}Z$

$(V_{max}/I_{max})\angle\phi = Z$

$V_{max} = I_{max}|Z|$

Series combination

$Z = Z_1 + Z_2$

Parallel combination

$1/Z = 1/Z_1 + 1/Z_2$

Basic component impedances

Resistor (all frequencies): $Z = R = R\angle 0$

Capacitor (sinusoidal steady state, frequency [itex]\omega[/itex>]): $Z = 1/(j\omega C) = (1/\omega C)\angle -\pi/2$

Inductor: $Z = j\omega L = \omega L\angle \pi/2$

Characteristic impedance of a transmission line

At angular frequency $\omega$ the characteristic impedance is:

$Z_0 = \sqrt{\dfrac{R_x + j\omega L_x}{G_x + j\omega C_x}}$

Here $R_x, L_x, G_x, C_x$ are the resistance, inductance, conductance (of the dielectric), and capacitance per unit length.

Intrinsic (wave) impedance of a medium

The intrinsic impedance of a medium is

$Z = \sqrt{\dfrac{j\omega\mu}{\sigma + j\omega\varepsilon}} = \sqrt{\dfrac{\mu}{\varepsilon – j\sigma/\omega}}$

for angular frequency $\omega$, where $\mu$ is permeability, $\varepsilon$ is permittivity, and $\sigma$ is conductivity.

The intrinsic impedance of vacuum is

$Z_0 = \sqrt{\dfrac{\mu_0}{\varepsilon_0}} = \mu_0 c = \dfrac{1}{\varepsilon_0 c}$

which is defined exactly as $119.9169832\pi\ \Omega$, or approximately $376.73\ \Omega$.

Extended explanation

Introduction

In resistors voltage and current are proportional: $V = IR$. In capacitors and inductors the relationship involves time derivatives: for a capacitor $dV/dt = I/C$, and for an inductor $V = L\,dI/dt$.

In steady-state sinusoidal (AC) circuits, the rates of change of voltage and current are proportional to the original signals and to the frequency, but shifted by 90° in phase. A convenient mathematical device replaces the time-varying sinusoidal quantities with constant complex numbers (phasors), so that $\boldsymbol{V}$ and $\boldsymbol{I}$ are complex constants and $\boldsymbol{V} = \boldsymbol{I}Z$.

Note: actual time-domain voltage and current are sinusoidal functions; the complex (phasor) representations are constant complex amplitudes whose real parts — when multiplied by $e^{j\omega t}$ and taking the real part — give the physical time-domain signals.

Complex voltage and current

In a steady sinusoidal circuit of frequency $\omega$, the instantaneous voltage and current can be written as

$V = V_x\cos\omega t + V_y\sin\omega t$ and $I = I_x\cos\omega t + I_y\sin\omega t$.

Equivalently,

$V = V_{max}\cos(\omega t + \phi_V) = \Re\{V_{max}e^{j\phi_V}e^{j\omega t}\}$

$I = I_{max}\cos(\omega t + \phi_I) = \Re\{I_{max}e^{j\phi_I}e^{j\omega t}\}$

The complex (phasor) voltage and current are the complex amplitudes:

$\boldsymbol{V} = V_x + jV_y = V_{max}e^{j\phi_V}$, $\boldsymbol{I} = I_x + jI_y = I_{max}e^{j\phi_I}$, with $j^2 = -1$. The impedance is $Z = \boldsymbol{V}/\boldsymbol{I}$.

Time derivatives correspond to multiplication by $j\omega$ for phasors: $\boldsymbol{V}’ = j\omega\boldsymbol{V}$ and $\boldsymbol{I}’ = j\omega\boldsymbol{I}$.

Resistors, capacitors and inductors (phasor form)

Time-domain laws:

$V = RI,\ \ dV/dt = I/C,\ \ V = L\,dI/dt$

In phasor form (no time derivatives):

$\boldsymbol{V} = R\boldsymbol{I},\ \ \boldsymbol{V} = \boldsymbol{I}/(j\omega C),\ \ \boldsymbol{V} = j\omega L\,\boldsymbol{I}$

Hence the impedances are $Z = R,\ Z = 1/(j\omega C),\ Z = j\omega L$.

For circuits not in a single fixed frequency, replace $\omega$ with a complex Laplace frequency $s$ to analyze transient behavior.

Series law

For components in series (no junctions between them), the current is the same through each component and total voltage is the sum of component voltages. Thus the total impedance is the sum of the individual impedances:

$\boldsymbol{V}_{total} = \sum \boldsymbol{V} = (\sum Z)\,\boldsymbol{I}$

Parallel law

For sections in parallel (joined at the same two nodes), the voltage across each section is the same and the total current is the sum of the currents through each branch. Thus the total impedance satisfies:

$\boldsymbol{I}_{total} = \sum \boldsymbol{I} = \left(\sum 1/Z\right)\boldsymbol{V},\quad\text{so}\quad \boldsymbol{V} = \left(1\Big/\sum 1/Z\right)\boldsymbol{I}_{total}$

Phasor form and arithmetic

Any complex number, such as $\boldsymbol{V} = V_x + jV_y$, can be written as $V_{max}(\cos\phi + j\sin\phi)$ where $\tan\phi = V_y/V_x$. In polar form this is $(V_{max},\phi)$ or phasor notation $V_{max}\angle\phi$.

Phasor multiplication and division are simple:

$A\angle\phi \times B\angle\psi = AB\angle(\phi+\psi),\quad A\angle\phi / B\angle\psi = (A/B)\angle(\phi-\psi)$

Addition and subtraction of phasors require converting to rectangular form and are less straightforward.

Power in AC circuits

Instantaneous power is voltage times current. For sinusoids with phase difference $\phi$:

$P(t) = V_{max}I_{max}\cos(\omega t + \phi/2)\cos(\omega t – \phi/2) = \dfrac{V_{max}I_{max}}{2}(\cos\phi + \cos 2\omega t)$

In RMS terms:

$P(t) = V_{rms}I_{rms}(\cos\phi + \cos 2\omega t)$

The average (real) power is $P_{av} = V_{rms}I_{rms}\cos\phi$. Apparent power is $S = V_{rms}I_{rms}$, reactive power $Q = S\sin\phi$, and complex power $S e^{j\phi} = P_{av} + jQ$. The power factor is $\cos\phi = P_{av}/S$.

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