Impedance Explained: Formulas, Equations & AC Circuit Basics

Impedance is the AC (alternating current) equivalent of resistance. It is a complex number, Z = R + jX, combining resistance R and reactance X, and it is used in the AC form of Ohm’s Law, V = IZ, where voltage and current are represented as complex phasors. Impedance is measured in ohms (Ω) and, except for pure resistances, depends on frequency.

Key Takeaways

• Impedance combines resistance and reactance into a single complex quantity, Z = R + jX, and can also be written in polar form as Z = |Z|e^jφ.
• A resistor’s impedance is Z = R at every frequency, while a capacitor’s impedance is Z = 1/(jωC) and an inductor’s is Z = jωL, both of which change with angular frequency ω.
• Impedances in series simply add: Z = Z₁ + Z₂; impedances in parallel combine as 1/Z = 1/Z₁ + 1/Z₂.
• The intrinsic impedance of vacuum is defined exactly as 119.9169832π ohms, approximately 376.73 Ω.
• Average real power in an AC circuit is P_av = V_rmsI_rmscos φ, where cos φ is the power factor.

What Is Impedance in an AC Circuit?

The impedance of a load in an alternating current (AC) circuit is a complex number $Z = R + jX$, where $R$ is resistance and $X$ is reactance. The same impedance can be written in polar form as $Z = |Z|e^{j\phi}$, or as the phasor $|Z|\angle\phi$.

Impedance is the AC equivalent of resistance and appears in the AC form of Ohm’s Law: $V_{complex} = I_{complex}Z$, or equivalently $(V_{max}/I_{max})\angle\phi = Z$, 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 ($\Omega$).

What Are the Core Impedance Equations?

General Form and Ohm’s Law

For a load across which voltage leads current by 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 in AC form: $V_{complex} = I_{complex}Z$, and $(V_{max}/I_{max})\angle\phi = Z$, giving the magnitude relation $V_{max} = I_{max}|Z|$.

Series and Parallel Combination Rules

For impedances in series: $Z = Z_1 + Z_2$.

For impedances in parallel: $1/Z = 1/Z_1 + 1/Z_2$.

Impedance of Basic Components

Impedance formulas for a resistor, capacitor, and inductor at angular frequency ω

Component	Impedance formula	Phasor form
Resistor (all frequencies)	$Z = R$	$R\angle 0$
Capacitor (sinusoidal steady state)	$Z = 1/(j\omega C)$	$(1/\omega C)\angle -\pi/2$
Inductor	$Z = j\omega L$	$\omega L\angle \pi/2$

Characteristic Impedance of a Transmission Line

At angular frequency $\omega$, the characteristic impedance of a transmission line 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 of the line.

Intrinsic (Wave) Impedance of a Medium

The intrinsic impedance of a medium at angular frequency $\omega$ is:

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

where $\mu$ is permeability, $\varepsilon$ is permittivity, and $\sigma$ is conductivity.

The intrinsic impedance of vacuum is defined exactly as 119.9169832π ohms, approximately 376.73 Ω, given by:

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

How Does Impedance Work in Resistors, Capacitors, and Inductors?

In resistors, voltage and current are directly proportional: $V = IR$. In capacitors and inductors, the voltage-current 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 frequency, but shifted by 90 degrees in phase. Representing time-varying sinusoidal quantities as constant complex numbers, called phasors, allows the relationship to be written as $\boldsymbol{V} = \boldsymbol{I}Z$, where $\boldsymbol{V}$ and $\boldsymbol{I}$ are complex constants.

The actual time-domain voltage and current are sinusoidal functions. The phasor representations are constant complex amplitudes; multiplying a phasor by $e^{j\omega t}$ and taking the real part recovers the physical time-domain signal.

Complex Voltage and Current

In a steady sinusoidal circuit of frequency $\omega$, 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: $\boldsymbol{V} = V_x + jV_y = V_{max}e^{j\phi_V}$ and $\boldsymbol{I} = I_x + jI_y = I_{max}e^{j\phi_I}$, with $j^2 = -1$. Impedance is then defined as $Z = \boldsymbol{V}/\boldsymbol{I}$. Time derivatives correspond to multiplication by $j\omega$ for phasors: $\boldsymbol{V}’ = j\omega\boldsymbol{V}$ and $\boldsymbol{I}’ = j\omega\boldsymbol{I}$.

Phasor Form of Component Laws

The time-domain laws for a resistor, capacitor, and inductor are $V = RI$, $dV/dt = I/C$, and $V = L\,dI/dt$ respectively. Converted into phasor form, with no time derivatives, these become $\boldsymbol{V} = R\boldsymbol{I}$, $\boldsymbol{V} = \boldsymbol{I}/(j\omega C)$, and $\boldsymbol{V} = j\omega L\,\boldsymbol{I}$. The corresponding impedances are $Z = R$, $Z = 1/(j\omega C)$, and $Z = j\omega L$.

For circuits that are not operating at a single fixed frequency, the angular frequency $\omega$ can be replaced with a complex Laplace frequency $s$ to analyze transient behavior.

How Do Series and Parallel Impedance Laws Work?

Series Combinations

For components connected in series, with no junctions between them, the same current flows through each component and the total voltage equals the sum of the individual component voltages. The total impedance is therefore the sum of the individual impedances: $\boldsymbol{V}_{total} = \sum \boldsymbol{V} = (\sum Z)\,\boldsymbol{I}$.

Parallel Combinations

For sections connected in parallel, joined at the same two nodes, the voltage across each section is identical and the total current equals the sum of the branch currents. The total impedance satisfies $\boldsymbol{I}_{total} = \sum \boldsymbol{I} = \left(\sum 1/Z\right)\boldsymbol{V}$, so that $\boldsymbol{V} = \left(1\Big/\sum 1/Z\right)\boldsymbol{I}_{total}$.

How Is Phasor Arithmetic Performed?

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

Phasor multiplication and division are straightforward: $A\angle\phi \times B\angle\psi = AB\angle(\phi+\psi)$ and $A\angle\phi / B\angle\psi = (A/B)\angle(\phi-\psi)$. Addition and subtraction of phasors require converting to rectangular form first and are less straightforward than multiplication and division.

How Is Power Calculated in AC Circuits?

Instantaneous power in an AC circuit equals voltage multiplied by current. For sinusoidal voltage and current 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 (root mean square) terms: $P(t) = V_{rms}I_{rms}(\cos\phi + \cos 2\omega t)$.

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

Glossary

• Impedance (Z) — the complex AC equivalent of resistance, combining resistance and reactance.
• Reactance (X) — the imaginary part of impedance, arising from capacitors and inductors.
• Phasor — a constant complex number representing the amplitude and phase of a sinusoidal signal.
• RMS (root mean square) — a way of expressing an effective, time-averaged value of a varying voltage or current.
• Power factor — the ratio of average (real) power to apparent power, equal to cos φ.
• Characteristic impedance — the impedance a transmission line presents to a wave traveling along it.
• Intrinsic impedance — the ratio of electric to magnetic field in a wave propagating through a medium.

Frequently Asked Questions

What is the difference between impedance and resistance?

Resistance is a real number that opposes current flow the same way regardless of frequency. Impedance is a complex number, $Z = R + jX$, that includes both resistance and reactance, and except for pure resistances, it changes with frequency.

What is the impedance of a capacitor?

In sinusoidal steady state at angular frequency ω, a capacitor’s impedance is $Z = 1/(j\omega C)$, equivalent to $(1/\omega C)\angle -\pi/2$ in phasor notation.

What is the impedance of an inductor?

An inductor’s impedance is $Z = j\omega L$, equivalent to $\omega L\angle \pi/2$ in phasor notation, where ω is angular frequency and L is inductance.

How do you combine impedances in series and parallel?

Impedances in series add directly: $Z = Z_1 + Z_2$. Impedances in parallel combine by adding their reciprocals: $1/Z = 1/Z_1 + 1/Z_2$.

What is the intrinsic impedance of free space?

The intrinsic impedance of vacuum is defined exactly as 119.9169832π ohms, approximately 376.73 Ω, and is given by $Z_0 = \sqrt{\mu_0/\varepsilon_0} = \mu_0 c$.

How is average power calculated in an AC circuit?

Average, or real, power is $P_{av} = V_{rms}I_{rms}\cos\phi$, where φ is the phase difference between voltage and current. This differs from apparent power, $S = V_{rms}I_{rms}$, by the power factor cos φ.

For further discussion, see the original Physics Forums thread on impedance.