Electrical impedance

Electrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal alternating current (AC). Electrical impedance extends the concept of resistance to AC circuits, describing not only the relative amplitudes of the voltage and current, but also the relative phases. In general impedance is a complex quantity $$\scriptstyle{\tilde{Z}}$$ and the term complex impedance may be used interchangeably; the polar form conveniently captures both magnitude and phase characteristics,


 * $$\tilde{Z} = Z e^{j\theta} \quad$$

where the magnitude $$\scriptstyle{Z}$$ gives the change in voltage amplitude for a given current amplitude, while the argument $$\scriptstyle{\theta}$$ gives the phase difference between voltage and current. In Cartesian form,


 * $$\tilde{Z} = R + j\Chi \quad$$

where the real part of impedance is the resistance $$\scriptstyle{R}$$ and the imaginary part is the reactance $$\scriptstyle{\Chi}$$. Dimensionally, impedance is the same as resistance; the SI unit is the ohm. The term impedance was coined by Oliver Heaviside in July 1886.

Ohm's law


We can understand this by substituting it into Ohm's law.


 * $$\tilde{V} = \tilde{I}\tilde{Z} = \tilde{I} Z e^{j\theta} \quad$$

The magnitude of the impedance $$\scriptstyle{Z}$$ acts just like resistance, giving the drop in voltage amplitude across an impedance $$\scriptstyle{\tilde{Z}}$$ for a given current $$\scriptstyle{\tilde{I}}$$. The phase factor tells us that the current lags the voltage by a phase of $$\theta$$ (i.e. in the time domain, the current signal is shifted $$\frac{\theta T}{2 \pi}$$ to the right with respect to the voltage signal).

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis such as voltage division, current division, Thevenin's theorem, and Norton's theorem, can also be extended to AC circuits by replacing resistance with impedance.

Complex voltage and current


In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as $$\scriptstyle{\tilde{V}}$$ and $$\scriptstyle{\tilde{I}}$$.


 * $$\ \tilde{V} = V_0e^{j(\omega t + \phi_V)}$$


 * $$\ \tilde{I} = I_0e^{j(\omega t + \phi_I)}$$

Impedance is defined as the ratio of these quantities.


 * $$\ \tilde{Z} = {\tilde{V} \over \tilde{I}}$$

Substituting these into Ohm's law we have



\begin{align} V_0e^{j(\omega t + \phi_V)} &= I_0e^{j(\omega t + \phi_I)} Z e^{j\theta}   \\ &= I_0 Z e^{j(\omega t + \phi_I + \theta)} \end{align} $$

Noting that this must hold for all $$t$$, we may equate the magnitudes and phases to obtain


 * $$\ V_0 = I_0 Z \quad$$


 * $$\ \phi_V = \phi_I + \theta \quad$$

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

Validity of complex representation
This representation using complex exponentials may be justified by noting that (by Euler's formula):


 * $$\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]$$

i.e. a real-valued sinusoidal function (which may represent our voltage or current waveform) may be broken into two complex-valued functions. By the principle of superposition, we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term; the results will be identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that


 * $$\ \cos(\omega t + \phi) = \Re \Big\{ e^{j(\omega t + \phi)} \Big\}$$

In other words, we simply take the real part of the result.

Phasors
A phasor is a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from Ohm's law given above, recognising that the factors of $$\scriptstyle{e^{j\omega t}}$$ cancel.

Device examples


The impedance of a resistor is purely real and is referred to as a resistive impedance.


 * $$\tilde{Z}_R = R \quad$$

Inductors and capacitors have a purely imaginary reactive impedance.


 * $$\tilde{Z}_L = j\omega L \quad$$


 * $$\tilde{Z}_C = {1 \over j\omega C}$$

Note the following identities for the imaginary unit and its reciprocal.


 * $$j = \cos{\left({\pi \over 2}\right)} + j\sin{\left({\pi \over 2}\right)} = e^{j{\pi \over 2}}$$


 * $${1 \over j} = -j = \cos{\left(-{\pi \over 2}\right)} + j\sin{\left(-{\pi \over 2}\right)} = e^{j(-{\pi \over 2})}$$

Thus we can rewrite the inductor and capacitor impedance equations in polar form


 * $$\tilde{Z}_L = \omega Le^{j{\pi \over 2}}$$


 * $$\tilde{Z}_C = {1 \over \omega C}e^{j(-{\pi \over 2})}$$

The magnitude tells us the change in voltage amplitude for a given current amplitude through our impedance, while the exponential factors give the phase relationship.

Resistance vs Reactance
It is important to realise that resistance and reactance are not individually significant; together they determine the magnitude and phase of the impedance, through the following relations.


 * $$|\tilde{Z}| = \sqrt{\tilde{Z}\tilde{Z}^*} = \sqrt{R^2 + \Chi^2}$$


 * $$\theta = \arctan{\left({\Chi \over R}\right)}$$

In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

Resistance
Resistance $$\scriptstyle{R}$$ is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.


 * $$R = Z \cos{\theta} \quad$$

Reactance
Reactance $$\scriptstyle{\Chi}$$ is the imaginary part of the impedance; a component with a finite reactance induces a phase shift $$\theta$$ between the voltage across it and the current through it.


 * $$\Chi = Z \sin{\theta} \quad$$

A reactive component is distinguished by the fact that the sinusoidal voltage across the component is in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance will not dissipate any power.

Capacitive reactance
A capacitor has a purely reactive impedance which is inversely proportional to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

At low frequencies a capacitor is open circuit, as no current flows in the dielectric. A DC voltage applied across a capacitor causes charge to accumulate on one side, the electric field due to the accumulated charge is the source of the opposition to the flow of current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the flow of current.

Inductive reactance
An inductor has a purely reactive impedance which is proportional to the signal frequency. An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf $$\scriptstyle{\mathcal{E}}$$ (voltage opposing current) due to a rate-of-change of magnetic field $$\scriptstyle{B}$$ through a current loop.


 * $$\mathcal{E} = -{{d\Phi_B} \over dt}.$$

For an inductor consisting of a coil with $$N$$ loops this gives.


 * $$\mathcal{E} = -N{d\Phi_B \over dt}.$$

The back-emf is the source of the opposition to current flow. A constant direct current has a zero rate-of-change, and sees an inductor as a short-circuit (it is typically made from a material with a low resistivity). An alternating current has a time rate-of-change that is proportional to frequency and so the inductive reactance is proportional to frequency.

Combining impedances
The total impedance of any network of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those used for combining resistances, although they require some familiarity with complex numbers.

Series combination
For components connected in series, the current through each circuit element is the same; the ratio of voltages across any two elements is the inverse ratio of their impedances.




 * $$\tilde{Z}_{eq} = \tilde{Z}_1 + \tilde{Z}_2 = (R_1 + R_2) + j(\Chi_1 + \Chi_2) \quad$$

Parallel combination
For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.


 * [[Image:Impedances in parallel.svg]]


 * $$\tilde{Z}_{eq} = \tilde{Z}_1 \| \tilde{Z}_2 = \left(\tilde{Z}_1^{-1} + \tilde{Z}_2^{-1}\right)^{-1} = {\tilde{Z}_1 \tilde{Z}_2 \over \tilde{Z}_1 + \tilde{Z}_2} \quad$$

The equivalent impedance $$\scriptstyle{\tilde{Z}_{eq}}$$ can be calculated in terms of the equivalent resistance $$\scriptstyle{R_{eq}}$$ and reactance $$\scriptstyle{\Chi_{eq}}$$.


 * $$\tilde{Z}_{eq} = R_{eq} + j \Chi_{eq} \quad$$


 * $$R_{eq} = { (\Chi_1 R_2 + \Chi_2 R_1) (\Chi_1 + \Chi_2) + (R_1 R_2 - \Chi_1 \Chi_2) (R_1 + R_2) \over (R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}$$
 * $$\Chi_{eq} = {(\Chi_1 R_2 + \Chi_2 R_1) (R_1 + R_2) - (R_1 R_2 - \Chi_1 \Chi_2) (\Chi_1 + \Chi_2) \over (R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}$$