Electrical impedance

From Wikipedia, the free encyclopedia

Electromagnetism
Electricity · Magnetism
Electrostatics
 · Electric charge · Coulomb’s law · Electric field · Electric flux · Gauss’ law · Electric potential · Electrostatic induction · Electric dipole moment ·
Magnetostatics
 · Ampère’s law · Electric current · Magnetic field · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss’s law for magnetism ·
Electrodynamics
 · Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potentials · Maxwell tensor · Eddy current ·
Electrical Network
 · Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides ·
Covariant formulation
 · Electromagnetic tensor · EM Stress-energy tensor · Four-current · Four-potential ·
Scientists
 · Ampere · Coulomb · Faraday · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Weber ·
This box: view • talk • edit
\scriptstyle{\tilde{Z}}
\tilde{Z} = Z e^{j\theta} \quad
\scriptstyle{Z}
\tilde{Z} = R + j\Chi \quad
\scriptstyle{R}

The reciprocal of impedance is admittance.

A graphical representation of the complex impedance plane.  Note that while reactance  can be either positive or negative, resistance  is always positive. \scriptstyle{\Chi}

Contents

Ohm's law

An AC supply applying a voltage , across a load , driving a current . \scriptstyle{V}
Main article: Ohm's law

The meaning of electrical impedance can be understood by substituting it into Ohm's law.

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

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

Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box. Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box. \scriptstyle{\tilde{V}}
\ \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

Main article: Phasor (electronics)

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.

\scriptstyle{e^{j\omega t}}

Device examples

Main article: Impedance of different devices (derivations)
The phase angles in the equations for the impedance of inductors and capacitors indicate that the voltage across a capacitor lags the current through it by a phase of , while the voltage across an inductor leads the current through it by .  The identical voltage and current amplitudes tell us that the magnitude of the impedance is equal to one. \scriptstyle{\pi/2}

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.

Deriving the device specific impedances

What follows below is a derivation of impedance for each of the three basic circuit elements, the resistor, the capacitor, and the inductor. Although the idea can be extended to define the relationship between the voltage and current of any arbitrary signal, these derivations will assume sinusoidal signals, since any arbitrary signal can be approximated as a sum of sinusoids through Fourier Analysis.

  Resistor
For a resistor, we have the relation:
v_\mathrm{R} \left( t \right) = {i_\mathrm{R} \left( t \right)}R

This is simply a statement of Ohm's Law.

Considering the voltage signal to be

v_\mathrm{R}(t) = V_p \sin(\omega t) \,

it follows that

\frac{v_\mathrm{R} \left( t \right)}{i_\mathrm{R} \left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \left( \omega  t \right)} = R.

This tells us that the ratio of AC voltage amplitude to AC current amplitude across a resistor is R, and that the AC voltage leads the AC current across a resistor by 0 degrees.

This result is commonly expressed as

Z_\mathrm{resistor} = R \,
  Capacitor
For a capacitor, we have the relation:
i_\mathrm{C}(t) = C \frac{dv_\mathrm{C}(t)}{dt}.

Considering the voltage signal to be

v_\mathrm{C}(t) = V_p \sin(\omega t) \,

it follows that

\frac{dv_\mathrm{C}(t)}{dt} = \omega  V_p \cos \left( \omega  t \right).

And thus

\frac{v_\mathrm{C} \left( t \right)}{i_\mathrm{C} \left( t \right)} = \frac{V_p \sin(\omega t)}{\omega V_p C \cos \left( \omega  t \right)}= \frac{\sin(\omega t)}{\omega C \sin \left( \omega  t + \frac{\pi}{2}\right)}.
\frac{1}{\omega C}

This result is commonly expressed in polar form, as

\tilde{Z}_\mathrm{capacitor} = \frac{1}{\omega C} e^{-j \frac{\pi}{2}}

or, more simply, using of Euler's formula, as

\tilde{Z}_\mathrm{capacitor} = \frac{1}{j \omega C}
  Inductor

For the inductor, we have the relation:

v_\mathrm{L}(t) = L \frac{di_\mathrm{L}(t)}{dt}.

This time, considering the current signal to be

i_\mathrm{L}(t) = I_p \sin(\omega t) \,

it follows that

\frac{di_\mathrm{L}(t)}{dt} = \omega  I_p \cos \left( \omega  t \right).

And thus

\frac{v_\mathrm{L} \left( t \right)}{i_\mathrm{L} \left( t \right)} = \frac{\omega I_p L \cos(\omega t)}{I_p \sin \left( \omega  t \right)}= \frac{\omega L \sin \left( \omega  t + \frac{\pi}{2}\right)}{\sin(\omega t)}.

This tells us that the ratio of AC voltage amplitude to AC current amplitude across an inductor is ωL, and that the AC voltage leads the AC current across an inductor by 90 degrees.

This result is commonly expressed in polar form, as

\tilde{Z}_\mathrm{inductor} = \omega L e^{j \frac{\pi}{2}}

or, more simply, using Euler's formula, as

\tilde{Z}_\mathrm{inductor} = j \omega L. \,

Resistance vs reactance

It is important to realize 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

Main article: Electrical resistance
\scriptstyle{R}
R = Z \cos{\theta} \quad

Reactance

Main article: Reactance (electronics)
\scriptstyle{\Chi}
\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

Main article: Capacitor

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 charge 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 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 current.

Inductive reactance

Main article: Inductor
\scriptstyle{\mathcal{E}}
\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

Main article: Series and parallel circuits

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.

Image:Impedances in series.svg
\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
\scriptstyle{\tilde{Z}_{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}

Measuring Impedance

According to Ohm’s law the impedance of a device can be calculated by complex division of the voltage and current. The impedance of the device can be calculated by applying a sinusoidal voltage to the device in series with a resistor, and measuring the voltage across the resistor and across the device. Performing this measurement by sweeping the frequencies of the applied signal provides the impedance phase and magnitude.

Impulse impedance spectroscopy

The use of an impulse response may be used in combintation with the fast Fourier transform (FFT) to rapidly measure the electrical impedance of various electrical devices. The technique compares well to other methodologies such as network and impedance analyzers while providing additional versatility in the electrical impedance measurement. The technique is theoretically simple, easy to implement and completed with ordinary laboratory instrumentation for minimal cost.