Electrical impedance: Wikis

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A graphical representation of the complex impedance plane.

Electrical impedance, or simply impedance, describes a measure of opposition to 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. When the circuit is driven with direct current (DC) there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.

The symbol for impedance is usually $\scriptstyle Z$ and it may be represented by writing its magnitude and phase in the form $\scriptstyle Z \angle \theta$. However, complex number representation is more powerful for circuit analysis purposes. The term impedance was coined by Oliver Heaviside in July 1886.[1][2] Arthur Kennelly was the first to represent impedance with complex numbers in 1893.[3]

Impedance is defined as the frequency domain ratio of the voltage to the current. In other words, it is voltage–current ratio for a single complex exponential at a particular frequency ω. In general, impedance will be a complex number, but this complex number has the same units as resistance, for which the SI unit is the ohm. For a sinusoidal current or voltage input, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular,

• The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude.
• The phase of the complex impedance is the phase shift by which the current is ahead of the voltage.

The reciprocal of impedance is admittance (i.e., admittance is the current-to-voltage ratio, and it conventionally carries mho or Siemens units).

Complex impedance

Impedance is represented as 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 $\scriptstyle Z$ represents the ratio of the voltage difference amplitude to the current amplitude, while the argument $\scriptstyle \theta$ gives the phase difference between voltage and current and $\scriptstyle j$ is the imaginary unit. 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 $\scriptstyle \Chi$.

Where it is required to add or subtract impedances the cartesian form is more convenient, but when quantities are multiplied or divided the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal conversion rules of complex numbers.

Ohm's law

An AC supply applying a voltage $\scriptstyle V$, across a load $\scriptstyle Z$, driving a current $\scriptstyle I$.

The meaning of electrical impedance can be understood by substituting it into Ohm's law.[4][5]

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

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 $\scriptstyle \theta$ (i.e. in the time domain, the current signal is shifted $\scriptstyle \frac{\theta}{2 \pi} T$ to the right with respect to the voltage signal).[6]

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.

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 $\scriptstyle \tilde{I}$.[7][8]

$\ \tilde{V} = V_0e^{j(\omega t + \phi_V)}$
$\ \tilde{I} = I_0e^{j(\omega t + \phi_I)}$

Impedance is defined as the ratio of these quantities.

$\ \tilde{Z} = \frac{\tilde{V}}{\tilde{I}}$

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 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 π / 2, while the voltage across an inductor leads the current through it by π / 2. The identical voltage and current amplitudes tell us that the magnitude of the impedance is equal to one.

The impedance of an ideal resistor is purely real and is referred to as a resistive impedance:

$\tilde{Z}_R = R.$

Ideal inductors and capacitors have a purely imaginary reactive impedance:

$\tilde{Z}_L = j\omega L,$
$\tilde{Z}_C = \frac{1}{j\omega C} \, .$

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

$j = \cos{\left(\frac{\pi}{2}\right)} + j\sin{\left(\frac{\pi}{2}\right)} = e^{j\frac{\pi}{2}},$
$\frac{1}{j} = -j = \cos{\left(-\frac{\pi}{2}\right)} + j\sin{\left(-\frac{\pi}{2}\right)} = e^{j(-\frac{\pi}{2})}.$

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

$\tilde{Z}_L = \omega Le^{j\frac{\pi}{2}},$
$\tilde{Z}_C = \frac{1}{\omega C}e^{j(-\frac{\pi}{2})}.$

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:

$v_{\text{R}} \left( t \right) = {i_{\text{R}} \left( t \right)}R.$

This is simply a statement of Ohm's Law.

Considering the voltage signal to be

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

it follows that

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

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

This result is commonly expressed as

$Z_{\text{resistor}} = R \, .$

Capacitor

For a capacitor, we have the relation:

$i_{\text{C}}(t) = C \frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t}.$

Considering the voltage signal to be

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

it follows that

$\frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t} = \omega V_p \cos \left( \omega t \right).$

And thus

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

This tells us that the ratio of AC voltage amplitude to AC current amplitude across a capacitor is $\scriptstyle \frac{1}{\omega C}$, and that the AC voltage leads the AC current across a capacitor by -90 degrees (or the AC current leads the AC voltage across a capacitor by 90 degrees).

This result is commonly expressed in polar form, as

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

or, by applying Euler's formula, as

$\tilde{Z}_{\text{capacitor}} = -j\frac{1}{\omega C} = \frac{1}{j \omega C} \, .$

Inductor

For the inductor, we have the relation:

$v_{\text{L}}(t) = L \frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t}.$

This time, considering the current signal to be

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

it follows that

$\frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t} = \omega I_p \cos \left( \omega t \right).$

And thus

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

This tells us that the ratio of AC voltage amplitude to AC current amplitude across an inductor is $\scriptstyle \omega L$, and that the AC voltage leads the AC current across an inductor by 90 degrees.

This result is commonly expressed in polar form, as

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

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

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

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:

$|\tilde{Z}| = \sqrt{\tilde{Z}\tilde{Z}^*} = \sqrt{R^2 + \Chi^2}$
$\theta = \arctan{\left(\frac{\Chi}{R}\right)}$

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 $\scriptstyle \theta$ between the voltage across it and the current through it.

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

A purely 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 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

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.

$\mathcal{E} = -\frac{{\operatorname{d}\Phi_B}}{\operatorname{d}t}.$

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

$\mathcal{E} = -N\frac{\operatorname{d}\Phi_B}{\operatorname{d}t}.$

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 many simple networks 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, except that the numbers in general will be complex numbers. In the general case however, equivalent impedance transforms in addition to series and parallel will be required.

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.

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

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.

$\tilde{Z}_{\text{eq}} = \tilde{Z}_1 \| \tilde{Z}_2 \,\stackrel{\text{def}}{=}\, \left(\tilde{Z}_1^{-1} + \tilde{Z}_2^{-1}\right)^{-1} = \frac{\tilde{Z}_1 \tilde{Z}_2}{\tilde{Z}_1 + \tilde{Z}_2} \quad$

The equivalent impedance $\scriptstyle \tilde{Z}_{\text{eq}}$ can be calculated in terms of the equivalent resistance $\scriptstyle R_{\text{eq}}$ and reactance $\scriptstyle \Chi_{\text{eq}}$.[9]

$\tilde{Z}_{\text{eq}} = R_{\text{eq}} + j \Chi_{\text{eq}} \quad$
$R_{\text{eq}} = \frac{(\Chi_1 R_2 + \Chi_2 R_1) (\Chi_1 + \Chi_2) + (R_1 R_2 - \Chi_1 \Chi_2) (R_1 + R_2)}{(R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}$
$\Chi_{\text{eq}} = \frac{(\Chi_1 R_2 + \Chi_2 R_1) (R_1 + R_2) - (R_1 R_2 - \Chi_1 \Chi_2) (\Chi_1 + \Chi_2)}{(R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}$

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.[10]

Impulse impedance spectroscopy

The use of an impulse response may be used in combination 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.[10]

Variable impedance

In general, neither impedance nor admittance can be time varying as they are defined for complex exponentials for –∞ < t < +∞. If the complex exponential voltage–current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many systems (e.g., varicaps that are used in radio tuners) may exhibit non-linear or time-varying voltage–current ratios that appear to be LTI for small signals over small observation windows; hence, they can be roughly described as having a time-varying impedance. That is, this description is an approximation; over large signal swings or observation windows, the voltage–current relationship is non-LTI and cannot be described by impedance.

References

1. ^ Science, p. 18, 1888
2. ^ Oliver Heaviside, The Electrician, p. 212, 23rd July 1886 reprinted as Electrical Papers, p64, AMS Bookstore, ISBN 0821834657
3. ^ Kennelly, Arthur. Impedance (IEEE, 1893)
4. ^ AC Ohm's law, Hyperphysics
5. ^ Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 32–33. ISBN 0-521-37095-7.
6. ^ Capacitor/inductor phase relationships, Yokogawa
7. ^ Complex impedance, Hyperphysics
8. ^ Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 31–32. ISBN 0-521-37095-7.
9. ^ Parallel Impedance Expressions, Hyperphysics
10. ^ a b Lewis Jr., George; George K. Lewis Sr. and William Olbricht (August 2008). "Cost-effective broad-band electrical impedance spectroscopy measurement circuit and signal analysis for piezo-materials and ultrasound transducers". Measurement Science and Technology 19: 105102. doi:10.1088/0957-0233/19/10/105102. Retrieved 2008-09-15.

Study guide

Up to date as of January 14, 2010

From Wikiversity

Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. The concept of electrical impedance generalises Ohm's law to AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit can be a complex number, but the same unit, the ohm, is used for both quantities. Oliver Heaviside coined the term "impedance" in July of 1886.

Generalized impedances in a circuit can be drawn with the same symbol as a resistor or with a labeled box.

In general, the solutions for the voltages and currents in a circuit containing resistors, capacitors and inductors (in short, all linearly behaving components) are solutions to a linear ordinary differential equation. It can be shown that if the voltage and current sources in the circuit are sinusoidal and of constant frequency, the solutions take a form referred to as AC steady state. Thus, all of the voltages and currents in the circuit are sinusoidal and have constant amplitude, frequency and phase.

In AC steady state, v(t) is a sinusoidal function of time with constant amplitude Vp, constant frequency f, and constant phase $\varphi$:

$v(t) = V_\mathrm{p} \cos \left( 2 \pi f t + \varphi \right) = \Re \left( V_\mathrm{p} e^{j 2 \pi f t} e^{j \varphi} \right)$

where

j represents the imaginary unit ($\sqrt{-1}$)
$\Re (z)$ means the real part of the complex number z.

The phasor representation of v(t) is the constant complex number V:

$V = V_\mathrm{p} e^{j \varphi} \,$

For a circuit in AC steady state, all of the voltages and currents in the circuit have phasor representations as long as all the sources are of the same frequency. That is, each voltage and current can be represented as a constant complex number. For DC circuit analysis, each voltage and current is represented by a constant real number. Thus, it is reasonable to suppose that the rules developed for DC circuit analysis can be used for AC circuit analysis by using complex numbers instead of real numbers.

Definition of electrical impedance

The impedance of a circuit element is defined as the ratio of the phasor voltage across the element to the phasor current through the element:

$Z_\mathrm{R} = \frac{V_\mathrm{r}}{I_\mathrm{r}}$

It should be noted that although Z is the ratio of two phasors, Z is not itself a phasor. That is, Z is not associated with some sinusoidal function of time.

For DC circuits, the resistance is defined by Ohm's law to be the ratio of the DC voltage across the resistor to the DC current through the resistor:

$R = \frac{V_\mathrm{R}}{I_\mathrm{R}}$

where

VR and IR above are DC (constant real) values.

Just as Ohm's law is generalized to AC circuits through the use of phasors, other results from DC circuit analysis such as voltage division, current division, Thevenin's theorem, and Norton's theorem generalize to AC circuits.

Impedance of different devices

For a resistor:

$Z_\mathrm{resistor} = \frac{V_\mathrm{R}}{I_\mathrm{R}} = R \,$

For a capacitor:

$Z_\mathrm{capacitor} = \frac{V_\mathrm{C}}{I_\mathrm{C}} = \frac{1}{j \omega C} \ = \frac{-j}{\omega C} \,$

For an inductor:

$Z_\mathrm{inductor} = \frac{V_\mathrm{L}}{I_\mathrm{L}} = j \omega L \,$

For derivations, see Impedance of different devices (derivations).

Reactance

Main article: Reactance

The term reactance refers to the imaginary part of the impedance. Some examples:

A resistor's impedance is R (its resistance) and its reactance is 0.

A capacitor's impedance is j (-1/ωC) and its reactance is -1/ωC.

An inductor's impedance is j ω L and its reactance is ω L.

It is important to note that the impedance of a capacitor or an inductor is a function of the frequency ω and is an imaginary quantity - however is certainly a real physical phenomenon relating the shift in phases between the voltage and current phasors due to the existence of the capacitor or inductor. Earlier it was shown that the impedance of a resistor is constant and real, in other words a resistor does not cause a phase shift between voltage and current as do capacitors and inductors.

When resistors, capacitors, and inductors are combined in an AC circuit, the impedances of the individual components can be combined in the same way that the resistances are combined in a DC circuit. The resulting equivalent impedance is in general, a complex quantity. That is, the equivalent impedance has a real part and an imaginary part. The real part is denoted with an R and the imaginary part is denoted with an X. Thus:

$Z_\mathrm{eq} = R_\mathrm{eq} + jX_\mathrm{eq} \,$

where

Req is termed the resistive part of the impedance
Xeq is termed the reactive part of the impedance.

It is therefore common to refer to a capacitor or an inductor as a reactance or equivalently, a reactive component (circuit element). Additionally, the impedance for a capacitance is negative imaginary while the impedance for an inductor is positive imaginary. Thus, a capacitive reactance refers to a negative reactance while an inductive reactance refers to a positive reactance.

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. That is, unlike a resistance, a reactance does not dissipate power.

It is instructive to determine the value of the capacitive reactance at the frequency extremes. As the frequency approaches zero, the capacitive reactance grows without bound so that a capacitor approaches an open circuit for very low frequency sinusoidal sources. As the frequency increases, the capacitive reactance approaches zero so that a capacitor approaches a short circuit for very high frequency sinusoidal sources.

Conversely, the inductive reactance approaches zero as the frequency approaches zero so that an inductor approaches a short circuit for very low frequency sinusoidal sources. As the frequency increases, the inductive reactance increases so that an inductor approaches an open circuit for very high frequency sinusoidal sources.

Combining impedances

Main article: Series and parallel circuits

Combining impedances in series, parallel, or in delta-wye configurations, is the same as for resistors. The difference is that combining impedances involves manipulation of complex numbers.

In series

Combining impedances in series is simple:

$Z_\mathrm{eq} = Z_1 + Z_2 = (R_1 + R_2) + j(X_1 + X_2) \!\ .$

In parallel

Combining impedances in parallel is much more difficult than combining simple properties like resistance or capacitance, due to a multiplication term.

$Z_\mathrm{eq} = Z_1 \| Z_2 = \left( {Z_\mathrm{1}}^{-1} + {Z_\mathrm{2}}^{-1}\right) ^{-1} = \frac{Z_\mathrm{1}Z_\mathrm{2}}{Z_\mathrm{1}+Z_\mathrm{2}} \!\ .$

In rationalized form the equivalent resistance is:

$Z_\mathrm{eq} = R_\mathrm{eq} + j X_\mathrm{eq} \!\ .$
$R_\mathrm{eq} = { (X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2) \over (R_1 + R_2)^2 + (X_1 + X_2)^2}$
$X_\mathrm{eq} = {(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2) \over (R_1 + R_2)^2 + (X_1 + X_2)^2}$

Circuits with general sources

Impedance is defined by the ratio of two phasors where a phasor is the complex peak amplitude of a sinusoidal function of time. For more general periodic sources and even non-periodic sources, the concept of impedance can still be used. It can be shown that virtually all periodic functions of time can be represented by a Fourier series. Thus, a general periodic voltage source can be thought of as a (possibly infinite) series combination of sinusoidal voltage sources. Likewise, a general periodic current source can be thought of as a (possibly infinite) parallel combination of sinusoidal current sources.

Using the technique of Superposition, each source is activated one at a time and an AC circuit solution is found using the impedances calculated for the frequency of that particular source. The final solutions for the voltages and currents in the circuit are computed as sums of the terms calculated for each individual source. However, it is important to note that the actual voltages and currents in the circuit do not have a phasor representation. Phasors can be added together only when each represents a time function of the same frequency. Thus, the phasor voltages and currents that are calculated for each particular source must be converted back to their time domain representation before the final summation takes place.

This method can be generalized to non-periodic sources where the discrete sums are replaced by integrals. That is, a Fourier transform is used in place of the Fourier series.

Magnitude and phase of impedance

Complex numbers are commonly expressed in two distinct forms. The rectangular form is simply the sum of the real part with the product of j and the imaginary part:

$Z = R + jX \,$

The polar form of a complex number the real magnitude of the number multiplied by the complex phase. This can be written with exponentials, or in phasor notation:

$Z = \left|Z\right| e^ {j \varphi} = \left|Z\right|\angle \varphi$

where

$\left|Z\right| = \sqrt{R^2+X^2} = \sqrt{Z Z^*}$ is the magnitude of Z (Z* denotes the complex conjugate of Z), and
$\varphi = \arctan \bigg(\frac{X}{R} \bigg)$ is the angle.

Peak phasor versus rms phasor

A sinusoidal voltage or current has a peak amplitude value as well as an rms (root mean square) value. It can be shown that the rms value of a sinusoidal voltage or current is given by:

$V_\mathrm{rms} = \frac{V_\mathrm{peak}}{\sqrt{2}}$
$I_\mathrm{rms} = \frac{I_\mathrm{peak}}{\sqrt{2}}$

In many cases of AC analysis, the rms value of a sinusoid is more useful than the peak value. For example, to determine the amount of power dissipated by a resistor due to a sinusoidal current, the rms value of the current must be known. For this reason, phasor voltage and current sources are often specified as an rms phasor. That is, the magnitude of the phasor is the rms value of the associated sinusoid rather than the peak amplitude. Generally, rms phasors are used in electrical power engineering whereas peak phasors are often used in low-power circuit analysis.

In any event, the impedance is clearly the same. Whether peak phasors or rms phasors are used, the scaling factor cancels out when the ratio of the phasors is taken.

Matched impedances

Main article: Impedance matching

When fitting components together to carry electromagnetic signals, it is important to match impedance, which can be achieved with various matching devices. Failing to do so is known as impedance mismatch and results in signal loss and reflections. The existence of reflections allows the use of a time-domain reflectometer to locate mismatches in a transmission system.

For example, a conventional radio frequency antenna for carrying broadcast television in North America was standardized to 300 ohms, using balanced, unshielded, flat wiring. However cable television systems introduced the use of 75 ohm unbalanced, shielded, circular wiring, which could not be plugged into most TV sets of the era. To use the newer wiring on an older TV, small devices known as baluns were widely available. Today most TVs simply standardize on 75 ohm feeds instead.

Inverse quantities

The reciprocal of a non-reactive resistance is called conductance. Similarly, the reciprocal of an impedance is called admittance. The conductance is the real part of the admittance, and the imaginary part is called the susceptance. Conductance and susceptance are not the reciprocals of resistance and reactance in general, but only for impedances that are purely resistive or purely reactive; in the latter case a change of sign is required.

Origin of impedances

The origin of j was found by calculating an electrical circuit by the direct method, without using impedances or phasors. The circuit is formed by a resistance an inductance and a capacitor in series The circuit is connected to a sinusoidal voltage source and we have waited long enough so that all the transitory phenomena have faded away. It is now in steady sinusoidal state. As the system is linear, the steady state current will be also sinusoidal and of the same frequency of the voltage source. The only two quantities that we ignore are the amplitude of the current and its phase relative to the voltage source. If the voltage source is $\scriptstyle{V=V_\circ\cos(\omega t)}$ the current will be of the form $\scriptstyle{I=I_\circ\cos(\omega t+\varphi)}$, where $\scriptstyle{\varphi}$ is the relative phase of the current, which is unknown. The equation of the circuit is:

$V_\circ\cos(\omega t)= V_R+V_L+V_C$

where

$\scriptstyle{V_R}$, $\scriptstyle{V_L}$ and $\scriptstyle{V_C}$ are the voltages across the resistance, the inductance and the capacitor.
$V_R\,$ is equal to $RI_\circ\cos(\omega t+\varphi)$

The definition of inductance says:

$V_L=L\textstyle{{dI\over dt}}= L\textstyle{{d\left(I_\circ\cos(\omega t+\varphi)\right)\over dt}}= -\omega L I_\circ\sin(\omega t+\varphi)$.

The definition of capacitance says that $\scriptstyle{I=C{dV_C\over dt}}$. It is easy to verify (taking the expression derivative) that:

$V_C=\textstyle{{1\over \omega C}} I_\circ\sin(\omega t+\varphi)$.

Then the equation to solve is:

$V_\circ\cos(\omega t)= RI_\circ\cos(\omega t+\varphi) -\omega L I_\circ\sin(\omega t+\varphi)+ \textstyle{{1\over \omega C}} I_\circ\sin(\omega t+\varphi)$

That is, we have to find the two values $\scriptstyle{I_\circ}$ and $\scriptstyle{\varphi}$ that makes this equation true for all values of time $\scriptstyle{t}$.

To do this, another circuit must be considered, identical to the former and fed by a voltage source whose only difference with the former is that it started with a lag of a quarter of a period. The voltage of this source is $\scriptstyle{V=V_\circ\cos(\omega t - {\pi \over 2} ) = V_\circ\sin(\omega t) }$. The current in this circuit will be the same as in the former one but for a lag of a quarter of period:

$I=I_\circ\cos(\omega t + \varphi - {\pi \over 2})= I_\circ\sin(\omega t + \varphi) \,$.

The voltage is given by:

$V_\circ\sin(\omega t)= RI_\circ\sin(\omega t+\varphi) +\omega L I_\circ\cos(\omega t+\varphi)- \textstyle{{1\over \omega C}} I_\circ\cos(\omega t+\varphi)$

Some of the signs have changed because a cosine becomes a sine, and a sine becomes a negative cosine.

The first equation is added with the second one multiplied by j, to try to replace expressions with the form $\scriptstyle{\cos x+j\sin x}$ by $\scriptstyle{e^{jx} }$, using the les Euler's formula. This gives:

$V_\circ e^{j\omega t} =RI_\circ e^{j\left(\omega t+\varphi\right)}+j\omega LI_\circ e^{j\left(\omega t+\varphi\right)} +\textstyle{{1\over j\omega C}}I_\circ e^{j\left(\omega t+\varphi\right)}$

As $\scriptstyle{e^{j\omega t} }$ is not zero we can divide all the equation by this factor:

$V_\circ =RI_\circ e^{j\varphi}+j\omega LI_\circ e^{j\varphi} +\textstyle{{1\over j\omega C}}I_\circ e^{j\varphi}$

This gives:

$I_\circ e^{j\varphi}= \textstyle{V_\circ \over R + j\omega L + \scriptstyle{{1 \over j\omega C}}}$

The left side of the equation contains the two values we are trying to deduce: the modulus and the phase of the current. The amplitude is the modulus of the complex number at the right and its phase is the argument of the complex number at the right.

The formula at right is the habitual formula which is written when doing circuit equations using phasors and impedances. The denominator of the equation is the impedances of the resistance, inductor and capacitor.

Even though the formula

$I= \textstyle{V_\circ \over R + j\omega L + \scriptstyle{{1 \over j\omega C}}}$

contains imaginary parts, at least some of the imaginary numbers will become real in the circuit (j*j = -1), which means that the previously stated formula can not be simplified to just

$I= \textstyle{V_\circ \over R}$

Analogous impedances

Electromagnetic impedance

In problems of electromagnetic wave propagation in a homogeneous medium, the intrinsic impedance of the medium is defined as:

$\eta = \sqrt{\frac{\mu}{\varepsilon}}$

where

μ and ε are the permeability and permittivity of the medium, respectively.

Acoustic impedance

Main article: Acoustic impedance

In complete analogy to the electrical impedance discussed here, one also defines acoustic impedance, a complex number which describes how a medium absorbs sound by relating the amplitude and phase of an applied sound pressure to the amplitude and phase of the resulting sound flux.

Data-transfer impedance

Another analogous coinage is the use of impedance by computer programmers to describe how easy or difficult it is to pass data and flow of control between parts of a system, commonly ones written in different languages. The common usage is to describe two programs or languages/environments as having a low or high impedance mismatch.

Application to physical devices

Note that the equations above only apply to theoretical devices. Real resistors, capacitors, and inductors are more complex and each one may be modeled as a network of theoretical resistors, capacitors, and inductors. Rated impedances of real devices are actually nominal impedances, and are only accurate for a narrow frequency range, and are typically less accurate for higher frequencies. Even within its rated range, an inductor's resistance may be non-zero. Above the rated frequencies, resistors become inductive (power resistors more so), capacitors and inductors may become more resistive. The relationship between frequency and impedance may not even be linear outside of the device's rated range.

• Antenna tuner
• Characteristic impedance
• Balance return loss
• Balancing network
• Bridging loss
• Damping factor
• Electrical characteristics of a dynamic loudspeaker
• Electromagnetic impedance
• Forward echo
• Harmonic oscillator
• Impedance bridging
• Impedance cardiography
• Impedance matching
• Log-periodic antenna
• Physical constants
• Reflection coefficient
• Reflection loss, Reflection (electrical)
• Resonance
• Return loss
• Sensitivity
• Signal reflection
• Smith chart
• Standing wave
• Time-domain reflectometer
• Voltage standing wave ratio
• Wave impedance
• Reactance
• Inductance
• nominal impedance
• Mechanical impedance

References

• Pohl R. W., Electrizitätslehre, Berlin-Göttingen-Heidelberg: Springer-Verlag, 1960.
• Popov V. P., The Principles of Theory of Circuits, – M.: Higher School, 1985, 496 p. Template:Ru icon.
• Küpfmüller K., Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.