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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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tilde{Z}}
and the term complex impedance may be used interchangeably; the polar form conveniently captures both magnitude and phase characteristics,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z} = Z e^{j\theta} \quad
where the magnitude Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{Z}
gives the change in voltage amplitude for a given current amplitude, while the argument Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\theta}
gives the phase difference between voltage and current. In Cartesian form,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z} = R + j\Chi \quad
where the real part of impedance is the resistance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{R}
and the imaginary part is the reactance Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
Image:Complex impedance plane.png
A graphical representation of the complex impedance plane. Note that while reactance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\Chi}
can be either positive or negative, resistance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{R}
is always positive.
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Ohm's law
Image:General AC circuit.png
An AC supply applying a voltage Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{V}
, across a load Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{Z} , driving a current Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{I}
.
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We can understand this by substituting it into Ohm's law.[1][2]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{V} = \tilde{I}\tilde{Z} = \tilde{I} Z e^{j\theta} \quad
The magnitude of the impedance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{Z}
acts just like resistance, giving the drop in voltage amplitude across an impedance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tilde{Z}}
for a given current Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tilde{I}}
. The phase factor tells us that the current lags the voltage by a phase of Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
(i.e. in the time domain, the current signal is shifted Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\theta T}{2 \pi}
to the right with respect to the voltage signal).[3]
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tilde{V}}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tilde{I}}
.[4][5]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ \tilde{V} = V_0e^{j(\omega t + \phi_V)}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ \tilde{I} = I_0e^{j(\omega t + \phi_I)}
Impedance is defined as the ratio of these quantities.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ \tilde{Z} = {\tilde{V} \over \tilde{I}}
Substituting these into Ohm's law we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): t
, we may equate the magnitudes and phases to obtain
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ V_0 = I_0 Z \quad
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ \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):
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ \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
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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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{e^{j\omega t}}
cancel.
Device examples
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Image:VI phase.png
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\pi/2}
, while the voltage across an inductor leads the current through it by Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\pi/2}
. The identical voltage and current amplitudes tell us that the magnitude of the impedance is equal to one.
The impedance of a resistor is purely real and is referred to as a resistive impedance.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z}_R = R \quad
Inductors and capacitors have a purely imaginary reactive impedance.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z}_L = j\omega L \quad
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z}_C = {1 \over j\omega C}
Note the following identities for the imaginary unit and its reciprocal.
- Failed to parse (Missing texvc executable; please see math/README to configure.): j = \cos{\left({\pi \over 2}\right)} + j\sin{\left({\pi \over 2}\right)} = e^{j{\pi \over 2}}
- Failed to parse (Missing texvc executable; please see math/README to configure.): {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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z}_L = \omega Le^{j{\pi \over 2}}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 realize that resistance and reactance are not individually significant; together they determine the magnitude and phase of the impedance, through the following relations:
- Failed to parse (Missing texvc executable; please see math/README to configure.): |\tilde{Z}| = \sqrt{\tilde{Z}\tilde{Z}^*} = \sqrt{R^2 + \Chi^2}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
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Resistance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{R}
is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.
- Failed to parse (Missing texvc executable; please see math/README to configure.): R = Z \cos{\theta} \quad
Reactance
-
Reactance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\Chi}
is the imaginary part of the impedance; a component with a finite reactance induces a phase shift Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
between the voltage across it and the current through it.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
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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
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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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\mathcal{E}}
(voltage opposing current) due to a rate-of-change of magnetic field Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{B}
through a current loop.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E} = -{{d\Phi_B} \over dt}.
For an inductor consisting of a coil with Failed to parse (Missing texvc executable; please see math/README to configure.): N
loops this gives.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
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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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\tilde{Z}_{eq}}
can be calculated in terms of the equivalent resistance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{R_{eq}}
and reactance Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle{\Chi_{eq}}
.[6]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{Z}_{eq} = R_{eq} + j \Chi_{eq} \quad
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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}
See also
External links
References
- ^ AC Ohm's law, Hyperphysics
- ^ Horowitz, Paul; Hill, Winfield (1989). "1", The Art of Electronics. Cambridge University Press, 32-33. ISBN 0-521-37095-7.
- ^ Capacitor/inductor phase relationships, Yokogawa
- ^ Complex impedance, Hyperphysics
- ^ Horowitz, Paul; Hill, Winfield (1989). "1", The Art of Electronics. Cambridge University Press, 31-32. ISBN 0-521-37095-7.
- ^ Parallel Impedance Expressions, Hyperphysics
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