Electrical Impedance

ilde{Z} = Z e^{j	heta} \quad

where the magnitude

\scriptstyle{Z}

gives the change in voltage amplitude for a given current amplitude, while the argument

\scriptstyle{	heta}

gives the phase difference between voltage and current. In Cartesian Form ,

:

ilde{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 .

. Note that while ]]

OHM'S LAW

applying a voltage

\scriptstyle{V}

, across a Load

\scriptstyle{Z}

, driving a current

\scriptstyle{I}

.]]
See Also: Ohm's law

We can understand this by substituting it into Ohm's Law . AC Ohm's law , Hyperphysics

:

ilde{V} = 	ilde{I}	ilde{Z} = 	ilde{I} Z e^{j	heta} \quad

The magnitude of the impedance

\scriptstyle{Z}

acts just like resistance, giving the drop in voltage amplitude across an impedance

\scriptstyle{	ilde{Z}}

for a given current

\scriptstyle{	ilde{I}}

. The phase factor tells us that the current lags the voltage by a Phase of

heta

(i.e. in the time domain, the current signal is shifted

rac{	heta T}{2 \pi}

to the right with respect to the voltage signal). Capacitor/inductor phase relationships , Hyperphysics

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, Sinusoid al voltage and current waves are commonly represented as complex-valued functions of time denoted as

\scriptstyle{	ilde{V}}

and

\scriptstyle{	ilde{I}}

.

:

\ 	ilde{V} = V_0e^{j(\omega t + \phi_V)}

\ 	ilde{I} = I_0e^{j(\omega t + \phi_I)}

Impedance is defined as the ratio of these quantities.

:

\ 	ilde{Z} = {	ilde{V} \over 	ilde{I}}

Substituting these into Ohm's law we have

:

\ \Rightarrow V_0e^{j(\omega t + \phi_V)} = I_0e^{j(\omega t + \phi_I)} Z e^{j	heta}

\ \Rightarrow V_0\cos(\omega t + \phi_V) = I_0 Z \cos(\omega t + \phi_I + 	heta)

Equating the magnitudes and phases we have

:

\ V_0 = I_0 Z \quad

\ \phi_V = \phi_I + 	heta \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) = rac{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 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

See Also: 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.

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

See Also: Impedance of different devices (derivations)

]]

The impedance of a Resistor is purely Real and is referred to as a ''resistive impedance''.

:

ilde{Z}_R = R \quad

Inductor s and Capacitor s have a purely Imaginary ''reactive impedance''.

:

ilde{Z}_L = j\omega L \quad

ilde{Z}_C = {1 \over j\omega C}

Note the following identities for the Imaginary Unit and its Reciprocal .

:

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

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

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

:

ilde{Z}_L = \omega Le^{j{\pi \over 2}}

ilde{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.

:<math> Ilde{Z} {eq}

ilde{Z}_1 \ ilde{Z}_2 = \left( ilde{Z}_1^{-1} + ilde{Z}_2^{-1} ight)^{-1} = { ilde{Z}_1 ilde{Z}_2 \over ilde{Z}_1 + ilde{Z}_2} \quad</math>

ilde{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}

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Information About

Electrical Impedance