# Electrical Impedance - Device Examples

Device Examples

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

In this case, the voltage and current waveforms are proportional and in phase.

Ideal inductors and capacitors have a purely imaginary reactive impedance:

the impedance of inductors increases as frequency increases;

the impedance of capacitors decreases as frequency increases;

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in quadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is lagging; in a capacitor the current is leading.

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

begin{align} j &equiv cos{left( frac{pi}{2}right)} + jsin{left( frac{pi}{2}right)} equiv e^{j frac{pi}{2}} \ frac{1}{j} equiv -j &equiv cos{left(-frac{pi}{2}right)} + jsin{left(-frac{pi}{2}right)} equiv e^{j(-frac{pi}{2})} end{align}

Thus the inductor and capacitor impedance equations can be rewritten in polar form:

begin{align} Z_L &= omega Le^{jfrac{pi}{2}} \ Z_C &= frac{1}{omega C}e^{jleft(-frac{pi}{2}right)} end{align}

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

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... This result is commonly expressed in polar form, as or, using Euler's formula, as. ...

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