impedance

From Physics wiki

Jump to: navigation, search

Given a sinusoidal voltage V(t)\, and current I(t)\,

V(t) = V_0 e^{i(\omega t + \phi_V)},
I(t) = I_0e^{j(\omega t + \phi_I)},

across a circuit element, the impedance is defined to be the ratio

Z \equiv \frac{V}{I}\,.

It is generally a complex quantity

Z = Z_0 e^{i \phi}\,,

given by

Z = R + i X\,,

where R\, measures the (frequency-independent) resistance of the element and X\, measures the (frequency-dependent) reactance. The phase difference between V\, and I\, are given by

\phi = \phi_V - \phi_I = \arctan\left(\frac{X}{R}\right)\,,

while

Z_0 = \frac{V_0}{I_0} = \sqrt{R^2 + X^2}\,.

The quantity

Y \equiv \frac{1}{Z} = \frac{Z^*}{|Z|^2}\,

is termed the admittance.

Examples

A resistor is purely resistive, and has

Z_R = R\,.

An ideal capacitor is purely capactive, has capacitive reactance X_C = \frac{1}{\omega C}\,, and

Z_C = X_C e^{-i \frac{\pi}{2}} = \frac{-i}{\omega C}\,.

An ideal inductor is purely inductive, has inductive reactance X_L = \omega L\, and

Z_L = X_L e^{i \frac{\pi}{2}} = i \omega L\,.

Combining impedances

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Analog_circuits/impedance"
Personal tools