From the phasor voltage-current relations of circuit elements.
| Element | Impedance | Admittance |
|---|---|---|
| Resistor \(R\) | \(\displaystyle \mathbf{Z} = R\) | \(\displaystyle \mathbf{Y} = \frac{1}{R}\) |
| Inductor \(L\) | \(\displaystyle \mathbf{Z} = j\omega L\) | \(\displaystyle \mathbf{Y} = \frac{1}{j\omega L}\) |
| Capacitor \(C\) | \(\displaystyle \mathbf{Z} = \frac{1}{j\omega C}\) | \(\displaystyle \mathbf{Y} = j\omega C\) |
Impedance
The impedance \(\mathbf{Z}\) of a circuit is the ratio of the phasor voltage \(\mathbf{V}\) to the phasor current \(\mathbf{I}\), measured in ohms \(\mathit{\Omega}\).
\(\boxed{\mathbf{Z} = \frac{\mathbf{V}}{\mathbf{I}}}\)
The impedance represents the opposition that the circuit exhibits to the flow of sinusoidal current. Although the impedance is the ratio of two phasors, it is not a phasor, because it does not correspond to a sinusoidally varying quantity.
From the table, \(\mathbf{Z}_L = j\omega L\) and \(\mathbf{Z}_C = 1/\left(j\omega C\right)\). When angular frequency \(\omega = 0\) (for dc sources), \(\mathbf{Z}_L = 0\) (short circuit) and \(\mathbf{Z}_C \rightarrow \infty\) (open circuit). When \(\omega \rightarrow \infty\) (for high frequencies), \(\mathbf{Z}_L \rightarrow \infty\) (open circuit) and \(\mathbf{Z}_C = 0\) (short circuit).
As a complex quantity, the impedance may be expressed in rectangular form and polar form.
\(\boxed{\mathbf{Z} = R + jX = \left|\mathbf{Z}\right|\angle\theta}\)
where
\(\displaystyle \left|\mathbf{Z}\right| = \sqrt{R^2 + X^2}\)
\(\displaystyle \theta = \tan^{-1}\left(\frac{X}{R}\right)\)
\(\displaystyle R = \left|\mathbf{Z}\right|\cos\left(\theta\right)\)
\(\displaystyle X = \left|\mathbf{Z}\right|\sin\left(\theta\right)\)
where \(R = \textrm{Re}\left(\mathbf{Z}\right)\) is the resistance and \(X = \textrm{Im}\left(\mathbf{Z}\right)\) is the reactance. The impedance is inductive when \(X\) is positive or capacitive when \(X\) is negative. Thus, impedance \(\mathbf{Z} = R + jX\) is said to be inductive or lagging since current lags voltage, while impedance \(\mathbf{Z} = R - jX\) is capacitive or leading bacause current leads voltage. The impedance, resistance, and reactance are all measured in ohms \(\mathit{\Omega}\).
For capacitive reactance \(X_C\)
\(\boxed{X_C = \frac{1}{\omega C}}\)
For inductive reactance \(X_L\)
\(\boxed{X_L = \omega L}\)
Admittance
The admittance \(\mathbf{Y}\) is the reciprocal of impedance \(\mathbf{Z}\), measured in siemens (S).
\(\boxed{\mathbf{Y} = \frac{1}{\mathbf{Z}} = \frac{\mathbf{I}}{\mathbf{V}}}\)
As a complex quantity, the impedance may be expressed in rectangular form.
\(\boxed{\mathbf{Y} = G + jB}\)
where \(G = \textrm{Re}\left(\mathbf{Y}\right)\) is the conductance and \(B = \textrm{Im}\left(\mathbf{Y}\right)\) is the susceptance. The admittance, conductance, and susceptance are all expressed in the unit of siemens (or mhos).
Showing that conductance \(G \ne 1/R\) as it is in resistive circuits.
\(\displaystyle G + jB = \frac{1}{R + jX}\)
By rationalization
\(\displaystyle G + jB = \frac{1}{R + jX}\cdot\frac{R - jX}{R - jX} = \frac{R - jX}{R^2 + X^2}\)
Equating the real and imaginary parts gives
\(\displaystyle G = \frac{R}{R^2 + X^2}\)
\(\displaystyle B = -\frac{X}{R^2 + X^2}\)