Impedance & Admittance

Impedance (Z)

$$Z=\frac{V}{I}=R+jX$$

Here:

• $R$: Resistance
• $X$: Reactance

Note

When mentioning the reactance of an element, the $j$ should not be included. $$

Admittance (Y)

Inverse of impedance.

$$Y=\frac{1}{Z}=\frac{I}{V}=G+jB$$

Here:

• $G$: Conductance
• $B$: Susceptance

From the definitions:

$$G=\frac{R}{R^2+X^2} \;\land B=-\frac{X}{R^2+X^2}$$

For simple circuit elements

Resistor

Let $i=I_m\sin{(\omega t + \phi_0)}$ is applied across a resistor with resistance $R$. From Ohm’s law:

$$v=RI_m\sin{(\omega t + \phi_0)} \implies Z_R = R$$

No changes in frequency, phase angle. $v$ is in phase with $i$. $R$ doesn’t have reactance.

Inductor

Let $i=I_m\sin{(\omega t + \phi_0)}$ is applied across an inductor with inductance $L$.

$$v=L\omega I_m\sin{\bigg(\omega t + \Big(\phi_0+\frac{\pi}{2}\Big)\bigg)} \implies Z_L = j\omega L$$

Reactance of the inductor is $X_L=L\omega$. $$

Note

$v$ leads $i$ by $\frac{\pi}{2}$. No changes in frequency.

Capacitor

Let $i=I_m\sin{(\omega t + \phi_0)}$ is applied across an capacitor with capacitance $c$.

$$v=\frac{I_m}{c\omega} \sin{(\omega t + (\phi_0 - \frac{\pi}{2}))} \implies Z_C = -j\frac{1}{c\omega}$$

Reactance of the capacitor (capacitive reactance) is $X_c=-\frac{1}{c\omega}$. $$

Note

$v$ lags $i$ by $\frac{\pi}{2}$. No changes in frequency.

Note

If $v$: $$

• lags $i$ - circuit is capacitive
• leads $i$ - circuit is inductive

For complex circuit elements

For a series circuit

Resultant impedance is the sum of each component’s impedance.

For a parallel circuit

Resultant admittance is the sum of each component’s admittance.