Impedance & Admittance | Sahithyan's Notes

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

Here:

Inverse of impedance.

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

Here:

From the definitions:

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

For Simple Circuit Elements

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.

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 $X_L=L\omega$ .

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 $X_c=-\frac{1}{c\omega}$ .

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

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