Impedance

Voltage and current phasors are related through the impedance of element $Z$, based on Ohm’s law, such that:

$$\vec{V}=\vec{I}Z$$

The impedance is a complex number that can be written as:

$$\begin{array}{rl}Z& =|Z|\angle \varphi \\ & =R+jX\end{array}$$

where $Z$ is impedance, $R$ is the resistance, and $X$ is the reactance, all in $\Omega$

The magnitude of $Z$ can be found in the following way:

$$Z=\frac{\vec{V}}{\vec{I}}⟹|Z|=\frac{V}{I}$$

and the phase angle can be found with:

$$\begin{array}{rl}\varphi & ={\theta }_{\vec{V}}-{\theta }_{\vec{I}}\\ & ={\tan }^{-1}\frac{X}{R}\end{array}$$

The reactance $X$ can be thought of as the imaginary part of $Z$, while $R$ is the real part.

Understanding resistors, inductors and capacitors in the context of impedance:

• In a resistor, the voltage and current are in phase.
• In an inductor and capacitor, there is a 90-degree phase shift between the voltage and current.
  • Inductor – voltage leads current by $90°$
  • Capacitor – current leads voltage by $90°$

$$\begin{array}{rl}\text{Resistor:}& \vec{V}=Z_R\vec{I},Z_R=R\angle 0°\\ \text{Inductor:}& \vec{V}=Z_L\vec{I},Z_L=j{X}_L=j\omega L=\omega L\angle 90°\\ \text{Capacitor:}& \vec{V}=Z_C\vec{I},Z_C=j{X}_C=-j\frac{1}{\omega C}=\frac{1}{\omega C}\angle -90°\end{array}$$

Series and Parallel Impedances

Impedances in series:

$$Z_s=Z_1+Z_2+\cdots +Z_n$$

Impedances in parallel:

$$\frac{1}{Z_p}=\frac{1}{Z_1}+\frac{1}{Z_2}+\cdots +\frac{1}{Z_N}$$

The above can alternatively be written in terms of parallel admittances $Y_p=Y_1+Y_2+\cdots +Y_n$

Admittance

The admittance of an element is defined as the inverse of the impedance, or the ratio of current phasor to voltage phasor:

$$Y=\frac{\vec{I}}{\vec{V}}=\frac{1}{Z}$$

The unit of admittance is siemens $S$.