# Complex Impedance

Complex impedances are commonly used quantities in the analysis of AC power systems. A complex impedance is represented by the following relation:

$Z=R+jX\,$ Where $Z\,$ is the complex impedance ($\Omega$ )

$R\,$ is the resistance ($\Omega$ )
$X\,$ is the reactance ($\Omega$ )
$j\,$ is the complex component, i.e. ${\sqrt {-1}}\,$ )

For more details about why complex quantities are used in electrical engineering, see the article on complex electrical quantities.

## Complex Arithmetic

The manipulation of complex impedances follow the rules of complex arithmetic.

### Series Impedances

Two impedances in series can be combined by simply adding the individual real and complex terms (i.e. resistance and reactance components). For example, given:

$Z_{1}=R_{1}+jX_{1}\,$ $Z_{2}=R_{2}+jX_{2}\,$ Then,

$Z_{1}+Z_{2}=R_{1}+R_{2}+j\left(X_{1}+X_{2}\right)\,$ ### Parallel Impedances

Two impedances in parallel can be combined according to the following standard relation:

$Z_{1}||Z_{2}={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}\,$ However, note that the multiplication and division of complex numbers is more involved than simply multiplying or dividing the real and complex terms:

• Multiplication: involves multiplying cross-terms, i.e.
$Z_{1}\times Z_{2}=\left(R_{1}+jX_{1}\right)\left(R_{2}+jX_{2}\right)\,$ $=R_{1}R_{2}+j^{2}X_{1}X_{2}+j\left(R_{1}X_{2}\right)+j\left(X_{1}R_{2}\right)\,$ $=R_{1}R_{2}-X_{1}X_{2}+j\left(R_{1}X_{2}+X_{1}R_{2}\right)\,$ • Division: involves multiplying by the complex conjugate of the denominator, i.e
${\frac {Z_{1}}{Z_{2}}}={\frac {\left(R_{1}+jX_{1}\right)}{\left(R_{2}+jX_{2}\right)}}\,$ $={\frac {\left(R_{1}+jX_{1}\right)}{\left(R_{2}+jX_{2}\right)}}\times {\frac {\left(R_{2}-jX_{2}\right)}{\left(R_{2}-jX_{2}\right)}}\,$ $={\frac {R_{1}R_{2}+X_{1}X_{2}+j\left(R_{1}X_{2}-X_{1}R_{2}\right)}{\left(R_{2}^{2}+X_{2}^{2}\right)}}\,$ 