# Complex Impedance

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

${\displaystyle Z=R+jX\,}$

Where ${\displaystyle Z\,}$ is the complex impedance (${\displaystyle \Omega }$)

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

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:

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

Then,

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

### Parallel Impedances

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

${\displaystyle Z_{1}||Z_{2}={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}\,}$

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