# Complex Impedance

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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
$\frac{Z_{1}}{Z_{2}} = \frac{\left( R_{1} + jX_{1} \right)}{\left( R_{2} + jX_{2} \right)} \,$
$= \frac{\left( R_{1} + jX_{1} \right)}{\left( R_{2} + jX_{2} \right)} \times \frac{\left( R_{2} - jX_{2} \right)}{\left( R_{2} - jX_{2} \right)} \,$
$= \frac{R_{1} R_{2} + X_{1} X_{2} +j \left( R_{1} X_{2} - X_{1} R_{2} \right)}{\left( R_{2}^{2} + X_{2}^{2} \right)} \,$