Complex Impedance

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z=R+jX\,}

Where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z\,} is the complex impedance ($\Omega$)

$R\,$ is the resistance ($\Omega$)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X\,} is the reactance ($\Omega$)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle j\,} is the complex component, i.e. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{1}=R_{1}+jX_{1}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{2}=R_{2}+jX_{2}\,}

Then,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{1}\times Z_{2}=\left(R_{1}+jX_{1}\right)\left(R_{2}+jX_{2}\right)\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {Z_{1}}{Z_{2}}}={\frac {\left(R_{1}+jX_{1}\right)}{\left(R_{2}+jX_{2}\right)}}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)}}\,}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)}}\,}