Complex Impedance
Complex impedances are commonly used quantities in the analysis of AC power systems. A complex impedance is represented by the following relation:
- [math] Z = R + jX \, [/math]
Where [math] Z \, [/math] is the complex impedance ([math] \Omega [/math])
- [math] R \, [/math] is the resistance ([math] \Omega [/math])
- [math] X \, [/math] is the reactance ([math] \Omega [/math])
- [math] j \, [/math] is the complex component, i.e. [math] \sqrt{-1} \, [/math])
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:
- [math] Z_{1} = R_{1} + jX_{1} \, [/math]
- [math] Z_{2} = R_{2} + jX_{2} \, [/math]
Then,
- [math] Z_{1} + Z_{2} = R_{1} + R_{2} + j \left( X_{1} + X_{2} \right) \, [/math]
Parallel Impedances
Two impedances in parallel can be combined according to the following standard relation:
- [math] Z_{1} || Z_{2} = \frac{Z_{1} Z_{2}}{Z_{1} + Z_{2}} \, [/math]
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.
- [math] Z_{1} \times Z_{2} = \left( R_{1} + jX_{1} \right) \left( R_{2} + jX_{2} \right) \, [/math]
- [math] = R_{1} R_{2} + j^{2} X_{1} X_{2} +j \left( R_{1} X_{2} \right) +j \left( X_{1} R_{2} \right) \, [/math]
- [math] = R_{1} R_{2} - X_{1} X_{2} +j \left( R_{1} X_{2} + X_{1} R_{2} \right) \, [/math]
- Division: involves multiplying by the complex conjugate of the denominator, i.e
- [math] \frac{Z_{1}}{Z_{2}} = \frac{\left( R_{1} + jX_{1} \right)}{\left( R_{2} + jX_{2} \right)} \, [/math]
- [math] = \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)} \, [/math]
- [math] = \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)} \, [/math]