Complex Impedance - Revision history

Jules: Created page with "Complex impedances are commonly used quantities in the analysis of AC power systems. A complex impedance is represented by the following relation: : $Z = R + jX \,$
2020-11-16T13:50:54Z

Created page with "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 \, </ma..."

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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:

: <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 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>

[[Category:Fundamentals]]