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Mathematics-Online course: Basic Mathematics - Complex Numbers

Complex Resistance in A.C. Networks

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For the analysis of linear a.c. networks (alternating current network) it is advantageous to use polar coordinates. With the voltage $ U$ and the current $ I$ given as

$\displaystyle U(t) = U_0 e^{\mathrm{i}(\omega t+\varphi)}\,, \quad I(t) = I_0 e^{\mathrm{i}(\omega t+\psi)}\,, $

the complex resistance

$\displaystyle Z = U(t)/I(t) $

is independent of time. For the basic elements

resistor $ R$   coil $ L$   capacitor $ C$        
\includegraphics[width=.2\moimagesize]{komplexe_zahlen_schaltelement_widerstand.eps}      \includegraphics[width=.2\moimagesize]{komplexe_zahlen_schaltelement_induktivitaet.eps}      \includegraphics[width=.2\moimagesize]{komplexe_zahlen_schaltelement_kapazitaet.eps}        
$ Z=R$   $ Z=\mathrm{i}\omega L$   $ Z=(\mathrm{i}\omega C)^{-1}$        
the complex resistances are added when elements are serially connected:

$\displaystyle Z_{\text{total}} = Z_1 + Z_2 $

On the other hand, when elements are connected in parallel, the resistance is calculated as the sum of their reciprocals:

$\displaystyle \frac{1}{Z_{\text{total}}} = \frac{1}{Z_1} + \frac{1}{Z_2} \quad \Rightarrow \quad Z_{\text{total}}=\frac{Z_1 Z_2}{Z_1+Z_2}\, . $

$ \operatorname{Re} Z$ is referred to as effective resistance, $ \operatorname{Im} Z$ as reactance, and $ \vert Z\vert$ as impedance.

\includegraphics[width=.6\moimagesize]{komplexe_zahlen_schaltkreis.eps}

As an example, the (total) resistance of the circuit, shown in the figure, is

$\displaystyle Z_{\text{total}}=\mathrm{i}\omega L+ \frac{R(\mathrm {i}\omega C... ...a)}{300\Omega-200\mathrm {i}\Omega}\approx (92.31 -38.46\mathrm{i})\Omega \,. $

Moreover, with an a.c. voltage of $ U_{\text{effective}}=220$V,

$\displaystyle I_{\text{effective}}=\frac{U_{\text{effective}}}{\vert Z\vert}=\frac{220\mathrm{V}}{100\Omega}=2.2\mathrm{A} $

is the effective current.
(Authors: Höllig/Wipper/Abele)
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  automatically generated 10/31/2008