Circuit Theory/Transform Domain

Impedance

Let's recap: In the transform domain, the quantities of resistance, capacitance, and inductance can all be combined into a single complex value known as "Impedance". Impedance is denoted with the letter Z, and can be a function of s or jω, depending on the transform used (Laplace or Fourier). This impedance is very similar to the phasor concept of impedance, except that we are in the complex domain (laplace or fourier), and not the phasor domain.

Impedance is a complex quantity, and is therefore comprised of two components: The real component (resistance), and the complex component (reactance). Resistors, because they do not vary with time or frequency, have real values. Capacitors and inductors however, have imaginary values of impedance. The resistance is denoted (as always) with a capital R, and the reactance is denoted with an X (this is common, although it is confusing because X is also the most common input designator). We have therefore, the following relationship between resistance, reactance, and impedance:

[Complex Laplace Impedance]

$Z=R+jX$

Susceptance and Admittance

The inverse of resistance is a quantity called "Conductance". Similarly, the inverse of reactance is called "Susceptance". The inverse of impedance is called "Admittance". Conductance, Susceptance, and Admittance are all denoted by the variables Y or G, and are given the units Siemens. This book will not use any of these terms again, and they are just included here for completeness.

Parallel Components

Once in the transform domain, all circuit components act like basic resistors. Components in parallel are related as follows:

$Z_{1}||Z_{2}={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}$

Series Components

Series components in the transform domain all act like resistors in the time domain as well. If we have two impedances in series with each other, we can combine them as follows:

$Z_{1}{\mbox{ in series with }}Z_{2}=Z_{1}+Z_{2}$

Solving Circuits

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