Dual impedance and dual network are terms used in electronic network analysis. The dual of an impedance is its algebraic inverse . Note that and are the duals of each other, that is, they are reciprocal. For this reason the dual impedance is also called the inverse impedance. The dual of a network of impedances is that network whose impedance is . In the case of a network with more than one port the impedance looking into each of the ports must simultaneously be dual.
Another way of stating this is that the dual of is the admittance .
This is consistent with the definition of dual as being that circuit whose voltages and currents are interchanged since and [1]
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In a real design situation it is usually desired to find the dual of an impedance with respect to some nominal or characteristic impedance. To do this, Z and Z' are scaled to the nominal impedance Z0 so that;
Z0 is usually taken to be a purely real number R0, so Z' is only changed by a real factor of R02. In other words, the dual remains qualitatively the same circuit but all the component values must be scaled quantitively by R02.[2]
Element | Z | Dual | Z' |
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There is a graphical method of obtaining the dual of a network which is often easier to use than the mathematical expression for the impedance. Starting with a circuit diagram of the network in question, Z, the following steps are drawn on the diagram to produce Z' superimposed on top of Z. Typically, Z' will be drawn in a different colour to help distinguish it from the original, or, if using CAD, Z' can be drawn on a different layer.
This completes the drawing of Z'. This method also serves to demonstrate that the dual of a mesh transforms in to a node and the dual of a node transforms in to a mesh. Two useful examples are given below, both to illustrate the process and to give some further examples of dual networks.
It is now clear that the dual of a star network of inductors is a delta network of capacitors. This dual circuit is not the same thing as a star-delta (Y-Δ) transformation. A Y-Δ transform results in an equivalent circuit, not a dual circuit.
Filters designed using Cauer's topology of the first form are low-pass filters consisting of a ladder network of series inductors and shunt capacitors.
It can now be seen that the dual of a Cauer low-pass filter is still a Cauer low-pass filter. It does not transform into a high-pass filter as might have been expected. Note, however, that the first element is now a shunt component instead of a series component.