Dual Impedance - Graphical Method

Graphical Method

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.

1. A generator is connected to each port of the original network. The purpose of this step is to prevent the ports from being "lost" in the inversion process. This happens because a port left open circuit will transform into a short circuit and disappear.
2. A dot is drawn at the centre of each mesh of the network Z. These dots will become the circuit nodes of Z'.
3. A conductor is drawn which entirely encloses the network Z. This conductor also becomes a node of Z'.
4. For each circuit element of Z, its dual is drawn between the nodes in the centre of the meshes either side of Z. Where Z is on the edge of the network, one of these nodes will be the enclosing conductor from the previous step.

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.