Dual impedance

Dual impedance and dual network are terms used in electronic network analysis. The dual of an impedance formula_1 is its reciprocal, or algebraic inverse formula_2. For this reason the dual impedance is also called the inverse impedance. Another way of stating this is that the dual of formula_1 is the admittance formula_4.
The dual of a network is the network whose impedances are the duals of the original impedances. In the case of a black-box network with multiple ports, the impedance looking into each port must be the dual of the impedance of the corresponding port of the dual network.
This is consistent with the general notion duality of electric circuits, where the voltage and current are interchanged, etc., since formula_5 yields formula_6
Scaled and normalised duals.
In physical units, the dual is taken 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 changed by a real factor of R02. In other words, the dual circuit is qualitatively the same circuit but all the component values are scaled by R02. The scaling factor R02 has the dimensions of Ω2, so the constant 1 in the unitless expression would actually be assigned the dimensions Ω2 in a dimensional analysis.
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 computer-aided design, 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 into a node and the dual of a node transforms into a mesh. Two examples are given below.
Example: star network.
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.
Example: Cauer network.
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.