Dual Network

Two electrical network are said to be dual, if the mesh equations of one is the node equation of other.

The dual network are based on Kirchhoff Current Law and Kirchhoff Voltage Law.
electric mesh network
Applying Kirchhoff Voltage Law in the network A, above we get,

electric node network
Applying Kirchhoff Current Law in the network B, above we get,

Here we have found that equation (i) and (ii) are similar in their mathematical form. Equation (i) is in mesh form and equation (ii) is in nodal form. Here, left side variable of equation (i) is voltage and left side variable of equation (ii) is current. Similarly, right side of equation (i) is product of current and total impedance of the circuit. Similarly, right side of equation (ii) is the product of voltage and admittance of the circuit. So, it is needless to say these two networks are dual network. From, the examples it is also clear that dual network may not be equivalent network. The circuit equation of two dual network are similar in form but the variable are interchanged.

Construction of Dual Network

Let us consider series RLC circuit as shown below.
series rlc circuit
Applying Kirchhoff Voltage Law in this circuit, we get,

Let us replace all the variables and constants by their dual in the equation.By doing that, we get,

The electrical network drawn by the circuit equation (iv), would be
parallel rlc circuit

It is needless to say

This is nothing but Kirchhoff Current Law. As per definition of dual network, the network C and network D are dual to each other.

Table of Dual Elements

ElementElement
ResistanceConductance
InductanceCapacitance
Service BranchParallel Branch
Switch ClosedSwitch Open
ChargeFlux Linkage
MeshNode

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