Dual Network Circuit Analysis (Theory And Examples)

Contents

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Key learnings:

Dual Network Definition: A dual network is defined as two electrical networks where the mesh equations of one match the node equations of the other.
Kirchhoff’s Laws: Dual networks are based on Kirchhoff’s Current Law and Kirchhoff’s Voltage Law.
Equation Comparison: Mesh equations use voltage, and node equations use current, showing duality in variables.
Construction of Dual Networks: A series RLC circuit and its corresponding parallel RLC circuit illustrate dual networks.
Practical Example: Replacing variables in a series RLC circuit to form a parallel RLC circuit demonstrates dual network construction.

What is a Dual Network?

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

Applying Kirchhoff Voltage Law in the network A, above we get,

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

We find that equations (i) and (ii) are similar mathematically. Equation (i) is in mesh form, while equation (ii) is in node form.

In equation (i), the left side variable is voltage, while in equation (ii), it is current.

Similarly, the right side of equation (i) is a product of the current and total impedance of the circuit.

Similarly, the right side of equation (ii) is the product of voltage and admittance of the circuit.

These two networks are dual networks. However, examples show that dual networks are not necessarily equivalent in function.

The circuit equation of the two dual networks are similar in form but the variable is interchanged.

Let us consider the series RLC circuit as shown below.

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

Hence:

This follows Kirchhoff’s Current Law. According to the definition, network C and network D are dual to each other.