Duality concept

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Circuit and networkActive Learning Assignment

Index1.Duality

2.Network convention for determining source orientation

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DUALITY

• Several analogous situations will have been noted in the preceding discussions, while analysing the networks. The statements of two Kirchhoff’s laws are almost similar word of word with ‘voltage’ substituted for ‘current’, ‘independent loop’ for ‘independent node pair’, etc. Like wise the integral-differential equations that resulted from the applications of two Kirchhoff's laws are similar in appearance. This repeated similarity is only part of larger pattern of identical behaviour patterns in the roles played by voltage and current in network analysis.

• L di/dt + Ri + 1/C∫idt = v(t) (1)

• C’ dv’/dt + G’v’ + 1/L’∫v’dt = I’(t) (2)

• The role of current and voltage in the two networks have been interchanged.

• The solution of one equation is also the solution of the other.

• This similarity is termed the principle of duality.

• An inspection of the terms of equations (1) and (2) shows that the following are the analogous quantities(with primes omitted)

• Ri and Gv• L di/dt and C dv/dt• 1/C∫ idt and 1/L∫ vdt

Evidently, the following pairs are dual quantities.• R and G• L and C• Loop current, I and node voltage v • Q or Ψ or• ∫ idt ∫ vdt• Loop and node-pair• Short circuit and open circuit• Switch open and switch closed• Current source and voltage source

The following procedure may be followed in constructing the dual of a given network:

1. Inside each loop place a node and give it a number for all loops for convenience. Place an extra node, the datum node, external to the network say 0 node. Place an extra the same numbered nodes on a separate space on the paper for construction of the dual.

2. Draw a line between two nodes of the original network traversing only one elements at a time. Thus draw lines from node to node through the elements in the original network. For each elements traversed in the original network, connect the dual elements from the listing just given – on dual network being constructed.

3. Continue this process until the number of possible paths through single elements is exhausted. If we go through a connecting leads assumed to be a short, the dual elements is an open circuit.

4. The network so constructed is a dual network. This construction may be verified by witting the mesh current equations for the one network and nodal equations for the other network.

Example- 1

Ans - After duality concept theorem

Example - 2

Ans -After duality concept theorem

Network conventional for determining source orientation

• In a linear bilateral network containing a single energy source , the magnitude of the branch voltage or current is fix and does not depend upon the orientation of the source.

• However , if the network contains two or more energy source then the branch voltage or current depends on the orientation of the source .

• We are already aware that the resistances , inductances , and capacitances in the original network appear in dual network as conductance , capacitances and inductances respectively with the same numerical values .

• The voltage source in the original network are replace by the current sources in the dual network and vice – versa.

• The most important task is to decide the orientation of these sources in the dual network it is determine as follows:

• First of all , we assume a clockwise direction for all mesh currents is original network N1. each mesh current enclosing a dot(node) of the dual network say N2.

• It is important to note that the orientation of the voltage source is identified with the direction of the mesh current whereas the orientation of the current source is identified with the node which is enclosed by mesh current.

• If the direction of the voltage source and clockwise mesh current is same in the network N1, then orientation of the current source in the dual network N2 is towards the node which is enclosed by the said mesh current in N1 .

• Conversely if the directions of the voltage source and the clockwise mesh current are opposite then the current of the current source in N2 is directed away from the node which is enclosed by the said mesh current in N1 .

• The principle is valid for the original network N1 as well as the dual network N2.

Example-3

Ans- After duality concept

Example-4

Ans- After duality concept

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