Duality In Electric Circuits

Principle of Duality In Electric Circuits
It is interesting to know how systems relate to one another. How a mechanical system can be modelled as an electrical system and observed. The concept of duality in electrical circuits is of great importance. Two phenomena are said to be dual if they can be expressed by same form of mathematical equations. This topic is usually covered under the network topology or graph theory.

Key Questions:
  • What is principle of duality in electric circuits?
  • List of dual pairs and their explanation
  • Formation of dual networks

Principle of Duality:

Principle of duality in context of electrical networks states that
  • A dual of a relationship is one in which current and voltage are interchangeable
  • Two networks are dual to each other if one has mesh equation numerically identical to others node equation

List of Dual Pairs:

For evaluating a dual network, you should follow these points
  1. The number of meshes in a network is equal to number of nodes in its dual network
  2. The impedance of a branch common to two meshes must be equal to admittance between two nodes in the dual network
  3. Voltage source common to both loops must be replaced by a current source between two nodes
  4. Open switch in a network is replaced by a closed switch in its dual network or vice versa


Elements
Dual Elements
1
Voltage (v)
v = iR
Current (i)
i = vG
2
Short Circuit
Open Circuit
3
Series
Parallel
4
Norton
Thevenin
5
Resistance (R)
Conductance (G)
6
Impedance
Admittance
7
KVL
KCL
8
Capacitance
Inductance

Formation of Dual Networks:

The principle of duality is applicable to planar circuits only. Carefully read the points stated below, follow each step and draw the dual circuit

  1. Place a dot within each loop, these dots will become nodes of the dual network
  2. Place a dot outside of the network, this dot will be the ground/datum node of the dual network
  3. Carefully draw lines between nodes such that each line cuts only one element
  4. If an element exclusively present in a loop, then connect the dual element in between node and ground/datum node
  5. If an element is common in between two loops, then dual element is placed in between two nodes
  6. Branch containing active source, consider as a separate branch
  7. Now to determine polarity of voltage source and direction of current sources, consider voltage source producing clockwise current in a loop. Its dual current source will have a current direction from ground to non-reference node

Example#01: Draw dual of the given circuit.

Graphical method of drawing a dual network
Graphical method of drawing dual network

Example#02: Draw dual of the given circuit 

Find out dual of the given network (solved example)

Reference:

  1. Fundamentals of Electric Circuits by Alexander
  2. Circuit Theory by A.V. Bakshi and U.A. Bakshi

4 comments:

  1. So, when you obtain, say, the tension in a given element, does it represent the current in his respective dual?

    ReplyDelete
    Replies
    1. I am considering electrical networks only. Under this context the dual of current is voltage.

      Delete
  2. As an electrician technician, I always strive to provide my clients with the best possible service. Whether it's a minor repair or a major installation, I approach every job with care and attention to detail.

    ReplyDelete
  3. Bro you could have used red marker for the nodes instead of a yellow one

    ReplyDelete

Popular Posts