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 dual network

Example#02: Draw dual of the given circuit 

Reference:

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

Network Equilibrium Equations - Electrical Network Graphs

This is my fifth article on electrical network graphs. The article covers the topic  network equilibrium equations. Network equilibrium equations completely determine the state of the network at any moment.

Key Concept

• Generalized form of network equilibrium equations for circuits having sources

Branch Current and Loop Current:

Let IB be the branch currents vector and IL be the loop currents vector. BT is the transpose matrix of the fundamental loop matrix or tie set matrix. Then IB branch current can be written as follows

IB = BT.IL

Nothing is new. Actually we have to analyse the network. In actual network if there is a current source in parallel with a passive element let's consider a resistor for simplicity. Then the total branch current will either be the sum or difference of the two currents.

• If current source and resistor current is in same direction, then branch current is the difference of two currents. IB = IS - IR
• If the current source and resistor current is in opposite direction, then branch current is the sum of two currents. IB = IS + IR

In this case the relationship between branch current and loop current can be modified as follows

IB = BT.IL + IS

Where,

Is is the column matrix of order bx1 representing current source across each branch.

Branch Voltage and Node Voltage:

VB = QTVn

Where

VB = branch voltage matrix

QT = transpose of fundamental cut set matrix

Vn = node voltage matrix

Let's consider, a voltage source is present in series branch, then total voltage across this branch will be algebraic sum of source voltage and branch voltage.

• If voltage source in series with a branch with similar polarity then total branch voltage will be the sum of two voltages. V = VS + Vb
• If voltage source in series with a branch with opposite polarity then total branch voltage will be the difference of two voltages. V = VS - Vb

So the above relation between node and branch voltages can be modified as

VB = QT Vn + VS

Where,

Vs = column matrix of order bx1 representing voltage sources in series with each branch

Generalized Form of Network Equilibrium Equations:

I am not going to formulate the network equilibrium equations, I am providing reference books, from where you can get proof of these equations.

Node Equation

YVn = A[YbVs - Is]

The above equation represents (n-1) node equations

Where

Y = AYbAT  is the nodal admittance matrix. This is (n-1) X (n-1) matrix.

A = Reduced incidence matrix

Yb = branch impedance matrix, order (bxb)

Loop Equation

ZIL = B[ZbIs - VS]

The above equation represents (b+n-1) loop equations

E = Z IL

Where,

E = B(VS - ZBIS)

Z = BZbBT is the loop impedance matrix of order (b-n+1)X(b-n+1) matrix

B = fundamental loop or tie set matrix

Zb = is the branch impedance matrix, order (bxb)

E = column matrix of order (b+n-1)x1 representing loop emf

Cut-Set Equation

YCVt = QC[YbVS - IS ]

The above equation represents (n-1) cut-set equations.

J = YC Vt

Where

J = QC(IS - YbVS)

YC = QCYbQCT , is Cut-Set impedance matrix of order (n-1) X (n-1)

QC = fundamental cut-set matrix

Vt = tree voltages

References:

1. Electrical Network Analysis and Synthesis
   By U.A.Bakshi, A.V.Bakshi

Network Analysis And Synthesis
By J.S.Chitode Dr.R.M.Jalnekar