Power Sources - Current & Voltage Sources

To derive an electronic circuitry there has to be source of energy. A voltage source and a current source is an example of energy source. An electrical energy source is a device that is capable of converting non-electrical energy into electrical energy. For example a battery which converts chemical energy into electrical energy. A dynamo which converts mechanical energy into electrical energy. It delivers power to rest of the circuit elements which are passive in nature. Our aim in this article is to explain some important concepts regarding voltage sources and current sources.

Key Concepts

• Ideal Vs practical sources (voltage and current sources)
• Independent voltage and current sources
• Dependent voltage and current sources

Concept Of An Ideal Voltage Source:

As the name implies it is responsible for delivering voltage across the circuit. It is a two terminal device. There are various definitions found for an ideal voltage source. An ideal voltage source is the one which maintains prescribed/constant voltage across its terminals regardless of the load connected to its terminals. Or we can say an ideal voltage source maintains a constant voltage across its terminals regardless of the current flowing through its terminals. I will explain it with the help of Ohm's law. Or in other words an ideal voltage source has 'zero’ internal resistance. If zero internal resistance then you can definitely have no voltage drop across the terminals.

According to Ohm's law V=IR

Consider

• V=12V, and R=10Ω (load resistor) it can not maintain unlimited current as per Ohm's law. As the load varies current varies accordingly
• V=12V, R=0Ω (internal resistance)   I = 12/0 = ∞ , it was is capable of supplying infinite current
• For ideal voltage source terminal voltage (V) remains same regardless of the load connected. E=V, where E=EMF and V=Terminal Voltage.

Concept Of A Practical Voltage Source:

Do you think an ideal voltage source exists?? Ideal voltage sources are not practical. For a practical voltage source there's a finite internal resistance that's associated with it's internal connection and terminals. This causes voltage drop across the terminals.

• All practical voltage sources have small internal resistance Ri
• Internal resistance Ri is connected in series with any load resistance RL
• There's a voltage drop of I*Ri because of this internal resistance
• Terminal voltage (V) which appears across load is V=E-I*Ri.
• Because of this small internal resistance load voltage is always less than the source voltage

Concept Of An Ideal Current Source:

As the name suggests it is the source of current, that supplies constant current to the circuit. An ideal current source has infinite internal resistance. Or an ideal current source maintains a constant current flow regardless of the voltage drop across its terminals.

• It provides constant current to the load irrespective of other conditions in the circuit
• It has 100% efficiency
• Because of infinite internal resistance all the current supplies to the load
• Ri = internal resistance = ∞ From Ohm's law I=V/Ri , because of such high internal resistance all the current will flow through the load irrespective of the voltage

Concept Of A Practical Current Source:

As the ideal voltage source doesn't exist, in the same way ideal current source also doesn't exist. No current source can maintain constant current. We have seen a practical voltage source can be modelled by a voltage source and a small resistance in series with it. Whereas a practical current source is represented a shunt resistor (Rs) connected in parallel with an ideal current source. Rs should be very large. It should be large enough so that small load resistances have no effect on the load current.

Reason for larger value of Rs

Do you remember, the current finds the lowest resistance path. The path which has lowest resistance, maximum current will flow through that path. From Ohm's law,  I=V/R lower the resistance higher will be the current value.

• Rs should be large enough so that all current flows through the load resistor

Independent Voltage Source:

Independent voltage sources that have voltage (fixed or time variant) which is not affected by any other current or voltage elsewhere in the circuit.

Independent Current Sources:

Independent current sources that delivers or absorbs current (fixed or time variant) at its terminals which is not affected by any other voltage or current elsewhere in the circuit.

Dependent Voltage Sources:

Dependent sources are also called controlled sources. A dependent voltage source can either be controlled by voltage or current elsewhere in the circuit. Input and output are linearly dependent. The equation for current and voltage is a linear equation.

There are two possible types of dependent voltage sources:

• Voltage Controlled Voltage Source (VCVS)
• Current Controlled Voltage Source (CCVS)

Voltage Controlled Voltage Source is a voltage source controlled by a voltage vc. vc= controlled voltage.

V= K*vc

Where K is the constant of proportionality

Current Controlled Voltage Source is a voltage source controlled by current ic. ic= controlled current.

V = K*ic

Where K is the constant of proportionality

Dependent Current Sources:

There are two possible types of dependent current sources:

• Voltage Controlled Current Source (VCCS)
• Current Controlled Current Source (CCCS)

Voltage Controlled Current Source is a current source controlled by voltage vc. vc is the controlled voltage.

I = K*vc

Where K is the constant of proportionality.

Current Controlled Current Source is a current source controlled by current ic. ic= controlled current.

I = K*ic

Where K is the constant of proportionality.

Circuit Analysis - Methods, Laws & Theorems

This tutorial is intended to provide detailed information about the circuit or network analysis, methods of analysis and network theorems. It is just an introductory article, in which I gather all the laws, methods and theorems. This topic is usually a part of the ECE basic course on circuits. The term solving a circuit means determining all the voltage across each element and currents through each element present in a circuit.

Key Concepts:

• What are fundamental laws for solving circuits?
• Methods used for network/circuit analysis
• Network theorems
• What is the purpose of network theorems or why do we study network theorems

Fundamental Circuit Laws:

To determine current and voltage in an electric circuit, we need to gain knowledge of fundamental laws that govern electronic circuits.

1. Ohm's law
2. Kirchhoff's Voltage Law (KVL)
3. Kirchhoff's Current Law (KCL)

Ohm's Law defines a linear relationship between voltage and current in an ideal conductor. This is one of the most important and fundamental laws of electric circuits.

Kirchhoff’s Current Law is used when analysing a parallel circuit. It is based on the idea of conservation of charge. According to KCL 'current entering the node is equal to current leaving the node’.

Kirchhoff's Voltage Law is used when analysing a series circuit. It is based on the idea of conservation of potential. According to KVL 'the sum of voltage drop or potential difference across a closed loop is zero’.

Methods of Analysis:

Analysis of a circuit is the determination of output response. These methods of analysis are based on basic circuit laws. These methods are restricted to linear circuits only. The two most common methods are

1. Mesh analysis
2. Nodal analysis

Nodal analysis is nothing but an application of KCL. In the nodal analysis, we apply KCL to each node. If we have 'n’ number of nodes, then there are (n-1) linearly independent node equations. These equations are in terms of node potential.

Mesh | loop analysis is nothing but an application of KVL. In mesh analysis, we apply KVL around each loop in a circuit. If we have 'b’ branches and 'n’ nodes in a circuit then we have b-(n-1) linearly independent equations. These equations are in terms of mesh currents.

Why do we study network theorems?

I suppose you are familiar with basic network analysis methods. Like node analysis and mesh analysis. When dealing with KCL and KVL we will have a fairly large number of independent equations. As I discussed in methods of analysis, we get (n-1) node equations and b-(n-1) mesh current equations, so total ‘b’ number of equations. Did you find them time-consuming, you have to deal with so many equations (the number of node and mesh equations are equal to the number of elements/branches present in a network). For example, you have a network/circuit having 4 passive elements, then there are 4 current or voltage equations. And if there are 5 passive elements you have to solve 5 equations and so on. Analysing more complex circuits by using these methods seem to be ridiculous. What do you think?? So instead of applying tedious mesh and nodal analysis to complex networks, we develop network theorems.

Network theorems

1. Superposition theorem
2. Reciprocating theorem
3. Thevenin’s theorem
4. Norton's theorem
5. Compensation theorem
6. Millman's theorem
7. Substitution theorem
8. Maximum power transfer theorem
9. Tellegen's theorem

Applications

• Some of these apply to linear as well as non-linear circuits
• With the help of these theorems, we can solve DC as well as AC Circuits
• Easy comprehension of complex circuits

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