MESH ANALYSIS

Mesh analysis is used to find the voltages and currents in a circuit. Usually mesh analysis is employed when there are more than one meshes/loops and many voltage sources.

Procedure:

1. Identify the number of meshes in a circuit.
2. Assign the mesh currents. The assignment of mesh currents is arbitrary and consistent.
3. Apply Kirchoff's Voltage Law (KVL) to all meshes and obtain the mesh equations.
4. Solve the equations and obtain the mesh currents.

Mesh Analysis with Current Sources:

1. If the current source is in only one mesh, then write KVL as usual and substitute value from the current source and solve. 
2. If current source is between two meshes, then eliminate the current source and element in series with it (like open circuiting the branch) and form a super mesh.
3. Write KVL and solve the equations. 
4. Write Nodal equation for the node where these elements where eliminated.
5. On solving all the equations mesh currents will be obtained.

Example:

Find the mesh/loop currents in the following circuit.

The circuit has a current source in between two meshes. So we eliminate the current source and the resistor in series with it, to form a super mesh.

KVL for the super mesh,

–8 + 2i1 – 2i3 + 12i2 – 4i3 = 0 or i1 + 6i2 – 3i3 = 4                               (1)

KVL for mesh 3,

8i3 – 2i1 – 4i2 = 0 or –i1 – 2i2 + 4i3 = 0                                                (2)

Kirchoff's Current Law (KCL) at node 0 in Fig. (a),

i1 = 4 + i2                     i1 – i2 = 4

Solving (1) to (3) we obtain the mesh currents,

i1 =  4.632 A, i2 =  631.6 mA, i3 =  1.4736 A

Reference:

Fundamentals of Electric Circuits( Fifth Edition) by Charles K Alexander and Matthew N O Sadiku

SIGN CONVENTION FOR CIRCUIT ANALYSIS

There are two types of sign conventions. They are:

1. Active Sign Convention
2. Passive Sign Convention

Active Sign Convention:

This sign convention is to be followed for active elements. When positive current enters the element through negative terminal energy is dissipated by the element. It is an energy source. The sign convention is as follows:

P=V(-I )

Passive Sign Convention:

This sign convention is followed for passive elements. When positive current enters the element through positive terminal energy is absorbed by the element. It is a passive element like resistor, capacitor, inductor etc. The sign convention is as follows:

P=V(I )

Reference:

Fundamentals of Electric Circuits (Fifth Edition) by Charles K Alexander and Matthews N O Sadiku

FUNDAMENTAL LAWS FOR CIRCUIT ANALYSIS

Ohm's law:

At a constant temperature, the electrical current (I) flowing through a fixed linear resistance (R) is directly proportional to the voltage (V) applied across it, and also inversely proportional to the resistance.

V=IR

Kirchoff's Current law: 

The algebraic sum of currents at a particular node is zero. In other words, the sum of currents entering a node is equal to the sum of currents leaving the node.

Kirchoff's Voltage law:

The algebraic sum of voltages in a closed loop is zero. In other words, voltage drop in a loop is equal to the voltage raise in the loop.

Voltage Division rule for resistors connected in series: 

A voltage source V and two resistors R1 and R2 are in series.

Volatge V1 through resistor R1 is:

V1=(R1*V)/(R1+R2)

Volatge V2 through resistor R2 is:

V2=(R2*V)/(R1+R2)

Current Division rule for resistors connected in parallel: 

A current source I and two resistors R1 and R2 are in parallel.

Current I1 through resistor R1 is:

I1=(R2*I)/(R1+R2)

Current I2 through resistor R2 is:

I2=(R1*V)/(R1+R2)

Equivalent Resistance for parallel resistors:

Two resistors R1 and R2 are in parallel. Their equivalent resistance R is:

R=(R1*R2)/(R1+R2)

Equivalent Resistance for series resistors:

Two resistors R1 and R2 are in series. Their equivalent resistance R is:

R=R1+R2

Reference:

Fundamentals of Electric Circuits (Fifth Edition) by Charles K Alexander and Matthews N O Sadiku

DIFFERENT TYPES OF CIRCUITS

Classification of circuits based on order:

• First Order Circuit: Circuit having differential equation of order one is called first order circuit. Also circuit having single storage element is called first order circuit. Examples: RL, RC circuits.

• Second Order Circuit: Circuit having differential equation of order two is called second order circuit. Also circuit having two storage elements is called second order circuit. Examples: RLC circuits.

Classification of circuits based on linearity:

• Linear Circuit: Value of resistance, inductance and capacitance are constant.They do not vary with voltage or current. In other words, output is proportional to input. Example: RLC circuits.

• Non-linear Circuit: Value of resistance, inductance and capacitance are not constant.They vary with voltage or current. In other words, output is not proportional to input. Examples: Diodes, Transistors.

Reference:
Fundamentals of Electric Circuits (Fifth Edition) by Charles K Alexander and Matthews N O Sadiku

FUNDAMENTAL CIRCUIT TERMINOLOGIES

Branch (b) - Branch represents a single element.

Node (n) - Node is the point of connection for two or more branches.

Loop (l) - Any closed path in a circuit is called a loop or mesh.

Fundamental theorem of network topology:

In any circuit the following equation must hold true:

b = n + l - 1

Reference:

Fundamentals of Electric Circuits (Fifth Edition) by Charles K Alexander and Matthews N O Sadiku

ELECTRIC SOURCES

Definition: A source of electrical power (electrical energy), a device that supplies electrical current. It can be electrochemical (battery or fuel cell) or an electromechanical device (dynamo) or a specialized electronic instrument. Also called "power source (supply)".

There are two types of electric sources namely voltage source and current source.

Voltage source:

A voltage source is a two terminal device which can maintain a fixed voltage.

•         Time Invariant Voltage Source: the output terminal voltage of the source changes with time.

•         Time Variant Voltage Source: the output terminal voltage of the source does not change with time.

•         Ideal Voltage Source: Delivers constant voltage irrespective of the current drawn through its terminals. So it produces infinite energy. It has no internal resistance. So no loss.

•         Practical Voltage Source: Has internal resistance. Hence has losses.

Current source:

The current source is a simple circuit, which will provide a current which remains constant regardless of the load placed at its output.

•         Time Invariant Current Source: the source in which current is not varying with time.

•         Time Variant Current Source: the source in which current is varying with time.

•         Ideal Current Source: Delivers constant current irrespective of the voltage drawn across its terminals. So it produces infinite energy. There is no internal resistance. So no loss.

•         Practical Current Source: Has internal resistance. So has losses.

Dependent and Independent Sources:

Dependent Sources: Produces current or voltage dependent on other circuit elements.

•         Voltage Dependent Voltage Source: Produces output voltage as a function of another voltage value.

•         Voltage Dependent Current Source: Produces output current as a function of a voltage value.

•         Current Dependent Current Source: Produces output current as a function of another current value.

•         Current Dependent Voltage Source: Produces output current as a function of a voltage value.

Independent Sources:

Independent sources produce current or voltage independent of other circuit elements.

·         Independent Voltage Source: produces current necessary to maintain constant voltage output independent of other circuit elements.

·         Independent Current Source: produces voltage necessary to maintain constant current output independent of other circuit elements.