METHODS OF CIRCUIT ANALYSIS

Methods of Circuit Analysis Mesh Analysis Nodal Analysis

Mesh Analysis Kirchhoff’s Voltage Law (KVL) forms the basis of mesh analysis. This technique is applicable to Basic circuit Circuit with dependent source Circuit with current source Case 1: Current source at the outer most boundary (known as mesh current) Case 2: Current source in between two loops (known as supermesh)

Step to determine Mesh Current Assign mesh currents I1, I2…, In to the n meshes Apply KVL to each of n meshes. Use Ohm’s Law to express voltages in terms of mesh currents. Solve the resulting n simultaneous equation to get the mesh current,

Example 10.3 For the circuit below, find Io using mesh analysis

Solution Applying KVL to Mesh 1 Mesh 2 …(1) Mesh 2 …(2) Substitute (I3=5) into meshes (1) and (2) …(3) …(4)

Solution Put equation (3) and (4) in matrix form Find determinant for the matrix (Cramer’s Rule)

Solution Use Cramer’s rule to solve for I2 Hence Io = (-I2) =

Practice Problem 10.3 For the circuit below, find Io using mesh analysis

Solution

Solution Mesh 1 …(1) Mesh 2 Mesh 3 Insert Mesh 3 into Mesh 2 …(2)

Solution Simplify Equation (1) …(3) Substitute equation (3) into (2)

Solution Hence

Example 10.4 For the circuit below, find Vo using mesh analysis

Solution

Solution Mesh 1 Mesh 2 Supermesh …(1) Mesh 2 Supermesh …(2) Due to current source between meshes 3 and 4 at node A …(3)

Solution Combine I2 = -3 into equation (1) …(4) Combine I2 = -3 into equation (2) and (3) …(5) Put equation (4) and (5) into matrix

Solution Use Cramer’s Rule to solve for I1

Solution Solve for Vo

Practice Problem 10.4

Solution

Solution Mesh 1 Supermesh …(1) Supermesh …(2) Also the current source between meshes 2 and 3 …(3)

Solution Eliminating I3 from equation (1) and (2) …(4) …(5) Put equation (4) and (5) into matrix

Solution Use Cramer’s Rule to solve for I1 and then Io

Exercise III (Problem 10.38) Using mesh analysis, find Io

Solution

Solution Mesh 1 …(1) Mesh 2 …(2) Substitute (1) into (2) …(3)

Solution Supermesh …(4) …(5) Substitute (1) and (5) into (4) …(6)

Solution Put equation (3) and (6) into matrix Use Cramer’s Rule to solve for I2

Nodal Analysis The basis of nodal analysis is Kirchhoff’s Current Law (KCL). This technique is applicable to Basic Circuit Circuit with dependent source Circuit with voltage source Case 1: Voltage source in between reference node and essential node Case 2: voltage source in between two nodes

Step to determine Node Voltages Select a node as the reference node. Assign voltages V1,V2…,Vn-1 to the remaining n-1 nodes. Apply KCL to each of the n-1 nonreference node. Use Ohm’s Law to express the branch currents in term of node voltages. Solve the resulting simultaneous equation to obtain the unknown node voltage.

Example 10.1 Find Ix in the circuit using nodal analysis

Solution Convert the circuit into frequency domain

Solution Applying KCL at node 1 Iin = Ix + I2 …(1)

Solution Applying KCL at node 2 Ix + I2 = I3 But Hence …(2)

Solution Put equation (1) and (2) into matrix Find determinant

Solution Solve for V1 and V2 using Cramer’s Rule Solve for Ix

Practice Problem 10.1 Find V1 and V2 usind nodal analysis

Solution Convert into frequency domain

Solution At node 1 …(1) At node 2 where …(2)

Solution Put equation (1) and (2) into matrix Solving for V1 and V2 using Cramer’s Rule

Example 10.2 Compute V1 and V2 in the circuit

Solution

Solution Nodes 1 and 2 form a supernode. Applying KCL to the supernode gives …(1) But a voltage source is connected between nodes 1 and 2 …(2)

Solution Substitute equation (2) in (1) result in

Practice Problem 10.2 Calculate V1 and V2 in the circuit using nodal analysis

Solution The only non-reference node is supernode The supernode gives …(1) The supernode gives …(2)

Solution Substitute (2) into (1) gives Therefore

Exercise III (Problem 10.9) Find Vo in the circuit using nodal analysis

Solution Convert into frequency domain

Solution Node 1 …(1) Node 2 Substitute …(2)

Solution Divide both equation (1) and (2) with 100 to simplify the equations and put into matrix

Solution Solve for V2 using Cramer’s Rule Solve for Vo by using voltage divider rule