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METHODS OF CIRCUIT ANALYSIS
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Methods of Circuit Analysis
Mesh Analysis Nodal Analysis
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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)
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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,
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Example 10.3 For the circuit below, find Io using mesh analysis
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Solution Applying KVL to Mesh 1 Mesh 2
…(1) Mesh 2 …(2) Substitute (I3=5) into meshes (1) and (2) …(3) …(4)
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Solution Put equation (3) and (4) in matrix form
Find determinant for the matrix (Cramer’s Rule)
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Solution Use Cramer’s rule to solve for I2 Hence Io = (-I2) =
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Practice Problem 10.3 For the circuit below, find Io using mesh analysis
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Solution
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Solution Mesh 1 …(1) Mesh 2 Mesh 3 Insert Mesh 3 into Mesh 2 …(2)
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Solution Simplify Equation (1) …(3) Substitute equation (3) into (2)
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Solution Hence
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Example 10.4 For the circuit below, find Vo using mesh analysis
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Solution
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Solution Mesh 1 Mesh 2 Supermesh
…(1) Mesh 2 Supermesh …(2) Due to current source between meshes 3 and 4 at node A …(3)
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Solution Combine I2 = -3 into equation (1)
…(4) Combine I2 = -3 into equation (2) and (3) …(5) Put equation (4) and (5) into matrix
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Solution Use Cramer’s Rule to solve for I1
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Solution Solve for Vo
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Practice Problem 10.4
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Solution
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Solution Mesh 1 Supermesh
…(1) Supermesh …(2) Also the current source between meshes 2 and 3 …(3)
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Solution Eliminating I3 from equation (1) and (2)
…(4) …(5) Put equation (4) and (5) into matrix
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Solution Use Cramer’s Rule to solve for I1 and then Io
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Exercise III (Problem 10.38) Using mesh analysis, find Io
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Solution
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Solution Mesh 1 …(1) Mesh 2 …(2) Substitute (1) into (2) …(3)
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Solution Supermesh …(4) …(5) Substitute (1) and (5) into (4) …(6)
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Solution Put equation (3) and (6) into matrix
Use Cramer’s Rule to solve for I2
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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
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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.
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Example 10.1 Find Ix in the circuit using nodal analysis
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Solution Convert the circuit into frequency domain
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Solution Applying KCL at node 1 Iin = Ix + I2 …(1)
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Solution Applying KCL at node 2 Ix + I2 = I3 But Hence …(2)
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Solution Put equation (1) and (2) into matrix Find determinant
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Solution Solve for V1 and V2 using Cramer’s Rule Solve for Ix
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Practice Problem 10.1 Find V1 and V2 usind nodal analysis
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Solution Convert into frequency domain
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Solution At node 1 …(1) At node 2 where …(2)
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Solution Put equation (1) and (2) into matrix
Solving for V1 and V2 using Cramer’s Rule
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Example 10.2 Compute V1 and V2 in the circuit
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Solution
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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)
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Solution Substitute equation (2) in (1) result in
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Practice Problem 10.2 Calculate V1 and V2 in the circuit using nodal analysis
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Solution The only non-reference node is supernode The supernode gives
…(1) The supernode gives …(2)
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Solution Substitute (2) into (1) gives Therefore
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Exercise III (Problem 10.9) Find Vo in the circuit using nodal analysis
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Solution Convert into frequency domain
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Solution Node 1 …(1) Node 2 Substitute …(2)
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Solution Divide both equation (1) and (2) with 100 to simplify the equations and put into matrix
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Solution Solve for V2 using Cramer’s Rule
Solve for Vo by using voltage divider rule
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