Mesh-Current Method (Loop Analysis)

Mesh-Current Method (Loop Analysis)• Nodal analysis was developed by applying KCL at each non-referencenode.• Mesh-Current method is developed by applying KVL aroundmeshes in the circuit.• A mesh is a loop which doesn't contain any other loops within it.• Loop (mesh) analysis results in a system of linear equations whichmust be solved for unknown currents.• Reduces the number of required equations to the number of meshes• Can be done systematically with little thinking• As usual, be careful writing mesh equations – follow sign convention.Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering1

•Powerful analysis method which applies KVL to find unknown currents.•It is applicable to a circuit with no branches crossing each other.no branchesCrossingDr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering2

DefinitionsA mesh is a loop which does not contain any other loops within it. A+-i 4i 5 i 1i 2Bi gCi 3 Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering3

Steps of Mesh AnalysisStep1Identify the number ofbasic meshes.Step 2Assign a current to each mesh.Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering4

Step 3Apply KVL around each loop to getan equation in terms of the loop currents.Step 4Solve the resulting system of linear equations.As an exampleV 1 + I 1 R 1 + (I 1 - I 2 ) R 3 = 0I 2 R 2 + V 2 + (I 2 -I 1 ) R 3 = 0Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering5

ExampleUse mesh analysis to find the current i oSolutionWe apply KVL to the three meshesFor mesh 1Divide by 2For mesh 2Divide by 2Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering6

For mesh 3at node ABy SubstitutionDivide by 2Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering7

Cases to be considered for Mesh AnalysisCase IA current source exists only in one meshset i 2 = −5A and write a mesh equation for the othermeshFor Loop 1-10 + 4i 1 + 6(i 1 -i 2 ) = 0For Loop 2i 2 = - 5A(No need to write a loop equation)Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering8

Case IIA current source exists between two meshesExclude this partDr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering9

a supermesh was created byexcluding the current source and anyelements connected in series with it,Why???Because mesh analysis applies KVL—we do not know thevoltage across a current source in advance.A supermesh results when two meshes have a (dependentor independent) current source in common.Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering10

Applying KVL to the supermeshi 1 i 2Apply KCL to a node in the branch where the two meshes intersect.Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering11

Case IIIMesh with Dependent SourcesSolve forI xi x = i 1 - i 2-75 + 5i 1 + 20(i 1 -i 2 ) = 010i x + 20(i 2 -i 1 ) + 4i 2 = 0Solve the two equations-75 + 5i 1 + 20(i 1 -i 2 ) = 0 …………………(1)10(i 1 -i 2 ) + 20(i 2 -i 1 ) + 4i 2 = 0 …………….(2)theni 2 = 5Ai 1 = 7ADr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering12

Example:Use the mesh-current method to find i oDr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering13

Solutionmeshes 1 and 2 form a supermeshmeshes 2 and 3 form a supermeshDr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering14

Applying KVL to the larger supermesh,Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering15

For the independent current source, we apply KCL to node PDr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering16

For the dependent current source, we apply KCL to node Q:Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering17

Applying KVL in mesh 4,Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering DepartmentELCT608, Electrical Engineering18