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Loop (Mesh) Analysis

Loop (Mesh) Analysis. Loop (mesh) analysis is the systematic application of KVL around various loops in a circuit. The KVL equations are written in terms of loop currents, common to all elements in a loop. The result will be a system of equations in which the unknowns are these loop currents.

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Loop (Mesh) Analysis

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  1. Loop (Mesh) Analysis • Loop (mesh) analysis is the systematic application of KVL around various loops in a circuit. • The KVL equations are written in terms of loop currents, common to all elements in a loop. • The result will be a system of equations in which the unknowns are these loop currents. • The solution of these equations will, therefore, yield values for the loop currents. ENGR201 - Loop Analysis

  2. Definition of a loop a b c d R1 R2 R3 + - R4 R5 R6 Vs e • Definition: • A loop is a path which satisfies both of the following conditions: • the path begins and ends at the same point (node) • no element is traversed more than once. • There are six (6) loops in the circuit below. Three loops are shown, each marked by a #1: a-b-e-a #2: b-c-e-b #3: c-d-e-c ENGR201 - Loop Analysis

  3. More Loops a b c d R1 R2 R3 Vs R4 R5 R6 + - e The remaining three are loops shown below #4: a-b-c-d-e-a #5: a-b-c-e-a #6: b-c-d-e-b ENGR201 - Loop Analysis

  4. Clockwise Convention a b c d R1 R2 R3 Vs + - R4 R5 R6 e The diagram is rather “busy” when all six loops are shown. Although loops can be drawn in either direction, the convention is to use a clockwise direction, as shown. ENGR201 - Loop Analysis

  5. Mesh Versus Loop a b c d R1 R2 R3 + - Vs R4 R5 R6 e A mesh is a special type of loop, one which contains no other loops inside. Below, meshes are marked by a Loops are marked by a ENGR201 - Loop Analysis

  6. Mesh Analysis a b c d R1 R2 R3 + - R4 R5 R6 Vs e • To keep the analysis as systematic as possible, we use (if possible) only meshes to perform loop analysis (mesh analysis). • There are cases (when a mesh traverses a current source) when we must use a loop in place of a mesh. ENGR201 - Loop Analysis

  7. Mesh Currents a b c d R1 R2 R3 I3 I2 I1 + - R4 R5 R6 Vs e • A mesh current is assigned to each mesh as shown. • A mesh current is the amount of current common to all elements in the mesh. • Mesh current is not (necessarily) the same as branch current. • Branch current is the actual current in an element and may be made up of more than one mesh current. ENGR201 - Loop Analysis

  8. Mesh vs. Branch Currents Iab a b c d Icb Icd Iec R1 R2 R3 I3 + - I2 I1 R4 R5 R6 Vs Ibe e Branch (element) currents can be written in terms of mesh currents: Iab = I1 Icb = -I2 Icd = I3 Ibe = I1 - I2 Iec = I3 - I2 ENGR201 - Loop Analysis

  9. Loop (Mesh) Analysis and KVL + - a b c d R1 R2 R3 + - + - I3 I2 I1 R4 R5 R6 Vs e Loop analysis applies KVL around each loop writing the voltage drops in terms of the loop currents. The convention is to assume all resistive voltages are voltage drops as one follows a loop current around the loop. The resulting equation for loop #1 is: Vs = R1I1 + R4(I1 -I2) = I1 (R1+R2) - I2  R4 ENGR201 - Loop Analysis

  10. Loop #2 + - a b c d R1 R2 R3 + - - + I3 I2 I1 + - R4 R5 R6 Vs e The resulting equation for mesh #2 is: 0 = R2I2 + R5(I2 -I3) + R4 (I2-I1) 0 = - I1 R4 + I2  (R2+R4+R5) - I3 R5 ENGR201 - Loop Analysis

  11. Loop #3 + - a b c d R1 R2 R3 - + + - I3 I2 I1 + - R4 R5 R6 Vs e The resulting equation for mesh #3 is: 0 = R3I3 + R6I3 + R5(I3 -I2) 0 = - I2 R5 + I3  (R3+R5+R6) ENGR201 - Loop Analysis

  12. Loop Equations – Matrix Form + - The equations represented in matrix form are: a b c d R1 R2 R3 I3 I2 I1 R5 R4 R6 Vs e ENGR201 - Loop Analysis

  13. Matrix Structure + - Notice the structure: the coefficients of I1, I2, and I3 in their respective mesh equations are the sum of the resistors in that mesh. a b c d R1 R2 R3 I3 I2 I1 R5 R4 R6 Vs e ENGR201 - Loop Analysis

  14. Matrix Structure + - The off-diagonal elements are equal to the negative of the resistor shared by two meshes: a b c d R1 R2 R3 I3 I2 I1 R5 R4 R6 Vs e ENGR201 - Loop Analysis

  15. Matrix Structure a b c d R1 R2 R3 I3 I2 I1 R5 R4 R6 + - Vs e The elements on the forcing function matrix (on the right-hand side) are the voltage source values; positive if the mesh current travels through a voltage rise and negative if it is a voltage drop. ENGR201 - Loop Analysis

  16. Using Mesh Currents a b c d R1 R2 R3 + Vde - + Vce - + - I3 I2 I1 R5 R6 Vs R4 e Mesh (loop) analysis generates N independent equations, one for each mesh (loop) current. These equations can be solved for the mesh currents. Any voltage or branch current can be found from the values of the mesh currents. For example, Vce = R5(I2 - I3) and Vde = I3 R5 ENGR201 - Loop Analysis

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