1. ECE201 Lect-101 Loop (Mesh) Analysis (3.2) Dr. Holbert February 27, 2006

2. ECE201 Lect-102 Loop Analysis Nodal analysis was developed by applying KCL at each non-reference node. Loop analysis is developed by applying KVL around loops in the circuit. Loop (mesh) analysis results in a system of linear equations which must be solved for unknown currents.

3. ECE201 Lect-103 Example: A Summing Circuit The output voltage V of this circuit is proportional to the sum of the two input voltages V 1 and V 2. This circuit could be useful in audio applications or in instrumentation. The output of this circuit would probably be connected to an amplifier.

4. ECE201 Lect-104 Summing Circuit Solution: V out = (V 1 + V 2 )/3 + – V out 1k  V1V1 V2V2 +–+– +–+–

5. ECE201 Lect-105 Steps of Mesh Analysis 1.Identify mesh (loops). 2.Assign a current to each mesh. 3.Apply KVL around each loop to get an equation in terms of the loop currents. 4.Solve the resulting system of linear equations.

6. ECE201 Lect-106 Mesh 2 1k  Identifying the Meshes V1V1 V2V2 Mesh 1 +–+– +–+–

7. ECE201 Lect-107 Steps of Mesh Analysis 1.Identify mesh (loops). 2.Assign a current to each mesh. 3.Apply KVL around each loop to get an equation in terms of the loop currents. 4.Solve the resulting system of linear equations.

8. ECE201 Lect-108 1k  Assigning Mesh Currents V1V1 V2V2 I1I1 I2I2 +–+– +–+–

9. ECE201 Lect-109 Steps of Mesh Analysis 1.Identify mesh (loops). 2.Assign a current to each mesh. 3.Apply KVL around each loop to get an equation in terms of the loop currents. 4.Solve the resulting system of linear equations.

10. ECE201 Lect-1010 Voltages from Mesh Currents R I1I1 +– VRVR V R = I 1 R R I1I1 +– VRVR I2I2 V R = (I 1 - I 2 ) R

11. ECE201 Lect-1011 KVL Around Mesh 1 -V 1 + I 1 1k  + (I 1 - I 2 ) 1k  = 0 I 1 1k  + (I 1 - I 2 ) 1k  = V 1 1k  V1V1 V2V2 I1I1 I2I2 +–+– +–+–

12. ECE201 Lect-1012 KVL Around Mesh 2 (I 2 - I 1 ) 1k  + I 2 1k  + V 2 = 0 (I 2 - I 1 ) 1k  + I 2 1k  = -V 2 1k  V1V1 V2V2 I1I1 I2I2 +–+– +–+–

13. ECE201 Lect-1013 Steps of Mesh Analysis 1.Identify mesh (loops). 2.Assign a current to each mesh. 3.Apply KVL around each loop to get an equation in terms of the loop currents. 4.Solve the resulting system of linear equations.

14. ECE201 Lect-1014 Matrix Notation The two equations can be combined into a single matrix/vector equation.

15. ECE201 Lect-1015 Solving the Equations Let:V 1 = 7V and V 2 = 4V Results: I 1 = 3.33 mA I 2 = -0.33 mA Finally V out = (I 1 - I 2 ) 1k  = 3.66V

16. ECE201 Lect-1016 Another Example 1k  2k  12V4mA 2mA I0I0 +–+–

17. ECE201 Lect-1017 Mesh 2 Mesh 3 Mesh 1 1. Identify Meshes 1k  2k  12V4mA 2mA I0I0 +–+–

18. ECE201 Lect-1018 2. Assign Mesh Currents I1I1 I2I2 I3I3 1k  2k  12V4mA 2mA I0I0 +–+–

19. ECE201 Lect-1019 Current Sources The current sources in this circuit will have whatever voltage is necessary to make the current correct. We can’t use KVL around the loop because we don’t know the voltage. What to do?

20. ECE201 Lect-1020 Current Sources The 4mA current source sets I 2 : I 2 = -4 mA The 2mA current source sets a constraint on I 1 and I 3 : I 1 - I 3 = 2 mA We have two equations and three unknowns. Where is the third equation?

21. ECE201 Lect-1021 1k  2k  12V4mA 2mA I0I0 I1I1 I2I2 I3I3 The Supermesh surrounds this source! The Supermesh does not include this source! +–+–

22. ECE201 Lect-1022 KVL Around the Supermesh -12V + I 3 2k  + (I 3 - I 2 )1k  + (I 1 - I 2 )2k  = 0 I 3 2k  + (I 3 - I 2 )1k  + (I 1 - I 2 )2k  = 12V

23. ECE201 Lect-1023 Matrix Notation The three equations can be combined into a single matrix/vector equation.

24. ECE201 Lect-1024 Solve Using MATLAB >> A = [0 1 0; 1 0 -1; 2e3 -1e3-2e3 2e3+1e3]; >> v = [-4e-3; 2e-3; 12]; >> i = inv(A)*v i = 0.0012 -0.0040 -0.0008

25. ECE201 Lect-1025 Solution I 1 = 1.2 mA I 2 = -4 mA I 3 = -0.8 mA I 0 = I 1 - I 2 = 5.2 mA

26. ECE201 Lect-1026 Class Example Learning Extension E3.8 Learning Extension E3.9 Learning Extension E3.11