Slide 1Circuit Analysis – A Systematic ApproachLearning by Examples: A Summing CircuitMesh Analysis: The RecipeStep 1: Identifying the MeshesSlide 6Step 2: Assigning Mesh CurrentsSlide 8Step 3: Voltages from Mesh CurrentsKVL Around Mesh 1KVL Around Mesh 2Slide 12Step 4: Solve the EquationsUsing MATLABAnother ExampleIdentify Mesh’sAssign Mesh CurrentsHow to Deal with Current SourcesSuperMeshKVL Around the SupermeshSolve the EquationsSolve Using MATLAB• Meshes and Loops• Steps of Mesh Analysis• Supermesh• ExamplesLecture 6. Mesh Analysis12Circuit Analysis – A Systematic Approach•Mesh Analysis is another general method that is almost as powerful as Nodal 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 analysis results in a system of linear equations which must be solved for unknown currents.3Learning by Examples: A Summing Circuit•The output voltage V of this circuit is proportional to the sum of the two input voltages V1 and V2.•This circuit could be useful in audio applications or in instrumentation.•The output of this circuit would probably be connected to an amplifier.+-Vout1k1k1kV1+-V2+-Vout = (V1 + V2)/34Mesh Analysis: The Recipe1. 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.5Mesh 21k1k1kStep 1: Identifying the MeshesV1+-V2+-Mesh 16Mesh Analysis: The Recipe1. Identify mesh (loops).2. Assign a current to each mesh.3. Apply KVL around each mesh to get an equation in terms of the mesh currents.4. Solve the resulting system of linear equations.71k1k1kStep 2: Assigning Mesh CurrentsV1+-V2+-I1I28Mesh Analysis: The Recipe1. Identify mesh (loops).2. Assign a current to each mesh.3. Apply KVL around each mesh to get an equation in terms of the mesh currents.4. Solve the resulting system of linear equations.9Step 3: Voltages from Mesh CurrentsRI1+ -VRVR = I1 RRI1+ -VRI2VR = (I1 - I2 ) R101k1k1kV1+-V2+-I1I2KVL Around Mesh 1-V1 + I1 1k + (I1 - I2) 1k = 0I1 1k + (I1 - I2) 1k = V1111k1k1kV1+-V2+-I1I2KVL Around Mesh 2(I2 - I1) 1k + I2 1k + V2 = 0(I2 - I1) 1k + I2 1k = -V212Mesh Analysis: The Recipe1. 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.13Step 4: Solve the Equations •The two equations can be combined into a single matrix/vector equation.2121k1k1k1k1k1k1VVIII1 + (I1 - I2) 1k = V1 (I2 - I1) 1k + I2 1k = -V2 •Re-organize the equations:- 1kI1 + (1k + 1k ) I2 = -V2 (1k + 1k )I1 - 1k I2 = V114Using MATLAB>> A = [1e3+1e3 -1e3; -1e3 1e3+1e3];>> v = [7; -4];>> i = inv(A)*vi = 0.00333 -0.00033I1 = 3.33mAI2 = -0.33mAVout = (I1 - I2) 1k = 3.66V15Another Example1k2k2k12V+-4mA2mAI016Mesh 2Mesh 3Mesh 1Identify Mesh’s1k2k2k12V+-4mA2mAI017Assign Mesh Currents1k2k2k12V+-4mA2mAI0I1I2I318How to Deal with 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?•The 4mA current source sets I2:I2 = -4mA•The 2mA current source sets a constraint on I1 and I3:I1 - I3 = 2mA•We have two equations and three unknowns. Where is the third equation?191k2k2k12V+-4mA2mAI0I1I2I3The Supermesh surrounds this source!The Supermesh does not include this source!SuperMesh20KVL Around the Supermesh-12V + I3 2k + (I3 - I2)1k + (I1 - I2)2k = 0I3 2k + (I3 - I2)1k + (I1 - I2)2k = 12V21Solve the Equations•The three equations can be combined into a single matrix/vector equation.V12mA2mA41k2k2k1k2k101010321III22Solve Using MATLAB>> A = [0 1 0; 1 0 -1; 2e3 -1e3-2e3 2e3+1e3];>> v = [-4e-3; 2e-3; 12];>> i = inv(A)*vi =0.0012 -0.0040 -0.0008I1 = 1.2mAI2 = -4mAI3 = -0.8mAI0 = I1 - I2 =
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