INC 111 Basic Circuit Analysis Week 4 Mesh Analysis
Mesh Analysis (Loop Analysis) Mesh = A closed loop path which has no smaller loops inside
Mesh Analysis • Procedure • Count the number of meshes. Let the number equal N. • Define mesh current on each mesh. Let the values beI1, I2, I3, … • Use Kirchoff’s voltage law (KVL) on each mesh, generating N equations • Solve the equations
I2 I1 Mesh current (loop current) Example Use mesh analysis to find thepower consumption inthe resistor 3 Ω
I2 I1 Loop 1 Equation 1 Loop 2 Equation 2 I1 = 6A, I2 = 4A, The current that pass throughR 3Ω is6-4 = 2A (downward) Power = 12 W
I2 I1 I3 Example Use Mesh analysis to findVx
I2 Equation 1 I1 I3 Equation 2 Equation 3 I1 = 3A, I2 = 2A, I3 = 3A Vx = 3(I3-I2) = 3V
Supermesh When there is a current source in the mesh path, we cannot use KVL because we do not know the voltage across the current source. We have to use supermesh, which is a combination of 2 meshes to be a big mesh, and avoid the inclusion of the current source in the mesh path.
I2 I1 I3 Example UseMesh analysis to findVx
I2 I1 I3 Equation from 2nd loop
Supermesh I2 I1 I3 Equation 2 Equation 3
I1 = 9A I2 = 2.5A I3 = 2A Vx = 3(I3-I2) = -1.5V
How to choose betweenNode and Mesh Analysis The hardest part in analyzing circuits is solving equations. Solving 3 or more equations can be time consuming. Normally, we will count the number of equations according to each method and select the method that have lesser equations.
Example From the previous example, if we want to use Nodal Analysis 7V V2 V1 V3 0V
I2 I1 I3 Example: Dependent Source Find Vx
I2 I1 I3 Equation 1 Equation 2 Equation 3 Equation 4 I1=15A, I2=11A, I3=17A Vx = 3(17-11) = 18V
Special Techniques • Superposition Theorem • Thevenin’s Theorem • Norton’s Theorem • Source Transformation
Superposition Theorem In a linear circuit, we can calculate the value of current (or voltage) that is the result from each voltage source and current source independently. Then, the real value is the sum of all current (or voltage) from the sources.
Linearity V and I have linear relationship I V
Implementation • When calculating the effect of a source, the other sources will be set to zero. • For voltage sources, when set as 0V, it will be similar to short circuit • For current sources, when set as 0A, it will be similar to open circuit
Example I1 = 1A I2 = 2A I total = 1+2 = 3A
Example I1 = 1A I2 = 0A I total = 1+0 = 1A
Example Find voltage Vx
Superposition andDependent Source Dependent sources cannot be used with superposition. They have to be active all the time.
Example Usesuperposition to findIx
KVL FindIx byeliminating the current source 3A
Ix+3 KVL outer loop FindIx by eliminating the voltage source10V