Circuit Analysis

How do you solve this circuit using mesh and nodal analysis?

Mesh analysis:

analysis
Mesh Analysis

There are 4 meshes in this circuit. Each mesh namely I1, I2, I3, I4 are the currents in the corresponding mesh. The mesh equations of the system are based on KVL (Kirchoff’s Voltage Law) that voltages around the mesh sums up to 0 (zero). Organize the equation will lead to the R matrix:

R’s in I1R1+R5 -mutual R between I1 and I2-R5 -mutual R between I1 and I30 -mutual R between I1 and I40
-mutual R between I1 and I2-R5 R’s in I2R2+R5+R7 -mutual R between I2 and I3-R7 -mutual R between I2 and I4-R2
-mutual R between I1 and I30 -mutual R between I2 and I3-R7 +R’s in I3R3+R6+R7 -mutual R between I3 and I4-R3
-mutual R between I1 and I40 -mutual R between I2 and I4-R2 -mutual R between I3 and I4-R3 R’s in I4R2+R3+R4

You can visually identify the correctness of this matrix and then solve for it based on

V=RI matrices and get

I=inv(R)×V

Here is the matrix R:

Mesh Matrix
Mesh Resistance Matrix

You can do the same for nodal analysis except to convert the voltage sources to current sources using ohm’s law:

IA=V1/R1 and IC=V2/R6 where IA is current going to node A and IC is current going to node C.

Voltage sources are converted to current sources for Nodal Analysis
Voltage sources are converted to Current sources for Nodal Analysis

Now you can set up the nodal equation similar to that of the mesh equation except now for VA, VB and VC, only 3×3 matrix.

V=RI matrices becomes I=GV

and to solve for V=inv(G)xI

G’s connected to VAG1+G5+G2+G4 -mutual G between VA and VB-G2 -mutual G between VA and VC-G4
-mutual G between VA and VB-G2 G’s connected to VBG2+G3+G7 -mutual G between VB and VC-G3
-mutual G between VA and VC-G4 -mutual G between VB and VC-G3 +G’s connected to VCG4+G3+G6
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