Mesh Analysis/Answers

Model Answer

KVL arround abca loop:

I1 * R1 + (I1 − I3) * R2 + (I1 − I2) * R3 =  − Vs

Therefore

I1(R1 + R2 + R3) − I2(R3) − I3(R2) =  − Vs   ...............   (1)

KVL arround abda loop:

 (I2 − I1) * R3 + (I2 − I3) * R4 + I2 * R5 = Vs

Therefore

 − I1(R3) + I2(R3 + R4 + R5) − I3(R4) = Vs   ...............   (2)

KVL arround bcdb loop:

 I3 * R6 + (I3 − I2) * R4 + (I3 − I1) * R2 = 0

Therefore

 − I1(R2) − I2(R4) + I3(R2 + R4 + R6) = 0   ...............   (3)

Now we can create a matrix with the above equations as follows:

The following matrix is the above with values substituted:

→

Now that we have arranged equations 1; 2 & 3 into a matrix we need to get Determinants of the General matrix, and Determinants of alterations of the general matrix as follows:

Solving determinants of:

• Matrix A  : General matrix A from KVL equations

• Matrix A1 : Genral Matrix A with Column 1 substituted by .

• Matrix A2 : Genral Matrix A with Column 2 substituted by .

• Matrix A3 : Genral Matrix A with Column 3 substituted by .

As follows:

detA1 =  − 857700000

detA2 = 11016000

detA3 = 6300000

Now we can use the solved determinants to arrive at solutions for Mesh Currents I₁;I₂andI₃ as follows:

1.

2.

3.

Now we can solve for the current through R₃ as follows:

The negative sign means that is flowing in the direction of I₂.