Electronic – Mesh Analysis on CCT with Voltage and Current Source

current measurementkirchhoffs-lawsvoltage

I'm not sure where to apply the supermesh on this problem, would it be over the top mesh and the one containing the current source? I have only tried doing the problem without a supermesh but am not able to get an answer. Without using a supermesh I used KVL in each mesh and got an answer, however this is clearly wrong since I did not take the 6mA current source into account. The trouble I am having is identifying whether or not I need to just set the mesh current in the bottom left mesh to 6mA or to create a supermesh.

Assuming that the mesh with the current source has a mesh current of 6mA resulted in an answer of 3V on for V0

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Best Answer

Here's the schematic, redrawn slightly to my own taste:

schematic

simulate this circuit – Schematic created using CircuitLab

Starting at the lower right-hand corner of each loop:

$$\begin{align*} 0\:\text{V}+12\:\text{V}-R_3\left(I_1-I_2\right)-V_{I_1}-R_4\left(I_1-I_3\right) &= 0\:\text{V}\tag{$I_1$}\\\\ 0\:\text{V}+V_{I_1}-R_3\left(I_2-I_1\right)-R_1\: I_2 -R_5\left(I_2-I_3\right) &= 0\:\text{V}\tag{$I_2$}\\\\ 0\:\text{V}-R_4\left(I_3-I_1\right)-R_5\left(I_3-I_2\right)-R_2\: I_3 &= 0\:\text{V}\tag{$I_3$}\\\\ I_0=I_1-I_2&=6\:\text{mA}\tag{Known} \end{align*}$$

If you look closely, the variables you need to solve for are \$I_1\$, \$I_2\$, \$I_3\$, and \$V_{I_1}\$. You have four equations and four unknowns.

No supermesh stuff, either.

Once you solve for those values, just work out the current in \$R_5\$. From that, you can figure out the voltage drop magnitude and also the polarity (from the current direction.)

Using Sympy/Sage, it's just:

 i1, i2, i3, vi0 = S('i1 i2 i3 vi0'.split())
 e1 = Eq( 0 + 12 - 4E3*(i1-i2) - vi0 - 2E3*(i1-i3), 0 )
 e2 = Eq( 0 + vi0 - 4E3*(i2-i1) - 8E3*i2 - 6E3*(i2-i3), 0 )
 e3 = Eq( 0 - 2E3*(i3-i1) - 6E3*(i3-i2) - 8E3*i3, 0 )
 e4 = Eq( i1 - i2, 6E-3 )
 r = solve( [e1, e2, e3, e4], [i1, i2, i3, vi0] )
 r
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