# Mesh Method with current source

I found an exercise with the circuit below and for which I'm asked to use Mesh Method for every essencial mesh (explicitly the loops I1, I2, I3) in order to find the equations that allow me to solve it.

I know how to apply the Mesh Analysis method, but when it comes to current sources within the loops I really don't have any idea how to proceed and write the equations. What I normally do in Mesh with current sources is to "supermesh" and try not to write de current directly, but now the problem specifically asks for the mesh equations for every one of the three essential loops.

How can I obtain the equations for the loops I2 and I3 in this case? Since IA is a current I cannot add it to the Mesh equation. And because it doesn't even have Resistor I cannot use something like (I3-I2)*R.

• Assign a voltage across IA. This is an unknown voltage, by the way. However, you do know that IA = I2-I3. So that's a new equation you can use.
– jonk
Aug 23, 2022 at 22:49

One way to redraw this is the following (I picked a ground reference):

simulate this circuit – Schematic created using CircuitLab

From the above it is clear that $$\I_{_\text{B}}=-I_2\$$ and that $$\I_{_\text{A}}=I_2-I_3\$$ (put another way, $$\I_3=I_2-I_{_\text{A}}\$$.)

Keeping that in mind we can write:

\begin{align*} 0\:\text{V} +\left(-I_2\right)\cdot a - \left(I_1-I_2\right)\cdot R_{_\text{C}} &= 0\:\text{V} \\\\ 0\:\text{V} - \left(I_2-I_1\right)\cdot R_{_\text{C}} - I_2\cdot R_{_\text{B}} - V_{_\text{B}} &= 0\:\text{V} \\\\ 0\:\text{V} + V_{_\text{B}} - \left(I_2-I_{_\text{A}}\right)\cdot R_{_\text{D}} &= -V_{_\text{E}} \end{align*}

Three mesh (KVL) equations with just three unknowns, $$\I_1\$$, $$\I_2\$$, and $$\V_{_\text{B}}\$$ (which is the voltage across the current source called $$\I_{_\text{A}}\$$), assuming you can provide $$\a\$$, $$\R_{_\text{B}}\$$, $$\R_{_\text{C}}\$$, $$\R_{_\text{D}}\$$, $$\I_{_\text{A}}\$$, and $$\V_{_\text{E}}\$$.