Finding a current in a circuit with a dependent source

[TheCanadian]

Homework Statement

My problem is in the images attached. Essentially, I just want to find $I_s$.

Homework Equations

$V = IR$[/B]

The Attempt at a Solution

I already have the original solution using KCL, which tells me that $I_s = 4 A$ and that $I = 1 A$, but I was trying to do it using KVL instead. After trying (as shown in the image), I keep getting a different answer. Do you see any errors in my methods? Is there anything extra I have to do when dealing with dependent voltage sources like in this example?

 

[gneill]

Is it possible that you've conflated the current $I$ used to control the dependent source with the loop current in the third loop?

Note that the branch current through the 12 Ω resistor is made up of the two loop current that pass through it.

 

[TheCanadian]

Is it possible that you've conflated the current $I$ used to control the dependent source with the loop current in the third loop?

Note that the branch current through the 12 Ω resistor is made up of the two loop current that pass through it.

Okay. Didn't I account for the fact that there are two currents through that branch by subtracting the mutual voltages in [3]? Is the 3rd loop's current not simply I in this case?

 

[gneill]

If the third loop mesh current is $I$, then $I_s = I$, but the current in the branch (that is the current through the 12 Ohm resistor) is $I - I_2$

 

[TheCanadian]

If the third loop mesh current is $I$, then $I_s = I$, but the current in the branch (that is the current through the 12 Ohm resistor) is $I - I_2$

Ok, I'll try it again. Also, when dealing with circuits involving both current and voltage sources, either dependent or independent, how exactly can one apply mesh analysis? Since you end up having to ascribe a voltage to the current sources, but the resistance of one should be infinite.

 

[gneill]

Ok, I'll try it again. Also, when dealing with circuits involving both current and voltage sources, either dependent or independent, how exactly can one apply mesh analysis? Since you end up having to ascribe a voltage to the current sources, but the resistance of one should be infinite.

You'll learn about something called a supermesh. Essentially you draw a loop surrounding the current source, not passing through it, and add a constraint equation that links the two "merged" mesh's currents with the current source that they pass through.

 

[TheCanadian]

You'll learn about something called a supermesh. Essentially you draw a loop surrounding the current source, not passing through it, and add a constraint equation that links the two "merged" mesh's currents with the current source that they pass through.

I'll certainly look into it. Is this supposed to work for dependent voltage sources, too?

 

[gneill]

I'll certainly look into it. Is this supposed to work for dependent voltage sources, too?

The supermesh is generally invoked to deal with current sources (dependent or independent) that border loops. Voltage sources that border loops don't pose a problem since they provide voltage values for the KVL equations.

A current source that does not border loops is trivial to deal with, since it effectively "solves" the mesh current for the loop it belongs to.