Thevenin equivalent with dependent current source

[gfd43tg]

For this problem, I added in an external source, and found an expression for the external current in terms of ix. Then I divided Vex/iex to get the Rth. Then I did superposition to find the Vth. I had some troubles getting rid of the extra currents, particularly i3 and i2. I was wondering if this is right or if I am incorrect. I noticed there is a supermesh.

 

[gneill]

Just how "dependent" is the current source? Vs is a fixed source. What determines ix? Can it ever change?

 

[gfd43tg]

I don't understand your hint. It is a dependent current source. My interpretation is that for a circuit with a dependent source, one must add a external source to find the external current. Then divide by the external voltage to get the thevenin resistance

 

[gneill]

Can ix EVER vary due to any external stimulus? In what way is ix dependent on anything external to the circuit as shown?

 

[gfd43tg]

I'm still not understanding the hint. Why wouldn't ix change if I added an external source? I don't see it.

 

[gfd43tg]

Okay, a few revelations here.

First, I want to show with the second image describing the external source. The textbook says that when dependent sources are in the circuit, one must employ an external source to find the equivalent circuit. I just want to prove that I am not crazy for using an external source method.

The first image is doing what I would do for a circuit normally, without a dependent source.

Thirdly, I finally think I am understanding your hint. If I use the external source method, I will short out of where the voltage supply Vs previously existed, and ix = 0, no matter what the external source is.

part (b) asks what value I would put a load resistor at terminals (a,b) to get maximum power transfer. That is just at Rth, R1 + R3, right?

 

[gneill]

View attachment 69651

View attachment 69652

Okay, a few revelations here.

First, I want to show with the second image describing the external source. The textbook says that when dependent sources are in the circuit, one must employ an external source to find the equivalent circuit. I just want to prove that I am not crazy for using an external source method.

The external source method works anytime, there doesn't have to be a dependent source.

On the other hand, the external source method is not required if the dependent source's value is set by fixed sources inside the circuit (as is the case in this problem). A "dependent" source that cannot vary is identical to a fixed source.

The first image is doing what I would do for a circuit normally, without a dependent source.

Except that I don't see a valid calculation for Vth in the first image.

Thirdly, I finally think I am understanding your hint. If I use the external source method, I will short out of where the voltage supply Vs previously existed, and ix = 0, no matter what the external source is.

part (b) asks what value I would put a load resistor at terminals (a,b) to get maximum power transfer. That is just at Rth, R1 + R3, right?

Sure. Refer to the Maximum Power Transfer theorem.

 

[gfd43tg]

All I did for the thevenin voltage was turn off both sources and did superposition. So I did it incorrectly?

 

[gneill]

All I did for the thevenin voltage was turn off both sources and did superposition. So I did it incorrectly?

The problem occurred when you suppressed the voltage source which supplies the stimulus to the dependent current source. Effectively you suppressed both sources at the same time.

Two possible approaches come to mind. First, use superposition but replace the "3ix" for the current source with its value it normally has due to Vs driving ix. Second, leave both sources active and use nodal analysis.

 

[gfd43tg]

So I'm allowed to deactivate the voltage source (which inadvertently deactivates the current source) to find the thevenin resistance, but I am not allowed to do the same to find the thevenin voltage?

 

[gneill]

So I'm allowed to deactivate the voltage source (which inadvertently deactivates the current source) to find the thevenin resistance, but I am not allowed to do the same to find the thevenin voltage?

The correct concept is that to apply superposition you need to suppress all but one source at a time. In this circuit if you suppress Vs you also suppress the current source by proxy. That's not good. What you need to do is have the current source behave as if Vs were still there, which you can do because for this circuit Vs is fixed and causes a fixed "sampling" current through R4, thus making the current source "fixed" also. Note that this will not be the case in general! It's a quirk of this particular circuit.

To find the Thevenin resistance you want to suppress all sources at once, so it's not a problem here.

 

[gfd43tg]

Here is my next attempt. This time, I employed the short circuit technique, creating a short between terminals (a,b) in order to calculate the short circuit current. I noticed that when I shorted the terminals, I had a supermesh and I wrote my KVL equations and did some substitutions to find Vth. Hopefully this time it is right !

 

[gneill]

Your supermesh equation doesn't look right. Surely i2 is flowing in the same loop direction through R4 as it is through R3. Why would their terms have different signs?

But with a little thought you should be able to simplify your analysis greatly. You know that Vs fixes the current through R4 and thus fixes ix. You can do away with "ix" as a variable immediately by using this relationship.

You can also forget about the third loop... You know that you can switch the order of parallel components (it doesn't matter how they're depicted graphically as long as their connections remain the same). So swap the positions of Vs and R4 on your diagram. Now your "i3" becomes identical to ix and it's already solved for. Helpful hint: a voltage source that borders two loops effectively isolates them (makes the effects of their currents independent of each other). No matter what happens in either loop, the shared KVL term is fixed for all time by that voltage source.

Another hint: Find the open circuit Thevenin voltage using nodal analysis. There's only one node since the voltage source makes the junction of R3, R4, and Vs into a supernode with the bottom rail.

 

[gfd43tg]

Is this what you mean? I don't understand the term bottom rail

 

[gneill]

Is this what you mean? I don't understand the term bottom rail
View attachment 69720

Getting there. But you haven't written the node equation properly. Look here:

There are two nodes (labeled V1 and Vs in red). But Vs is already solved (it's fixed by the voltage source). So only V1 is left to solve for. Note that ix has been done away with as a variable. It's not required if it's a fixed value depending on the fixed voltage source. Now, write the node equation for node V1 and solve for V1.

 

[gfd43tg]

I am hesitant to write a nodal equation at the node with the current source. I thought you weren't allowed to do that. There is a voltage drop across the current source that cannot be calculated??

 

[gfd43tg]

This is another way I may be interpreting what you are saying

 

[gfd43tg]

Sorry I am having a brain meltdown the final exam is tomorrow. Is this what you meant?

 

[gneill]

View attachment 69728

I am hesitant to write a nodal equation at the node with the current source. I thought you weren't allowed to do that. There is a voltage drop across the current source that cannot be calculated??

No, that's incorrect. A node equation is just KCL, a sum of currents at a node. A current source makes for a trivial term in the equation! Current sources go very well indeed with nodal analysis.

Also note that R2 plays no role whatsoever in the nodal equation: the current source sets the current in the branch no matter what value R2 has. Forget R2.

 

[gneill]

Sorry I am having a brain meltdown the final exam is tomorrow. Is this what you meant?

View attachment 69730

Yes! Much better