circuit analysisthevenin
I have this problem where I need to find the Thévenin resistance. However I have a bit of a trouble of thinking how I should do it with a dependent source and that everything is unknown. How should I go about this problem?
Thanks in advance!
Edit:
I'm trying to figure out the third equation that involves V1. Could some try to give a hint on how to manage that?
Since \$i_x \$ is not associated in any way with the output port AB, \$i_x\$ is an independent variable and thus, as seen from the port AB, the CCVS is an independent source. The Thevenin equivalent is:
simulate this circuit – Schematic created using CircuitLab
It's actually quite straightforward to find the Thevenin impedance of this circuit.
The equivalent impedance looking into the port ab is defined by:
$$Z_{ab} \equiv \frac{V_{ab}}{I_{ab}} = \frac{V_{ab}}{i_{\Delta}} = Z_{th}$$
But you can write by inspection a simple KVL equation for \$V_{ab}\$ in terms of \$i_{\Delta}\$:
$$V_{ab} = i_{\Delta}(-jX_{1uF} + 10k\Omega) + 200 i_{\Delta} \cdot 100 \Omega) = i_{\Delta}(-jX_{1uF} + 30k)\Omega$$
Generally speaking, to find the Thevenin equivalent of a circuit with only dependent source(s), you must be sure to "activate" the dependent source(s) with a test source.
This is what was done above. We solved for the voltage across the port due to a test current source, \$I_{ab}\$ which, in this case, equals the controlling variable thus making this problem particularly easy to solve.
Best Answer
There are several ways to determine the Thévenin equivalent circuit of a given electrical diagram. In the present case, there is one fixed source and a voltage-controlled current source. According to a paper written by Mr. Leach, it is possible to apply the superposition theorem and alternatively turn sources on and off, even the controlled one. The trick, in this case, is to start by determining the control variable, \$V_1\$ in this exercise. Here we go:
Once the control variable is obtained, then you can apply superposition again to obtain the Thévenin voltage:
Then, to determine the Thévenin resistance, short the \$V_A\$ source and install a test generator across the output from which you want to determine \$R_{th}\$. This is exactly as what is described following the fast analytical techniques to determine time constants. Here, you can simplify the analysis because \$R_3\$ is in parallel with the intermediate result which itself is in series with \$R_2\$. The circuit to solve ends up being the simple below one:
Once this is done, a quick Mathcad sheet and a SPICE simulation tell you if this is good or not. In SPICE, a .TF (transfer function) calculates the output impedance at the considered node, directly giving the small-signal resistance \$R_{th}\$ you want:
Applying superposition to controlled sources represents a valid extension of the original theorem as this example shows.