I think Thevenin (or Norton) equivalent circuits do not consider variable sources. The same refers to non-linear resistors (and other elements in AC scope). But I understand what you mean: you would like to have something like these.
In your case you should first select all the elements that are not dependent on other and do not alter other elements, and simplify them. The next step is to find all independent voltage/current sources.
Now combine non-linear static elements, like resistors. The combination of a linear object and a non-linear object is also non-linear object (but there is a theoretical possibility that two non-linear functions make a linear one).
At this moment you get: combined resistances that are (generally speaking) non-linear and do not alter anything and independent and dependent sources, and the elements that alter sources. If possible, combine independent sources.
That's the hardest task now: to combine independent sources with dependent. The Kirchoff's laws might be necessary here.
According to your circuit, this is not that difficult as it seems on the first sight. Please forgive me there are no exact calculations as I did them last time almost 20 years ago...
First of all, take a look at the non-ideal current source
I1. Because it has
R1 in parallel you can convert it to a non-ideal voltage source, which has resistance in series. This voltage source would have internal resistance 1 Ohm too and voltage
R1 * 4Ix that is
4*Ix volts as
R1 = 1 Ohm. I will name this new source as
At the moment on the left side of the circuit you have non-ideal voltage source
V2 (equivalent to
I1 current source), its internal resistance (equivalent to
R1), than voltage source
R1 resistance is gone as it became internal load of voltage source. More reading about source transformation.
Because in the same branch there are two voltage sources you can combine them. So it is
E = V1 + V2 which leads to
(4 Ix - 10) V (
V1 is in opposition to
Now we have the first part of our task, the source. Now we're going to find equivalent resistance, and, moreover, we need to drive out
Ix from source equation, because after combining resistances to one there will be no
As we know from Mr. Kirchoff, the load current (the one in
I, divides in two:
IL flows through
U2 / R2 and
U2 / (R3 + RL). You can write down proper equations yourself :).
Now you can find relation between
IL (you need
IL in equation of voltage source) and make
E function of
IL. If this source is no more function of
Ix, you can combine other resistances to one equivalent. Do not forget source
E internal resistance (the one driven from
Please note that this method will lead you to have voltage source that is a function of load current (so in fact load resistance
RL). This is normal as
U2 depends on this load (that's why I've written at the very beginning it is not true Thevenin method).