# Circuit without Thevenin equivalent?

(I edited heavily my original post because it was based on my serious misunderstanding of a textbook exercise. However the main question remains valid as explained below.)

It appears to me (and also pointed out in comments below) that the following circuit does not have a Thevenin equivalent.

simulate this circuit – Schematic created using CircuitLab

I take branch directions as follows: $$\V_1\$$ from 0 to 1, $$\R_2\$$ from 2 to 1, $$\R_3\$$ from 0 to 2, $$\I_4\$$ from 2 to 3, $$\R_5\$$ from 3 to 4. For clarity, the CCCS is $$\I_4=8i_{R_2}\$$.

To find the Thevenin equivalent I connect as load a voltage source $$\V_L\$$ from 4 to 0. To solve the circuit I take loops $$\V_1R_2R_3\$$, $$\R_3I_4R_5V_L\$$; call the loop currents $$\p_1,p_2\$$. In $$\V_1R_2R_3\$$ we have: $$V_1+R_2p_1+R_3(p_1-p_2)=0\Rightarrow 5+10p_1+10(p_1-8i_2)=0\Rightarrow 5+10p_1+10(p_1+8p_1)=0\Rightarrow p_1=-0.05.$$ Then $$i_L=i_5=i_4=8i_2=-8p_1=0.40.$$ In other words, the current through the load $$\V_L\$$ is $$\i_L=0.40\$$, independent of $$\V_L\$$. This cannot be attained by any Thevenin equivalent, because a Thevenin equivalent is a practical voltage source and must have characteristic $$\i_{Th}=\frac{V_{Th}}{R_{Th}}\$$.

Question 1: So does it follow that the circuit has no Thevenin equivalent?

Question 2: More generally, what conditions must a circuit satisfy to have a Thevenin equivalent?

PS2: The analysis and conclusion remain valid if we use a resistor load $$\R_L\$$ instead of $$\V_L\$$. We cannot use a current source load $$\I_L\$$, because then we would get $$\I_L=i_4=0.4\$$ (by the same analysis as above). But $$\i_L\$$ can be arbitrary; it is not obliged to be $$\I_L=0.4\$$. In other words, the circuit with a current source load has no solution (or an infinity of solutions).

PS2: The circuit does have a Norton equivalent with $$\I_{Nr}=0.4\$$, $$\R_{Nr}=\infty\$$. In other words, it seems to me that the original can be replaced by an ideal current source.

• @All - Although well-intentioned, the long comment chain has exceeded what is reasonable for comments. Therefore comments so far have been moved to chat & should be continued there (see link in the comment below). || As this bulk moving of comments to chat can only be done once per question, any further comments posted here may be deleted without notice. Keep it in chat now, please! || Any improvement / clarification to the question decided during the chat, should be made via an edit, not as a comment. When someone has enough information to post an answer, please answer it as usual. TY Commented Apr 3 at 9:33
• Comments have been moved to this chatroom and discussion should be continued there. Please do not continue the discussion in comments here. Commented Apr 3 at 9:34