Is the direction of current flow through a variable resistor constant?

Question

Question: We have a closed circuit composed of multiple DC sources and resistors. We also have a variable resistor in the circuit. Can we prove that the direction of current through this resistor will be the same, irrespective of the value of the resistor?

(The above fact may not even be true. It's just that I haven't been able to find a counterexample yet)

Inspiration: A highly ideal model of P-N junctions treats P-N junctions as constant resistance resistors when forward biased, and open junctions (resistance = infinity) when reverse biased. I was just curious if in this model it was possible to create a paradoxical circuit (of DC sources, resistors and P-N junctions) - as in one that has either no or more than one mathematical solution for the currents and potentials in the circuit.

Answer 1

Can we prove that the direction of current through this resistor will be the same, irrespective of the value of the resistor?

Since a circuit with multiple DC sources and resistors is a linear circuit, we can apply Thevenin's theorem and replace all of the circuit elements connected to the rheostat (variable resistor) by a single DC source with voltage across $V_{TH}$ in series with a resistor with resistance $R_{TH}$.

It follows that the current through $I$ is given by

$$I = \frac{V_{TH}}{R_{TH} + R}$$

where $R$ is the resistance of the rheostat. If $R_{TH} \ge 0$, then the sign of the current through is independent of $R$.

However, if $R_{TH} \lt 0$, then the sign of the current through may depend on $R$.

For an example of a circuit that can have $R_{TH} \lt 0$, consider the following that uses a voltage controlled voltage source (VCVS):

You should find that

$$R_{TH} = \frac{R_1}{1 -A}$$

which is negative for $A \gt 1$. Also find that the current through is

$$I = \frac{I_1R_1}{R_1 + (1 - A)R}$$

Answer 2

About your question: As indicated in Thevenin's theorem or Norton's theorem, the network, comprises only accurate ideal model of DC sources and non-variable resistors, could get simplified into an ideal model of voltage source(or current source) with a inner resistance in series(or in parallel). After this simplification, or I should say mathematically-equivalent transformation, it's easy to find that the direction of current through your variable resistor cannot change, if only the resistance is negative.

About your inspiration: Your model for diode is one of the simplest. If you want an exact mathematical proof about existence of your "paradoxical circuit", then I have no idea actually. But it is not that important. It is because that this model, as well as many more complicated models using in analog circuit design, is only an approximation to real devices, and models must not give out paradoxes in calculation, otherwise there must exist some unconsidered effects to rule out redundant solutions. Actually, we should choose these models that not only give out only one solution, but also guarantee the solution is true and usable, as close to experiment results and your design destination as the design requirements require. The experiment results always exist and are unique. (And sometimes it just boomed directly......)

From perspective of classical electromagnetism, the distribution of electromagnetic field around a circuit always exists and is unique at a certain time.