Behavior of two hypothetical generators in parallel powering the same circuit?

Question

We can't mathematically describe what happens when two generators(simple batteries ) in parallel power the same component with two different voltages, as anomalies arise in the study of this circuit.

In reality, one or both generators will stop functioning, and this circuit is forbidden in electronics because it can destroy the generators.

But imagine that we replace the exact voltages of the two generators with two isolated moving magnets.

What would happen to the magnets?

Can we study the real behavior of a circuit powered by two different voltages in parallel from two moving magnets in this case, without risk?

Here is an explanatory diagram. The orange outline represents the area where the two magnets move; we can even add a resistor if a short circuit is observed. The rectangle in the middle is the light bulb. We assume that the two magnets are very far apart and apply different voltages at each end.

Answer 1

Two ideal voltages sources in parallel will certainly break a simulator because there's no element in the system to relate voltage to current, and current can't be resolved. However, doing this in real life will work just fine, because there's always some resistance present in series with each source:

^{simulate this circuit – Schematic created using CircuitLab}

As long as a source's internal resistance, and the resistance of wiring to the outside world is small compared to load impedance, and as long as the sources have the same voltage, then things will be close to "ideal", and current will be shared equally between the two sources, as shown on the ammeters above.

You are referring to generators, though, which produce AC, but the same principle applies. As long as the generators produce equal voltages, and are completely in phase, load current will be shared equally between them, and the voltage across the load will be the same as generator voltage. I've changed R3 (V2's internal resistance) slightly to make both source current curves visible on the same plot, and to show how current symmetry is highly dependent upon symmetry of the resistances present:

^{simulate this circuit}

Any asymmetry in the system will result in huge current's flowing, and not through the load. If I make the sources slightly out of phase with respect to each other, by only 5°, load current (in R1) will be mostly unchanged, and you might think that everything's OK:

^{simulate this circuit}

However, take a look at current through the generators V1 and V2:

The generators pass huge currents, ±400A here. Maybe in reality they can't produce that much current for any number of reasons, but this is likely to burn them out. As far as the magnets are concerned, they would experience huge alternating forces on them, likely to vibrate the structure to pieces, if the windings didn't melt first.

Answer 2

Ideal voltage sources in parallel are mathematically "impossible" to analyze, as you say. What I think you are proposing is to replace them with models of real-world generators, which is absolutely doable. You could start by using an ideal voltage source (representing the no-load voltage of a real-world generator) in series with small resistance (representing the generator's internal resistance). Of course in the real world things are more complicated and generators are usually controlled with an external circuit to provide a predictable voltage droop as their loading increases.

Answer 3

Consider a situation in which two different moving magnets induce two different open circuit voltages in two different wires, and then these wires are tied together. One may naively assume that we are connecting two voltage sources together, and are facing a mathematical paradox. In reality, no mathematical paradox is posed, even if we assume that the wires are ideal, (i.e. without resistance).

The reason is that not only do the magnets create magnetic flux around the wires, but current also creates magnetic flux around the wires. Thus, when connected, and current is allowed to flow, the two wires no longer form ideal voltage sources.

The flow of current is associated with a proportional magnetic flux. This magnetic flux must be considered along with the flux of the magnets in order to determine the voltages induced in the wires. It will turn out that the voltage induced in both wires, when they are connected to form a closed circuit, is the same.

In the following model, I have replaced the moving magnets with coupled inductors fed by ideal current sources. These current sources produce (or appear with) a certain amount of flux in the coupled inductors. But current flowing through the wires between the left and right hand side of the circuit also produce (or appear with) a certain amount of flux in the coupled inductors. The two flux fields superpose, just as the flux from permanent magnets and from current superpose.

I have chosen current sources with different frequencies and different amplitudes, to model the magnets being of different strengths and not moving in unison.

^{simulate this circuit – Schematic created using CircuitLab}

Here are the currents through the "primaries" of the coupled inductors:

Here is the current that goes between the two "secondaries" of the coupled inductors.

And here is the voltages across all of the inductors. Notice that the voltages are all the same, even though the primary currents, are different. (Remember that the primary currents represent the magnetic flux induced by the moving magnets, but not the total flux, because some of the flux is induced by the current flowing between the pairs of coupled inductors)

Can we study the real behavior of a circuit powered by two different voltages in parallel from two moving magnets in this case, without risk?

We can model the behavior, (as demonstrated above). We can also study the real behavior provided the current between the generators is not too high. The risk involved is that the current circulating between the generators will be excessive and cause them to burn out. However, if the difference in instantaneous open circuit voltages is small (which requires that they have the same frequency and nearly the same phase), then the current circulating between the generators will be correspondingly small.