What if we connect two different voltage sources in parallel in ideal condition?

From the question here that is assuming internal resistors, Different voltage sources in parallel, I thought I can analyze ideal circuit if internal resistors all goes zero ohm.

$${V_{out}=V_1\cdot{R_{i1}\over R_{i1}+R_{i2}}+V_2\cdot {R_{i2}\over R_{i1}+R_{i2}}}$$

but wolframalpha says limit does not exist

Is there any way to analyze this circuit? And if we can't, how do we know what would happen if we connect two ideal different voltage sources?

enter image description here

  • 2
    \$\begingroup\$ A circuit consisting of two ideal voltage sources in parallel has two problems: 1) there are infinite valid solutions for the current, since there’s no equation that relates it to the circuit elements; 2) violates KVL if the sources have different voltage. Thus we say such connection is invalid in circuit theory. \$\endgroup\$
    – alejnavab
    Sep 23, 2021 at 5:49

2 Answers 2


You can't.

An (ideal) voltage source defines the voltage between its terminals. With 2 in parallel (in fact any amount in a loop), there is a conflict between each one trying to force a different voltage.

SPICE and other simulators won't allow this because the matrix that it needs to solve becomes indeterminate. It's essentially the same reason as your limit above doesn't exist.

Two identical voltage sources in parallel doesn't create a conflict in determing the voltage between the two nodes, but the current that flows (out from one voltage source and into the other) is indeterminate (could be any value from -infinity to +infinity).

In practice, voltage sources (e.g. batteries or power supplies) have some limit of current that could flow (in or out), or a definite output impedance so that you don't get two 'pure' voltage sources in parallel; If they don't get damaged, the voltage will be something between the two values, depending on the relative strengths of each.


Two voltage sources in parallel, like a 12V automobile battery in parallel with a wire, is a short circuit. Blowing a fuse is the nicest way the situation can be resolved, but boiling acid could also erupt from the battery.

Ideal voltage sources simply do not exist in the real world; close to zero impedance is achievable, exactly zero is not. So, there will never be two ideal voltage sources connected in this fashion, nor will an irresistible force ever be applied to an immovable object.

  • \$\begingroup\$ Exactly zero is possible (we believe) for superconductors, giving you an indeterminate current in a loop of the stuff, but it's only up to a finite current. \$\endgroup\$
    – Neil_UK
    Sep 23, 2021 at 8:14
  • \$\begingroup\$ @Neil_UK True, a block of superconductor is intrinsically a left-side zero impedance source in parallel with a right-side zero impedance source; zero voltage for each. Amusingly, if left side is a different superconductor than the right side, and near edge is different temperature than far edge, the Peltier cooling/heating effect should be enormous! \$\endgroup\$
    – Whit3rd
    Sep 23, 2021 at 19:17

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