# Voltage across a resistor?

simulate this circuit – Schematic created using CircuitLab

What would the voltage across the resistor be? What would its polarity be? I don't see anyway for KVL to not be violated.

• You are right, KVL cannot be satisfied. Two unequal ideal voltage sources in parallel with each other is a self-contradictory model. – The Photon Apr 10 '14 at 23:23
• That configuration would result in a BIG short-circuit! :D – Ricardo Apr 11 '14 at 0:15
• @ThePhoton But then if certain combinations of voltage sources and current sources create contradictory models, when given a complicated circuit how do you check if the model is contradictory or not? My textbook immediately starts trying to solve the circuit instead of checking for possible contradictions. – dfg Apr 11 '14 at 1:17
• @dfg, One very general way is to write down the equations for mesh or nodal analysis, and if you get an inconsistent set of equations, then there's a contradiction in your model. – The Photon Apr 11 '14 at 16:54

An ideal circuit diagram has an associated set of equations. In this particular case, the resistor is irrelevant to the KVL equation which is $$1V = -1V$$

which is nonsense.

Just as one can write inconsistent mathematical equations, one can draw inconsistent ideal circuit diagrams (which are simply a different representation of a system of equations).

Here's another example which is, in fact, the dual of the given circuit:

simulate this circuit – Schematic created using CircuitLab

The KCL equation for this circuit is

$$1A = -1A$$

which is, again, nonsense.

So, there are rules for ideal circuit diagrams including but not limited to

• don't place ideal voltage sources in parallel (or short-circuit a voltage source)

• don't place ideal current sources in series (or open-circuit a
current source)

• Thanks Alfred. One last question: given a complex random circuit, is there a way to check if the circuit is nonsense/contains contradictions? I know you listed 2 "rule" for checking for contradictions, but how do we know that these are the only ways a contradiction can exist? – dfg Apr 11 '14 at 1:51
• @dfg, for the KCL equation to be consistent, there must a dependent current variable to solve for. For a KVL equation to be consistent, there must be a dependent voltage variable to solve for. So, for example, if three independent current sources (and only those sources) connect to a node, there is no dependent variable to solve for and, in general, there is no solution to the KCL equation. – Alfred Centauri Apr 11 '14 at 2:28

If you assumed a resistor was in series with each voltage source the resultant voltage across R1 is zero. The series resistor could be nano ohms or tera ohms and the voltage across R1 would still be zero.

• But if the voltage across R1 is zero, KVL is not satisfied. – dfg Apr 11 '14 at 1:16

This configuration is incorrect. You can not put two different voltages sources in parallel. It violates the Kirchhoff Voltage Law.