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Resistors in Parallel

Resistors can be combined in parallel and in series to create new resistances. Two resistors are connected in parallel in this LTspice figure. In this combination these two resistors act like a single resistor with a resistance that depends on the two individual resistances:

where R₁ is the value of R1 and R₂ is the value of R2. This is useful when you don’t have the value resistor that you need: you may be able to construct from resistors that you do have. For example, two 1K resistors in parallel provide the resistance

and a 1K and a 3K in parallel provide

Kirchoff’s Current Law

The formula for the resistance of parallel resistors is implied by another basic electronic relationship called Kirchoff’s Current Law, or KCL for short.

Kirchoff’s Current Law: The sum of currents flowing towards a point in an electrical circuit equals to the sum of currents flowing away from that point.

In our circuit with parallel resistors, KCL says that the currents flowing through the two resistors sum to the current flowing from the power supply:

where I_S is the power supply current, I₁ is the current through R1, and I₂ is the current through R2.

As the schematic above shows, the potential across the two parallel resistors equals the voltage of the power supply. So we can also write

where V_S is the power supply voltage, V₁ is the voltage across R1, and V₂ is the voltage across R2.

If we combine these relationships with Ohm’s law, then we can figure out the currents through each resistor and the effective resistance of the resistors in parallel from V_S and the resistors’ values. Ohm’s law applies to each resistor:

where R₁ is the resistance of resistor R1 and R₂ is the resistance of resistor R2. Substituting V_S for the voltage values, these equations imply that

and substituting these values into the current equation gives

This last equation can be reinterpreted as telling us the combined resistance of the parallel resistors. If a single resistor were in the place of the parallel resistors and we measured a voltage supply equal to V_S and a current equal to I_S then we could compute the value of that resistor using Ohm’s law as

This is the resistance of two parallel resistors predicted by Ohm’s law and KCL.

This part of a circuit is sometimes called a current divider. The parallel resistors effectively divide the current of the circuit I_S into V_S/R₁ and V_S/R₂.