# Parallel Resistors

### Parallel Resistors

In series circuits, current is the same through all components, but the voltage across the components can vary.  In contrast, parallel circuits have the same voltage across all components, but the current through them can vary.  Two or more components are in parallel if they are connected at the same two nodes.

Recalling the definition of a node (see the KCL and KVL article), if multiple components are connected at the same two nodes, they must have the same voltage across them.  Only wires separate the three voltages in Figure 7.  An ideal wire has zero resistance.  Plugging zero into Ohm’s law results in no voltage drop between parallel components, hence the voltage must be the same.

Figure 8 shows a completely different configuration in which none of the resistors are in parallel.  None of the resistors are connected at the same two nodes, meaning that voltage can vary from one resistor to the next.  (Figures 5 and 8 are neither in series nor parallel.  You will learn how to work with them in Circuit Theory I.)

### Parallel vs. Series configurations

In one respect, a parallel configuration is similar to a series configuration in that the current going into the beginning node is equal to the current leaving the ending node.  Figure 9 contrasts the current flow in the circuits from Figure 7 and Figure 8, respectively.

Denoting parallel components in circuit theory is similar to denoting parallel lines in geometry.  Simply write R1||R2||R3 to represent the parallel configuration in Figure 7.  Formulas for combining parallel resistors are given below (where N is the number of parallel resistors).  Notice that for N > 2, you must invert everything inside the parentheses.  Don’t forget this inversion! If your answer is smaller than 1 Ω, you’ve probably forgotten to perform this operation.

### Parallel Resistance Formulas

Two Resistors (N = 2)

$R_{T} = R_{1} || R_{2} = (\frac{1}{R_{1}} + \frac{1}{R_{2}})^-1 = \frac{R_{1}R_{2}}{R_{1} + R_{2}}$

N > 2

$R_{T} = R_{1} || R_{2} ||$$|| R_{N} = (\frac{1}{R_{1}} + \frac{1}{R_{N}} +$$+ \frac{1}{R_{N}})^-1$

You’ve seen that adding a resistor in series increases the overall resistance of the circuit.  This increase in resistance makes it more difficult for current to flow, resulting in a decrease in current through that circuit.  Parallel resistors are the opposite.  Adding a resistor in parallel opens up a new path for current to flow through, allowing more current to flow.  In contrast to series resistors, placing a resistor in parallel with another resistor always decreases the overall resistance of the circuit.  When using the parallel resistance formulas, RT should always be smaller than the smallest parallel resistor. Written by Ryan Eatinger (reatinge@ksu.edu).  Thank you!