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Parallel Combination Rules

Module by: Don Johnson

Summary: A discussion of combination rules for circuit elements connected in series and of the voltage divider rule.

One interesting simple circuit (Figure 1) has two resistors connected side-by-side, what we will term a parallel connection, rather than in series.

Figure 1: A simple parallel circuit.
Figure 1 (circuit25.png)

Here, applying KVL reveals that all the voltages are identical: v 1 =v v 1 v and v 2 =v v 2 v . This result typifies parallel connections. To write the KCL equation, note that the top node consists of the entire upper interconnection section. The KCL equation is i in - i 1 - i 2 =0 i in i 1 i 2 0 . Using the v-i relations, we find that

i out = R 1 R 1 + R 2 i in i out R 1 R 1 R 2 i in (1)

Exercise 1

Suppose that you replaced the current source in Figure 1 by a voltage source. How would i out i out be related to the source voltage? Based on this result, what purpose does this revised circuit have?

Solution 1

Replacing the current source by a voltage source does not change the fact that the voltages are identical. Consequently, v in = R 2 i out v in R 2 i out or i out = v in R 2 i out v in R 2 . This result does not depend on the resistor R 1 R 1 , which means that we simply have a resistor ( R 2 R 2 )across a voltage source. The two-resistor circuit has no apparent use.

This circuit highlights some important properties of parallel circuits. You can easily show that the parallel combination of R 1 R 1 and R 2 R 2 has the v-i relation of a resistor having resistance 1 R 1 +1 R 2 -1= R 1 R 2 R 1 + R 2 1 R 1 1 R 2 -1 R 1 R 2 R 1 R 2 . A shorthand notation for this quantity is R 1 ∥ R 2 ∥ R 1 R 2 . As the reciprocal of resistance is conductance, we can say that for a parallel combination of resistors, the equivalent conductance is the sum of the conductances.

Figure 2
Figure 2 (parallelR.png)

Similar to voltage divider for series resistances, we have current divider for parallel resistances. The current through a resistor in parallel with another is the ratio of the conductance of the first to the sum of the conductances. Thus, for the depicted circuit, i 2 = G 2 G 1 + G 2 i i 2 G 2 G 1 G 2 i . Expressed in terms of resistances, current divider takes the form of the resistance of the other resistor divided by the sum of resistances: i 2 = R 1 R 1 + R 2 i i 2 R 1 R 1 R 2 i .

Figure 3
Figure 3 (circuit14.png)

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