Fundamental Engineering Concepts - Mathematics and Statistics for the Environment - page 282

■ EXAMPLE 11.42
Problem: What is the equivalent resistance of a 20-ohm and a 30-ohm resistor connected in parallel?

Solution:
Given:
R 1 = 20 ohms
R 2 = 30 ohms

RR
RR

×
+

20 ×

+

30

1 2

R

T =

= = 12 ohms

20 30

1 2

11.7.7.8 Power in Parallel Circuits
As in the series circuit, the total power consumed in a parallel circuit is equal to the sum of the
power consumed in the individual resistors.

Note: Because power dissipation in resistors consists of a heat loss, power dissipations are additive
regardless of how the resistors are connected in the circuit.

P T = P 1 + P 2 + P 3 + … + P n (11.39)

where P T is the total power, and P 1 , P 2 , P 3 , … P n are the branch powers.
Total power can also be calculated by the following equation:

P T = E × I T (11.40)

where P T is the total power, E is the voltage source across all parallel branches, and I T is the total
current. The power dissipated in each branch is equal to E × I and equal to V 2 / R .

Note: In both parallel and series arrangements, the sum of the individual values of power dissi-
pated in the circuit equals the total power generated by the source. The circuit arrangements
cannot change the fact that all power in the circuit comes from the source.

11.7.8 s eries -p arallel C irCuits
So far, we have discussed series and parallel DC circuits; however, operators will seldom encounter
a circuit that consists solely of either type of circuit. Most circuits consist of both series and parallel
elements. A circuit of this type is referred to as a series-parallel circuit or as a combination circuit .
Analyzing a series-parallel (combination) circuit is simply a matter of applying the laws and rules
discussed up to this point.

11.7.8.1 Solving a Series-Parallel Circuit
At least three resistors are required to form a series-parallel circuit: two parallel resistors connected
in series with at least one other resistor. In a circuit of this type, the current ( I T ) divides after it flows
through R 1 ; part of it flows through R 2 , and part flows through R 3 . Then, the current joins at the junc-
tion of the two resistors and flows back to the positive terminal of the voltage source ( E ) and through
the voltage source to the positive terminal.
When solving for values in a series-parallel circuit (current, voltage, and resistance), follow the
rules that apply to a series circuit for the series part of the circuit, and follow the rules that apply to
a parallel circuit for the parallel part of the circuit. Solving series-parallel circuits is simplified if
all parallel and series groups are first reduced to single equivalent resistances and the circuits are
redrawn in simplified form. Recall that the redrawn circuit is called an equivalent circuit .