boylistad

176 ⏐⏐⏐ PARALLEL CIRCUITS
R T = 16 k�
R T
R 1
FIG. 6.16
Example 6.9.
R 1 30 � R 2
30 �
FIG. 6.17
Example 6.10: two equal, parallel resistors.
R 2
R 3
Substituting values:
(9 k�)(12 k�)
R2 ���
12 k��9 k�
108 k�
���36 k�
3
EXAMPLE 6.9 Determine the values of R1, R2, and R3 in Fig. 6.16 if
R2 � 2R1 and R3 � 2R2 and the total resistance is 16 k�.
Solution:
1 1 1 1
�� ������ RT
R1 R2 R3
1 1 1 1
� ������ 16 k� R1 2R1 4R1
since R 3 � 2R 2 � 2(2R 1) � 4R 1
1 1 1 1 1 1
and � ��� � � � ���� 16 k� R1 2 R1 4�R1�
1
� 16 k�
1
� 1.75 � �R1� with R 1 � 1.75(16 k�) � 28 k�
RT R1 30 � R2 30 � R3 30 �
Recall for series circuits that the total resistance will always increase
as additional elements are added in series.
For parallel resistors, the total resistance will always decrease as
additional elements are added in parallel.
The next example demonstrates this unique characteristic of parallel
FIG. 6.18
Adding a third parallel resistor of equal value
to the network of Fig. 6.17.
resistors.
EXAMPLE 6.10
a. Determine the total resistance of the network of Fig. 6.17.
b. What is the effect on the total resistance of the network of Fig. 6.17
if an additional resistor of the same value is added, as shown in Fig.
RT R1 30 � R2 30 � R3 1 k� 6.18?
c. What is the effect on the total resistance of the network of Fig. 6.17
if a very large resistance is added in parallel, as shown in Fig. 6.19?
d. What is the effect on the total resistance of the network of Fig. 6.17
FIG. 6.19
if a very small resistance is added in parallel, as shown in Fig. 6.20?
Adding a much larger parallel resistor to the
network of Fig. 6.17.
Solutions:
30 �
a. RT � 30 � � 30 ��� �15 �
2
RT R1 30 � R2 30 � R3 0.1 �
30 �
b. RT � 30 � � 30 � � 30 ��� �10 � � 15 �
3
RT decreased
c. RT � 30 � � 30 � � 1 k� �15 � � 1 k�
FIG. 6.20
Adding a much smaller parallel resistor to the
(15 �)(1000 �)
��� � 14.778 � � 15 �
15 ��1000 �
network of Fig. 6.17.
Small decrease in RT P