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EXAMPLE 6.4
a. Find the total resistance of the network of Fig. 6.9.
b. Calculate the total resistance for the network of Fig. 6.10.
Solutions:
a. Figure 6.9 is redrawn in Fig. 6.11:
R T
R1 12 � R2 12 � R3 12 �
FIG. 6.11
Redrawing the network of Fig. 6.9.
R
RT � �
12 �
���4 �
N
3
b. Figure 6.10 is redrawn in Fig. 6.12:
R 2 �
RT � � ���0.5 �
N 4
In the vast majority of situations, only two or three parallel resistive
elements need to be combined. With this in mind, the following equations
were developed to reduce the effects of the inverse relationship
when determining R T.
For two parallel resistors, we write
1 1 1
� ���� RT R1 R2
Multiplying the top and bottom of each term of the right side of the
equation by the other resistor will result in
1
� RT
� � � � � � � �
�
R2 �
R2
1
� R1
R 2 � R 1
��
R1R 2
R1 �
R1
R1R2
and RT � �� (6.5)
R1 �
In words,
the total resistance of two parallel resistors is the product of the two
divided by their sum.
For three parallel resistors, the equation for RT becomes
1
RT � ��
(6.6a)
1 1 1
�� � �� � �� R R R
requiring that we be careful with all the divisions into 1.
1
1
� R2
2
R 2
R 2
� R1R 2
3
R 1
� R1R 2
TOTAL CONDUCTANCE AND RESISTANCE ⏐⏐⏐ 173
R T
R T
RT R1 12 � R2 12 � R3 12 �
FIG. 6.9
Example 6.4: three parallel resistors
of equal value.
R1 2 � R2 2 � R3 2 � R4 2 �
FIG. 6.10
Example 6.4: four parallel resistors
of equal value.
R1 2 � R2 2 � R3 2 � R4 2 �
FIG. 6.12
Redrawing the network of Fig. 6.10.