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N A
Solution:
RB RC (20 �)(10 �) 200 � 1
� R1 ��� ���� ���33� RA � RB � RC 30 ��20 ��10 � 60
RARC (30 �)(10 �) 300 �
R2 ��� ��� ���5 �
RA � RB � RC 60 � 60
RARB (20 �)(30 �) 600 �
R3 ��� ��� ���10 �
RA � RB � RC 60 � 60
The equivalent network is shown in Fig. 8.77 (page 298).
EXAMPLE 8.28 Convert the Y of Fig. 8.78 to a D.
Solution:
R1R2 � R1R3 � R2R3 RA ����
R1
(60 �)(60 �) � (60 �)(60 �) � (60 �)(60 �)
������
60 �
3600 ��3600 ��3600 � 10,800 �
���� ���
60
60
RA � 180 �
However, the three resistors for the Y are equal, permitting the use of
Eq. (8.8) and yielding
R D � 3R Y � 3(60 �) � 180 �
and RB � RC � 180 �
The equivalent network is shown in Fig. 8.79.
EXAMPLE 8.29 Find the total resistance of the network of Fig. 8.80,
where RA � 3 �, RB � 3 �, and RC � 6 �.
Solution:
Two resistors of the D were equal;
therefore, two resistors of the Y will
be equal.
RBRC (3 �)(6 �) 18 �
R1 ��� ��� ���1.5 �←−
RA � RB � RC 3 ��3 ��6 � 12
RARC (3 �)(6 �) 18 �
R2 ��� ��� ���1.5 � ←⎯⎯⎯
RA � RB � RC 12 � 12
RARB (3 �)(3 �) 9 �
R3 ��� ��� ���0.75 �
RA � RB � RC 12 � 12
Replacing the D by the Y, as shown in Fig. 8.81, yields
(4 ��1.5 �)(2 ��1.5 �)
RT � 0.75 ������
(4 ��1.5 �) � (2 ��1.5 �)
(5.5 �)(3.5 �)
� 0.75 ����
5.5 ��3.5 �
� 0.75 ��2.139 �
R T � 2.889 �
Y-D (T-p) AND D-Y (p-T) CONVERSIONS ⏐⏐⏐ 299
b
a
c
b
a
c
a
a
R 1
60 � R 2
R 3
c
FIG. 8.78
Example 8.28.
R B
R C
180 �
60 �
180 � R A
c
60 �
b
180 �
FIG. 8.79
The D equivalent for the Y of Fig. 8.78.
R T
R T
a
a
4 � 2 �
R B
R C
3 � R 6 �
A
c
FIG. 8.80
Example 8.29.
R 1
3 �
4 � 2 �
1.5 � 1.5 �
R 3
c
R 2
0.75 �
b
b
“ ”
FIG. 8.81
Substituting the Y equivalent for the bottom D
of Fig. 8.80.
b