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296 ⏐⏐⏐ METHODS OF ANALYSIS AND SELECTED TOPICS (dc)
Let us first assume that we want to convert the D (RA,RB,RC)totheY (R1, R2, R3). This requires that we have a relationship for R1, R2, and R3 in terms of RA,RB, and RC.Ifthe resistance is to be the same between terminals
a-c for both the D and the Y, the following must be true:
Ra-c (Y) � Ra-c (D)
RB(RA
� RC) so that Ra-c � R1 � R3 ��� (8.5a)
RB � ( RA � RC)
Using the same approach for a-b and b-c, we obtain the following relationships:
RC(RA � RB) Ra-b � R1 � R2 ��� RC � ( RA � RB)
(8.5b)
RA(RB � RC) and Rb-c � R2 � R3 ��� (8.5c)
RA � ( RB � RC)
Subtracting Eq. (8.5a) from Eq. (8.5b), we have
RC RB � RC RA RB RA � RB RC �� ��
RA � RB � RC RA � RB � RC (R 1 � R 2) � (R 1 � R 3) � � � � � �
RA
RC
� RB
RA
so that R2 � R3 ��� (8.5d)
R � R � R
Subtracting Eq. (8.5d) from Eq. (8.5c) yields
R A R B � R A R C
(R2 � R3) � (R2 � R3) � �� � ���� � �
RA � R B � R C
2RBRA
so that 2R3 ��� RA � RB
� RC resulting in the following expression for R3 in terms of RA, RB, and RC: RA
R3 ��� RA � R � RC Following the same procedure for R 1 and R 2, we have
R C
R1 ��� RA � RB
� RC A
(8.6a)
(8.6b)
R C
and R2 ��� (8.6c)
RA � RB
� RC Note that each resistor of the Y is equal to the product of the resistors
in the two closest branches of the D divided by the sum of the resistors
in the D.
R B
B
B R
A R
B
C
R A R C � R B R A
RA � R B � R C
N A