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17.7 �-Y, Y-� CONVERSIONS
The �-Y, Y-� (or p-T, T-p as defined in Section 8.12) conversions for
ac circuits will not be derived here since the development corresponds
exactly with that for dc circuits. Taking the �-Y configuration shown
in Fig. 17.46, we find the general equations for the impedances of the Y
in terms of those for the �:
Z 1
Z 3
Z 2
ZBZC
Z1 � �� ZA � ZB
� ZC ZAZC
Z2 � �� ZA � ZB
� ZC ZAZB
Z3 � �� ZA � ZB
� ZC Z B
FIG. 17.47
The T and p configurations.
Z C
(17.13)
(17.14)
(17.15)
Forthe impedances of the � in terms of those for the Y, the equations are
Z1Z2 � Z1Z3 � Z2Z3 ZB � ���
(17.16)
Z1Z2 � Z1Z3 � Z2Z3 ZA � ���
(17.17)
Z1Z2 � Z1Z3 � Z2Z3 ZC � ���
(17.18)
Note that each impedance of the Y is equal to the product of the
impedances in the two closest branches of the �, divided by the sum
of the impedances in the �.
Further, the value of each impedance of the � is equal to the sum of the
possible product combinations of the impedances of theY, divided by the
impedances of theY farthest from the impedance to be determined.
Drawn in different forms (Fig. 17.47), they are also referred to as the T
and p configurations.
Z2
Z1
Z3
Z A
�-Y, Y-� CONVERSIONS ⏐⏐⏐ 771
a b
ZC Z 1
Z B
Z 3
c
FIG. 17.46
�-Y configuration.
Z 2
Z A