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766 ⏐⏐⏐ METHODS OF ANALYSIS AND SELECTED TOPICS (ac)
E
+
–
R 1
R 3
C1 R5 FIG. 17.40
Maxwell bridge.
R 4
R 2
L 4
Apply mesh analysis to the network of Fig. 17.40. The network is
redrawn in Fig. 17.41, where
BC Z1 � � � � j �2 G 1 � B 2 G1 �2 G 1 � B C
2 1 1
� �
Y1
G1 �j BC C
Z 2 � R 2 Z 3 � R 3 Z 4 � R 4 � j X L Z 5 � R 5
+
E
–
Applying the format approach:
(Z1 � Z3)I1 � (Z1)I2 � (Z3)I3 � E
(Z1 � Z2 � Z5)I2 � (Z1)I1 � (Z5)I3 � 0
(Z3 � Z4 � Z5)I3 � (Z3)I1 � (Z5)I2 � 0
which are rewritten as
I1(Z1 � Z3) � I2Z1 � I3Z3 � E
�I1Z1 � I2(Z1 � Z2 � Z5) � I3Z5 � 0
�I1Z3 � I2Z5 � I3(Z3 � Z4 � Z5) � 0
Note the symmetry about the diagonal of the above equations. For
balance, IZ5 � 0 A, and
IZ5 � I2 � I3 � 0
From the above equations,
⎪Z1 � Z3 E �Z3 ⎪
⎪ �Z1 0 �Z5 ⎪
⎪ �Z3 0 (Z3�Z4�Z5)⎪ I2 � ––––––––––––––––––––––––––––––––––––– –
⎪Z1 � Z3 �Z1 �Z3 ⎪
⎪ �Z1 (Z1 � Z2 � Z5) �Z5 ⎪
⎪ �Z3 �Z5 (Z3 � Z4 � Z5)⎪ E(Z1Z3 � Z1Z4 � Z1Z5 � Z3Z5) � ����
�
where � signifies the determinant of the denominator (or coefficients).
Similarly,
I 3 �
E(Z 1Z 3 � Z 3Z 2 � Z 1Z 5 � Z 3Z 5)
����
�
and IZ5 � I2 � I3 � E(Z1Z4 � Z3Z2) ���
I 1
FIG. 17.41
Assigning the mesh currents and subscripted impedances for the network of
Fig. 17.40.
Z 1
Z 3
I 2
Z 5
I 3
Z 2
Z 4
–
N A