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where M of Zm � qM �90° is positive, and
I1(Z1 � ZL1 � Z2) � I2(Z2 � Zm) � E1 Note in the above that the mutual impedance was treated as if it were
an additional inductance in series with the inductance L1 having a sign
determined by the dot convention and the voltage across which is determined
by the current in the magnetically coupled loop.
For the secondary loop,
�Z2(I2 � I1) � I2ZL2 � I1Zm � I2Z3 � 0
or I2(Z2 � ZL2 � Z3) � I1(Z 2 � Zm) � 0
For the network of Fig. 21.46, we find a mutual term between L1 and
L2 and L1 and L3, labeled M12 and M13, respectively.
For the coils with the dots (L1 and L3), since each current through the
coils leaves the dot, M13 is positive for the chosen direction of I1 and I3. However, since the current I1 leaves the dot through L1, and I2 enters the
dot through coil L2, M12 is negative. Consequently, for the input circuit,
E1 � I1Z1 � I1ZL1 � I2(�Zm12 ) � I3Zm13 � 0
or E1 � I1(Z1 � ZL1 ) � I2Zm12 � I3Zm13 � 0
For loop 2,
�I2Z 2 � I2ZL2 � I1(�Zm12 ) � 0
�I1Zm12 � I2(Z2 � ZL2 ) � 0
and for loop 3,
�I3Z3 � I3ZL3 � I1Zm13 � 0
or I1Zm13 � I3(Z3 � ZL3 ) � 0
In determinant form,
I1(Z1 � ZL1 ) � I2Zm12 � I3Zm13 � E1 �I1Zm12 � I2(Z2 � ZL12 ) � 0 � 0
I1Zm13 � 0 � I3(Z3 � Z13) � 0
21.14 APPLICATIONS
The transformer has appeared throughout the text in a number of
described applications, from the basic dc supply to the soldering gun to
the flyback transformer of a simple flash camera. Transformers were
used to increase or decrease the voltage or current level, to act as an
impedance matching device, or in some cases to play a dual role of
transformer action and reactive element. They are so common in such a
wide variety of systems that one should make an effort to be very familiar
with their general characteristics. For most applications their design
today is such that they can be considered 100% efficient. That is, the
power applied is the power delivered to the load. In general, however,
they are frequently the largest element of a design and because of the
nonlinearity of the B-H curve can cause some distortion of the transformed
waveform. By now it should be clear that they are useful only
in situations where the applied voltage is changing with time. The
application of a dc voltage to the primary will result in 0 V at the secondary,
but the application of a voltage that changes with time, no matter
what its general appearance, will result in a voltage on the sec-
+
E 1
–
Z 1
I 1
I 1
L 1
M 12
APPLICATIONS ⏐⏐⏐ 963
M 13
FIG. 21.46
Applying mesh analysis to a network with two
magnetically coupled coils.
L 2
I 2
I 3
L 3
I 2
I 3
Z 2
Z 3