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12.8 INITIAL VALUES
This section will parallel Section 10.9 (Initial Values—Capacitors) on
the effect of initial values on the transient phase. Since the current
through a coil cannot change instantaneously, the current through a
coil will begin the transient phase at the initial value established by
the network (note Fig. 12.23) before the switch was closed. It will
then pass through the transient phase until it reaches the steady-state
(or final) level after about five time constants. The steady-state level of
the inductor current can be found by simply substituting its shortcircuit
equivalent (or R l for the practical equivalent) and finding the
resulting current through the element.
Using the transient equation developed in the previous section, an
equation for the current i L can be written for the entire time interval of
Fig. 12.23; that is,
i L � I i � (I f � I i)(1 � e �t/t )
with (I f � I i) representing the total change during the transient phase.
However, by multiplying through and rearranging terms:
we find
i L � I i � I f � I f e �t/t � I i � I ie �t/t
� I f � I f e �t/t � I ie �t/t
i L � I f � (I i � I f)e �t/t
(12.12)
If you are required to draw the waveform for the current i L from initial
value to final value, start by drawing a line at the initial value and
steady-state levels, and then add the transient response (sensitive to the
time constant) between the two levels. The following example will clarify
the procedure.
EXAMPLE 12.5 The inductor of Fig. 12.24 has an initial current level
of 4 mA in the direction shown. (Specific methods to establish the initial
current will be presented in the sections and problems to follow.)
a. Find the mathematical expression for the current through the coil
once the switch is closed.
b. Find the mathematical expression for the voltage across the coil during
the same transient period.
c. Sketch the waveform for each from initial value to final value.
Solutions:
a. Substituting the short-circuit equivalent for the inductor will result in
a final or steady-state current determined by Ohm’s law:
E
16 V 16 V
If ����� ���1.78 mA
R1 � R2 2.2 k��6.8 k� 9 k�
The time constant is determined by
L 100 mH 100 mH
t ����� ���11.11 ms
RT 2.2 k��6.8 k� 9 k�
i L
INITIAL VALUES ⏐⏐⏐ 485
Ii initial
conditions
transient
response
steady-state
region
0 t
FIG. 12.23
Defining the three phases of a transient waveform.
R 1
2.2 k�
E 16 V L = 100 mH
R 2
6.8 k�
FIG. 12.24
Example 12.5.
I f
i L
4 mA
+
v L
–