Inductor transient response
Inductors have the exact opposite
characteristics of capacitors. Whereas capacitors store
energy in an electric field (produced by the voltage
between two plates), inductors store energy in a magnetic
field (produced by the current through wire). Thus, while
the stored energy in a capacitor tries to maintain a
constant voltage across its terminals, the stored energy in
an inductor tries to maintain a constant current through its
windings. Because of this, inductors oppose changes in
current, and act precisely the opposite of capacitors, which
oppose changes in voltage. A fully discharged inductor (no
magnetic field), having zero current through it, will
initially act as an open-circuit when attached to a source
of voltage (as it tries to maintain zero current), dropping
maximum voltage across its leads. Over time, the inductor's
current rises to the maximum value allowed by the circuit,
and the terminal voltage decreases correspondingly. Once the
inductor's terminal voltage has decreased to a minimum (zero
for a "perfect" inductor), the current will stay at a
maximum level, and it will behave essentially as a
short-circuit.
When the switch is first closed, the voltage
across the inductor will immediately jump to battery voltage
(acting as though it were an open-circuit) and decay down to
zero over time (eventually acting as though it were a
short-circuit). Voltage across the inductor is determined by
calculating how much voltage is being dropped across R,
given the current through the inductor, and subtracting that
voltage value from the battery to see what's left. When the
switch is first closed, the current is zero, then it
increases over time until it is equal to the battery voltage
divided by the series resistance of 1 Ω. This behavior is
precisely opposite that of the series resistor-capacitor
circuit, where current started at a maximum and capacitor
voltage at zero. Let's see how this works using real values:
---------------------------------------------
| Time | Battery | Inductor | Current |
|(seconds) | voltage | voltage | |
|-------------------------------------------|
| 0 | 15 V | 15 V | 0 |
|-------------------------------------------|
| 0.5 | 15 V | 9.098 V | 5.902 A |
|-------------------------------------------|
| 1 | 15 V | 5.518 V | 9.482 A |
|-------------------------------------------|
| 2 | 15 V | 2.030 V | 12.97 A |
|-------------------------------------------|
| 3 | 15 V | 0.747 V | 14.25 A |
|-------------------------------------------|
| 4 | 15 V | 0.275 V | 14.73 A |
|-------------------------------------------|
| 5 | 15 V | 0.101 V | 14.90 A |
|-------------------------------------------|
| 6 | 15 V | 37.181 mV | 14.96 A |
|-------------------------------------------|
| 10 | 15 V | 0.681 mV | 14.99 A |
---------------------------------------------
Just as with the RC circuit, the inductor
voltage's approach to 0 volts and the current's approach to
15 amps over time is asymptotic. For all practical
purposes, though, we can say that the inductor voltage will
eventually reach 0 volts and that the current will
eventually equal the maximum of 15 amps.
Again, we can use the SPICE circuit analysis
program to chart this asymptotic decay of inductor voltage
and buildup of inductor current in a more graphical form
(inductor current is plotted in terms of voltage drop across
the resistor, using the resistor as a shunt to measure
current):
inductor charging
v1 1 0 dc 15
r1 1 2 1
l1 2 0 1 ic=0
.tran .5 10 uic
.plot tran v(2,0) v(1,2)
.end
legend:
*: v(2) Inductor voltage
+: v(1,2) Inductor current
time v(2)
(*+)------------ 0.000E+00 5.000E+00 1.000E+01 1.500E+01
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 1.500E+01 + . . *
5.000E-01 9.119E+00 . . + * . .
1.000E+00 5.526E+00 . .* +. .
1.500E+00 3.343E+00 . * . . + .
2.000E+00 2.026E+00 . * . . + .
2.500E+00 1.226E+00 . * . . + .
3.000E+00 7.429E-01 . * . . + .
3.500E+00 4.495E-01 .* . . +.
4.000E+00 2.724E-01 .* . . +.
4.500E+00 1.648E-01 * . . +
5.000E+00 9.987E-02 * . . +
5.500E+00 6.042E-02 * . . +
6.000E+00 3.662E-02 * . . +
6.500E+00 2.215E-02 * . . +
7.000E+00 1.343E-02 * . . +
7.500E+00 8.123E-03 * . . +
8.000E+00 4.922E-03 * . . +
8.500E+00 2.978E-03 * . . +
9.000E+00 1.805E-03 * . . +
9.500E+00 1.092E-03 * . . +
1.000E+01 6.591E-04 * . . +
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Notice how the voltage decreases (to the
left of the plot) very quickly at first, then tapering off
as time goes on. Current also changes very quickly at first
then levels off as time goes on, but it is approaching
maximum (right of scale) while voltage approaches minimum.
-
REVIEW:
-
A fully "discharged" inductor (no current
through it) initially acts as an open circuit (voltage
drop with no current) when faced with the sudden
application of voltage. After "charging" fully to the
final level of current, it acts as a short circuit
(current with no voltage drop).
-
In a resistor-inductor "charging" circuit,
inductor current goes from nothing to full value while
voltage goes from maximum to zero, both variables changing
most rapidly at first, approaching their final values
slower and slower as time goes on.
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