Download Inductor Transient Analysis

Lesson 12
Inductors Transient Analysis
Learning Objectives

Calculate inductor voltage and current as a function of
time.

Explain inductor DC characteristics.
Inductor Charging



Inductor: Oppose Changes (Choking effects)
Inductor is initially fully discharged
 acts like a open circuit
When switch is closed, the changing current across the inductor
immediately induces a voltage that opposes that change, which
keeps the current near zero:
vL  E

Inductor charging
iL (t )  I 0 (1  e  tR1 / L )
As current iL builds up, the voltage
across the R1 resistor increases.
E
I0 
R1
Inductor Charging Equations

Voltages and currents in a charging circuit change
exponentially over time
Steady State Conditions

Circuit is at steady state: voltage and current reach their final
values and stop changing

There is no change in current in the circuit, so the inductor has
zero voltage induced across it. Inductor current will be steady:
E
100V

 100mA
R1 1000
Inductor then looks like a short circuit
iL 

The Time Constant

Rate at which an inductor charges depends on R and L,
which is called the TIME CONSTANT:
 ch arg ing

L

R1
Transients can be considered
to last for five time constants
Example Problem 1
iL (t )  I 0 (1  e  tR1 / L )
E
R1
In the circuit below, the switch is initially open and conditions are
at steady-state.
I0 
 ch arg ing 
L
R1
After the switch is shut, determine:
a.how long it will take for the inductor to reach a steady-state condition
(>99% of final current).
b. Write the equation for the VL(t) & iL(t). Sketch the transient.
c. Find the Energy stored in the Inductor.
Interrupting Current in an Inductive
Circuit

When switch opens in an RL circuit
 Energy
is released in a short time
 This may create a large voltage
 Induced voltage is called an inductive kick

Opening of inductive circuit may cause
voltage sparks of thousands of volts
Interrupting a Circuit

Switch flashovers are generally undesirable
 They can be controlled with proper engineering design

These large voltages can be useful
 Such as in automotive ignition systems
Inductor Discharging


Inductor is initially fully charged with constant 100 ma
current through it. It acts like a short circuit
When switch is opened, the inductor will immediately
induce a voltage to keep the 100 mA current constant.


KVL can be used to calculate this induced voltage
Notice the polarity of the induced voltage!
vR1  I 0 R1  100mA 1000   100V
vR 2  I 0 R2  100mA 10000   1000V
vL  vR1  vR 2  1100V
Inductor Discharging i (t )  I et R  R / L
L
0
1

As stored energy is released, the
induced voltage across the inductor drops.

This makes the voltage drop across the
resistor drop, so current in the circuit
drops
2
Inductor Discharging Equations

Voltages and currents in a discharging circuit also change
exponentially over time
The Time Constant

Rate at which an inductor discharges depends on R and L,
which is called the TIME CONSTANT:
 disch arg ing 

L
 R1  R2 
Transients can be considered
to last for five time constants
Example Problem 2

The circuit shown below has been in operation with the switch shut
for a long time. The switch opens at time t = 0, determine:
how long it will take for the inductor to discharge.
b. Write the equation for the VL, iL,. Sketch the transient.
a.
 disch arg ing 
L
 R1  R2 