ECE 201 Circuit Theory I 2
The Natural Response of a Circuit
• The currents and voltages that arise when energy stored in an inductor or capacitor is suddenly released into a resistive circuit.
• These “signals” are determined by the circuit itself, not by external sources!
ECE 201 Circuit Theory I 3
Step Response
• The sudden application of a DC voltage or current source is referred to as a “step”.
• The step response consists of the voltages and currents that arise when energy is being absorbed by an inductor or capacitor.
ECE 201 Circuit Theory I 4
Circuits for Natural Response
• Energy is “stored” in an inductor (a) as an initial current.
• Energy is “stored” in a capacitor (b) as an initial voltage.
ECE 201 Circuit Theory I 5
General Configurations for RL
• If the independent sources are equal to zero, the circuits simplify to
ECE 201 Circuit Theory I 6
Natural Response of an RL Circuit
• Consider the circuit shown.
• Assume that the switch has been closed “for a long time”, and is “opened” at t=0.
ECE 201 Circuit Theory I 7
What does “for a long time” Mean?
• All of the currents and voltages have reached a constant (dc) value.
• What is the voltage across the inductor just before the switch is opened?
ECE 201 Circuit Theory I 8
Just before t = 0
• The voltage across the inductor is equal to zero.
• There is no current in either resistor.
• The current in the inductor is equal to IS.
ECE 201 Circuit Theory I 9
Just after t = 0• The current source and its parallel resistor
R0 are disconnected from the rest of the circuit, and the inductor begins to release energy.
ECE 201 Circuit Theory I 11
0di
L Ridt
A first-order ordinary differential equation with constant coefficients.
How do we solve it?
di Rdt idt
dt L
ECE 201 Circuit Theory I 12
0 0
( )
( )
0
0
( )
( )ln ( )
( )
( )ln
(0)
( ) (0)
i t t
i t t
Rt
L
di Rdt idt
dt Ldi R
dti Ldx R
dyx Li t R
t ti t L
i t Rt
i L
i t i e
ECE 201 Circuit Theory I 13
The current in an inductor cannot change instantaneously
• Let the time just before switching be called t(0-).
• The time just after switching will be called t(0+).
• For the inductor,
0(0 ) (0 )i i I
ECE 201 Circuit Theory I 15
The voltage drop across the resistor
0
0
, 0 .
(0 ) 0
(0 )
Rt
L
v iR
v I Re t
v
v I R
ECE 201 Circuit Theory I 16
The Power Dissipated in the Resistor
2
2
22
0, 0
Rt
L
vp vi i R
R
p I Re t
ECE 201 Circuit Theory I 17
The Energy Delivered to the Resistor
22
00 0
22
0
2
0
1(1 ), 0.
2
1,
2
Rt t xL
Rt
L
w pdx I Re dx
w I R e tRL
t w LI
ECE 201 Circuit Theory I 18
Time Constant
• The rate at which the current or voltage approaches zero.
LR
ECE 201 Circuit Theory I 19
Rewriting in terms of Time Constant
0
0
22
0
22
0
( )
( )
1(1 )
2
t
t
t
t
i t I e
v t I Re
p I Re
w LI e
ECE 201 Circuit Theory I 21
Graphical Interpretation of Time Constant
• Determine the time constant from the plot of the circuit’s natural response.
0
0
0
0
0
( )
1
(0 )
( )
t
t
i t I e
diI e
dtdi Idt
Ii t I t
Straight Line Approximation
ECE 201 Circuit Theory I 22
Graphical Interpretation
Tangent at t = 0 intersects the time axis at the time constant