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Natural and step response
Ahsan Khawaja
Cont. inductor
• Where V= voltage between inductor in (volt)L= inductor in (henery) = rate of change of current flow in amper = rate of change of time in second
Vt
Ldi
d
dtdi
Cont. capacitor
• Mathematical relation
• Wherei= current that in capacitor in (ampere)C= capacitance in (farad) = rated of change of voltage. = rate of change of time in second
it
Cdv
d
.
dv
dvdt
First-Order Circuits• A circuit that contains only sources, resistors and an
inductor is called an RL circuit.• A circuit that contains only sources, resistors and a
capacitor is called an RC circuit.• RL and RC circuits are called first-order circuits
because their voltages and currents are described by first-order differential equations.
–+
vs L
R
–+
vs C
R
i i
6 Different First-Order CircuitsThere are six different STC circuits. These
are listed below.• An inductor and a resistance (called RL
Natural Response).• A capacitor and a resistance (called RC
Natural Response).• An inductor and a Thévenin equivalent
(called RL Step Response). • An inductor and a Norton equivalent
(also called RL Step Response).• A capacitor and a Thévenin equivalent
(called RC Step Response). • A capacitor and a Norton equivalent
(also called RC Step Response).
These are the simple, first-order cases. Many circuits can be reduced to one of these six cases. They all have solutions which are in similar forms.
LX RX RXCX
+
-
vS
RXLX
+
-
vS
RXCX
LXRXiS
RXiS CX
6 Different First-Order CircuitsThere are six different STC circuits. These
are listed below.• An inductor and a resistance (called RL
Natural Response).• A capacitor and a resistance (called RC
Natural Response).• An inductor and a Thévenin equivalent
(called RL Step Response). • An inductor and a Norton equivalent
(also called RL Step Response).• A capacitor and a Thévenin equivalent
(called RC Step Response). • A capacitor and a Norton equivalent
(also called RC Step Response). These are the simple, first-order cases. They all have solutions which are in similar forms.
LX RX RXCX
+
-
vS
RXLX
+
-
vS
RXCX
LXRXiS
RXiS CX
These are the simplest cases, so we handle them first.
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!
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.
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.
General Configurations for RL
• If the independent sources are equal to zero, the circuits simplify to
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.
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?
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.
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.
The Source-Free RL Circuit
• A first-order RL circuit consists of a inductor L (or its equivalent) and a resistor (or its equivalent)
0 RL vvBy KVL
0 iRdt
diL
Inductors law Ohms law
dtL
R
i
di LtReIti /
0 )(
The Source-FreeRC Circuit
• A first-order circuit is characterized by a first-order differential equation.
• Apply Kirchhoff’s laws to purely resistive circuit results in algebraic equations. • Apply the laws to RC and RL circuits produces differential equations.
Ohms law Capacitor law
0 dt
dvC
R
v0 CR iiBy KCL
Natural Response of an RL Circuit• Consider the following circuit, for which the switch is closed
for t < 0, and then opened at t = 0:
Notation:0– is used to denote the time just prior to switching0+ is used to denote the time immediately after switching
• The current flowing in the inductor at t = 0– is Io
LRo RIo
t = 0 i +
v
–
Solving for the Current (t 0)• For t > 0, the circuit reduces to
• Applying KVL to the LR circuit:
• Solution:
LRo RIo
i +
v
–
tLReiti )/()0()( = I0e-(R/L)t
Solving for the Voltage (t > 0)
• Note that the voltage changes abruptly:
tLRoeIti )/()(
LRo RIo
+
v
–
I0Rv
ReIiRtvt
vtLR
o
=Þ
==>
=
+
-
-
)0(
)( 0,for
0)0()/(
Time Constant t
• In the example, we found that
• Define the time constant
– At t = t, the current has reduced to 1/e (~0.37) of its initial value.
– At t = 5t, the current has reduced to less than 1% of its initial value.
R
L
tLRo
tLRo ReItveIti )/()/( )( )( and
(sec)
The Source-Free RL Circuit
/0)( teIti
R
L
A RL source-free circuit
where /0)( teVtv RC
A RC source-free circuit
where
Comparison between a RL and RC circuit
The Complete Solution
0( ) , 0
Rt
Li t I e t
The voltage drop across the resistor
0
0
, 0 .
(0 ) 0
(0 )
Rt
L
v iR
v I Re t
v
v I R
The Power Dissipated in the Resistor
2
2
22
0, 0
Rt
L
vp vi i R
R
p I Re t
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
The Source-Free RL Circuit
• The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.
• i(t) decays faster for small t and slower for large t.• The general form is very similar to a RC source-free circuit.
/0)( teIti
R
L
A general form representing a RL
where
The Source-Free RC Circuit
• The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
• The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.
• v decays faster for small t and slower for large t.
CRTime constantDecays more slowly
Decays faster
Natural Response SummaryRL Circuit
• Inductor current cannot change instantaneously
• time constant
RC Circuit
• Capacitor voltage cannot change instantaneously
• time constantR
L
/)0()(
)0()0(teiti
ii
R
i
L
+
v
–
RC
/)0()(
)0()0(tevtv
vv
RC
General Solution for Natural and Step Responses of RL and RC Circuits
0( )
0( ) [ ( ) ]t t
f fx t x x t x e
Final Value
Initial Value
Time Constant
Determine the initial and final values of the variable of interest and the time constant of the circuit.
Substitute into the given expression.
Example
V190 V
R1
400kOhm
C0.5uF
R260 Ohm
R3
20 Ohm
V240 V
J1
Key = Space
b a
• What is the initial value of vC?
• What is the final value of vC?• What is the time constant when the switch is in
position b?• What is the expression for vC(t) when t>=0?
+
vC(t)
-
Initial Value of vC
V190 V
R1
400kOhm
C0.5uF
R260 Ohm
R3
20 Ohm
V240 V
J1
Key = Space
b a
• The capacitor looks like an open circuit, so the voltage @ C is the same as the voltage @ 60Ω.
60(0) 40 30
20 60Cv V V
+
vC(0)
-
+
V60
-
Final Value of vC
• After the switch is in position b for a long time, the capacitor will look like an open circuit again, and the voltage @ C is +90 Volts.
V190 V
R1
400kOhm
C0.5uF
R260 Ohm
R3
20 Ohm
V240 V
J1
Key = Space
b a
+
vC(∞)
-
The time constant of the circuit when the switch is in position b
• The time constant τ = RC = (400kΩ)(0.5μF)• τ = 0.2 s
V190 V
R1
400kOhm
C0.5uF
R260 Ohm
R3
20 Ohm
V240 V
J1
Key = Space
b a
The expression for vC(t) for t>=0
0.2
5
( ) ( ) [ (0) ( )]
( ) 90 [ 30 90]
( ) 90 120
t
C C C C
t
C
tC
v t v v v e
v t e
v t e V
The expression for i(t) for t>=0
• Initial value of i is (90 - - 30)V/400kΩ = 300μA• Final value of i is 0 – the capacitor charges to +90 V
and acts as an open circuit• The time constant is still τ = 0.2 s
V190 V
R1
400kOhm
C0.5uF
R260 Ohm
R3
20 Ohm
V240 V
J1
Key = Space
b a
i(t)-
30V
+
The expression for i(t) (continued)
6 0.2
5
( ) ( ) [ (0 ) ( )]
( ) 0 [300 10 0]
( ) 300
t
t
t
i t i i i e
i t e
i t e A
How long after the switch is in position b does the capacitor voltage equal 0?
5
5
5
( ) 90 120 0
120 90
90
12090
5 ln 0.28768120
0.05754 57.54
tC
t
t
v t e
e
e
t
t s ms
Plot vC(t)
Plot i(t)