K. N. Toosi University of Technology
Electric Circuits
Chapter 7. Response of First-Order
RL and RC Circuits
By:
B
y: FARHAD
FARHAD FARADJI,
FARADJI, Ph.D.
Ph.D.
Assistant
Assistant Professor,
Professor
Electrical and Computer Engineering,
K. N. Toosi University of Technology
http://wp.kntu.ac.ir/faradji/ElectricCircuits1.htm
Reference:
ELECTRIC CIRCUITS, 9th edition, 2011,
James W. Nilsson, Susan A. Riedel
1
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
2
7.0. Introduction
KNTU
‰ Inductors and capacitors are able to store energy.
‰ We now determine currents and voltages that arise when energy is either
released or acquired by an inductor or capacitor in response to an abrupt
change in a dc voltage or current source.
‰ In this chapter,
of sources,
r, we will focus on circuits that consistt only o
resistors, and eithe
either
not
inductors
orr ccapacitors.
er ((but
butt n
ot both)
both
h) in
nductors o
apacitors
‰ Such circuits are
RLL ((resistor-inductor)
RC
arre called
called R
resistor-inductor) or
or R
C ((resistor-capacitor)
resissto
circuits.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
3
7.0. Introduction
KNTU
™ Our analysis of RL and RC circuits will be divided into 3 phases.
™ In 1st phase, we consider currents and voltages that arise when stored
energy in an inductor or capacitor is suddenly released to a resistive
network.
™ This happens when inductor or capacitor is abruptlyy disco
disconnected from its
dc source.
™ We can reduce
one
off 2 eequivalent
ce ccircuit
ircuit tto
oo
ne o
quivvalent fforms
orms shown
shown in Fig. 7.1.
™ Currents and vo
voltages
oltages tthat
hat arise
arise iin
n this
this configuration
configuration
n are
are re
referred to as
natural response off circuit.
i i
™ Nature of circuit itself, not external
sources of excitation, determines
its behavior.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
4
7.0. Introduction
KNTU
¾ In 2nd phase, we consider currents and voltages that arise when energy is
being acquired by an inductor or capacitor due to sudden application of a
dc voltage or current source.
¾ This response is referred to as step response.
¾ Process for finding
same.
nding both natural and step responsess is sam
¾ In 3rd phase, wee d
develop
evelop a ggeneral
eneral method
method tthat
hat can
can be
be u
used
se to find
response of RL aand
RC
dcc voltage or
nd R
C ccircuits
ircuits to
to any
any aabrupt
brupt cchange
hange iin
nad
current source.
e..
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
5
7.0. Introduction
KNTU
9 Figure 7.2 shows 4 possibilities for general
configuration of RL and RC circuits.
9 When there are no independent sources in circuit:
– Thevenin voltage or Norton current is zero.
– Circuit reduces
duces to one of those shown in Fig. 7.1.
7.1
1.
– We have a n
natural-response
problem.
atural-rresponse p
roblem.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
6
7.0. Introduction
KNTU
‰ RL and RC circuits are also known as 1st order circuits.
‰ Their voltages and currents are described by 1st order
differential equations.
‰ No matter how complex a circuit may appear.
‰ If circuit can be reduced to a TThevenin
Norton
heveniin orr N
ortton
equivalent connected
onn
nected to
to an
an equivalent
equivalent inductor
inductor or
or
capacitor, it iss a 1sstt o
order
rder ccircuit.
ircuit.
‰ If multiple inductors
du
uctors o
orr ccapacitors
apacitors eexist
xist iin
n original
original
circuit, they must be
b iinterconnected
d so that
h they
h can
be replaced by a single equivalent element.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
7
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
8
7.1. The Natural Response of an RL Circuit
KNTU
™ We assume that independent current
source generates a dc current of Is A.
™ Switch has been in a closed position for
a long time.
™ We define phrase
rase a long time
more accurately
ely later
latter in this
this section.
sectio
on.
™ For now it means
have
ean
ns tthat
hat aallll ccurrents
urrents aand
nd vvoltages
oltages h
ave rreached
each a constant
value.
™ Only constant (d
(dc)) currents can exist
i just
j prior
i to switch's
i h' being opened.
™ Inductor appears as a short circuit (Ldi/dt = 0) prior to release of stored
energy.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
9
7.1. The Natural Response of an RL Circuit
KNTU
¾ Because L appears as a short circuit,
voltage across it is zero.
¾ There can be no current in either R0 or
R.
¾ All source current
rent Is appears in L.
¾ Finding natural
al rresponse
esponse rrequires
equires
finding voltage
ge aand
nd ccurrent
urrent aatt tterminals
erminals
of R after switch
has
opened
tcch h
as been
been o
pened ((after
after
source has been
disconnected
een d
isconnected aand
nd L
begins releasing energy).
¾ If we let t = 0 denote instant when
switch is opened, problem becomes one
of finding v(t) and i(t) for t ≥ 0.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
10
7.1. The Natural Response of an RL Circuit
KNTU
Deriving Expression for Current
9 Equation is an ordinary 1st order
differential equation
quation with constant
con
nstant
coefficients.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
11
7.1. The Natural Response of an RL Circuit
KNTU
Deriving Expression for Current
‰ An instantaneous change of current
cannot occur in L.
‰ In 1st instant after
affter switch
switch has
has been
been
opened, current
en
nt iin
n L rremains
emains
unchanged.
‰ If we use 0- to denote time just prior to
switching, and 0+ for time immediately
following switching:
‰ Initial current in L is oriented in same direction as direction of i.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
12
7.1. The Natural Response of an RL Circuit
KNTU
Deriving Expression for Current
™ i(t) starts from an initial value I0 and decreases exponentially toward 0 as
t increases.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
13
7.1. The Natural Response of an RL Circuit
KNTU
Deriving Expression for Current
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
14
7.1. The Natural Response of an RL Circuit
KNTU
Deriving Expression for Current
¾ v is defined only
nlyy for
for t > 0
0,, n
not
ot aatt t = 0
0..
¾ A step changee o
occurs
ccurs iin
n v aatt zzero.
ero.
¾ For t < 0, di/dt = 0, so v = Ldi/dt = 0.
¾ v at t = 0 is unknown.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
15
7.1. The Natural Response of an RL Circuit
KNTU
Deriving Expression for Current
¾ As t becomes infinite, energy
dissipated in R approaches initial
energy stored in L.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
16
7.1. The Natural Response of an RL Circuit
KNTU
Time Constant
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
17
7.1. The Natural Response of an RL Circuit
KNTU
Time Constant
™ Time constantt is an
n important
impo
ortantt p
parameter
arrameter ffor
or 1sstt o
order
rder ccircuits.
irc
™ Several of its ch
characteristics
haracteristics iiss worthwhile.
worthwhilee.
™ First, it is convenient
off ttime
veenient tto
o tthink
hi nk o
ime eelapsed
lapsed aafter
fter sswitching
witchi in terms of
integral multiples
l off τ.
™ One time constant after L has begun to release its stored energy to R,
current has been reduced to e-1, or approximately 0.37 of its initial value.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
18
7.1. The Natural Response of an RL Circuit
KNTU
Time Constant
‰ When elapsed time exceeds 5τ, current is less than 1% of its initial value.
‰ We sometimes say that after 5τ, currents and voltages have, for most
practical purposes, reached their final values.
‰ For single τ circuits (1st order circuits) with 1% accuracy, phrase a long
time implies that 5 or more τ have elapsed.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
19
7.1. The Natural Response of an RL Circuit
KNTU
Time Constant
™ Existence of current in RL circuit shown in Fig. 7.4 is a momentary event
and is referred to as transient response of circuit.
™ Response that exists a long time after switching has taken place is called
steady-state response.
™ Phrase a long time then also means time it takes circuit to reach its
steady-state value.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
20
7.1. The Natural Response of an RL Circuit
KNTU
Time Constant
9 Any 1st order circuit is characterized, in part, by value of its τ.
9 If we have no method for calculating τ of such a circuit (perhaps because
we don't know values of its components), we can determine its value from
a plot of circuit's
uit s natural response.
9 τ gives time required
to change at
eq
quired ffor
or i tto
o reach
reach iits
ts ffinal
inal vvalue
alu
ue iiff i ccontinues
ontin
its initial rate..
9 Assume that i continues to change at
this rate:
9 i would reach its final value of zero in
τ seconds.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
21
7.1. The Natural Response of an RL Circuit
KNTU
¾ Calculating natural response of an RL circuit is summarized as follows:
1. Find initial current, I0 , through inductor.
2. Find time constant of circuit, τ = L/R.
3. Use equation I0e-t/τ, to generate i(t) from I0 and τ.
ulattio
ons of interest
intereest follow
follow from
from kn
nowing ii(t).
(t).
¾ All other calculations
knowing
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
22
7.1. The Natural Response of an RL Circuit
9
9
9
9
9
KNTU
Switch has been
een
n closed
closed for
for a llong
ong ttime
ime p
prior
rior to
to t = 0
0..
Voltage acrosss L m
must
bee 0 aatt t = 0-.
ust b
iL(0-) = 20 A.
iL(0+) = 20 A.
We replace resistive circuit connected to terminals of L with a single
resistor of 10 Ω:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
23
7.1. The Natural Response of an RL Circuit
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
24
7.1. The Natural Response of an RL Circuit
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
25
7.1. The Natural Response of an RL Circuit
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
26
7.1. The Natural Response of an RL Circuit
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
27
7.1. The Natural Response of an RL Circuit
KNTU
9 This result is dif
difference between
initially stored energy (320 J) and
energy trapped in parallel inductors
(32 J).
9 Leq for parallel inductors (predicting
terminal behavior of parallel
combination) has an initial energy
of 288 J.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
28
7.1. The Natural Response of an RL Circuit
KNTU
9 EEnergy
nergy sstored
tored in
in Leq represents
amount of energy that will be
delivered to resistive network at
terminals of original inductors.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
29
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
30
7.2. The Natural Response of an RC Circuit
KNTU
‰ Natural response of an RC circuit is
developed from circuit shown in Fig. 7.10.
‰ Switch has been in position “a” for a long
time, allowing loop made up of dc voltage
source Vg , R1 ,and C to reach a steady-state
condition.
‰ A capacitor behaves
open
eh
haves aass aan
no
pen ccircuit
ircuit iin
n
presence of a cconstant
onsttant vvoltage.
oltage.
‰ Voltage source
ce ccannot
annot ssustain
ustain a ccurrent.
urrent.
‰ Source voltage appears across capacitor
terminals.
‰ When switch is moved from position “a” to
position “b” (at t = 0), voltage on C is Vg.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
31
7.2. The Natural Response of an RC Circuit
KNTU
‰ Because there can be no instantaneous
change in voltage at terminals C, problem
reduces to solving circuit shown in Fig.
7.11.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
32
7.2. The Natural Response of an RC Circuit
KNTU
Deriving the Expression for the Voltage
‰ Because there can be no instantaneous
change in voltage at terminals C, problem
reduces to solving circuit shown in Fig.
7.11.
‰ We can easily find voltage
volttagge v(t)
v(t)) by
by thinking
thin
nking
in terms of node
od
de voltages.
voltages.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
33
7.2. The Natural Response of an RC Circuit
KNTU
Deriving the Expression for the Voltage
™ Natural response of an RC circuit is an
exponential decay of initial voltage.
™ Time constant RC governs rate of decay.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
34
7.2. The Natural Response of an RC Circuit
KNTU
Deriving the Expression for the Voltage
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
35
7.2. The Natural Response of an RC Circuit
KNTU
¾ Calculating natural response of an RC circuit is summarized as follows:
1. Find initial voltage, V0 , across capacitor.
2. Find time constant of circuit, τ = RC.
3. Use equation V0e-t/τ, to generate v(t) from V0 and τ.
ulattio
ons of interest
intereest follow
follow from
from kn
nowing vv(t).
(t).
¾ All other calculations
knowing
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
36
7.2. The Natural Response of an RC Circuit
KNTU
9 Switch has been
n in
in position
position “x”
“x” ffor
or a long
long time.
time.
9 C will charge to
bee positive
upper
o 100
100 V aand
nd b
positive aatt u
pper terminal.
terminall.
9 We can replace resistive
i ti network
t
k connected
t d tto C att t = 0+ with
ith an equivalent
resistance of 80 kΩ.
9 τ = (0.5 X 10-6)(80 X 103) = 40 ms.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
37
7.2. The Natural Response of an RC Circuit
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
38
7.2. The Natural Response of an RC Circuit
KNTU
9 Once we know v(t), we can obtain i(t)
from Ohm's law.
9 After determining i(t), we can calculate
v1(t) and v2(t).
9 To find v(t), we replace series-connected
capacitors with an equivalent capacitor.
has
capacitance
9 IItt h
as a ca
apacitance of 4 μF and is
ccharged
harged to
to a voltage
voltag of 20 V.
Circuit
9 C
irccuit shown
sh
hown in
in Fig.
Fig 7.14 reduces to one
sshown
hown in FFig.
i g. 7
7.15.
.1
15.
9 τ = ((4)(250)
4))(250)) X 10-33 = 1 s.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
39
7.2. The Natural Response of an RC Circuit
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
40
7.2. The Natural Response of an RC Circuit
KNTU
Energy stored in C1 and C2:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
41
7.2. The Natural Response of an RC Circuit
KNTU
Energy delivered to R:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
42
7.2. The Natural Response of an RC Circuit
KNTU
9 EEnergy
nergy sstored
tored iin
n Ceq
e is 800 μJ.
Because
9 B
ecause Ceqq p
predicts
redict terminal behavior
o
off original
oriigin
nall series-connected
series-co
capacitors,
eenergy
nergy stored
d in Ceqq is energy delivered
to R.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
43
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
44
7.3. Step Response of RL and RC Circuits
KNTU
™ We now find currents and voltages generated in 1st order RL or RC circuits
when either dc voltage or current sources are suddenly applied.
™ Response of a circuit to sudden application of a constant voltage or
current source is referred to as step response of circuit.
™ We show how
stored in L or C.
w circuit responds when energy is being
bein
ng store
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
45
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
‰ Energy stored in L at time switch is
closed is given in terms of a nonzero
initial current i(0).
‰ Task is to find expressions for i(t)
and v(t) after switch
h has
has been
been
n
closed.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
46
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
¾ Algebraic sign of I0 is positive if I0 is in same direction as i.
¾ Otherwise, I0 carries a negative sign.
¾ When initial energy in L is 0, I0 = 0:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
47
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
™ 1τ after switch has been closed,
current will have
ave reached
approximatelyy 63%
63% of
of its
its final
final value:
value:
™ If current were to continue to increase at its initial rate, it would reach its
final value at t = τ:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
48
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
9 If current were
re tto
o ccontinue
ontinue tto
o
increase at itss iinitial
nitial rrate,
ate, iitt would
would
reach its finall vvalue
alue at
at t = ττ::
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
49
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
™ Voltage acrosss L iiss 0 b
before
efore sswitch
witch iiss cclosed
losed ( t < 0 ))..
™ v jumps to Vs - l0R when switch is closed and decays exponentially to 0.
™ Initial current is I0 and L prevents an instantaneous change in current.
™ Current is I0 in instant after switch has been closed.
™ Voltage drop across resistor is I0R.
™ Voltage across L is source voltage (Vs) minus voltage drop (I0R).
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
50
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
¾ When initial inductor
nd
ductor ccurrent
urrent iiss 0
0::
¾ Initial current is 0.
¾ v jumps to Vs.
¾ v approaches 0 as t increases.
¾ Current is approaching constant value of Vs/R.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
51
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
9 When I0 = 0:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
52
7.3. Step Response of RL and RC Circuits
KNTU
9 Switch has been in
n position
position a for
for a long
long time.
time.
9 L is a short circuit across 8 A current source.
9 L carries an initial current of 8 A.
9 This current is oriented opposite to reference direction for i: I0 = - 8 A.
9 When switch is in position b: final value of i will be 24/2 = 12 A.
τ = L/R = 200/2 = 100 ms.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
53
7.3. Step Response of RL and RC Circuits
KNTU
9 In instant after switch has been moved to position b, L sustains a current of 8 A
counterclockwise around newly formed closed path.
9 This current causes a 16 V drop across 2 Ω resistor.
9 This voltage drop adds to drop across source, producing a 40 V drop across L.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
54
7.3. Step Response of RL and RC Circuits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
55
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
‰ We can also describe v(t) directly, not
just in terms of i(t):
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
56
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RL Circuit
9 2 equations have
avve ssame
ame form.
form.
9 Each equates su
sum
um of
of 1sstt d
derivative
erivative of
of variable
variable and
and a constant
consta times
variable to a constant value.
9 In 1st equation, constant on right-hand side happens to be 0.
9 This equation takes on same form as natural response equations.
9 In both Equations, constant multiplying dependent variable is R/L = 1/τ.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
57
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RC Circuit
¾ For mathematical convenience, we
choose Norton equivalent of network
connected to equivalent capacitor.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
58
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RC Circuit
V0 is initial value of vC .
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
59
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RC Circuit
‰ We obtained solutions by using a
mathematical analogy to solution for
step response of RL circuit.
‰ Let's see whether
ther these solutions for
RC circuit make
ke sense
sen
nse in terms
teerm
ms of
known circuit b
behavior.
ehavior.
‰ Initial voltage across C is V0 .
‰ Final voltage across C is ISR .
‰ τ = RC.
‰ Solution for vC is valid for t > 0.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
60
7.3. Step Response of RL and RC Circuits
KNTU
The Step Response of an RC Circuit
™ Current in C at t = 0+ iss Is – V0//R.
R.
™ This prediction
n makes
makes sense.
sense.
™ Capacitor voltage
tagge ccannot
annot cchange
hange
instantaneously.
l
™ Initial current in R is V0/R.
™ Capacitor branch current changes
instantaneously from 0 at t = 0- to Is –
V0/R at t = 0+.
™ Capacitor current is 0 at t = λ.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
61
7.3. Step Response of RL and RC Circuits
KNTU
9 We find Norton equivalent
eq
quivalent with
with respect
respect tto
o te
terminals
erminals of
of C ffor
or t > 0
0::
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
62
7.3. Step Response of RL and RC Circuits
KNTU
We check consistency of solutions:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
63
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
64
7.4. Solution for Step & Natural Responses
KNTU
‰ To generalize solution of these 4 possible circuits, we
let x(t) represent unknown quantity.
‰ x(t) can have 4 possible values.
‰ x(t) can represent i(t) or v(t) of an L or C.
‰ Constant K can
n be
be 0.
0.
‰ Sources are constant
and/or
t t voltages
lt
d/ currents.
t
‰ Final value of x, (xf), will be constant.
‰ Final value must satisfy above equation.
‰ When x reaches its final value, dx/dt = 0:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
65
7.4. Solution for Step & Natural Responses
KNTU
™ To obtain a general solution, we use time t0 as lower
limit and t as upper limit.
™ Time t0 corresponds to time of switching or other
change.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
66
7.4. Solution for Step & Natural Responses
KNTU
™ Previously we assumed that t0 = 0.
™ u and v are symbols of integration.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
67
7.4. Solution for Step & Natural Responses
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
68
7.4. Solution for Step & Natural Responses
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
69
7.4. Solution for Step & Natural Responses
KNTU
Calculating natural or step response of RL or RC circuits
1. Identify variable of interest for circuit.
9 For RC circuits, it is best to choose vC.
9 For RL circuits, it is best to choose iL.
2. Determine variable initial value (its value at t0).
9 If variable
ab
ble is
is vC o
orr iL, iitt iiss n
not
ot n
necessary
ecessary to
to
distinguish
gu
uish between
between t = t0- aand
nd t = t0+. They
They
both are
arre ccontinuous
ontinuous variables.
variables.
9 If another
th
h variable
i bl iis chosen,
h
iits
t iinitial
itii l value
l is
i
defined at t = t0+.
3. Calculate variable final value (its value as t ՜ λ).
4. Calculate τ for circuit.
5. Use:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
70
7.4. Solution for Step & Natural Responses
KNTU
Calculating natural or step response of RL or RC circuits
™ You can then find equations for other circuit variables
using:
– circuit analysis techniques or
– by repeatingg preceding
p
g steps
p for other variables.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
71
7.4. Solution for Step & Natural Responses
KNTU
¾ Switch has been
n in
n position
position a for
for a long
long time.
time.
¾ C looks like an open
opeen circuit.
circuit.
¾ vC = v60Ω =
=
¾ After switch has been in position b for a long time, C will look like an open circuit in
terms of 90 V source.
¾ Final value of vC is + 90 V.
¾
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
72
7.4. Solution for Step & Natural Responses
KNTU
¾ τ doesn't changee for
for ii(t).
(t).
¾ We need to find
finall values
for
d only
l initial
i i i l and
d fi
l
f i.i
¾ When obtaining initial value, we must get i(0+).
¾ Current in capacitor can change instantaneously.
¾ This current is equal to current in resistor:
¾ Final value of i(t) = 0.
¾ Alternative solution: i(t) = CdvC(t)/dt.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
73
7.4. Solution for Step & Natural Responses
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
74
7.4. Solution for Step & Natural Responses
KNTU
9 Magnetically coupled
pled coils can be replaced
by a single inductor
or havin
having
ng an in
inductance
nducctancee of:
of:
See Problem 6.41.
9 By hypothesis initial
ti l value
l off io iis 0.
0
9 Final value of io will be 120/7.5 or 16 A.
9 τ of circuit is 1.5/7.5 or 0.2 s.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
75
7.4. Solution for Step & Natural Responses
KNTU
i2(0) = 0
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
76
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switchin
Switching
ng
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
77
7.5. Sequential Switching
KNTU
‰ Whenever switching occurs more than once in a circuit, we have
sequential switching.
‰ For example, a single 2-position switch may be switched back and forth, or
multiple switches may be opened or closed in sequence.
‰ Time reference
ce for all switchings cannot be t = 0.
‰ We determine
position
off switch
e vv(t)
(t) aand
nd ii(t)
(t) for
for a given
given p
osition o
switch or
o switches and
then use these
e ssolutions
olutions to
to determine
determine initial
initial cconditions
onditions for
fo next position
of switch or switches.
witches.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
78
7.5. Sequential Switching
KNTU
™ With sequential switching problems, obtaining initial value x(t0) is
important.
™ Anything but inductive currents and capacitive voltages can change
instantaneously at time of switching.
™ Solving first for
or inductive currents and capacitive vvoltages
oltages is even more
pertinent.
™ Drawing circuit
pertaining
helpful.
uitt p
ertaining to
to each
each time
time interval
interval is
is often
often h
e
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
79
7.5. Sequential Switching
KNTU
¾ At instant switch is moved to position b,
initial voltage on C is 0.
¾ If switch were to remain in position b, C
would eventually charge to 400 V.
¾ τ when switch is in position b is 10 ms.
¾ Switch
Switch remains in position
pos
b for only 15
ms.
¾ This expression is valid for 0 < t < 15 ms.
¾ When switch is moved to position c,
initial voltage on C is 310.75 V.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
80
7.5. Sequential Switching
KNTU
¾ With switch in position c, final value of C
voltage is 0.
¾ τ is 5 ms.
¾ t0 = 15 ms.
¾ TThis
his expression
expresssion iiss vvalid
al for t > 15 ms.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
81
7.5. Sequential Switching
KNTU
¾ C voltage will equal 200 V at 2 different
times:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
82
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
83
7.6. Unbounded Response
KNTU
‰ A circuit response may grow, rather than decay, exponentially with time.
‰ This type of response, called an unbounded response.
‰ Unbounded response is possible if circuit contains dependent sources.
‰ Thevenin equivalent resistance with respect to terminals of either an
inductor or a capacitor may bee negat
negative.
tive
e.
‰ This negative re
resistance
negative
esistance ggenerates
enerates a n
egative ττ..
‰ Resulting currents
without
ren
nts aand
nd vvoltages
oltages iincrease
ncrease w
ithout limit.
limit.
‰ In an actual circuit,
i it response eventually
t ll reaches
h a llimiting
i itii value when a
component breaks down or goes into a saturation state.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
84
7.6. Unbounded Response
KNTU
™ Here, concept of a final value is confusing.
™ Hence, rather than using step response solution, we
– derive differential equation containing negative resistance
– solve it using separation of variables technique.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
85
7.6. Unbounded Response
KNTU
9 To find Thevenin equivalent resistance
resistancce with
respect to C terminals,
nalls, w
wee use
use test
test ssource
ource
method:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
86
7.6. Unbounded Response
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
87
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
ingg A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
88
7.7. The Integrating Amplifier
KNTU
‰ Output voltage is proportional to
integral of input voltage.
‰ We assume that op amp is ideal.
‰ t0 represents instant in time when we begin integration.
‰ vo(t0) is value of output voltage at that time: vo(t0) = vCf(t0)
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
89
7.7. The Integrating Amplifier
KNTU
‰ Output voltage equals initial value
of voltage on Cf plus an inverted
(minus sign), scaled (1/RsCf) replica
of integral of inp
input
putt vvoltage.
oltagge.
‰ If no energy iss stored
stored in
i n Cf w
when
hen
integration starts:
arrts:
‰ Output voltage is proportional to integral of input voltage only if op amp
operates within its linear range (if it doesn't saturate).
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
90
7.7. The Integrating Amplifier
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
KNTU
91
7.7. The Integrating Amplifier
KNTU
vo(0.009) = -5 + 9 = 4 V
During this time interval, vo is decreasing, and op amp eventually saturates at -6 V:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
92
7.7. The Integrating Amplifier
KNTU
‰ Integrating amplifier can perform
integration function very well, but
only within specified limits that
avoid saturating op amp.
‰ Op amp saturates due to
accumulation
n of charge on
feedback capacitor.
accitor.
‰ We can prevent
entt ffrom
rom ssaturating
aturating by
by p
placing
laccing aan
n R in
np
parallel
arallel with Cf.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
93
7.7. The Integrating Amplifier
KNTU
‰ We can convert integrating amplifier to a differentiating amplifier by
interchanging Rs and Cf :
‰ We can design
differentiating-amplifier
n both
both iintegratingntegrating- aand
nd d
ifferentiating-amplif circuits by
using an L instead
C..
teead of
of a C
apacitors ffor
or iintegrated-circuit
ntegrated-ccircuit devices
devices iiss much
m
‰ Fabrication off ccapacitors
easier.
‰ Inductors are rarely used in integrating amplifiers.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
94
Chapter Contents
KNTU
7.0. Introduction
7.1. The Natural Response of an RL Circuit
7.2. The Natural Response of an RC Circuit
7.3. The Step Response of RL and RC Circuits
7.4. A General Solution for Step
p and Natural Responses
p
7.5. Sequential Switch
Switching
hing
7.6. Unbounded Re
Response
esponse
7.7. The Integrating
Amplifier
in
ng A
mplifier
7.8. More on First-order
st order Circuits
Circuiits
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
95
7.8. More on First-order Circuits
KNTU
‰ In circuit analysis, we are almost always interested in the behavior of a
particular network variable called the response (or output).
‰ A network variable is either:
– a branch voltage,
– a branch curren
current,
nt,
– a linear combination
branch
om
mbination of
of branch
branch vvoltages
olltages aand
nd b
ranch currents,
cu
– a charge on
naC(
– a flux in an L (
,
,
),
),
).
‰ Usually the responses are due:
– to either initial condition (zero-input response), or
– to the independent sources as inputs (zero-state response), or
– to both (complete response).
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
96
7.8. More on First-order Circuits
KNTU
Zero-input response:
9 The initial conditions
nditions are
are sspecified
peciified by
by V0 aand
nd I 0 .
9 The term s0 = -1/τ:
™ has a dimension of reciprocal time or frequency and
™ is measured in radians per second,
™ is called the natural frequency of the circuit.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
97
7.8. More on First-order Circuits
KNTU
Zero-input response:
™ The zero-input response of the first-order circuit is:
¾ an exponential curve,
¾ completely specified by:
9 the natural frequency -1/τ and
9 the initial condition.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
98
7.8. More on First-order Circuits
KNTU
Zero-input response:
o The initial conditions
nditions are
are aalso
lso called
caalled the
the initial
initial sstate.
tate
o For first-order linear time-invariant circuits,
¾ the zero-input response considered as a waveform in 0 ≤ t < λ is a
linear function of the initial state.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
99
7.8. More on First-order Circuits
KNTU
Zero-state response:
¾ The initial condition in the circuit is 0.
9 Voltage across the capacitor
is 0
p
before application
plicatiion off input.
in
npu
ut.
¾ A circuit is in the
th
he zzero
ero state
state iiff aallll tthe
he
initial conditions
ons in
in the
the circuit
circuit are
are 0
0..
¾ The response of a circuit, which starts from the zero state, is due
exclusively to the input.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
100
7.8. More on First-order Circuits
KNTU
Zero-state response:
™ By definition, zero-s
zero-state
state response
respo
onse
e is
is
¾ response off a circuit
time (t0)
circuit to
to an
an input
input applied
applied at
at ssome
ome aarbitrary
rbit
subject to
o ccondition
ondition tthat
hat
o circuitt be
(t0-).
be in
in zzero
ero state
state jjust
ust prior
prior to
to application
application of
of iinput
n
™ Our interest is the response for t > t0.
¾ Thus, we adopt following convention:
o input and zero-state response are taken to be 0 for t < t0.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
101
7.8. More on First-order Circuits
KNTU
Zero-state response:
¾ Consider any circuit
cirrcuit that
that contains
contains llinear
inear ((time-invariant
time-invariant o
orr time-varying)
elements.
¾ Let the circuitt be
driven
be d
riven by
by a ssingle
ingle iindependent
ndependent ssource.
ource
¾ Then zero-state response is a linear function of the input.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
102
7.8. More on First-order Circuits
KNTU
Complete response:
‰ Let vi be the zero-input response:
respon
nse:
‰ Let vo be the zero-state
ze
ero-state response:
response:
‰ We obtain by addition:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
103
7.8. More on First-order Circuits
KNTU
Complete response:
¾ If we assume that input is a constant
co
onstant current
currrent source
so
ourrce (i
(is = I) applied at t = 0,
complete response
po
onse of
of tthe
he ccircuit
ircuit is:
is:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
104
7.8. More on First-order Circuits
KNTU
Complete response:
™ Complete response is not a linear function of the input
– unless, of course, the circuit starts from the zero state.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
105
7.8. More on First-order Circuits
KNTU
Complete response:
arrtition the
the ccomplete
omplete rresponse
esponse in a different
diffferen
n way:
9 We can also p
partition
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
106
7.8. More on First-order Circuits
KNTU
Complete response:
o Transient is contributed by both zero-input response and zero-state
response.
o The steady state is contributed only by zero-state response.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
107
7.8. More on First-order Circuits
KNTU
Complete response:
‰ Physically, transient is a result of 2 causes:
– initial conditions in the circuit and
– sudden application of the input.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
108
7.8. More on First-order Circuits
KNTU
Complete response:
9 If circuit is well behaved as time goes on, transient eventually dies out.
9 Steady state is a result of only input and has a waveform closely related to
that of input.
¾ For example, if input is a constant, steady-state is also a constant.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
109
7.8. More on First-order Circuits
KNTU
Step response:
o Step response of a circuit its zero-state response to unit step input u(t).
o We denote the step response by ः(t).
ly, ः(t)
ः(tt) iiss res
sponsse at
at time
time t of
of circuit
circcuit provided
provid
de that:
o More precisely,
response
1. its input iss step
step ffunction
unction u(t),
u(t), and
and
2. circuit is in
unit
n zero
zero state
state just
just prior
prior tto
o aapplication
pplication of
of u
nit step.
st
o As mentioned before, we adopt the convention that ः(t) = 0 for t < 0.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
110
7.8. More on First-order Circuits
KNTU
Step response:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
111
7.8. More on First-order Circuits
KNTU
Time-invariance property:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
112
7.8. More on First-order Circuits
KNTU
Impulse response:
o Impulse response of a circuit is its zero-state response to unit impulse
input ࢾ(t).
o We denote the impulse response by h(t).
o More precisely,
ly,, h(t)
h(t) is
is response
response aatt ttime
ime t of
of circuit
circuitt provided
provide that:
1. its input iss unit
uniit iimpulse,
mpulse, and
and
2. circuit is in
n zero
zero state
state just
just prior
prior to
to application
application of
of impulse.
impul
o As mentioned before, we adopt the convention that h(t) = 0 for t < 0.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
113
7.8. More on First-order Circuits
KNTU
Impulse response:
First method
‰ We approximate impulse
byy p
pulse function pΔ.
p
‰ Solution will be obtained by:
¾ approximating unit impulse Ɂ by pulse function pΔ,
¾ computing the resulting solution hΔ, and then
¾ letting Δ՜0.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
114
7.8. More on First-order Circuits
KNTU
Impulse response:
First method
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
115
7.8. More on First-order Circuits
KNTU
Impulse response:
First method
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
116
7.8. More on First-order Circuits
KNTU
Impulse response:
First method
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
117
7.8. More on First-order Circuits
KNTU
Impulse response:
™ Let hΔ be zero-state response to input pΔ:
™ The operator Աȟ iiss called
called a sshift
hift o
operator.
perattor.
zeero state response,
response wee have:
havee:
™ By linearity off zero-state
™ Since circuit is linear and time-invariant, ࣴͲ operator and shift operator
commute:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
118
7.8. More on First-order Circuits
KNTU
Impulse response:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
119
7.8. More on First-order Circuits
KNTU
Impulse response:
¾ Impulse response
onse o
off a lin
linear
near time-invariant
time-invariant ccircuit
irrcuit is
is time
time derivative of its
step response:
e:
¾ For linear time-varying circuits, time derivative of step response is not
impulse response.
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
120
7.8. More on First-order Circuits
KNTU
Impulse response:
Second method
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
121
7.8. More on First-order Circuits
KNTU
Impulse response:
Third method
he solution.
9 Let us call y the
fo
or t > 0
9 Since Ɂ(t) = 0 for
0::
9 Since Ɂ(t) = 0 for t < 0 and circuit is in zero state at time 0-:
9 Thus:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
122
7.8. More on First-order Circuits
KNTU
Impulse response:
Third method
caalculate yy(0
(0+))::
¾ It remains to calculate
¾ Substituting in differential equation, we obtain:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
123
7.8. More on First-order Circuits
KNTU
Impulse response:
Third method
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
124
7.8. More on First-order Circuits
KNTU
Impulse response:
‰ We have just shown that solution of differential equation
for t > 0 is identical with solution of
‰ By integrating both sides:
‰ Since v is finite:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
125
7.8. More on First-order Circuits
KNTU
Step and Impulse Responses for Simple Linear Time-invariant Circuits:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
126
7.8. More on First-order Circuits
KNTU
Step and Impulse Responses for Simple Linear Time-invariant Circuits:
Electric Circuits
Chapter 7. Response of First-Order RL and RC Circuits
127