Module
3
DC Transient
Version 2 EE IIT, Kharagpur
Lesson
11
Study of DC transients
in R-L-C Circuits
Version 2 EE IIT, Kharagpur
Objectives
•
•
•
•
Be able to write differential equation for a dc circuits containing two storage
elements in presence of a resistance.
To develop a thorough understanding how to find the complete solution of second
order differential equation that arises from a simple R − L − C circuit.
To understand the meaning of the terms (i) overdamped (ii) criticallydamped, and
(iii) underdamped in context with a second order dynamic system.
Be able to understand some terminologies that are highly linked with the
performance of a second order system.
L.11.1 Introduction
In the preceding lesson, our discussion focused extensively on dc circuits having
resistances with either inductor ( L ) or capacitor ( C ) (i.e., single storage element) but not
both. Dynamic response of such first order system has been studied and discussed in
detail. The presence of resistance, inductance, and capacitance in the dc circuit introduces
at least a second order differential equation or by two simultaneous coupled linear first
order differential equations. We shall see in next section that the complexity of analysis
of second order circuits increases significantly when compared with that encountered
with first order circuits. Initial conditions for the circuit variables and their derivatives
play an important role and this is very crucial to analyze a second order dynamic system.
L.11.2 Response of a series R-L-C circuit due to a dc
voltage source
Consider a series R − L − C circuit as shown in fig.11.1, and it is excited with a dc
voltage source Vs . Applying KVL around the closed path for t > 0 ,
di (t )
(11.1)
L
+R i (t ) + vc (t ) = Vs
dt
The current through the capacitor can be written as
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dvc (t )
dt
Substituting the current ‘ i (t ) ’expression in eq.(11.1) and rearranging the terms,
i (t ) = C
LC
d 2 vc (t )
dv (t )
+R C c + vc (t ) = Vs
2
dt
dt
(11.2)
The above equation is a 2nd-order linear differential equation and the parameters
associated with the differential equation are constant with time. The complete solution of
the above differential equation has two components; the transient response vc n (t ) and the
steady state response vc f (t ) . Mathematically, one can write the complete solution as
(
)
vc (t ) = vc n (t ) + vc f (t ) = A1 eα1 t + A2 eα 2 t + A
(11.3)
Since the system is linear, the nature of steady state response is same as that of forcing
function (input voltage) and it is given by a constant value A . Now, the first part vc n (t ) of
the total response is completely dies out with time while R > 0 and it is defined as a
transient or natural response of the system. The natural or transient response (see
Appendix in Lesson-10) of second order differential equation can be obtained from the
homogeneous equation (i.e., from force free system) that is expressed by
d 2 vc (t )
dvc (t )
d 2 vc (t ) R dvc (t ) 1
=
+
R
C
+
v
(
t
)
0
⇒
+
+
vc (t ) = 0
c
dt 2
dt
dt 2
L dt
LC
R
1
d 2 vc (t )
dv (t )
)
a
+ b c + c vc (t ) = 0 (where a = 1, b = and c =
2
L
LC
dt
dt
LC
(11.4)
The characteristic equation of the above homogeneous differential equation (using the
d
d2
2
operator α =
, α = 2 and vc (t ) ≠ 0 ) is given by
dt
dt
R
1
R
1
α2 + α +
= 0 ⇒ a α 2 + b α + c = 0 (where a = 1, b = and c =
)
(11.5)
L
LC
L
LC
and solving the roots of this equation (11.5) one can find the constants α1 and α 2 of the
exponential terms that associated with transient part of the complete solution (eq.11.3)
and they are given below.
2
2
⎛ R
⎞
1 ⎞⎟ ⎛⎜ b 1 ⎛ b ⎞
⎛ R⎞
+ ⎜ ⎟ −
= −
+ ⎜ ⎟ − ac ⎟ ;
α1 = ⎜ −
⎜ 2 L ⎝ 2L ⎠ LC ⎟ ⎜ 2a a ⎝ 2 ⎠
⎟
⎝
⎠ ⎝
⎠
(11.6)
2
2
⎛ R
⎞
⎛
⎞
1 ⎟ ⎜ b 1 ⎛b⎞
⎛ R⎞
− ⎜ ⎟ −
= −
− ⎜ ⎟ − ac ⎟
α2 = ⎜ −
⎜ 2 L ⎝ 2L ⎠ LC ⎟ ⎜ 2a a ⎝ 2 ⎠
⎟
⎝
⎠ ⎝
⎠
R
1
where, b =
.
and c =
L
LC
The roots of the characteristic equation (11.5) are classified in three groups depending
upon the values of the parameters R , L , and C of the circuit.
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2
1
⎛ R ⎞
> 0 , this implies that the roots are
⎜
⎟ −
⎝ 2 L ⎠ LC
distinct with negative real parts. Under this situation, the natural or transient part of the
complete solution is written as
vc n (t ) = A1 eα1 t + A2 eα 2 t
(11.7)
Case-A (overdamped response): When
and each term of the above expression decays exponentially and ultimately reduces to
zero as t →∞ and it is termed as overdamped response of input free system. A system
that is overdamped responds slowly to any change in excitation. It may be noted that the
α
α
exponential term A1 e 1 t takes longer time to decay its value to zero than the term A1 e 2 t .
One can introduce a factor ξ that provides an information about the speed of system
response and it is defined by damping ratio
(ξ ) =
R
Actual damping
b
L >1
=
=
2
critical damping 2 ac
LC
(11.8)
2
1
⎛ R ⎞
Case-B ( critically damped response): When ⎜
= 0 , this implies that the roots
⎟ −
⎝ 2 L ⎠ LC
of eq.(11.5) are same with negative real parts. Under this situation, the form of the
natural or transient part of the complete solution is written as
R
(where α = −
)
(11.9)
vc n (t ) = ( A1 t + A2 ) e α t
2L
where the natural or transient response is a sum of two terms: a negative exponential and
a negative exponential multiplied by a linear term. The expression (11.9) that arises from
the natural solution of second order differential equation having the roots of characteristic
equation are same value can be verified following the procedure given below.
The roots of this characteristic equation (11.5) are same α = α1 = α 2 =
2
R
when
2L
2
1
1
⎛ R ⎞
⎛ R ⎞
and the corresponding homogeneous equation (11.4)
=0 ⇒ ⎜
⎜
⎟ −
⎟ =
LC
⎝ 2 L ⎠ LC
⎝ 2L ⎠
can be rewritten as
d 2 vc (t )
R dvc (t ) 1
+2
+
vc (t ) = 0
2
dt
2 L dt
LC
d 2 vc (t )
dv (t )
+ 2 α c + α 2 vc (t ) = 0
or
2
dt
dt
d ⎛ dv (t )
⎞
⎛ dv (t )
⎞
or ⎜ c
+ α vc (t ) ⎟ + α ⎜ c
+ α vc (t ) ⎟ = 0
dt ⎝ dt
⎠
⎝ dt
⎠
dvc (t )
df
or
+ α f = 0 where f =
+ α vc (t )
dt
dt
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The solution of the above first order differential equation is well known and it is given
by
f = A1 eα t
dv (t )
Using the value of f in the expression f = c
+ α vc (t ) we can get,
dt
dvc (t )
dv (t )
d αt
e vc (t ) = A1
+ α vc (t ) = A1 e−α t ⇒ e α t c
+ eα t α vc (t ) = A1 ⇒
dt
dt
dt
Integrating the above equation in both sides yields,
vc n (t ) = ( A1 t + A2 ) e α t
(
)
R
) decays exponentially with the time and tends to
2L
R
zero as t →∞ . On the other hand, the value of the term A1 t e α t (with α = −
) in
2L
2L −1
e at a time
equation (11.9) first increases from its zero value to a maximum value A1
R
1
⎛ 2L ⎞ 2L
t =− = −⎜−
and then decays with time, finally reaches to zero. One can
⎟=
α
⎝ R ⎠ R
easily verify above statements by adopting the concept of maximization problem of a
single valued function. The second order system results the speediest response possible
without any overshoot while the roots of characteristic equation (11.5) of system having
the same negative real parts. The response of such a second order system is defined as a
critically
damped
system’s
response.
In
this
case
damping
ratio
R
Actual damping
b
L =1
(11.10)
(ξ ) =
=
=
2
critical damping 2 ac
LC
In fact, the term A2 eα t (with α = −
2
1
⎛ R ⎞
Case-C (underdamped response): When ⎜
< 0 , this implies that the roots of
⎟ −
⎝ 2 L ⎠ LC
eq.(11.5)
are
complex
conjugates
and
they
are
expressed
as
2
2
⎛ R
⎛ R
1 ⎛ R ⎞ ⎞⎟
1 ⎛ R ⎞ ⎞⎟
⎜
α1 = ⎜ −
β
j
γ
α
j
;
+j
−⎜
=
+
=
−
−
−⎜
2
⎟
⎟ = β − j γ . The
⎜ 2L
⎜ 2L
LC ⎝ 2 L ⎠ ⎟
LC ⎝ 2 L ⎠ ⎟
⎝
⎠
⎝
⎠
form of the natural or transient part of the complete solution is written as
β + jγ )
β − jγ )
vc n (t ) = A1 eα1 t + A2 eα 2 t = A1 e(
+ A2 e(
(
)
(
)
= e β t ⎡⎣ A1 + A2 cos ( γ t ) + j A1 − A2 sin ( γ t ) ⎤⎦
= e β t ⎡⎣ B1 cos ( γ t ) + B2 sin ( γ t ) ⎤⎦ where B1 = A1 + A2 ; B2 = j ( A1 − A2 )
(11.11)
For real system, the response vcn (t ) must also be real. This is possible only if A1 and A2
conjugates. The equation (11.11) further can be simplified in the following form:
e β t K sin ( γ t + θ )
(11.12)
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β = real
where
part
of
the
root
,
γ = complex
part
of
the
root,
⎛B ⎞
and θ = tan −1 ⎜ 1 ⎟ . Truly speaking the value of K and θ can be
⎝ B2 ⎠
calculated using the initial conditions of the circuit. The system response exhibits
oscillation around the steady state value when the roots of characteristic equation are
complex and results an under-damped system’s response. This oscillation will die down
with time if the roots are with negative real parts. In this case the damping ratio
2
K = B1 + B2
2
(ξ ) =
R
Actual damping
b
L <1
=
=
2
critical damping 2 ac
LC
(11.13)
Finally, the response of a second order system when excited with a dc voltage source is
presented in fig.L.11.2 for different cases, i.e., (i) under-damped (ii) over-damped (iii)
critically damped system response.
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Example: L.11.1 The switch S1 was closed for a long time as shown in fig.11.3.
Simultaneously at t = 0 , the switch S1 is opened and S 2 is closed
Find
(a) iL (0+ ); (b) vc (0+ );
(c) iR (0+ ); ( d ) vL (0 + ); (e) ic ( 0+ ) ; ( f )
dvc (0+ )
.
dt
Solution: When the switch S1 is kept in position ‘ 1’ for a sufficiently long time, the
circuit reaches to its steady state condition. At time t = 0− , the capacitor is completely
charged and it acts as a open circuit. On other hand,
the inductor acts as a short circuit under steady state condition, the current in inductor can
be found as
50
iL (0− ) =
×6 = 2 A
100 + 50
Using the KCL, one can find the current through the resistor iR (0− ) = 6 − 2 = 4 A and
subsequently the voltage across the capacitor vc (0− ) = 4 × 50 = 200 volt.
Note at t = 0+ not only the current source is removed, but 100 Ω resistor is shorted or
removed as well. The continuity properties of inductor and capacitor do not permit the
current through an inductor or the voltage across the capacitor to change instantaneously.
Therefore, at t = 0+ the current in inductor, voltage across the capacitor, and the values of
other variables at t = 0+ can be computed as
iL (0+ ) = iL (0− ) = 2 A ; vc (0+ ) = vc (0− ) = 200 volt.
Since the voltage across the capacitor at t = 0+ is 200 volt , the same voltage will appear
across the inductor and the 50 Ω resistor. That is, vL (0+ ) = vR (0+ ) = 200 volt. and hence,
200
the current ( iR (0+ ) ) in 50 Ω resistor =
= 4 A . Applying KCL at the bottom terminal
50
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of
the
capacitor
+
we
obtain
ic (0+ ) = − (4 + 2) = − 6 A
and
subsequently,
+
dvc (0 ) ic (0 ) − 6
=
=
= − 600 volt./ sec.
dt
C
0.01
Example: L.11.2 The switch ‘ S ’ is closed sufficiently long time and then it is opened at
time
‘t =0’
as
shown
in
fig.11.4.
Determine
dv (t )
dv (t )
di (t )
(i) v0 (0+ ) (ii) c
(iii) iL (0+ ), and (iv) L
(v ) 0
when
dt t =0+
dt t =0+
dt
+
t =0
R1 = R2 = 3 Ω .
Solution: At t = 0− (just before opening the switch), the capacitor is fully charged and
current flowing through it totally blocked i.e., capacitor acts as an open circuit). The
voltage across the capacitor is vc (0− ) = 6V = vc (0+ ) = vbd (0+ ) and terminal ‘ b ’ is higher
potential than terminal ‘ d ’. On the other branch, the inductor acts as a short circuit (i.e.,
voltage across the inductor is zero) and the source voltage 6V will appear across the
6
resistance R2 . Therefore, the current through inductor iL (0− ) = = 2 A = iL (0+ ) . Note at
3
+
+
t = 0 , vad (0 ) = 0 (since the voltage drop across the resistance R1 = 3 Ω = vab = − 6V )
and vcd (0+ ) = 6V and this implies that vca (0+ ) = 6V = voltage across the inductor ( note,
terminal ‘ c ’ is + ve terminal and inductor acts as a source of energy ).
Now, the voltage across the terminals ‘ b ’ and ‘ c ’ ( v0 (0+ ) ) = vbd (0+ ) − vcd (0+ ) = 0 V .
The following expressions are valid at t = 0+
dv
dv
C c
= ic (0+ ) = 2 A ⇒ c
=1 volt / sec. (note, voltage across the capacitor will
dt t =0+
dt t =0+
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decrease with time i.e.,
dvc
dt
= − 1 volt / sec ). We have just calculated the voltage
t = 0+
across the inductor at t = 0+ as
vca (0+ ) = L
Now,
diL (t )
di (t )
6
= 6V ⇒ L
=
= 12 A / sec.
dt t =0+
dt t =0+
0.5
dv0 (0+ ) dvc (0 + )
di (0+ )
=
− R2 L
= 1 − (12 × 3) = − 35 volt / sec.
dt
dt
dt
Example: L.11.3 Refer to the circuit in fig.11.5(a). Determine,
(i) i (0+ ), iL (0+ ) and v(0+ ) (ii)
(assumed vc (0) = 0 ; iL (0) = 0 )
di ( 0+ )
dt
and
dv(0+ )
dt
(iii)
i ( ∞ ) , iL ( ∞ )
and v ( ∞ )
Solution: When the switch was in ‘off’ position i.e., t < 0
i(0- ) = i L (0- ) = 0, v(0- ) = 0 and v C (0- ) = 0
The switch ‘ S1 ’ was closed in position ‘1’ at time t = 0 and the corresponding circuit is
shown in fig 11.5 (b).
(i) From continuity property of inductor and capacitor, we can write the following
expression for t = 0+
1
i L (0+ ) = i L (0- ) = 0, vc (0+ ) = vc (0- ) = 0 ⇒ i (0+ ) = vc (0+ ) = 0
6
+
+
v (0 ) = i L (0 ) × 6 = 0 volt .
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(ii) KCL at point ‘a’
8 = i (t ) + ic (t ) + iL (t )
At t = 0 + , the above expression is written as
8 = i (0+ ) + ic (0+ ) + iL (0+ )
⇒ ic (0+ ) = 8 A
We know the current through the capacitor ic (t ) can be expressed as
dv (t)
i c (t) = C c
dt
dv c (0 + )
+
i c (0 ) = C
dt
+
dv c (0 )
1
= 8 × = 2 volt./sec. .
∴
dt
4
Note the relations
dvc ( 0+ )
= change in voltage drop in 6 Ω resistor = change in current through 6 Ω
dt
di ( 0+ ) 2
di ( 0+ )
1
= amp./ sec.
resistor ×6 = 6 ×
⇒
=
dt
6
3
dt
Applying KVL around the closed path ‘b-c-d-b’, we get the following expression.
vc (t ) = vL (t ) + v(t )
At, t = 0 + the following expression
vc (0+ ) = vL (0+ ) + iL (0+ ) × 12
diL (0+ )
di (0+ )
=0 ⇒ L
=0
dt
dt
di L (0+ )
di (0+ )
dv(0 + )
= 0 and this implies 12 L
= 12 × 0 = 0 v/sec =
=0
dt
dt
dt
0 = vL (0+ ) + 0 ×12 ⇒ vL (0+ ) = 0 ⇒ L
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Now, v (t ) = R iL (t ) also at t = 0 +
dv (0 + )
di (0 + )
di (0 + )
=R L
= 12 L
= 0 volt / sec.
dt
dt
dt
(iii) At t = α , the circuit reached its steady state value, the capacitor will block the flow of
dc current and the inductor will act as a short circuit. The current through 6 Ω and 12
Ω resistors can be formed as
12×8 16
i(∞) =
=
= 5.333A, i L (∞) = 8 -5.333 = 2.667 A
18
3
vc (∞) = 32 volt.
Example: L.11.4 The switch S1 has been closed for a sufficiently long time and then it is
opened at t = 0 (see fig.11.6(a)). Find the expression for (a) vc (t ) , (b) ic (t ), t > 0 for
inductor values of (i ) L = 0.5 H (ii ) L = 0.2 H (iii ) L = 1.0 H and plot vc (t ) − vs − t and
i (t ) − vs − t for each case.
Solution: At t = 0 − (before the switch is opened) the capacitor acts as an open circuit or
block the current through it but the inductor acts as short circuit. Using the properties of
inductor and capacitor, one can find the current in inductor at time t = 0+ as
12
iL (0+ ) = iL (0− ) =
= 2 A (note inductor acts as a short circuit) and voltage across the
1+ 5
5Ω resistor = 2 × 5 = 10 volt. The capacitor is fully charged with the voltage across the
5Ω resistor and the capacitor voltage at t = 0+ is given by
vc (0+ ) = vc (0− ) =10 volt. The circuit is opened at time t = 0 and the corresponding circuit
diagram is shown in fig. 11.6(b).
Case-1: L = 0.5H , R = 1Ω and C = 2 F
Let us assume the current flowing through the circuit is i(t ) and apply KVL equation
around the closed path is
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dv (t )
dvc (t )
d 2 vc (t )
di (t )
Vs = R i (t ) + L
+ vc (t ) ⇒ Vs = R C
+ LC
+ vc (t ) (note, i (t ) = C c )
2
dt
dt
dt
dt
2
d vc (t ) R dvc (t )
1
(11.14)
Vs =
+
++
vc (t )
2
dt
L dt
LC
The solution of the above differential equation is given by
vc (t ) = vcn (t ) + vcf (t )
(11.15)
The solution of natural or transient response vcn (t ) is obtained from the force free
equation or homogeneous equation which is
d 2 vc (t ) R dvc (t )
1
(11.16)
+
+
vc (t ) = 0
2
dt
L dt
LC
The characteristic equation of the above homogeneous equation is written as
R
1
α2 + α +
=0
(11.17)
L
LC
The roots of the characteristic equation are given as
2
2
⎛ R
⎛ R
1 ⎞⎟
1 ⎞⎟
⎛ R ⎞
⎛ R ⎞
⎜
;
+ ⎜
−
=
−
α1 = ⎜ −
α
=
−
−
−
= − 1.0
1.0
2
⎟
⎜
⎟
⎜ 2L
⎜ 2L
2 L ⎠ LC ⎟
2 L ⎠ LC ⎟
⎝
⎝
⎝
⎠
⎝
⎠
and the roots are equal with negative real sign. The expression for natural response is
given by
(11.18)
vcn (t ) = ( A1 t + A2 ) eα t (where α = α1 = α 2 = − 1 )
The forced or the steady state response vcf (t ) is the form of applied input voltage and it
is constant ‘ A ’. Now the final expression for vc (t ) is
vc (t ) = ( A1 t + A2 ) eα t + A = ( A1t + A2 ) e − t + A
(11.19)
The initial and final conditions needed to evaluate the constants are based on
vc (0+ ) = vc (0− ) = 10 volt ; iL (0+ ) = iL (0− ) = 2 A (Continuity property).
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At t = 0+ ;
vc (t ) t =0+ =A2 e−1× 0 + A = A2 + A
Forming
⇒ A2 + A = 10
(11.20)
dvc (t )
(from eq.(11.19)as
dt
dvc (t )
= α ( A1t + A2 ) eα t + A1 eα t = − ( A1t + A2 ) e − t + A1 e− t
dt
dvc (t )
= A1 − A2 ⇒ A1 − A2 = 1
dt t =0+
(11.21)
dvc (0+ )
dv (0+ )
= ic (0 + ) = iL (0+ ) = 2 ⇒ c
= 1 volt / sec. )
dt
dt
It may be seen that the capacitor is fully charged with the applied voltage when t = ∞ and
the capacitor blocks the current flowing through it. Using t = ∞ in equation (11.19) we
get,
vc (∞) = A ⇒ A =12
Using the value of A in equation (11.20) and then solving (11.20) and (11.21) we
get, A1 = − 1; A2 = − 2 .
(note, C
The total solution is
vc (t ) = − ( t + 2 ) e − t + 12 = 12 − ( t + 2 ) e − t ;
i (t ) = C
dvc (t )
= 2 × ⎡⎣( t + 2 ) e − t − e − t ⎤⎦ = 2 × ( t + 1) e − t
dt
(11.22)
The circuit responses (critically damped) for L = 0.5 H are shown fig.11.6 (c) and
fig.11.6(d).
Case-2: L = 0.2 H , R = 1Ω and C = 2 F
It can be noted that the initial and final conditions of the circuit are all same as in case-1
but the transient or natural response will differ. In this case the roots of characteristic
equation are computed using equation (11.17), the values of roots are
α1 = − 0.563; α 2 = − 4.436
The total response becomes
vc (t ) = A1 eα1 t + A2 eα 2 t + A = A1 e− 4.436 t + A2 e− 0.563t + A
(11.23)
dvc (t )
= α1 A1 eα1 t + α 2 A2 eα 2t = − 4.435 A1 e− 4.436 t − 0.563 A2 e− 0.536t
(11.24)
dt
dv (0+ )
Using the initial conditions( vc (0+ ) =10 , c
= 1 volt / sec . ) that obtained in case-1 are
dt
used in equations (11.23)-(11.24) with A = 12 ( final steady state condition) and
simultaneous solution gives
A1 = 0.032; A2 = − 2.032
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The total response is
vc (t ) = 0.032 e − 4.436 t − 2.032 e − 0.563t + 12
(11.25)
dvc (t )
= 2 ⎡⎣1.14 e − 0.563t − 0.14 e − 4.436 t ⎤⎦
dt
The system responses (overdamped) for L = 0.2 H are presented in fig.11.6(c) and
fig.11.6 (d).
i (t ) = C
Case-3: L = 8.0 H , R = 1Ω and C = 2 F
Again the initial and final conditions will remain same and the natural response of the
circuit will be decided by the roots of the characteristic equation and they are obtained
from (11.17) as
α1 = β + jγ = −0.063 + j 0.243; α 2 = β − jγ = − 0.063 − j 0.242
The expression for the total response is
vc (t ) = vcn (t ) + vcf (t ) = e β t K sin ( γ t + θ ) + A
(11.26)
(note, the natural response vcn (t ) = e β t K sin ( γ t + θ ) is written from eq.(11.12) when
roots are complex conjugates and detail derivation is given there.)
dvc (t )
(11.27)
= K e β t ⎡⎣ β sin ( γ t + θ ) + γ cos ( γ t + θ ) ⎤⎦
dt
dv (0+ )
Again the initial conditions ( vc (0+ ) =10 , c
= 1 volt / sec . ) that obtained in case-1 are
dt
used in equations (11.26)-(11.27) with A = 12 (final steady state condition) and
simultaneous solution gives
K = 4.13; θ = − 28.980 ( deg ree )
The total response is
vc (t ) = e β t K sin (γ t + θ ) + 12 = e−0.063t 4.13sin ( 0.242 t − 28.990 ) + 12
(
vc (t ) =12 + 4.13 e−0.063t sin 0.242 t − 28.990
)
(11.28)
dvc (t )
= 2e−0.063t ⎡⎣0.999*cos 0.242 t − 28.990 − 0.26sin 0.242 t − 28.990 ⎤⎦
dt
The system responses (under-damped) for L = 8.0 H are presented in fig.11.6(c) and fig.
11.6(d).
i(t ) = C
(
)
(
)
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Version 2 EE IIT, Kharagpur
Remark: One can use t = 0 and t = ∞ in eq. 11.22 or eq. 11.25 or eq. 11.28 to verify
whether it satisfies the initial and final conditions ( i.e., initial capacitor voltage
vc (0+ ) =10 volt. , and the steady state capacitor voltage vc (∞) =12 volt. ) of the circuit.
Example: L.11.5 The switch ‘ S1 ’ in the circuit of Fig. 11.7(a) was closed in position ‘1’
sufficiently long time and then kept in position ‘2’. Find (i) vc (t ) (ii) ic (t ) for t ≥ 0 if C
1
1
1
is (a) F (b) F (c) F .
9
4
8
Solution: When the switch was in position ‘ 1’, the steady state current in inductor is
given by
30
i L (0- ) =
= 10A, v c (0- ) = i L (0- ) R = 10× 2 = 20 volt.
1+ 2
Using the continuity property of inductor and capacitor we get
i L (0 + ) = i L (0 - ) = 10, v c (0 + ) = v c (0 - ) = 20 volt.
The switch ‘ S1 ’ is kept in position ‘2’ and corresponding circuit diagram is shown in
Fig.11.7 (b)
Applying KCL at the top junction point we get,
vc (t)
+ ic (t) + i L (t) = 0
R
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or
vc (t)
dv (t)
+ C c + i L (t) = 0
R
dt
di (t )
d 2i L (t)
L di L (t)
+ C.L
+ i L (t) = 0 [note: vc (t ) = L L ]
dt
R dt
dt 2
d 2i L (t) 1 di L (t) 1
(11.29)
+
+
i L (t) = 0
RC dt
LC
dt 2
The roots of the characteristics equation of the above homogeneous equation can
1
obtained for C = F
9
2
2
2
2
1
9
4×9
1 ⎞
9
+ ⎜⎛
−
− + ⎜⎛ ⎟⎞ −
⎟ − 4 LC
RC
2
2
⎝ RC ⎠
⎝ 2⎠
= −1.5
α1 =
=
2
2
1
9
1 ⎞
⎛ 9 ⎞ 4×9
−
− ⎜⎛
⎟ − 4 LC − − ⎜ ⎟ −
RC
2
2
⎝ RC ⎠
⎝ 2⎠
α2 =
=
= − 3.0
2
2
1
Case-1 ( ξ =1.06, over damped system ) : C = F , the values of roots of characteristic
9
equation are given as
α1 = − 1.5 , α 2 = − 3.0
The transient or neutral solution of the homogeneous equation is given by
i L (t) = A1e- 1.5t + A2e-3.0t
(11.30)
To determine A1 and A2 , the following initial conditions are used.
At t = 0+ ;
i L (0 + ) = i L (0- ) = A1 + A2
(11.31)
10 = A1 + A2
v c (0+ ) = vc (0- ) = v L (0+ ) = L
di L (t)
dt t = 0+
20 = 2× ⎡⎣ A1 × -1.5 e- 1.5t - 3.0 × A2 e- 3.0t ⎤⎦
(11.32)
= 2 [ -1.5A1 - 3A 2 ] = - 3A1 - 6A 2
Solving equations (11.31) and (11,32) we get , A2 = −16.66 , A1 = 26.666 .
The natural response of the circuit is
80
50
i L = e −1.5 t − e−3.0 t = 26.66e−1.5 t − 16.66e−3.0 t
3
3
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di L
= 2 ⎡⎣ 26.66 × −1.5e −1.5 t − 16.66 × −3.0 e−3.0 t ⎤⎦
dt
vL (t ) = vc (t) = ⎡⎣100 e- 3.0 t - 80 e- 1.5 t ⎤⎦
L
ic (t ) = c
dvc (t ) 1
= ( −300.0 e- 3.0 t + 120 e- 1.5t ) = (13.33e- 1.5t − 33.33e- 3.0 t )
dt
9
Case-2 (ξ = 0.707,under damped system ) : For C =
equation are
1
F , the roots of the characteristic
4
α1 = − 1.0 + j 1.0 = β + j γ
α 2 = − 1.0 − j 1.0 = β − j γ
The natural response becomes 1
i L (t) = k e β t sin(γ t + θ )
Where k and θ are the constants to be evaluated from initial condition.
(11.33)
At t = 0+ , from the expression (11.33) we get,
i L ( 0+ ) = k sin θ
10 = k sin θ
(11.34)
di(t)
= 2 × k ⎡⎣β eβ t sin(γ t + θ ) + eβ t γ cos(γ t + θ ) ⎤⎦
t=0+
dt t = 0 +
Using equation (11.34) and the values of β and γ in equation (11.35) we get,
20 = 2 k ( β snθ + γ cos θ ) = k cos θ (note: β = −1, γ = 1 and k sin θ = 10 )
L
(11.35)
(11.36)
From equation ( 11.34 ) and ( 11.36 ) we obtain the values of θ and k as
1
10
⎛1⎞
= 22.36
tan θ = ⇒ θ = tan -1 ⎜ ⎟ = 26.56o and k =
2
sin θ
⎝2⎠
∴ The natural or transient solution is
i L (t) = 22.36 e - t sin ( t + 26.56o )
L
di(t)
= vc (t) = 2 × k × [β sin (γ t + θ) + γ cos (γ t + θ)] e β t
dt
= 44.72 ⎡⎣ cos (t + 26.56o ) - sin (t + 26.56o ) ⎤⎦ × e − t
{
dvc (t ) 1
d
⎡⎣cos (t + 26.56o ) - sin (t + 26.56o ) ⎤⎦ e -t
= × 44.72
dt
4
dt
= − 22.36 cos(t + 26.56) e− t
ic (t ) = c
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1
Case-3 (ξ =1, critically damped system ) : For C = F ; the roots of characteristic
8
equation are α1 = − 2; α 2 = − 2 respectively. The natural solution is given by
iL (t ) = ( A1 t + A2 ) eα t
(11.37)
where constants are computed using initial conditions.
At t = 0+ ; from equation ( 11.37) one can write
i L (0+ ) = A2 ⇒ A2 = 10
L
di(t)
= 2 × ⎡⎣ A2 α e α t + α A1 t e α t + A1 e α t ⎤⎦ +
t =0
dt t = 0+
= 2 × ⎡⎣( A1 + A2α ) e α t + α A1 t eα t ⎤⎦
L
t = 0+
di(t)
= vc (0+ ) = 20 = 2 ( A1 − 2 A2 ) ⇒ A1 = 30
dt t = 0+
The natural response is then
iL (t ) = (10 + 30 t ) e − 2 t
di L (t)
d
= 2 × ⎡⎣(10 + 30t ) e −2t ⎤⎦
dt
dt
di (t)
L L
= vc (t ) = 2 [10 − 60t ] e − 2 t
dt
dv (t ) 1
d
ic (t ) = c c = × 2 × ⎡⎣(10 − 60t ) e −2t ⎤⎦ = ⎡⎣ −20 e −2t + 30t e−2t ⎤⎦
dt
8
dt
1
Case-4 (ξ = 2, over damped system ) : For C =
F
32
Following the procedure as given in case-1 one can obtain the expressions for (i) current
in inductor iL (t ) (ii) voltage across the capacitor vc (t )
L
iL (t ) = 11.5 e −1.08t −1.5 e −14.93t
di (t )
L
= vc (t ) = ⎡⎣ 44.8 e −14.93 t − 24.8 e −1.08 t ⎤⎦
dt
dv (t ) 1 d
ic (t ) = c c = × ⎡⎣ 44.8 e −14.93 t − 24.8 e−1.08 t ⎤⎦
dt
32 dt
−1.08t
= 0.837 e
− 20.902 e −14.93t
L.11.3 Test your understanding
(Marks: 80)
T.11.1 Transient response of a second-order ------------------ dc network is the sum of two
real exponentials.
[1]
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T.11.2 The complete response of a second order network excited from dc sources is the
sum of -------- response and ---------------- response.
[2]
T.11.3 Circuits containing two different classes of energy storage elements can be
described by a ------------------- order differential equations.
[1]
T.11.4 For the circuit in fig.11.8, find the following
[6]
dvc (0 − )
dvc (0 + )
di (0 − )
di (0 + )
(d )
(e) L
(f) L
dt
dt
dt
dt
(Ans. (a ) 6 volt. (b) 6 volt. (c) 0 V / sec. ( d ) 0 V / sec. (e) 0 amp / sec. ( f ) 3 amp./ sec. )
( a ) vc (0− ) (b) vc (0 + ) (c)
T.11.5 In the circuit of Fig. 11.9,
Find,
Version 2 EE IIT, Kharagpur
dvR (0+ )
dvL (0+ )
(a ) vR (0 ) and vL (0 ) (b)
and
(c) vR (∞) and vL (∞)
dt
dt
+
+
[8]
(Assume the capacitor is initially uncharged and current through inductor is zero).
(Ans. (a ) 0 V , 0 V (b) 0 V , 2 Volt./ Sec. (c) 32 V , 0 V )
T.11.6 For the circuit shown in fig.11.10, the expression for current through inductor
is given by iL (t ) = (10 + 30t ) e−2t
for t ≥ 0
Find, (a ) the values of L , C (b) initial condition vc (0− ) (c) the expression for vc (t ) > 0 .
1
[8]
(Ans. (a) L = 2 H , C = F (b) vc (0− ) = 20V (c) vc (t ) = ( 20 − 120 t ) e −2t V . )
8
T.11.7 The response of a series RLC circuit are given by
vc (t ) =12 + 0.032 e−4.436t − 2.032 e−0.563t
iL (t ) = 2.28 e−0.563t − 0.28 e−4.436t
where vc (t ) and iL (t ) are capacitor voltage and inductor current respectively. Determine
[4+4]
(a) the supply voltage (b) the values R, L, C of the series circuit.
(Ans. (a ) 12 V (b) R = 1 Ω , L = 0.2 H and C = 2 F )
T.11.8 For the circuit shown in Fig. 11.11, the switch ‘ S ’was in position ‘1’ for a long
time and then at t = 0 it is kept in position ‘2’.
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Find,
(a) iL (0− ); (b) vc (0+ ); (c) vR (0+ ); (d ) iL (∞);
[8]
Ans.
(a ) iL (0− ) = 10 A ; (b) vc (0+ ) = 400 V ;
(c) vR (0+ ) = 400 V (d ) iL (∞) = − 20 A
T.11.9 For the circuit shown in Fig.11.12, the switch ‘ S ’ has been in position ‘1’ for a
long time and at t = 0 it is instantaneously moved to position ‘2’.
Determine i (t ) for t ≥ 0 and sketch its waveform. Remarks on the system’s
response.
1 ⎞
⎛7
(Ans. i (t ) = ⎜ e −7 t − e− t ⎟ amps. )
3 ⎠
⎝3
[8]
T.11.10 The switch ‘ S ’ in the circuit of Fig.11.13 is opened at t = 0 having been closed
for a long time.
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Determine (i) vc (t ) for t ≥ 0 (ii) how long must the switch remain open for the voltage
vc (t ) to be less than 10% ot its value at t = 0 ?
[10]
(Ans. (i) (i) vc (t ) = (16 + 240t ) e −10t (ii) 0.705sec. )
T.11.11 For the circuit shown in Fig.11.14, find the capacitor voltage vc (t ) and inductor
current iL (t ) for all t (t < 0 and t ≥ 0).
[10]
Plot the wave forms vc (t ) and iL (t ) for t ≥ 0 .
(Ans. vc (t ) = 10 e −0.5t sin(0.5 t ); iL (t ) = 5 ( cos(0.5 t ) − sin(0.5 t ) ) e −0.5t )
T.11.12 For the parallel circuit RLC shown in Fig.11.15, Find the response
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of iL (t ) and vc (t ) respectively.
[10]
(Ans. iL (t ) = ⎡⎣ 4 − 4 e −2t (1 + 2 t ) ⎤⎦ amps. ; vc (t ) = 48 t e −2t volt. )
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