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12.3 
the SerIeS Lc cIrcuIt wIth SmALL dAmpIng – Another SpecIAL cASe
Now we need the full expression for v
C
(t) given in Eqn. 12.1-7 to proceed with our analysis.
v t
e
V
I
t V
t
C
t
o
o
L
C
n
o
n
n
( )
sin
cos
=
+
−








−
+
−



−
xw
x
x
x w
x w
1
1
1
2
2
2





≥
=
=
+
for
where
t
LC
R
L
C
n
0
1
2
w
x
,
(12.3-1) 

---

12.14


SeriesandParallel
RLC
Circuits
The resistance is small since damping is taken to be small. We note from the equation
above that, for given L and C values the number 
x
decides the rate of loss of amplitude of oscillations. 
Therefore, 
x
has to be small in a case in which damping is small. Therefore, the assumption of small 
damping is equivalent to the assumption that R
L
C
<<
2
, i.e., small resistance compared to 2 L C.
This resistance, 2 L C, which is a characteristic value for a given series RLC circuit is termed as its 
critical resistance. Since amplitude loss is basically due to damping and since the factor 
x
governs the 
rate of loss of amplitude, this factor 
x
is aptly termed as damping factor in circuit studies. Thus, we 
are dealing with series RLC circuit with 
x
 <<1 in this section. We make use of this fact to simplify the 
expression in Eqn. 12.3-1 considerably.
We employ the following approximations in this context.
V
I
L
C
I
L
C
o
o
o
d
n
n
x
x
x
w
x w w
x
+
−
≈
<<
= −
≈
<<
1
2
1
1
2
1
for
for
∴
≈
+




=
=
−
−
v t
e
I
t V
t
i t
C
dv t
dt
e
C
t
o
L
C
n
o
n
C
n
( )
sin
cos
( )
( )
xw
x
w
w
and 
w
w
w
w
n
t
o
n
o
C
L
n
I
t V
t
cos
sin
−




(12.3-2)
The algebra involved in the second step in Eqn. 12.3-2 has been skipped. Reader is urged to verify 
the derivation of i(t). 
Expressing the expressions for v
C
(t) and i(t) in Eqn. 12.3-2 as single sinusoids, we have,
v t
V
LI
C
e
t
t
i t
I
CV
L
C
o
o
t
n
o
o
n
( )
cos(
)
( )
≈
+
−
≥
≈ −
+
−
+
2
2
2
2
0
xw
w
f
V for
ee
t
t
I
V
n
t
n
o
L
C
o
−
+
−
−
≥
=




xw
w
sin(
)
tan
f
f
A for
where 
0
1
(12.3-3)
We see that the expressions for v
C
(t) and i(t) in the case of series RLC circuit with small damping is 
different from corresponding functions in the case of an undamped LC circuit only by the factor e
n
t
-
xw
appearing as a multiplier. Compare Eqns 12.2-3 and 12.3-3.
Let us examine how the total stored energy in the circuit evolves in time in this case. 

---

TheSeriesLCCircuitwithSmallDamping–AnotherSpecialCase

12.15
Total stored energy
( )
( )
=
+
=
+
−
0 5
0 5
0 5
2
2
2
2
. Li t
. Cv t
e
L I
CV
C
t
o
n
xw
.
oo
n
o
o
n
L
t
C V
LI
C
t
2
2
2
2
2
0 5




− +
+




−






cos (
)
.
sin (
)
w
w
f
f


=
+
−
( .
.
)
0 5
0 5
2
2
2
LI
CV
e
o
o
t
n
xw
It is not a constant anymore. Starting at the initial value (equal to the total initial energy storage in 
L and C), it decreases exponentially with time and goes to zero as t 
→
∞
. Once again, the damping 
factor 
x
governs the rate at which the total stored energy decreases with time.
It is instructive to look at the amount of energy lost in one oscillation period. However, is there a 
period of oscillation in this case?
The product of exponential function and sinusoidal function is not a sinusoid. It will not be a 
periodic waveform at all. However, the time interval between the successive zero-crossings of the 
waveform will remain constant since this is decided by the sinusoid in the product. Therefore, we 
can not talk about a period in the case of damped oscillations; but we can talk about the time interval 
between two successive zero-crossings in the same direction. Sometimes this interval is called ‘period’ 
of damped oscillation in loose manner.
Let us find out the energy loss incurred over one such interval and express it as a fraction of the 
total energy storage at the beginning of that interval.
Let the variable 
w
n
t be k
p
at the beginning of one such interval ( i.e., at a zero-crossing) where k 
is an integer. The value of total energy storage at this point is 0 5
0 5
2
2
2
.
.
LI
CV
e
o
o
k
+
(
)
−
px
. The next 
similar zero crossing will take place at 
w
n
t
+
2
p
rad. The value of total energy storage at that time point 
will be ( .
.
)
(
)
0 5
0 5
2
2
2
2
LI
CV
e
o
o
k
+
−
+
px
J.
∴
=
−
Fractional loss of stored energy over one oscillation
e
2
kk
k
k
e
e
e
px
px
px
px
px
px
px
−
= −
= − −
+
+




=
−
+
−
−
2
2
2
4
2
1
1 1 4
4
2
4
(
)
(
)
!
ffor
1
x
<<
By a similar reasoning one can see that the fractional loss of amplitude of voltage and current 
oscillations in the circuit over one oscillation period 
≈
2
px
, i.e., the amplitude of oscillation in the 
(n 
+ 
1)
th
cycle of oscillation will be 
≈
(1 
- 
2
px
) times the amplitude of n
th
oscillation.
All the voltage variables and the circuit current variable during the zero-input response of a weakly 
damped series RLC circuit with L 
=
1 H, C 
=
1 F, R 
=
0.08 
W
, V
o
=
2 V and I
o
=
1 A are plotted in 
Fig. 12.3-1. The damping factor 
x
in this case is 0.04 and the undamped natural frequency 
w
n
is 
1 rad/s. The successive positive peaks of v
C
(t) are 2.22 V and 1.71V and negative peaks are 1.95 V and 
1.51 V. The fractional loss of amplitude over first cycle of oscillation calculated from these numbers 
are 0.23 and 0.226 and are almost equal to 2
px
=
0.25. The exponential envelope function has a time 
constant of 1/0.04 
=
25 s. One full oscillation takes about 6.3 s and hence there will be roughly about 
9 oscillations before the amplitude of oscillation get damped down to less than 10% of their starting 
values.

---

12.16


SeriesandParallel
RLC
Circuits
2.5
2
1.5
1
0.5
2
4
Volts
Amps
6
8
10
12
–
+
+
+
–
–
R
= 0.08 
Ω
L
= 1 H
C
= 1 F
–0.5
–1
–1.5
–2
–2.5
v
R
(
t
)
v
R
(
t
)
v
L
(
t
)
v
L
(
t
)
I
0
= 1 A
V
0
= 2 V
v
C
(
t
)
v
C
(
t
)
i
(
t
)
Time (s)
e
–0.04
t
Envelope function
i
(
t
)
Fig. 12.3-1 

Zero-inputresponseofaweaklydamped
RLC
circuit
(L
=
1H,C
=
1F,R
=
0.08
W
,V
o

=
2VandI
o

=
1A)
The time-variation of various energy storage functions for zero-input response of this circuit is 
plotted in Fig. 12.3-2.
2.5
Energy storage in capacitor
Stored
energy
(J)
Energy storage in inductor
Average energy
storage in 
L
and 
C
0.5
2
4
6
8
10
12
Time (s)
1
1.5
2
Total circuit energy store = 2.5 e
–0.08
t
Fig. 12.3-2 

Energystoragefunctionsofaweaklydamped
RLC
circuit
(L
=
1H,C
=
1F,R
=
0.08,V
o

=
2VandI
o

=
1A)
This exponential decay of total energy storage in a series RLC circuit is applicable only when 
x
<< 1. Total energy storage will come down with time for other values of 
x
 too; but not as per a clean 
exponential law. This is illustrated in Fig. 12.3-3 for a circuit with 
x
 
=
0.25. v
C
(t) and i(t) for the circuit 
referred in this figure was already plotted in Fig. 12.1-4. 
The nature of zero-input response when 
x
is not small is qualitatively similar to that of the weakly 
damped case. However, the time-interval in the zero crossings (period of oscillation in a loose sense) 
is not 2
p
/
w
n
– it is rather 2
p
/
w
d
where 
w
x w
d
n
=
−
1
2
. This number is termed as ‘damped natural 
frequency’ in circuit studies.

---

StandardFormatsforSecond-OrderCircuitZero-InputResponse

12.17
4
Stored
energy
(J)
Energy storage in capacitor
Energy storage in inductor
Total circuit energy storage
Time (s)
3.5
3
2.5
2
1.5
1
0.5
2
4
Fig. 12.3-3 

Energystoragefunctionsofamoderatelydamped(
x
 
=
0.25)
RLC
circuit
(L
=
1H,C
=
1F,R
=
0.5
W
,V
o

=
2VandI
o

=
2A)