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Study of transient response of RLC circuit.
Typology: Lab Reports
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Lab 3 : Study of the transient response of an RLC circuit.
3.1 Lab no.3 : Study of the Transient Response of an RLC Circuit_________________________________ 3
3.2 Abstract________________________________________________________________________ 3
3.3 Introduction:____________________________________________________________________ 3
3.3.1 Objective:_________________________________________________________________ 3
3.3.2 Background:_______________________________________________________________ 3
3.4 Equipment:_____________________________________________________________________ 3
3.5 Procedure:______________________________________________________________________ 3
3.6 Results:________________________________________________________________________ 4
3.6.1 Overdamped response of RL circuit, with R = 1.2KΩ:______________________________ 4
3.6.2 Underdamped response with R = 47 Ω:__________________________________________ 5
3.7 Discussion:_____________________________________________________________________ 9
3.8 Conclusion:_____________________________________________________________________ 9
3.9 Appendix: Pre-Lab#3:Study of the transient response of an________________________________ 9
3.9.1 Theoretical Results:_____________________________________________________________ 9
3.9.2 Simulation Results:_____________________________________________________________ 11
Figure 3. 1 : Schematic diagram of series RLC circuit.___________________________________________ 4
Figure 3. 2 : Plot of V C
with R = 1.2K Ω, V C
(1ms) = 3V
Table 3. 1 : Series RLC Circuit Response with R1=1.2kΩ_________________________________________ 5
Figure 3. 3 : Plot of voltage across capacitor with R = 47 Ω, V C
(0.5ms) = 6.6V__________________________ 5
Table 2 : Series RLC Circuit Response with R1=47_____________________________________________ 6
Figure 4 : Observation table from lab experiment, with R = 47ohm_________________________________ 7
Figure 5 : Observation table from lab experiment, with R = 1.2Kohm_______________________________ 8
Figure 3. 6 :Input(green) and Output(red) voltages, R = 1.2K_____________________________________ 11
Figure 3. 7 : Measurement Results__________________________________________________________ 11
Figure 3. 8 :Input(green) and Output(red) voltages_____________________________________________ 12
Figure 3. 9 : Measurement Results__________________________________________________________ 13
Lab 3 : Study of the transient response of an RLC circuit.
Figure 3. 1 : Schematic diagram of series RLC circuit.
C
3.6.1 Overdamped response of RL circuit, with R = 1.2KΩ:
Figure 3. 2 : Plot of V C
with R = 1.2KΩ, V C
(1ms) = 3V.
Quantity
Value from
Theory
Value from
Simulation
Value from
Experiment
% Deviation
of
Simulation
from Theory
% Deviation
of
Experiment
from Theory
α
10909
X X X X
ω
0
4264
X X X X
Type of
response
Overdamped Overdamped Overdamped
X X
s 1 , 2
-20950, -867.
X X X X
v c
(
)
-5 -5 -4.8 0 4
v c
( ∞ )
5 5 5.2 0 4
A
I
/ B
I
/ D
I
1 1 1
-10.
X X X X
A
I
/ B
I
/ D
I
2 2 2
X X X X
v c
(0_._ 5ms)
-1.732 -1.741 -2.4 0.57 25
vc (1ms)
0.626 0.620 0.240 0.37 58
v c
(2ms)
3.15 3.15 2.8 0 11.
Table 3. 1 : Series RLC Circuit Response with R1=1.2kΩ
3.6.2 Underdamped response with R = 47 Ω:
Figure 3. 3 : Plot of voltage across capacitor with R = 47 Ω, V C
(0.5ms) = 6.6V
Figure 4 : Observation table from lab experiment, with R = 47ohm
Lab 3 : Study of the transient response of an RLC circuit.
Figure 5 : Observation table from lab experiment, with R = 1.2Kohm
Lab 3 : Study of the transient response of an RLC circuit.
dv
c
dt
= s
1
'
1
2
'
2
'
1
'
2
Solving eq(3)and eq.(4) simultaneously gives:
1
2
Hence eq.(2) becomes:
v c
(t) = 5 − 10.362e
−862.467t
−24669.447t
Now,
v c
v c
(0.5m) = − 1.732V
v c
(1m) = 0.626V
v c
(2m) = 3.154V
2. For R = 47 Ω:
R = 47 Ω, C = 1 μ, L = 47m
Now, initial condition is:
v c
The characteristic equation is:
s
2
o
2
Now, first calculating neper’s frequency:
α =
2 × 47m
= 500rad/sec
Now, Resonant frequency is:
ω o
47m × 1 μ
= 4612.656rad/sec
From the above calculations we can see that the response of the circuit is underdamped i.e. ω o
2
α
2
Hence the voltage solution will;
v c
(t) = V
f
'
1
e
−αt
csc ω
d
t + B
'
2
e
−αt
sin ω
d
t −−−−−−− (2)
where
ω
d
= ω
o
2
− α
2
2
2
= 4585.476 rad/sec
The roots of the eq.(1) are
s
1,
=− α ± ω
d
i =− 500 ± 4585.476i
s 1
= − 500 + 4585.476i
s 2
= − 500 − 4585.476i
Roots are real and not equal hence the response is underdamped.
In order to calculate B1 and B2 we use initial conditions:
At t = 0+:
v
c
f
'
1
e
−α(0)
csc (ω
d
'
2
e
−α(0)
sin (ω
d
1
1
Now;
dv
c
dt
= −αB
1
d
2
2
2
Hence eq.(2) becomes:
v
c
(t) = 5 − 10 e
−500t
csc (4585.476t) − 1.09e
−500t
sin (4585.476t)
Now,
v c
v c
(0.5m) = 9.509V
v c
(1m) = 6.424V
v
c
(2m) = 8.460V
1. For R = 1.2KΩ:
Circuit:
Results:
Figure 3. 6 :Input(green) and Output(red) voltages, R = 1.2K
Figure 3. 7 : Measurement Results
Figure 3. 9 : Measurement Results
