Physics 401, Spring 2014.
Eugene V. Colla
Transients and Oscillations in RLC Circuits.
Outline
Main goals of this week Lab:
Transients. Definition.
transient ( physics ) a short-lived oscillation in a
system caused by a sudden change of voltage or
current or load
a transient response or natural response is the
response of a system to a change from equilibrium.
System under
study
Transients in RLC circuit.
Resistance R [Ohm]
Capacitance C [uF] (10-6F)
Inductance L [mH] (10-3H)
R
L
scope
V
C
Transients in RLC circuit.
According the Kirchhoff’s law
*
VR+VL+VC=V(t)
d2
d
q(t)
L 2 q(t) + R q(t) +
=0
dt
dt
C
(1)
*See Lab write-up for details
R
L
VL
1.0
scope
V(t)
V0
V(t)
VR
0.5
C
VC
0.0
-1.0
-0.5
0.0
time
0.5
1.0
Transients in RLC circuit. Three solutions
The solution of this differential
equation can be found in the form
This will convert (1) in
quadratic equation
s1,2
with solutions:
a
R
s 
L
2
q (t )  A est
1

0
s 
LC

2
R
 R   1 

 
 
  a  b
2L
 2L   LC 
2
R
 R   1 
, b 
 

2L
2
L
LC

 

b2>0 over-damped solution
b2=0 critically damped solution
b2<0 under-damped solution
Transients in RLC circuit. Over-damped solution:
b2>0
In this case the solution will be aperiodic
exponential decay function with no
oscillations:
i (t ) 
q (t )  e  at  A1e bt  B1e  bt 
dq
 ae  at ( A1e bt  B1e  bt )  be  at ( A1e bt  B 1e  bt )
dt
Transients in RLC circuit. Over-damped solution:
b2>0
Taken in account the initial conditions: q(0)=q0 and i(0)=0
q (t )  q 0e
 at

( a -b )t 1
a


 cosh bt  sinh bt 
b


q0 
a  -(a -b )t
1


e
2 
b
q 0  a 2  b 2   (a b )t
i (t )  

e
2 
b

This is exponential decay function
Transients in RLC circuit. Critically-damped
solution b2=0
For this case the general solution can be found as
q(t)=(A2+B2t)e-at. Applying the same initial condition
the current can be written as i=–a2q0te-at
4L
b 0 R 
C
2
2
and
R
a
2L
Critically-damped
conditions for our
network
Critical damped case shows the
fastest decay with no oscillations
Transients in RLC circuit. Critically-damped
solution. Real data analysis.
2
b =0
In this experiment R=300 ohms,
C=1mF, L=33.43mH.
The output resistance of Wavetek is 50 ohms and
resistance of coil was measured as 8.7 ohms, so actual
resistance of the network is Ra=300+50+8.7=358.7
Decay coefficient
𝒂=
𝑹
𝟑𝟓𝟖.𝟕
=
𝟐𝑳 𝟐∗𝟑𝟑.𝟒𝟑𝑬−𝟑
≈ 𝟓𝟑𝟔𝟓
Transients in RLC circuit. Critically-damped
solution. Real data analysis. b2=0
Vc ~q, fiiting function: Vc=Vco(1+at)e-at
Now the experimental results:
Calculated decay
coefficient ~5385,
Obtained from fitting ~5820.
Possible reason – it is
still slightly over damped
Calculated b2 is
b2=2.99e7-2.90e7>0
Transients in RLC circuit. Under-damped
solution.
If b2<0 we will have oscillating solution. Omitting the details (see
Lab write-up) we have the equations for charge and current as:
a
a


q(t)  q 0e  at  cos bt  sin bt   q 0e  at 1  2 sin(bt   )
b
b


2
2
 at  a  b 
i(t)  q 0e 
 sin bt
b


2
2
R
 R   1
a
, b 
 
2L
 2L   LC
1

;
f


2

 1

 LC
  R 


  2L 
2
Transients in RLC circuit. Under-damped solution.
Log decrement. Quality factor.
Log decrement can be defined as 𝜹 = 𝒍𝒏
6
Quality factor can be
𝑬
defined as 𝑸 = 𝟐𝝅 ,
∆𝑬
For RLC
𝑸=
=
= 𝒂𝑻𝟏 , where T1=1/f1
𝝎𝟏 𝑳
𝑹
=
𝝅
𝜹
From this plot d≈0.67
Q≈4.7
VC (q/C) (V)
𝒍𝒏
𝒆−𝒂𝒕𝒎𝒂𝒙
𝒆−𝒂𝒕𝒎𝒂𝒙 +𝑻𝟏
𝒒(𝒕𝒎𝒂𝒙
𝒒(𝒕𝒎𝒂𝒙 +𝑻𝟏
3.529
3
1.809
0.92962
0.47494
0
-3
-6
-1
0
1
2
3
4
5
6
7
time (ms)
8
9
10
Transients in RLC circuit. Data analysis. Using
Origin software.
VC (q/C) (V)
6
0.00115
3
0.0023
0.00346
0.00463
0
f=862Hz
-3
-6
-1
0
1
2
3
4
5
6
7
8
9
10
9
10
time (ms)
VC (q/C) (V)
6
3
0
f=862Hz
-3
-6
-1
0
1
2
3
4
5
6
7
time (ms)
8
1.
2.
3.
4.
Pick peaks
Envelope
Exponential term
Nonlinear fitting
Transients in RLC circuit. Under-damped solution.
Log decrement. Quality factor.
Transients in RLC circuit. Data analysis. Log
decrement. Using Origin software. Results.
Time domain trace
Points found using “Find peaks”
Envelope curve
Transients in RLC circuit. Data analysis. Log
decrement. Using Origin software. Results.
Fitting the “envelope data” to
exponential decay function
Transients in RLC circuit. Data analysis.
(1/T)2 vs 1/C experiment.
q(t)  Ae  at s in(t   )  offset
0ffset
Manual evaluation of the
period of the oscillations
Limited accuracy
Results can be
effected by DC offset
Zero crossing points
Transients in RLC circuit. Data analysis.
(1/T)2 vs 1/C experiment. Using Origin software.
q(t)  Ae  at s in(t   )
Use Origin standard
function
Category:
Limited Waveform
Function: SineDamp
Fitting function ; y0,A,t0 xc, w – fitting parameters
Transients in RLC circuit. Data analysis.
(1/T)2 vs 1/C experiment. Using Origin software.
q(t)  Ae  at s in(t   )
Limited
Data plot + fitting curve
Residuals - criteria
of quality of fitting
Transients in RLC circuit. Data analysis.
(1/T)2 vs 1/C experiment. Using Origin software.
q(t)  Ae  at s in(t   )
f
2
2
1  1
 1
  
 
T
2

 
  LC
Final results
  R 


  2L 
2



Resonance in RLC circuit.
1904.83204
14
R2
12
f=1500Hz
VC
UC
C
V(t)
10
L
8
6
4
2
0
𝑸=
𝒇 𝟏𝟗𝟎𝟒
=
𝜟𝒇 𝟏𝟓𝟎𝟎
= 𝟏. 𝟐𝟔
100
1000
f (Hz)
10000
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