physics 401, spring 2013. · physics 401, spring 2013. eugene v. colla . transients and...
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![Page 1: Physics 401, Spring 2013. · Physics 401, Spring 2013. Eugene V. Colla . Transients and Oscillations in RLC Circuits. Outline . transient ( physics ) a short-lived oscillation in](https://reader035.vdocuments.net/reader035/viewer/2022063015/5fd377c340852c7efe5da965/html5/thumbnails/1.jpg)
Physics 401, Spring 2013. Eugene V. Colla
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Transients and Oscillations in RLC Circuits.
Outline
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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
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R
L
C
V scope
Resistance R [Ohm]
Capacitance C [uF] (10-6F)
Inductance L [mH] (10-3H)
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R
L
C
V(t) scope
VR VL
VC
According the Kirchhoff’s law VR+VL+VC=V(t)
-1.0 -0.5 0.0 0.5 1.00.0
0.5
1.0
V(t
)
time
V0
2
02
d d q(t) q(t)L q(t) +R q(t) + = 0 , = V
dt dt C C*See Lab write-up for details
*
(1)
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The solution of this differential
equation can be found in the form ( ) est
q t A
This will convert (1) in
quadratic equation
Rs s
L LC
21
0
,
,
R Rs a b
L L LC
R Ra b
L L LC
2
1 2
2
1
2 2
1
2 2
with solutions:
b2>0 over-damped solution
b2=0 critically damped solution
b2<0 under-damped solution
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b2>0
( ) e at bt bt
q t A e B e1 1
In this case the solution will be aperiodic
exponential decay function with no
oscillations:
( ) ( ) ( )at bt bt at bt btdq
i t ae A e B e be A e B e
dt1 1 1 1
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b2>0
Taken in account the initial conditions: q(0)=q0 and i(0)=0
( ) cosh sinhat
aq t q e bt bt
b0
( )( ) a b tq a b
i t e
b
2 2
0
2
This is exponential decay function
-( - )
( - )
a b t
a b t
q ae
b
0
11
2
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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
Critical damped case shows the
fastest decay with no oscillations
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b2=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
𝟐𝑳=
𝟑𝟓𝟖.𝟕
𝟐∗𝟑𝟑.𝟒𝟑𝑬−𝟑≈ 𝟓𝟑𝟔𝟓
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b2=0
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
Vc ~q, fiiting function: Vc=Vco(1+at)e-at
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If b2<0 we will have oscillating solution. Omitting the details (see
Lab write-up) we have the equations for charge and current as:
,
cos sin sin( )
sin
;
at at
at
a aq(t) q e bt bt q e bt
b b
a bi(t) q e bt
b
R R Ra b f
L L LC LC L
2
0 0 2
2 2
0
2 2
1
1 1 1
2 2 2 2
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Log decrement can be defined as 𝜹 = 𝒍𝒏𝒒(𝒕𝒎𝒂𝒙
𝒒(𝒕𝒎𝒂𝒙+𝑻𝟏=
𝒍𝒏𝒆−𝒂𝒕𝒎𝒂𝒙
𝒆−𝒂𝒕𝒎𝒂𝒙+𝑻𝟏= 𝒂𝑻𝟏, where T1=1/f1
Quality factor can be
defined as 𝑸 = 𝟐𝝅𝑬
∆𝑬,
For RLC 𝑸 =𝝎𝟏𝑳
𝑹=
𝝅
𝜹
From this plot d≈0.67
Q≈4.7 -1 0 1 2 3 4 5 6 7 8 9 10
-6
-3
0
3
6
3.529
1.809
0.929620.47494
VC (
q/C
) (V
)
time (ms)
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1. Pick peaks
2. Envelope
3. Exponential term
4. Nonlinear fitting
-1 0 1 2 3 4 5 6 7 8 9 10-6
-3
0
3
6
0.00115
0.0023
0.003460.00463
VC (
q/C
) (V
)
time (ms)
f=862Hz
-1 0 1 2 3 4 5 6 7 8 9 10-6
-3
0
3
6
VC (
q/C
) (V
)
time (ms)
f=862Hz
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Time domain trace
Points found using “Find peaks”
Envelope curve
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Fitting the “envelope data” to
exponential decay function
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Zero crossing points
0ffset Manual evaluation of the
period of the oscillations
in( )at
q(t) Ae s t offset
Limited accuracy
Results can be
effected by DC offset
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in( )at
q(t) Ae s t
Limited
Use Origin standard
function
Category: Waveform
Function: SineDamp
Fitting function ; y0,A,t0 xc, w – fitting parameters
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in( )at
q(t) Ae s t
Limited
Data plot + fitting curve
Residuals - criteria
of quality of fitting
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in( )at
q(t) Ae s t
Final results
Rf
T LC L
2 2
21 1 1
2 2
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R2
L
C VC V(t)
100 1000 100000
2
4
6
8
10
12
14 1904.83204
UC
f (Hz)
f=1500Hz
𝑸 =𝒇
𝜟𝒇=
𝟏𝟗𝟎𝟒
𝟏𝟓𝟎𝟎= 𝟏. 𝟐𝟔
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One important parameter for any electrical measuring equipment
(DMM, scope, amplifier) is its input resistance
Rin≠∞;
Im=I0
RRin
1+ RRin
;
𝑉=I0𝑅
1+R
Rin
Real situation:
For correct voltage measurements
it necessary to have Rin>>R
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The same about current measurement, but for ampere meter the
requirement for Rin is much different.
𝑰 =𝑽
𝑹 + 𝑹𝒊𝒏
I≈I0 if Rin →0
Real situation:
For correct current measurements
it necessary to have Rin<<R
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Why?
FET The current applied
by ohmmeter to the
input of amplifier or
DMM can destroy
the electronics
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Correct idea Ignoring battery output
resistance Vb we can
measure if Rs=0. Next
we have to vary Rs until
DMM reading becomes
equal 0.5 Vb
𝑽 = 𝟎. 𝟓𝑽𝒃 =𝑹𝒊𝒏
𝑹𝒔 + 𝑹𝒊𝒏𝑽𝒃
𝑹𝒔 = 𝑹𝒊𝒏
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+
-
V Rout RL VL
Voltage on the load:
VL = 𝑽 ×𝑹𝑳
𝑹𝑳+𝑹𝒐𝒖𝒕
VL≈V if Rout<<RL
Power on the load:
𝑷 =𝑽𝟐
𝑳
𝑹𝑳=
𝑽𝟐𝑹𝑳
𝑹𝑳+𝑹𝒐𝒖𝒕𝟐
P=Pmax= 𝐕𝟐
𝟒𝐑𝐨𝐮𝐭
if RL=Rout Voltage source circuit
diagram symbol
Equivalent circuit of voltage source
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V Rout
RL
IL Ideal current source – current
in the load should not depend
on the load resistance.
IL = 𝐕
𝐑𝐋+𝐑𝐨𝐮𝐭=
𝐈𝟎
𝟏+𝐑𝐋
𝐑𝐨𝐮𝐭
,
where
𝐈𝟎 = 𝐕
𝐑𝐨𝐮𝐭
So, for ideal current
source Rout →∞ Current source circuit
diagram symbol
Equivalent circuit of current source
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Varying the resistance of the parallel resistor Rp we
have to find the value of Rp corresponding the
reading of current equal I0/2. In this case
Rin=Rp
Current from some current source
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1/14/2013 Physics 403 30
If the sensor is mounted in cryostat the
overall leads resistance could reach a
couple of ohms. This will In case of RTD100
the resistance at 0oC is 100W give an error
of a couple percent!
Most of DMM’s have four probe
option for resistance
measurements.
I1=Io*RT/(r2+r3+Rin) ~0
Rin~1010W
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Open template
button
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Very short and simple manual
which covers only main general
operations with Origin.
Document located on server
(\\Phyaplportal\PHYCS401\Comm
on\Origin manuals) and there is a
link from P401 WEB page
There are (\\Phyaplportal\PHYCS401\Common\Origin manuals) also manuals
from OriginLab.
Video Tutorials at the
site of the company
http://www.originlab.com/index.aspx?go=SUPPORT/VideoTutorials
Do not forget about
Origin Help