Physics 401, Spring 2015....If b2
Transcript of Physics 401, Spring 2015....If b2
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Physics 401, Spring 2015.Eugene V. Colla
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• Transients. Definition.
• Transients in RLC
• Resonance in RLC
• Data analysis. Origin. Fitting.
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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
Vscope
Resistance R [Ohm]
Capacitance C [mF] (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
2
d d q(t)L q(t) +R q(t) + = 0
dt dt 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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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
9
b2>0
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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
2 2 40
Lb R
C
Critically-damped
conditions for our
network2
Ra
Land
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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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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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b2=0
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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)14
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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
0ffsetManual 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
VCV(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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Open template
button
\\engr-file-03\phyinst\APL Courses\PHYCS401\Common\Origin templates
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Very short and simple manual which covers
only main general operations with Origin.
Document located on server and there is a
link from P401 WEB page
There are 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
\\engr-file-03\phyinst\APL Courses\PHYCS401\Common\Origin manuals
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