Transients and Oscillations in RLC Circuits · Driven Systems and Transients •Consider a...
Transcript of Transients and Oscillations in RLC Circuits · Driven Systems and Transients •Consider a...
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Transients and Oscillations in RLC Circuits
Professor Jeff FilippiniPhysics 401Spring 2020
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Goals of this Lab•Concepts: Oscillators in the time domain• Transients•Resonances•Damping regimes
• Implementation: RLC electrical circuits
•Data analysis using OriginPhysics 401 2
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Driven Systems and Transients• Consider a system that takes an input x(t) and yields output y(t)• We’ll focus mostly on (approx.) linear time-invariant (LTI) systems, governed
by constant-coefficient linear homogeneous differential equations
• The transient response of such a system is its (“short-lived”) response to a change in input from an equilibrium state• Commonly discuss impulse response and step response
Physics 401 3
System under
studyInput x(t)
Output y(t)
TransientStep response
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From Harmonic Oscillator to RLC Circuit• A good reference LTI system is a driven damped harmonic oscillator
𝑚𝑑#𝑥𝑑𝑡#
+ 𝑐𝑑𝑥𝑑𝑡+ 𝑘𝑥 = F t
• A useful implementation of this is an RLC circuit
Physics 401 4
Inertia
Damping force Restoring force
Driving force
R
L
C
V(t)scope
𝑉- + 𝑉. + 𝑉/ = V(t)
L𝑑#𝑞𝑑𝑡#
+ 𝑅𝑑𝑞𝑑𝑡+1𝐶𝑞 = 𝑉 t
Where… • q(t) is the charge on the capacitor• The scope measures 𝑉/ 𝑡 = 8(9)
/
𝑉- = 𝑅𝐼 𝑉. = 𝐿𝑑𝐼𝑑𝑡
𝑉/ =𝑞(𝑡)𝐶
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RLCs in the 401 Lab• Voltage V V (Volt)• Resistance R Ω (Ohm)• Inductance L mH (10-3 Henry)• Capacitance C μF (10-6 Farad)
Physics 401 5
R
L
C
V(t)scope
L𝑑#𝑞𝑑𝑡#
+ 𝑅𝑑𝑞𝑑𝑡+1𝐶𝑞 = 𝑉 t
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RLC Transients: Three Solutions• What happens after input voltage drops to zero?• Solutions have the form: 𝑞 𝑡 = 𝐴𝑒>9
• This turns our diff. eq. into a quadratic equation:
𝑠# +𝑅𝐿𝑠 +
1𝐿𝐶
= 0
• … with solutions:
𝑠± = −𝑅2𝐿
±𝑅2𝐿
#
−1𝐿𝐶
≡ −𝑎 ± 𝑏
• … and boundary conditions 𝑞 0 = 𝐶𝑉G, 𝑖 0 = �̇� 0 = 0
Physics 401 6
-1.0 -0.5 0.0 0.5 1.00.0
0.5
1.0
V(t)
time
a b
b2>0: Overdampedb2=0: Critically dampedExponential decay
b2<0: UnderdampedOscillation / ringing
V0
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RLC Transients: Over-Damped Solutions• b2>0 (𝑹𝟐 > ⁄𝟒𝑳
𝑪): aperiodic exponential decay
• Solutions have the form:𝑞 𝑡 = 𝑒QR9 𝐴S𝑒T9 + 𝐴#𝑒QT9
i t = �̇� 𝑡 = −𝑎𝑒QR9 𝐴S𝑒T9 + 𝐴#𝑒QT9 + 𝑏𝑒QR9 𝐴S𝑒T9 − 𝐴#𝑒QT9
• Applying boundary conditions 𝑞 0 = 𝐶𝑉G, 𝑖 0 = �̇� 0 = 0
𝑞 𝑡 = 𝑞(0)𝑒QR9 cosh 𝑏𝑡 + sinh 𝑏𝑡
𝑞(𝑡)RQT 9≫S
𝑞(0)2
1 +𝑎𝑏𝑒Q(RQT)9
Physics 401 7
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RLC Transients: Critical Damping• b2=0 (𝑹𝟐 = ⁄𝟒𝑳
𝑪): fastest possible exponential decay
• Solutions have the form:𝑞 𝑡 = 𝑒QR9 𝐴S + 𝐴#𝑡
i t = �̇� 𝑡 = −𝑎𝑒QR9 𝐴S + 𝐴#𝑡 + 𝐴#𝑒QR9
• Applying boundary conditions 𝑞 0 = 𝐶𝑉G, 𝑖 0 = �̇� 0 = 0
𝑞 𝑡 = 𝑞(0)𝑒QR9 1 + 𝑎𝑡𝑖 𝑡 = −𝑎#𝑞 0 𝑡𝑒QR9
Physics 401 8
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Ex: Real Data Analysis for Critical Damping
Physics 401 9
Waveteksignal
generator
In this experiment:• R = 300 ohm• C = 1 μF• L = 33.43 mH
… plus practical reality:• Wavetek has 50 ohm
output resistance• Inductor coil has 8.7
ohm measured R=> Loop Rtot = 358.7 ohm
Decay coefficient𝑎 = -\]\
#.= ^_`.b
#∗^^.d^×SGfg≈ 5365 𝑠QS ≈ S
G.# l>
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Ex: Real Data Analysis for Critical Damping
Physics 401 10
Fitting function for measured 𝑉/ ∝ 𝑞:𝑉/ = 𝑽𝒄𝟎(1 + 𝒂𝑡)𝑒Q𝒂9
Delay coefficient:• Calculated: a=5385 s-1
• Fitted: a= 5820 s-1 (+8%)
Possible cause for discrepancy?Perhaps slightly overdamped?Calculated b2 = 2.99e7 - 2.90e7 > 0
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RLC Transients: Underdamped Case• b2<0 (𝑹𝟐 < ⁄𝟒𝑳
𝑪): decaying oscillation
• Solutions (see write-up!):
𝑞 𝑡 = 𝑞(0)𝑒QR9 1 −𝑎#
𝜔# sin 𝜔𝑡 + 𝜑
i t = 𝑞(0)𝑒QR9𝑎# − 𝜔#
𝜔sin𝜔𝑡
𝑎 =𝑅2𝐿; 𝜔 = 2𝜋𝑓 =
1𝐿𝐶
−𝑅2𝐿
#
Physics 401 11
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Quantifying Damping
• Log-decrement can be defined as:𝛿 ≡ ln
𝑞(𝑡lRz)𝑞(𝑡lRz + 𝑇)
= ln𝑒QR9|}~
𝑒QR(9|}~��)=𝑎𝑇
• More commonly, define Quality Factor
𝑄 ≡ 2𝜋𝐸Δ𝐸
=𝜋𝛿
• From this plot, 𝛿 ≈ 0.67, 𝑄 ≈ 4.7Physics 401 12
-1 0 1 2 3 4 5 6 7 8 9 10-6
-3
0
3
6
3.529
1.8090.92962
0.47494
V C (q
/C) (
V)
time (ms)
General idea: How many oscillation periods (𝑇 ≡ 1/𝑓) does it take for the oscillation amplitude to decay “substantially”?
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Analysis Using OriginKeep in mind:• Fitting multi-parameter linear models
to data is generally pretty robust• Fitting non-linear models to data is all
about making good initial guesses
Practical procedure:1. Identify peaks2. Fit “envelope”3. Perform nonlinear fit
Physics 401 13
-1 0 1 2 3 4 5 6 7 8 9 10-6
-3
0
3
6
0.00115
0.00230.00346
0.00463
V C (q
/C) (
V)
time (ms)
f=862Hz
-1 0 1 2 3 4 5 6 7 8 9 10-6
-3
0
3
6
V C (q
/C) (
V)
time (ms)
f=862Hz
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Analysis Using Origin: Identify Peaks
Physics 401 14
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Analysis Using Origin: Fit Decay Envelope
Physics 401 15
Time-domain trace
Points found using “Find Peaks”
Envelope curve
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Analysis Using Origin: Fit Decay Envelope
Physics 401 16
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Analysis Using Origin: Periods and Offsets𝑞 𝑡 = 𝐴𝑒QR9 sin 𝜔𝑡 + 𝜑 + 𝐾
• Manually evaluate period of the oscillations
• Limited accuracy!
• Answer can be biased by DC offset
Physics 401 17
Offset
Zero-Crossings
Offset
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Analysis Using Origin: Non-Linear Fit𝑞(𝑡) = 𝐴𝑒QR9 sin 𝜔𝑡 + 𝜑
𝑈/ 𝑡 =𝑞 𝑡𝐶
• Use Origin standard function• Category: Waveform• Function: SineDamp
• Fit parameters: y0, A, t0, xc, w
• From fit we can obtain:𝒂 = S
9�; 𝑻 = S
�= 2𝑤
Physics 401 18
Offset
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Analysis Using Origin: Evaluating the Fit𝑞(𝑡) = 𝐴𝑒QR9 sin 𝜔𝑡 + 𝜑
Physics 401 19
Data plotted + fitted curve Residuals (data – fit): metric for
quality of fit
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Analysis Using Origin: Interpreting Results𝑞(𝑡) = 𝐴𝑒QR9 sin 𝜔𝑡 + 𝜑
Physics 401 20
Final Results Not the fit itself, but constraints on the physical model parameters!
pæ öæ ö æ ö æ ö æ ö= = -ç ÷ç ÷ ç ÷ ç ÷ ç ÷ç ÷è ø è ø è ø è øè ø
RfT LC L
2 2 22 1 1 1
2 2
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100 1000 100000
2
4
6
8
10
12
14 1904.83204
UC
f (Hz)
Df=1500Hz
Foreshadowing: Resonance in RLC Circuits
Physics 401 21
R2
L
CVCV(t)
Parallel config:Energy can “slosh” from C to L and back
Resonance: Amplified response when a system is driven at (or near) one of its “natural” frequencies
𝑄 =𝑓∆𝑓
=19041500
= 1.26
Full-width at half maximum
(FWHM)
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Origin Templates for This Week’s Lab
Physics 401 22
Open Template button
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Origin Manuals
Short, simple manual covering only basic operations with Origin (linked from P401 webpage)
Physics 401 23
Don’t forget about Origin help!
Video tutorial library on company website