Transients And Oscillations In RLC Circuits

Transients and Oscillations inRLC CircuitsProfessor Jeff FilippiniPhysics 401Spring 2020

Goals of this Lab Concepts: Oscillators in the time domain Transients Resonances Damping regimes Implementation: RLC electrical circuits Data analysis using OriginPhysics 4012

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, governedby constant-coefficient linear homogeneous differential equations The transient response of such a system is its (“short-lived”) responseto a change in input from an equilibrium state Commonly discuss impulse response and step responseTransientStep responseSystem understudyInput x(t)Physics 401Output y(t)3

From Harmonic Oscillator to RLC Circuit A good reference LTI system is a driven damped harmonic oscillator𝑑# 𝑥𝑑𝑥DrivingInertia 𝑚 𝑐 𝑘𝑥 F t force#𝑑𝑡𝑑𝑡Damping force Restoring force A useful implementation of this is an RLC circuitR𝑉- 𝑉. 𝑉/ V(t)#𝑑𝑞𝑑𝑞 1𝑑𝐼𝑉- 𝑅𝐼L # 𝑅 𝑞 𝑉 t𝑉 𝐿.L𝑑𝑡𝑑𝑡 𝐶𝑑𝑡V(t)scope𝑞(𝑡) C𝑉/ 𝐶Where q(t) is the charge on the capacitor The scope measures 𝑉/ 𝑡 Physics 40148(9)/

RLCs in the 401 Lab Voltage Resistance Inductance CapacitanceVRLCV (Volt)Ω (Ohm)mH (10-3 Henry)μF (10-6 Farad)RLscopeV(t)CPhysics 401𝑑# 𝑞𝑑𝑞 1L # 𝑅 𝑞 𝑉 t𝑑𝑡𝑑𝑡 𝐶5

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:𝑅1#𝑠 𝑠 0𝐿𝐿𝐶 with solutions:𝑅𝑠 2𝐿a𝑅2𝐿#1 𝑎 𝑏𝐿𝐶b1.0V(t)RLC Transients: Three SolutionsV00.50.0-1.0-0.50.00.51.0timeb2 0: Overdampedb2 0: Critically dampedExponential decayb2 0: UnderdampedOscillation / ringing and boundary conditions 𝑞 0 𝐶𝑉G , 𝑖 0 𝑞̇ 0 0Physics 4016

RLC Transients: Over-Damped Solutions b2 0 (𝑹𝟐 𝟒𝑳 𝑪): aperiodic exponential decay Solutions have the form:𝑞 𝑡 𝑒 QR9 𝐴S 𝑒 T9 𝐴# 𝑒 QT9i t 𝑞̇ 𝑡 𝑎𝑒 QR9 𝐴S 𝑒 T9 𝐴# 𝑒 QT9 𝑏𝑒 QR9 𝐴S 𝑒 T9 𝐴# 𝑒 QT9 Applying boundary conditions 𝑞 0 𝐶𝑉G , 𝑖 0 𝑞̇ 0 0𝑞 𝑡 𝑞(0)𝑒 QR9 cosh 𝑏𝑡 sinh 𝑏𝑡𝑞(0)𝑎 Q(RQT)9𝑞(𝑡)1 𝑒RQT 9 S 2𝑏Physics 4017

RLC Transients: Critical Damping b2 0 (𝑹𝟐 𝟒𝑳 𝑪): fastest possible exponential decay Solutions have the form:𝑞 𝑡 𝑒 QR9 𝐴S 𝐴# 𝑡i t 𝑞̇ 𝑡 𝑎𝑒 QR9 𝐴S 𝐴# 𝑡 𝐴# 𝑒 QR9 Applying boundary conditions 𝑞 0 𝐶𝑉G , 𝑖 0 𝑞̇ 0 0𝑞 𝑡 𝑞(0)𝑒 QR9 1 𝑎𝑡𝑖 𝑡 𝑎# 𝑞 0 𝑡𝑒 QR9Physics 4018

Ex: Real Data Analysis for Critical DampingIn this experiment: R 300 ohm C 1 μF L 33.43 mH plus practical reality: Wavetek has 50 ohmoutput resistance Inductor coil has 8.7ohm measured R Loop Rtot 358.7 ohmWaveteksignalgeneratorDecay coefficient-\]\ .b𝑎 #.Physics 401# .d SGfg 5365 𝑠QS SG.# l 9

Ex: Real Data Analysis for Critical DampingFitting function for measured 𝑉/ 𝑞:𝑉/ 𝑽𝒄𝟎 (1 𝒂𝑡)𝑒 Q𝒂9Delay 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 0Physics 40110

RLC Transients: Underdamped Case b2 0 (𝑹𝟐 𝟒𝑳 𝑪): decaying oscillation Solutions (see write-up!):#𝑎𝑞 𝑡 𝑞(0)𝑒 QR9 1 # sin 𝜔𝑡 𝜑𝜔##𝑎 𝜔i t 𝑞(0)𝑒 QR9sin 𝜔𝑡𝜔𝑅𝑎 ; 𝜔 2𝜋𝑓 2𝐿Physics 4011𝑅 𝐿𝐶2𝐿#11

Quantifying DampingGeneral idea: How many oscillation periods (𝑇 1/𝑓) does it take forthe oscillation amplitude to decay “substantially”? Log-decrement can be defined as: More commonly, define Quality Factor𝐸𝜋𝑄 2𝜋 Δ𝐸 𝛿 From this plot, 𝛿 0.67, 𝑄 4.7Physics 401VC (q/C) (V)𝑞(𝑡lRz )𝑒 QR9 } 𝛿 ln ln QR(9 ) 𝑎𝑇 } 𝑞(𝑡lRz time (ms)891210

Analysis Using OriginVC (q/C) (V)60.0011530.00230.003460.004630f 862Hz-3-6-1012345678910Keep in mind: Fitting multi-parameter linear modelsto data is generally pretty robust Fitting non-linear models to data is allabout making good initial guessestime (ms)Practical procedure:1. Identify peaks2. Fit “envelope”3. Perform nonlinear fitVC (q/C) (V)630f 862Hz-3-6-101234567time (ms)8910Physics 40113

Analysis Using Origin: Identify PeaksPhysics 40114

Analysis Using Origin: Fit Decay EnvelopePoints found using “Find Peaks”Time-domain traceEnvelope curvePhysics 40115

Analysis Using Origin: Fit Decay EnvelopePhysics 40116

Analysis Using Origin: Periods and Offsets𝑞 𝑡 𝐴𝑒 QR9 sin 𝜔𝑡 𝜑 𝐾OffsetOffset Manually evaluate period of theoscillations Limited accuracy!Zero-Crossings Answer can be biased by DC offsetPhysics 40117

Analysis Using Origin: Non-Linear Fit𝑞(𝑡) 𝐴𝑒 QR9 sin 𝜔𝑡 𝜑𝑞 𝑡𝑈/ 𝑡 𝐶 Use Origin standard function Category: Waveform Function: SineDamp Fit parameters: y0, A, t0, xc, w From fit we can obtain:SS𝒂 ; 𝑻 2𝑤Offset9‰Physics 401‹18

Analysis Using Origin: Evaluating the Fit𝑞(𝑡) 𝐴𝑒 QR9 sin 𝜔𝑡 𝜑Data plotted fitted curvePhysics 401Residuals (data – fit): metric forquality of fit19

Analysis Using Origin: Interpreting Results𝑞(𝑡) 𝐴𝑒 QR9 sin 𝜔𝑡 𝜑2ææ1öæ 1 ö æ 1 ö æ R ö ö ç ç çç ç -ç èT øè 2 p ø è è LC ø è 2 L ø ø2f22Final ResultsNot the fit itself, but constraints onthe physical model parameters!Physics 40120

Foreshadowing: Resonance in RLC Circuits1904.8320414CV(t)12VCL10UCParallel config:Energy can“slosh” from Cto L and backR28Physics 401Df 1500Hz64Resonance: Amplified response whena system is driven at (or near) one ofits “natural” frequenciesFull-width athalf maximum(FWHM)20𝑓1904𝑄 1.26 𝑓 1500100100010000f (Hz)21

Origin Templates for This Week’s LabOpen Template buttonPhysics 40122

Origin ManualsShort, simple manual covering onlybasic operations with Origin(linked from P401 webpage)Don’t forget about Origin help!Video tutorial library oncompany websitePhysics 40123

From Harmonic Oscillator to RLC Circuit A good reference LTI system is a driven damped harmonic oscillator! "# "%# ’ " "% ( Ft A useful implementation of this is an RLC circuit Physics 401 4 Inertia Damping forceRestoring force Driving force R L C V(t) scope,- ,. , / V(t) L "#4 "%# 5 "4 "% 1 7 4 ,t Where q(t) is the charge .