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Lecture 5: Physics 2CL 20 August 2007 • Experiment 4: Filters – Band-Pass – High-Pass • Rejection of Data/Theory – Chauvenet Criterion − χ2 Test • Quiz 4 Summer II 2007 Extra Credit: Nikola Tesla! 5 × 1015 % inflation! Experiment 3: Filters • Band-Pass – passes frequencies within a specific range and attenuates those outside of that range • High-Pass – passes frequencies as high as a specific threshold frequency and attenuates any signal of lesser frequency • Band-Pass = Low-Pass + High- Pass Experiment 3: Resonance Definition: Tendency of system to oscillate with a maximum amplitude at a specific frequency Other Examples: Acoustical, Mechanical (e.g. mass on a spring, pendulum), Tidal, Orbital Electrical Resonance: Occurs in an electric circuit when impedance between the input and the output is minimum Impedance - Resistance in AC (with phase) Experiment 3: Electrical Resonance Time-dependent signal V (t ) = Vo sin(ωt ) 1 Resistor V = IR IR = Vo sin(ωt ) ⇒ I = Vo sin(ωt ) R dI VoC Inductor V = − L dI − L = Vo sin(ωt ) ⇒ I = cos(ωt ) dt dt L Voltage leads current by π/2 Q Capacitor V = = ∫ Idt Idt C C ∫ C = Vo sin(ωt ) ⇒ I = VoC cos(ωt ) Current leads voltage by π/2 Experiment 3: Impedance Re-visited V = Vo eiωt e iωt = cos(ωt ) + i sin(ωt ) VR VR = I R R ⇒ I R = R dVC iωt IC = VC 0iωe = ⇒ I C = iCωVC dt C 1 iωt i ∫ VL dt = VL0 iω e = LI L ⇒ I L = − Lω VL Experiment 3: Electrical Resonance • Impedance – Series: Z Σ = Z1 + Z 2 ZR = R – Parallel: 1 Z L = i ωL 1 1 = + ZΣ Z1 Z 2 ZC = 1 =− i iCω Cω 1 − iCω − iCω −i +++ • = = iCω − iCω − (i ) (Cω ) 2 2 Cω • Maximize parallel impedance/Minimize series impedance • In RLC, electric energy oscillates from magnetic field (L) to electrical field (C) Experiment 3: Resonance (Frequency Domain) VR = RI m sin(ω t − ϕ ) Loop rule: 1 VC = − I m cos(ω t − ϕ ) V = VR + VC + VL ωC all have different VL = ω LI m cos(ω t − ϕ ) phases… V 2 = I m R 2 + (I m X L − I m X C )2 2 X L ≡ ωL V Imω L Im = 1 Z XC ≡ Z ≡ R2 + (X L − X C ) 2 ωC φ V φ 2 φ Impedance: Z = R 2 + ωL − 1 Im ωC ωC Im R Courtesy of Professor Basov Experiment 3: Resonance (Frequency Domain) Resistance of Inductor V out Z (ω ) G= = Vin R in Plot Gain vs. Frequency Experiment 3: Resonance vertical horizontal trigger Ext input Ch 1 Ch2 ext scale scale coupling V/div Sec/div DC AC LF rej HF rej Ch1 Ch2 frequency Bandwidth: difference between frequencies at which amplitude is 1 2 ωo ∆ω = Q Experiment 3: Resonance (Band-Pass) • Plot Phase vs. Frequency − 1 Im • Fit to expression ϕ = tan Re −1 x 2 ϕ = tan ax 1 − − bx c Experiment 3: High Pass • Plot Gain vs. Frequency • Fit the data to an appropriate form of Z fb G= Z = x + iy Z in Z = x2 + y2 Rejection of Data: Chauvenet’s Criterion suspicious x − x suspect = tσ x − x suspect = 1 .6 = 2σ P ( X − 2σ > x ∩ X + 2σ < x ) = 1 − P ( X − 2σ < x < X + 2σ ) P ( X − 2σ > x ∩ X + 2σ < x ) = 1 − 0 .95 n = N × P ( X − 2σ > x ∩ X + 2σ < x ) 1 n< suspicious value can be rejected 2 n = 0.3 1.8 is improbable and rejectable +++ The probability that a measurement If the method/technique is suspicious, will be as deviant as the suspect value must data are disposable without comparison be scaled by the total number of measurements to other data. in order to determine the expected fraction of such deviants in the relevant sample size. χ2 ∑ (y − f (x j )) N Test j 2 j =1 χ = 2 σ 2 y 30 20 y(x) 10 How good is the agreement between theory and data? 0 0 5 10 15 20 25 x Courtesy of Professor Basov χ2 Test (continued - 2) ∑ (y − f (x j )) N 2 j Nσ 2 ≅ =N j =1 y χ = 2 σ 2 y σ 2 y ~2 = χ ≅ 1 2 30 χ d # of degrees of freedom 20 d=N-c y(x) 10 # of data points # of parameters 0 computed from data 0 5 10 15 20 25 x (# of constraints) Courtesy of Professor Basov χ2 TEST for FIT 1 − x2 d −1 χ2 distribution: H d , χ 2 ~ ~ = x e 2 const ~ ~ Prob( χ 2 ≥ χ 02 ) Hd,χ2 GX,σ tσ 12 20 28 36 44 52 60 0.0 1.0 2.0 3.0 4.0 ~ χ 2 0 Prob outside tσ from data Courtesy of Professor Basov (y − f (x ) ) Probability of χ2 2 n χ =∑ 2 i i i =1 σ2 y Quantitative measure of agreement between χ 2 χ = %2 observed data and their expected distribution nd.o.f. Table D Probability of obtaining a value of χ2 greater or equal to χ02 , assuming the measurements are governed by the expected distribution ~ ~ Probability that for d degrees of freedom reduced χ2 is as Prob d ( χ ≥ χ )2 2 o large as the reduced observed χ2 disagreement is “significant” if ProbN(χ2 = χ02) < 5 % reject the expected disagreement is “highly significant” if ProbN(χ2 = χ02) < 1 % distribution Courtesy of Professor Avi Yagil Example: Application of χ2 Test Die is tossed 600 x Expectation: each face has equivalent likelihood of showing up v 1 2 3 4 5 6 _ Verification of expectation n 91 137 111 87 80 94 by computing the χ2 exp 100 100 100 100 100 100 ∆2 81 1369 121 169 400 36 σ 10 10 10 10 10 10 This term is the squared difference χ2i 0.81 13.7 1.21 1.69 4.0 0.36 between observation and expectation. Total χ2 21.76 In computation of χ2 the ∆2 term is divided by ndof 5 expectation. σ is square root of expectation reduced χ2 4.35 (Ey = σy2) Adapted from presentation by Professor Avi Yagil Application of χ2 Test: Usage of Table D Just calculated: Total χ2 21.77 ndof 5 Reduced χ2 4.35 Die is loaded at 99.9% Confidence Level Courtesy of Professor Avi Yagil

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posted: | 6/1/2010 |

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