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I = (1/Z)Vext Z must have LRC circuit amplitude & phase L I C Vext R Imag |I| = |1/Z| |V| f V Real Usually write: Vext Z is IMPEDANCE = IZ(w ) (generalized resistance) But Z is ratio of applied voltage to resulting current, or "in/out" 1 I= Vext Z(w ) 1/Z (also Y) is ADMITTANCE Is a ratio of "out/in" 1/Z is a frequency-dependent, complex quantity that describes the system's response to a driving voltage. It is a "response function". Is there a phase shift? R circuit I V=IR Vext R Vext = V0 e iwt Vext -V = 0 Look for Vext = IR iw t iw t I = I0e if iw t = I0 e e V0 I 0 = ;f = 0 R independent of w V0 e I= R Purely resistive circuit: • Current in phase with driving voltage at all frequencies • Magnitude indep. of frequency R circuit Vext = V0 e I V=IR Vext R iwt V0 i 0 iw t I= e e R drive response Phasor diagram Vext =R I Impedance, Z I 1 = Vext R Admittance, 1/Z C circuit Vext = V0 e I V=q/C Vext I C Look for iwt = I0e if iwt = I0 e e iwt Vext -V = 0 q = Vext C I = Vext = iwV0 eiw t C i p2 iw t I = w CV0 e e p I 0 = wCV0 ; f = 2 dependent on w Purely capacitive circuit: • Current leads driving voltage (ICE) •Magnitude depends on frequency C circuit Vext = V0 e I iwt input V=q/C Vext C Phasor diagram ICE i p2 iw t I = w CV0 e e output Vext 1 -i p2 = e I wC Impedance, Z I i p2 = wCe Vext Admittance, 1/Z (sometimes Y) L circuit VL =LdI/dt I + Vext - L V Look for I = I 0 eiw t = I 0 eif eiw t V0 p I0 = ;f = wL 2 dependent on w Vext = V0 e iwt Vext -VL = 0 LI = Vext iw LI = V0 e iw t V0 iw t V0 -i p2 iw t I= e = e e iw L wL Purely inductive circuit: • Current lags driving voltage (ELI) • Magnitude depends on frequency L circuit Vext = V0 e I + Vext - L V Phasor diagram ELI iwt V0 -i p2 iw t I= e e wL input output Vext i p2 = wLe I Impedance, Z I 1 -i p2 = e Vext wL Admittance, 1/Z LRC circuit L I dI q VL = L ;VR = IR;VC = dt C C R L (inductance), C (capacitance), cause oscillation, R (resistance) causes damping 2 q˙˙ + 2bq˙ + w 0 q = 0 dI q -L - IR - = 0 dt C q Lq + Rq + = 0 C R 1 q+ q+ q=0 L LC I = (1/Z)V LRC circuit L I C Vext R Imag |I| = |1/Z| |V| f V Real q ( t ) = Re éë q0 e ùû iw t iw t Vext = Re éëV0 e ùû i (fq + p /2 ) iw t ù é I(t) = q ( t ) = Re w q0 e e ë û I0 = wV0 L ( é w2 -w 0 ë ) 2 2 1/2 + 4b w ù û 2 -2bw f I = + arctan 2 2 2 w0 - w 2 p 12 I0 = wV0 L ( é w -w ë 2 0 ) 2 2 1/2 + 4b w ù û 2 2 What is the best variable to plot for the LRC lab? ADMITTANCE Amplitude of I/Vapp Frequency, w -> p -2bw f I = + arctan 2 2 2 w0 - w 1 -wL tan f = w C R Phase of I (rel to Vapp) Frequency, w -> Q factor of an underdamped oscillator Damping time or "1/e" time is t = 1/b > 1/w0 (>> 1/w0 if b is very small) How many T0 periods elapse in the damping time? This number (times π) is the Quality factor or Q of the system. t w0 Q=p = T0 2 b large if b is small compared to w0 Max Amplitude Current Amplitude |I0| 1 of max amp 2 Dw w0 Q= Dw Driving Frequency------> Find frequencies where POWER drops to half maximum (current drops to 0.707 of max). These define w. Find resonant frequency, w0 Homework problem to show the two definitions are the same. QUETSION: FM radio stations have broadcast frequencies of approximately 100 MHz. Most radios use a series LRC circuit similar to the one you used in the lab as part of the receiver electronics. Estimate the spacing of the broadcast frequencies of FM stations if typical receivers have a Q of 500 or better. Explain your reasoning, and include a graph. station 1 w station 2 w w0 f0 Qº = Dw Df QUETSION: FM radio stations have broadcast frequencies of approximately 100 MHz. Most radios use a series LRC circuit similar to the one you used in the lab as part of the receiver electronics. Estimate the spacing of the broadcast frequencies of FM stations if typical receivers have a Q of 500 or better. Explain your reasoning, f0 and include a graph. Q= station 1 w Df = 500 station 2 f0 100 MHz Df = = = 0.2 MHz Q 500 Therefore, stations 99.3 and 99.5 FM are allowed, but 99.3 and 99.4 FM are not! They have cross-talk! You should be able to: • Calculate & plot the magnitude and phase of 1/Z • Convert between the mag/phase and Re/Im forms • Draw phasor diagrams of Vext, I, 1/Z (or Z) • Express 1/Z (or Z) in terms of R, L, C or w0, b You should be able to discuss: • The amplitude of the response and resonance • The phase of the response • The nature of the behavior at all frequencies • The transfer of the series LCR circuit analysis to analogous oscillatory systems