Plumber s LCR Analogy Valve V 1 V 2 P 1 P 2 The plumber s analogy of an LC circuit is a rubber diaphragm that has been stretched by pressure on the top (P 1 ) side. When the valve starts the flow, the diaphragm forces water past the flywheel, which begins to spin. After the diaphragm has become flat, the momentum of the flywheel forces the diaphragm to be stretched in the other direction, and the cycle repeats. V 3 Rubber Diaphragm Valve = Switch P 3 Flywheel Rubber Diaphragm = Capacitor Flywheel = nductor Pressure = Potential Water Flow = Current
0 = wv 0 L é( w 2 0 - w 2 ) 2 + 4b 2 w 2 ë What is the best variable to plot for the LRC lab? ADMTTANCE 1/2 ù û Amplitude of / w - tanf = 1 wc - wl w w f 2 2 o w f w 0 1 LC The undriven system will oscillate at Errata: in general, the driven 2 2 resonance frequency w f wo (see PS #2b to solve for it, it will be close)
C circuit = V 0 e iwt C V=q/C -V = 0 Look for = 0 e iwt = 0 e if e iwt q C = C = = iwv 0 e iwt = wcv 0 e i p 2 eiwt 0 = wcv 0 ;f = p 2 dependent on w Purely capacitive circuit: Current leads driving voltage (CE) Magnitude depends on frequency
C circuit C V=q/C = V 0 e iwt = wcv 0 e i p 2 eiwt input output Phasor diagram CE = 1 wc e-i p 2 mpedance, Z = wce i p 2 Admittance, 1/Z
The AC current through a capacitor leads the capacitor voltage by p/2 rad or 90 0. The phasors for V C and C are perpendicular, with the C phasor ahead of the V C phasor. Capacitor AC Circuits 5
L circuit + - V L =Ld/dt L V = V 0 e iωt V L = 0 Look for Lɺ = iω L = V 0 e iωt = 0 e iωt = 0 e iφ e iωt = V 0 iω L eiωt = V 0 ω L e i π 2 eiωt 0 = V 0 ω L ;φ = π 2 dependent on ω Purely inductive circuit: Current lags driving voltage (EL) Magnitude depends on frequency
L circuit + - L V = V 0 = V 0 e iωt π ωl e i 2 eiωt input output Phasor diagram EL = ωle i π 2 mpedance, Z = 1 π ωl e i 2 Admittance, 1/Z
R circuit s there a phase shift? R V=R = V 0 e iωt Look for = 0 e iωt = 0 e iφ e iωt V = 0 = R = V 0e iωt R 0 = V 0 R ;φ = 0 independent of ω Purely resistive circuit: Current in phase with driving voltage at all frequencies Magnitude indep. of frequency
R circuit = V 0 e iωt drive V=R R = V 0 R ei0 e iωt response Phasor diagram = R = 1 R mpedance, Z Admittance, 1/Z
Analyzing an RC Circuit Draw the current vector at some arbitrary angle. All elements of the circuit will have this current. Draw the resistor voltage V R in phase with the current. Draw the capacitor voltage V C 90 0 behind the current. Make sure all phasor lengths scale properly. Draw the emf E 0 as the vector sum of V R and V C. The angle of this phasor is ωt, where the timedependent emf is E 0 cos ωt. The phasors V R and V C form the sides of a right triangle, with E 0 as the hypotenuse. Therefore, V 0 2 = V R2 +V C2.
Analyzing an LRC Circuit Draw the current vector at some arbitrary angle. All elements of the circuit will have this current. Draw the resistor voltage V R in phase with the current. Draw the inductor and capacitor voltages V L and V C 90 0 before and behind the current, respectively. Draw the emf E 0 as the vector sum of V R and V L -V C. The angle of this phasor is ωt, where the time-dependent emf is E 0 cos ωt. The phasors V R and V L -V C form the sides of a right triangle, with E 0 as the hypotenuse. Therefore, E 0 2 = V R2 +(V L -V C ) 2.
The Series RLC Circuit The figure shows a resistor, inductor, and capacitor connected in series. The same current i passes through all of the elements in the loop. From Kirchhoff s loop law, = V R + V L + V C. Because of the capacitive and inductive elements in the circuit, the current i will not in general be in phase with E, so we will have i= cos(ωt-φ) where φis the phase angle between current and voltage. f V L >V C then the current i will lag Eand φ>0. E0 = Vo 2 2 2 2 2 2 E0 = VR + ( VL VC ) = R + ( Z L Z C ) V0 V0 o( ω) = = R + ( Z Z ) R + ( ωl 1/ ωc) 2 2 2 2 L C
mpedance and Phase Angle We can define the TOTAL circuit impedancezof the circuit as: = / Z Z = R + ( ωl 1/ ωc) 2 2 1 ωl 1/ ωc φ = tan R
Resonance = V R + ( ωl 1/ ωc) 0 2 2 The current will be a maximum when ωl=1/ωc. This defines the resonant frequency of the system ω 0 : ω = 0 1 LC = R V 2 2 0 0 ω + ( Lω ) 1 ω 2 2 9
You should be able to: Calculate & plot the magnitude and phase of 1/Z Convert between the mag/phase and Re/m forms Draw phasor diagrams of,, 1/Z (or Z) Express 1/Z (or Z) in terms of R, L, C or ω 0, β 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