Electricity and Magnetism--Driven LCR circuits

1P26-Class 26: Outline Hour 1: Driven Harmonic Motion (RLC) Hour 2: Experiment 11: Driven RLC Circuit 2P26-Last Time: Undriven RLC Circuits 3P26-LC Circuit It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass) 4P26-Damped LC Oscillations Resistor dissipates energy and system rings down over time 5P26-Mass on a Spring: Simple Harmonic Motion` A Second Look 6P26-Mass on a Spring 2 2 2 2 0 dxF kx ma mdt dx m kxdt =− = = + = 0 0() cos( )xt x tωφ= + (1) (2) (3) (4) We solved this: Simple Harmonic Motion What if we now move the wall? Push on the mass? Moves at natural frequency 7P26-Demonstration: Driven Mass on a Spring Off Resonance 8P26-Driven Mass on a Spring () () 2 2 2 2 dxFF t kx ma mdt dx m kx F t dt = − = = + = max () cos( )xt x tω φ= + Now we get: Simple Harmonic Motion F(t) Assume harmonic force: 0() cos( )Ft F tω= Moves at driven frequency 9P26-Resonance Now the amplitude, xmax, depends on how close the drive frequency is to the natural frequency max () cos( )xt x tωφ= + ω ω0 xmax Let’s See… 10P26-Demonstration: Driven Mass on a Spring 11P26-Resonance xmax depends on drive frequency max () cos( )xt x tωφ= + ω ω0 xmax Many systems behave like this: Swings Some cars Musical Instruments … 12P26-Electronic Analog: RLC Circuits 13P26-Analog: RLC Circuit Recall: Inductors are like masses (have inertia) Capacitors are like springs (store/release energy) Batteries supply external force (EMF) Charge on capacitor is like position, Current is like velocity – watch them resonate Now we move to “frequency dependent batteries:” AC Power Supplies/AC Function Generators 14P26-Demonstration: RLC with Light Bulb 15P26-Start at Beginning: AC Circuits 16P26-Alternating-Current Circuit • sinusoidal voltage source 0() sin Vt V tω= 0 2 : angular frequency : voltage amplitude f V ω π= • direct current (dc) – current flows one way (battery) • alternating current (ac) – current oscillates 17P26-AC Circuit: Single Element 0 sin V V V tω= = 0() sin( )It I tωφ= − Questions: 1. What is I0? 2. What is φ? 18P26-AC Circuit: Resistors R RV I R = ( ) 0 0 sin sin 0 R R VVI tR R I t ω ω = = = − 0 0 0 VI R ϕ = = IR and VR are in phase 19P26-AC Circuit: Capacitors C QV C = 0 20 () cos sin( ) C dQIt dt CV t I t π ω ω ω − = = = − 0 0 2 I CVω πϕ = =− 0() sin CQ t CV CV tω= = IC leads VC by π/2 20P26-AC Circuit: Inductors L L dIV L dt = ( ) 0 0 20 () sin cos sin L VIt t dt L V tL I t π ω ωω ω = =− = − ∫ 0 sinL L VdI V tdt L L ω= = 0 0 2 VI Lω πϕ = = IL lags VL by π/2 AC Circuits: SummaryElement I0 Current vs. Voltage Resistance Reactance Impedance Resistor V0R R In Phase R = R Capacitor ωCV0C Leads 1XC ωC = Inductor 0LV ωL Lags XL =ωL Although derived from single element circuits, these relationships hold generally! 21P26-22P26-PRS Question: Leading or Lagging? 23P26-Phasor Diagram Nice way of tracking magnitude & phase: tω( )0() sin Vt V tω= 0V ω Notes: (1) As the phasor (red vector) rotates, the projection (pink vector) oscillates (2) Do both for the current and the voltage 24P26-Demonstration: Phasors 25P26-Phasor Diagram: Resistor 0 0 0 V I R ϕ = = IR and VR are in phase 26P26-Phasor Diagram: Capacitor IC leads VC by π/2 0 0 0 1 2 CV I X I Cω πϕ = = =− 27P26-Phasor Diagram: Inductor IL lags VL by π/2 0 0 0 2 LV IX ILω πϕ = = = 28P26-PRS Questions: Phase 29P26-Put it all together: Driven RLC Circuits 30P26-Question of Phase We had fixed phase of voltage: It’s the same to write: 0 0sin ( ) sin( )VV t It I tω ω φ= = − 0 0sin( ) ( ) sin VV t It I tω φ ω= + = (Just shifting zero of time) 31P26-Driven RLC Series Circuit ( )0 sinS SV V tωϕ= + 0() sin( )It I tω= ( )0 sinR RV V tω= ( )0 2sinL LV V t πω= + ( )0 2sinC CV V t πω −= + Must Solve: S R L CV V V V= + + 0 0 0 0 0 0 0What is (and , , )? R L L C CI V IRV IX V IX = = = What is ? Does the current lead or lag ?sVϕ 32P26-Driven RLC Series Circuit Now Solve: S R L CV V V V= + + I(t) 0I 0RV0LV0CV0SVNow we just need to read the phasor diagram! VS 33P26-Driven RLC Series Circuit 0I 0RV0LV0CV0SV2 2 2 2 0 0 0 0 0 0( ) ( )S R L C L CV V V V I R X X IZ = + − = + − ≡ ϕ 1tan L CX X Rφ − −⎛ ⎞ = ⎜ ⎟⎝ ⎠ 2 2( )L CZ R X X= + −0 0 SVI Z = ( )0 sinS SV V tω= 0() sin( )It I tωϕ= − Impedance 34P26-Plot I, V’s vs. Time 0 1 2 3 0 +φ -π/2 +π/2 VS Time (Periods) 0VC 0VL 0VR 0I( ) ( ) ( ) ( ) ( ) 0 0 0 2 0 2 0 () sin () sin () sin () sin () sin R L L L C S S It I t Vt IR t Vt IX t Vt IX t Vt V t π π ω ω ω ω ω ϕ = = = + = − = + 35P26-PRS Question: Who Dominates? 36P26-RLC Circuits: Resonances 37P26-Resonance 0 0 0 2 2 1; , ( ) L C L C V VI X L XZ CR X X ω ω = = = = + − I0 reaches maximum when L CX X= 0 1 LC ω = At very low frequencies, C dominates (XC>>XL): it fills up and keeps the current low At very high frequencies, L dominates (XL>>XC): the current tries to change but it won’t let it At intermediate frequencies we have resonance 38P26-Resonance 0 0 0 2 2 1; , ( ) L C L C V VI X L XZ CR X X ω ω = = = = + − C-like: φ < 0 I leads 1tan L CX X Rφ − −⎛ ⎞ = ⎜ ⎟⎝ ⎠ 0 1 LCω = L-like: φ > 0 I lags 39P26-Demonstration: RLC with Light Bulb 40P26-PRS Questions: Resonance 41P26-Experiment 11: Driven RLC Circuit 42P26-Experiment 11: How To Part I • Use exp11a.ds • Change frequency, look at I & V. Tryto find resonance – place where I is maximum Part II • Use exp11b.ds • Run the program at each of the listed frequencies to make a plot of I0 vs. ω