Physics 102: Lecture 13 RLC circuits & Resonance

Transcription

1 Physics 102: Lecture 13 L circuits & esonance L Physics 102: Lecture 13, Slide 1

2 I = I max sin(2pft) V = I max sin(2pft) V in phase with I eview: A ircuit L V = I max X sin(2pft p/2) V lags I IE I V t V L = I max X L sin(2pft+p/2) V L leads I ELI V V L Physics 102: Lecture 13, Slide 2

3 L Peak & MS values in A ircuits (EVIEW) When asking about MS or Maximum values relatively simple expressions V,max = I max V,max = I max X V L,max = I max X L X = 1 2πf = 1 ω X L = 2πfL = ωl Physics 102: Lecture 13, Slide 3

4 Kirchhoff: generator voltage Instantaneous voltage across generator (V gen ) must equal sum of voltage across all of the elements at all times: L I V V gen (t) = V (t) +V (t) +V L (t) t V gen,max V L,max +V,max +V,max What is V gen,max? V V L Define impedance Z: V gen,max I max Z Like: V=I Z = 2 + X L X 2 Physics 102: Lecture 13, Slide 4

5 One last ingredient: is the generator voltage leading or lagging the current? I = I max sin(2pft) V gen = I max Z sin(2pft + f) Phase angle: tan(φ) = X L X V gen,max =I max Z I max I V gen t φ < 0, voltage is lagging Physics 102: Lecture 13, Slide 5

6 Power in A circuits The voltage generator supplies power. Only resistor dissipates power. apacitor and Inductor store and release energy. P(t) = I(t)V (t) oscillates so sometimes power loss is large, sometimes small. Average power dissipated by resistor: Physics 102: Lecture 13, Slide 6 P = ½ I max V,max = ½ I max V gen,max cos(f) = I rms V gen,rms cos(f) If there is only a resistor, φ = 0

7 Problem Time! An A circuit with = 2 W, = 15 mf, and L = 30 mh is driven by a generator with voltage V(t)=2.5 sin(8pt) Volts. alculate the maximum current in the circuit, and the phase angle. L I max = V gen,max /Z Z = 2 + X L X 2 Z = I max = = Amps tan(φ) = X L X Physics 102: Lecture 13, Slide 7 = φ =

8 AT: Power dissipation Which one of these circuits dissipates the most power? V gen,max =10 V V gen,max =100 V e=1 V L (A) (B) () Physics 102: Lecture 13, Slide 8

9 Kirchhoff: generator voltage V gen L Write down Kirchhoff s Loop Equation: V gen (t) = V L (t) + V (t) + V (t) at every instant of time I V If the frequency is just right t V gen (t) = V (t) V L esonance! V Physics 102: Lecture 13, Slide 9

10 Examples of mechanical resonance Pendulum: ω 0 = g/l Mass on a spring: ω 0 = k/m ommon features: Energy converts between two forms (kinetic & potential) Weak input leads to a big response! Physics 102: Lecture 13, Slide 10

11 esonance! Tacoma Narrows Bridge Millennium Bridge Physics 102: Lecture 13, Slide 11

12 Electric energy Magnetic energy L circuit: Electrical resonance Electric energy Magnetic energy Physics 102: Lecture 13, Slide 12

13 Impedance Impedance Impedance Impedance esonance is independent of f X L increases with f X L = 2pfL X decreases with f X = 1/(2pf) Z: X L and X subtract Z = 2 + X L X 2 esonance: X L = X f 0 = esonance in A ircuits Z is minimum at resonance frequency! X L f 0 2π 1 L Z X frequency Physics 102: Lecture 13, Slide 13

14 esonance is independent of f X L increases with f X L = 2pfL X decreases with f X = 1/(2pf) Z: X L and X subtract esonance in A ircuits urrent is maximum at resonance frequency! Z I max = V gen,max /Z urrent Z = 2 + X L X 2 f 0 frequency esonance: X L = X f 0 = 2π 1 L Physics 102: Lecture 13, Slide 14

15 L AT: esonance The A circuit to the right is being driven at its resonance frequency. ompare the maximum voltage across the capacitor with the maximum voltage across the inductor. 1) V,max > V L,max 2) V,max = V L,max 3) V,max < V L,max 4) Depends on Physics 102: Lecture 13, Slide 15

16 heckpoint 14.1 L As the frequency of the circuit is either raised above or lowered below the resonant frequency, the impedance of the circuit: Always increases Only increases for lowering the frequency Only increases for raising the frequency Z Physics 102: Lecture 13, Slide 16

17 What is it good for? L urrent through circuit depends on frequency (maximum at resonance frequency f o ) adio receiver Stereo equalizer NM/MI Physics 102: Lecture 13, Slide 17 Picks out radio station freq f o Picks out music freq f o Picks out signal from protons at f o

18 esonance in adios An A circuit with = 2 W, L = 0.30 mh and variable capacitance is connected to an antenna to receive radio signals at the resonance frequency. If you want to listen to music broadcasted at 96.1 MHz, what value of should be used? f 0 = 2π 1 L L = = = F Physics 102: Lecture 13, Slide 18

19 AT: adios Your radio is tuned to FM 96.1 MHz and want to change it to FM MHz, which of the following will work. 1. Increase apacitance 2. Decrease apacitance 3. Neither, you need to change Physics 102: Lecture 13, Slide 19

20 Another use for L circuits: traffic sensors Physics 102: Lecture 13, Slide 20

21 A Summary esistors: V,max =I max In phase with I apacitors: V,max =I max X X c = 1/(2pf ) Lags I Inductors: V L,max =I max X L X L = 2pf L Leads I Generator: V gen,max =I max Z Z = 2 +(X L -X ) 2 an lead or lag I tan(f) = (X L -X )/ Power is only dissipated in resistor: P = ½I max V gen,max cos(f) Physics 102: Lecture 13, Slide 21