Oscillating Currents. Ch.30: Induced E Fields: Faraday s Law Ch.30: RL Circuits Ch.31: Oscillations and AC Circuits

Transcription

1 Oscillating urrents h.30: Induced E Fields: Faraday s aw h.30: ircuits h.31: Oscillations and A ircuits

2 eview: Inductance If the current through a coil of wire changes, there is an induced emf proportional to the rate of change of the current. Define the proportionality constant to be the inductance : ε di dt SI unit of inductance is the henry (H).

3 ircuit Oscillations Suppose we try to discharge a capacitor, using an inductor instead of a resistor: At time t0 the capacitor has maximum charge and the current is zero. ater, current is increasing and capacitor s charge is decreasing

4 Oscillations (cont d) What happens when q0? Does I0 also? No, because inductor does not allow sudden changes. In fact, q 0 means i maximum! So now, charge starts to build up on again, but in the opposite direction!

5 Textbook Figure 31-1 Energy is moving back and forth between, U U B 1 i U U E 1 q /

6 Textbook Figure 31-1

7 Mechanical Analogy ooks like SHM (h. 15) Mass on spring. Variable q is like x, distortion of spring. Then idq/dt, like vdx/dt, velocity of mass. By analogy with SHM, we can guess that q ω Q cos( t) i dq dt ωq sin( ω t)

8 ook at Guessed Solution q Q cos( ω t) i dq dt ωq sin( ω t) q i

9 Mathematical description of oscillations Note essential terminology: amplitude, phase, frequency, period, angular frequency. You MUST know what these words mean! If necessary review hapters 10, 15.

10 Get Equation by oop ule If we go with the current as shown, the loop rule gives: q di 0 dt Now replace i by dq/dt to get: d dt q 1 q

11 Guess Satisfies Equation! Start with q dq ω Q cos( t) so that ω Q sin( t) dt ω d dt And taking one more derivative gives us d q ω Q cos( ω t) ω q dt q So the solution is correct, provided our angular frequency satisfies ω 1 1 q

12 ircuit Example Given an inductor with 8.0 mh and a capacitor with.0 nf, having initial charge q(0) 50 n and initial current i(0) 0. (a) What is the frequency of oscillations (in Hz)? (b) What is the maximum current in the inductor? (c) What is the capacitor s charge at t 30 µs?

13 Example: Part (a) 8 mh, nf, q(0) 50 n, i(0) 0 (a) What is the frequency of the oscillations? ω rad s ω π rad / s f rad / cycle khz

14 Example: Part (b) ( 8 mh, nf, q(0) 50 n, i(0) 0 ) (b) What is the maximum current? E Q ( ) J But E 1 I so I E E So I ma

15 Example: Part (c) ( 8 mh, nf, q(0) 50 n, i(0) 0 ) (c) What is the charge at t 30 µs? q( t) Q cos( ω t) (50n)cos( ) (50n)cos(7.5 rad ) (50n)cos(430 ) n

16 A Voltage Sources For an A circuit, we need an alternating emf, or A power supply. This is characterized by its amplitude and its frequency. ε ε sinωt m amplitude angular frequency

17 Notation for oscillating functions Note that the textbook uses lower-case letters for oscillating time-dependent voltages and currents, with upper-case letters for the corresponding amplitudes. v V sinωt i I sinωt

18 A urrents and Voltages ε ε m sin ωt v V sinωt Ohm s aw gives: i v ε / ( / ) sinωt m So the A current is: i I sinωt So the amplitudes are related by: V I

19 A Voltage-urrent elations First apply an alternating emf to a resistor, a capacitor, and an inductor separately, before dealing with them all at once. For,, define reactance X analogous to resistance for resistor. Measured in ohms. V V I I X V I X

20 Summary for,, Separately { V I EI the IE man v and i are in phase { V I X with X i leads v by ω { V I X with X v leads i by 90. ω

21 Q.31-1 Which of the following is true about the phase relation between the current and the voltage for an inductor? (1) The current is in phase with the voltage. () The current is ahead of the voltage by 90º. (3) The current is behind the voltage by 90º. (4) They are out of phase by 180º.

22 Q.31-1 Which of the following is true about the phase relation between the current and the voltage for an inductor? emember EI the IE man! Inductor: voltage leads current. (1) The current is in phase with the voltage. () The current is ahead of the voltage by 90º. (3) The current is behind the voltage by 90º. (4) They are out of phase by 180º.

23 Q.31-1 What is the phase relation between the current and the voltage for an inductor? Proof: i sin( ωt) et i I sin( ω t) di Drop is v dt I ω cos( ω t) v hits peak before i v cos( ωt)

24 Q.31- An inductor carries a current with amplitude I at angular frequencyω. What is the amplitude V of the voltage across this inductor? I 3.0 A ω 00 rad.03 H / s (1) V (4) V 0.5 V 1V () V (5) V.0 V 18 V (3) V (6) V 6.0 V 600 V

25 Q.31- The reactance is: X ω Ω I 3.0 A ω 00 rad / s Thus the voltage amplitude is:.03 H V I X 3.0 A 6.0Ω 18 V (1) V (4) V 0.5 V 1V () V (5) V.0 V 18 V (3) V (6) V 6.0 V 600 V

26 Impedance The A Ohm s aw is: ε m IZ where Z is called the impedance. Obviously, for a resistor, Z for a capacitor Z X and for an inductor Z X But if you have a combination of circuit elements, Z is more complicated.

27 Impedance and Phase Angle General problem: if we are given ε ε ( t) sinω t m can we find i(t)? We can always write i( t) I sin t ( ω φ) By definition of impedance ε I / m Z Given ε m and ω, find Z and φ.??

28 Series ircuit 1. The currents are all equal.. The voltage drops add up to the applied emf as a function of time: ε v + v + BUT: Because of the phase differences, the amplitudes do not add: So the impedance is not just a sum: ε m v V + V + V Z + X + X

29 ε esults for series circuits ε t) sinω t m ( i( t) I sin t ( ω φ) It turns out that for this particular circuit Z + ( X X ) X X tanφ Z φ X X

30 Series ircuit Example Given 50 mh, 60 µf, ε m 10V, f60hz, and I4.0A. (a) What is the impedance? (b) What is the resistance? Solution to (a) is easy: Z ε m I 4 Ω

31 Example (part b) (a) Z 30Ω 50 mh, 60 µf, ε m 10V, f60hz, I4.0A (b) What is the resistance? ω π f rad / s 3 X ω Ω 1 1 X 44. Ω ω X X Ω Z ( X X ) Ω

32 A ircuits h.30: Faraday s aw h.30: Inductors and ircuits: h.31: A ircuits

33 eview: Inductance If the current through a coil of wire changes, there is an induced emf proportional to the rate of change of the current. Define the proportionality constant to be the inductance : ε di dt SI unit of inductance is the henry (H).

34 eview: ircuits q Q cos( ω t) dq i ωq sin( ω t) dt oop rule gives differential equation: d dt q 1 q Solution if frequency is correct: ω 1 Natural frequency of oscillations!

35 eview: A Voltage Sources For an A circuit, we need an alternating emf, or A power supply. This is characterized by its amplitude and its frequency. ε ε sinωt m amplitude angular frequency

36 eview:,, Separately { V I EI the IE man v and i are in phase { V I X with X i leads v by ω { V I X with X v leads i by 90. ω

37 ε eview: Simple series circuit ε t) sinω t m ( i( t) I sin t ( ω φ) It turns out that for this particular circuit Z + ( X X ) X X tanφ Z φ X X

38 Series ircuit Example Given 50 mh, 60 µf, ε m 10V, f60hz, and I4.0A. (a) What is the impedance? (b) What is the resistance? Solution to (a) is easy: Z ε m I 4 Ω

39 Example (part b) (a) Z 30Ω 50 mh, 60 µf, ε m 10V, f60hz, I4.0A (b) What is the resistance? ω π f rad / s 3 X ω Ω 1 1 X 44. Ω ω X X Ω Z ( X X ) Ω

40 Q.31-3 A certain series circuit is driven by an applied emf with angular frequency 0 radians per second. If the maximum charge on the capacitor is 0.03 coulomb, what is the maximum current in the circuit? (1) 6 A () 1.5 A (3) 0.6 A (4) 0.15 A (5) Not enough information

41 Q.31-3 A certain series circuit is driven by an applied emf with angular frequency 0 radians per second. If the maximum charge on the capacitor is 0.03 coulomb, what is the maximum current in the circuit? q I Q sin ωq dq dt 0.6 ( ωt) i ωq cos( ωt) (1) 6 A () 1.5 A (3) 0.6 A (4) 0.15 A (5) Not enough information A

42 Q.31-4 In a series circuit, find the resistance, given the impedance and phase constant: Z 500 Ω φ 60? (1) 400 Ω () 350 Ω (3) 300 Ω (4) 50 Ω

43 In an circuit: Z 500 Ω φ 60? Q.31-4 Z φ X X Z cosφ cos ( 60 ) sin( 30 ) Z / 50Ω 1 (1) 400 Ω () 350 Ω (3) 300 Ω (4) 50 Ω

44 esonance For a given series circuit, what applied frequency will give the biggest current? ε m I + ( X X ) I is biggest when denominator is smallest, which is when reactances cancel: 1 ω Which gives ω ω 1 Same as natural frequency for oscillations!

45 esonance Plot: I vsω Peak at ω 1/ Becomes sharper as 0

46 Power in A ircuits To know how much power will be consumed in an A circuit we need more than the impedance Z; we also need the phase angle φ.

47 A Power What is the power provided by an A source? Only interested in the time average! Time average power to and is zero. So P ave (from source) P ave (to resistor). P ave v i ( V sin ω t)( I sin ω t) V I sin ωt 1 V I

48 MS Values 1 V v V sin ωt V V MS / ikewise I i MS I / So for a resistor: P ave V MS I MS V I 1 VI

49 Power and Phase Angle Power to resistor P P I ( t ) i I sin ω φ avg sin avg 1 I But we want result in terms of impedance and phase angle: Z cos φ So P avg 1 I Z cos φ Power factor

50 What about more complicated circuits? Method of Phasors Useful mathematical trick for working with oscillating functions having different phase relationships. We ll use it to derive the known result for the series circuit, to show how it works. Good for general case. i I sin ( ω t) I i ength of phasor Vertical component of phasor

51 1. The currents are all equal.. The voltage drops add up to the applied emf: Series ircuit ε v + v + Because of the phase differences, the amplitudes do not add, but in the phasor diagram, this means that the vertical components do add, so we get the right answer if we add the phasors as vectors. v

52 Phasors for Series ircuit ε i v + v + v ε m i i I sin t sin ω ( ω φ) t i I ( ω t φ)

53 Adding the Voltage Phasors ε v + v + v ε m sin ω t ε I

54 Adding the Voltage Phasors ε v + v + v ε m sin ω t ε m V + ( V V ) tanφ V V V

55 ε m Impedance and Phase Angle V + ( V V ) tanφ Divide by the common current amplitude I: V V V ε m IZ, V I, V IX, V IX Z + ( X X ) tanφ X X