Review: Inductance. Oscillating Currents. Oscillations (cont d) LC Circuit Oscillations

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Lionel Collins
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Define the proportionality constant to be the inductance : di SI unit of inductance is the henry (H).

1 Oscillating urrents h.30: Induced E Fields: Faraday s aw h.30: ircuits h.3: Oscillations and A ircuits eview: Inductance If the current through a coil of wire changes, there is an induced ef proportional to the rate of change of the current. Define the proportionality constant to be the inductance : di SI unit of inductance is the henry (H). ircuit Oscillations Suppose we try to discharge a capacitor, using an inductor instead of a resistor: At tie t0 the capacitor has axiu charge and the current is zero. ater, current is increasing and capacitor s charge is decreasing Oscillations (cont d) What happens when q0? Does I0 also? No, because inductor does not allow sudden changes. In fact, q 0 eans i axiu! So now, charge starts to build up on again, but in the opposite direction! Textbook Figure 3- Textbook Figure 3- Energy is oving back and forth between, U UB i U UE q /

2 Mechanical Analogy ooks like SHM (h. 5) Mass on spring. Variable q is like x, distortion of spring. Then idq/, like vdx/, velocity of ass. By analogy with SHM, we can guess that i q Q cos( ω t) dq ωq sin( ω t) ook at Guessed Solution dq q Q cos( ω t) i ωq sin( ω t) q i Matheatical description of oscillations Note essential terinology: aplitude, phase, frequency, period, angular frequency. You MUST know what these words ean! If necessary review hapters 0, 5. Get Equation by oop ule If we go with the current as shown, the loop rule gives: q di Now replace i by dq/ to get: 0 d q q Guess Satisfies Equation! Start with q Q cos( ω t) d q dq q so that ω Q sin( ω t) And taking one ore derivative gives us d q ω Q cos( ω t) ω q So the solution is correct, provided our angular frequency satisfies ω ircuit Exaple Given an inductor with 8.0 H 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 axiu current in the inductor? (c) What is the capacitor s charge at t 30 µs?

3 Exaple: Part (a) 8 H, nf, q(0) 50 n, i(0) 0 (a) What is the frequency of the oscillations? ω rad s 5 ω.5 0 rad / s 4 f khz π 6.8 rad / cycle Exaple: Part (b) ( 8 H, nf, q(0) 50 n, i(0) 0 ) (b) What is the axiu current? 9 Q (50 0 ) E.0 0 But E I so I E 7 7 E So I. 5 A J Exaple: Part (c) ( 8 H, nf, q(0) 50 n, i(0) 0 ) (c) What is the charge at t 30 µs? q( t) Q cos( t) ω (50n)cos(.5 0 (50n)cos(7.5 rad) (50n)cos(430 ) n ) A Voltage Sources For an A circuit, we need an alternating ef, or A power supply. This is characterized by its aplitude and its frequency. aplitude angular frequency Notation for oscillating functions Note that the textbook uses lower-case letters for oscillating tie-dependent voltages and currents, with upper-case letters for the corresponding aplitudes. v V i I A urrents and Voltages Oh s aw gives: sin ωt v V i v So the A current is: / ( So the aplitudes are related by: i I / ) V I

4 A Voltage-urrent elations First apply an alternating ef to a resistor, a capacitor, and an inductor separately, before dealing with the all at once. For,, define reactance X analogous to resistance for resistor. Measured in ohs. V I V I X V I X Suary for,, Separately { { { V I v and i are in phase V I X with X i leads v by 90. V I EI the IE an X with X v leads i by 90. ω ω Q.3- Which of the following is true about the phase relation between the current and the voltage for an inductor? Q.3- Which of the following is true about the phase relation between the current and the voltage for an inductor? eeber EI the IE an! () 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 80º. Inductor: voltage leads current. () 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 80º. Q.3- Proof: What is the phase relation between the current and the voltage for an inductor? et i I sin( ωt) Drop is v i sin( ωt) di Iω cos( ωt) Q.3- An inductor carries a current with aplitude I at angular frequency ω. What is the aplitude V of the voltage across this inductor? I 3.0 A ω 00 rad / s.03 H v cos( ωt) v hits peak before i () V 0.5 V (4) V V () V (5) V.0 V 8 V (3) V (6) V 6.0 V 600 V

Q.3- The reactance is: X ω 00.03 6. 0 Ω Thus the voltage aplitude is: V I X 3.0 A 6.0 Ω 8 V I 3.0 A ω 00 rad / s.

5 Q.3- The reactance is: X ω Ω Thus the voltage aplitude is: V I X 3.0 A 6.0 Ω 8 V I 3.0 A ω 00 rad / s.03 H The A Oh s aw is: Obviously, for a resistor, for a capacitor Z Ipedance IZ where Z is called the ipedance. X Z and for an inductor Z X () V (4) V 0.5 V V () V (5) V.0 V 8 V (3) V (6) V 6.0 V 600 V But if you have a cobination of circuit eleents, Z is ore coplicated. Ipedance and Phase Angle General proble: if we are given ( t) sinω t can we find i(t)? We can always write By definition of ipedance i( t) I sin t Given and ω, find Z and φ. ( ω φ ) I / Z?? Series ircuit. The currents are all equal.. The voltage drops add up to the applied ef as a function of tie: v + v + v BUT: Because of the phase differences, the aplitudes do not add: So the ipedance is not just a su: V + V + V Z + X + X esults for series circuits ( t) sinω t 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 Series ircuit Exaple Given 50 H, 60 µf, 0V, f60hz, and I4.0A. (a) What is the ipedance? (b) What is the resistance? Solution to (a) is easy: 0 Z 30Ω I 4

6 Exaple (part b) (a) Z 30Ω 50 H, 60 µf, 0V, f60hz, I4.0A (b) What is the resistance? ω π f rad / s 3 X ω Ω X 44. Ω ω Ω X X Z ( X X ) Ω Oscillating urrents h.30: Induced E Fields: Faraday s aw h.30: ircuits h.3: Oscillations and A ircuits hapter 30 Hoework for Today: Questions, 3, 7 Probles 3, 5, 9, 44 hapter 3 Hoework for Toorrow: Questions 3, 4, 7 Probles 5, 9, 39 WileyPlus chapters 30, 3.

Chapter 28: Alternating Current

Chapter 28: Alternating Current hapter 8: Alternating urrent Phasors and Alternating urrents Alternating current (A current) urrent which varies sinusoidally in tie is called alternating current (A) as opposed to direct current (D).