# Physics 115. Inductors, Capacitors, and RLC circuits. General Physics II. Session 34

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1 Physics 115 General Physics II Session 34 Inductors, Capacitors, and RLC circuits R. J. Wilkes Home page: 06/05/13 1

2 Lecture Schedule Today 2

3 Announcements Final exam is 2:30 pm, Monday 6/9, here 2 hrs allowed for exam (really: 1 to 1.5 hr), comprehensive, but with extra items on material covered after exam 3 Usual arrangements If you took midterms with section B please do NOT do that for final everyone takes it with our group on Monday Final exam will contain ONLY Ch. 24 topics covered in class I will be away all next week Final exam will be hosted by Dr. Scott Davis If you need to see me, do so this week... Exam scores and grade data will be posted before the end of next week, final grades before Tuesday 6/17 TA Songci Li will have office hour MONDAY 12:30-1:30, B-442 PAB Homework set 9 is due Friday 6/6 11:59pm 3

4 How best to study for final? Announcements Review and work to understand what you did not get right when you did HW problems, quizzes, or mid-term exam questions. Final Exam will not go into tricky details or fine points! Focus on main ideas A few practice questions for ch. 24 will be posted tonight, reviewed Friday in class 4

5 Inductors can make sparks Electromagnet circuit If we quickly interrupt DC current flow through an inductor, the back- EMF may cause a very large voltage (L di/dt) across its terminals. The induced V typically causes an arc (spark) across the switch or broken wire that is breaking the current. Example: large electric motors act like inductors a simple on/off switch would pull a spark when opened Sparks can damage switches or cause fires, so we use special switch arrangements in such circuits 5

6 Make-Before-Break Switches Sliding contacts: d-f is closed before d-e opens Special make-before-break switches are used for inductive circuits: the inductor is shorted across a resistor before the switch actually opens the circuit. R dissipates the current generated by back-emf, and R 1 keeps the EMF source from being shorted out. 6

7 Example: Large Voltage across an Inductor A 1.0 A current passes through a 10 mh inductor coil. What potential difference is induced across the coil if the current drops to zero in 5 µs? di dt ΔI (-10. A) = = = -6 Δ t ( s) A/s di Δ V = L == = dt 5 L (0.010 H)( A/s) 2000 V Big jolt from a small current and inductance! Where does the energy for this come from? 7

8 Example: Inductive reactance A 10 H inductor is connected to a 1000 Hz oscillator with a peak emf of 5.0 V. What is the RMS current in the inductor? X L (1000 Hz) = 2π fl = 2π (1000 s -1 )(10 H) = 6.28 x 10 4 Ω I L( PEAK ) = V PEAK X L = (5.0 V) (6.28 x 10 4 Ω) = 8 x 10 5 A (80 µa) I RMS = I PEAK 2 = 0.707I PEAK = 56 µa Remember: Reactance does not dissipate energy like a resistor: energy is stored in electromagnetic fields 8

9 Capacitors and springs AC voltage and current in reactance are related like position and velocity in a spring+mass system: when one is max the other is zero AC current through a capacitor leads the capacitor voltage by π/2 rad or This is just like the relationship of the position and velocity for a mass + spring, or a pendulum. 9

10 For an LC circuit, suppose we put charge Q on the capacitor initially. Once the switch closes, charge flows from C through L (E field decreases, B field increases) and back again: oscillation of current flow (AC). (if we really had no R, it would go on forever) It can be shown that for this situation, Q varies sinusoidally: Q(t) = Q PEAK cos(ωt) Calculus fact: for this Q(t), I(t) = ωq PEAK sin(ωt) LC circuits - resonance where ω = 1 LC = 2π f f = 1 2π LC ω = rad / s ( ) ( f =1/ s = Hz) Oscillation frequency depends only on L and C This is called the resonant frequency for the LC combination 10

11 Example: An AM Radio tuning circuit You have a 10mH inductor. What capacitor do you need with it to make resonant circuit with a frequency of 920 khz? (This frequency is near the center of the AM radio band.) ω= 2π f = 2 π( s ) = s C 1 1 = = = = ω L ( s ) ( H) F 30 pf Such circuits were used to tune in on desired stations in old radios: now tuners are built into complex microchips (integrated circuits) for radio receivers 11

12 The Series RLC Circuit Now add a resistor in series with the inductor and capacitor. The same current i passes through all of the components. Fact: The C and L reactances create currents with +90 o phase shifts, so their contributions end up 180 o out of phase tending to cancel each other. So the net reactance is X = (X L X C ) I = E 0 R 2 + (X L X C ) 2 = E 0 R 2 + (ωl 1/ ωc) 2 R 2 + (X L X C ) 2 = Z Z = Impedance : resistance and/or reactance E 2 0 =V 2 R + (V L V C ) 2 = " # R 2 + (X L X C ) 2 \$ % I 2 12

13 Impedance and resonance for RLC We define the impedance Z of the circuit as: Z R + ( X X ) 2 2 L C = R + ( ωl 1/ ωc) Then I = E / Z 2 2 (Peak, or RMS here we mean peak values ) If circuit includes no C or L, then Z is just the resistance. If t frequency f is just such that X L =X C, we get resonance: minimum possible Z. Then the circuit looks like only the resistor. Current is maximum. Notice: if there are reactances in addition to R, they do not contribute to RMS power dissipation but the circuit has to handle the reactive currents they produce (eg, wire sizes may need to be be larger) 13

14 Series RLC Resonance I E0 E0 = = R 2 + ( ωl 1/ ωc) 2 Z ( ω ) The current I will be a maximum when ωl=1/ωc. This defines the resonant frequency ω 0 : Note ( cultural comment, not on test ): Resonance is an important phenomenon in physics! (Example: Tacoma Narrows Bridge*) Off-resonance, the current is given by E0 I = ω 0 R + ( Lω ) 1 ω ω = 0 1 LC The resonance is sharper if the resistance is smaller. (analogy: mass + spring with friction: greater friction diminishes the amplitude of motion rapidly. * 14

15 quiz Which of the following is TRUE when a circuit with R, L, C in series is at its resonant frequency? A. Net impedance = 0 B. Capacitive reactance = Inductive reactance C. EMF source sees only reactance, not R D. The capacitor explodes E. None of the above 15

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