1 / 66

Chapter 30

Chapter 30. Inductance. Energy through space for free??. A puzzler!. Creates increasing flux INTO ring. Increasing current in time. Energy through space for free??. A puzzler!. Creates increasing flux INTO ring. Increasing current in time.

alvis
Download Presentation

Chapter 30

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 30 Inductance

  2. Energy through space for free?? • A puzzler! Creates increasing flux INTO ring Increasing current in time

  3. Energy through space for free?? • A puzzler! Creates increasing flux INTO ring Increasing current in time Induce counter-clockwise current and B field OUT of ring

  4. Energy through space for free?? • But… If wire loop has resistance R, current around it generates energy! Power = i2/R!! Increasing current in time Induced current i around loop of resistance R

  5. Energy through space for free?? • Yet…. NO “potential difference”! Increasing current in time Induced current i around loop of resistance R

  6. Energy through space for free?? • Answer? Energy in B field!! Increased flux induces EMF in coil radiating power Increasing current in time

  7. Goals for Chapter 30 - Inductance • To learn how current in one coil can induce an emf in another unconnected coil • To relate the induced emf to the rate of change of the current • To calculate the energy in a magnetic field

  8. Goals for Chapter 30 • Introduce circuit components called INDUCTORS • Analyze circuits containing resistors and inductors • Describe electrical oscillations in circuits and why the oscillations decay

  9. Introduction • How does a coil induce a current in a neighboring coil. • A sensor triggers the traffic light to change when a car arrives at an intersection. How does it do this? • Why does a coil of metal behave very differently from a straight wire of the same metal? • We’ll learn how circuits can be coupled without being connected together.

  10. Mutual inductance • Mutual inductance:A changing current in one coil induces a current in a neighboring coil. Increase current in coil 1

  11. Mutual inductance • Mutual inductance:A changing current in one coil induces a current in a neighboring coil. Increase B flux Increase current in coil 1

  12. Mutual inductance • Mutual inductance:A changing current in one coil induces a current in a neighboring coil. Increase B flux Induce flux opposing change Increase current in coil 1 Induce current in loop 2

  13. Mutual inductance • EMF induced in single loop 2 = - d (B2 )/dt • Caused by change in flux through second loop of B field • Created by the current in the single loop 1 • So… B2 is proportional to i1 • What does that proportionality constant depend upon?

  14. Mutual inductance • B2 is proportional to i1 and is affected by: • # of windings in loop 1 • Area of loop 1 • Area of loop 2 • Define “M” as mutual inductance of coil 1 on coil 2

  15. Mutual inductance • Define “M” as mutual inductance on coil 2 from coil 1 • B2 = M21 I • But what if loop 2 also has many turns? Increase #turns in loop 2? => increase flux!

  16. Mutual inductance • IF you have N turns in coil 2, each with flux B , total flux is Nx larger • Total Flux = N2B2

  17. Mutual inductance • B2 proportional to i1 So N2B = M21i1 M21 is the “Mutual Inductance” [Units] = Henrys = Wb/Amp

  18. Mutual inductance • EMF2 = - N2 d [B2 ]/dt and • N2B = M21i1 so • EMF2 = - M21d [i1]/dt • Because geometry is “shared” M21 = M12 = M

  19. Mutual inductance • Define Mutual inductance:A changing current in one coil induces a current in a neighboring coil. • M = N2B2/i1M = N1B1/i2

  20. Mutual inductanceexamples • Long solenoid with length l, area A, wound with N1 turns of wire • N2 turns surround at its center. What is M?

  21. Mutual inductanceexamples • M = N2B2/i1 • We need B2 from the first solenoid (B1 = moni1) • n = N1/l • B2 = B1A • M = N2moi1 AN1/li1 • M = moAN1 N2/l • All geometry!

  22. Mutual inductanceexamples • M = moAN1 N2/l • If N1 = 1000 turns, N2 = 10 turns, A = 10 cm2, l = 0.50 m • M = 25 x 10-6 Wb/A • M = 25 mH

  23. Mutual inductanceexamples • Using same system (M = 25 mH) • Suppose i2 varies with time as = (2.0 x 106 A/s)t • At t = 3.0 ms, what is average flux through each turn of coil 1? • What is induced EMF in solenoid?

  24. Mutual inductanceexamples • Suppose i2 varies with time as = (2.0 x 106 A/s)t • At t = 3.0 ms, i2 = 6.0 Amps • M = N1B1/i2 = 25 mH • B1= Mi2/N1 = 1.5x10-7 Wb • Induced EMF in solenoid? • EMF1 = -M(di2/dt) • -50Volts

  25. Self-inductance • Self-inductance: A varying current in a circuit induces an emfin that same circuit. • Always opposes the change! • Define L = N B/i • Li = N B • If i changes in time: • d(Li)/dt = NdB/dt = -EMF or • EMF = -Ldi/dt

  26. Inductors as circuit elements! • Inductors ALWAYS oppose change: • In DC circuits: • Inductors maintain steady current flow even if supply varies • In AC circuits: • Inductors suppress (filter) frequencies that are too fast.

  27. Potential across an inductor • The potential across an inductor depends on the rate of change of the current through it. • The self-induced emf does not oppose current, but opposes a change in the current. • Vab = -Ldi/dt

  28. Magnetic field energy • Inductors store energy in the magnetic field: U = 1/2 LI2 • Units: L = Henrys (from L = N B/i ) • N B/i = B-field Flux/current through inductor that creates that fluxWb/Amp = Tesla-m2/Amp • [U] = [Henrys] x [Amps]2 • [U] = [Tesla-m2/Amp] x [Amps]2 = Tm2Amp • But F = qv x B gives us definition of Tesla • [B] = Teslas= Force/Coulomb-m/s = Force/Amp-m

  29. Magnetic field energy • Inductors store energy in the magnetic field: U = 1/2 LI2 • [U] = [Tesla-m2/Amp] x [Amps]2 = Tm2Amp • [U] = [Newtons/Amp-m] m2Amp = Newton-meters = Joules = Energy!

  30. Magnetic field energy • The energy stored in an inductor is U = 1/2 LI2. • The energy density in a magnetic field (Joule/m3) is • u = B2/20 (in vacuum) • u = B2/2 (in a magnetic material) • Recall definition of 0 (magnetic permeability) • B = 0 i/2pr (for the field of a long wire) • 0 = Tesla-m/Amp • [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter

  31. Magnetic field energy • The energy stored in an inductor is U = 1/2 LI2. • The energy density in a magnetic field (Joule/m3) is • u = B2/20 (in vacuum) • [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter • [U] = Tm2Amp = Joules • So… energy density [u] = Joules/m3

  32. Calculating self-inductance and self-induced emf • Toroidal solenoid with area A, average radius r, N turns. • Assume B is uniform across cross section. What is L?

  33. Calculating self-inductance and self-induced emf • Toroidal solenoid with area A, average radius r, N turns. • L = N B/i • B= BA = (moNi/2pr)A • L = moN2A/2pr (self inductance of toroidal solenoid) • Why N2 ?? • If N =200 Turns, A = 5.0 cm2, r = 0.10 m L = 40 mH

  34. Potential across an inductor • The potential across a resistor drops in the direction of current flowVab = Va-Vb > 0 • The potential across an inductor depends on the rate of change of the current through it.

  35. Potential across an inductor • The potential across an inductor depends on the rate of change of the current through it. • The self-induced emf does not oppose current, but opposes a change in the current.

  36. Potential across an inductor • The potential across an inductor depends on the rate of change of the current through it. • The self-induced emf does not oppose current, but opposes a change in the current. • The inductor acts like a temporary voltage source pointing OPPOSITE to the change.

  37. Potential across an inductor • The inductor acts like a temporary voltage source pointing OPPOSITE to the change. • This implies the inductor looks like a battery pointing the other way! • Note Va > Vb! Direction of current flow from a battery oriented this way

  38. Potential across an inductor • What if current was decreasing? • Same result! The inductor acts like a temporary voltage source pointing OPPOSITE to the change. • Now inductor pushes current in original direction • Note Va < Vb!

  39. The R-L circuit • An R-L circuit contains a resistor and inductor and possibly an emf source. • Start with both switches open • Close Switch S1: • Current flows • Inductor resists flow • Actual current less than maximum E/R • E – i(t)R- L(di/dt) = 0 • di/dt = E /L – (R/L)i(t)

  40. The R-L circuit • Close Switch S1: • E – i(t)R- L(di/dt) = 0 • di/dt = E /L – (R/L)i(t) Boundary Conditions • At t=0, di/dt = E /L • i() = E /R Solve this 1st order diff eq: • i(t) = E /R (1-e -(R/L)t)

  41. Current growth in an R-L circuit • i(t) = E /R (1-e -(R/L)t) • The time constant for an R-L circuit is  = L/R. • [ ]= L/R = Henrys/Ohm • = (Tesla-m2/Amp)/Ohm • = (Newtons/Amp-m) (m2/Amp)/Ohm • = (Newton-meter) / (Amp2-Ohm) • = Joule/Watt • = Joule/(Joule/sec) • = seconds! 

  42. Current growth in an R-L circuit • i(t) = E /R (1-e -(R/L)t) • The time constant for an R-L circuit is  = L/R. • [ ]= L/R = Henrys/Ohm • EMF = -Ldi/dt • [L] = Henrys = Volts /Amps/sec • Volts/Amps = Ohms (From V = IR) • Henrys = Ohm-seconds • [ ]= L/R = Henrys/Ohm = seconds!  • Vab = -Ldi/dt

  43. The R-L circuit • E = i(t)R+ L(di/dt) • Power in circuit = E I • E i = i2R+ Li(di/dt) • Some power radiated in resistor • Some power stored in inductor

  44. The R-L circuit example • R = 175 W; i = 36 mA; current limited to 4.9 mA in first 58 ms. • What is required EMF • What is required inductor • What is the time constant?

  45. The R-L circuit Example • R = 175 W; i = 36 mA; current limited to 4.9 mA in first 58 ms. • What is required EMF • What is required inductor • What is the time constant? • EMF = IR = (0.36 mA)x(175W ) = 6.3 V • i(t) = E /R (1-e -(R/L)t) • i(58ms) = 4.9 mA • 4.9mA = 6.3V(1-e -(175/L)0.000058) • L = 69 mH •  = L/R= 390 ms

  46. Current decay in an R-L circuit • Now close the second switch! • Current decrease is opposed by inductor • EMF is generated to keep current flowing in the same direction • Current doesn’t drop to zero immediately

  47. Current decay in an R-L circuit • Now close the second switch! • –i(t)R - L(di/dt) = 0 • Note di/dt is NEGATIVE! • i(t) = -L/R(di/dt) • i(t) = i(0)e -(R/L)t • i(0) = max current before second switch is closed

  48. Current decay in an R-L circuit • i(t) = i(0)e -(R/L)t

  49. Current decay in an R-L circuit • Test yourself! • Signs of Vab and Vbc when S1 is closed? • Vab >0; Vbc >0 • Vab >0, Vbc <0 • Vab <0, Vbc >0 • Vab <0, Vbc <0

  50. Current decay in an R-L circuit • Test yourself! • Signs of Vab and Vbc when S1 is closed? • Vab >0; Vbc >0 • Vab >0, Vbc <0 • Vab <0, Vbc >0 • Vab <0, Vbc <0 • WHY? • Current increases suddenly, so inductor resists change

More Related