1. Chapter 32 Inductance

2. Introduction In this chapter we will look at applications of induced currents, including: – Self Inductance of a circuit – Inductors as circuit elements. – Mutual Inductance between two circuits – RL, LC, and RLC Combinations

3. 32.1 Self-Inductance It is important in this section that we make sure to distinguish between the physical source of emf and current in a circuit, and the emf and current that are induced from magnetic fields. – emf and current for those caused by a battery or power supply – Induced emf and induced current for those caused by changing magnetic fields.

4. 32.1 Consider a simple switched resistor circuit. – When the switch is closed, the current does not immediately flow at its maximum value, but increases with time. – This creates a magnetic flux increasing with time, and therefore an induced emf.

5. 32.1 – From Lenz’s law we know the induced emf will oppose the changing flux, and so is opposite to the direction of the battery emf. – This is also called a “back emf” similar to what is found in a motor coil. – This self induced emf is given the symbol ε L

6. 32.1 Consider a second example of wire coil wrapped around a cylindrical core. With the direction of the current, the B field points to the left.

7. 32.1 As the current increases, so does the B Field, giving an induced emf in the opposite direction.

8. 32.1 As the current and B field decrease, inducing an emf with the direction of the current.

9. 32.1 A self-induced emf is always proportional to the time rate of change of current. L is the proportionality constant called “inductance” and depends on the geometry and physical properties of the coil.

10. 32.1 By combining this expression with Faraday’s Law We get an expression for the inductance L, of a coil (solenoid/toroid) assuming the same magnetic flux passes through each turn

11. 32.1 We can also describe it simply as Remember that Resistance is a measure of opposition to current (R = Δ V/I) Inductance is a measure of opposition to change in current.

12. 32.1 The unit for inductance is the henry (H) equal to a volt-second per amp. Quick Quiz p. 1005 Examples 32.1-32.2

13. 32.2 RL Circuits If a circuit contains a coil (ex: solenoid) the inductance of the coil prevents the current from changing instantaneously. If this element has a large inductance, it is called and inductor and shows up in circuit schematics as a coil. An inductor acts to resist changes to the current.

14. 32.2 Consider a circuit with a battery, resistor, inductor, and switch. An RL Circuit. What happens when the switch is closed at t = 0? The current increases, and a back emf is induced in the inductor.

15. 32.2 Going around the circuit (Kirchoff’s Loop rule), the potential difference is given as The solution to this differential equation gives I as a function of time. or

16. 32.2 Where I = 0 and t = 0 and the time constant τ = L/R This exponential function increases asymptotically to the maximum current ε /R.

17. 32.2 Taking the first derivative of the equation gives the rate of change of current. We see that this is at its maximum value at t = 0. (Also from the slope of the current plot)

18. 32.2

19. ADD SWITCH EXAMPLE

22. 32.2 Quick Quizzes p. 1009 Example 32.3

23. 32.3 Energy in a Magnetic Field Since the induced emf resists an instantaneous current, the battery must provide more energy in circuits with an inductor. – Part of that energy is dissipated in the restistor. – The remainder is stored in the magnetic field within the inductor.

24. 32.3 The rate at which energy stored in a inductor is given as We can integrate to get the total energy stored Note this is similar to the energy in a capacitor We can also determine the energy density for a given inductor geometry, like a solenoid.

25. 32.3 In example 32.1 we found the inductance of a solenoid to be And the B field in a solenoid is given as By substituting into the energy equation for L and I

26. 32.3 This can be simplified to And since Al is the volume of the solenoid, the energy per unit volume is

27. 32.3 Again we see similarities with E-fields Quick Quiz p. 1012 Example 32.4

28. 32.4 Mutual Inductance Often times, an emf can be induced in a circuit because of changing currents in other nearby circuits, a process called mutual induction. Let’s look at two wound coils of wire. The current in one wire creates a magnetic field that pass through the other coil.

29. 32.4 The magnetic flux through coil two from coil one is given as Φ 12. We can define the mutual inductance of the two coils as This property of the pair of coils depends on both geometries and their orientations with respect to each other.

30. 32.4 If the I 1 varies with time, then the emf induced in coil 2 is given as Similarly, if the current I 2 varies with time, then the emf induced in coil 1 is

31. 32.4 It can be shown that So the emf induced in each coil can be written and In mutual induction, the emf induced in one coil is always proportional to the rate of change of current in the other coil.

32. 32.4 Quick Quiz p. 1014 Example 32.6

33. 32.5 Oscillations in an LC Circuit An simple LC circuit consists of a initially charged capacitor, an inductor, and a switch. When the switch is closed, we find that both the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values.

34. 32.5 The oscillations should continue indefinitely if we assume – Zero resistance in the circuit (zero loss to internal energy) – Zero loss to the radiation of energy. If we look at the energy involved, we can see a lot of similarities to a simply harmonic oscillator.

35. 32.5 The harmonic oscillator transitions from U elastic with +A, to K, back to U elastic with -A. The LC oscillator transitions from fully charged capacitor To maximum current and therefore magnetic field in the inductor Back to a fully charged capacitor with opposite polarity.

36. 32.5 The circuit swings past a “simple discharge” because of the inductor’s resistance to changing current. – At Qmax, I = zero – At Imax, Q = zero – At –Qmax. I = zero

37. 32.5 Capacitor is fully charged, switch is closed.

38. 32.5 Current reaches maximum value as capacitor is fully discharged.

39. 32.5 Current continues to flow recharging the Cap with opposite polarity.

40. 32.5 Capacitor discharges to maximum current flowing in the opposite direction.

41. 32.5 Current slows to zero as the circuit reaches its original state.

42. 32.5 An “oscillation” in the charge on the capacitor of this type is represented by a differential equation, the solution to which is a trig function (sine/cosine). Where ω is the angular frequency and is the phase constant.

43. 32.5 ω represents the natural frequency of the circuit, and as such has applications in resonance. Determining the phase angle based on initial conditions. – At t = 0 I = 0 Q = Qmax – So the phase angle must be, = 0.

44. 32.5 Since current is the rate of charge flow, we can take dQ/dt to find it as a function of time.

45. 32.5

46. Again recognize that the total energy in the system is the sum of electrical energy and magnetic energy stored in the cap and inductor.

47. 32.5

48. See Table 32.1 p 1021 for analogies between physical oscillation and electrical oscillation. Quick Quizzes p. 1019 Example 32.7

49. 32.6 The RLC Circuit When studying the LC circuit, we recognize the analogy to an harmonic oscillator for ideal conditions (no resistance). With a resistor added to the circuit, we get a dissipation of energy, and so the oscillations will not continue indefinitely.

50. 32.6 The analogue to the RLC circuit is a damped oscillator.

51. 32.6 The presence of resistance in the circuit adds an additional term to the differential equation describing the circuit. When R = 0, we have the ideal LC circuit. As the value of R increases we go through the various types of damping.

52. 32.6 When R is small, the damping is light. The solution to the differential is Where ω d is Note the value of ω, when R = 0.

53. 32.6 Note the combination of the exponential decay and the sinusoidal oscillation. The amplitude of the maximum charge decreases within the decay envelope.

54. 32.6 As the value of R increases we approach critical damping where the circuit reaches equilibrium as fast as possible without oscillating. For values of R > Rc, we have overdamped conditions, returning slowly to equilibrium.

55. 32.6 Overdamped

56. 32.6 Understanding how the presence of resistance affects the circuit is essential because the LC circuit is just an ideal case, all “LC” circuits have some damping. Applications – Variable Tuning (ie. Radio station frequencies) – Signal Filtering

57. The End… No seriously, its over… Don’t make this weird…