CHAPTER 6. Inductance, Capacitance, and Mutual Inductance

Transcription

1 CHAPTER 6 Inductance, Capacitance, and Mutual Inductance

2 6.1 The Inductor Inductance is symbolized by the letter L, is measured in henrys (H), and is represented graphically as a coiled wire. The inductor v - i equation Figure 6.1 (a) The graphic symbol for an inductor with an inductance of L henrys. (b) Assigning reference voltage and current to the inductor, following the passive sign convention.

3 Example 6.1 The independent current source in the circuit shown in Fig. 6.2 generates zero current for t < 0 and a pulse 10te 5t A, for t > 0. Figure 6.2 The circuit for Example 6.1.

4 Example 6.1 a) Sketch the current waveform. b) At what instant of time is the current maximum? c) Express the voltage across the terminals of the 100 mh inductor as a function of time. d) Sketch the voltage waveform. e) Are the voltage and the current at a maximum at the same time? f) At what instant of time does the voltage change polarity? g) Is there ever an instantaneous change in voltage across the inductor? If so, at what time?

5 Example 6.1

6 Example 6.1 Figure 6.3 The current waveform for Example 6.1. Figure 6.4 The voltage waveform for Example 6.1.

7 Current in an Inductor in Terms of the Voltage Across the Inductor The inductor i - v equation

8 Example 6.2 The voltage pulse applied to the 100 mh inductor shown in Fig. 6.5 is t < 0 for and is given by the expression v(t) = 20te 10t V for t > 0. Also assume i = 0 for t 0. a) Sketch the voltage as a function of time. b) Find the inductor current as a function of time. c) Sketch the current as a function of time. Figure 6.5 The circuit for Example 6.2.

9 Example 6.2

10 Example 6.2 Figure 6.6 The voltage waveform for Example 6.2. Figure 6.7 The current waveform for Example 6.2.

11 Power and Energy in the Inductor If the current reference is in the direction of the voltage drop across the terminals of the inductor, the power is Power in an inductor Energy in an inductor

12 Example 6.3 a) For Example 6.1, plot i, v, p, and versus time. Line up the plots vertically to allow easy assessment of each variable s behavior. b) In what time interval is energy being stored in the inductor? c) In what time interval is energy being extracted from the inductor? d) What is the maximum energy stored in the inductor? e) Evaluate the integrals and comment on their significance. f) Repeat (a) (c) for Example 6.2. g) In Example 6.2, why is there a sustained current in the inductor as the voltage approaches zero?

13 Example 6.3

14 Example 6.3 Figure 6.8 The variables i, υ, p, and w versus t for Example 6.1.

15 Example 6.3

16 Example 6.3

17 Example 6.3

18 Example 6.3 Figure 6.9 The variables i, υ, p, and w versus t for Example 6.2.

19 6.2 The Capacitor The circuit parameter of capacitance is represented by the letter C, is measured in farads (F), and is symbolized graphically by two short parallel conductive plates. Because the farad is an extremely large quantity of capacitance, practical capacitor values usually lie in the picofarad (pf) to microfarad (μf) range.

20 Capacitor i - v equation Figure 6.10 (a) The circuit symbol for a capacitor. (b) Assigning reference voltage and current to the capacitor, following the passive sign convention.

21 Capacitor v - i equation Capacitor power equation Capacitor energy equation

22 Example 6.4 The voltage pulse described by the following equations is impressed across the terminals of a 0.5 mf capacitor: a) Derive the expressions for the capacitor current, power, and energy. b) Sketch the voltage, current, power, and energy as functions of time. Line up the plots vertically. c) Specify the interval of time when energy is being stored in the capacitor.

23 Example 6.4 d) Specify the interval of time when energy is being delivered by the capacitor. e) Evaluate the integrals and comment on their significance

24 Example 6.4

25 Example 6.4

26 Example 6.4

27 Example 6.4 Figure 6.11 The variables i, υ, p, and w versus t for Example 6.4.

28 Example 6.5 An uncharged 0.2 mf capacitor is driven by a triangular current pulse. The current pulse is described by a) Derive the expressions for the capacitor voltage, power, and energy for each of the four time intervals needed to describe the current. b) Plot i, v, p, and w versus t. Align the plots as specified in the previous examples. c) Why does a voltage remain on the capacitor after the current returns to zero?

29 Example 6.5

30 Example 6.5

31 Example 6.5

32 Example 6.5 Figure 6.11 The variables i, υ, p, and w versus t for Example 6.4.

33 6.3 Series-Parallel Combinations of Inductance and Capacitance Combining inductors in series Figure 6.13 Inductors in series. Figure 6.14 An equivalent circuit for inductors in series carrying an initial current i(t 0 ).

34 Figure 6.16 An equivalent circuit for three inductors in parallel. Figure 6.15 Three inductors in parallel.

35 Combining inductors in parallel Equivalent inductance initial current

36 Figure 6.17 An equivalent circuit for capacitors connected in series. (a) The series capacitors. (b) The equivalent circuit.

37 Combining capacitors in series Equivalent capacitance initial voltage

38 Combining capacitors in parallel Figure 6.18 An equivalent circuit for capacitors connected in parallel. (a) Capacitors in parallel. (b) The equivalent circuit.

39 6.4 Mutual Inductance Two circuits are linked by a magnetic field. The voltage induced in the second circuit can be related to the time-varying current in the first circuit by a parameter known as mutual inductance. The self-inductances of the two coils are labeled L 1 and L 2 and the mutual inductance is labeled M.

40 Figure 6.19 Two magnetically coupled coils. Figure 6.20 Coil currents i 1 and i 2 used to describe the circuit shown in Fig

41 Dot convention for mutually coupled coils When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal. Figure 6.21 The circuit of Fig with dots added to the coils indicating the polarity of the mutually induced voltages.

42 Dot convention for mutually coupled coils (alternate) When the reference direction for a current leaves the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is negative at its dotted terminal.

43 In Fig. 6.21, the dot convention rule indicates that the reference polarity for the voltage induced in coil 1 by the current i 2 is negative at the dotted terminal of coil 1.This voltage (Mdi 2 /dt) is a voltage rise with respect to The voltage induced in coil 2 by the current i 1 is Mdi 1 /dt and its reference polarity is positive at the dotted terminal of coil 2.This voltage is a voltage rise in the direction of i 2. Figure 6.22 shows the self- and mutually induced voltages across coils 1 and 2 along with their polarity marks.

44 Figure 6.22 The self- and mutually induced voltages appearing across the coils shown in Fig Now let s look at the sum of the voltages around each closed loop. In Eqs and 6.32, voltage rises in the reference direction of a current are negative:

45 The Procedure for Determining Dot Markings a) Arbitrarily select one terminal b) Assign a current into the dotted terminal and label it c) Use the right-hand rule1 to determine the direction of the magnetic field d) Arbitrarily pick one terminal of the second coil e) Use the right-hand rule to determine the direction of the flux f) Compare the directions of the two fluxes

46 Figure 6.23 A set of coils showing a method for determining a set of dot markings.

47 Figure 6.24 An experimental setup for determining polarity marks.

48 Example 6.6 a) Write a set of mesh-current equations that describe the circuit in Fig in terms of the currents i 1 and i 2. b) Verify that if there is no energy stored in the circuit at t = 0 and if i g = 16 16e 5t A, the solutions for i 1 and i 2 are Figure 6.25 The circuit for Example 6.6.

49 Example 6.6

50 Example 6.6

51 Example 6.6 Figure 6.26 The circuit of Example 6.6 when t =.

52 6.5 A Closer Look at Mutual Inductance A Review of Self-Inductance The voltage induced in the conductor is proportional to the number of lines that collapse into, or cut, the conductor. This image of induced voltage is expressed by what is called Faraday s law; Figure 6.27 Representation of a magnetic field linking an N-turn coil.

53 The flux linkage is the product of the magnetic field(f), measured in webers(wb), and the number of turns linked by the field (N): where N is the number of turns on the coil, and P is the permeance of the space occupied by the flux.

54 Here, we assume that the core material the space containing the flux is nonmagnetic. Then, substituting Eqs and 6.35 into Eq yields which shows that self-inductance is proportional to the square of the number of turns on the coil.

55 The Concept of Mutual Inductance Figure 6.28 Two magnetically coupled coils Figure 6.28 Two magnetically coupled coils.

56 The total flux linking coil 1 is f 1, the sum of f 11 and f 12 : The flux f 1 and its components f 11 and f 12 are related to the coil current i 1 as follows: where P 1 is the permeance of the space occupied by the flux f 1, P 11 is the permeance of the space occupied by the flux f 11, and P 21 is the permeance of the space occupied by the flux f 21.

57 Substituting Eqs. 6.38, 6.39, and 6.40 into Eq yields the relationship between the permeance of the space occupied by the total flux f 1 and the permeances of the spaces occupied by its components f 11 and f 21 :

58 We use Faraday s law to derive expressions for v 1 and v 2 : and

59 The coefficient of di 1 /dt in Eq is the selfinductance of coil 1. The coefficient of di 1 /dt in Eq is the mutual inductance between coils 1 and 2.Thus The subscript on M specifies an inductance that relates the voltage induced in coil 2 to the current in coil 1.

60 The coefficient of mutual inductance gives Note that the dot convention is used to assign the polarity reference to v 2 in Fig Figure 6.28 Two magnetically coupled coils.

61 For the coupled coils in Fig. 6.28, exciting coil 2 from a time-varying current source (i 2 ) and leaving coil 1 open produces the circuit arrangement shown in Fig Again, the polarity reference assigned to v 1 is based on the dot convention. Figure 6.29 The magnetically coupled coils of Fig. 6.28, with coil 2 excited and coil 1 open.

62 The total flux linking coil 2 is The flux f 2 and its components f 22 and f 12 are related to the coil current i 2 as follows: The voltages v 1 and v 2 are

63 The coefficient of mutual inductance that relates the voltage induced in coil 1 to the time-varying current in coil 2 is the coefficient of di 2 /dt in Eq. 6.51: For nonmagnetic materials, the permeances P 21 and P 12 are equal, and therefore Hence for linear circuits with just two magnetically coupled coils, attaching subscripts to the coefficient of mutual inductance is not necessary.

64 Mutual Inductance in Terms of Self- Inductance The value of mutual inductance is a function of the self-inductances. We derive this relationship as follows. From Eqs and 6.50, respectively. From Eqs and 6.55,

65 We now use Eq and the corresponding expression for P 2 to write But for a linear system, P 21 = P 12, so Eq becomes

66 Relating self-inductances and mutual inductance using coupling coefficient Replacing the two terms involving permeances by a single constant expresses Eq in a more meaningful form: Substituting Eq into Eq yields or where the constant k is called the coefficient of coupling.

67 Energy Calculations For linear magnetic coupling, If we reverse the procedure that is, if we first increase i 2 from zero to I 2 and then increase i 1 from zero to I 1 the total energy stored is

68 Figure 6.30 The circuit used to derive the basic energy relationships.

69 Energy stored in magnetically-coupled coils