Chapter 11. Inductors ISU EE. C.Y. Lee

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1 Chapter 11 Inductors

2 Objectives Describe the basic structure and characteristics of an inductor Discuss various types of inductors Analyze series inductors Analyze parallel inductors Analyze inductive dc switching circuits Analyze inductive ac circuits 2

3 The Basic Inductor When a length of wire is formed onto a coil, it becomes a basic inductor When there is current through the coil, a threedimensional electromagnetic field is created, surrounding the coil in all directions 3

4 Basics of a Inductance The unit of inductance is the henry (H), defined as the inductance when one ampere per second through the coil, induces one volt across the coil di( t ) v ( t ) = L dt In many practical applications, mh (10-3 H) or μh (10-6 H) are the more common units. 4

5 Basics of a Inductance Inductance is a measure of a coil s ability to establish an induced voltage as a result of a change in its current, and that induced voltage is in a direction to oppose that change in current An inductor stores energy in the magnetic field created by the current: 5 U = 1 LI 2 2 (J)

6 Physical Characteristics Inductance is directly proportional to the permeability of the core material Inductance is directly proportional to the cross-sectional area of the core Inductance is directly proportional to the square of the number of turns of wire Inductance is inversely proportional to the length of the core material L N μ l 2 A = (H) 6

7 Physical Characteristics Example: Determine the inductance of the coil μ = H/m A = πr 2 = π( ) 2 = m 2 L = μn 2 A/l = ( )(350) 2 ( )/(0.015) = 40 mh 7

8 Winding Resistance When many turns of wire are used to construct a coil, the total resistance may be significant The inherent resistance is called the dc resistance or the winding resistance (R W ) 8

9 Winding Capacitance When two conductors are placed side-by-side, there is always some capacitance between them When many turns of wire are placed close together in a coil, there is a winding capacitance (C W ) C W becomes significant at high frequencies 9

10 Faraday s Laws Faraday s law: The amount of voltage induced in a coil is directly proportional to the rate of change of the magnetic field with respect to the coil 10

11 Lenz s Laws Example of Lenz s law: 11

12 Lenz s Laws Example of Lenz s law: 12

13 Typical Inductors 13

14 Series Inductors When inductors are connected in series, the total inductance increases L T = L 1 + L 2 + L L n 14

15 Parallel Inductors When inductors are connected in parallel, the total inductance is less than the smallest inductance 1/L T = 1/L 1 + 1/L 2 + 1/L /L n 15

16 Inductors in DC Circuits When there is constant current in an inductor, there is no induced voltage There is a voltage drop in the circuit due to the winding resistance of the coil Inductance itself appears as a short to dc V=0 + I I 16

17 Inductors in DC Circuits Illustration of the exponential buildup of current in an inductor 17

18 Inductors in DC Circuits Illustration of the exponential decrease of current in an inductor 18

19 RL Time Constant Because the inductor s basic action opposes a change in its current, it follows that current cannot change instantaneously in an inductor The rate, a certain time is required for the current to make a change from one value to another, is determined by the RL time constant (τ = L/R) i + + i 19

20 Energizing Current in an Inductor In a series RL circuit, the current will increase to approximately 63% of its full value in one timeconstant (τ) interval after the switch is closed The current reaches its final value in approximately 5τ i = ( tr / L 1 ) I F e 20

21 De-energizing Current in an Inductor In a series RL circuit, the current will decrease to approximately 63% of its fully charged value one time-constant (τ) interval after the switch is closed The current reaches 1% of its initial value in approximately 5τ; considered to be equal to 0 i = I i e tr / L 21

22 RL Time Constant Example: Determine the inductor current 30 μs after the switch is closed τ = L/R = 100mH/2.2kΩ = 45.5 μs I F = V s /R = 12V/2.2kΩ = 5.45 ma i L = I F (1 e Rt/L ) = (5.45)(1 e 0.66 ) = 2.63 ma 22

23 RL Time Constant Example: Determine the inductor current 2 ms after the switch(1) is opened and switch(2) is closed τ = L/R = 200mH/56Ω = 3.57 ms I i = V s /R = 5V/56Ω = 89.3 ma i L = I i (e Rt/L ) = (89.3)(e 0.56 ) = 51.0 ma 23

24 Analysis of Inductive AC Circuit The current lags the voltage by 90 in a purely inductive ac circuit v ( t ) = L di( t dt ) 24

25 Ohm s Law in Inductive AC Circuits The reactance (X L ) of a inductor is analogous to the resistance (R) of a resistor V = IZ L = I(jωL) = I(jX L ) Magnitude V = IX L V = IR 25

26 Inductive Reactance, X L The relationship between inductive reactance, inductance and frequency is: X L = 2π f L (Ω) Example: Determine the inductive reactance 1 khz X L = 2π f L = 2π( )( ) = 31.4 kω 26

27 Inductive Reactance, X L I = V X L = V 2πfL 27

28 Inductive Reactance, X L I = V X L = V 2πfL 28

29 Inductive Reactance, X L Example: Determine the rms current in below circuit X L = 2π( )( ) = 6283 Ω V rms = I rms X L I rms = V rms /X L = (5V)/(6283Ω) = 796 μa 29

30 Power in a Inductor Instantaneous power (p) is the product of v and i True power (P true ) is zero, since no net energy is lost due to conversion to heat in the inductor 30

31 Power in a Inductor The rate at which an inductor stores or returns energy is called reactive power (P r ); units: (VAR) P = P r r P r = = I rms V X I 2 rms L 2 rms V X rms L These equations are of the same form as those introduced in Chapter 3 for true power in a resistor 31

32 Quality Factor (Q) of a Coil The quality factor (Q) is the ratio of the reactive power in the inductor to the true power in the winding resistance of the coil or the resistance in series with the coil Q = (reactive power) / (true power) Q = X L /R W 32

33 Inductor Applications An inductor can be used in the filter to smooth out the ripple voltage 33

34 Summary Self-inductance is a measure of a coil s ability to establish an induced voltage as a result of a change in its current Inductance is directly proportional to the square of the number of turns, the permeability, and the cross sectional area of the core. It is inversely proportional to the length of the core An inductor opposes a change in its own current 34

35 Summary Faraday s law states that relative motion between a magnetic field and a coil induces voltage across the coil Lenz s law states that the polarity of induced voltage is such that the resulting induced current is in a direction that opposes the change in the magnetic field that produced it 35

36 Summary The permeability of a core material is an indication of the ability of the material to establish a magnetic field The time constant for a series RL circuit is the inductance divided by the resistance In an RL circuit, the voltage and current in an energizing or de-energizing inductor make a 63% change during each time-constant interval 36

37 Summary Inductors add in series Total parallel inductance is less than that of the smallest inductor in parallel Current lags voltage by 90 in an inductor Inductive reactance (X L ) is directly proportional to frequency and inductance An inductor blocks high-frequency ac 37

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