1. DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
David M. Buchla
Inductors
Chapter 11

2. The Basic Inductor Ch.11 Summary
When a length of wire is formed into a coil., it becomes an inductor. When there is current in the inductor, a three-dimensional magnetic field is created.
A change in current causes the magnetic field to change. This in turn induces a voltage across the inductor that opposes the original change in current. S N

3. The Basic Inductor Ch.11 Summary
One henry (H) is the inductance of a coil when a current, changing at a rate of one ampere per second, induces one volt across the coil. Most coil values are far less than 1 H.
Large inductors and transformers are wound around an iron core to increase inductance.
Iron core

4. Faraday’s Law Ch.11 Summary
The voltage induced in a coil is directly proportional to the rate of change of the magnetic field with respect to the coil.

5. Ch.11 Summary
Lenz’s Law
When the current through a coil changes, an induced voltage is created as a result of the changing magnetic field. The direction of the induced voltage is such that it always opposes the change in the current.

6. Ch.11 Summary
Lenz’s Law
A basic circuit to demonstrate Lenz’s law is shown.
Initially, the SW is open and there is a small current in the circuit through L and R1.

7. Ch.11 Summary
Lenz”s Law
SW closes and immediately a voltage appears across L that tends to oppose any change in current.
Initially, the meter reads same current as before the switch was closed.

8. Ch.11 Summary
Lenz’s Law
After a time, the current stabilizes at a higher level (due to I2) as the voltage across the coil decays.
Later, the meter reads a higher current because of the load change.

9. Inductor Characteristics
Ch.11 Summary
Inductor Characteristics
In addition to inductance, inductors have winding resistance (RW), which is the resistance of the wire, and winding capacitance (CW) between the turns. An equivalent circuit for a practical inductor that includes these effects is shown:
Notice that the winding resistance is in series with the coil and the winding capacitance is in parallel with both.

10. Types of Inductors Ch.11 Summary
There are a variety of inductors, depending on the amount of inductance required and the application. Some, with fine wires, are encapsulated and may appear like a resistor.
Common symbols for inductors (coils) are
Air core
Iron core
Ferrite core
Variable

11. Factors Affecting Inductors
Ch.11 Summary
Factors Affecting Inductors
Four factors affect the amount of inductance for a coil. The equation for the inductance of a coil is
where
L = inductance in henries
N = number of turns of wire
m = permeability in Wb/At-m
l = coil length in meters

12. Ch.11 Summary
Example
What is the inductance of a 2 cm long, 150 turn coil wrapped on an low carbon steel core that is 0.5 cm diameter? The permeability of low carbon steel is 2.5 x10-4 H/m (Wb/At-m).
22 mH

13. Common Inductors Ch.11 Summary
Inductors come in a variety of types and sizes. A few common ones are illustrated here.

14. Series Inductors Ch.11 Summary
When inductors are connected in series, the total inductance is the sum of the individual inductors. The general equation for inductors in series is
If a 1.5 m inductor is connected in series with an 680 mH inductor, the total inductance is
2.18 mH

15. Parallel Inductors Ch.11 Summary
When inductors are connected in parallel, the total inductance is smaller than the smallest one. The general equation for inductors in parallel is
The total inductance of two inductors is
…or you can use the product-over-sum rule.

16. Parallel Inductors Ch.11 Summary
The total inductance in the parallel circuit shown is
468 mH

17. Inductors in DC Circuits
Ch.11 Summary
Inductors in DC Circuits
Vinitial
When an inductor is connected in series with a resistor and a dc source, current changes at an exponential rate.
Ifinal

18. Inductors in DC Circuits
Ch.11 Summary
Inductors in DC Circuits VS
Exponential waveforms are also generated when a square wave source is connected to a series RL circuit. VL VR

19. Universal Exponential Curves
Ch.11 Summary
Universal Exponential Curves
The exponential curves show how the current in an RL circuit increases (or decreases) over five equal periods, called time constants. For an RL circuit, the length of a time constant is
Rising exponential
Falling exponential

20. Universal Exponential Curves
Ch.11 Summary
Universal Exponential Curves
The universal curves can be applied to general formulas for the current (or voltage) curves for RL circuits. The general current formula is
where
IF = final value of current
Ii = initial value of current
i = instantaneous value of current
e = Napier’s constant (approximately )
The final current is greater than the initial current when the inductive field is building, and less than the initial current when the field is collapsing.

21. Inductive Reactance Ch.11 Summary
Inductive reactance (XL) is the opposition of an inductor to alternating current (ac). The equation for inductive reactance is
The reactance of a 33 mH inductor that is operated at 550 kHz is
114 W

22. Inductive Reactance Ch.11 Summary 1.89 kW
When inductors are in series, the total reactance is the sum of the individual reactances. That is,
Assume three 220 mH inductors are in series with a 455 kHz ac source. What is the total reactance?
The reactance of each inductor is
1.89 kW

23. Inductive Reactance Ch.11 Summary
When inductors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactances. That is,
If the three 220 mH inductors from the last example are placed in parallel with the 455 kHz ac source, what is the total reactance?
The reactance of each inductor is 629 W. Using these values:
210 W

24. Inductive Phase Shift Ch.11 Summary
When a sine wave is applied to an inductor, there is a phase shift between voltage and current such that voltage always leads the current by 90o.

25. Power in an Inductor Ch.11 Summary
True Power: The power that is dissipated in the winding resistance of an inductor. One form of the true power equation is:
Ptrue = (Irms)2RW
The unit of measure for true power is the volt-ampere (VA).
Reactive Power: The rate at which the inductor stores and returns energy. One form of the reactive power equation is:
Pr = Vrms  Irms
The unit for reactive power is the volt-ampere-reactive (VAR).

26. Ch.11 Summary
Q of a Coil
The quality factor (Q) of a coil equals the ratio of reactive power to true power. or
Since I2 appears in both the numerator and the denominator of the right-hand fraction, it cancels, leaving:

27. Key Terms Ch.11 Summary Inductor Winding Induced voltage Inductance
An electrical device formed by a wire wound around a core having the property of inductance; also known as a coil.
Winding
The loops or turns of wire in an inductor.
Induced voltage
Voltage produced as a result of a changing magnetic field.
Inductance
The property of an inductor whereby a change in current causes the inductor to produce a voltage that opposes the change in current.

28. Key Terms Ch.11 Summary Henry (H) RL time constant Inductive reactance
The unit of inductance.
RL time constant
A fixed time interval set by the L and R values, that determines the time response of a circuit. It equals the ratio of L/R.
Inductive reactance
The opposition of an inductor to sinusoidal current, measured in ohms.
Quality factor (Q)
The ratio of reactive power to true power for an inductor.

29. Ch.11 Summary
Quiz
1. Assuming all other factors are the same, the inductance of an inductor will be larger if
a. more turns are added
b. the area is made larger
c. the length is shorter
d. all of the above

30. Ch.11 Summary
Quiz
2. The henry is defined as the inductance of a coil when
a constant current of one amp develops one volt.
one volt is induced due to a change in current of one amp per second.
one amp is induced due to a change in voltage of one volt.
the opposition to current is one ohm.

31. Quiz Ch.11 Summary 3. The symbol for a ferrite core inductor is a. b.

32. Quiz Ch.11 Summary 4. The symbol for a variable inductor is a. b. c.

33. Quiz Ch.11 Summary 5. The total inductance of a 270 mH inductor
connected in series with a 1.2 mH inductor is
a. 220 mH
b. 271 mH
c. 599 mH
d mH

34. Quiz Ch.11 Summary 6. The total inductance of a 270 mH inductor
connected in parallel with a 1.2 mH inductor is
a. 220 mH
b. 271 mH
c. 599 mH
d mH

35. Ch.11 Summary
Quiz
7. When an inductor is connected through a series resistor and switch to a dc voltage source, the voltage across the resistor after the switch closes has the shape of
a. a straight line
b. a rising exponential
c. a falling exponential
d. none of the above

36. Quiz Ch.11 Summary 8. For circuit shown, the time constant is
a ns
b ms
c ms
d s

37. Ch.11 Summary
Quiz
9. For circuit shown, assume the period of the square wave is 10 times longer than the time constant. The shape of the voltage across L is a. b. c. d.

38. Ch.11 Summary
Quiz
10. If a sine wave from a function generator is applied
to an inductor, the current will
a. lag voltage by 90o
b. lag voltage by 45o
c. be in phase with the voltage
d. none of the above

39. Answers Ch.11 Summary 1. d 2. b 6. a 3. d 7. b 4. c 8. a 5. d 9. c