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

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

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

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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.

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

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

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

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

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 2.71828) 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.

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

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

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

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.

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).

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:

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.

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.

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

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.

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

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

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. 1.47 mH

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. 1.47 mH

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

Quiz Ch.11 Summary 8. For circuit shown, the time constant is a. 270 ns b. 270 ms c. 270 ms d. 3.70 s

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.

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

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