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• Course overview and information 09/16/2010 © 2010 NTUST

Magnetic Qualities

• Magnetic fields are described by drawing flux lines that represent the magnetic field. • Where lines are close together, the flux density is higher.

• Where lines are further apart , the flux density is lower.

The Magnetic Field

• Magnetic fields are composed of invisible lines of force that radiate from the north pole to the south pole of a magnetic material. • Field lines can be visualized with the aid of iron filings sprinkled in a magnetic field.

Relative Motion Relative motion

• When a wire is moved across a magnetic field , there is a relative motion between the wire and the magnetic field.

• When a magnetic field is moved past a stationary wire , there is also relative motion.

• In either case, the relative motion results in an induced voltage in the wire.

Induced Voltage

• The induced voltage due to the relative motion between the conductor and the magnetic field when the motion is perpendicular to the field is dependent on three factors: • the relative velocity (motion is perpendicular) • the length of the conductor in the magnetic field • the flux density

Faraday’s Law

• Faraday experimented with generating current by relative motion between a magnet and a coil of wire . The amount of voltage induced across a coil is determined by two factors: S

1. The rate of change of the

N

magnetic flux with respect to the coil .

-

V

+

Voltage is indicated only when magnet is moving.

Faraday’s Law

• Faraday factors: also experimented generating current by relative motion between a magnet and a coil of wire.

The amount of voltage induced across a coil is determined by two S 1. The rate of change of the N to the coil.

2. The number of turns of wire in the coil.

-

V

+

Voltage is indicated only when magnet is moving.

Magnetic Field around a Coil

• Just as a moving magnetic field electromagnet a permanent magnet.

induces a voltage, current in a coil causes a magnetic field. The coil acts as an , with a north and south pole as in the case of

South North

The Basic

• One henry 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 coils are much smaller than 1 H. • The effect of inductance is greatly magnified by adding turns and winding them on a magnetic material . Large inductors and transformers are wound on a core to increase the inductance.

Magnetic core

The Basic Inductor

• When a length of wire is formed into a coil., it becomes a basic 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

Faraday’s Law

• Faraday’s law was introduced in Chapter 7 and repeated here because of its importance to inductors. • 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.

Lenz’s Law

• Lenz’s law was also introduced in Chapter 7 and is an extension of Faraday’s law, defining the direction of the induced voltage: • When the current through a coil changes and 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.

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

R

1 .

L

SW

V

S + 

R

1

R

2  +

Lenz’s Law

SW closes and immediately a voltage appears across

L

that tends to oppose any

change

in current.

+

L

 SW

V

S + 

R

1

R

2  + Initially, the meter reads same current as before the switch was closed.

Lenz’s Law

After a time, the current stabilizes at a higher level (due to

I

2 ) as the voltage decays across the coil.

L

SW

V

S + 

R

1

R

2  + Later, the meter reads a higher current because of the load change.

Practical

• • In addition to inductance , actual inductors have winding resistance (

R

W ) due to the resistance of the wire and winding capacitance (

C

W ) between turns. An equivalent circuit for a practical inductor including these effects is shown:

C

W Notice that the winding resistance is in series with the coil and the winding capacitance is in parallel with both.

R

W

L

Types of Inductors

• 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

• Four factors affect the amount of inductance for a coil. The equation for the inductance of a coil is

L



N

2 

A l

where

L

= inductance in henries

N

 = number of turns of wire = permeability in H/m (same as Wb/At-m)

l

= coil length on meters

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).

N

2 

A L

 

l

 150 t  2  

22 mH

 4 0.02 m   5 2 

Practical

• Inductors come in a variety of sizes. A few common ones are shown here. Encapsulated Torroid coil Variable

Inductor

Series Inductors

Series Inductors

• When inductors are connected in series , the total inductance is the sum of the individual inductors. The general equation for inductors in series is

L

T   1

L

2 

L

3  ...

L n

If a 1.5 mH inductor is connected in series with an 680  H inductor, the total inductance is

2.18 mH

.

6 

Parallel inductors

• When inductors are connected in parallel , the total inductance is smaller than the smallest one. The general equation for inductors in parallel is

L

T  1 1

L

1  1

L

2  1

L

3 1

L

T The total inductance of two inductors is •

L

T  1 1

L

1  1

L

2 …or you can use the product-over-sum rule.

Parallel Inductors

Parallel Inductors

If a 1.5 mH inductor is connected in parallel with an 680  H inductor, the total inductance is

468



H

Charging

Time Constant

Inductors in DC Circuit

• When an inductor is connected in series with a resistor and dc source , the current change is exponential.

V

initial

0

t

Inductor voltage after switch closure

I

final

R L

0 Current after switch closure

t

Discharging

Inductor in DC Circuits

• The same shape curves are seen if a square wave is used for the source. Pulse response is covered further in Chapter 20.

R L V S

V S V R V L

Universal Exponential

• Specific values for current and voltage can be read from a universal curve. For an

RL

circuit, the time constant is τ 

L R

100% 80% 60% 40% 20% 0 0 63% 95% 98% 99% 86%

Rising exponential

1 t 37%

Falling exponential

14% 5% 2 t 3 t Number of time constants 4 t 2% 5 t 1%

Universal Exponential

• The curves can give specific information about an

RL

circuit.

In a series

RL

circuit, when is

V

R > 2

V

L ? Read the rising exponential at the 67% level.

After 1.1

t 100% 80% 60% 40% 20% 0 0 63% 86% 95% 98% 99% 37% 1 t 14% 5% 2 t 3 t Number of time constants 4 t 2% 5 t 1%

Universal Exponential

• The universal curves can be applied to general formulas for the current (or voltage) curves for

RL

circuits . The general current formula is

i =I

F + (

I i



I

F )

e



Rt/L i I

F

I i

= final value of current = initial value of current = instantaneous value of current • The final current is greater than the initial current when the inductive field is building, or less than the initial current when the field is collapsing.

Examples

Examples

Examples

Inductor Impedance

Inductive Reactance

• Inductive reactance is the opposition to ac by an inductor. The equation for inductive reactance is

X L

 2π

fL

The reactance of a 33  H inductor when a frequency of 550 kHz is applied is

114

W

Inductive Phase Shift

• 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 90 o .

V L

0

I

0 90 

Power in An Inductor

• True Power: Ideally, inductors do not dissipate power. However, a small amount of power is dissipated in winding resistance given by the equation: •

P

true = (

I

rms ) 2

R

W Reactive Power: Reactive power is a measure of the rate at which the inductor stores and returns energy. One form of the reactive power equation is: •

P r

=

V

rms

I

rms The unit for reactive power is the VAR.

Q of a Coil

• The quality factor (

Q

) of a coil is given by the ratio of reactive power to true power. • For a series circuit,

I

cancels, leaving

Selected Key Terms

Inductor

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.

Selected Key Terms

Henry (

H

)

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. The unit is the ohm.

Quality factor

The ratio of reactive power to true power for an inductor.

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

Quiz

2. The henry is defined as the inductance of a coil when a. a constant current of one amp develops one volt.

b. one volt is induced due to a change in current of one amp per second.

c. one amp is induced due to a change in voltage of one volt.

d. the opposition to current is one ohm.

Quiz

3. The symbol for a ferrite core inductor is a.

b.

c.

d.

Quiz

4. The symbol for a variable inductor is a.

b.

c.

d.

Quiz

5. The total inductance of a 270



H inductor connected in series with a 1.2 mH inductor is a. 220



H b. 271



H c. 599



H d. 1.47 mH

Quiz

6. The total inductance of a 270



H inductor connected in parallel with a 1.2 mH inductor is a. 220



H b. 271



H c. 599



H d. 1.47 mH

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

8. For circuit shown, the time constant is a. 270 ns

L

b. 270



s

2 0

c. 270 ms

1 V

d. 3.70 s

R

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.

L

b.

c.

d.

V

S

R

Quiz

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

Quiz

Answers: 1. d 6. a 2. b 7. b 3. d 8. a 4. c 9. c 5. d 10. a