chapter 11johnston/me204/ee204 power... · 2011-11-07 · chapter 11 inductors. the basic inductor...

Chapter 11Chapter 11Inductors

The Basic Inductor

• When a length of wire is formed onto a coil, it

becomes a basic inductor

• Magnetic lines of force around each loop in the

winding of the coil effectively add to the lines of winding of the coil effectively add to the lines of

force around the adjoining loops, forming a strong

electromagnetic field within and around the coil

• 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

Factors that determine the inductance of a coil

Thomas L. Floyd

Electronics Fundamentals, 6e

Electric Circuit Fundamentals, 6e

Upper Saddle River, New Jersey 07458

Inductors in DC Circuits

• When there is constant (DC) current in an

inductor, there is no induced voltage

• There is a slight voltage drop in the circuit • There is a slight voltage drop in the circuit

due to the winding resistance of the coil

• The coil basically appears as a short to DC

FIGURE 11-2 Symbol for the inductor.

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Basic Symbol for a Coil

FIGURE 11-9 Symbols for fixed and variable inductors.

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FIGURE 11-10 Inductor symbols.

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Typical Inductors

Winding resistance of a coil

•The small inherent wire resistance is called the dc

resistance or the winding resistance (RW)

•It is usually ignored

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Winding capacitance of a coil

•When many turns of wire are placed close

together in a coil, there is a winding capacitance

•CW may become significant at high

frequencies

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Self-Inductance

• Inductance is a measure of a coil’s ability to

establish an induced voltage as a result of a

change in its currentchange in its current

• The induced voltage opposes the change in

current (vind)

Faraday’s and Lenz’s Laws

• Recall 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 coilwith respect to the coil

• Recall Lenz’s law:

– When the current through a coil changes:

• An induced voltage is created as a result of the changing electromagnetic field

• The direction/polarity of the induced voltage is such that it always opposes the change in current

Induced Voltage in the Series RL

Circuit

• At the instant of switch closure, the inductor

effectively acts as an open with all the source

voltage across it

• During the first 5 time constants, the current is • During the first 5 time constants, the current is

building up exponentially, and the induced coil

voltage is decreasing

• The resistor voltage increases with current

• After 5 time constants, all of the source voltage is

dropped across the resistor and none across the coil

Coil appears

as an open

Resistor and Coil Voltages in an Energizing Circuit

Thomas L. Floyd

Coil appears

as a short

10V instantaneously

induced in the coil

trying to keep current

flowing in same

direction

Resistor and Coil Voltages in a De-Energizing Circuit

Thomas L. Floyd

After Vind decays,

there is no current

RL Time Constant

• Because the inductor’s basic action opposes

a change in its current, it follows that

current cannot change instantaneously in an current cannot change instantaneously in an

inductor

ττττ = L/R

where: τ is in seconds (s)

L is in henries (H)

R is in ohms (Ω)

Exponential Formulas

• RL Time Constant = L/R

• The special formula for determining instantaneous

current values for an RL circuit from zero is:

i =I (1 - e-Rt/L ) => I (1 - )i =IF (1 - e-Rt/L ) => IF (1 - )

• The special formula for determining instantaneous

current values an RL circuit to zero is:

i =Iie-Rt/L => Ii

Energizing Current in an Inductor

• In a series RL circuit, the current will increase to

approximately 63% of its full value in one time-

constant (τ) interval after the switch is closed

• The current reaches its final value in • The current reaches its final value in

approximately 5τ

Circuit Current in a an Energizing Circuit

Thomas L. Floyd

Energizing current in an inductor

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De-energizing Current in an

Inductor

• In a series RL circuit, the current will decrease to

approximately 37% of its fully charged value one

time-constant (τ) interval after the switch is closed

• The current reaches 1% of its initial value in • The current reaches 1% of its initial value in

approximately 5τ; considered to be equal to 0

Circuit Current in a De-Energizing Circuit

At t = 0,

instantaneous

current

remains the

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De-energizing current in an inductor

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Series Inductors

• When inductors are connected in series, the total

inductance increases

LT = L1 + L2 + L3 + … + Ln

Parallel Inductors

• When inductors are connected in parallel, the total

inductance is less than the smallest inductance

1/LT = 1/L1 + 1/L2 + 1/L3 + … + 1/Ln

Inductors in AC Circuits

• An increase in frequency induces more

voltage across the inductor in a direction to

oppose the current and causes it to decrease oppose the current and causes it to decrease

in amplitude

• 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

Inductive Reactance

• Inductive reactance is the opposition to

sinusoidal current, expressed in ohms

• The inductor offers opposition to current, • The inductor offers opposition to current,

and that opposition varies directly with

frequency

• The formula for inductive reactance, XL, is:

XL = 2πf L

The current in an inductive circuit varies inversely with the frequency of the source voltage

Thomas L. Floyd

Higher Frequency = Higher XL

For a fixed voltage and fixed frequency, the current varies inversely with the inductance value

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More Inductance = Higher XL

Phase Relationship of Current

and Voltage in an Inductor

• The voltage leads inductor current by 90° (ELI)

• The current lags inductor voltage by 90°

• The curves below are for a purely inductive circuit• The curves below are for a purely inductive circuit

Current is always lagging the inductor voltage by 90º

Current Lags

the Voltage by

90 Deg

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Voltage Leads

the Current by

90 Deg (ELI)

Power in an Inductor

• Instantaneous power (p) - the product of v and i

gives instantaneous power

• True Power (Ptrue) - ideally is zero, since all power

stored by an inductor in the positive portion of the true

stored by an inductor in the positive portion of the

power cycle is returned to the source during the

negative portion (is shifted back and forth). If

winding resistance is taken into account, the true

power is:

Ptrue = (Irms)2RW

Power curve for an inductor

Reactive

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•Power is maximum when I and V are

equal (either positive or negative)

•Power is zero when either I or V are

Instantaneous Values

Reactive Power

• The rate at which an inductor stores or

returns power is called its reactive power

(Pr), with units of VAR (volt-ampere (Pr), with units of VAR (volt-ampere

reactive)

Pr = VrmsIrms

Pr = V2rms/XL or

Pr = I2rmsXL

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

• A theoretically perfect coil would have a Q of 1• A theoretically perfect coil would have a Q of 1

Q = (reactive power) / (true power)

Q = XL/RW

Checking a coil by measuring the resistance

Troubleshooting:

•Coils usually open

•Some windings short out from time to time effecting performance

Thomas L. Floyd

Basic capacitor power supply filter with a series inductor

Power Supply Application

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An inductor used as an RF choke to minimize interfering signals on the power supply line

Power Supply Application

Thomas L. Floyd

chapter 11johnston/me204/ee204 power... · 2011-11-07 · chapter 11 inductors. the basic inductor...

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