ISU EE C.Y. Lee

Chapter 11Inductors

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Objectives

Describe the basic structure and characteristics of an inductorDiscuss various types of inductorsAnalyze series inductorsAnalyze parallel inductorsAnalyze inductive dc switching circuitsAnalyze inductive ac circuits

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The Basic Inductor

When there is current through the coil, a three- dimensional electromagnetic field is created, surrounding the coil in all directions

When a length of wire is formed onto a coil, it becomes a basic inductor

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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

In many practical applications, mH (10-3 H) or μH (10-6 H) are the more common units.

dt)t(diL)t(v =

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Basics of a Inductance

An inductor stores energy in the magnetic field created by the current:

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

2

21 LIU = (J)

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Physical Characteristics

Inductance is directly proportional to the permeability of the core materialInductance is directly proportional to the cross-sectional area of the coreInductance is directly proportional to the square of the number of turns of wireInductance is inversely proportional to the length of the core material

lANL

2

μ= (H)

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Physical CharacteristicsExample: Determine the inductance of the coil

A = πr2 = π(0.25×10-2)2 = 1.96×10-5 m2

L = μN2A/l = (0.25×10-3)(350)2(1.96×10-5)/(0.015) = 40 mH

μ

= 0.25×10-3 H/m

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Winding Resistance

When many turns of wire are used to construct a coil, the total resistance may be significantThe inherent resistance is called the dc resistance or the winding resistance (RW )

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Winding CapacitanceWhen two conductors are placed side-by-side, there is always some capacitance between themWhen many turns of wire are placed close together in a coil, there is a winding capacitance (CW )CW becomes significant at high frequencies

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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

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Lenz’s Laws

Example of Lenz’s law:

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Lenz’s Laws

Example of Lenz’s law:

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Typical Inductors

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Series Inductors

When inductors are connected in series, the total inductance increases

LT = L1 + L2 + L3 + … + Ln

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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

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Inductors in DC CircuitsWhen there is constant current in an inductor, there is no induced voltageThere is a voltage drop in the circuit due to the winding resistance of the coilInductance itself appears as a short to dc

I I

V=0 +−

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Inductors in DC CircuitsIllustration of the exponential buildup of current in an inductor

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Inductors in DC CircuitsIllustration of the exponential decrease of current in an inductor

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RL Time ConstantBecause the inductor’s basic action opposes a change in its current, it follows that current cannot change instantaneously in an inductorThe 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+

− +

−

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Energizing Current in an InductorIn 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 closedThe current reaches its final value in approximately 5τ

( )LtRF eIi /1 −−=

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De-energizing Current in an InductorIn 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 closedThe current reaches 1% of its initial value in approximately 5τ; considered to be equal to 0

LtRieIi /−=

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RL Time ConstantExample: Determine the inductor current 30 μs after the

switch is closed

τ

= L/R = 100mH/2.2kΩ

= 45.5 μsIF = Vs /R = 12V/2.2kΩ

= 5.45 mA

iL = IF (1−e−Rt/L) = (5.45)(1−e−0.66) = 2.63 mA

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RL Time ConstantExample: Determine the inductor current 2 ms after the

switch(1) is opened and switch(2) is closed

τ

= L/R = 200mH/56Ω

= 3.57 msIi = Vs /R = 5V/56Ω

= 89.3 mA

iL = Ii (e−Rt/L) = (89.3)(e−0.56) = 51.0 mA

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Analysis of Inductive AC Circuit

The current lags the voltage by 90°

in a

purely inductive ac circuit

dt)t(diL)t(v =

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Ohm’s Law in Inductive AC Circuits

The reactance (XL ) of a inductor is analogous to the resistance (R) of a resistor

V = IRV = IXL

V = IZL = I(jωL) = I(jXL )Magnitude →

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Inductive Reactance, XL

The relationship between inductive reactance, inductance and frequency is:

XL = 2π

f L (Ω)

Example: Determine the inductive reactance

XL = 2π

f L= 2π(1×103)(5×10-3)= 31.4 kΩ

1 kHz

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Inductive Reactance, XL

fLV

XVI

L π2==

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Inductive Reactance, XL

fLV

XVI

L π2==

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Inductive Reactance, XL

Example: Determine the rms current in below circuit

XL = 2π(10×103)(100×10-3) = 6283 ΩVrms = Irms XL

Irms = Vrms /XL = (5V)/(6283Ω) = 796 μA

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Power in a InductorInstantaneous power (p) is the product of v and i True power (Ptrue ) is zero, since no net energy is lost due to conversion to heat in the inductor

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Power in a InductorThe rate at which an inductor stores or returns energy is called reactive power (Pr ); units: (VAR)

These equations are of the same form as those introduced in Chapter 3 for true power in a resistor

rmsrmsr VIP =

L

rmsr X

VP2

=

Lrmsr XIP 2=

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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 = XL /RW

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Inductor Applications

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

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Summary

Self-inductance is a measure of a coil’s ability to establish an induced voltage as a result of a change in its currentInductance 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 coreAn inductor opposes a change in its own current

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Summary

Faraday’s law states that relative motion between a magnetic field and a coil induces voltage across the coilLenz’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

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Summary

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

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Summary

Inductors add in seriesTotal parallel inductance is less than that of the smallest inductor in parallelCurrent lags voltage by 90°

in an inductor

Inductive reactance (XL ) is directly proportional to frequency and inductanceAn inductor blocks high-frequency ac