Chapter 11
Inductors
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C.Y. Lee
Objectives
Describe the basic structure and characteristics of
an inductor
Discuss various types of inductors
Analyze series inductors
Analyze parallel inductors
Analyze inductive dc switching circuits
Analyze inductive ac circuits
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The Basic Inductor
When a length of wire is formed onto a coil, it
becomes a basic inductor
When there is current
through the coil, a threedimensional
electromagnetic field is
created, surrounding the
coil in all directions
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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
di ( t )
v( t ) = L
dt
In many practical applications, mH (10-3 H) or μH
(10-6 H) are the more common units.
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Basics of a Inductance
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
An inductor stores energy in the magnetic field
created by the current: U = 1 LI 2 (J)
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Physical Characteristics
Inductance is directly proportional to
the permeability of the core material
Inductance is directly proportional to
the cross-sectional area of the core
Inductance is directly proportional to
the square of the number of turns of
wire
Inductance is inversely proportional
to the length of the core material
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2
N A
L=μ
l
(H)
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Physical Characteristics
Example: Determine the inductance of the coil
μ = 0.25×10-3 H/m
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
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Winding Resistance
When many turns of wire are used to construct a
coil, the total resistance may be significant
The inherent resistance is called the dc resistance
or the winding resistance (RW)
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Winding Capacitance
When two conductors are placed side-by-side,
there is always some capacitance between them
When 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 Circuits
When there is constant current in an inductor, there is
no induced voltage
There is a voltage drop in the circuit due to the winding
resistance of the coil
Inductance itself appears as a short to dc
−
I
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Inductors in DC Circuits
Illustration of the exponential buildup of current in an inductor
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Inductors in DC Circuits
Illustration of the exponential decrease of current in an inductor
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RL Time Constant
Because the inductor’s basic action opposes a change in
its current, it follows that current cannot change
instantaneously in an inductor
The 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
−
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Energizing Current in an Inductor
In a series RL circuit, the current will increase to
approximately 63% of its full value in one timeconstant (τ) interval after the switch is closed
The current reaches its final value in approximately 5τ
(
i = I F 1 − e − tR / L
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De-energizing Current in an Inductor
In 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 closed
The current reaches 1% of its initial value in
approximately 5τ; considered to be equal to 0
i = I i e − tR / L
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RL Time Constant
Example: Determine the inductor current 30 μs after the
switch is closed
τ = L/R = 100mH/2.2kΩ = 45.5 μs
IF = 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 Constant
Example: Determine the inductor current 2 ms after the
switch(1) is opened and switch(2) is closed
τ = L/R = 200mH/56Ω = 3.57 ms
Ii = 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
di ( t )
v( t ) = L
dt
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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 = IZL = I(jωL) = I(jXL)
Magnitude → V = IX
L
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V = IR
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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
V
V
=
I=
X L 2πfL
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Inductive Reactance, XL
V
V
=
I=
X L 2πfL
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Inductive Reactance, XL
Example: Determine the rms current in below circuit
XL = 2π(10×103)(100×10-3) = 6283 Ω
Vrms = IrmsXL
Irms = Vrms/XL = (5V)/(6283Ω) = 796 μA
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Power in a Inductor
Instantaneous 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 Inductor
The rate at which an inductor stores or returns
energy is called reactive power (Pr); units: (VAR)
Pr = I rmsVrms
2
Vrms
Pr =
XL
2
Pr = I rms
XL
These equations are of the same form as those
introduced in Chapter 3 for true power in a resistor
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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 current
Inductance 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 core
An 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 coil
Lenz’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 field
The time constant for a series RL circuit is the
inductance divided by the resistance
In 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 series
Total parallel inductance is less than that of the
smallest inductor in parallel
Current lags voltage by 90° in an inductor
Inductive reactance (XL) is directly proportional to
frequency and inductance
An inductor blocks high-frequency ac
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