Chapter 11
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
S
N
field to change. This in
turn induces a voltage
across the inductor that
opposes the original
change in current.
The Basic Inductor
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
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 R1.
L
VS
SW
+
R1
−
−
+
R2
Lenz’s law
SW closes and immediately a voltage appears
across L that tends to oppose any change in current.
L
+
VS
−
+
SW
R1
R2
−
−
+
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 I2) as the voltage decays across the coil.
L
VS
SW
+
R1
R2
−
−
+
Later, the meter
reads a higher
current because of
the load change.
Practical inductors
In addition to inductance, actual inductors have
winding resistance (RW) due to the resistance of the
wire and winding capacitance (CW) between turns.
An equivalent circuit for a practical inductor
CW
including these effects is shown:
Notice that the winding resistance
is in series with the coil and the
winding capacitance is in parallel
with both.
RW
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 inductance
Four factors affect the amount of inductance for a
coil. The equation for the inductance of a coil is
N 2µ A
L=
l
where
L = inductance in henries
N = number of turns of wire
µ = permeability in H/m (same as Wb/At-m)
l = coil length on meters
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).
2
2
A = πr = π ( 0.0025 m ) = 7.85 × 10−5 m 2
N 2µ A
L=
l
2
(150 t ) ( 2.5 ×10−4 Wb/At-m )( 7.85 ×10−5 m 2 )
=
0.02 m
= 22 mH
Practical inductors
Inductors come in a variety of sizes. A few
common ones are shown here.
Encapsulated
Torroid coil
Variable
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
LT = L1 + L2 + L3 + ...Ln
If a 1.5 mH inductor is
connected in series with
an 680 µH inductor, the
total inductance is 2.18 mH
L1
1.5 mH
L2
680 µH
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
LT =
1
1 1 1
1
+ + + ... +
L1 L2 L3
LT
The total inductance of two inductors is
LT =
1
1 1
+
L1 L2
…or you can use the product-over-sum rule.
Parallel inductors
If a 1.5 mH inductor is connected in
parallel with an 680 µH inductor,
the total inductance is 468 µH
L1
1.5 mH
L2
680 µH
Inductors in dc circuits
When an inductor is connected
in series with a resistor and dc
source, the current change is
exponential.
Vinitial
t
0
Inductor voltage after switch closure
Ifinal
R
L
0
Current after switch closure
t
Inductors 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.
VS
VL
R
VS
L
VR
Universal exponential curves
L
τ=
R
100%
95%
98%
99%
86%
80%
Percent of final value
Specific values for
current and voltage
can be read from a
universal curve. For
an RL circuit, the
time constant is
Rising exponential
63%
60%
40%
37%
Falling exponential
20%
14%
5%
0
0
1τ
2%
2τ
3τ
4τ
Number of time constants
1%
5τ
Universal exponential curves
The curves can give
specific information
about an RL circuit.
Read the rising
exponential at the
67% level. After 1.1 τ
95%
99%
63%
60%
40%
37%
20%
14%
5%
0
0
98%
86%
80%
Percent of final value
In a series RL circuit,
when is VR > 2VL?
100%
1τ
2%
2τ
3τ
4τ
Number of time constants
1%
5τ
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
i =IF + (Ii − IF)e−Rt/L
IF = final value of current
Ii = initial value of current
i = 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.
Inductive reactance
Inductive reactance is the opposition to
ac by an inductor. The equation for
inductive reactance is
X L = 2πfL
The reactance of a 33 µH inductor when a
frequency of 550 kHz is applied is 114 Ω
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 90o.
VL 0
90°
I 0
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:
Ptrue = (Irms)2RW
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:
Pr=VrmsIrms
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.
I2XL
Q= 2
I RW
For a series circuit, I cancels, leaving
XL
Q=
RW