Passive Electronic Components
Lectures 10
Page 1 of 18
20-May-2016
Inductors
Lecture Plan
1.
2.
3.
4.
5.
Physical basics
Parameters specified
Typical constructions
Typical applications
Selection guidlines
1. Physical basics
Self-inductance.
Suppose that the coil composed of N identical turns has the same flux  through each of the turns.
Faraday's law of electromagnetic induction states that
E  N
d
,
dt
where E is EMF in the coil, t - time.
From other side EMF in the coil may be represented in the terms of self-inductance L and coil current i
as the following:
E  L
di
.
dt
It follows from two above equations after integration and taking in account that  = 0 when i = 0 that
N  Li .
(1)
The term N is called flux linkage. Similar to capacitor with capacitance C, in which the
generated charge q is proportional to the applied voltage V ( q  CV ), the coil with N turns
produces a certain amount of magnetic flux . At that, flux linkage N is proportional to the
passing current i. Proportionality factor L (self-inductance) may be represented as
L
i
E
i
N
.
i
(2)
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Magnetic flux  unit is 1 Wb (Weber). It is named after German physicist Wilhelm Eduard
Weber (1804 – 1891). Self-inductance L unit is 1 H (Henry). It is named after Joseph Henry
(1797-1878), an American physicist.
1H = 1 s = 1 Vs/A.
Note: It follows from (2) that any formula for flux  calculation related to some coil (solenoid,
toroid etc.) may be transformed to the formula for self-inductance calculation for the same coil
by multiplying the initial formula by N/i.

 
Let us express  in (1) through flux density B using definition of magnetic flux   S B  dS

1  
and express i in (1) through flux density B using Ampere’s law N  i  
B  dl :


C 0
 
 
N  B  dS
N 2  B  dS
S
S
L

.


1
1
1  
B

d
l
B

d
l
 
N C  0 
C 0
S
(3)
C
S - cross-section of each turn in the coil measured in m2;
C - closed path that all the turns of the coil pass through it as shown in above picture.

B - flux density measured in Tesla (T). 1 T = 1Wb/ m2,
 0 - permeability constant. µ0 = 4π10-7 Tm/A = 4π10-7 H/m,
 - relative permeability of core material inside the coil.
Compare (3) with very similar expressions (4) for conductance and capacitance:

1 I
 
R U
S
b
 
E
  dl
a


 E  dS
, C
q

V

  0E  dS
S
 
E
  dl
S
S
.
(4)
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It follows from (3) that self-inductance (like conductance and capacitance) depends on geometry
(form, dimensions) of passive component and respective physical property of its material.
Permeability and magnetic susceptibility.
Permeability constant µ0 plays the same role in calculating of magnetic fields as the permittivity
constant  0 in calculating electric fields (  0  8.85 10 12 F/m – permittivity of free space).
They are linked through the speed of light c  3108 m/s:
c
1
 0 0
.
If the environment is some media differs from vacuum then



B   0 1   H   0 H ,
where

H - magnetic field strength measured in A/m,
 - unitless scalar called magnetic susceptibility. It is a measure of the ease with which magnetic
dipoles are formed in the medium.
The product  0  is called permeability of the medium.
Values of Magnetic Constants at Room Temperature

  1 
Material
(magnetic
(relative
susceptibility)
permeability)
Initial/Maximum
 10 5
Paramagnetic
Chromium
1.00033
33
Platinum
1.00026
26
Tungsten
1.000068
6.8
Aluminum
1.000022
2.2
Magnesium
1.000012
1.2
Diamagnetic
Mercury
0.999968
-3.2
Silver
0.999974
-2.6
Carbon (diamond)
0.999979
-2.1
Lead
0.999982
-1.8
Bismuth
0.999983
-1.7
Copper
0.9999903
-0.97
Ferromagnetic
Iron, 99.8% pure, annealed
150…5,000
Iron, 99.95% pure, annealed in H2
10,000…200,000
Cobalt, 99% pure, annealed
70…250
Nickel, 99% pure, annealed
110…600
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Solenoid.
Let us calculate integral
1  
B  dl for the path C shown in Fig.1. (The path encloses N turns of
 
C
0
wiring). Suppose that the solenoid is infinitely long. The sum of the integrals along the vertical

segments of C is zero because of orthogonality of magnetic field B to the vertical segments of C.

The values 1 ,  2 are arbitrary. Infinitely increasing  2 we reduce to zero B in recessive points and
respectively the integral along upper horizontal part of C. Integration along residuary lower horizontal
part of C gives
1  
1
B  dl 
Bz ( 1 )l
 
0 
0
C
This expression has to be valid for all possible values of 1 . We have to conclude that inside the coil
Bz ( 1 )  B  const .
(5)

(6)
Finally
1 
   B  dl
C
0

1
0 
Bl .
Finally, for infinite solenoid the inductance of unit of its length will be
L  0 N 2 BS  0 N 2 S


  0 n 2 S ,
l
Bl 2
l2
where n  N l (number of turns per unit of length).
It may be shown that solenoid with radius r and finite length l has inductance
L
where w  r l .
0 N 2 S 
l
8w w2 w4 5w6 35w8 
1 




... ,
2
4
16
64 
 3
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It is clear that for finite l and small w values
L
0 N 2 S
l

0 N 2 S
l
 0 n 2 Sl  0 n 2V .
Here V is the volume of the solenoid.
Toroidal inductor.
i
i
r
Suppose that outline dimensions of core cross-section S are significantly less than mean radius

r of the core. In this case B may be regarded as approximately constant inside the core.
It follows from (4) that
L
 0 N 2 BS  0 N 2 S

  0 n 2 S  2r   0 n 2V
2rB
2r
where n  N 2r is the number of turns per unit of core length, V is the volume of the core. The final
result is the same as for solenoid because when r   toroidal inductor approaches to infinitely long
solenoid.
Manufacturers of inductors use toroid core to derive a magnetic material characteristics. The
reason is the uniformity of magnetic field in a toroid (no gap).
Inductance of non-closed segment of conductor.
If not to consider the closed current loop the inductance may be understood from the following

energy approach. Electrical current i in elementary part dl of conductor segment l sets up


magnetic field dB in the point defined by vector r that originates from the point of

elementary part dl (the Biot-Savart law):
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
  0 i dl  r
dB 
 3 ;
4
r


B   dB.
l
The energy of the magnetic field is
W
2
1  
1
B

H
dV

B
dV .
2 V
2 0 V
From the other side the energy of the same magnetic field is proportional to i 2 . The
responding proportionality factor L is inductance:
1 2
Li ;
2
2
2W
2 1  
1
L  2  2   B  HdV 
B
dV .

i
i 2V
 0i 2 V
W 
For example the inductance of wire (IC bond wire for example) per unit length may be evaluated as
approximately 1nH/mm.
Example 1. Let us evaluate lead wire inductance using relationship for inductance of two-wire line
with length l, wire diameter 2R, distance between wires d:
 0 l d
ln ;

R
L  0 d

ln .
l

R
L
2R
d
Suppose that  = 1; d / R = 20:
L 4  10 7

ln 20  1.2  10 6 (H/m)  1.2nH/mm
l

Example 2. Suppose that the major part of parasitic inductance L in some leaded capacitor with
capacitance C is related to its leads and may be evaluated as 1nH/mm multiplied by total length of the
leads. Then self-resonant frequency f of the capacitor may be calculated as f  1 2 LC .
Calculation results are presented in the table below.


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CAPACITOR SELF-RESONANT FREQUENCY, MHz

f  1 2 LC
C, nF
0.5
1
10
100
300
500

Total lead length
6
12
25
mm
mm
mm
92
65
45
65
46
32
21
15
10
7
5
3
4
3
2
3
2
1
2. Equivalent circuit of inductor.
Fig. 2
Real inductor and its equivalent circuits
L – Inductance.
R – Effective resistance including DC resistance of wire, skin effect and core loses.
C – Parasitic capacitance.
Circuit A is used in the case of a ferromagnetic core material (ferrite, powdered iron).
Circuit B is used in the case of ferromagnetic core absence (“air core”).
Inductor equivalent circuits are parallel resonant circuits. Capacitor and inductor in the equivalent
circuit are connected in parallel to the voltage source. It is convenient to consider the combined
admittance.
Y  Z 1 ;
Z  R  iX ;
Y  G  iB.
Z – impedance;
R – resistance;
X – reactance;
Y – admittance;
G – conductance;
B – susceptance.
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C
L
Reactance
i

C
iL
Susceptance
iC

i
L
Suppose that inductor equivalent circuit is Circuit B (see Fig.2). Then
Y  iC 
1
R  iL
.
 iC  2
R  iL
R   2 L2
Parallel resonant frequency may be defined in two ways:
1. The frequency at which parallel impedance is a maximum.
2. The frequency at which current is in phase with the voltage.
In the second case resonance condition ImY  0 for circuit B (Fig.2) takes place when
  0 
1 
R 2C 
.
 1 
LC 
L 
Note that in contrast to series resonance frequency the parallel resonant frequency depends on the
effective resistance. But if R is relatively small the approximate equation for resonant frequency will
be the same as for series resonant circuit:
0 
1
.
LC
Parallel resonant results in the low combined admittance (high impedance).
Impedance/Frequency curves of real and ideal 10H inductor
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There are two kinds of inductors:
 High-Q type. The applications are oscillators, filters, highly resonant circuits.
 Low-Q type (“ferrite beads”, “RF chokes”). The application is EMI (electro-magnetic
interference) filtering in power line. High frequency EMI is dissipated in ferrite core as heat
loss.
2.1. Parameters of high Q inductors.
2.1.1. Inductance is a measure of the property of a circuit element which tends to oppose any
change in the current flowing through it. The inductance for a given inductor is influenced
by the core material, core shape and size, the turns count and the shape of the coil. Most
often the inductance is expressed in H.
2.1.2. Inductance tolerance. Standard inductance tolerances are typically designated by a
tolerance letter. Standard inductance tolerance letters include:
Letter
Tolerance, %
F
1
G
2
H
3
J
5
K
10
L
15 *)
M
20
*)
For some military products L designates 20%.
2.1.3. Test frequency of inductance - the frequency at which inductance is measured. The test
frequencies that are most widely used in the industry are the following:
Most of these test frequencies have been designated by military specifications. However, there are
some conflicting frequency assignments among the military specifications. There is a present trend to
assign test frequencies that match the user frequencies. This is particularly true for very low values.
These user frequencies do not match those listed above.
2.1.4. Quality factor. The quality factor or Q of an inductor is a measure of the relative losses in
an inductor (the ratio between stored and dissipated energy). It may be presented as the
ratio of inductor reactance to its effective resistance:
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Q
X X L L


R
R
R
Q-meter is a standard instrument used to measure both inductance and quality factor of small
RF inductors. The Q-meter is based on a stable, continuously variable oscillator and a resonant
circuit which is connected to the part to be tested. The quality factor is proportional to the voltage
across the internal calibrated variable capacitor. The voltage is measured by an internal RF
voltmeter. Test frequency range is approximately 22kHz…70MHz.
Since X and R are functions of frequency, the test frequency must be given when specifying Q.
X typically increases with frequency at a faster rate than R at lower frequencies, and vice versa at
higher frequencies. This results in a bell shaped curve for Q versus frequency. R is defined by (a)
DC resistance of the wire, (b) skin effect in the wire, (c) core losses.
Methods to increase Q:
 Use a larger diameter of wire. This decreases AC and DC resistance of the windings.
 Increase the permeability of the flux linkage path. This is most often done by winding the
inductor around a magnetic core material, such as iron or ferrite. A coil made in this
manner consists of fewer turns for a given inductance. It results in lower resistance of wire
and lower interwinding capacitance. But at high frequencies the high permeability
magnetic core has high internal loss.
2.1.5. Test frequency of quality factor. The frequency at which inductors are tested for Q.
2.1.6. Self-Resonant Frequency (SRF). It is parallel resonant frequency of inductor. At SRF the
inductor acts as a purely resistive high impedance element. It is desirable to increase
inductor’s SRF. SRF may be increased for example by spreading the winding. Air has a
lower dielectric constant than most insulators. Thus, an air gap between the windings
decreases the interwinding capacitance.
2.1.7. Direct Current Resistance (DCR). It is resistance of the inductor winding measured using
direct current. The DCR is most often minimized in the design of an inductor and specified
as a maximum rating.
2.1.8. Rated Direct Current. The level of continuous direct current that can be passed through
the inductor with no damage. This DC level is based on a maximum temperature rise in the
inductor at the maximum rated ambient temperature. The rated current can be increased by
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reducing the DC resistance or increasing the inductor size. For low frequency current
waveforms, the RMS AC can be substituted for the DC rated current.
2.2. Parameters of low-Q inductors (ferrite beads).
A ferrite bead may be regarded as a frequency dependent resistor. At high frequencies the
permeability and losses of ferrite vary with frequency (the permeability declines while the losses rise
to a broad peak as shown in the left plot below). This property can be used as a broad band filter.
2.2.1. Impedance. An equivalent circuit for a bead consists of a resistor and inductor in series.
The resulting change (of impedance over frequency) is directly associated with the
frequency dependent complex impedance of the ferrite material. As shown in example
(right plot below) at frequencies below 10MHz the inductive impedance is 10 or less. At
higher frequencies, the impedance increases to over 100, and becomes mostly resistive
above 100MHz. The impedance of the ferrite beads is the total resistance to the flow of
current, including equivalent resistance (skin effect, core losses, ohmic resistance) and
reactance. In a high-frequency range the equivalent resistance is dominant. In a lowfrequency range the inductance is dominant.
2.2.2. Impedance tolerance.
2.2.3. Test frequency of impedance.
2.2.4. Rated DC current.
3.
Typical constructions.
Typical constructions
of inductors
Wirewound
Leaded
Axial
Radial
Chip
Multilayer
Planar
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3.1.
Wirewound inductors.
3.2.
Multilayer inductors.
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3.3.
Planar inductors.
High Q planar inductors are manufactured using MEMS processes. There is a gap between the
winding and the substrate to prevent the energy loss in conductive silicon substrate.
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4. Typical applications.
Inductor applications
Analog signal
processing
Oscillatory circuits
EMI supression
Noise suppression in
signal line
(Common-mode choke)
Impedance matching
Power management
Energy accumulation
(DC/DC, AC/DC
converters)
Noise suppression in
power line
(Ferrite bead)
5. Selection guidelines.
5.1. Noise (EMI) suppression in DC circuits. (Murata)



DC power supply input section. A common mode choke coil is installed in the input
section of the DC power supply line to suppress common mode noise. (This coil can be
replaced with two ferrite bead inductors.)
Differential mode noise is suppressed by installing a three-terminal capacitor and ferrite bead
inductor in the supply line.
Video signal output section. Common mode noise transmitted to the video signal output
section is suppressed by using a common mode choke coil.
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5.2. Noise (EMI) suppression in AC power supply line. (Murata)
Common mode noise is suppressed by using a common mode choke coil and capacitor (line bypass
capacitor or Y-capacitor) installed between each line and the metallic casing. The Y-capacitor returns
noise to the noise source in the following order; Y-capacitor  metallic casing  stray capacitance 
noise source.
Differential mode noise is suppressed by installing capacitors (X capacitors) across the supply line.
5.3. Power management
BOOST REGULATOR (DC-DC). A basic DC-DC switching converter topology that takes an
unregulated input voltage, and produces a higher, regulated output voltage. This higher output voltage
is achieved by storing energy in an input inductor and then transferring the energy to the output by
turning a shunt switch (transistor) on and off.
BUCK REGULATOR (DC-DC). A basic DC-DC switching converter topology that takes an
unregulated input voltage, and produces a lower, regulated output voltage. This output voltage is
achieved by chopping the input voltage with a series connected switch (transistor) which applies
pulses to an averaging inductor and capacitor circuit.
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2.4. Analog signal processing.
Front end filter
5.4. Impedance matching.
Example: LC impedance matching network matches RC load to 50  source and line.
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Appendix 1
Magnetic materials
Iron Powder Cores. The material is based on finely defined particles of iron which are insulated from
each other but bound together with a binding compound. This mixture may be formed by molding
using high pressure and temperature. Powdered iron cores do not saturate easily, and have high core
temperature stability and Q. However, it is only in low relative permeability (below 75).
Ferrites. Ferrites are ceramics materials made of iron oxide combined with binders: nickel,
manganese, zinc or magnesium. Two major categories of ferrites are: (a) manganese zinc (low volume
resistivity, the core need external electrical isolation), (b) nickel zinc (high volume resistivity). Ferrites
are manufactured by homogeneously mixing the iron oxide with the binder, and heating the mixture
up to 1000°C. This causes partial decomposition of the carbonates and oxides. Then the mixture is dry
pressed into a core configuration, and finally sintered at 1500°C. Ferrites can be manufactured to
relative permeability of over 15, 000 with little eddy current losses. However, the high permeability of
the ferrite makes it unstable at high temperatures, and saturates easily. Driving ferrites with excessive
current may cause permanent damage to the core.
Ferrites are widely used as suppressors of unwanted high frequency signals. These ferrites are known
as EMI/RFI suppressors.
Laminated or Tape Wound Cores. These cores are manufactured by using different steel grades.
Tape wound cores have very high permeability and are used primarily in power transformers, reactors
in 60 Hz to 400 Hz. It provides very high flux densities and good temperature stabilities but is most
costly core to manufacture.
Appendix 2
Units of Magnetic Physical Quantities
Quantity name
Magnetic field strength
Magnetic flux
Magnetic flux density
Inductance
Permeability of free space
Quantity
symbol
H
Φ
B
L
µ0
Unit name
Ampere per meter
Weber
Tesla
Henry
Henry per meter
Unit
symbol
A m-1
Wb
T
H
H m-1
Base units
Am-1
kgm2 s-2  A-1
kgs-2 A-1
kgm2s-2A-2
kgms-2A-2