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Lecture 10
Coils (Inductors)
Passive Electronic Components and Circuits (PECC)
V. Bande, Applied Electronics Department
www.ael.utcluj.ro (English version)-> Information for students
1
Coils
Coils
Short history
Electrical Properties
Constructive elements for coils
Parameters
Categories
Transformers
Short history
• 1821 – Michael Faraday reveals the magnetic field lines which occur
around a conductive material through which an electric current flows.
• 1825 – William Sturgeon builds the very first electromagnet.
• 1831 – independently, Michael Faraday and Joseph Henry discover the
magnetic induction law.
 Afterwards, Faraday builds the first electrical engine, the first
electrical generator and the first transformer.
 Henry builds the first telegraph improved later by Morse.
• 1876 – Bell invents the first telephone and the first electromagnetic
phonograph.
Coils
Coils
Short history
Electrical Properties
Constructive elements for coils
Parameters
Categories
Transformers
Electrical properties
• The inductance of a coil is strongly
dependent of the coil’s geometry and the
magnetic properties of the environment in
which it is placed.
• Equation 1 is suitable for the case in which
the length “l” of the coil is higher then its
diameter “2rc”.
A 2
L  0   r  N
l
(  0 N 2 rc )
L
Henries
l
8r
L   0 N 2 rc {ln( c )  2}
rw
2
• Equation 2 is suitable for the case in which
the length “l” of the coil is lower then its
diameter “2rc”. The “rw” quantity
represents the winding wire’s diameter.
(1)
(2)
0  4 10 7 [Wb  A1  m 1 ]
Electrical properties
• The inductance is dependent of the coil’s geometry (l, d=2r, h in mm).
All the above formulas are available when the environment in which
the coil is placed is the air.
L  0,001
N 2d
l
0,44 
d
[μH]
N 2d 2
L  0,008
[μH]
3d  9l  10h
Electrical properties
• The inductance is dependent of the
distance between its turns.
• The inductance is dependent of the
magnetic
properties
of
the
environment in which the coil is
placed – magnetic permeability, μ.







Air: 1.257x10-6H/m.
Ferrite U M33: 9.42x10-4H/m.
Nickel: 7.54x10-4H/m.
Iron: 6.28x10-3H/m.
Ferrite T38: 1.26x10-2H/m.
Orr: 5.03x10-2H/m.
Super Malloy: 1.26H/m.
Electrical properties
 Equivalent circuit
ZL 
R p  jL
1  2LC p  jR pCp
Electrical properties
 Frequency characteristic
ZL
Rp  Q
L
zona inductiva
10%
1
C
Rp
10%
2, 2 R p
L
0,3  0
0 
1
LCp

Electrical properties
 Dimensioning the inductance:
km
l
8
6
D
4
2
p/d
0
p
d
(ND) 2
L0 
 107 [H]
l  0,45  D
L1  L0  L
1
2
3
4
L  k m  N  D  107 [H]
Electrical properties
 The parasitic capacitance value calculus:
Cp[pF]
30
20
10
7
5
D=10cm
3
D=8cm
2
D=6cm
D=4cm
1
D=2cm
0,7
0,5
p/d
1 1,1
1,3 1,5
1,7
2
2,5
3
3,5
4
Electrical properties
 Steps in designing a coil:
• The procedure starts from the
desired value for the inductance –
“L”, its diameter “D” and the
frequency domain in which it will
work – “ω0”.
L,  0 , D

C p max 
1
L02
• From the last slide characteristic
the maximum value of the
parasitic capacitance is chosen.

• The turns number can be
calculated in respect with the coil’s
geometrical
dimensions
by
resolving the adjacent equation.
N 2D
L[μH] 
 0,1k m ND
dp 
0,44  N   1
Dd 
p / d  km

Calculate the length of a coil with a 2 cm diameter and an inductance
of 50 H which is executed in a single layer and will have a desirable
parasitic capacitance lower then 2pF.
Coils
Coils
Short history
Electrical Properties
Constructive elements for coils
Parameters
Categories
Transformers
Constructive elements for coils
• The winding (the turns).
• The casing.
• The impregnation (soak) material.
• The core.
without a core
Iron core
Ferrite core
Constructive elements for coils
 The coil’s winding
• The most common used material for the conductive winding wire
is the copper (due to its electrical and mechanical properties) and
rarely aluminum.
• The conductive wires are being insulated to avoid short-circuits
between two adjacent turns. The materials used for the insulation
are enamels (ro: emailuri) – different composition varnishes, textile
fibers (silk, cotton) or mineral fibers (glass fiber). The insulating
material type is generally chosen in strong dependency with the
conductive wire’s estimated maximum temperature. The most
thermal resistant materials are the glass fibers, the most
susceptible ones are the textile fibers.
Constructive elements for coils
 The coil’s winding
• The coil’s winding diameter can be estimated following 2 criteria:
 The maximum estimated value of the current that passes through
the conductive materials is inferiorly limiting this parameter due
to excessive heating possibility.
 The parasitic resistance maximum value introduces a
supplementary limitation for the windings diameter.
• At high frequencies, due to the pellicle effect, stranded (ro: litat) wires
are used (thin bundles of wires – ro: manunchiuri de fire) or silvered
copper wires.
• The conductive winding wires are being delivered by the producers
in a standardized fashion: 0.05 mm, 0.07 mm, 0.1 mm, … , 2 mm. The
thickness of the insulating material is not included in the above
values.
Constructive elements for coils
 The coil’s casing
• The coil’s casing fulfill the role of keeping the stiffness of the
winding.
• The materials used must have adequate electrical (dielectric rigidity,
low dielectric losses) and mechanical properties (thermal and
humidity stability).
• Examples – ascending order of theirs performances: electroinsulating carton, pertinax, textolit, thermo-rigid materials (Bakelite),
thermo-plastic materials (polystyrene, polyethylene, Teflon) or
ceramic materials.
• From the geometrically point of view the material can be of different
cross-sections: circular, square, rectangular.
• At very high frequencies, the coils can be made without a casing.
Constructive elements for coils
 The coil’s impregnation (soak) material
• The impregnation material has the role to protect the coil against
humidity and also realizes a supplementary stiffness (especially for
the cases when the coils are not using casings).
• The impregnation advantages:
 Winding stiffness.
 Improves heat dissipation.
 Improves dielectric properties of the turns insulation.
 Avoids humidity penetration between turns.
• The impregnation disadvantages: can concur to a higher parasitic
capacitance (by growing the electrical permittivity of the insulating
material between the turns).
Constructive elements for coils
 The coil’s core
• To increase the obtained inductance, magnetic cores are being
displaced inside coil’s winding. So a magnetic circuit is created
which has the major contribution in concentrating the magnetic field
lines. In this way the magnetic flux increases, almost all the magnetic
lines intersect the turns, in conclusion the coil’s inductance increases.
• The magnetic materials have a non-linear behavior when are being
placed in an exterior magnetic field. This non-linearity is related
with the dependency of the magnetic induction “b” and the
magnetic field intensity “H”. The ratio between the above two
quantities is the environment's magnetic permeability:

B
1 B
; r 
H
0 H
Constructive elements for coils
 The magnetic materials properties – Hysteresis phenomena
• Hc – coercive field – cancels the
magnetic induction.
• Br
–
residual
induction.
B
Bs
Bm
magnetic
• Hs – the magnetic field intensity
at which saturation phenomena
occurs.
• Bs – the magnetic induction at
saturation point.
Br
-Hs -Hm
0
-Hc
H c Hm H s
-Br
-Bm
-Bs
H
Constructive elements for coils
 The magnetic materials properties – Hysteresis phenomena
• The magnetic materials have atoms with an own magnetic
moment, and the neighbor atomic moments are being orientated
identically, so the material will have a residual magnetization.
• When applying an exterior magnetic field, a reorientation of the
magnetic domains occurs. The exterior field intensity at which
the magnetic induction is canceling is called coercive field. When
“H” increases, the saturation phenomena occurs (“B” remains
constant).
• The phenomena are dependent with the direction on which the
magnetic field changes (hysteresis).
• The residual magnetization exhibits until a certain temperature
(Curie temperature) at which the thermal agitation destroys the
well-ordered orientation domains.
Constructive elements for coils
 Magnetic materials applications
• Soft magnetic materials – Hc<80A/m (narrow hysteresis)
• Rough magnetic materials – Hc>80A/m (wide hysteresis)
• Soft magnetic materials with the ratio Br/Bm (ratio which
characterizes the hysteresis inclination) lower then 0.5 are used
for constant inductances, the ones with 0.5< Br/Bm<0.8 for
common used cores, the ones with Br/Bm>0.8 (rectangular
hysteresis) are used in switching and memorizing applications.
• Rough magnetic materials with the ratio Br/Bm<0.4 are used for
magnetically recording the information and the ones with
Br/Bm>0.4 for permanent magnets fabrication.
Constructive elements for coils
 Core constructive types
• Core-trays (ro: tole), bands, columns, coatings for transformers
magnetic circuits.
• Cylindrical bars for high frequencies inductances (also for
adjustable inductances).
• Torus (ro: miezuri toroidale) and pots (miezuri tip “oala”) for high
frequency and pulses applications.
• Different forms of yokes (ro: miezuri tip “jug”)
• For high frequency applications, the cores are being obtained by
compressing magnetic powders. That results in obtaining
magneto-electric cores (the magnetic powder is a ferromagnetic
material) or magneto-ceramic cores (ferrites).
Constructive elements for coils
 Designing a core coil
• If a coil without a core has the “L0” inductance, the core
displacements inside its windings changes its inductance:
L   ef L0
• The effective magnetic permeability, μeff, is dependent with the
material’s relative permeability, with its geometry and with the
relative position in respect with the winding.
• The ferrite producers indicate in the datasheets a so called
inductance factor, AL, having the following meaning: the
inductance factor is the obtain inductance if on the ferrite core is
made only 1 turn (nH/turn or μH/turn). Using this parameter ,
the total inductance can be obtained:
L  AL  N 2
Coils
Coils
Short history
Electrical Properties
Constructive elements for coils
Parameters
Categories
Transformers
Parameters
• The inductance and its tolerance.
• The coil’s own resistance.
• The loss-angle tangent.
• The quality factor.
• The temperature coefficient.
tg L 
vRL
vL
QL 

RL
L
L
RL
1 dL
L 
L dT
Coils
Coils
Short history
Electrical Properties
Constructive elements for coils
Parameters
Categories
Transformers
Categories
 Constructive types
• Toroidal (A).
• Cylindrical (B).
• Encapsulated (C).
• Adjustable (D, E).
Categories
 Spiral circular plane coil
3,2 10 2  a 2  N 2
L
[ μH ]; where
6a  10c
d a  di
d a  di
a
[ m] c 
[ m]
4
2
Categories
 Spiral square plane coil
0,0122  l 2  N 2
L
[μH];
4l  11c
c – trace width
Coils
Coils
Short history
Electrical Properties
Constructive elements for coils
Parameters
Categories
Transformers
Transformers
• A transformer is an electrical component which consists on two
coils mounted on the same magnetic core.
• The magnetic core links the magnetic flux,ФB , from the two coils.
• Taking into consideration the Faraday law:
d B
VP   N P
dt
d B
VS   N S
dt
VS
NS

The transformer’s equation
VP N P
N S  N P Voltage increasing transformers
N S  N P Voltage decreasing transformers
Transformers
 Ideal transformer
• An ideal transformer doesn’t have losses, so:
Input Power = Output Power
VP I P  VS I S
I S VP N P


I P VS
NS
Real transformers can reach an efficiency
higher then 99%
• A transformer makes its job only if the voltage/current varies
through one of its windings. This variation will generate a variable
flux which will lead to a variable voltage into the second winding.
Transformers
 Mutual inductance
• The time variation of the current from the
primary winding determines the occurrence of
a induced voltage in the secondary winding.
Obviously, the current through the second
winding appears only if at the second
windings terminals, a load is connected.
• Let us consider the coil 1 with N1 – turns number and coil 2 with N2
– turns number and Ф21 – the magnetic flux in the second coil
occurred due to the current that flows through the first coil:
N221  i1  N221  (constanta ) i1  M 21i1
N 2  21
M 21 
i1
Mutual inductance
Unit – Henry
1H=Vs/A=Ωs
Transformers
 Mutual inductance
• The induced voltage in the second coil can be expressed:
d 21
v2   N 2
dt
 21 
M 21
i1
N2
d 21 M 21 di1

dt
N 2 dt
M 21 di1
di1
v2   N 2
  M 21
N 2 dt
dt
Transformers
 Mutual inductance
• Similar, it can be shown that the induced voltage into the first coil
by the second coil’s current variation is:
v1   M 12
di2
dt
• But, generally M21=M12 and can be written as follows, where k –
coupling factor:
M  M  M  k Lp Ls
2
21
2
12
2
2
• In conclusion:
di2
v1   M
dt
di1
v2   M
dt
Transformers
 The transformer – circuit analysis
Ro
vo
ip
vp
vs
Primar
Secundar
NP
LP
NS
LS
is
Rl
• The transformer has two coils (as the symbol from the circuit above).
The one from the circuit where the Vo source is applied is called
primary coil and has the Lp inductance, and the one from the circuit
where the Rl – load resistance is , is called secondary coil and has the
Ls inductance.
• Both inductances work on theirs designated circuit as was presented
before, but supplementary the inductances are being magnetically
coupled through the mutual inductance, M.
Transformers
 The transformer – circuit analysis
Ro
vo
ip
vp
vs
Primar
Secundar
NP
LP
NS
LS
• The voltage across the primary coil will be:
vp  jLP ip  jMis
• The voltage across the secondary coil will be:
vs  jLSis  jMip
is
Rl
Transformers
 The transformer – circuit analysis
Ro
vo
ip
vp
vs
Primar
Secundar
NP
LP
NS
LS
is
• Applying the KVL in the primary winding:
vo  Ro  ip  jLPip  jMis
• Applying the KVL in the secondary winding:
0  Rl  is  jLSis  jMip
Rl
Transformers
 The transformer – circuit analysis
Ro
vo
ip
vp
vs
Primar
Secundar
NP
LP
NS
LS
is
Rl
• Extracting, from the second equation, the is current
dependent of the ip current and introducing it in the first
equation (KVL for the primary winding), and also
taking into consideration that:
• It can be obtained that:
LP  N P 

 
LS  N S 
2




N
vo  Ro  ip   jLP || Rl   S    ip

NP  



2
Transformers
 The transformer – constructive types
• Cylindrical (solenoidal)
• Torus
• Yoke
Important formulas
 Resistors
 Capacitors
l
l
R     Cu   r
S
S
A
C  0  r
d
A
2
L  0   r  N
l
Cu=5,344 x 10-7 -cm
0=8,8542·10-12 F/m
0=4·π·10-7 H/m
 Coils
Problem
Using a copper wire (φCo = 5.344x10-7 Ωm) with 1 mm diameter, a
winding of 40 turns is being executed on insulated cylindrical support
with the diameter of 10 mm.
 Please determine the electrical parameters of the realized coil.
 What is the modulus of the coil’s impedance at 50Hz? What is
the modulus of the coil’s impedance at 500kHz?