Transcript Lecture10 - madalina

Passive components and circuits - CCP
Lecture 10
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Content
 Coils
 Short history
 Electrical properties
 Constructive elements of a coil
 Parameters
 Categories
 Transformers
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The coil – history
 1821 – Michael Faraday highlights the magnetic field lines
that appear around a conductor through electric current
flows.
 1825 – William Sturgeon builds the first electromagnet
 1831 – independently, Michael Faraday and Joseph Henry
discover the law of the magnetic induction
 Faraday is the one who built the first electric engine, the first
electric generator and the first transformer.
 Henry is the one who built the first telegraph then improved by
Morse
 1876 – Bell invents the first telephone and electromagnetic
phonograf
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Electrical properties
 The inductance depends on
the geometry of the coil and
on the magnetic properties
of the media where the coil
is placed.
 Formula (1) is valid for a
length l of the coil greater
than its diameter 2rc.
 Formula (2) is valid for a coil
of length l smaller than its
diameter 2rc. rw represents
the diameter of the reeling
string.
(  0 N 2 rc )
L
Henries
l
2
L   0 N 2 rc {ln(
8rc
)  2}
rw
(1)
( 2)
0  4 10 7 [Wb  A1  m 1 ]
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Electrical properties
 The inductance depends on the geometry of the coil (l,
d=2r, h in mm). The formulas are valid for air.
L  0,001
N 2d
l
0,44 
d
[μH]
N 2d 2
L  0,008
[μH]
3d  9l  10h
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Electrical properties
 The inductance is dependent on the
distance between the whirls (turns).
 The inductance is dependent on the
magnetic property of the media in
which the coil is placed, property
characterized by the magnetic
permeability, .
 air 1.257x10-6 H/m
 ferrite U M33 9.42x10-4 H/m
 nickel 7.54x10-4 H/m
 iron 6.28x10-3 H/m
 ferrite T38 1.26x10-2 H/m
 steell 5.03x10-2 H/m
 supermalloy 1.26 H/m
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The coil – equivalent circuit
ZL 
R p  jL
1  2LC p  jR pCp
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The coil – the frequency characteristic
ZL
Rp Q
L
Inductive area
10%
1
C
Rp
10%

2 ,2 R p
L
0 ,3   0
0 
1
LC p
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Dimensioning the inductance when the wires
are remoted
km
l
8
6
D
4
2
p/d
0
p
d
( ND ) 2
L0 
 107 [H]
l  0,45  D
1
2
3
4
7
L  k m  N  D  10 [H]
L1  L0  L
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Calculus of the parasitic capacitance
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
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Designing steps for a coil
 It starts from the value of the
needed inductance L, its
diameter D and from the
domain in which it is going to
be used.
 From these it can be deduced
the maximum value for Cp.
 Then, calculate the number of
wires depending on the
geometric dimensions of the
coil by solving the equation
on the right.
L,  0 , D

C p max 
1
L02

p / d  km

N 2D
L[μH] 
 0,1k m ND
dp 
0,44  N   1
Dd 
Dimension the length of a coil with diameter of 2cm and the inductivity
of 50 H that is executed in one layer and for which it is desired a
parasite capacitance lower that 2 pF.
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Constructive elements of a coil
 The coil turns (whirls)
 The body
 The impregnating material
 The core
No core
Iron core
Ferrite core
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Coil turns
 The most frequently used material for reeling conductors is
cooper (due to its electrical and mechanical properties) but the
aluminum is also used.
 The used conductors are isolated in order to avoid shortcircuits between the adjoined turns.
 The materials used for isolation are enamels (lacquers of
different compositions), fabrics (silk, cotton) or inorganic fibers
(glass fiber).
 The type of isolating material is chosen depending on the
reached estimated temperature of the conductor. The
materials with the lower thermal resistance are the textiles
and the materials with the higher thermal resistance are the
glass fibers.
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Coil turns
 The diameter of the conductor is chosen depending on two
criteria:
 The intensity of the current that flows through the conductor,
gives the inferior limit of the diameter in order to avoid
overheating.
 The maximum value accepted for the resistance of the coil
(parasite parameter) can furthermore limit the dimension of
the diameter.
 At high frequencies, due to the pelicular effect, stranded wired
conductors or silvered cooper conductors are used.
 The conductors for reeling are delivered by producers with
diameters of standardized dimensions: 0,05mm, 0,07mm,
0,1mm, ... 2mm. These diameters do not include the
thickness of the isolator layer.
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The carcass of the coil
 Its role is to insure the stiffening of the reeling (and through
this keep the electric properties of the coil ).
 The materials used must have adequate properties both
electrical (dielectric stiffness, small dielectric loses ) and
mechanical (thermal stability and resistance to the action of
humidity).
 Examples in increasing order of performances: cardboard, electro
isolating cardboard, pertinax, textolite, thermorigide materials
(bakelite), thermoplastic materials ( polystyrene, polyethylene, teflon),
ceramic materials.
 Constructively, they can have different sections: circular,
square, rectangular, with or without flanges.
 At very high frequencies, the coils can be manufactured
without a body (carcass).
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The impregnating material
 It has the role of increasing protection against humidity and a
role in extra stiffening (especially when the coils are not
placed in carcasses).
 Advantages of impregnating:
 Stiffening of the wires;
 Improves the heat dissipation;
 Improves the electrical properties of the isolation between the
wires;
 Avoids the humidity access between the wires;
 Disadvantages of impregnating: can lead to the increase of
the parasite capacities (by increasing the relative permeability
of the dielectric between the wires).
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The core of the coil
 For increasing the usual inductance, magnetic cores are
introduced in the coil. They make up a magnetic circuit
(sometimes with interruptions)
that has the purpose of
concentrating the magnetic field lines. In this way, the magnetic
flux increases, most of the lines intersecting the surface of the
wires, and so the inductance of the coil increases too.
 Magnetic materials have a nonlinear behavior when placed into
an exterior magnetic field. This nonlinearity refers to the
dependence of the magnetic induction B on the intensity of the
magnetic field H. The ratio between the two represents the
magnetic permeability of that media:

B
1 B
; r 
H
0 H
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The properties of the magnetic materials –
the histerezis phenomena
 Hc – coercive field, anules
the magnetic induction;
 Br – remanent magnetic
induction;
 Hs – the intensity of the
magnetic field to which
the saturation phenomena
appear;
 Bs – the magnetic
induction at saturation .
B
Bs
Bm
Br
-Hs -Hm
0
-Hc
H c Hm H s
H
-Br
-Bm
-Bs
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The properties of the magnetic materials –
the histerezis phenomena
 The magnetic materials have atoms with their own magnetic
moments, and neighboring atomic moments are oriented
identically, the material presenting a remanent
magnetization.
 When applying an exterior field, the magnetic domains are
reoriented.
 The intensity of the exterior field at which the magnetic inductance is
zero is called coercive field. When H increases at a certain moment,
there appears a saturation phenomena (B no longer modifies).
 The phenomena are dependent on the sense in which the
magnetic field modifies (histerezis).
 The remanent magnetization manifests itself up to a certain
temperature (Curie temperature) at which the thermal
agitation destroys the domains of ordered orientation.
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The use of magnetic materials
 Clasification:
 Soft magnetic materials – Hc<80 A/m (narrow histeresis )
 Rough magnetic naterials – Hc>80 A/m (broad histeresis)
 The soft magnetic materials with the ratio:
 Br/Bm (ratio that characterizes the histeresis inclination) <0,5 for
inductances approximately constant ;
 0,5<Br/Bm<0,8 for common cores;
 Br/Bm>0,8 (rectangular histeresis ) in memorization and commutation
circuits.
 The rough magnetic materials with the ratio:
 Br/Bm<0,4 are used for the magnetic recording of the information;
 Br/Bm>0,4 for manufacturing permanent magnets.
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Constructive shapes for cores
 Sheet plates, bands, cloaks for manufacturing the magnetic circuit
for transformers;
 Cylindrical bars for inductances used in high frequencies
(sometimes they are adjustable);
 Toroidal coils and pot cores used in high frequency and
pulses;
 Yokes of different shapes in circuits of magnetic deflection;
 The cores for high frequencies are obtained by pressing
magnetic powders. This is how are obtained magnetoelectric cores (powder is a ferromagnetic material), but also
the magneto-ceramic ones (also called ferrites).
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Dimensioning coils with core
 If a coil without a core has the inductance L0 by introducing
a core it becomes:
L   ef L0
The actual permeability, ef is dependent on the relative
permeability of the material, on its geometry and on its position
relative to the reeling.
 The ferrites manufacturers indicate, in catalogs, an
inductance factor for them, AL meaning the inductance that is
obtained if a single wire is made on the ferrite (in nH/wire or
H/wire). Using this parameter, the total inductance is
obtained with the formula:
L  AL  N 2
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Parameters of coils
 Its inductance and tolerance
 The parasitic resistance
 The loss angle tangent
 The quality factor
 The temperature coefficient
tg L 
QL 
vRL
vL

RL
L
L
RL
1 dL
L 
L dT
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A few categories of coils
 Toroidal (A)
 Cylindrical (B)
 Incapsulated (C)
 Adjustable (D,E)
Color
Digit
Multiplier
Black
Brown
Red
Orange
Yellow
Green
Blue
Purple
Gray
White
None
Silver
Gold
0
1
2
3
4
5
6
7
8
9
1
10
100
1000
.
Tolerance
Code of colors for
encapsulated coils
±20%
±10%
±5%
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Transformers
 This component consists of two coils manufactured on the
same magnetic support (core), like iron.
 The magnetic core couples the magnetic flux, B, between
the two coils.
 In accordance with Faraday’s induction law:
d B
VP   N P
dt
d B
VS   N S
dt
VS
NS

VP N P
The ecuation of the transformer
NS  NP
Step-up transformer
NS  NP
Step-down transformer
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The ideal transformer
 An ideal transformer has no loses, therefore:
The input power = The output power
VP I P  VS I S
I S VP N P


I P VS
NS
The real transformers that are
well manufactured can have
the efficiency over 99%.
A transformer achieves its function only if the
voltage/current varies through one of the wires. This will
generate a variable voltage in the second wire.
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The mutual inductance
 The variation in time of the current in
circuit 1 causes an induced voltage in
circuit 2, v2. The curent through circuit
2 apares only if it is closed on a load.
Let coil 1 have N1 wires and coil 2 with N2 wires.
Φ21 = the magnetic flux in coil 2 due to the current i1 from coil 1
N221  I1
M 21 
N 2 21
i1
 N221  (constant) i1  M 21i1
Mutual inductance Unit = Henry
1 H = V.s/A = Ω.s
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The mutual inductance

The induced voltage in coil 2 can be expressed:
d 21
v2   N 2
dt
M 21
 21 
I1  d 21  M 21 dI1
N2
dt
N 2 dt
M 21 di1
di1
v2   N 2
  M 21
N 2 dt
dt
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The mutual inductance

Similary, it can be proved that the induced
voltage in coil 1 by the variation of the current
in coil 2 is:
v1   M12
di2
dt
It can be proved that: M21 = M12 = M=k2LPLS
In an ideal case, k, the coupling factor, is 1.

di2
v1   M
dt
di1
v2   M
dt
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The transformer – circuit functioning
Ro
vo
ip
vp
vs
Primar
Secundar
NP
LP
NS
LS
is
Rl
 As the symbol shows, the transformer has two coils. The
one in the circuit where the voltage source is, is called
primary (Ls), and the one found in the other circuit where
the load RL is called secondary (LL).
 Each inductance functions in the circuit where it is placed
in accordance with the properties studied, and moreover
they are coupled by the mutual inductance, M.
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The transformer – circuit functioning
Ro
vo
ip
vp
vs
Primar
NP
LP
is
Rl
Secundar
NS
LS
 The voltage on the primary coil will be:
v p  jLpi p  jMis
 The voltage on the secondary coil will be: vs  jLsis  jMi p
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The transformer – circuit functioning
Ro
vo
ip
vp
vs
Primar
NP
LP
is
Rl
Secundar
NS
LS
 The sum of the voltages from the primary circuit obeys TKV:
vo  Roi p  jLpi p  jMis
 The sum of the voltages from the secondary circuit obeys TKV:
0  R is  jLsis  jMi p
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The transformer – circuit functioning
Ro
vo
ip
vp
vs
Primar
Secundar
NP
LP
NS
LS
 Taking into consideration that:
It is obtained:
LP  N P 

 
LS  N S 
is
Rl
2
2




N
vo  Ro  ip   jLP || Rl   S    ip

NP  



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Transformers – constructive variants
 Cylindrical (solenoidal)
 Toroidal
 Yoke
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Basic formulas for dimensioning the studied
components
l
l
R     Cu   r
S
S
Cu=5,344 x 10-7 -cm
A
C  0 r
d
0=8,8542·10-12 F/m
A
L  0   r  N 2
l
0=4·π·10-7 H/m
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Problems
 Using a 1mm diameter Copper wire (=5,344 x 10-7 -cm)
40 turns are made on a cylindrical insulating substrate, with
10 mm diameter.
 Determine the electrical parameters of the coil.
 How is the value of modulus of coil impedance at 50 Hz and
500KHz frequencies?
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