11. inductors. the transformer (lecture 11)

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Passive components and circuits - CCP

Lecture 11

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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 Henrydiscover 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 electromagneticphonograf

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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.

)1()(

22

0 Henriesl

rNL c

)2(}2)8

{ln(20 w

cc

r

rrNL

][104 1170 mAWb

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The inductance depends on the geometry of the coil (l, d=2r, h in mm). The formulas are valid for air.

][

44,0

001,02

H

d

l

dNL

][1093

008,022

Hhld

dNL

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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

ppp2

pL

CRjLC1

LjRZ

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The coil the frequency characteristic

10%

10%

Inductive area

pR

QR p

p

0LC

1

03,0 L

R2,2 p

L

C

1

LZ

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Dimensioning the inductance when the wires

are remoted

]H[10DNkL7

m

LLL 01

]H[10D45,0l

)ND(L

72

0

l

dp

D

0

2

4

6

8

1 2 3 4

p/d

km

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Calculus of the parasitic capacitanceCp[pF]

p/d

1 1,3 1,5 1,7 2 2,5 3 3,5 4

0,5

0,7

1

2

3

5

7

10

20

30

1,1

D=10cm

D=8cm

D=6cm

D=4cm

D=2cm

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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.

NDk

d

p

D

dN

DNL

kdp

LC

DL

m

m

p

1,0

144,0

][

/

1

,,

2

2

0

max

0

H

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 short-circuits 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 twocriteria: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 ofthe 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 withdiameters of standardized dimensions: 0,05mm, 0,07mm,0,1mm, ... 2mm. These diameters do not include thethickness 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 areintroduced in the coil. They make up a magnetic circuit(sometimes with interruptions) that has the purpose ofconcentrating the magnetic field lines. In this way, the magneticflux increases, most of the lines intersecting the surface of thewires, 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:

H

B

H

Br

0

1;

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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 .

H

B

Hs

BsBm

Hm

Hc

-Hc

-Hm

-Hs

B r

-Br

-Bm-Bs

0

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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 arereoriented. 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 themagnetic field modifies (histerezis).

The remanent magnetization manifests itself up to a certaintemperature (Curie temperature) at which the thermalagitation destroys the domains of ordered orientation.

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The use of magnetic materials

Clasification:

Soft magnetic materials Hc80 A/m (broad histeresis)

The soft magnetic materials with the ratio: Br/Bm (ratio that characterizes the histeresis inclination)

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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 magneto-

electric 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:

0LL ef

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 orH/wire). Using this parameter, the total inductance is obtained with the formula:

2NAL L

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Parameters of coils

Its inductance and tolerance

The parasitic resistance

The loss angle tangent

The quality factor

The temperature coefficient

L

R

v

vtg L

L

RL

L

L

LR

LQ

dT

dL

LL

1

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A few categories of coils

Toroidal (A)

Cylindrical (B)

Incapsulated (C)

Adjustable (D,E)

Color

Black

Brown

Red

Orange

Yellow

Green

Blue

Purple

Gray

White

None

Silver

Gold

Digit

0

1

2

3

4

5

6

7

8

9

.

Multiplier

1

10

100

1000

Tolerance

20%

10%

5%

Code of colors for encapsulated coils

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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 Faradays induction law:

dt

dNV

dt

dNV BSS

BPP

P

S

P

S

N

N

V

V The ecuation of the transformer

PS NN Step-up transformer

PS NN Step-down transformer

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The ideal transformer

An ideal transformer has no loses, therefore:

The input power = The output power

SSPP IVIV

S

P

S

P

P

S

N

N

V

V

I

I

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.

The real transformers that are

well manufactured can have

the efficiency over 99%.

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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.

21 = the magnetic flux in coil 2 due to the current i1 from coil 1

2 21 1 2 21 1 21 1 (constant)N I N i M i

2 2121

1

NM

i

Mutual inductance Unit = Henry

1 H = V.s/A = .s

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The induced voltage in coil 2 can be expressed:

212 2

dv N

dt

1

2

2121 I

N

M

dt

dI

N

M

dt

d 1

2

2121

21 1 12 2 21

2

M di div N M

N dt dt

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21 12

div M

dt

2 11 2

di div M v M

dt dt

Similary, it can be proved that the induced voltage in coil 1 by the variation of the current

in coil 2 is:

It can be proved that: M21 = M12 = M=k2LPLS

In an ideal case, k, the coupling factor, is 1.

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The transformer circuit functioning

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.

Ro

Rl

Primar Secundar

N P N S

L P L S

vo vp vsi p

i s

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The voltage on the primary coil will be:

The voltage on the secondary coil will be: s s s pv j L i j Mi

p p p sv j L i j Mi

Ro

Rl

Primar Secundar

N P N SL P L S

vo vp vsi p

i s

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The sum of the voltages from the primary circuit obeys TKV:

The sum of the voltages from the secondary circuit obeys TKV:

o o p p p sv R i j L i j Mi

0 s s s pR i j L i j Mi

Ro

Rl

Primar Secundar

N P N SL P L S

vo vp vsi p

i s

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Taking into consideration that:

It is obtained:

Ro

Rl

Primar Secundar

N P N S

L P L S

vo vp vsi p

i s

2

S

P

S

P

N

N

L

L

p

2

P

SPpo || i

N

NRlLjiRov

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Transformers constructive variants

Cylindrical (solenoidal)

Toroidal

Yoke

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Basic formulas for dimensioning the studied

components

S

l

S

lR rCu

d

AC r 0

2

0 Nl

AL r

Cu=5,344 x 10-7 -cm 0=8,854210

-12 F/m 0=410-7 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 the coils impedance modulus at 50 Hz

and 500KHz frequencies?

11. inductors. the transformer (lecture 11)

Documents

coil of length

coil greater

magnetic properties

hm iron

diameter d

used material

hm supermalloy

hm steell