1/36

Passive components and circuits - CCP

Lecture 10

2/36

Content Coils

Short history Electrical properties Constructive elements of a coil Parameters Categories

Transformers

3/36

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

4/36

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.

)1()( 22

0 Henriesl

rNL c

)2(}2)8

{ln(20

w

cc r

rrNL

][104 1170

mAWb

5/36

Electrical properties 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

μH

dl

dNL

][

1093008,0

22

μHhld

dNL

6/36

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

7/36

The coil – equivalent circuit

ppp2p

LCRjLC1

LjRZ

8/36

The coil – the frequency characteristic

10%

10%

Inductive area

pR

QRp

p0

LC

1

03,0 L

R2,2 p

L

C

1

LZ

9/36

Dimensioning the inductance when the wires are remoted

]H[10DNkL 7m

LLL 01

]H[10D45,0l

)ND(L 7

2

0

l

dp

D

0

2

4

6

8

1 2 3 4

p/d

km

10/36

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

11/36

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

dp

Dd

N

DNL

kdp

LC

DL

m

m

p

1,0144,0

][

/

1

,,

2

20

max

0

μH

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.

12/36

Constructive elements of a coil

The coil turns (whirls) The body The impregnating material The core

No core Iron core Ferrite core

13/36

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.

14/36

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.

15/36

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).

16/36

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).

17/36

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:

H

B

H

Br

0

1;

18/36

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

H s

B sBm

HmH c-Hc

-Hm-Hs

B r

-B r

-Bm-Bs

0

19/36

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.

20/36

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.

21/36

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).

22/36

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

2NAL L

23/36

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

RLL

LL R

LQ

dT

dL

LL

1

24/36

A few categories of coils Toroidal (A) Cylindrical (B) Incapsulated (C) Adjustable (D,E)Color

BlackBrownRedOrangeYellowGreenBluePurpleGrayWhiteNoneSilverGold

Digit

0123456789

.

Multiplier

1101001000

Tolerance

±20%±10%±5%

Code of colors for encapsulated coils

25/36

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:

dt

dNV

dt

dNV B

SSB

PP

P

S

P

S

N

N

V

V The ecuation of the transformer

PS NN Step-up transformer

PS NN Step-down transformer

26/36

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%.

27/36

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

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 = Ω.s

28/36

The mutual inductance The induced voltage in coil 2 can be expressed:

212 2

dv N

dt

12

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

29/36

The mutual inductance

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.

30/36

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 SecundarN P N S

L P L S

vo vp vsi p

i s

31/36

The transformer – circuit functioning

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 SecundarN P N S

L P L S

vo vp v si p

i s

32/36

The transformer – circuit functioning

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 SecundarN P N S

L P L S

vo vp v si p

i s

33/36

The transformer – circuit functioning

Taking into consideration that:

It is obtained:

Ro

Rl

Primar SecundarN 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

34/36

Transformers – constructive variants Cylindrical (solenoidal)

Toroidal

Yoke

35/36

Basic formulas for dimensioning the studied components

S

l

S

lR rCu

d

AC r 0

20 N

l

AL r

Cu=5,344 x 10-7 -cm 0=8,8542·10-12 F/m 0=4·π·10-7 H/m

36/36

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?