Introduction to Electrical Engineering - Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.)
 5. Magnetic Field
 5.1. Magnetic Phenomena 5.2. Force Actions in a Magnetic Field 5.3. Electromagnetic Induction

5.1. Magnetic Phenomena

Magnetic phenomena are caused, by magnets. These are bodies which exert forces of attraction on iron and steel.

A distinction is made between

· permanent magnets; these magnets are made of steel or special materials for permanent magnets and they maintain their magnetic properties in almost unchanged strength for a long period.

· electromagnets; these are coils which usually have an iron core. Magnetic actions only occur when a current passes through the coils.

The ends of the magnets exert the greatest force of attraction and are called poles. Under the influence of the terrestrial magnetic field, a rod-shaped magnet is adjusted in north-south direction. The magnetic end which points to north is called north pole, the opposite end is called south pole. The connecting line between the two poles is called magnetic axis.

When we place two magnets opposite to each other (see Fig. 5.1.), then we will find:

Non-homonymous poles attract each other, homonymous poles repel each other.

Fig. 5.1. Force actions between magnets opposing each other

1 - North pole
2 - south pole

The effects of the force of magnets can be represented by force action lines which are called magnetic field lines. Fig. 5.2. shows the course taken by the magnetic field lines of a rod magnet. Their direction has been fixed arbitrarily.

Obvious is the following:

Fig. 5.2. Magnetic field of a rod magnet

1 - North pole
2 - South pole

The magnetic field lines are closed in themselves; outside of the magnet they run from the north pole to the south pole and inside the magnet from the south pole to the north pole.

In Section 2.1. we have already said that every electrical current is associated with a magnetic field. This tact is described by the so-called circulation law, which is also known as Biot-Savart’s law. To give proof of it, one passes a straight conductor through a sheet of paper or a glass plate, in accordance with Fig. 5.5., and then allows a current to flow through the conductor. Then fine iron powder is sprinkled on the paper or plate of glass and immediately the iron particles will adjust themselves in the form of concentric circles around the conductor, thus, forming a typical field line pattern. The direction of the field lines can be determined by means of a magnetic needle and has been entered in Fig. 5.5.

Fig. 5.3. Magnetic field of a current carrying conductor

a) Pattern of field lines

b) Direction of field lines

c) Direction of field lines after reversal of current direction

The correlation between current direction and field line direction is defined by the screw rule:

When one imagines a right-handed screw to be moved in a conductor in the direction of the current, the sense of rotation required for this longitudinal motion indicates the direction of the field lines.

When one intends to mark the electrical current direction in a conductor, the symbols shown in Fig. 5.4. have to be used. Basically, the direction of current is indicated by an arrow. When looking at the cross-section of the conductor, one sees the arrow-head as a point when the current flows in the direction toward the observer. When the current flows away from the observer while he continues to look in the same direction, he sees the feathers of the arrow-head in the form of a cross on the cross-section of the conductor.

Fig. 5.4. Symbolising the current direction in conductors

When a conductor is wound on a cylinder in the form of a spiral, one obtains a coil. When current passes through it, a magnetic field is brough about which is made up of the fields of the individual turns of the coil. A long, cylindrically wound coil has a field which resembles that of a rod magnet (Fig. 5-5.). Which end of the coil forms the north pole and which the south pole is dependent on the direction of current and can easily be found with the help of the pole determination rule:

When the current flows clockwise through the coil, one looks at a south pole; when the current flows anti-clockwise through the coil, one looks at a north pole.

Fig. 5.5. Magnetic fields of a current-carrying coil and a rod magnet

1 - North pole
2 - South pole

When placing a piece of iron in the field of a magnet, the field line distribution will change considerably (B.5.6.). Like nickel, cobalt and their alloys, iron also is a very good conductor for magnetic field lines. Therefore, these materials are called ferromagnetica. Fig. 5.6. shows how the parallel course of the field lines (homogeneous, i.e. uniformly built up field) is changed by a ferromagneticum. In general, the following holds:

A ferromagneticum bunches magnetic field lines together.

Fig. 5.6. Changing the course of the field by ferromagnetic bodies

1 - North pole
2 - South pole

When the ferromagnetic body has the shape of a ring, inside the ring the space is field-free because the field lines run in the wall of the ring. This fact is used for magnetic screening. Components which must be protected from disturbing magnetic fields are surrounded by sheaths of ferromagnetic material.

As has already been said above, every current is associated with a magnetic field. A coil through which current flows is also surrounded by a magnetic field. In the coil, electrical energy is converted into magnetic energy. Since the field lines are closed in themselvers, we speak of a circle in which the magnetic field is caused by a magnetomotive force. The coil through which current flows is the seat of this magnetomotive force Q 1) It is the greater, the greater the current is and the more turns the coil includes. Fig. 5.7. shows a magnetic basic circuit. This circuit consists of a coil with N turns through which a current I passes. The ferromagnetic conductor is not closed but separated by an air gap.

1) Q Greek letter theta

Fig. 5.7. Magnetic fundamental circuit

Analogously to the electric basic circuit, the following occurs: When an electrical current flows through the coil, a mangetomotive force Q is produced. It drives a magnetic flux F 2) whose direction can be found with the help of the pole determination rule. Since air conducts the magnetic flux much worse than a ferromagnetic material, the air gap represents the decisive magnetic resistance or reluctance Rm in the magnetic circuit. The magnitude of the magnetic resistance is dependent on the geometric dimensions of the air gap. The magnetic flux now causes a magnetic potential drop V at the magnetic reluctance Rm; this potential drop is - with a given F - the greater, the greater Rm is. The magnetic potential difference, i.e. the magnetic potential related to the length of a magnetic resistance or reluctance, is called magnetic field-strength H. The latter is closely related with the magnetic flux density B (also known as magnetic induction) which indicates the magnetic flux which vertically penetrates a certain area. The material-specific conductivity for the magnetic flux is called permeability µ. Mostly, the permeability is given as the product of the absolute permeability µ0 times relative permeability µr.

2) F Greek letter phi

µ = µ0 · µr

where:

 µ permeability µ0 absolute permeability (induction constant) µr relative permeability

The absolute permeability applies to vacuum and amounts to

µ0 = 1.256 · 10-6 V.s/(A.m)

Hence, the relative permeability µr, is a numerical factor without a unit and indicates how many times the magnetic flux is conducted better by a certain material than by a vacuum. For electrical engineering, a classification of materials with respect to permeability into two types will provide sufficient accuracy:

1. non-ferromagnetic materials = vacuum, air, copper, wood, water
For these materials we have µr = const. »1

2. ferromagnetic materials = iron, nickel, cobalt and certain alloys
For these materials we have µr = const. >> 1 (100 to 10,000)

Below, the above mentioned magnetic quantities are presented, in the form of a list; further their mathematical correlations and their units are given. The shown fundamental equations can be applied easily. In analogy to the electrical circuits one has to observe that, for example, the magnetic flux in a branched magnetic circuit is divided into individual fluxes and the total magnetic resistance Rmers is smaller than the smallest individual resistance. In an unbranched magnetic circuit with several magnetic resistances connected in series, the total resistance Rmers is equal to the sum of the individual resistances; the sum of all magnetic potential drops is equal to the total magnetomotive force. This shows that, analogously. Ohm’s law and the 1st and 2nd Kirchhoff’s laws are applicable. Example 5.1 deals with an unbranched magnetic circuit with two magnetic resistances RmL and RmFe connected in series.

Survey of important magnetic quantities

Formula signs and notations

 Q magnetomotive force V magnetic potential drop F magnetic flux Rm magnetic resistance, also known as relustance µ permeability µ0 absolute permeability µr relative permeability H magnetic field-strength B magnetic flux density

 Equations Units Q = N · I (5.1) A V = Rm · F (5.2) A F = V/Rm (5.3) Wb = V · s Wb = Weber in honour of the German physicist Wilhelm Eduard Weber (1804 - 1891) Rm = V/F (5.4) Rm = 1/µ · 1/A (5.5) 1/H = A/Wb = A/(V · s) V · s/A = H = Henry in honour of the American physicist Josef Henry (1797 - 1878) µ = µ0 · µr (5.6) H/m = Wb/(A · m) = V ··s/(A · m) H = V/I (5.7) A/m B = F/A (5.8) B = µH (5.9) T = Wb/m2 = (V · s)/m2 T = tesla in honour of the Yugoslavian physicist Nicola Tesla (1856 - 1943)

Example 5.1

For the magnetic circuit shown in Fig. 5.7., the magnetic flux F, the magnetic flux density B, and the magnetic field-strengths in the ferromagnetic conductor HFe, and in the air gap HL have to be found. The 250 turns of the coil carry 2 A; the length of the ferromagnetic conductor lFe = 80 cm and its cross-sectional area A = 4. 4 cm2. The relative permeability is assumed to be µr 600. The air gap length is lL = 2 mm.

Given:

N = 250
I = 2 A
lFe = 80 cm
lL = 2 mm
A = 4 · 4 cm2 = 16 cm2
µr = 600
µ0 = 1.256 · 10-6 V · s/(A · m)

To be found:

F
B
HFe

Solution: According to equation (5.3)

F = Q/Rmers

According to equation (5.1)

Q = N · I = 250 · 2 A = 500 A
Rmers = RmL + RmFe

According to equation (5.5)

Pay attention to the tact that the ferromagnetic conductor has a smaller magnetic resistance than the air gap although the length of the former is 400 times that of the latter.

Rmers = 106 A/(V·· s) + 0.665 · 106 A/(V·· s)
Rmers = 1.665 · 106 A/(V·· s)

F = 300 µWb

B is in the air gap as great as in the ferromagnetic conductor because F (and A) is equal everywhere.

According to equation (5.8)

B = F/A

B = (300 · 10-6 Wb)/(16 · 10-4 m2) = 18.8 · 10-2 Wb/m2
B » 190 mT

According to equation

HFe = B/(µ0µr)
rHFe = B/µ0)

HFe = 250 A/m

and

HL = B/µ0 = µrHFe
HL = 600 · 250 A/m = 150,000 A/m
HL = 150,000 A/m

Magnetic phenomena are caused by magnets. A distinction is made “between permanent magnets and electromagnets. They exert forces of attraction on iron and steel.

The lines of force action are called magnetic field lines; they are closed in themselves, take their course from the north pole to the south pole outside of the magnet and run from the south pole to the north pole inside the magnet.

Every electrical current is associated with a magnetic field. It whirls around the current-carrying conductor. Its direction can be determined with the help of the screw rule.

A conductor wound in several turns is called coil. When it carries an electrical current, a magnetic field is also built up. Its direction can be found with the help of the pole determination rule.

A ferromagnetic body bunches up the field lines of a magnetic field. When this body has the shape of a ring, the space inside the ring is field-free. Consequently, magnetic screening is possible.

A coil energised by an electrical current in connection with a magnetic conductor and the magnetic resistances forms a magnetic circuit. By means of the defined quantities, this circuit can be dealt with mathematically like an electrical circuit.

Questions and problems:

1. Quote examples of magnetic phenomena in electrical components and devices!

2. Explain why in the air gap of the ferromagnetic circuit the field strength increases while the magnetic flux density remains constant!

3. What is the formal analogy between the electrical circuit and the magnetic circuit? Compare wiring diagrams and circuit diagrams, characteristics and fundamental equations!