electromagnetic systems

7/29/2019 Electromagnetic Systems

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Electromagnetic Basics

1,2,3,4

In this section we will look at the general electromagnetic principles which are

widely employed in engineering. This is a very short introduction to a complex

subject. You should find yourself a good book on magnetism andelectromagnetism if you want to develop a better understanding in this area.You can also find most of these concepts examined in detail atFizzics Fizzle.

Electromagnetic Fields and Forces

Before we look specifically at the case of the coilgun it will be beneficial tobriefly examine the fundamentals of electromagnetic fields and forces.

Whenever there is charge in motion there is a corresponding magnetic fieldassociated with it. This motion can take the form of current in a wire, orbitalelectrons in a molecule or the flow of a plasma etc. To help us with our

understanding of electromagnetics we employ the concepts of the

electromagnetic field and magnetic poles. The differential vector equationswhich describe this field were developed by James Clark Maxwell.

1. Systems of Units -

Just to make life difficult, there are three systems of units in popular use,namely the Sommerfield, Kennely and Gaussian systems. Since each system

has different units for many of the quantities things can become confusing. I'llbe using the Sommerfield System as outlined below:

Quantity Unit

Field H Am-

Flux weber (W)Induction B tesla (T)

Magnetisation M Am-1

Intensity of

MagnetisationI -

Moment m Am
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Table 1. Sommerfield System of Units

2. Biot-Savart Law -

It is possible to determine the magnetic field generated by a current elementusing the Biot-Savart Law.

Fig 2.1

Eqn 2.1

where His the field component at a distance r generated by the current i

flowing in the elemental length l . u is a unit vector directed radially from

l.

We can determine the magnetic field generated by some basic currentconfigurations using this law. Consider an infinitely long wire carrying acurrent i. We can use the Biot-Savart Lawto derive a general solution for the

field at any distance from the wire. I won't repeat the derivation here, anybook on electromagnetism will detail this. The general solution is:


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Eqn 2.2

Fig 2.2

The field is circular and concentric with the current.

Another configuration which has an analytical solution is the axial field of acurrent loop. While we can develop an analytical solution for the axial field,

it's not possible to do this for the field in general. In order to find the field at

some arbitrary point we need to solve complex integral equations, this bestdone with numerical techniques..

3. Ampere's Law -

This is an alternative method of determining the magnetic field due to a groupof current carrying conductors. The law can be stated as:

Eqn 3.1


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where N is the number of conductors carrying a current i and l is a line

vector. The integration must form a closed line around the current. Looking atthe infinite wire again we can apply Ampere's Law as follows:

Fig 3.1

We know that the field is circular and concentric with the current so H can beintegrated around the current at a distance r to give:

Eqn 3.2

The integration is very straightforward and shows how Ampere's Law can beapplied to provide quick solutions in some types of geometry. A knowledge ofthe field pattern necessary before this Law can be applied.

4. Field of a Solenoid


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When charge flows in a coil, it generates a magnetic field whose direction is

given by the right hand convention - Take your right hand and curl your

fingers in the direction of the current while extending your thumb, thedirection of your thumb now points to the magnetic north end of the coil. The

convention for the direction of flux has the flux emerging from a north poleand terminating on a south pole. The field and flux lines form closed loops

around the coil. Remember that these lines don't actually exist as such, theysimply connect points of equal value. It's a bit like the contours on a map

where the lines represent points of equal height. The ground height variescontinuously between these contours, in the same way the field and flux from

a coil are continuous (the continuum isn't necessarily smooth - a disctete

change in permeability will cause field values to change sharply, a bit like acliff face in the map analogy).

Fig 4.1

If the solenoid is long and thin then the field inside the solenoid can beconsidered almost uniform.

5. Ferromagnetic Materials

Probably the most well known ferromagnetic material is iron but there are

other elements such as cobalt and nickel, as well as numerous alloys like

silicon steel. Each material has a particular property which makes it suitablefor its application. So what do we mean by a ferromagnetic material? Put

simply, a ferromagnetic material is attracted by a magnet. While this iscorrect it's hardly a useful definition and it doesn't tell us why the attraction

occurs. The detailed theory of the magnetism of materials is quite a complexsubject involving quantum mechanics so we'll stick to a simpler conceptual

description. As you know, the flow of charge generates a magnetic field sowhenever we find charge in motion we should expect an associated field. In a

ferromagnetic material the orbiting electrons are arranged in such a way as to


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generate a small magnetic field. Now this means that the material is

effectively composed of many tiny current loops which have their own

magnetic field. Normally the atoms are orientated in small groups calleddomains, these are directed randomly throughout the material so there is no

net magnetic field. However if we apply an external field to the ferromagneticmaterial from a coil or permanent magnet, the current loops try and align

with this field - the domians which are most aligned with the field 'grow' atthe expense of the less well aligned domians. When this occurs it results in anet magnetisation and attraction between the material and the magnet/coil.

6. Magnetic Induction and Permeability -

The production of a magnetic field has an associated magnetic flux density,also known as magnetic induction. The induction B is linked to the field by theermeabilityof the medium through which the field penetrates.

Eqn 6.1

where 0 is the permeability of free space and r is the relativepermeability. The unit of induction is the tesla.

7. Magnetisation -

The magnetisation of a material is a measure of it's magnetic 'strength'.

Magnetisation can be inherent in the material, such as in a permanentmagnet or it can be generated by external fields sources such as a solenoid.The magnetic induction in a material can be expressed as the vector sum ofthe magnetisation, M, and the magnetic field, H.

Eqn 7.1

8. Magnetomotive Force (mmf)-

This is analogous to electromotive force (emf) and is used in magnetic circuitsto determine the flux densities in the various circuit paths. Mmf is measured

in ampere-turns or just amperes. The magnetic circuit equivalent of


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resistance is reluctance which is defined as

Eqn 8.1

where l is the length of the circuit path, is the permeability, and A is thecross-sectional area.

Let's look at a simple magnetic circuit:

Fig 8.1

The torus has a mean radius of r and a cross-sectional area A. The mmf is

supplied from a coil with N turns carrying a current i. The calculation ofreluctance is complicated by the nonlinearity of the materials permeability.

Eqn 8.2

If the reluctance can be determined then we can calculate the flux whichexists in the magnetic circuit.


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9. Demagnetising Fields -

If a piece of ferromagnetic material such as a rod becomes magnetised, polesform at the ends. These poles generate an internal field which tries to

demagnetise the material - it acts in the opposite direction to any field which

is creating the magnetisation. The result of this is that the net internal field

can be much smaller than the external field. The shape of the material has alarge effect on this demagnetising field, a long slender rod (high

length/diameter ratio) has a small demagnetising field compared to say, astubby shape like a sphere. From the perspective of coilgun design, thismeans that projectiles with a small length/diameter ratio require a stronger

external field to achieve a given state of magnetisation. Take a look at the

graph below, it shows the net internal field along the axis of two projectiles -one 20mm long x 10mm diameter and the other 10mm long x 10mm

diameter. For the same external magnetic field there is a large difference in

the internal fields, the shorter projectile has a peak field about 40% that ofthe longer projectile. This would very likely result in a significant difference inthe performance of the projectiles.

Fig 9.1

It should be noted that poles only form when there is a discontinuity in the

permeability of the material. In a closed magnetic path like a torus, no polesform and so there is no demagnetising field.

Sources:

Introduction to Magnetism and Magnetic Materials, David Jiles


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Electromagnetics (Schaum's Outline Series), J.A. Edminister

Introduction to Electromagnetism, M. Sibley

University Physics, Harris Benson

Introductory Circuit Analysis, Boylestad
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