In this chapter, a reappraisal on additive generators is presented. The treatment focuses on generators that are cannular and equipped with traveling lasting magnets. Detailss on the machine building and its constituents are briefed. This is followed by a survey on the basicss of the magnetic field analysis which forms the footing of the mathematical theoretical account and the FEA.

## Introduction

A additive generator is an electromechanical energy convertor driven by a premier mover which converts the kinetic energy of additive gesture into electrical energy. The gesture can be in one way or reciprocating. In a system where the gesture is in one way and has a long shot such as in the magnetic levitated train ( MAGLEV ) , the rule of operation of the machine is indistinguishable to that of a rotary machine. Thus it can bring forth a three-phase electrical power. In a reciprocating additive generator, the three-phase end product can non be produced since the electromotive force will change by reversal as the mover reverses its way of the gesture after making the terminal point of the shot. Indeed, the reciprocating machine can merely bring forth a individual stage electrical power. However, the three-phase term is used in the additive generator to depict a three consecutive flux linkage as a map of gesture supplanting generated by the machine.

A additive generator converts the mechanical energy to electrical energy utilizing electromagnetic initiation. The beginning of the mechanical energy is a premier mover which can be a steam engine, H2O falling through a turbine or water wheel, a Stirling engine, an internal burning engine, a air current turbine, a manus grouch, tidal moving ridge, compressed air or any mechanisms that produce additive gesture.

Another term normally used for a additive generator is a additive alternator. The simplest type of a additive alternator is the Faraday flashlight or the shingle torchlight. It contains a spiral and a lasting magnet. When the contraption is shaken back and Forth, the magnet oscillates through the spiral and induced an electric current. This current charges a capacitance that shops the electrical energy. The capacitance is discharged when used to illume up a light breathing rectifying tube and it can be re-charged by farther shaking.

Figure 2. 1. Hand agitate flashlight [ Wikipedia ]

Most additive generators are proposed to be used as a base entirely power supply [ Parviz Famouri, William R. Cawthorne, 1998 ] and as a portion of a particular purpose equipments like a gun [ J.Li, et. al,1999 ; Peter Mongeau, 1997 ; Duan Xiaojun, 2005 ] or a intercrossed application [ William R. Cawthorne,1999 ] . An illustration of a additive alternator used for distant electrical power coevals is presented in the work of Famouri et. Al. [ Parviz Famouri, William R. Cawthorne, 1998 ] . The machine is straight coupled to an internal burning engine with double Pistons to organize a free Piston generator. Each terminal of the transcriber is connected straight to a Piston. The burning procedure at each terminal moves the Pistons in the opposite waies alternately therefore bring forthing a reciprocating gesture.

## Machine Construction

An fanciful description of the transmutation of a rotary machine into a additive machine is shown in Figure 2. 2 and Figure 2. 3. In rule, for every rotary electric machine constellation a additive antagonistic portion can be imagined. The rotary machine is cut and unrolled to obtain a level or cannular additive initiation machine. The same procedure may be imagined for level and cannular linear synchronal PM motors. [ Boldea, Nasar, 1997 ] .

Figure 2. 2. Double and individual sided level additive machine

[ Boldea, Nasar, 1997 ]

Figure 2. 3. Flat and cannular additive machine

[ Boldea, Nasar, 1997 ]

Construction-wise, a additive generator consists of a inactive portion and a movable portion. The inactive portion, known as the stator is fixed at a certain place, while the movable portion, known as the transcriber moves along the machine axis. An illustration of additive generator is shown in Figure 2. 4. The stator comprises the armature twists, the dorsum Fe, and the twist spacers. The transcriber is the traveling portion of the machine which is formed by lasting magnets, non-magnetic spacers and a shaft.

Figure 2. 4. The free-piston generator

[ Cawthorne, 1999 ]

Most additive generators are cannular or planar, although other forms are possible depending on the applications and demands of the system. Tubular machines are symmetric in the radial way and the flux escape in cannular machines occurs merely at the machine ‘s terminals.

Unlike rotary machines which produce angular torsion, a additive generator produces a high minute when the transcriber stops at the shot ends before traveling back for a complete rhythm. Therefore, it requires a minimum transcriber weight. If the machine is running at high velocity, the weight of the transcriber will be critical. In general, the transcriber weight should be kept every bit light as possible to cut down mechanical quiver.

Current rare-earth lasting magnets produce high flux denseness with less weight. The lasting magnets can be mounted on the transcriber surface. The flux of an axial lasting magnet flows parallel with the machine axis, so that a permeable medium is needed to impart the flux radially to associate with the twists. On the other manus, no medium is needed to link the flux of a radial lasting magnet to the twists.

It is found that the surface mounted lasting magnet is the most normally used constellation due to the application of the radial magnetic lasting magnets. The tubing shaped lasting magnet offers a surface mounted building for RMPM or embedded building for AMPM. The AMPM building is really surface mounted except for the demand of spacer or permeable stuff interpolation between lasting magnets.

A particular building known as the Halbach array lasting magnet can be considered as a surface mounted. The machine offers a high power denseness, and a high efficiency and low traveling mass [ Jiabin, et all, 2005 ] . Detailss of different types of buildings for additive machines will be given in chapter 3.

## Fundamentalss of the Magnetic Field Analysis

In electromagnetic field analysis, all cardinal Fieldss are governed by the Maxwell ‘s equations. In this chapter, individual scalar parametric quantities are represented as normal characters, individual vector parametric quantities are represented as bold face characters and matrices as characters in brackets. The four Maxwell ‘s equations that govern electromagnetic and electrostatic Fieldss are given by

where

[ H ] : magnetic field strength vector,

[ J ] : entire current denseness vector,

[ Js ] : applied beginning current denseness vector,

[ Je ] : induced eddy current denseness vector and

[ Jv ] : speed current denseness vector.

[ D ] : electric flux denseness vector.

[ E ] : electric field strength vector,

[ B ] : magnetic flux denseness vector and ? is electric charge denseness.

By using the preservation jurisprudence, the divergency of both sides of Equation gives the continuity equation,

The behaviour of electromagnetic stuffs besides contributes to a constituent relation in the field equations. The constituent relation for the magnetic Fieldss affecting saturable stuffs is given by

where [ ? ] is the magnetic permeableness matrix which may be a map of temperature or field, or even both. If [ ? ] is merely a map of field, so

where ?h is the permeableness derived from the input B versus H curve. This equation is valid for a system without lasting magnets. When lasting magnets exist, the constituent relation becomes:

where [ Mo ] is the remanent intrinsic magnetisation vector. Rewriting the general constituent equation in footings of reluctivity it becomes:

where [ ? ] = [ ? ] -1 is reluctivity matrix. The reluctivity of free infinite is.

The solution of the magnetic field jobs is normally obtained utilizing possible maps. Two sorts of possible maps, the magnetic vector potency and the magnetic scalar potency are used depending on the job to be solved. Factors impacting the pick of possible include: field kineticss, field dimensionality, beginning current constellation, sphere size and discretization. The applicable parts for all sort of stuffs are shown in Figure 4. 1. These will be referred to with each solution process discussed in the undermentioned subdivisions.

Figure 4. 1. Electromagnetic Field Regions

where:

a„¦0: free infinite part

a„¦1: nonconductive permeable part

a„¦2: conducting part

? : permeableness of Fe

?o: permeableness of air

Moment: lasting magnets

S1: boundary of a„¦ 1

? : conduction

a„¦ = a„¦1 + a„¦2 + a„¦0

Magnetic Scalar Potential

The magnetic scalar potency B is defined in a part of infinite in the absence of currents. In the magnetostatic jobs, the clip changing effects are ignored. This reduces the Maxwell ‘s equations for magnetic Fieldss to Equation and Equation below:

Solution Schemes

Mentioning to Figure 4. 1, in the sphere of a„¦0 and a„¦1 ( excepting a„¦2 ) a solution is required which satisfies the relevant Maxwell ‘s Equation and Equation, and the constituent relation Equation in the undermentioned signifiers:

where [ Hg ] is known as the preliminary or initial magnetic field and ?g is the generalised potency. The development of [ Hg ] varies depending on the job and the preparation. Basically [ Hg ] must fulfill Ampere ‘s jurisprudence in Equation so that the staying portion of the field can be derived as the gradient of the generalised scalar possible ?g. This ensures that ?g is individual valued. In add-on, the absolute value of [ Hg ] must be greater than that of ??g. In other words, [ Hg ] should be a good estimate of the entire field to avoid troubles with cancellation mistakes.

As mentioned above, the choice of [ Hg ] is of import to the development of any of the undermentioned scalar possible schemes. The development of [ Hg ] involves the Biot-Savart field [ Hs ] which satisfies Ampere ‘s jurisprudence and is a map of the beginning current [ Js ] . [ Hs ] is obtained by measuring the built-in:

where [ Js ] is current beginning denseness vector at dv, [ R ] is place vector from current beginning to node point and V is volume of current beginning.

The above volume built-in can be reduced to the undermentioned surface built-in:

where S is the surface of the current beginning

The solution of this built-in should be suited for the initial status. The values of [ Js ] are obtained either straight from the current beginnings or as the consequence of an external electric field computation.

Magnetic Vector Potential

The vector possible method is implemented for both 2-D and 3-D electromagnetic Fieldss. Sing the inactive and dynamic Fieldss and pretermiting displacement currents ( quasi-stationary bound ) , the undermentioned subset of Maxwell ‘s equations apply.

In the full sphere a„¦ of an electromagnetic field job, a solution is required which satisfies the above Maxwell ‘s equations ( Equation through Equation ) .

Since the magnetic field denseness B is divergenceless it can be expressed as the coil of the magnetic potency Angstrom as

Therefore, replacing B with -xA in Equation 4.37, the electric field [ E ] can be written as

where:

[ A ] is magnetic vector potency

V is electric scalar potency

Electromagnetic Field Evaluations

The basic magnetic analysis consequences include magnetic field strength, magnetic flux denseness, magnetic forces and current densenesss. Whereas, the basic electric analysis consequences include electric field strength, electric current densenesss, electric flux denseness, Joule heat and stored electric energy.

Magnetic Scalar Potential Results

The magnetic field strength can be divided into two parts [ Hg ] and [ H? ] which are the generalised field and as the gradient of the generalised possible -?g severally. [ H? ] is evaluated at the integrating points utilizing the component form map as:

where [ N ] is shape maps and [ i?¦g ] is nodal generalised possible vector.

Then the magnetic field strength is defined as

where [ H ] is magnetic field strength ( end product as H )

The magnetic flux denseness is calculated from the field strength as

where [ B ] is magnetic flux denseness ( end product as B )

Nodal values of the field strength and flux denseness are computed from the integrating point ‘s values.

Magnetic Vector Potential Results

As noted earlier the magnetic flux denseness is defined as the coil of the magnetic vector potency. This rating is performed at the integrating points utilizing the component form maps:

where [ B ] is the magnetic flux denseness, [ NA ] is the form maps and [ Ae ] is the nodal magnetic vector potency

Then the magnetic field strength is computed from the flux denseness as

The current densenesss are besides related to the vector potency and is given as

where: [ Jt ] is entire current denseness

where:

[ Je ] : current denseness constituent due to [ A ] ,

[ NA ] : element form maps for [ A ] evaluated at the integrating points,

[ Ae ] : clip derived function of magnetic vector potency,

[ Js ] : current denseness constituent due to V,

[ Ve ] : electric scalar potency

[ N ] : element form maps for V evaluated at the integrating points

[ Jv ] : speed current denseness vector

[ V ] : applied speed vector

Magnetic Forces

The magnetic forces are computed utilizing the vector possible method or the scalar possible method. Three different techniques are normally used to cipher the magnetic forces at the elemental degree.

Lorentz forces

The Lorentz forces are the magnetic forces generated in current carrying music directors. They are calculated utilizing numerical integrating by the undermentioned expression

Where Fjb is the Lorentz force.

For 2-D analysis, the corresponding electromagnetic torsion Tjb about +Z is given by

where [ Z ] is the unit vector along +Z axis and [ R ] is the place vector in the planetary Cartesian co-ordinate system.

Maxwell Forces

The Maxwell emphasis tensor is a 2nd rank tensor used to stand for the interaction between electric or magnetic force and mechanical impulse [ ] . The tensor is used to calculate forces on ferromagnetic parts. This force computation is performed on the surfaces of the air stuff elements which have a nonzero face burden. For the 2-D applications, this method uses extrapolated field values and consequences in the undermentioned numerically incorporate surface built-in given below.

where

Virtual Work Forces

Another method used for calculating the magnetic force is known as the practical work force. The magnetic forces calculated utilizing the practical work method are obtained as the derived function of the energy versus the supplanting of the movable portion. This computation is valid for a bed of air elements environing a movable portion. To find the entire force moving on the organic structure, the forces in the air bed environing it is summed. The basic equation for the force of an air material component in the s way is:

where Fs is the force in the component in the s way, is the derived function of the field strength with regard to displacement, s is the practical supplanting of the nodal coordinates taken alternately in the X, Y, Z planetary waies and dv is the elemental volume of the environing air.

## Decision

Ringing lasting magnets and pealing spiral are frequently used in cannular additive machine. Taking advantage of its symmetricalness, a cannular machine can be represented by 2D layout. The planar simulation of the machine is advantageous since it is simpler and faster than the normal 3D simulation.

Like rotary machines, all cardinal Fieldss that govern the behaviour of additive machines are given by the Maxwell ‘s equations. These include the vector potency A and the flux denseness B from which other parametric quantities such as flux linkage, force and the induced electromotive force are derived.

[ finish ]