Year2b Design Moving-coil Driver

Design of a Moving-Coil Driver

Year 2

Electrical design project module

©2004 J D Edwards

DESIGN OF A MOVING-COIL DRIVER This document is based on a guide prepared for a Year 2 electrical design project module in the Department of Engineering and Design at the University of Sussex. The module runs for ten weeks, and absorbs 25% of student time. The aim of the project is to compare several different designs for a moving-coil driver of the kind used in loudspeakers and vibration generators. A simplified design theory has been developed for the project, based on a simple model for heat transfer from the moving coil. Three different kinds of magnet structure are considered:

• Ceramic ferrite ring magnet: a low-cost option, with the disadvantage of a high external magnetic field

• Neodymium iron boron centre-pole magnet: a high performance option, with a low external magnetic field

• Shielded ceramic: a modification of the simple low-cost design, with the addition of shield components to reduce the external magnetic field

In each case, two different kinds of design are to be considered: high performance and low cost. The high-performance design aims to make the device non-linearity as small as possible, subject to a cost constraint. The low-cost design aims to minimise the cost, regardless of the penalty in non-linearity. Initial designs can be produced quite easily from a table of pre-defined magnets. These serve as the starting points for a design study that uses MagNet linked to an Excel spreadsheet. This combination forms a most effective design tool for systematically exploring design changes. For the project, students were given individual device specifications that had been pre-tested to ensure they would result in sensible designs. It is an effective way of eliminating plagiarism and collusion in work of this kind. The outline spreadsheet described in the document is available from Infolytica Corporation. Design of a Moving-Coil Driver Copyright © 2004 J D Edwards

CONTENTS

1 INTRODUCTION .................................... 1

1.1 The design topic ...............................................1

2 DESIGN CONCEPTS.............................. 2

2.1 Driver properties .............................................2

2.2 Coil properties .................................................2

3 INITIAL DESIGN PROCEDURE................. 4

3.1 Introduction .....................................................4

3.2 Design with a spreadsheet ...............................4

3.3 NdFeB magnet design......................................9

3.4 Material quantities and costs..........................9

4 INITIAL DESIGNS ................................ 10

4.1 Design specification .......................................10

4.2 Design procedure ...........................................10

5 FINAL DESIGNS.................................. 11

5.1 Introduction ...................................................11

5.2 Non-linearity ..................................................11

5.3 External magnetic field .................................12

6 ANALYSIS WITH MAGNET.................... 13

6.1 Introduction ................................................... 13

6.2 Using MagNet ................................................ 13

6.3 Using the spreadsheet.................................... 13

7 MODIFYING THE DESIGNS ................... 16

7.1 Introduction ................................................... 16

7.2 Current non-linearity.................................... 16

7.3 Position non-linearity.................................... 16

7.4 Saturation ...................................................... 17

7.5 Ceramic magnet............................................. 17

7.6 NdFeB magnet ............................................... 18

8 MAGNETIC SHIELDING ........................ 19

8.1 Introduction ................................................... 19

8.2 Modifying the workbook .............................. 19

8.3 Designing the shielded magnet ..................... 22

8.4 Design refinement.......................................... 22

9 MAGNET DATA................................... 23

9.1 Ceramic magnet data .................................... 23

9.2 NdFeB magnet data....................................... 24

List of Symbols Ac cross-sectional area of wire in the coil Ag cross-sectional area of the airgap Am cross-sectional area of the magnet B airgap flux density C material unit cost Cc ceramic unit cost Cn NdFeB unit cost Cs steel unit cost d axial depth of a cylinder da active axial depth of the coil dbp depth of the magnet bottom plate dc total axial depth of the coil dcm minimum total axial depth of the coil dcn nominal total axial depth of the coil dcp practical total axial depth of the coil dm maximum axial movement of the coil do overall diameter of the wire dmm depth of the permanent magnet dpm minimum depth of the magnet centre pole dpo axial position offset of the coil dpp total depth of the magnet centre pole dsbp depth of the shield bottom plate dsp depth of the steel centre pole dtp depth of the magnet top plate dw diameter of the wire conductor dwp practical diameter of the wire conductor e voltage induced in the coil f force on the coil G transducer constant or coefficient Gav average value of the transducer constant Gc calculated value of the transducer constant Gmax maximum value of the transducer constant Gmin minimum value of the transducer constant Gs specified value of the transducer constant i coil current I RMS coil current Im maximum coil current kc cooling coefficient ki wire insulation factor l length of wire in the active part of the coil lc total length of wire in the coil lct radial thickness of the coil lg radial length of the airgap lm radial mechanical clearance of the coil lmc main cylinder wall thickness lsc shield cylinder wall thickness lsg shield cylinder radial gap M mass of a cylinder

N number of turns per layer in the coil Na number of turns in the active part of the coil Ne effective number of turns in the coil Na number of turns in the active part of the coil Nl number of layers in the coil Np practical number of turns per layer in the coil P power dissipated in the coil Q non-linearity Qav average value of the non-linearity Qi current non-linearity Qy position non-linearity r radius of a cylinder ri inner radius of a cylinder ro outer radius of a cylinder rbp radius of the magnet bottom plate rcm mean radius of the coil rmmi inner radius of the main magnet rmmo outer radius of the main magnet rsmi inner radius of the shield magnet rsmo outer radius of the shield magnet rsbp radius of the shield bottom plate R reluctance of the magnetic circuit Rg reluctance of the airgap Rm reluctance of the magnet Rp reluctance of the saturated centre pole R resistance of the coil Rc calculated value of the resistance of the coil Rs specified value of the resistance of the coil S area of the curved surfaces of the coil Ta temperature of the ambient air Tc temperature of the coil Tcm maximum temperature of the coil u velocity of the coil V volume of a cylinder y axial displacement of the coil from its rest position ΔG deviation of G from the average value ΔT temperature rise above ambient ΔTm maximum temperature rise above ambient α temperature coefficient of resistance μ mass density μc ceramic mass density μn NdFeB mass density μs steel mass density φ magnet airgap flux φcm maximum flux from the coil φm magnet airgap flux from table φs required magnet airgap flux ρ resistivity at the coil temperature Tc ρm resistivity at the maximum coil temperature Tcm ρ0 resistivity at 20ºC

Section 1 – Introduction 1

1 INTRODUCTION

1.1 The design topic The primary objective of the project is to explore different designs for a moving-coil driver of the kind used in vibration generators and loudspeakers, as described below. Secondary objectives are to develop skills in using a spreadsheet for engineering design, using a professional electromagnetics software package, and linking the two packages to form a powerful design tool. Figure 1 shows a half section of the magnet and coil for a typical moving-coil driver. This diagram shows only the parts that affect the electromagnetic action of the device; mechanical details such as the supports for the coil are omitted. A ring of permanent-magnet material is the source of the magnetic field. The steel top plate, bottom plate and centre pole act as flux guides, which concentrate the magnetic flux in the narrow gap where the coil moves.

Figure 1: Moving-coil driver: ceramic magnet.

This form of magnet uses a low-cost ceramic ferrite material with a low remanence – typically Br = 0.4 T – so the magnet ring has a large cross-sectional area. Although inexpensive and simple to manufacture, this form of magnet has the serious disadvantage of a significant external magnetic field, so it requires magnetic shielding if it is to be placed close to a video monitor or TV set. High-performance magnetic materials such as neodymium iron boron (NdFeB) have a much higher remanence – typically 1.2 T – so a different kind of magnet design is possible, as shown in figure 2.

Figure 2: Moving-coil driver: NdFeB magnet.

In this form of magnet, the permanent-magnet material is effectively a continuation of the centre pole. The steel top plate, cylinder, and bottom plate complete the magnetic circuit. These parts form a shield around the permanent-magnet material, so the external field is greatly reduced, but the permanent-magnet material is much more expensive than ceramic ferrite. Three different kinds of design for a moving-coil driver are to be investigated, based on these two magnet configurations: • Unshielded ceramic magnet • NdFeB magnet • Shielded ceramic magnet In each case, two variants of the design will be considered: (a) low cost, (b) high performance. Initial designs of the unshielded ceramic magnet and the NdFeB magnet will make use of tabulated magnet data. Final designs for these magnets, and the design of the shielded magnet, require an electromagnetics software package to determine the magnetic field and the force on the coil.

moving coil centre pole

top plate

permanent magnet

bottom plate

moving coil centre pole

top plate

permanent magnet

bottom plate

cylinder

2 Design of a Moving-Coil Driver

2 DESIGN CONCEPTS

2.1 Driver properties

Transducer constant Part of the coil moves in the magnet gap where the average magnetic flux density is B. This is the active part of the coil. If the total length of wire in the active part is l, and the coil carries a current i, there will be a force f exerted on the coil, given by: GiBlif == (1)

where G = Bl. If the coil moves with velocity u, there will be an EMF e induced in the coil, given by: GuBlue == (2) These are the classical equations of the moving-coil transducer. The coefficient G is known as the force constant or the transducer constant.

Active depth and magnet flux The depth of the active part of the coil will be greater than the depth of the top plate in figure 1 or figure 2, because the field is not confined to this region. Beyond a distance equal to the length of the airgap, the fringing field decays very rapidly, so an active depth da may be defined by: gtpa ldd 2+= (3)

where dtp is the depth of the top plate or outer pole, and lg is the radial length of the airgap. See figure 3 (page 3) for key dimensions of the coil and magnet. The average flux density B is defined as follows. We suppose that an idealised field of constant magnitude B, confined to the active depth da, replaces the actual field. The value of B is chosen so that equation 1 gives the true value for the total force on the coil. If the active part of the coil has Na turns and a mean radius rcm, the length of conductor is: acm Nrl π2= (4)

If the coil is made from Nl layers of round wire with an overall diameter do (including the insulation thickness), then the number of active turns in each layer is da / do. The total number of active turns is therefore:

o

ala d

dNN = (5)

Similarly, the total number of turns in the coil is:

o

clc d

dNN = (6)

where dc is the total axial depth of the coil. Calculation of the coil resistance (see below) requires the wire conductor diameter dw. Let the overall diameter of the wire be: wio dkd = (7)

where ki is an insulation factor. Equation 5 becomes:

wi

al

o

ala dk

dNd

dNN == (8)

The transducer constant is then given by:

wi

l

wi

alcmacm dk

Ndk

BdNrBNrBlG φππ ====22 (9)

where φ is the airgap flux, given by: 2 cm aAB r d Bφ π= = (10)

where A is the area of a cylindrical surface of radius rcm and depth da. If the flux φ is taken to be the total flux that would be obtained with a very long coil, then equation 10 gives the equivalent uniform flux density over the depth da that would have the same effect.

2.2 Coil properties

Coil resistance If lc is the total length of the wire in the coil, and Ac is its cross-sectional area, the resistance of the coil is given by:

32288

4/2

wwwdk

dNrdd

dNrd

NrAlR

i

clcm

o

clcmccm

c

c ρρππρρ ====

(11) where ρ is the resistivity. The value of the resistivity depends on the coil temperature, as follows: )}20(1{0 −+= cTαρρ (12)

where ρ0 is the resistivity at 20ºC, Tc is the coil temperature, and α is the temperature coefficient of resistance.

Coil heating Current flowing in the coil will result in power dissipation in the form of heat, with a consequent rise in the coil temperature. The cooling conditions are very complex, so an exact calculation of the temperature rise is difficult. For the purposes of this design project, it will be assumed that the temperature rise ΔT above ambient is given by SPkT c /=Δ (13)

where P is the power dissipated in the coil, S is the area of the two curved surfaces, and kc is a cooling

Section 2 – Design Concepts 3

coefficient with a typical value of 0.04 Km2/W. The temperature rise is thus:

ccm

cdrRIkT

π4

2=Δ (14)

where I is the RMS current in the coil. Substituting for the coil resistance R from equation 11 gives

3

2

3

22

wi

lmc

wi

lc

dkNIk

dkNIkT

πρ

πρ ==Δ (15)

where Im = √2I is the maximum coil current, and ρ is the resistivity at the coil temperature Tc. The temperature rise ΔT is defined as: ac TTT −=Δ (16)

where Ta is the ambient air temperature.

Coil movement If the coil movement is very small, the minimum depth of the coil is simply the active depth da. In practice the coil is required to move an axial distance of ±dm from its rest position. If this movement is not to take the coil out of the active region of the magnet, the minimum depth of the coil must be increased to: mgtpmacm dldddd 222 ++=+= (17)

So far, it has been assumed that the rest position of the coil is symmetrical with respect to the top plate of the magnet, as shown in figures 1 and 2. However, with some designs of magnet, this gives a force/displacement characteristic that is not symmetrical: see section 5.2. The performance can be improved if the rest position of the coil is offset by a distance dpo from this symmetrical position. The minimum coil depth is then given by:

2 2 2cm a m po tp g m pod d d d d l d d= + + = + + + (18)

The offset dpo is positive if the coil is displaced downwards from its symmetrical position, so that more of the coil projects below the top plate. The magnet centre pole must be deep enough to accommodate the coil when it moves. The required condition can be found as follows. If there is no offset, the coil in its rest position has equal amounts projecting above and below the top plate. Given that dtp is the depth of the top plate, the centre of the coil is a distance dtp / 2 below the top of the pole. The bottom of the coil is therefore a distance dc / 2 + dtp / 2 below the top of the pole. To allow for coil movement and a position offset, the minimum depth of the pole is therefore:

2

cm tppm m po

d dd d d

+= + + (19)

In practice, the total depth of the centre pole dpp must be greater than this, since the coil must not make contact with the bottom plate of the magnet. For a ceramic magnet, this depth is just the steel pole depth, but for a NdFeB magnet, it is the sum of the steel pole depth dsp and the magnet depth dmm (see figure 3).

Magnet airgap The radial thickness of the coil is given by wilolct dkNdNl == (20)

The magnet airgap length lg must accommodate this thickness plus a mechanical clearance lm on each side to ensure that the coil does not touch the magnet. The minimum length of the airgap is therefore: mwilmctgm ldkNlll 22 +=+= (21)

In practice, the coil is usually wound on a rigid cylindrical former, so the thickness of the former must be included. The gap between the coil and the centre pole will then be larger than the gap between the coil and the top plate. This lack of symmetry makes very little difference to the design, so it will be ignored in this project. The value of lm given in the design specification (section 4.1) should be understood to mean the air space plus half the thickness of the coil former.

lct

lg

rcm

dtp

dpp

dc

dmm

dbprbp

(a)

lct

lg

rcm

dtpdsp dc

dmm

dbprbp

lmc

(b)

Figure 3: Moving-coil driver main dimensions:

(a) ceramic magnet, (b) NdFeB magnet.


3 INITIAL DESIGN PROCEDURE

3.1 Introduction The specification of a moving-coil driver will usually include the required values of the following quantities: • The transducer constant Gs • The coil resistance Rs • The maximum coil current Im • The coil axial movement ±dm The notation Gs and Rs indicates the specified values of G and R, which must be distinguished from the calculated values Gc and Rc for a particular design. In addition, other quantities such as the maximum coil temperature Tcm will be specified. For the designer, the problem is to determine the dimensions of the coil and magnet that will satisfy this specification. As with most engineering design problems, there are many possible solutions that will satisfy this specification. Choosing the “best” solution will involve other factors such as the quantities of materials used in the magnet, and measures of performance such as linearity and the external magnetic field. Any design must satisfy the equations for the coil developed in section 2, and the design will also involve the magnet through the relationship of the airgap flux φ to the magnet dimensions. The final design will require electromagnetic simulation software to explore the effects of changing the magnet dimensions. However, a useful initial design can be based on tabulated results for a few representative magnet configurations. Section 3.2 describes a systematic way of doing this.

Magnet tables The tables in section 9 give values of the airgap flux for the following ranges of variables: • Airgap length values of 3 mm, 4 mm and 6 mm. • Pole depth values of 20 mm and 30 mm for the

smaller airgaps, and 25 and 35 mm for an airgap of 6 mm.

• Mean coil radius values that give a useful range of magnet flux values.

The other magnet dimensions have been chosen to give a reasonably good magnet design. For simplicity, the top plate thickness is held constant at 6 mm. These magnet dimensions will accommodate a wide range of coil designs, so they are suitable for the initial driver design. The NdFeB designs use more permanent-magnet material than is common in commercial designs, so the resulting magnet costs are much higher than for

equivalent ceramic designs. However, these designs give much improved device performance, as will be seen in section 6. One of the aims of the project is to explore the trade-off between performance and cost.

3.2 Design with a spreadsheet An Excel spreadsheet will be used for the initial design, so that the effects of changes can be explored systematically. The sections below explain the design sequence and the procedure for developing the spreadsheet. When this spreadsheet is working correctly with the example data, it can be used for the initial designs as specified in section 4. An outline of a spreadsheet is supplied in the Excel workbook file Driver Design Outline.xls. This has the descriptive text entered and certain cells named, but it will need to be completed as described below.

Opening the workbook This workbook uses macros. When you open the file, the following dialog may be displayed:

Click Enable Macros to continue. However, you may see a different dialog telling you that macros have been disabled:

In this case, click OK to continue. From the Tools menu, select Options / Security / Macro Security, and change the level to Medium. Close the workbook and re-open it.

Workbook versions During the project, you will need to save several different versions of the Excel workbook file. It is helpful to have a logical sequence of names and version numbers, so that it is easy to keep track of the work. The instructions in this document give preferred file names for the different stages of the project. However, you are free to use different names, provided there is a logical sequence. Begin by saving a copy of the outline workbook, with the following file name:

Driver Design Example 1a.xls

Section 3 – Initial Design Procedure 5

As you develop the spreadsheet, it is a good idea to save successive stages with different suffix letters such as:

Driver Design Example 1b.xls

Do this frequently, to minimise loss of work in the event of mistakes or computer failure.

Spreadsheet layout The design spreadsheet is in the Design sheet of the workbook. Ignore the Analysis sheet at this stage – it is required later for the final design. The spreadsheet is set out with data in clearly marked areas, separate from the calculations. Data values that may be changed for the designs are shown in red. Calculated values that need to be inspected as part of the design process are grouped together and shown in blue. It is essential not to alter any of the cell locations for the coil data and magnet data on the spreadsheet. The workbook includes a link to the MagNet electromagnetic simulation software that takes data values from these cells. This link will fail if the locations are changed. Also, the names of the worksheets (Design and Analysis) must not be changed.

Enter your name in cell C1 of the Design Sheet. This helps to identify the file when you submit it for comment or assessment.

Magnet Data Cells B23 – D38 contain data for the magnets for each design. For the initial design, values will be taken from the tables in section 9. Subsequently, these values will be modified when the driver is modelled with MagNet. Example data values for two magnets have already been entered in the outline spreadsheet. These values are required for developing the rest of the spreadsheet. Once the development is complete, it will be necessary to enter different data values for the initial designs. Other magnet dimensions are calculated from the magnet data by formulae in cells G31 to I42.

Constants and Specification Enter data in column B of the Design sheet as follows:

A B 4 Specification 5 Transducer constant Gs (N/A) 8.00

6 Coil resistance Rs (ohms) 5.00

7 Maximum current Im (A) 2.00

8 Axial movement dm (mm) 6.00

9 Max. coil temperature Tc (ºC) 120

10 Ambient temperature Ta (ºC) 30.0

11 Cooling coefficient kc (Km2/W) 0.0400

12 Coil insulation factor ki 1.10

13 Coil mechanical clearance lm (mm) 1.00

The first four values are examples only, and are different from the individual specifications issued to students for this project.

Enter data in column G of the Design sheet as follows:

F G 1 Constants 2 Copper cold resistivity ρ0 (nΩ.m) 17.2

3 Temperature coefficient α 0.00390

4 Steel mass density μs (kg/m3) 7600

5 Ceramic mass density μc (kg/m3) 4900

6 NdFeB mass density μn (kg/m3) 7390

7 Steel unit cost Cs (£/kg) 2.00

8 Ceramic unit cost Cc (£/kg) 1.00

9 NdFeB unit cost Cn (£/kg) 50.0

Cell names Cells B5 – B13 and G2 – G9 contain values that will be the same for all designs. These cells have been given names, so that they can be referenced by name in the formulae for the design calculations. Cells G12 and G13 also have names, but all other cells in the spreadsheet are referenced by their column letter and row number. With this form of reference, formulae created in column G for the ceramic design can be copied to column H for the NdFeB design. When a formula is copied, any column references will be changed automatically to refer to cells in the next column, but cell names will continue to refer to the original cells.

Inspect the cell names by selecting the cells in turn and viewing the name in the text box to the left of the Formula Bar above the spreadsheet grid. If the cell does not have a name, its column/row reference will be shown instead.


Calculated values Enter formulae in cells G12 and G13 to calculate the maximum coil temperature rise (equation 16) and the resistivity of copper at the maximum coil temperature (equation 12) as follows:

G12 =Tcm-Ta G13 =Rho0*(1+Alpha*(Tcm-20))

These formulae should give the following results:

F G 11 Calculated values

12 Max. temperature rise ΔTm (K) 90.0

13 Copper hot resistivity ρm (nΩ.m) 23.9

Coil Data The Coil Data section is used for entering data values as the design progresses. Initially, a value must be specified for the number of layers – see (a) below – and for the coil position offset.

Enter the value 4 in cell B16 and 3.0 in cell B20:

A B 15 Coil Data Ceramic

16 Number of layers Nl 4

17 Practical wire diameter dwp (mm)

18 Nominal coil depth dc (mm)

19 Coil mean radius rcm (mm)

20 Coil position offset dpo (mm) 3.0

(a) Starting point for the design Most of the design equations in section 2 involve the number of layers Nl. A value must be assumed for Nl at the outset. Other quantities that depend on Nl may be calculated, to see whether this assumption leads to a feasible design. If not, the value of Nl can be changed and the process repeated. For this design project Nl can be 2, 4 or 6, so a sensible starting point is to try Nl = 4.

(b) Wire diameter and magnet flux Consider equation 15:

3

2

wi

lmc

dkNIkT

πρ

=Δ [15]

For a given coil current Im, the temperature rise depends on the number of layers Nl and the wire diameter dw. It does not depend on the depth or the radius of the coil. Therefore, once the number of layers has been chosen, the wire diameter can be calculated. If the maximum allowable value ΔTm is chosen for the temperature rise, re-arranging equation 15 gives the minimum wire diameter:

3/12

⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ=

mi

lmcmwm Tk

NIkdπ

ρ (22)

To calculate this value, enter a formula in cell G16 as follows: =(Rho*Kc*Im^2*B16/(Pi*Ki*DTm))^(1/3) • Rho, Kc, Im, Ki, and DTm are the names of cells

containing the values of ρm, kc, Im, ki and ΔTm respectively.

• Pi is a constant that has been defined in the spreadsheet to have the value of π.

• B16 is the cell containing the value of the number of layers Nl.

All of the numerical values displayed on the spreadsheet are in convenient SI multiples, such as μWb for flux values and mm for lengths. This makes the spreadsheet easy to use, but some formulae must include the corresponding factors to convert these values to SI units. In the formula for wire diameter above, the resistivity is in nΩm, which must be multiplied by 10–9 to get the SI value in Ωm. When the cube root is taken, the resulting factor of 10–3 is cancelled by the factor of 103 for converting the result from m to mm, so no conversion factors appear in the final version of the formula. It could have been entered as: =1e3*(1e-9*Rho*Kc*Im^2*B16/(Pi*Ki*DTm))^(1/3)

The result should be:

F G 15 Coil Calculation Ceramic

16 Minimum wire diameter dwm (mm) 0.366

The value for the minimum wire diameter must be rounded up to the nearest 0.01 mm for a practical design: call this value dwp. A larger diameter may be used if required by other design considerations – see (e) below.

Enter the value of the practical wire diameter (0.37 mm) in cell B17.

Once the wire diameter has been chosen, the required magnet airgap flux φs can be determined from equation 9:

wi

ldk

NG φ= [9]

If Gs is the specified value of the transducer constant, rearranging this equation gives:

l

swpis N

Gdk=φ (23)


Enter a formula in cell G19 to calculate the value of φs from equation 23, using cell names for ki and Gs, and column/row references for dwp and Nl. Unit conversions are required. The result should be 814 μWb.

Before a magnet can be selected from the tables, however, the airgap length must be calculated.

(c) Airgap length and coil depth Equations 20 and 21 can be used to determine the minimum airgap length for the magnet: wpilct dkNl = [20]

mctgm lll 2+= [21]

where lm is the specified mechanical clearance.

Enter formulae in cells G23 and G20 to calculate the value of lct and lgm from these equations. The results should be 1.63 mm and 3.63 mm respectively.

From the magnet tables, a practical value of airgap length lg larger than lgm must be selected. The nearest available airgap length is 4 mm, which has already been entered in cell B24. Now that the airgap length is known, equation 3 gives the active coil depth da, and equation 18 gives the minimum coil depth dcm: gtpa ldd 2+= [3]

2cm a m pod d d d= + + [18]

where dm is the specified axial movement of the coil and dpo is the coil position offset (cell B20).

Enter formulae in cells G22 and G17 to calculate the value of da and dcm from these equations. The results should be 14.0 mm and 29.0 mm respectively.

(d) Coil mean radius To determine the coil mean radius rcm, a magnet must be selected from the tables that will give a value of airgap flux φm close to the required value φs found in (b) above. It is now possible to find two ceramic magnets from the table that will give a flux density close to the required value. One has a pole depth of 20 mm, and the other has a pole depth of 30 mm. Since the required depth is unknown at this stage, a value of 30 mm is selected. The required minimum pole depth will be found later, and a different magnet can be chosen if necessary. From the tables of ceramic magnet data, a magnet designed for a mean coil radius of 14 mm will be suitable. This gives an airgap flux value of 782 μWb, which is within 4% of the required value of 814 μWb. The data for this magnet has already been entered in cells B23 to B31. Other magnet

dimensions are calculated by the formulae already entered in cells G31 to G34.

Enter the coil mean radius (14 mm) in cell B19.

(e) Coil resistance Once the coil mean radius rcm has been found, equation 11 with ρ = ρ0 gives the coil resistance at 20ºC in terms of the coil depth:

03

8

wp

cm l cs

i

r N dRk d

ρ= [11]

This equation determines the coil depth dcr that will give the specified coil resistance Rs. Re-arranging equation 11 gives:

3

08wpi s

crcm l

k d Rd

r Nρ= (24)

Enter a formula in cell G18 to calculate the value of dcr from equation 24. Unit conversions are required. The result should be 36.2 mm.

The minimum coil depth dcm has already been found from equation 18 – see (c) above. If dcr > dcm, as it is for the example data, then setting dc = dcr will meet both the coil resistance specification and the coil movement specification. A possible design has been achieved, provided that the pole depth requirement is also satisfied – see (f) below. The tolerance of ±10% in the coil resistance can be exploited to make dc < dcr, which is beneficial because it requires less pole depth. A suitable choice is 34 mm.

Enter a value of 34 mm in cell B18.

This is the nominal coil depth dcn, which must be modified by practical considerations.

(f) Practical coil depth and pole depth There must be a whole number of turns in each layer of the coil. A practical coil depth dcp that will achieve this can be calculated as follows. • Calculate the exact number of turns per layer as:

wpi

cdkdN = (25)

• Round this value to the nearest integer Np. See below for the Excel function to do this.

• Calculate the practical coil depth from: wpipcp dkNd = (26)


• Calculate the minimum practical pole depth by substituting the value of dcp in equation 19:

2

cp tppm m po

d dd d d

+= + + (27)

where dtp is the depth of the top plate and dm is the coil axial movement. In the spreadsheet, to round a number to the nearest integer, use the Excel function ROUND:

ROUND(number, num_digits) number – the value to be rounded num_digits – the number of digits. Setting this to zero will round the value to the nearest integer.

Enter formulae in cells G24, G25, G26 and G21 to calculate the values of N, Np, dcp and dpm from these equations. The results should be 83.5, 84, 34.2 mm and 29.1 mm respectively. Note that the ROUND function must be used in a formula in cell G25. Do not enter the numerical value 84, since this will give wrong results with other data.

The depth of the centre pole for the ceramic design, or the total depth of the steel centre pole and the magnet for the NdFeB design, must exceed the minimum value dpm. For the example ceramic design, the selected pole depth of 30 mm exceeds the minimum of 29.1 mm. In this case, it is not possible to use a magnet with a pole depth of 20 mm.

(g) Calculated results From the final values of the coil dimensions, it is necessary to calculate the transducer constant G using equation 9 with the actual magnet flux φm:

wpi

mlc dk

NG φ= (28)

It is also necessary to calculate the coil resistance R using equation 11 with ρ = ρ0:

308

wpdk

dNrR

i

cplcmc

ρ= (29)

Enter formulae in cells G27 and G28 to calculate the value of Gc and Rc from these equations. Unit conversions are required. The results should be 7.69 N/A and 4.73 Ω respectively.

These values are within ±10% of the specified values of 8 N/A and 5 Ω. If either quantity had been outside the permitted range, the design would have had to be changed.

Example results The coil data values for the example specification should be as follows:

A B 15 Coil Data Ceramic

16 Number of layers Nl 4

17 Practical wire diameter dwp (mm) 0.370

18 Nominal coil depth dc (mm) 34.0

19 Coil mean radius rcm (mm) 14.0

20 Coil position offset dpo (mm) 3.0

The corresponding calculated results are:

F G 15 Coil Calculation Ceramic 16 Minimum wire diameter dw (mm) 0.366

17 Minimum coil depth dcm (mm) 29.0

18 Coil depth dcr for given Rs (mm) 36.2

19 Required airgap flux φs (μWb) 814

20 Minimum airgap length lgm (mm) 3.63

21 Minimum pole depth dpm (mm) 29.1

22 Active coil depth da (mm) 14.0

23 Coil radial thickness lct (mm) 1.63

24 Number of turns per layer N 83.5

25 Practical no. of turns per layer Np 84

26 Practical coil depth dcp (mm) 34.2

27 Calculated trans. const. Gc (N/A) 7.69

28 Calculated coil resistance Rc (Ω) 4.73

Note that cells G16 – G28 contain formulae, which display the values shown above. Do not proceed to the next section until the spreadsheet is giving correct results for the ceramic magnet.


3.3 NdFeB magnet design Open the workbook Driver Design Example 1x, where x is the last suffix letter used in section 3.2. Save the file with the following file name:

Driver Design Example 2a.xls

The spreadsheet can be extended as follows for the NdFeB magnet design.

Copy the contents of cells G16–G28 to H16–H28. This will give several #DIV/0! errors, because coil data values are missing from column C.

Enter values for the Coil Data in column C, using a procedure similar to that for the ceramic magnet:

• Start by setting the number of layers to 4 and the coil position offset to 0 for this design.

• Use a value of 18 mm for the coil mean radius.

• The nominal coil depth dc can be set to the minimum value of 26 mm.

Results for Gc and Rc should be 7.71 N/A and 4.63 Ω respectively.

3.4 Material quantities and costs An important aspect of a design is the quantity of material used. All of the components of the magnets are solid or hollow cylinders, so the following equations can be used. The mass of a solid cylinder is given by

drVM 2μπμ == (30)

where μ is the mass density, V is the volume, r is the radius and d is the depth. It follows that the mass of a hollow cylinder is

drrVM io )( 22 −== μπμ (31)

where ri and ro are the inner and outer radii. The cost of the material is then given by CM, where C is the unit cost in £/kg.

Enter formulae in the relevant cells in the range G45 to H58 to calculate the mass and cost values. For the component dimensions, use cell references from the Magnet Data area (B23 to C32) and the Magnet Dimensions area (G31 to H42). For the density and unit cost values, use cell names from the Constants area (G2 to G9). Use the constant Pi for the value of π. The results should be as shown below. Note that all of these cells contain formulae that produce the numerical results shown. The formula for a total just needs to refer to the values that have already been calculated in other cells; for example, the formula for the total mass of steel in G51 is =G45+G46+G47. The total mass of PM (permanent magnet) material will be equal to the mass of the main magnet for these two designs, but for the shielded design it will include the mass of the shield magnet. Ignore the shielded design at this stage.

F G H Magnet Calculation Ceramic NdFeB Mass of centre pole (kg) 0.103 0.037 Mass of top plate (kg) 0.216 0.099 Mass of bottom plate (kg) 0.253 0.156 Mass of cylinder (kg) 0.142 Mass of shield cylinder (kg) Mass of shield bottom plate (kg) Total mass of steel (kg) 0.572 0.433 Mass of main magnet (kg) 0.603 0.143 Mass of shield magnet (kg) Total mass of PM material (kg) 0.603 0.143 Total mass of magnet (kg) 1.175 0.576 Cost of steel (£) 1.14 0.87 Cost of PM material (£) 0.60 7.13 Total cost of material (£) 1.75 8.00

When the spreadsheet is working correctly, ensure that you save the final version of the file.


4 INITIAL DESIGNS

4.1 Design specification Two initial designs are required for a moving-coil driver. The first design is to use a ceramic magnet, and the second design is to use a NdFeB magnet. Both designs are to meet the following specification.

Transducer constant: ** N/A ±10%. Coil resistance at 20ºC: ** Ω ±10%. Maximum current: ** A. Coil axial movement: ±** mm. Maximum coil temperature: 120ºC. Ambient temperature: 30ºC. Cooling coefficient: 0.04 Km2/W. Coil insulation factor: 1.1. Coil mechanical clearance: 1.0 mm. Number of layers: 2, 4 or 6.

** Individual specification for the first four items.

The resistivity of copper at 20ºC is 17.2 nΩm, and the temperature coefficient of resistance is 0.0039. Properties of the magnet materials are as follows:

Material Mass density Unit cost Steel 7600 kg/m3 2.0 £/kg Ceramic ferrite 4900 kg/m3 1.0 £/kg NdFeB 7390 kg/m3 50 £/kg

The unit cost figures are for the material alone, and do not include the cost of manufacture. For each design, a suitable magnet is to be selected from the tables in section 9.

4.2 Design procedure

Ceramic magnet Open the completed version of the project workbook containing the example data. This should be named Driver Design Example 2x, where x is the suffix letter of the last saved version. Before starting the design work, save the workbook with the following file name:

Driver Design Initial 1a.xls

Enter your individual specification in cells B5 – B8 of the Design sheet. Start with the ceramic design, following the procedure described in section 3. In summary, the steps are as follows.

1 Start with a value of 4 for the number of layers.

2 Enter the practical wire diameter in cell B17, by rounding up the minimum wire diameter (cell G16) to the nearest 0.01 mm.

3 Scan the magnet data tables in section 9 for a magnet with (a) an airgap length greater than the minimum, (b) a flux close to the required value (cell G19), with a pole depth of 30 mm.

4 Enter the coil mean radius in cell B19. Leave the coil position offset set to 3.0 mm in cell B20.

5 Compare the two calculated values of coil depth (cells G17 and G18). If dcr > dcm, you can use dcr for the nominal coil depth. However, if dcr < dcm, the following procedure is required. • Check whether it is possible to meet the coil

resistance specification by setting the nominal coil depth to the minimum dcm.

• If the resulting value of R is outside the 10% tolerance, increase the practical wire diameter dwp by 0.01 mm, since this will increase the value of dcr. The spreadsheet will re-calculate the flux φ.

• If the new value of φ requires it, select a different magnet from the data tables, and enter the new value of coil mean radius rcm.

• Compare the new values of coil depth. • Repeat this process as required. • Alternatively, choose a different value for Nl

and repeat the design process.

6 Check whether the magnet pole depth could be reduced to 20 mm. If so, select a different magnet from the tables.

7 When you have a design that meets the specification, enter all the magnet data in cells B23 – B31.

8 Save the workbook.

NdFeB magnet Open the project workbook containing the initial ceramic design. This should be named Driver Design Initial 1x, where x is the suffix letter of the last saved version. Save the workbook with the following file name:

Driver Design Initial 2a.xls

Follow a similar design procedure with this magnet, but with the coil position offset set to 0. Note that the total pole depth is the sum of the depths of the steel pole and the central magnet, so the data tables for NdFeB have entries for total pole depths of 20 mm and 30 mm (or 25 mm and 35 mm with an airgap of 6 mm).

Section 5 – Final Designs 11

5 FINAL DESIGNS

5.1 Introduction Final designs are required for the drivers, where the ceramic and NdFeB magnets have been redesigned using the MagNet electromagnetic simulation software. In addition, designs for a third driver are required, using a ceramic magnet with shielding to reduce the external magnetic field. All of these designs must meet the performance specification. Section 5.2 discusses non-linearity, which is a feature of most moving-coil devices. The NdFeB magnet of the initial design has better linearity than the ceramic magnet, but it is much more costly. One of the aims of the project is to investigate the trade-off between performance and cost, by considering alternative designs as follows:

Ceramic magnet (a) Low cost, based on the initial design. (b) Non-linearity improved as much as possible.

NdFeB magnet (a) Good linearity, based on the initial design. (b) Cost reduced as much as possible.

Shielded ceramic magnet Low external magnetic field together with: (a) Low values of weight, size and cost. (b) Low values of non-linearity.

Workbook versions The Excel workbook for the project has columns for all three magnet configurations: ceramic, NdFeB and shielded. For each configuration there are two variants: low cost, and high performance (low values of non-linearity). An effective way of organising the workbooks is to use one sequence of files for the low-cost designs, and another for the high-performance designs. This principle will be followed for the recommended file names in sections 6, 7 and 8.

Practical considerations For ease of manufacture, all of the designs must use simple shapes such as cylinders and flat plates. Other shapes are unsuitable. For example, the thickness of the bottom plate could be tapered so that the magnitude of the flux density was uniform. This would reduce the volume of the plate, and apparently save material. In practice the tapered plate would have to be made from a flat plate by machining it on a lathe. This would increase the cost of manufacture, and the material removed would be wasted.

Similarly, a hollow centre pole is not a useful design change.

5.2 Non-linearity The force on the moving coil is given by equation 1: GiBlif == [1]

In section 2, G was termed the transducer constant. In practice, however, G is not constant. Its value depends on the coil current i and the axial displacement y of the coil from its rest position. The quantity G should therefore be called the transducer coefficient, rather than the transducer constant. Consider the effect of the coil current. So far, it has been assumed that the flux density B in equation 1 is the value created by the permanent magnet. However, current flowing in the coil will create an additional component of flux in the magnet gap, with a value proportional to the current. Since the force depends on the product of the current and the flux density, there will be an additional component of force, with a magnitude that varies as the square of the current, and a direction that remains constant – it does not reverse when the current reverses. The main flux in the airgap is the constant quantity produced by the permanent magnet; for a given position of the coil, this will give a force proportional to the current. Since the additional component of force varies as the square of the current, the total force on the moving coil will not be a linear function of the coil current. Moreover, since the additional component adds to the main force for one direction of current, and subtracts for the other direction, the characteristic is not symmetrical. However, the effect is usually small, so we can continue to use equation 1 if we specify that the value of G is no longer a constant, but varies with the coil current. This variation in the value of G is a measure of the non-linearity of the characteristic. Coil displacement also affects the value of G. If the total coil depth dc is not much greater than the active depth da, it will take only a small displacement to move part of the coil out of the magnet gap. When this happens, the effective value of da will be reduced and the value of G will fall. A further source of non-linearity is a lack of symmetry in the magnetic field above and below the top plate of the magnet. In most applications both types of non-linearity are undesirable, so a designer may need to minimise the non-linearity. This is particularly important for hi-fi loudspeakers, where non-linearity will result in distortion of the sound output. Numerical measures of non-linearity can be defined as follows in terms of the transducer


coefficient G. If the maximum and minimum values of G are Gmax and Gmin, then the average value is

2

minmax GGGav+

= (32)

The maximum deviation from the average is

max min

max min

2

av avG G G G GG G

Δ = − = −−=

(33)

The non-linearity Q is defined as the maximum deviation divided by the average:

max min

max min

max min max min

2

2av

G GG GGQ G GG G G

−−Δ= = =+ +

(34)

The current non-linearity Qi is the value given by equation 34 when Gmax and Gmin are the values of G determined with the coil in its normal rest position, with currents of +Im and –Im, where Im is the maximum rated current. The position non-linearity Qy is the value given by equation 34 when Gmax and Gmin are the maximum and minimum values of G in the working position range, with the current equal to its rated value and acting in a direction to give the highest value of G. The average non-linearity Qav is the average of Qi and Qy. It is a useful general measure of performance when comparing different designs. The transducer constant is the value of G when the current is small and the coil is in its rest position. In practice it may be found by taking the average of the values calculated for small positive and negative currents, and it will be very close to the value of Gav given by equation 32 for currents of +Im and –Im.

Determining the values of G The values of G can be determined as follows, using the electromagnetic simulation package MagNet. A model of the device is created, and the y-component of the force on the coil is determined. This force is equal to Gi, where i is the coil current. For the current non-linearity, it is sufficient to find the values of G for (a) i = +Im, (b) i = –Im. The transducer constant is the mean of these values. For the position non-linearity, the current direction that gives the largest value of G is used. Values of G are then found for several displacement steps in each direction, and the lowest and highest values are selected.

5.3 External magnetic field All driver magnets will produce some external magnetic field. Since one of the project objectives is to reduce the external field by screening, a numerical measure of the field is required so that designs can be compared. For this project, a measure of the external field is defined as follows. It is the maximum value of the flux density magnitude over the surface of a sphere of radius 200 mm, when there is no current in the coil. The centre of the sphere is at the centre of the top face of the centre pole. The required flux density value is determined from the field solution generated by the MagNet package, by sampling the field at regularly spaced angles and taking the maximum.

Section 6 – Analysis with MagNet 13

6 ANALYSIS WITH MAGNET

6.1 Introduction MagNet is a professional electromagnetic simulation software package that is widely used as an industrial design tool. For this project, it will be used as a “virtual laboratory” to try out the effects of design changes on the performance of the moving-coil driver. With MagNet, a model of the device is built and then solved to determine the magnetic field. Flux plots and colour maps of the flux density can be displayed to guide the designer, and the force on the moving coil can be calculated. The Excel workbook for this project can interact with the MagNet software, as described in section 6.3. This enables the model to be created automatically from the magnet and coil data, with the resulting force values imported into the workbook. Before using the workbook in this way, however, it is necessary to gain experience of using MagNet interactively, as described in section 6.2. The Excel link exploits a powerful feature of MagNet known as scripting. Scripts are text files containing commands that control MagNet. These commands can reproduce actions that the user would have taken when running the package interactively. Through the Microsoft Automation interface, other packages such as Excel can control MagNet with scripts. The workbook contains procedures, written in Visual Basic for Applications, that use scripting commands to interact with MagNet. For the ceramic and NdFeB drivers, the Visual Basic procedures are fully functional, and can be used to develop the final designs. However, for the shielded driver the procedures are incomplete, so one of the tasks is to complete them, using the working procedures as examples.

6.2 Using MagNet The MagNet package is described in the document An Introduction to MagNet for Static 2D Modeling.

1 Read at least the first part of chapter 1, so that you understand the principles of representing a 3D object by a 2D model.

2 Log on to one of the computers in the laboratory where MagNet is installed.

3 Double-click the MagNet icon. • Wait for the program to load,

which may take up to 40 seconds.

4 Work through the whole of the tutorial in chapter 2, even though the device is quite different from a moving-coil driver, because it introduces essential features of the package.

5 Read the introduction to chapter 4.

6 Work through the case study on the moving-coil transducer in chapter 4. This is very similar to the driver with a ceramic magnet in this project.

6.3 Using the spreadsheet The case study on the moving-coil transducer should have shown two things: the power of MagNet to produce useful results, and the need for a faster method of constructing the model if many different designs are to be analysed. The project workbook meets this need.

Analysis The Analysis sheet of the workbook handles the link between Excel and MagNet. Buttons on this sheet activate the main Visual Basic procedures for starting and closing MagNet, and for building and solving models. Results for forces and magnetic flux density values are displayed in the lower part of the sheet; calculated driver performance values are displayed in the upper part. Calculations are carried out in a Visual Basic procedure and the results displayed on the sheet, so there are no formulae in any cells on the Analysis sheet. See section 8.2 for information about the Visual Basic procedures. The driver performance is calculated from the force values returned by MagNet, using the methods described in section 5.2. The transducer constant Gav is used to update the value for the magnet flux φm on the Design sheet as follows. Equation 9 gives a relationship between G and the flux:

wi

l

wi

alcmacm dk

Ndk

BdNrBNrBlG φππ ==== 22 [9]


Re-arranging this equation gives:

l

avwim N

Gdk=φ (35)

Consequently, the calculation of the transducer constant on the Design sheet gives the same result as Gav, since the formula is derived from equation 9.

Starting

1 Log on to one of the computers in the laboratory (ENGG III F7) where MagNet is installed.

2 Open the version of the project workbook containing the example data. • This should be named

Driver Design Example 2x, where x is the suffix letter of the last saved version.

3 Save the file with the following file name:

Driver Design Example 3a

4 In the Design sheet, change the value of the coil position offset (cell B20) from 3.0 to 0.

5 Select the Analysis sheet, and inspect its contents. • Take care not to alter the names of the

materials in cells B11, B12 and B13.

6 Click the Start MagNet button. • Wait for the normal mouse pointer to return.

7 Click the MagNet Visibility button.

8 In the dialog box, click Yes to make MagNet visible. This should display the MagNet main window and the MagNet icon on the task bar. • If the MagNet main window is not displayed,

click the MagNet icon on the task bar.

Building the ceramic model.

1 Click the Excel icon on the task bar.

2 In the Excel window, click Build Ceramic.

3 When the normal pointer returns, click the MagNet icon on the task bar. This should display the MagNet main window, with the model visible in the View window.

4 Maximise the MagNet window.

5 Inspect the model of the device. It should resemble the model created in the case study.

6 Minimise the MagNet window.

Analysing the ceramic model

1 In the Solution Data for the Ceramic model, enter the following values: • Number of current increments 1 • Number of position increments 5

2 Click Solve Ceramic. • Click the MagNet icon on the task bar to

display the Solver Progress dialog.

3 When the solutions are complete, click the MagNet icon to display the MagNet window.

4 In the MagNet window, display the flux plot and the shaded plot of |B| smoothed.

5 Use the Field Probe to investigate the values of flux density in the centre pole and the bottom plate. • Confirm that the values are above 1.8 T in

some parts of these components.

6 Display the B/H curve for the steel as follows: • In the Project bar, click the Material tab. • If necessary, expand the Model Materials tree. • Right-click CR10: Cold rolled 1010 steel,

and select Properties. • In the Properties dialog, click the Magnetic

Permeability tab. • Re-size the Properties dialog to display the

curve more clearly.

7 Observe the shape of the curve, which is approaching saturation at 1.8 T. • Close the Properties dialog.

Driver performance Minimise the MagNet window. In the Excel Analysis sheet, inspect the computed performance figures for the ceramic magnet. These have been calculated from the MagNet results listed in the lower part of the spreadsheet.

Modifying the example design The MagNet results show a marked asymmetry in the variation of force with coil position, which partly accounts for the position non-linearity value of about 5.1%. Notice that the maximum force is obtained with a coil position between –4.8 mm and –3.6 mm. If a position offset of about 4 mm is introduced, the maximum force should be developed in the rest position. However, this would require a minimum pole depth of 30.1 mm; since the actual pole depth is only 30 mm, the coil would hit the bottom plate of the magnet when it moved below the rest position.

Section 6 – Analysis with MagNet 15

To allow some mechanical clearance, the pole depth should be increased to about 32 mm. This will give a corresponding increase in the depth of the permanent magnet material. Proceed as follows.

1 Select the Design sheet.

2 In the Coil Data, change the coil position offset to 4.0 mm.

3 In the Magnet Data, change the total centre pole depth in cell B25 to 32 mm.

4 Select the Analysis sheet.

5 Copy the Ceramic Driver Performance results to column H, so that they can be compared with the new results.

6 Click MagNet Visibility, and click Yes in the dialog box to make MagNet invisible.

7 Click Build Ceramic, and wait for the normal mouse pointer to return.

8 Click Solve Ceramic. Results should appear in the lower part of the spreadsheet as the model is solved successively.

9 When the solution is complete, compare the new performance figures in column D with the previous values in column H.

10 Observe that the force/position characteristic is nearly symmetrical.

11 Change the number of position increments from 5 to 1, and click Solve Ceramic. • The solution is much faster, but the position

non-linearity value is virtually unchanged. Provided the force/position characteristic is nearly symmetrical, only one position increment is required for calculating the non-linearity.

NeFeB magnet It is instructive to compare the performance of the ceramic magnet with that of the NdFeB magnet.

1 In the Solution Data for the NdFeB design, set the Number of Position Increments to 5.

2 Click Build NdFeB, and wait for the normal mouse pointer to return.

3 Click Solve NdFeB.

4 Compare the non-linearity values for the two designs. The position non-linearity for NdFeB is somewhat smaller, because the fringing field is more symmetrical, and the current non-linearity is very much smaller.

5 Save the final version of the workbook.

6 Click Close Magnet before closing the workbook.


7 MODIFYING THE DESIGNS

7.1 Introduction Both types of non-linearity are important properties. Design changes that improve the position non-linearity may have an adverse effect on the current non-linearity, and vice versa, so we require a constraint on the relative magnitudes. For the purpose of this project, the ratio of one type of non-linearity to the other should not exceed 3:1. In principle, it would be possible to improve the designs by making random changes to the design parameters and selecting the best results. However, a better approach is to consider the origin of the non-linearity and take appropriate counter-measures.

7.2 Current non-linearity The origin of the current non-linearity is the variable component of airgap flux that results from the coil current, which aids or opposes the constant flux from the permanent magnet. Making the variable component of flux small in comparison with the constant component will reduce the non-linearity. The magnetic circuit concept gives a simple way of deciding which parameters to change. If the magnetic circuit has reluctance R, the maximum flux from the coil is given by:

R

mecm

IN=φ (36)

where Ne is the effective number of coil turns linking the magnetic circuit, and Im is the maximum coil current. Reducing the coil depth dc will reduce the value of Ne, and therefore reduce φcm, but this will increase the position non-linearity. Increasing the reluctance R will also reduce φcm. If the steel is unsaturated, the reluctance is given by:

0 0

0 0

g mm

g r m

g mm

g m

l dA A

l dA A

μ μ μ

μ μ

= + = +

≈ +

g mR R R

(37)

where Rg is the reluctance of the airgap, Rm is the reluctance of the permanent magnet, dmm is the depth of the permanent magnet, and Ag and Am are the corresponding cross-sectional areas. The value of μr for ceramic or neodymium magnets is close to 1. From equation 37, increasing the depth dmm of the permanent magnet will increase the reluctance. This change will also tend to increase the permanent-magnet flux φm. Increasing lg will not help because it also affects the permanent-magnet flux φm.

With some designs, however, part of the centre pole may be driven into saturation when the depth of the permanent magnet is increased. The centre pole then has significant reluctance, so equation 37 must be modified:

0 0

g mm

g m

l dA Aμ μ

= + + ≈ + +g m p pR R R R R (38)

where Rp is the reluctance of the centre pole. In this case, saturation may have a beneficial effect on the performance. Note, however, that the bottom plate must be deep enough to avoid saturation. Although saturation of this plate would increase the reluctance, it would also increase the external magnetic field. With a ceramic magnet, the area Am represents the effective area of the top plate, not the permanent-magnet material, so altering the inner and outer radii of the magnet will not change the value of Am.

7.3 Position non-linearity As the example in section 6.3 has shown, the first step towards reducing the position non-linearity is to introduce an offset in the coil position that will make the force/position characteristic symmetrical. The position non-linearity has two sources: the finite depth of the coil, and the fringing field above and below the top plate of the magnet. Increasing the coil depth dc will reduce the position non-linearity, but it will tend to increase the current non-linearity. To compensate for the increase in coil resistance when dc is changed, the wire diameter can be increased, but this may require an increase the magnet airgap length. The airgap length has only a limited effect on the fringing field, so changing this length will not usually make much difference to the position non-linearity.

Section 7 – Modifying the Designs 17

7.4 Saturation If the flux density in parts of the steel is too high, the material will be saturated magnetically. This will increase the reluctance of the magnetic circuit, and therefore reduce the airgap flux. Electromagnetic devices are normally designed to avoid saturation, but saturation can be beneficial in reducing the current non-linearity of the moving-coil driver if the size of the magnet is increased to compensate for the increased steel reluctance. See section 7.2. Small regions of high flux density (above 1.8 T) will probably have little effect. A simple way of assessing whether saturation is significant is to set an option in MagNet so that the solver treats each material as linear, with a constant permeability equal to the initial slope of the B/H characteristic. This is equivalent to ignoring saturation in all parts of the model. Proceed as follows.

1 Start MagNet and build the model as usual.

2 Copy the current solution results to a free part of the Analysis sheet.

3 Click the Material Type button and select Linear.

4 Solve the model.

5 Compare the new values with the original values. A large difference in the value of the transducer constant indicates that saturation is significant.

6 Restore the non-linear solution option by clicking the Material Type button again.

7.5 Ceramic magnet

Design for low cost Open the workbook Driver Design Initial 2x, where x is the last suffix letter used in section 4.2. Save the file with the following file name: Driver Design Low Cost 1a.xls After each design change, save the file with a different suffix: 1b, 1c, etc., so that it is always possible to go back to a previous design change. • All trial designs should have the position non-

linearity minimised by adjusting the coil position offset. If necessary, increase the depth of the centre pole.

• See whether there it is possible to reduce the dimensions of any of the components, including the depth of the centre pole, and still meet the specification.

• If you change the magnet airgap length, ensure that it is not less than the minimum value given in cell G16 of the Design Sheet.

• Build and solve the ceramic model after each design change.

When exploring design changes, you can speed up the solution by using the Solution Accuracy button on the Analysis sheet to change the adaption tolerance from 0.5% to 1%, but it must be restored to 0.5% or less for the final results. Note that the top and bottom plates do not completely cover the magnet. If the plates are extended to cover the magnet, the airgap flux will decrease, with a resulting reduction in the G value. However, the initial design does not necessarily have the optimum plate size. Although the non-linearity values are of secondary importance when considering a low-cost design, the average non-linearity should not be too large. You can use the non-linearity to compare alternative low-cost designs. Do not allow the ratio of the non-linearity values to exceed 3:1. With each successive design change, save a new version of the workbook, and record the results in your logbook. Do not spend too long on the low-cost design at this stage; the initial design is already quite a good low-cost design.


Design for high performance Start a new file series for the high performance designs as follows. Open the workbook Driver Design Initial 2x, where x is the last suffix letter used in section 4.2. Save the file with the following file name: Driver Design High Perf 1a.xls Try the strategies discussed in sections 7.2 to 7.4 for reducing the non-linearity. A different number of layers for the coil may help. To explore this option, begin by following the initial design process with a new magnet selected from the table in section 9. Ensure that the magnet airgap length is not less than the minimum value given in cell G16 of the Design Sheet. To compare different designs, use the average non-linearity value as a figure of merit. Do not allow the ratio of the non-linearity values to exceed 3:1. In principle, it is possible to keep reducing the non-linearity by increasing the magnet size. To set a limit to this process, do not allow the cost to be more than 2 times the cost of the initial design.

7.6 NdFeB magnet

Design for high performance Start a new high-performance file series as follows. Open the workbook Driver Design High Perf 1x, where x is the last suffix letter used for the high-performance ceramic design. Save the file with the following file name:

Driver Design High Perf 2a.xls

Save new versions of the file as the work progresses. The initial design should have a low current non-linearity, but further work may be needed to improve the position non-linearity. Begin by adjusting the coil position offset, if necessary increasing the depth of the centre pole. If the ratio of position non-linearity to current non-linearity is less than 3:1, this will be a satisfactory high-performance design. Do not attempt any further improvement at this stage, but go on to the low-cost design. If the ratio of position non-linearity to current non-linearity is greater than 3:1, you will need to make design changes that improve the position non-linearity. As with the ceramic magnet, use the average non-linearity value to compare different designs. Do not allow the cost to be more than 1.5 times the cost of the initial design.

Design for low cost Start a new low-cost file series as follows. Open the workbook Driver Design Low Cost 1x, where x is the last suffix letter used for the low-cost ceramic design. Save the file with the following file name:

Driver Design Low Cost 2a.xls

Save new versions of the file as the work progresses. To reduce the cost, it is necessary to reduce the volume of expensive permanent-magnet material. Try increasing the depth of the steel centre pole, which will result in a corresponding reduction in the depth of the permanent magnet. This will reduce the magnet flux, so the driver may not meet the G specification. It will be necessary to make other changes such as increasing the radius of the centre pole, with a corresponding change to the coil mean radius. The coil depth may need to be adjusted to meet the coil resistance specification. As with the ceramic design, it may be worth selecting a different number of layers for the coil. Begin by following the initial design process with a new magnet selected from the table in section 9. To compare different designs with similar costs, use the average non-linearity value as a figure of merit. Do not allow the ratio of the non-linearity values to exceed 3:1. Aim for a cost which is no more than half of the cost of the high-performance design, although it may be difficult to achieve this with some driver specifications.

Section 8 – Magnetic Shielding 19

8 MAGNETIC SHIELDING

8.1 Introduction There is a significant external magnetic field around a moving-coil driver with a simple ceramic magnet. The external field can be greatly reduced by an external steel shield together with a shield magnet, which counteracts the external field of the main magnet. Figure 4 shows the cross-section of this type of shielded magnet, together with the key dimensions of the shield components, and figure 5 shows the corresponding magnetic flux plot.

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Figure 4: Moving-coil driver: shielded magnet.

Figure 5: Shielded magnet flux plot.

The shield magnet is magnetised in the opposite direction to the main magnet. Together with the steel shield, its effect is to cancel the external field of the main magnet, and to increase the flux crossing the airgap. In this example, the shield magnet is identical to the main magnet. This reduces the manufacturing cost, but it is unlikely to give the best performance. In contrast to the unshielded design, the magnets do not project beyond the top and bottom plates. Note that the top of the shield cylinder is in line with the upper surface of the main magnet top plate.

8.2 Modifying the workbook

Workbook file Start a new low-cost file series as follows. Open the workbook Driver Design Low Cost 2x, where x is the last suffix letter used for the low-cost NdFeB design. Save the file with the following file name:

Driver Design Low Cost 3a.xls

Save new versions of the file as the work progresses.

Design sheet The Design sheet of the Excel workbook will need to be completed for the shielded magnet, in a similar way to the NdFeB magnet as described in section 3.3. Initially the main part of the shielded magnet will be identical to the ceramic magnet. Copy the magnet data, coil data and coil calculation formulae for the ceramic magnet to the corresponding columns for the shielded magnet. If this is done correctly, the numerical results should be identical. In a similar way, the ceramic magnet calculations can be copied to the shielded column. Enter test data for the shield as follows: • Make the dimensions of the shield magnet the

same as those of the main magnet. • Use a value of 10 mm for the shield cylinder

radial gap lsg. • Use a value of 3 mm for the bottom plate depth

and the cylinder wall thickness. Add formulae to calculate the mass and cost figures for the shielded magnet. Check the results of these formulae by hand calculation.

Analysis sheet It is necessary to complete the Visual Basic procedures for analysing the shielded design. The instructions below give the outline of what is required. Refer to the document An Introduction to MagNet for Static 2D Modeling for background information. The sequence described below will develop the new procedures for the shielded design by adapting the procedures for the ceramic design. Initially, this should produce exactly the same MagNet model, with the same results in the spreadsheet. When this part is working correctly, further changes can be made to analyse the shielded design.

O


Preliminary

1 Begin by saving a new version of the workbook file.

2 In the Tools menu, click Macro and select Visual Basic Editor.

3 Maximise the Visual Basic window and the code window for Module 1.

4 Scroll through the code, noting the following points. • The Option Explicit statement means that all

variables must be declared. • Constants are set at the start of the module to

specify the rows and columns used in the worksheets for data and results.

• Global variables are declared with Dim statements at the start of the module.

• The first 11 subroutines are public subroutines, linked to buttons in the Analysis sheet.

• Of these subroutines, the first five control the operation of MagNet.

• The next four call other subroutines to build and solve the ceramic and NdFeB models.

• The subroutines BuildShielded and SolveShielded are incomplete.

5 Study the BuildCeramic subroutine, and see how each component is constructed by calling the private subroutine BuildComponent.

Building the shielded model – 1

The first stage is to build a duplicate of the ceramic model.

1 Copy the contents of the BuildCeramic subroutine to the incomplete BuildShielded subroutine.

2 Change the last statement from Model = 1 to Model = 3.

3 Select Excel and test the subroutine as follows: • Start MagNet and make it visible. • Click Build Shielded. • Select MagNet, and check that the model is

the same as the ceramic model.

Solving the shielded model

1 Select Excel and click Close MagNet.

2 Select Visual Basic.

3 Copy the contents of the SolveCeramic subroutine to the incomplete SolveShielded subroutine.

4 Change the test of the Model variable to use a value of 3 instead of 1.

5 Change the call to SolveModel to put the results in the correct column by changing CeramicCol to ShieldCol.

6 Select Excel and test the subroutine as follows: • Click Start MagNet. • Click Build Shielded • Click Solve Shielded. • The model should be solved and results

displayed in column D. • Check that the results are the same as for the

ceramic model.


The next stage is to get data from the correct column in the Analysis sheet.

1 Click Close MagNet and select Visual Basic.

2 In the BuildShielded subroutine, change the call to GetCoilData to get the data from the correct column by changing CeramicCol to ShieldCol.

3 In the BuildShielded subroutine, change the statement GetCeramicData to GetShieldedData.

4 Copy the contents of the GetCeramicData subroutine to the incomplete GetShieldedData subroutine.

5 Make the following changes to the copy: • Change all the column references from

CeramicCol to ShieldCol.

6 Select Excel and test the subroutine as follows: • Click Build Shielded and then Solve

Shielded. • If the values for the main magnet in column

D of the Design sheet are the same as for the ceramic magnet in column B, the results in the Analysis sheet should be the same.

• Reduce the outer radius of the main magnet in column D of the Design Sheet to match the top and bottom plates, and check that the model is changed when you click Build Shielded in the Analysis Sheet.

Section 8 – Magnetic Shielding 21


The final stage is to get data for the shield components and add them to the model.

1 Select Excel, click Close MagNet, and select Visual Basic.

2 At the beginning of the module, add a Dim statement after the comment line to declare variables for the data and dimensions of the shield components. • See the text in the Analysis sheet for suitable

variable names.

3 In the GetShieldedData subroutine, add statements to get the data for the shield components from the Analysis sheet.

4 In the GetShieldedData subroutine, add statements to calculate the dimensions of the shield components.

5 In the BuildShielded subroutine, add new calls to BuildComponent that will build the new components. • The parameters of BuildComponent are

defined as follows: BuildComponent(Offset, Inner, Outer, Depth, Name, Material, Direction) Offset: the distance of the top of the component below the x-axis. Inner: the inner radius of the component. Outer: the outer radius of the component. Depth: the axial depth of the component. Name: the name of the component. Material: the name of the material of the component. Direction: this specifies the magnetisation direction, with one of the following values: 0 for a non-magnetic material, 1 for the positive direction, –1 for the negative direction.

• Use the variable name CeramicMat for the name of the material for the permanent magnet.

• The direction of magnetisation of the shield magnet must be the reverse of the main magnet, so the last parameter must be 1 instead of –1 for this component.

• All components must have different names.

6 Select Excel and test the subroutines.

Debugging

It is likely that there will be errors at some stage, particularly the last. A Visual Basic error may produce a dialog box similar to this:

If this happens, proceed as follows.

1 Click Debug. This should take you to the line in the Visual Basic code where the error occurred.

2 Identify the source of the error, and correct it.

3 Press F5 to continue.

If there are no Visual Basic errors, but the model is not built or solved correctly, the mistakes may be hard to find. A useful technique is to single-step through the Visual Basic code:

1 Start MagNet and make it visible.

2 Select the Visual Basic editor, and place the insertion point anywhere in the subroutine BuildShielded.

3 Press F8. This will begin single-step debugging, where the program code is executed one line at a time.

4 Press F8 repeatedly until you reach a line of code that you want to examine.

5 Inspect the value of any variable by pausing the mouse pointer over the variable name.

6 Continue single-stepping and inspecting variable values as required.

7 Press F5 to continue running the program.


8.3 Designing the shielded magnet

Design for low cost Save the modified Excel workbook with the following file name:

Driver Design Low Cost 3x.xls

where x is the next suffix letter in sequence after the workbook modifications. Save new versions of the file as the work progresses. The low-cost ceramic design has already been used for developing the shielded parts of the workbook. This is a suitable starting point for the low-cost shielded design. Experiment with the dimensions of the shield magnet, and the dimensions of the steel shield, to achieve the following:

• External magnetic field comparable with the value in the NdFeB design.

• Minimum total cost.

It may be necessary to alter the coil position offset to minimise the position non-linearity. The main magnet dimensions will also need to be changed if the value of the transducer constant is outside the permitted range.

Design for high performance It is necessary to merge two workbook files to start a new high-performance file series. Proceed as follows.

1 Open the workbook Driver Design Low Cost 3x, where x is the last suffix letter used above.

2 Save this file with a new name:

Driver Design High Perf 3a

3 Open the workbook Driver Design High Perf 2y, were y is the last suffix letter used for the high performance NdFeB design.

4 Select the Design sheet in this workbook.

5 Select all the coil and magnet data cells in this sheet: the range B16 to C32.

6 Copy the contents to the clipboard by pressing Ctrl+C.

7 In the Window menu, click Driver Design High Perf 3a.

8 Select the Design sheet, then: • Select cell B16. • Press Ctrl+V to insert the copied cells.

9 Select the Analysis sheet, then: • Start MagNet • Build and solve the ceramic design • Build and solve the NdFeB design

10 Save the modified workbook.

At the end of this sequence, the workbook named Driver Design High Perf 3a should contain the Visual Basic procedures for the shielded design, together with the data and results for (a) the high-performance ceramic and NdFeB designs, (b) the low-cost shielded design. Modify the magnet and coil dimensions, using the high-performance ceramic design as a guide, and make corresponding changes to the shield components. Work on the design to achieve the following:

• An external magnetic field comparable with the value in the NdFeB design.

• A low value for the average non-linearity, with a ratio of not more than 3:1 in the two types of non-linearity.

• A total cost that is not more than 3 times the cost for the low-cost shielded design.

8.4 Design refinement The workbooks Driver Design Low Cost 3x and Driver Design High Perf 3y should now contain the latest designs for all three magnets: ceramic, NdFeB and shielded. Compare these designs, and see whether any further design improvement can be achieved.

Section 9 – Magnet Data 23

23

9 MAGNET DATA

9.1 Ceramic magnet data Airgap flux (μWb) 612 698 776 851 923 1155 633 826 962 1071 1141 1264

Airgap length (mm) 3 3 3 3 3 3 3 3 3 3 3 3 Coil mean radius (mm) 12.5 14.5 16.5 18.5 20.5 22.5 11.5 13.5 15.5 17.5 19.5 21.5 Centre pole radius (mm) 11 13 15 17 19 21 10 12 14 16 18 20 Centre pole depth (mm) 20 20 20 20 20 20 30 30 30 30 30 30 Top plate inner radius (mm) 14 16 18 20 22 24 13 15 17 19 21 23 Top plate outer radius (mm) 41 43 45 47 49 51 40 42 44 46 48 50 Top plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Bottom plate radius (mm) 41 43 45 47 49 51 40 42 44 46 48 50 Bottom plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 8 Magnet inner radius (mm) 21 23 25 27 29 31 20 22 24 26 28 30 Magnet outer radius (mm) 45 47 49 51 53 55 44 46 48 50 52 54 Magnet depth (mm) 14 14 14 14 14 14 24 24 24 24 24 24

Airgap flux (μWb) 626 696 763 827 892 955 616 782 898 995 1086 1173

Airgap length (mm) 4 4 4 4 4 4 4 4 4 4 4 4 Coil mean radius (mm) 15 17 19 21 23 25 12 14 16 18 20 22 Centre pole radius (mm) 13 15 17 19 21 23 10 12 14 16 18 20 Centre pole depth (mm) 20 20 20 20 20 20 30 30 30 30 30 30 Top plate inner radius (mm) 17 19 21 23 25 27 14 16 18 20 22 24 Top plate outer radius (mm) 43 45 47 49 51 53 40 42 44 46 48 50 Top plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Bottom plate radius (mm) 43 45 47 49 51 53 40 42 44 46 48 50 Bottom plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Magnet inner radius (mm) 23 25 27 29 31 33 20 22 24 26 28 30 Magnet outer radius (mm) 47 49 51 53 55 57 44 46 48 50 52 54 Magnet depth (mm) 14 14 14 14 14 14 24 24 24 24 24 24

Airgap flux (μWb) 602 688 764 837 908 976 627 789 907 1008 1102 1186

Airgap length (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Coil mean radius (mm) 14 16 18 20 22 24 13 15 17 19 21 23 Centre pole radius (mm) 11 13 15 17 19 21 10 12 14 16 18 20 Centre pole depth (mm) 25 25 25 25 25 25 35 35 35 35 35 35 Top plate inner radius (mm) 17 19 21 23 25 27 16 18 20 22 24 26 Top plate outer radius (mm) 41 43 45 47 49 51 40 42 44 46 48 50 Top plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Bottom plate radius (mm) 41 43 45 47 49 51 40 42 44 46 48 50 Bottom plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 8 Magnet inner radius (mm) 21 23 25 27 29 31 20 22 24 26 28 30 Magnet outer radius (mm) 45 47 49 51 53 55 44 46 48 50 52 54 Magnet depth (mm) 19 19 19 19 19 19 29 29 29 29 29 29


9.2 NdFeB magnet data Airgap flux (μWb) 590 670 753 840 930 1024 648 831 931 1035 1145 1258

Airgap length (mm) 3 3 3 3 3 3 3 3 3 3 3 3 Coil mean radius (mm) 15.5 16.5 17.5 18.5 19.5 20.5 15.5 17.5 18.5 19.5 20.5 21.5 Centre pole radius (mm) 14 15 16 17 18 19 14 16 17 18 19 20 Total centre pole depth (mm) 20 20 20 20 20 20 30 30 30 30 30 30 Steel centre pole depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Top plate inner radius (mm) 17 18 19 20 21 22 17 19 20 21 22 23 Top plate outer radius (mm) 31 32 33 34 35 36 31 33 34 35 36 37 Top plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Bottom plate radius (mm) 31 32 33 34 35 36 31 33 34 35 36 37 Bottom plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Magnet radius (mm) 14 15 16 17 18 19 14 16 17 18 19 20 Magnet depth (mm) 14 14 14 14 14 14 24 24 24 24 24 24 Cylinder inner radius (mm) 27 28 29 30 31 32 27 29 30 31 32 33 Cylinder outer radius (mm) 31 32 33 34 35 36 31 33 34 35 36 37 Cylinder depth (mm) 14 14 14 14 14 14 24 24 24 24 24 24

Airgap flux (μWb) 571 646 724 805 890 976 635 785 913 1011 1118 1225

Airgap length (mm) 4 4 4 4 4 4 4 4 4 4 4 4 Coil mean radius (mm) 16 17 18 19 20 21 16 18 19 20 21 22 Centre pole radius (mm) 14 15 16 17 18 19 14 16 17 18 19 20 Total centre pole depth (mm) 20 20 20 20 20 20 30 30 30 30 30 30 Steel centre pole depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Top plate inner radius (mm) 20 19 20 21 22 23 18 20 21 22 23 24 Top plate outer radius (mm) 31 32 33 34 35 36 31 33 34 35 36 37 Top plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Bottom plate radius (mm) 31 32 33 34 35 36 31 33 34 35 36 37 Bottom plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 8 Magnet radius (mm) 14 15 16 17 18 19 14 16 17 18 19 20 Magnet depth (mm) 14 14 14 14 14 14 24 24 24 24 24 24 Cylinder inner radius (mm) 27 28 29 30 31 32 27 29 30 31 32 33 Cylinder outer radius (mm) 31 32 33 34 35 36 31 33 34 35 36 37 Cylinder depth (mm) 14 14 14 14 14 14 24 24 24 24 24 24

Airgap flux (μWb) 603 682 765 852 942 1033 649 741 934 994 1140 1256

Airgap length (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Coil mean radius (mm) 17 18 19 20 21 22 17 18 20 21 22 23 Centre pole radius (mm) 14 15 16 17 18 19 14 15 17 18 19 20 Total centre pole depth (mm) 25 25 25 25 25 25 35 35 35 35 35 35 Steel centre pole depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Top plate inner radius (mm) 20 21 22 23 24 25 20 21 23 24 25 26 Top plate outer radius (mm) 31 32 33 34 35 36 31 32 34 35 36 37 Top plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Bottom plate radius (mm) 31 32 33 34 35 36 31 32 34 35 36 37 Bottom plate depth (mm) 6 6 6 6 6 6 6 6 6 6 6 6 Magnet radius (mm) 14 15 16 17 18 19 14 15 17 18 19 20 Magnet depth (mm) 19 19 19 19 19 19 29 29 29 29 29 29 Cylinder inner radius (mm) 27 28 29 30 31 32 27 28 30 31 32 33 Cylinder outer radius (mm) 31 32 33 34 35 36 31 32 34 35 36 37 Cylinder depth (mm) 19 19 19 19 19 19 29 29 29 29 29 29

J D Edwards Edp0506GuideExport_LtrSize.doc 10 March 2006 Revision 1

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Year2b Design Moving-coil Driver