This document contains the workbook of the DrillBits Team. It outlines various analyses they are conducting on a motor, including quantifying losses in the armature and field laminations due to eddy currents under different load points and material properties like resistivity. Code is provided to model the magnetic field produced by coils, calculate frequencies of field direction changes, and plot power lost versus resistivity for different materials and gauges. Sources are cited.

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The DrillBits Team Workbook
Author: Brian Avadikian
Date : 12/8/2014
Contents
Finding the Frequency of the Armature Lamination Magnetic Field Direction Change
Quantification of the Magnetic Field Produced by the Two Coils [2]
Quantifying the Armature Eddy Current Losses [1] [4] [5]
Quantifying the Field Eddy Current Losses [1] [4] [5]
Visualizing the loss vs restivity at different load points
Lamination Material Profiles [3]
Comparison of Core Loss Across Gauge and Grade
Sources Cited
Matlab housekeeping
Finding the Frequency of the Armature Lamination Magnetic Field Direction Change
clc
clear all
RPMtry=30000; %This is a loadpoint chosen to determine the frequency of the magnetic field switch
SecondsPerMinute=60;
I need to find the number of times that the magnetic field switches in one full rotation of the motor In one full rotation there are two pole swithes the magnetic field switches every time a new set of
tangs comes into contact with the brushes The number of Lamination Arms is also happens to be the number of SETs of tangs
LamArm=12;
The number of Tangs
Tangs= 2*LamArm;
The magnetic field switches direction 2 times for each and every Lamination Arms(Set of Tangs)for every full cycle (360 deg armature rotation)
frequency=RPMtry*2*LamArm/SecondsPerMinute; %This will give us cycles per one second Hz
RadFreq=frequency*pi/180; %This will be in radians/s
This is a range of restvities for many differeny materials with values within 1 order of magnitude of values for common electrical steel SOURCE: [6]

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resistivity=linspace(0,9.8E-7,400);
% This is the length of the field and is used to determine the number of
% laminations (assuming that during the material change, the overall length
% of the field stack does not change)
OverallFieldLength=30.29/1000; %(mm------->m)
%This is the length of the field and is used to determine the number of
%armature laminations(assuming that during the material change, the overall length
%of the armature stack does not change)
OverallArmLength=29.75/1000; %(mm--------->m)
Quantification of the Magnetic Field Produced by the Two Coils [2]
The properties of the field are as follows:
t=linspace(0,100E-3,100); % (milliseconds) This is a time span 0 to 100 for plotting
Vfield=120; % (Volts) This is the voltage applied to the field
Rfield=.9; % (Ohms) This is the resistance of the field windings
Nfield=93; % (Turns) This is the number of turns per field.
Ifield = Vfield/Rfield; % (Amps) This is the current through the field
Laverage=(38.5*2+36.025*2)/1000; % (mm----->m) this is the average length of the path that the current travels
% This is the equation for the magnetic field strength based on the
% parameters we have. Source: [2]
Hfield=.4*pi*Nfield*Ifield/Laverage; %Oersteds value based on the parameters above
% 10,000 Oersted = 1 Tesla
% Source:
Bmaxfield=Hfield/10000;
milliteslas=1000*Bmaxfield;
B=2*Bmaxfield*sin(RadFreq*t); % Assuming BMax=2*Bmaxfield because we have two loops
This next section plots the figure that names the cross-sectional dimensions that are used for the eddy current loss relationship
figure;
axis off
ax(1) = subplot(1,1,1);
rgb = imread('Figure3_4_1.PNG');
image(rgb)
title('Lamination Crossection')
% This plots the figure that shows the fluctuation frequency through the
% armature as a function of time and rpm
figure;
axis on

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axis([0 1.4E-6 0 600],'manual')
plot(B)
xlabel('Time (ms)');
ylabel('Magnetic Field Strength (mT)');
title('Change in Magnetic Field Intensity through Lamination ');
Quantifying the Armature Eddy Current Losses [1] [4] [5]
% ArmSection1=[t (Thickness), w(width), h(Height)]
ArmSection1=[.63;6.3875;1.85]./1000;
ArmSection1_Volume=ArmSection1(1)*ArmSection1(2)*ArmSection1(3);
ArmEddyLoss =(ArmSection1_Volume*(pi^2)*frequency*ArmSection1(1)^2*(2*Bmaxfield)^2)./(6*resistivity); % ()
TotalArmCoreLoss = ArmEddyLoss.*12.*48;
The losses in the main crossection of the armature lamination are determined and multiply it by 12 to get the power lost per full lamination. It is then then multplied it by the number of laminations
TotalArmCoreLoss = ArmEddyLoss.*12.*48;
Quantifying the Field Eddy Current Losses [1] [4] [5]

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% FieldSection1=[t (Thickness), **w(width), h(Height)]
FieldSection1=[.63;55.68;4.655]./1000;
FieldVolume=FieldSection1(1)*FieldSection1(2)*FieldSection1(3);
PowerLostFieldEddy =(FieldVolume*(pi^2)*60*FieldSection1(1)^2*(2*Bmaxfield)^2)./(6*resistivity); % ()
TotalFieldCoreLoss=2.*48.*PowerLostFieldEddy;
This next section plots the sum of the field and armature loss against resistivity in order to establish how resistivity effects core loss
figure;
plot(resistivity,(TotalArmCoreLoss+TotalFieldCoreLoss)) %This is a plot of the resistivity versus the total power lost to the eddy current
axis([0 1.4E-7 0 600],'manual')
xlabel('Restivity (Ohm*m)');
ylabel('Power Lost in Eddy (W)');
title('Power Lost in Eddy as a function of Restivity');
Visualizing the loss vs restivity at different load points
The load points are sourced from the xls. document motor options
RPM=[30337;29538;28246;27220;26131;25011;23992;22841;21776;20789;19692;18666;17569;16473;15408;14374;13316;12227;11146;10152];

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The following code displays the xls table that the rpm load points are sourced from
OPTIONS=imread('MotorOptions.PNG');
figure
imshow(OPTIONS)
Here we see that the RPM is now a range of 20 values (Load Points)
frequency=RPM.*2.*LamArm./SecondsPerMinute; %This will give us cycles per one second Hz
%Below I'm just picking a few values from the 20 load points, the lowest
%middle and highest RPM values respectively
The powerloss vs resistivity is then plotted at the three different RPM loadpoints to visualize how RPM effects power lost in the core

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LoadPoints=[frequency(1);frequency(10);frequency(20)]; %Choosing 3 load points
hold off
for Ticker=1:3 %This "Ticker" Runs the code through the three load point choices between 0 and 9.8E-7 [6]
for Ticker2=1:400 %This "Ticker" sweeps each loadpoint choice above through 400 restivity
%Loss Values for the armature are below, and are based on source [2]
ArmEddyLoss =(ArmSection1_Volume*(pi^2)*(LoadPoints(Ticker))*ArmSection1(1)^2*(2*Bmaxfield)^2)./(6*resistivity(Ticker2)); % ()
TotalArmCoreLoss = ArmEddyLoss.*12.*48; % there are 12 cross sections per lamination and 48 laminations
%Loss values for the field are below, and are based on source [2]
PowerLostFieldEddy =(FieldVolume*(pi^2)*60*FieldSection1(1)^2*(2*Bmaxfield)^2)./(6*resistivity(Ticker2)); % ()
TotalFieldCoreLoss=2.*48.*PowerLostFieldEddy; % there are 2 cross sections per lamination and 48 laminations
plot(resistivity(Ticker2),(TotalFieldCoreLoss+TotalArmCoreLoss)) %This plots the resistivity versus the power lost to the eddy current
hold on %hold on ensures that each for-loop plots over the for-loop before it (on the same graph)
axis([0 5E-7 0 600])
xlabel('Restivity (Ohm*m)');
ylabel('Power Lost in Eddy (W)');
title('Power Lost in Eddy as a function of Restivity');
pause(.00000000001)
end
end

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Lamination Material Profiles [3]
In this section, matricies that hold the material properties of each of the materials are built up so that the different properties can be easilly read out by matlab.
Columns DI-MAX M-15 FP, DI-MAX M-36 FP, DI-MAX M-47 FP Black Yellow Red
Rows # Restivity (Ohms*m) # Nominal Thickness # Hardness (rockwell b hardness) # Normalized Cost (a feature we didn't use matlab to explore)
Source: [3] The guages below are comprised of vertical matricies that hold the material properties for each of the three grades. The first and second entry from the top of each grade is what we'll use to determine power
loss (resistivity and nominal thickness)
Gauge_29=[[ 5.0E-7; .3556; 72; 1], [4.3E-7; .3556; 64; 1],[ 0;0; 64; 1]]; % Triangle 29 guage Ak Electrical Steel
Gauge_26=[[5.0E-7; .4699; 72; 1],[ 4.3E-7; .4699; 64; 1], [3.7E-7;.4699; 61; 1]] ; % Squares 26 guage Ak Electrical Steel
Gauge_24=[[ 5.0E-7; .6355; 72; 1], [4.3E-7; .6355; 64; 1],[ 3.7E-7;.6355; 64; 1]]; % Circles 24 guage Ak Electrical Steel
DeWalt=[5.7E-7;.63;64;1]; %the current design
% An all encompassing set of these three guages and grades is created so that matlab can
% access them faster. Over-all it is 9 different steels

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ComparisonMatrix=[Gauge_24,Gauge_26,Gauge_29,DeWalt]
ComparisonMatrix =
Columns 1 through 7
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.6355 0.6355 0.6355 0.4699 0.4699 0.4699 0.3556
72.0000 64.0000 64.0000 72.0000 64.0000 61.0000 72.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Columns 8 through 10
0.0000 0 0.0000
0.3556 0 0.6300
64.0000 64.0000 64.0000
1.0000 1.0000 1.0000
%During plotting, it will be helpful to have different colors and styles to
%differentiate between the grades and gauges
S={'ok', 'oy', 'or', 'xk', 'xy', 'xr', 'vk', 'vy', 'vr','og-'}; %this will hold the information accessed by the for loops to plot guages as symbols and grades as colors.
Comparison of Core Loss Across Gauge and Grade
He we set up routines that take the material profiles established above and run them through the governing equations also established above, and compares them across a range of operationl points (RPM). The
Operational points are taken from the motor comparison spreadsheet provided in elms.
The entire spectrum of operational RPM will be displayed.
%The goal is to plot Powerloss vs RPM for each grade on the same graph for
%each different Gague and Grade
hold off %This hold off makes sure that there's no leftover plot data on this new plot about to be produced
RANGEofLoadPoints=linspace(min(RPM),max(RPM),100);
RANGEofFREQUENCY=RANGEofLoadPoints.*2.*LamArm./SecondsPerMinute;
figure;
for TICKER2=1:10 % This is a "Ticker" that sweeps through the grades of the steel
SETTER1=ComparisonMatrix(:,TICKER2); %This is a "Setter" that reads in a grade of steel to be plotted
% This is where it gets cool! The setter allows the code to sweep through the 9 columns of the comparison
% matrix.
% The first 3 are gauge 24
% then next three are gauge 26

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% and the last three are gauge 29
for TICKER1=1:100; %This "Ticker" sweeps the chosen grade through the load-point and magnetic-field-switch-frequency values
% This is the loss in the field
% t (Thickness) **w(width) h(Height)
FieldSection1=[SETTER1(2); 55.68; 4.655]./1000;
FieldVolume=FieldSection1(1)*FieldSection1(2)*FieldSection1(3);
PowerLostFieldEddy =(FieldVolume*(pi^2)*60*FieldSection1(1)^2*(2*Bmaxfield)^2)./(6*SETTER1(1)); % ()
TotalFieldCoreLoss=2.*round(OverallFieldLength/FieldSection1(1)).*PowerLostFieldEddy;
% t (Thickness) w(width) h(Height)
ArmSection1=[SETTER1(2);6.3875;1.85]./1000;
ArmSection1_Volume=ArmSection1(1)*ArmSection1(2)*ArmSection1(3);
ArmEddyLoss =(ArmSection1_Volume*(pi^2)*RANGEofFREQUENCY(TICKER1)*ArmSection1(1)^2*(2*Bmaxfield)^2)./(6*SETTER1(1)); % ()
TotalArmCoreLoss = ArmEddyLoss.*12.*round(OverallArmLength/ArmSection1(1));
hold on
plot(RANGEofLoadPoints(TICKER1),(TotalFieldCoreLoss+TotalArmCoreLoss), S{TICKER2})
xlabel('Motor Speed (RPM)');
ylabel('Power Lost in Eddy (W)');
title('Power Lost vs Motor RPM');
pause(.000000001)
end
end

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Sources Cited
1. http://www.vias.org/matsch_capmag/matsch_caps_magnetics_chap3_27.html
2. http://socratic.org/questions/what-is-magnet-fields
3. http://www.aksteel.com/pdf/markets_products/electrical/Non_Oriented_Bulletin.pdf
4. https://www.navalengineers.org/SiteCollectionDocuments/2008%20Proceedings%20Documents/EMTS%202008/Graf%20Paper.pdf
5. F. Fiorillo, Measurement and characterization of magnetic materials, Elsevier Academic Press, 2004, ISBN 0-12-257251-3, page. 31
6. https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
Matlab housekeeping
%To close all the unwanted figures that happen when I publish
%Find all windows of type figure, which have an empty FileName attribute.
allPlots = findall(0, 'Type', 'figure', 'FileName', []);
% Close.
delete(allPlots);
Published with MATLAB® R2014a