Mathematical Model of an Eddy Current Brake

Table of Contents

Definitions Parameteres Equations Calculations Graphs

Based on: Wouterse, J.h. “Critical Torque and Speed of Eddy Current Brake with Widely Separated Soft Iron Poles.” IEE Proceedings B Electric Power Applications, vol. 138, no. 4, 1991, p. 153., doi:10.1049/ip-b.1991.0019.

Definitions

ρ, The specific resistance of the disk material/n
d, (m) The thickness of the disk
D, (m) The diameter of the Iron pole or a the Diameter of a circle of equal area if the pole used is not circular
χ, (m) The air gap between pole faces including disc thickness
ξ, The ratio of zone width, in asymptotic current distribution around poles, to air gap
c, The ratio of total contour resistance to resistance of contour part under pole; c = 0.5 if disc has infinite diameter
v, (m/s) Tangential speed, measured at centre of pole
, (m/s) The critical speed, i.e., speed at which exerted force is maximum
, (N) The force of electromagnetic origin exerted by the polea on the disk
, (N) The maximum value of the exerted force as function of v
, () The magnetic field when v = 0
R, (m) The distance from the centre of disc to centre of the magnet
A, (m) The radius of the disk
, (Nm) The torque of electro magnetic origin

Parameteres

%Constants
mu_0 = 1.256 * 10^-6
mu_0 = 1.2560e-06
xi = 1 % We cannot compute a value for the proportionality constant however it is close to unity
xi = 1
% Testing rig paramaters
rho = 1.782 * 10^-8
rho = 1.7820e-08
d = 4*10^-3
d = 0.0040
D = 0.001256
D = 0.0013
chi = 5 * 10^-3
chi = 0.0050
B_0 = 0.2344
B_0 = 0.2344
R = 0.025 + 0.075 / 2
R = 0.0625
A = 0.1
A = 0.1000

Equations

For lower speeds the

caused by the eddies created by the moving disk passing throught the magnetic field generated by the pole magnets follows the rule

. where,

However for higher velocities of the disk the induction of the magnetic field begins to lose effect as the eddies it generates in the disk reach the same Current. Thefore to model the falloff of

as v increases the systems can be represented by this rule.

assuming,

and,

Calculations

rpm = 0:1:180;
rpm = 1×181
     0     1     2     3     4     5     6     7     8     9    10    11    12    13    14    15    16    17    18    19    20    21    22    23    24    25    26    27    28    29    30    31    32    33    34    35    36    37    38    39    40    41    42    43    44    45    46    47    48    49
v = rpm * 2 * 3.145 * R;
v = 1×181
         0    0.3931    0.7863    1.1794    1.5725    1.9656    2.3588    2.7519    3.1450    3.5381    3.9312    4.3244    4.7175    5.1106    5.5038    5.8969    6.2900    6.6831    7.0762    7.4694    7.8625    8.2556    8.6487    9.0419    9.4350    9.8281   10.2212   10.6144   11.0075   11.4006   11.7937   12.1869   12.5800   12.9731   13.3663   13.7594   14.1525   14.5456   14.9388   15.3319   15.7250   16.1181   16.5113   16.9044   17.2975   17.6906   18.0837   18.4769   18.8700   19.2631
c = 1/2 * (1-1/4 * 1 / ((1+R/A)^2 * ((A-R)/D)^2));
c = 0.4999
F_e_slow = 1/4 * pi/rho * D^2 * d * B_0^2 * c * v;
F_e_slow = 1×181
         0    0.0030    0.0060    0.0090    0.0120    0.0150    0.0180    0.0210    0.0240    0.0270    0.0300    0.0330    0.0360    0.0390    0.0420    0.0450    0.0481    0.0511    0.0541    0.0571    0.0601    0.0631    0.0661    0.0691    0.0721    0.0751    0.0781    0.0811    0.0841    0.0871    0.0901    0.0931    0.0961    0.0991    0.1021    0.1051    0.1081    0.1111    0.1141    0.1171    0.1201    0.1231    0.1261    0.1291    0.1321    0.1351    0.1381    0.1412    0.1442    0.1472
v_k = 2 / mu_0 * (1/(c*xi))^1/2 * rho/d * (chi/D)^1/2;
v_k = 14.1216
F_e_hat = 1 / mu_0 * (c/xi)^1/2 * D^2 * B_0^2 * (chi/D)^1/2;
F_e_hat = 0.0343
F_e_fast = F_e_hat * (2./ (v_k./ v + v * v_k^-1));
F_e_fast = 1×181
         0    0.0019    0.0038    0.0057    0.0076    0.0094    0.0112    0.0129    0.0146    0.0162    0.0177    0.0192    0.0206    0.0220    0.0232    0.0244    0.0255    0.0266    0.0275    0.0284    0.0292    0.0299    0.0306    0.0312    0.0317    0.0322    0.0326    0.0330    0.0333    0.0336    0.0338    0.0340    0.0341    0.0342    0.0343    0.0343    0.0343    0.0343    0.0343    0.0342    0.0341    0.0340    0.0339    0.0338    0.0336    0.0335    0.0333    0.0331    0.0329    0.0327
Tau = F_e_fast * R;
Tau = 1×181
         0    0.0001    0.0002    0.0004    0.0005    0.0006    0.0007    0.0008    0.0009    0.0010    0.0011    0.0012    0.0013    0.0014    0.0015    0.0015    0.0016    0.0017    0.0017    0.0018    0.0018    0.0019    0.0019    0.0019    0.0020    0.0020    0.0020    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0021    0.0020

Graphs

plot(rpm, F_e_slow)

hold on

plot(rpm, F_e_fast)

hold off

xlabel('RPM');

ylabel('Torque');

title('Torque per RPM of a Eddy Break');

xlim([0 180]);

ylim([0 0.5]);

legend('Initial Model', 'Updated Model')