Contents

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% Power Transfer Efficiency Model
% 1/31/16
% Model references at bottom
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clear all
load('NTable.mat', 'N');

define constants

Uo = 4*pi*10^-7; % Uo = magnetic permeability of free space
Eo = 8.85*10^-12; % permittivity of free space
rho = 1.68 * 10^-8; % rho = resistivity of copper
sigma = 58.5 * 10^6; % conductivity of copper

Loop through wire gauges

i = 0;
for w1 = [.0255 .0286 .0321 .0361 .0405 .0455 .0511 .0573 .0644 .0723 .0644 .0723 .0812 .0912 .1024];
i = i + 1;
j = 0;
for w2 = [.0255 .0286 .0321 .0361 .0405 .0455 .0511 .0573 .0644 .0723 .0644 .0723 .0812 .0912 .1024];

j = j + 1;

define paramters (all lengths, widths, etc. in centimeters)

D = 0.5; % distance between coils --> 0.5 - 1 cm
n = [15 6]; % number of turns
s = [0 0]; % wire spacing
w = [w1 w2]; % wire thickness
DIn = [0.7 2.2]; % inner diameter
DOut = DIn + 2.*((n).*w + n.*s); % outer diameter
t = w+s; % layer thickness in cm wire + spacing
f = 3*10^6;
omega = 2*pi*f;
a = (DOut+DIn)/4; % Average radius in cm
b = n.*t - s; % thickness of total turns in cm
c = w; % thickness of total layers in cm
l = 2*pi*a.*n; % length of conductor in cm
skindepth = 100/sqrt(pi*f*sigma*Uo); %skin depth in cm
Ac = pi*(w/2).^2; % cross sectional area of conductor in coil
DAvg = (DOut + DIn)./2; % Davg = mean diameter
Area_eff = w*skindepth*pi; % effective area of conductor
RLoadAC = 1000; % hypothetical load on receiver

define inductance of multi-layered coil

Kim, J., Basham, E., Pedrotti, K. Geometry-based optimization of radio-frequency coils for powering neuroprosthetic implants International Federation for Medical and Biological Engineering October 21, 2012

Lself = 10^-6*(a.^2.*n.^2)./(2.54*(9*a+10*b)); % Inductance in henries

Effective Series Resistance

R_DC = rho*l/Ac;
R_Skin = w*(20.8*pi/rho)*sqrt(f)*10^-9;

zeta = sqrt(pi)*w/(2*skindepth);
R_Proximity = zeta .* ((sinh(2*zeta) + sin(2*zeta)) ./(cosh(2*zeta) - cos(2*zeta)) + ...
    (2/3)*(n.^2-1).*((sinh(2*zeta) - sin(2*zeta)) ./(cosh(2*zeta) + cos(2*zeta))));

R_ESR = R_DC * (1 + R_Skin + R_Proximity);

Quality factor

Q = omega * Lself./R_ESR;

Coupling coefficient of pair of coils

r2r1_ratio = sqrt((a(1)-a(2))^2 + D^2)/sqrt((a(1)+a(2))^2 + D^2); % radius and distance ratio
Nval = NLookup(r2r1_ratio, N);
k = 2.54*Nval*sqrt(((6*a(1)+9*b(1)+10*c(1))*(6*a(2)+9*b(2)+10*c(2)))/(0.64*a(1)*a(2)));
M = k * sqrt(Lself(1) * Lself(2));

Power Transfer Efficiency

PTE(i,j) = 100 *(k^2 * Q(1)*Q(2)^3*RLoadAC*R_ESR(2))/((k^2*Q(1)*Q(2)*RLoadAC + (1+100/R_ESR(1)) * ...
    (RLoadAC+Q(2)^2*R_ESR(2)))*(RLoadAC+Q(2)^2*R_ESR(2)));

wire(j,1) = w2;
wire(j,2) = w2;

end
end

subplot(2,2,1)
surf(wire(:,1),wire(:,2),PTE);
xlabel('Secondary wire thickness')
ylabel('Primary wire thickness')
zlabel('Power Transfer Efficiency %')
zlim([0 100])
title('PTE vs W')

Loop through coil turns

i = 0;
for n1 = [1:16];
i = i + 1;
j = 0;
for n2 = [1:.5:8.5];

j = j + 1;
s1 = (3.911 - n1*2*.0255 - 0.8) / n1;
s2 = (3.034 - n2*2*.1024 - 1.6) / n2;

define paramters (all lengths, widths, etc. in centimeters)

D = .5; % distance between coils --> 0.5 - 1 cm
n = [n1 n2]; % number of turns
s = [s1 s2]; % wire spacing
w = [.0255 .1024]; % wire thickness
DIn = [0.7 2.2]; % inner diameter
DOut = DIn + 2.*((n).*w + n.*s); % outer diameter
t = w+s; % layer thickness in cm wire + spacing
f = 3*10^6;
omega = 2*pi*f;
a = (DOut+DIn)/4; % Average radius in cm
b = n.*t - s; % thickness of total turns in cm
c = w; % thickness of total layers in cm
l = 2*pi*a.*n; % length of conductor in cm
skindepth = 100/sqrt(pi*f*sigma*Uo); %skin depth in cm
Ac = pi*(w/2).^2; % cross sectional area of conductor in coil
DAvg = (DOut + DIn)./2; % Davg = mean diameter
Area_eff = w*skindepth*pi; % effective area of conductor
RLoadAC = 1000; % hypothetical load on receiver

define inductance of multi-layered coil

Kim, J., Basham, E., Pedrotti, K. Geometry-based optimization of radio-frequency coils for powering neuroprosthetic implants International Federation for Medical and Biological Engineering October 21, 2012

Lself = 10^-6*(a.^2.*n.^2)./(2.54*(9*a+10*b)); % Inductance in henries

Effective Series Resistance

R_DC = rho*l/Ac;
R_Skin = w*(20.8*pi/rho)*sqrt(f)*10^-9;

zeta = sqrt(pi)*w/(2*skindepth);
R_Proximity = zeta .* ((sinh(2*zeta) + sin(2*zeta)) ./(cosh(2*zeta) - cos(2*zeta)) + ...
    (2/3)*(n.^2-1).*((sinh(2*zeta) - sin(2*zeta)) ./(cosh(2*zeta) + cos(2*zeta))));

R_ESR = R_DC * (1 + R_Skin + R_Proximity);

Quality factor

Q = omega * Lself./R_ESR;

Coupling coefficient of pair of coils

r2r1_ratio = sqrt((a(1)-a(2))^2 + D^2)/sqrt((a(1)+a(2))^2 + D^2); % radius and distance ratio
Nval = NLookup(r2r1_ratio, N);
k = 2.54*Nval*sqrt(((6*a(1)+9*b(1)+10*c(1))*(6*a(2)+9*b(2)+10*c(2)))/(0.64*a(1)*a(2)));
M = k * sqrt(Lself(1) * Lself(2));

Power Transfer Efficiency

PTE(i,j) = 100 *(k^2 * Q(1)*Q(2)^3*RLoadAC*R_ESR(2))/((k^2*Q(1)*Q(2)*RLoadAC + (1+100/R_ESR(1)) * ...
    (RLoadAC+Q(2)^2*R_ESR(2)))*(RLoadAC+Q(2)^2*R_ESR(2)));

turns(j,2) = n2;

end
turns(i,1) = n1;
end
subplot(2,2,2)
surf(turns(:,1),turns(:,2),PTE);
xlabel('Secondary # of turns')
ylabel('Primary # of turns')
zlabel('Power Transfer Efficiency %')
ylim([0,10])
xlim([0,20])
zlim([0,100])
title('PTE vs N')

Loop through inner diameter

i = 0;
for d1 = [.1:.1:1.2];
i = i + 1;
j = 0;
for d2 = [.2:.2:2.4];

j = j + 1;
s1 = (4 - 59*2*.0255 - d1) / 59;
s2 = (3 - 5*2*.1024 - d2) / 5;

define paramters (all lengths, widths, etc. in centimeters)

D = .5; % distance between coils --> 0.5 - 1 cm
n = [15 6]; % number of turns
s = [s1 s2]; % wire spacing
w = [.1024 .255]; % wire thickness
DIn = [d1 d2]; % inner diameter
DOut = DIn + 2.*((n).*w + n.*s); % outer diameter
t = w+s; % layer thickness in cm wire + spacing
f = 3*10^6;
omega = 2*pi*f;
a = (DOut+DIn)/4; % Average radius in cm
b = n.*t - s; % thickness of total turns in cm
c = w; % thickness of total layers in cm
l = 2*pi*a.*n; % length of conductor in cm
skindepth = 100/sqrt(pi*f*sigma*Uo); %skin depth in cm
Ac = pi*(w/2).^2; % cross sectional area of conductor in coil
DAvg = (DOut + DIn)./2; % Davg = mean diameter
Area_eff = w*skindepth*pi; % effective area of conductor
RLoadAC = 1000; % hypothetical load on receiver

define inductance of multi-layered coil

Kim, J., Basham, E., Pedrotti, K. Geometry-based optimization of radio-frequency coils for powering neuroprosthetic implants International Federation for Medical and Biological Engineering October 21, 2012

Lself = 10^-6*(a.^2.*n.^2)./(2.54*(9*a+10*b)); % Inductance in henries

Effective Series Resistance

R_DC = rho*l/Ac;
R_Skin = w*(20.8*pi/rho)*sqrt(f)*10^-9;

zeta = sqrt(pi)*w/(2*skindepth);
R_Proximity = zeta .* ((sinh(2*zeta) + sin(2*zeta)) ./(cosh(2*zeta) - cos(2*zeta)) + ...
    (2/3)*(n.^2-1).*((sinh(2*zeta) - sin(2*zeta)) ./(cosh(2*zeta) + cos(2*zeta))));

R_ESR = R_DC * (1 + R_Skin + R_Proximity);

Quality factor

Q = omega * Lself./R_ESR;

Coupling coefficient of pair of coils

r2r1_ratio = sqrt((a(1)-a(2))^2 + D^2)/sqrt((a(1)+a(2))^2 + D^2); % radius and distance ratio
Nval = NLookup(r2r1_ratio, N);
k = 2.54*Nval*sqrt(((6*a(1)+9*b(1)+10*c(1))*(6*a(2)+9*b(2)+10*c(2)))/(0.64*a(1)*a(2)));
M = k * sqrt(Lself(1) * Lself(2));

Power Transfer Efficiency

PTE(i,j) = 100 *(k^2 * Q(1)*Q(2)^3*RLoadAC*R_ESR(2))/((k^2*Q(1)*Q(2)*RLoadAC + (1+100/R_ESR(1)) * ...
    (RLoadAC+Q(2)^2*R_ESR(2)))*(RLoadAC+Q(2)^2*R_ESR(2)));

inner(j,2) = d2;

end
inner(i,1) = d1;
end
subplot(2,2,3)
surf(inner(:,1),inner(:,2),PTE);
xlabel('Primary Inner Diameter, centimeters')
ylabel('Secondary Inner Diameter, centimeters')
zlabel('Power Transfer Efficiency %')
ylim([0,2.5])
xlim([0,1.3])
zlim([50 100])
title('PTE vs DIn')

Loop through distances

for D = [0:.25:5];

i = i + 1;

w = [0.1024 0.0255];
n = [15 6];
DIn = [0.7 2.2];
s = [0 0];

define paramters (all lengths, widths, etc. in centimeters)

DOut = DIn + 2.*((n).*w + n.*s); % outer diameter
t = w+s; % layer thickness in cm wire + spacing
f = 3*10^6;
omega = 2*pi*f;
a = (DOut+DIn)/4; % Average radius in cm
b = n.*t - s; % thickness of total turns in cm
c = w; % thickness of total layers in cm
l = 2*pi*a.*n; % length of conductor in cm
skindepth = 100/sqrt(pi*f*sigma*Uo); %skin depth in cm
Ac = pi*(w/2).^2; % cross sectional area of conductor in coil
DAvg = (DOut + DIn)./2; % Davg = mean diameter
Area_eff = w*skindepth*pi; % effective area of conductor
RLoadAC = 1000; % hypothetical load on receiver

define inductance of multi-layered coil

Kim, J., Basham, E., Pedrotti, K. Geometry-based optimization of radio-frequency coils for powering neuroprosthetic implants International Federation for Medical and Biological Engineering October 21, 2012

Lself = 10^-6*(a.^2.*n.^2)./(2.54*(9*a+10*b)); % Inductance in henries

Effective Series Resistance

R_DC = rho*l/Ac;
R_Skin = w*(20.8*pi/rho)*sqrt(f)*10^-9;

zeta = sqrt(pi)*w/(2*skindepth);
R_Proximity = zeta .* ((sinh(2*zeta) + sin(2*zeta)) ./(cosh(2*zeta) - cos(2*zeta)) + ...
    (2/3)*(n.^2-1).*((sinh(2*zeta) - sin(2*zeta)) ./(cosh(2*zeta) + cos(2*zeta))));

R_ESR = R_DC * (1 + R_Skin + R_Proximity);

Quality factor

Q = omega * Lself./R_ESR;

Coupling coefficient of pair of coils

r2r1_ratio = sqrt((a(1)-a(2))^2 + D.^2)/sqrt((a(1)+a(2))^2 + D.^2); % radius and distance ratio
Nval = NLookup(r2r1_ratio, N);
k = 2.54*Nval*sqrt(((6*a(1)+9*b(1)+10*c(1))*(6*a(2)+9*b(2)+10*c(2)))/(0.64*a(1)*a(2)));
M = k * sqrt(Lself(1) * Lself(2));

Power Transfer Efficiency

PTE(i,:) = 100 *(k^2 * Q(1)*Q(2)^3*RLoadAC*R_ESR(2))/((k^2*Q(1)*Q(2)*RLoadAC + (1+100/R_ESR(1)) * ...
    (RLoadAC+Q(2)^2*R_ESR(2)))*(RLoadAC+Q(2)^2*R_ESR(2)));

end
D = [0:.25:5];
subplot(2,2,4)
plot(D,PTE,'r');
title('PTE vs D')
xlabel('Distance, centimeters')
ylabel('PTE, percent')
grid;

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%References
% [1] An Improved Calculation of Proximity-Effect Loss in
% High-Frequency Windings of Round Conductors (2003)
% Xi Nan, Charles R. Sullivan
% https://engineering.dartmouth.edu/inductor/papers/newcalc.pdf
% [2] Geometry-based optimization of radio-frequency coils for
% powering neuroprosthetic implants
% Jungsuk Kim, Eric Basham, Kenneth D. Pedrotti
%  Med. Biol. Engineering and Computing 51(1-2):123-134 (2013)
% [3] Radio Engineers' Handbook
% Frederick Emmons Terman
% McGraw Hill (1947)
% [4] P.L. Dowell, “Effects of eddy currents in transformer windings”
% Proceedings of the IEEE, vol. 113, no. 8, pp. 1387–1394, Aug. 1966
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Published with MATLAB® R2015b