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keisanHigh-precision calculations for everyday life and work

Nagaoka Coefficient Ks and Inductance L of a Square Loop Antenna

Author: kegasaExecutions: 443

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Square loop antenna (side a = p, side b = p)
L Rr Rac η
μ0 = 4*pi*10^(-7);
k0 = p/w;
k1 = w/p;
k2 = k1^2;
k3 = k1^3;
ksr1 = sqrt(k2+1);
ksr2 = sqrt(k2+2);
Ks = (2/pi)*((k1-1/k1)*ln(ksr1/(ksr2+1)) + k1*ln((ksr1+1)/k1) + 2*atan(k1/ksr1) +
(1/k1)*(ksr1 - ksr2 - ln(sqrt(2) + 1) + (1/3)*(k3 + (k2 + 2)^(3/2) - 2*(k2 + 1)^(3/2) + sqrt(2) - 1)));
L = Ks * μ0 * p^2 * N^2 / w;
LmicroH = L * 10^6;
S = p^2;
print(k0);
print(Ks);
print(LmicroH);
print(S);
ω = 2*pi*frequency;
XL = ω*L;
C = 1/(ω*XL);
CpF = C *10^12;
print(XL);
print(CpF);
LengthWire = 4*p*N;
print(LengthWire);
λ = (3*10^8)/frequency;
Rr = (8/3)*pi^3*120*pi*(N*S/λ^2)^2;
print(Rr);
ρ = 1.68*10^(-8);
δ = sqrt(2*ρ/(ω*μ0));
radius = radius_mm/1000;
Rdc = ρ*LengthWire /(pi*radius^2);
Rac = ρ * LengthWire / (π * (radius^2 - (radius - δ)^2));
Diameter_mm = 2 * radius_mm;
δmicron = δ * 10^6;
print(Diameter_mm);
print(Rdc);
print(δmicron);
print(Rac);
η = Rr/(Rr+Rac)*100;
print(η);
heff = 2*pi*p^2/λ;
print(heff);
Q = XL/Rac;
print(Q);
print(ρ);
Vo = Q*heff*Eradio;
print(Vo);
Rp = (Rr+Rac)*Q^2;
print(Rp);
w: width of coil
m
square(side a=b=p) p
m
N turns
frequency
Hz
ρ
radius_mm
mm
Eradio
mV/m
p/w 
Ks 
L μH
Area = p × p m²
XL ohm
C pF
Wire Length m
Rr ohm
Diameter_mm mm
Rdc ohm
δmicron μm
Rac ohm
η %
heff m
Q XL/Rac 
ρ Ω・m
Vo mV
Rp Ω
Comprehensive Collection of Nagaoka Coefficients for Integral Equations of Perfect Ellipses
December 2006
Author: “Minato”
Club: “Homeless Ko~yo~”
Nagaoka Coefficients for Coils with Square Cross-Sections
https://kouyoubako.web.fc2.com/docu/nagaoka_elliptic_02.pdf

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