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Addendum

Addendum: Babic, S., et al. Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities. Physics 2020, 2, 352–367

1
Independent Researcher, 53 Berlioz, 101, Montréal, QC H3E 1N2, Canada
2
École Polytechnique, C.P. 6079 Centre-Ville, Montréal, QC H3C 3A7, Canada
*
Author to whom correspondence should be addressed.
Submission received: 7 January 2021 / Accepted: 8 January 2021 / Published: 18 January 2021
(This article belongs to the Section Applied Physics)
In [1] some ambiguities and imperfections were subsequently identified and are correspondingly modified and simplified below so that they correspond to those given in our software package [2]. Thus, people that have an interest in this domain could easily calculate the self-inductances presented here using the Mathematica code for them.
On page 354, Equation (5) reads:
L R = 2 μ 0 N 2 R 1 b 2 ln 2 ( α ) n = 1 11 0 π 2 cos ( 2 β )   T n d β ,
where
T 1 = b 3 3 sin ( 2 β ) [ 2 arctan ( q ) arctan ( q 1 ) arctan ( q 2 ) ] ,
T 2 = 8 3 ( α 3 + 1 ) cos 3 ( β ) 4 3 cos 2 ( β ) ( α 2 r 2 + r 1 ) + 2 3 [ ( α 2 + 1 ) cos ( 2 β ) 2 α ] ( r r 0 ) ,
T 3 = 4 b α 2   cos 2 ( β ) arsinh [ b 2 α cos ( β ) ] + 4 b cos 2 ( β ) arsinh [ b 2 cos ( β ) ] 2 b [ ( α 2 + 1 ) cos ( 2 β ) + 2 α ] arsinh ( b r 0 ) ,
T 4 = 2 b 2 [ α   arsinh ( v 22 ) + arsinh ( v 11 ) ] ,
T 5 = 2 b 2 [ α   arsinh ( v 2 ) + arsinh ( v 1 ) ] ,
T 6 = 2 b sin ( 2 β ) [ α 2 arctan ( p 22 ) + arctan ( p 11 ) ] ,
T 7 = 2 b sin ( 2 β ) [ α 2 arctan ( p 2 ) + arctan ( p 1 ) ] ,
T 8 = α 3 3 sin 2 ( 2 β ) ln ( m 2 + 1 m 2 1 ) + 1 3 sin 2 ( 2 β ) ln ( m 1 + 1 m 1 1 ) ,
T 9 = α 3 3 sin 2 ( 2 β ) ln ( m 20 + 1 m 20 1 ) 1 3 sin 2 ( 2 β ) ln ( m 10 + 1 m 10 1 ) ,
T 10 = α 3 3 sin 2 ( 2 β ) ln ( m 22 + 1 m 22 1 ) 1 3 sin 2 ( 2 β ) ln ( m 11 + 1 m 11 1 ) ,
T 11 = 2 ( α 3 + 1 ) 3 sin 2 ( 2 β ) ln [ cos ( β / 2 ) cos ( β / 2 ) ] ,
where
r = b 2 + α 2 + 1 + 2 α cos ( 2 β ) ,         r 0 = α 2 + 1 + 2 α cos ( 2 β ) ,           r 1 = b 2 + 4 cos 2 ( β ) ,    
r 2 = b 2 + 4 α 2 cos 2 ( β )   ,       r 01 = 4 cos 2 ( β ) = 2 cos ( β ) ,     r 02 = 4 α 2 cos 2 ( β ) = 2 α cos ( β ) ,
q = α sin 2 ( 2 β ) b 2 cos ( 2 β ) b sin ( 2 β ) r ,             q 1 = α 2 sin 2 ( 2 β ) b 2 cos ( 2 β ) b sin ( 2 β ) r 2 ,
q 2 = sin 2 ( 2 β ) b 2 cos ( 2 β ) b sin ( 2 β ) r 1 ,     v 1 = α + cos ( 2 β ) b 2 + sin 2 ( 2 β ) ,     v 2 = 1 + α cos ( 2 β ) b 2 + α 2 sin 2 ( 2 β ) ,
v 11 = 1 + cos ( 2 β ) b 2 + sin 2 ( 2 β ) ,     v 22 = α + α cos ( 2 β ) b 2 + α 2 sin 2 ( 2 β ) ,     p 1 = b [ α + cos ( 2 β ) ] sin ( 2 β ) r , p 2 = b [ 1 + α cos ( 2 β ) ] α   sin ( 2 β ) r ,       p 11 = b [ 1 + cos ( 2 β ) ] sin ( 2 β ) r 1 ,       p 22 = b [ α + α cos ( 2 β ) ] α sin ( 2 β ) r 2   ,   m 1 = r α + cos ( 2 β ) ,       m 2 = r 1 + α cos ( 2 β ) ,       m 10 = r 0 α + cos ( 2 β ) ,       m 20 =   r 0 1 + α cos ( 2 β )   ,   m 11 = r 1 1 + cos ( 2 β )   ,     m 22 = r 2 α + α cos ( 2 β ) .
Expressions for T n ,   n = 1 , 2 , , 11 , are given as in [2] and Appendix A where we calculate the self-inductance for the coil of the rectangular cross-section with radial current in Example 2 [1].
On page 355, Equation (6) reads:
L R disk = 4 μ 0 N 2 R 1 ( α + 1 ) ln 2 α [ E ( k 0 ) 1 ] ,
where
k 0 2 = 4 α ( α + 1 ) 2 .
On page 355, Equation (7) reads:
L A = μ 0 N 2 R 1 15 b 2 ( α 1 ) 2 n = 1 6 0 π 2 cos ( 2 β )   S n d β ,
where
S 1 = b 4 sin 2 ( 2 β ) [ r 2 b cos ( 2 β ) sin ( 2 β ) arctan ( q 2 ) ] + b 4 sin 2 ( 2 β ) [ r 1 b cos ( 2 β ) sin ( 2 β ) arctan ( q 1 ) ] 2 b 4 sin 2 ( 2 β ) [ r b cos ( 2 β ) sin ( 2 β ) arctan ( q ) ] ,
S 2 = 9 a 2 b 2 r 2 + 9 b 2 r 1 9 ( a 2 + 1 ) b 2 r + 2 [ 6 a 4 cos 2 ( 2 β ) 2 a 4 cos ( 2 β ) 8 a 4 ] ( r 2 r 02 ) + 2 [ 6 cos 2 ( 2 β ) 2 cos ( 2 β ) 8 ] ( r 1 r 01 ) 4 [ 3 ( a 4 + 1 ) cos 2 ( 2 β ) α ( a 2 + 1 ) cos ( 2 β ) 2 ( α 2 + 1 ) 2 ] ( r r 0 )   ,
S 3 = 30 b sin ( 2 β ) cos ( 2 β ) [ a 4 arctan ( p 22 ) + arctan ( p 11 ) a 4 arctan ( p 2 ) arctan ( p 1 ) ] ,
S 4 = 30 b { α 4 sin 2 ( 2 β ) ln ( r 2 + b r 2 b ) + sin 2 ( 2 β ) ln ( r 1 + b r 1 b ) 1 2 [ ( α 2 + 1 ) 2 2 ( α 4 + 1 ) cos 2 ( 2 β ) ] ln ( r + b r b ) } ,
S 5 = 12 cos ( 2 β ) sin 2 ( 2 β ) { α 5 ln [ r 0 + 1 + α cos ( 2 β ) ] + ln [ r 0 + α + cos ( 2 β ) ] ( α 5 + 1 ) ln [ 4 cos ( β ) cos 2 ( β / 2 ) ] α 5 ln ( α ) } ,
S 6 = 4 cos ( 2 β ) [ 5 b 2 3 sin 2 ( 2 β ) ] ln [ r + α + cos ( 2 β ) r 1 + 1 + cos ( 2 β ) ] + 4 α 3 cos ( 2 β ) [ 5 b 2 3 α 2 sin 2 ( 2 β ) ] ln [ r + 1 + α cos ( 2 β ) r 2 + α + α cos ( 2 β ) ] .
Expressions for S n   , n = 1 , 2 , , 6 , are given as in [2] and Appendix B where we calculate the self-inductance for the coil of the rectangular cross-section with azimuthal current in Example 4 (case α = 3 ,   b = 2 ) [1].
On page 359, the self-inductance of the thin Bitter disk (pancake) is to be replaced by:
L R disk = 3.56991288673   H .
At the request of many interested people, who have contacted us, we give the Mathematica codes for calculating the self-inductances with modified expressions for T n and S n [2], which are more friendly for the calculations than those given in [1].

Appendix A

MATHEMATICA CODE (RADIAL CURRENT) - EXAMPLE 2
ClearAll["Global`*"]
$Messages = {};
Beep[[]
mu = 4Pi/10000000;
R1 = 25/1000;
R2 = 35/1000;
l = 4/100;
a = R2/R1;
b = l/R1;
n1 = 100;
r = (a^2 + 2 a Cos[2x] + 1 + b^2)^(1/2);
r0 = (a^2 + 2a Cos[2x] + 1)^(1/2);
r2 = (2a^2 + 2 a^2 Cos[2x] + b^2)^(1/2);
r1 = (2 + 2 Cos[2x] + b^2)^(1/2);
f11 = 2b^3/3/Sin[2x] ArcTan[(a Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r)];
f22 = b^3/3/Sin[2x](ArcTan[( Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r1)] +
+ ArcTan[(a ^2Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r2)]);
T1 = f11-f22;
T2 = 8/3(a^3 + 1) Cos[x]^3 -4/3Cos[x]^2(a^2r2 + r1) + 2/3((a^2 + 1)Cos[2x] + 2a)(r-r0);
T3 = 4b a^2 Cos[x]^2 ArcSinh[b/(2a Cos[x])] + 4b Cos[x]^2 ArcSinh[b/(2Cos[x])]-2b((a^2 + 1)Cos[2x] + 2a)ArcSinh[b/r0];
T4 = 2b^2 (a ArcSinh[(a + a Cos[2x])/(a^2Sin[2x]^2 + b^2)^(1/2)] + ArcSinh[(1 + Cos[2x])/(Sin[2x]^2 + b^2)^(1/2)]);
T5 = -2b^2 (a ArcSinh[(1 + a Cos[2x])/(a^2Sin[2x]^2 + b^2)^(1/2)] + ArcSinh[(a + Cos[2x])/(Sin[2x]^2 + b^2)^(1/2)]);
T6 = -2b Sin[2x](a^2ArcTan[b (a + a Cos[2x])/(a Sin[2x]r2)] + ArcTan[b (1 + Cos[2x])/( Sin[2x]r1)]);
T7 = 2b Sin[2x](a^2ArcTan[b (1 + a Cos[2x])/(a Sin[2x]r)] + ArcTan[b (a + Cos[2x])/( Sin[2x]r)]);
T8 = Sin[2x]^2/3(a^3Log[(r + 1 + a Cos[2x])/(r-1-a Cos[2x])] + Log[(r + a + Cos[2x])/(r-a- Cos[2x])]);
T9 = -Sin[2x]^2/3(a^3Log[(r0 + 1 + a Cos[2x])/(r0-1-a Cos[2x])] + Log[(r0 + a + Cos[2x])/(r0-a- Cos[2x])]);
T10 = -Sin[2x]^2/3(a^3Log[(r2 + a + a Cos[2x])/(r2-a-a Cos[2x])] + Log[(r1 + 1 + Cos[2x])/(r1-1- Cos[2x])]);
T11 = 2/3(a^3 + 1) Sin[2x]^2Log[Cos[x/2]/Sin[x/2]];
f = Cos[2x](T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 + T11);
A = NIntegrate[f,{x,0,Pi/2},WorkingPrecision->30, AccuracyGoal->30];
N[A,16];
B = -2mu n1^2 R1/(b^2Log[a]^2);
N[B,16];
L = A B;
N[L,16]
0.0004383988542717143
L = 0.0004383988542717143 (H) = 0.4383988542717143 (mH)

Appendix B

MATHEMATICA CODE (AZIMUTHAL CURRENT)-EXAMPLE 4 (case α   =   3 ,   b   =   2 ) .
ClearAll["Global`*"]
$Messages = {};
Beep[[]
mu = 4Pi/10000000;
n1 = 1;
R1 = 1;
R2 = 3;
l = 2;
a = R2/R1;
b = l/R1;
n1 = 1;
r = (a^2 + 2 a Cos[2x] + 1 + b^2)^(1/2);
r0 = (a^2 + 2 a Cos[2x] + 1)^(1/2);
r2 = (2a^2 + 2 a^2Cos[2x] + b^2)^(1/2);
r02 = (2a^2 + 2 a^2Cos[2x])^(1/2);
r1 = (2 + 2 Cos[2x] + b^2)^(1/2);
r01 = (2 + 2 Cos[2x])^(1/2);
f11 = b^4/Sin[2x]^2( r2-b Cos[2x]/Sin[2x] ArcTan[(a ^2Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r2)]);
f12 = b^4/Sin[2x]^2( r1-b Cos[2x]/Sin[2x] ArcTan[(Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r1)]);
f13 = -2b^4/Sin[2x]^2( r-b Cos[2x]/Sin[2x] ArcTan[(a Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r)]);
S1 = f11 + f12 + f13;
f21 = 9a^2b^2r2 + 2(6a^4Cos[2x]^2-2a^4Cos[2x]-8a^4)(r2-r02);
f22 = 9b^2r1 + 2(6Cos[2x]^2-2Cos[2x]-8)(r1-r01);
f23 = -9b^2(a^2 + 1)r -4(3(a^4 + 1)Cos[2x]^2-a(a^2 + 1)Cos[2x]-2(a^2 + 1)^2)(r-r0);
S2 = f21 + f22 + f23;
f31 = 30b Sin[2x]Cos[2x](a^4ArcTan[b (1 + Cos[2x])/( Sin[2x]r2)] + ArcTan[b (1 + Cos[2x])/( Sin[2x]r1)]);
f32 = -30b Sin[2x]Cos[2x](a^4ArcTan[b (1 + a Cos[2x])/( a Sin[2x]r)] + ArcTan[b (a + Cos[2x])/( Sin[2x]r)]);
S3 = f31 + f32;
S4 = 15b (a^4Sin[2x]^2Log[(r2 + b)/(r2-b)] + Sin[2x]^2Log[(r1 + b)/(r1-b)]-((a^2 + 1)^2-2(a^4 + 1)Cos[2x]^2)/2Log[(r + b)/(r-b)]);
S5 = 12Cos[2x]Sin[2x]^2(a^5Log[r0 + 1 + a Cos[2x]] + Log[r0 + a + Cos[2x]]-(a^5 + 1)Log[4Cos[x]Cos[x/2]^2]-a^5Log[a]);
f61 = 4Cos[2x](5b^2-3Sin[2x]^2)Log[(r + a + Cos[2x])/(r1 + 1 + Cos[2x])];
f62 = 4a^3Cos[2x](5b^2-3a^2Sin[2x]^2)Log[(r + 1 + a Cos[2x])/(r2 + a + a Cos[2x])];
S6 = f61 + f62;
f = Cos[2x](S1 + S2 + S3 + S4 + S5 + S6);
A = NIntegrate[f,{x,0,Pi/2},WorkingPrecision->30, AccuracyGoal->30];
N[A,16];
B = -mu n1^2 R1/15/b^2/(a-1)^2;
N[B,16];
L = A B;
N[L,16]
2.533006546891938*10−6
L = 2.533006546891938 (μH)

References

  1. Babic, S.; Akyel, C. Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities. Physics 2020, 2, 19. [Google Scholar] [CrossRef]
  2. Babic, S. Calculation of the Self, Mutual Inductances, and the Magnetic Forces between Circular Coils-Software Package. Available on the request.
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Babic, S.; Akyel, C. Addendum: Babic, S., et al. Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities. Physics 2020, 2, 352–367. Physics 2021, 3, 1-5. https://0-doi-org.brum.beds.ac.uk/10.3390/physics3010001

AMA Style

Babic S, Akyel C. Addendum: Babic, S., et al. Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities. Physics 2020, 2, 352–367. Physics. 2021; 3(1):1-5. https://0-doi-org.brum.beds.ac.uk/10.3390/physics3010001

Chicago/Turabian Style

Babic, Slobodan, and Cevdet Akyel. 2021. "Addendum: Babic, S., et al. Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities. Physics 2020, 2, 352–367" Physics 3, no. 1: 1-5. https://0-doi-org.brum.beds.ac.uk/10.3390/physics3010001

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