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| calculator:inductance_of_coaxial_cable [2025/02/08 10:30] – stan_zurek | calculator:inductance_of_coaxial_cable [2025/02/08 17:10] (current) – stan_zurek |
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| ==== Calculator of inductance of a straight coaxial cable ==== | ==== Calculator of inductance of a straight coaxial cable ==== |
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| [[/Inductance]] of a straight [[/coaxial cable]] with round (circular) cross-section (with return current path) can be calculated withe the equation as specified below. | [[/Inductance]] of a straight [[/coaxial cable]] with round (circular) cross-section (with return current path) can be calculated with the the equations as specified below. |
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| r2 = d/2 | r2 = d/2 |
| r1 = D/2 | r1 = D/2 |
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| // fix d in T(x) because it should be a radius??????????????????????? | |
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| // T(x) function | // T(x) function |
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| ^ Inductance of a straight round magnetic wire or conductor ^^^ | ^ Inductance of a straight round magnetic wire or conductor ^^^ |
| | // Sources: [1] [[https://isbnsearch.org/isbn/0876645570|Frederick W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570]] and \\ [2] [[https://isbnsearch.org/isbn/9780470461884|Clayton R. Paul. Inductance: Loop and Partial, Wiley-IEEE Press, 2009, New Jersey, ISBN 9780470461884]] \\ [3] [[https://archive.org/details/selfmu430134419088080unse|Edward B. Rosa, The self and mutual inductance of linear conductors, Department of Commerce and Labor, Bulletin of the Bureau of Standards, Volume 4, 1907-8, Washington, 1908]], {accessed 2025-01-06}// ||| | | // Sources: [1] [[https://isbnsearch.org/isbn/0876645570|Frederick W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570]] and \\ [2] [[https://isbnsearch.org/isbn/9780470461884|Clayton R. Paul. Inductance: Loop and Partial, Wiley-IEEE Press, 2009, New Jersey, ISBN 9780470461884]] \\ [3] [[https://archive.org/details/selfmu430134419088080unse|Edward B. Rosa, The self and mutual inductance of linear conductors, Department of Commerce and Labor, Bulletin of the Bureau of Standards, Volume 4, 1907-8, Washington, 1908]], {accessed 2025-01-06} \\ [4] B.C. Wadell, Transmission Line Design Handbook, Artech House, Norwood, 1991, ISBN 0890064369// ||| |
| | **(1)** \\ //[1] Grover, eq. (232), p. 280 \\ (assumes infinitely thin shield)// | $$ L_{AC} = \frac{μ_0 ⋅ l}{2⋅π} ⋅ \left( ln \left( \frac{d}{c} \right) + \frac{T(x)}{4} \right) $$ | (H) | | | **(1)** \\ //[1] Grover, eq. (232), p. 280 \\ (assumes infinitely thin shield)// | $$ L_{AC} = \frac{μ_0 ⋅ l}{2⋅π} ⋅ \left( ln \left( \frac{d}{c} \right) + \frac{T_{W,Z}(x)}{4} \right) $$ | (H) | |
| | **(2)** \\ //[2] Paul, eq. (4.29), p. 132 \\ [3] Rosa, eq. (58), p. 333 \\ (for high frequency limit)// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅ln \left( \frac{d}{c} \right) $$ | (H) | | | **(2)** \\ //[2] Paul, eq. (4.29), p. 132 \\ [3] Rosa, eq. (58), p. 333 \\ [1] Grover, eq. (25), p. 42 (corrected here) \\ (for high frequency limit)// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅ln \left( \frac{d}{c} \right) $$ | (H) | |
| | **(3.1)** \\ //[1] Grover, eq. (24), p. 42 \\ (original, with the discrepancy)// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{D}{c} \right) + \frac{2· \left( \frac{d}{D} \right) ^2}{1-\left( \frac{d}{D} \right) ^2} ⋅ ln \left( \frac{D}{d} \right) - \frac{3}{4} + ξ_Z \right) $$ | (H) | | | **(3.1)** \\ //[1] Grover, eq. (24), p. 42 \\ (original, with the discrepancy, see the note below)// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{D}{c} \right) + \frac{2· \left( \frac{d}{D} \right) ^2}{1-\left( \frac{d}{D} \right) ^2} ⋅ ln \left( \frac{D}{d} \right) - \frac{3}{4} + ξ_Z \right) $$ | (H) | |
| | **(3.2)** \\ //[1] Grover, eq. (24), p. 42 \\ (see the note below about the possible discrepancy)// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{d}{c} \right) + \frac{2· \left( \frac{d}{D} \right) ^2}{1-\left( \frac{d}{D} \right) ^2} ⋅ ln \left( \frac{D}{d} \right) - \frac{3}{4} + ξ_Z \right) $$ | (H) | | | **(3.2)** \\ //[1] Grover, eq. (24), p. 42 \\ (with Zurek's correction)// | $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{d}{c} \right) + \frac{2· \left( \frac{d}{D} \right) ^2}{1-\left( \frac{d}{D} \right) ^2} ⋅ ln \left( \frac{D}{d} \right) - \frac{3}{4} + ξ_Z \right) $$ | (H) | |
| | | where: $μ_0$ - [[/magnetic permeability of vacuum]] (H/m), $l$ - wire length (m), $D$ - outer diameter (m) of the shield conductor, $d$ - inner diameter (m) of the shield conductor, $c$ - outer diameter (m) of the solid inner conductor, $x$ - Grover's coefficient (unitless), $T_{W,Z}(x)$ - Wadell's approximation of Grover's function T(x) with Zurek's correction (unitless), $k_{SI}$ - factor scaling from SI units (1/(Ω·m)), $f$ - frequency (Hz), $ρ$ - resistivity of wire (Ω·m), $ξ_Z$ - Grover's function from Table 4 (unitless) approximated here by S. Zurek with a polynomial function, for ratio $z = c/d$ (unitless): ||| |
| | where: $μ_0$ - [[/magnetic permeability of vacuum]] (H/m), $l$ - wire length (m), $D$ - outer diameter (m) of the shield conductor, $d$ - inner diameter (m) of the shield conductor, $c$ - outer diameter (m) of the solid inner conductor, $ξ_Z$ - Grover's function from Table 4 (unitless) approximated here by S. Zurek with a polynomial function, for ratio $z = d/D$ (unitless): ||| | | (1.1) \\ Grover's coefficient $x$ | $$x = π⋅d⋅100⋅k_{SI}⋅ \sqrt{2⋅μ_r⋅μ_0⋅f⋅ρ} $$ | (unitless) | |
| | (1.1) \\ Grover's Table 4 with Zurek's approximation by polynomials | $$ ξ_Z = 0.1705⋅z^3 - 0.3979⋅z^2 - 0.0214⋅z + 0.25 $$ | (unitless) | | | (1.2) \\ Wadell's approximation of $T(x)$ with Zurek's correction (this reduces T(x) approximation difference near x = 0 from 6.5 % to below 1 %, [[approximation of Grover Tx|see more]]) | $$T_{W,Z}(x) = \sqrt{\frac{0.873011 + 0.00186128 ⋅ x}{1 - 0.279381⋅x + 0.127964⋅x^2}} + \frac{0.06}{(x+1)^6} $$ | (unitless) | |
| | //The original Grover's equation appears to be incorrect, with the first logarithm given as $ln(D/c)$ but it is clear from Grover's description that it should be $ln(d/c)$, as this is the limit for high frequency, and as listed for example by Paul [2]. Grover's and Paul's equations are based on radii, but all variables are used as ratios, so diameters can be used directly instead because ratio of radii is equal to the ratio of the corresponding diameters.// ||| | | (3.1) \\ Grover's Table 4 with Zurek's approximation by polynomials ([[Approximation of Grover zeta|see more]]) | $$ ξ_Z = 0.1705⋅z^3 - 0.3979⋅z^2 - 0.0214⋅z + 0.25 $$ | (unitless) | |
| | | //The original Grover's equation appears to be incorrect, with the first logarithm given as $ln(D/c)$ but it is clear from Grover's description that it should be $ln(d/c)$, as this is the limit for high frequency, and as listed for example by Paul [2] and Rosa [3]. Grover's and Paul's equations are based on radii, but all variables are used as ratios, so diameters can be used directly instead, because ratio of radii is equal to the ratio of the corresponding diameters.// ||| |
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