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--- calculator:inductance_of_coaxial_cable [2025/02/03 23:39] – [Calculator of inductance of a straight coaxial cable] stan_zurek
+++ calculator:inductance_of_coaxial_cable [2025/02/08 17:10] (current) – stan_zurek
@@ Line 1: / Line 1: @@
 ==== Calculator of inductance of a straight coaxial cable ====
-{{page>insert/todo}}
 <box 100% #efffef>
@@ Line 16: / Line 14: @@
 </box>
-[[/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.
 <HTML>
@@ Line 38: / Line 36: @@
 var D          = frm.D.value
 var l          = frm.l.value
-var result1    = "1"
+var freq       = frm.freq.value
+var rho        = frm.rho.value
 var c_unit     = getSelectedValue(frm.c_unit)
@@ Line 44: / Line 43: @@
 var D_unit     = getSelectedValue(frm.D_unit)
 var l_unit     = getSelectedValue(frm.l_unit)
+var freq_unit     = getSelectedValue(frm.freq_unit)
 var result1_unit    = getSelectedValue(frm.result1_unit)
+var result2_unit    = getSelectedValue(frm.result2_unit)
+var result31_unit    = getSelectedValue(frm.result31_unit)
+var result32_unit    = getSelectedValue(frm.result32_unit)
 const pi = 3.14159265358979
@@ Line 55: / Line 59: @@
 D = parseFloat(D)
 l = parseFloat(l)
+freq = parseFloat(freq)
-c = c * c_unit  //r3
+c = c * c_unit
-d = d * d_unit  //r2
+d = d * d_unit
-D = D * D_unit  //r1
+D = D * D_unit
+l = l * l_unit
+freq = freq * freq_unit
 r3 = c/2
@@ Line 64: / Line 72: @@
 r1 = D/2
-l = l * l_unit
+// T(x) function
+// calculate inductance
+// Wadell (1.1), * 100 because conversion from cm in Wadell
+ur = 1
+xSI = pi * d * 100 * Math.pow(2*ur*freq*u0*rho,0.5)
+Tx = Math.pow((0.873011+0.00186128*xSI)/(1-0.278381*xSI+0.127964*xSI*xSI),0.5)
+//Wadell's T(x) with Zurek's correction
+TxSZ = Tx + 0.06/Math.pow(xSI+1,6)
 // ratio z = r_inner / r_outer, in Grover logz = f (r2/r1)
@@ Line 70: / Line 85: @@
 logzpoly = 0.1705*z*z*z -0.3979*z*z -0.0214*z +0.25
-// calculate inductance
+// (1) Grover (232)
-result1 = result1_unit * u0 * l * ( Math.log(D/c) + 2*(d/D)*(d/D)/(1-(d/D)*(d/D) ) * Math.log(D/d) - 3/4 + logzpoly ) / (2 * pi )
+result1 = result1_unit * u0 * l * ( Math.log(d/c) + TxSZ/4 ) / (2 * pi )
+// (2) Paul/Rosa
+result2 = result2_unit * u0 * l * ( Math.log(d/c) ) / (2 * pi )
+// (3.1) Grover (24) as in the book, with discrepancy
+result31 = result31_unit * u0 * l * ( Math.log(D/c) + 2*(d/D)*(d/D)/(1-(d/D)*(d/D) ) * Math.log(D/d) - 3/4 + logzpoly ) / (2 * pi )
+// (3.2) Grover with Zurek correction
+result32 = result32_unit * u0 * l * ( Math.log(d/c) + 2*(d/D)*(d/D)/(1-(d/D)*(d/D) ) * Math.log(D/d) - 3/4 + logzpoly ) / (2 * pi )
-//result1 = result1_unit * u0 * l * ( Math.log(r1/r3) + 2*(r2/r1)*(r2/r1)/(1-(r2/r1)*(r2/r1) ) * Math.log(r1/r2) - 3/4 + logzpoly ) / (2 * pi )
 // format number to x digits precision, result will equal 1.234e+2
 result1  = result1.toPrecision(5)
+result2  = result2.toPrecision(5)
+result31  = result31.toPrecision(5)
+result32  = result32.toPrecision(5)
 // display result
 frm.result1.value = result1
+frm.result2.value = result2
+frm.result31.value = result31
+frm.result32.value = result32
 }
@@ Line 123: / Line 152: @@
            <OPTION value="1e-3" selected>(mm)</OPTION>
         </SELECT> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br><br>
+<b>Inner wire resistivity <i>ρ</i></b> = <input type="text" value="17.24e-9" name="rho" size="10" maxlength="10" onChange="calculate_function(this.form)"> (Ω·m)  <br><br>
+<b>Frequency <i>f</i></b> = <input type="text" value="30" name="freq" size="10" maxlength="10" onChange="calculate_function(this.form)">
+        <SELECT name="freq_unit" onChange="calculate_function(this.form)">
+           <OPTION value="1">(Hz)</OPTION>
+           <OPTION value="1e3">(kHz)</OPTION>
+           <OPTION value="1e6" selected>(MHz)</OPTION>
+           <OPTION value="1e9">(GHz)</OPTION>
+        </SELECT> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br><br>
 <input type="button" name="Button" value="== Calculate ==" onClick="calculate_function(this.form)"> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
@@ Line 129: / Line 169: @@
     <br><br>
-<b><i>L</i></b> = <input type="text" name="result1" size="10" maxlength="10">
+<b><i>L<sub>AC</sub></i></b> = <input type="text" name="result1" size="10" maxlength="10">
           <SELECT name="result1_unit" onChange="calculate_function(this.form)">
            <OPTION value="1">(H)</OPTION>
@@ Line 135: / Line 175: @@
            <OPTION value="1e6" selected>(μH)</OPTION>
            <OPTION value="1e9">(nH)</OPTION>
-        </SELECT> <i>from eq. (1) </i> <br>
+        </SELECT> <i>Eq. (1) Grover, thin shield, skin effect</i>    <br><br>
+<b><i>L<sub>HF</sub></i></b> = <input type="text" name="result2" size="10" maxlength="10">
+          <SELECT name="result2_unit" onChange="calculate_function(this.form)">
+           <OPTION value="1">(H)</OPTION>
+           <OPTION value="1e3">(mH)</OPTION>
+           <OPTION value="1e6" selected>(μH)</OPTION>
+           <OPTION value="1e9">(nH)</OPTION>
+        </SELECT> <i>Eq. (2) Paul / Rosa / Grover, high frequency limit</i> <br><br>
+<b><i>L<sub>DC</sub></i></b> = <input type="text" name="result31" size="10" maxlength="10">
+          <SELECT name="result31_unit" onChange="calculate_function(this.form)">
+           <OPTION value="1">(H)</OPTION>
+           <OPTION value="1e3">(mH)</OPTION>
+           <OPTION value="1e6" selected>(μH)</OPTION>
+           <OPTION value="1e9">(nH)</OPTION>
+        </SELECT> <i>Eq. (3.1) Grover, with possible discrepancy </i> <br><br>
+<b><i>L<sub>DC</sub></i></b> = <input type="text" name="result32" size="10" maxlength="10">
+          <SELECT name="result32_unit" onChange="calculate_function(this.form)">
+           <OPTION value="1">(H)</OPTION>
+           <OPTION value="1e3">(mH)</OPTION>
+           <OPTION value="1e6" selected>(μH)</OPTION>
+           <OPTION value="1e9">(nH)</OPTION>
+        </SELECT> <i>Eq. (3.2) Grover, with Zurek's correction</i>
  </form>
@@ Line 146: / Line 212: @@
 <WRAP lo>
-//Note: Several assumptions are made for these equations: 1) The return path is **not** considered so the total inductance of the complete circuit can be significantly different. 2) The length of the wire is assumed to be significantly longer than its diameter(s) (a,b << l), otherwise the calculation errors might be excessive. 3) The surrounding medium is assumed to be non-magnetic (μ<sub>r</sub> = 1). 4) The current is uniformly distributed inside the wire (no [[/skin effect]]). 5) The equations were converted here to be consistent with [[/SI units]].//
+//Note: Several assumptions are made for these equations: 1) The return path is considered so this equation represents a "closed-loop" electrical circuit. 2) The length of the wire is assumed to be significantly longer than its diameter(s) (c,d,D << l), otherwise the calculation errors might be excessive. 3) All the materials and the surrounding medium are assumed to be non-magnetic (μ<sub>r</sub> = 1). 4) The current is uniformly distributed inside the wire (no [[/skin effect]]) for the DC case. 5) The equations were converted here to be consistent with [[/SI units]].//
 </WRAP>
-<WRAP lo>
 |< 100% >|
 ^  Inductance of a straight round magnetic wire or conductor  ^^^
-| // Source: [1] F.W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570//  |||
+| // 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. (24), p. 42//  |  $$ L = \frac{μ_0 ⋅ l}{2⋅π}⋅\left( ln \left( \frac{4⋅l}{a+b} \right) - 0.05685 \right) $$  |  (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)  |
-| where: $μ_0$ - [[/magnetic permeability of vacuum]] (H/m), $l$ - wire length (m), $a$, $b$ - shorted and longer diameters of ellipse (m)  |||
+|  **(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)  |
-| //The original Grover's equation is expressed by "semiaxes" α, β  of the ellipse, with the under-logarithm coefficient as 2·l/(α+β), hence if the full "diameters" are used then this becomes 4·l/(a+b), as written above.//   |||
+|  **(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)  |
-</WRAP>
+|  **(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):    |||
+|  (1.1) \\ Grover's coefficient $x$  |  $$x = π⋅d⋅100⋅k_{SI}⋅ \sqrt{2⋅μ_r⋅μ_0⋅f⋅ρ}  $$  |  (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)  |
+|  (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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