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calculator:toroidal_coil_inductance [2025/02/25 23:45] – [Equations] stan_zurekcalculator:toroidal_coil_inductance [2025/04/05 21:22] (current) – [Calculator of inductance of a toroidal coil] stan_zurek
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 |  {{/wiki/logo.png?20&nolink}} //See more: [[/Calculators of inductance]]//  || |  {{/wiki/logo.png?20&nolink}} //See more: [[/Calculators of inductance]]//  ||
 </box> </box>
- 
-{{page>insert/todo}} 
- 
- 
  
 <box 35% right #f0f0f0> <box 35% right #f0f0f0>
-Toroidal coil dimensions: D - outer diameter, d - inner diameter, h - thickness, c - round wire diameter, p - average pitch between the turns+Toroidal coil dimensions: D - outer diameter, d - inner diameter, h - thickness, c - round wire diameter, p - average pitch between the turns; the core is not shows in this image but it is assumed to completely fill the inside of the coil
 [[file/toroidal_coil_dimensions_png|{{/toroidal_coil_dimensions.png}}]] [[file/toroidal_coil_dimensions_png|{{/toroidal_coil_dimensions.png}}]]
 {{page>insert/by_SZ}} {{page>insert/by_SZ}}
 </box> </box>
  
-[[/Inductance]] of an ideal [[/toroidal coil]] or winding can be calculated with this following equation.+[[/Inductance]] of an ideal [[/toroidal coil]] or winding can be calculated from the equations as specified below.
  
 <HTML> <HTML>
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 var ur          = frm.ur.value var ur          = frm.ur.value
 var c           = frm.c.value var c           = frm.c.value
-var pitch       = frm.pitch.value+// var pitch       = frm.pitch.value
  
 var result2           = "1" var result2           = "1"
Line 48: Line 44:
 var h_unit     = getSelectedValue(frm.h_unit) var h_unit     = getSelectedValue(frm.h_unit)
 var c_unit     = getSelectedValue(frm.c_unit) var c_unit     = getSelectedValue(frm.c_unit)
-var pitch_unit = getSelectedValue(frm.pitch_unit)+// var pitch_unit = getSelectedValue(frm.pitch_unit)
  
 var result1_unit     = getSelectedValue(frm.result1_unit) var result1_unit     = getSelectedValue(frm.result1_unit)
 var result2_unit     = getSelectedValue(frm.result2_unit) var result2_unit     = getSelectedValue(frm.result2_unit)
 var result3_unit     = getSelectedValue(frm.result3_unit) var result3_unit     = getSelectedValue(frm.result3_unit)
 +
  
 // read values and convert to numbers // read values and convert to numbers
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 h = parseFloat(h) * h_unit h = parseFloat(h) * h_unit
 c = parseFloat(c) * c_unit c = parseFloat(c) * c_unit
-pitch = parseFloat(pitch) * pitch_unit+// pitch = parseFloat(pitch) * pitch_unit
  
 const pi = 3.14159265358979 const pi = 3.14159265358979
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 result1 = ur * u0 * h * N * N * Math.log(D/d) / (pi * 2) * result1_unit result1 = ur * u0 * h * N * N * Math.log(D/d) / (pi * 2) * result1_unit
  
-// calculate eq. (2) +// calculate eq. (2), lc = average core length = magnetic path length 
-A = h * (D - d)/2  +lc = pi * (D + d)/2  
-result2 = ur * u0 * A * N * N / (pi * d) * result2_unit +A = h * (D - d)/2 
 +result2 = ur * u0 * A * N * N / lc * result2_unit
  
 // calculate eq. (3) // calculate eq. (3)
-G = 5/4 - Math.log(2*pitch/c)+pitch_avg = lc / N 
 +G = 5/4 - Math.log(2*pitch_avg/c)
 H = 0.33780 - 0.43478 * Math.pow(N,-0.8) + 0.096876 / (N * N) H = 0.33780 - 0.43478 * Math.pow(N,-0.8) + 0.096876 / (N * N)
 hg = h + c hg = h + c
-l = (D - d) + 2 * hg +lt = (D - d) + 2 * hg 
-result3 = u0 * N * (  hg * N * Math.log(D/d) - l * (G + H )   ) / (pi * 2) * result3_unit+result3 = u0 * N * (  hg * N * Math.log(D/d) - lt * (G + H )   ) / (pi * 2) * result3_unit
  
-console.log("G=", G, "    H=", H)+//console.log("G=", G, "    H=", H)
  
 // format number to x digits precision, result will equal 1.234e+2 // format number to x digits precision, result will equal 1.234e+2
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 result2 = result2.toPrecision(6) result2 = result2.toPrecision(6)
 result3 = result3.toPrecision(6) result3 = result3.toPrecision(6)
 +//result4 = result4.toPrecision(6)
  
 // display result // display result
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 frm.result2.value = result2 frm.result2.value = result2
 frm.result3.value = result3 frm.result3.value = result3
 +//frm.result4.value = result4
  
 +// inner perimeter
 +pitch_max = pi * d / N
 +
 +if (c > pitch_max  ) {
 +result3 = "N or c too big!"
 +frm.result3.value = result3
 } }
 +
 +} // end function?
  
 //--> //-->
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 <br><br> <br><br>
  
-relative permeability of the core <i>μ<sub>r</sub></i> = <input type="text" value="1" name="ur" size="10" maxlength="10" onChange="calculate_function2(this.form)"> +relative permeability of the core <i>μ<sub>r</sub></i> = <input type="text" value="1000" name="ur" size="10" maxlength="10" onChange="calculate_function2(this.form)"> 
 <br><br> <br><br>
  
 <br> <br>
    
-(optional) round wire diameter <i>c</i> = <input type="text" value="0" name="c" size="10" maxlength="10" onChange="calculate_function2(this.form)"> +(optional) round wire diameter <i>c</i> = <input type="text" value="1" name="c" size="10" maxlength="10" onChange="calculate_function2(this.form)"> 
           <SELECT name="c_unit" onChange="calculate_function2(this.form)">           <SELECT name="c_unit" onChange="calculate_function2(this.form)">
            <OPTION value="1">(m)</OPTION>            <OPTION value="1">(m)</OPTION>
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         </SELECT> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br><br>         </SELECT> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br><br>
  
 +<!-- 
 (optional) mean turn pitch <i>p</i> = <input type="text" value="0" name="pitch" size="10" maxlength="10" onChange="calculate_function2(this.form)">  (optional) mean turn pitch <i>p</i> = <input type="text" value="0" name="pitch" size="10" maxlength="10" onChange="calculate_function2(this.form)"> 
           <SELECT name="pitch_unit" onChange="calculate_function2(this.form)">           <SELECT name="pitch_unit" onChange="calculate_function2(this.form)">
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            <OPTION value="1e-3" selected>(mm)</OPTION>            <OPTION value="1e-3" selected>(mm)</OPTION>
         </SELECT> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br><br>         </SELECT> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br><br>
 +-->
  
 <input type="button" name="Button" value="== Calculate ==" onClick="calculate_function2(this.form)"> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  <input type="button" name="Button" value="== Calculate ==" onClick="calculate_function2(this.form)"> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
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            <OPTION value="1e3" selected="selected">(mH)</OPTION>            <OPTION value="1e3" selected="selected">(mH)</OPTION>
            <OPTION value="1e6">(μH)</OPTION>            <OPTION value="1e6">(μH)</OPTION>
-        </SELECT> [1] eq. (2), simplified (for d/D > 0.7)<br><br>+        </SELECT> [2] eq. (2) <br><br> 
 + 
 + 
 +<hr> 
                  
-<b><i>L<sub>3</sub></i></b> = <input type="text" name="result3" size="10" maxlength="10"> +<b><i>L<sub>3</sub></i></b> = <input type="text" name="result3" size="15" maxlength="10"> 
           <SELECT name="result3_unit" onChange="calculate_function2(this.form)">           <SELECT name="result3_unit" onChange="calculate_function2(this.form)">
            <OPTION value="1">(H)</OPTION>            <OPTION value="1">(H)</OPTION>
            <OPTION value="1e3" selected="selected">(mH)</OPTION>            <OPTION value="1e3" selected="selected">(mH)</OPTION>
            <OPTION value="1e6">(μH)</OPTION>            <OPTION value="1e6">(μH)</OPTION>
-        </SELECT> [2] eq. (3), no magnetic core, μ<sub>r</sub> = 1<br><br>      +        </SELECT> [3] eq. (3), no magnetic core, μ<sub>r</sub> = 1<br><br>      
  
 </form> </form>
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 <!-- ==================== calculator ends here ============== --> <!-- ==================== calculator ends here ============== -->
 </HTML> </HTML>
- 
- 
  
 === Equations === === Equations ===
-^   Approximate inductance of Helmholtz coil (two halves connected in series)  ^^^+^   Approximate inductance of a toroidal coil with a magnetic core  ^^^
 ^ //Source: [1] [[https://isbnsearch.org/isbn/9780470461884|Clayton R. Paul. Inductance: Loop and Partial, Wiley-IEEE Press, 2009, New Jersey, ISBN 9780470461884]]//  ^^^ ^ //Source: [1] [[https://isbnsearch.org/isbn/9780470461884|Clayton R. Paul. Inductance: Loop and Partial, Wiley-IEEE Press, 2009, New Jersey, ISBN 9780470461884]]//  ^^^
 |  **(1)** \\ //[1], eq. (4.39), p. 138//  |  $$ L = \frac{μ_r · μ_0 · h · N^2 }{2 · π} · ln \left( \frac{D}{d} \right)  $$  |  (H)  | |  **(1)** \\ //[1], eq. (4.39), p. 138//  |  $$ L = \frac{μ_r · μ_0 · h · N^2 }{2 · π} · ln \left( \frac{D}{d} \right)  $$  |  (H)  |
-|  **(2)** \\ //[1], eq. (4.40b), p. 139 \\ (valid only for d/D > 0.7 with 20 % error) //  |  $$ L ≈ \frac{μ_r · μ_0 · A · N^2 }{ π · d }   $$  |  (H)  | +| where: $μ_r$ - [[/relative permeability]] of the magnetic core (unitless), $μ_0$ - [[/permeability of vacuum]] (H/m), $h$ - thickness of the toroid (m), $N$ - number of turns in the coil (unitless), $D$ - outer diameter of the toroid (m), $d$ - inner diameter of the toroid (m)  ||| 
-| where: $μ_r$ - [[/relative permeability]] of the magnetic core (unitless), $μ_0$ - [[/permeability of vacuum]] (H/m), $h$ - thickness of the toroid (m), $N$ - number of turns in the coil (unitless), $D$ - outer diameter of the toroid (m), $d$ - inner diameter of the toroid (m), and $A = 0.5 · h · (D - d)$ - core cross-sectional area (m<sup>2</sup>)  ||| +^ //Source: [2] [[https://isbnsearch.org/isbn/9781118717790|Marian K. Kazimierczuk, High-Frequency Magnetic Components, Second edition, John Wiley & Sons, Chichester, 2014, ISBN 9781118717790]] //  ^^^ 
-^ //Source: [2] [[https://isbnsearch.org/isbn/0876645570|Frederick W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570]]//  ^^^ +|  **(2)** \\ //[2], eq. (1.330), p. 48//  |  $$ L = \frac{μ_r · μ_0 · N^2 · h · (D - d) }{π · (D + d)} = \frac{μ_r · μ_0 · A · N^2 }{ l_c }  $$  |  (H)  | 
-|  **(3)** \\ //[2], eq. (149), p. 170//  |  $$ L = \frac{ μ_0 · N }{2 · π}  \left( h_G · N · ln \left( \frac{D}{d} \right) - l_t · (G + H) \right) $$  |  (H)  | +| where other variables as above, and: $A = h · (D - d) / 2 $ - core cross-sectional area (m<sup>2</sup>), $ l_c =  π · (D + d) / 2 $ - core length (m) or [[/magnetic path length]]  ||| 
-| where other symbols as above, and: $G$ - first correction factor for  space between turns (unitless), $H$ - second correction factor for space between turns (unitless), $c$ - round wire diameter (m), $p$ - mean pitch between wire centres (m) with the turns assumed to be distributed uniformly in a single layer  |||+^ //Source: [3] [[https://isbnsearch.org/isbn/0876645570|Frederick W. Grover, Inductance Calculations: Working Formulas and Tables, ISA, New York, 1982, ISBN 0876645570]]//  ^^^ 
 +|  **(3)** \\ //[3], eq. (149), p. 170// \\ (valid only for a uniform single layer of turns)  |  $$ L = \frac{ μ_0 · N }{2 · π}  \left( h_G · N · ln \left( \frac{D}{d} \right) - l_t · (G + H) \right) $$  |  (H)  | 
 +| where other symbols as above, and: $G$ - first correction factor for  space between turns (unitless), $H$ - second correction factor for space between turns (unitless), $c$ - round wire diameter (m), $p$ - mean pitch between wire centres (m) with the turns assumed to be distributed uniformly in a single layer, and:  |||
 |  //toroid thickness measured between wire centres in the axial direction for a single-layer coils//  |  $$ h_G = h + c $$ |  (m)  | |  //toroid thickness measured between wire centres in the axial direction for a single-layer coils//  |  $$ h_G = h + c $$ |  (m)  |
-|  //turn length//  |  $$ l_t = D - d + 2 · h_g  $$ |  (m)  |+|  //turn length//  |  $$ l_t = D - d + 2 · h_G  $$ |  (m)  |
 |  //(first correction factor) \\ [2], Table 38, p. 148, \\ exact function//  |  $$ G = \frac{5}{4} - ln \left( 2 · \frac{p}{c} \right) $$ |  (unitless)  | |  //(first correction factor) \\ [2], Table 38, p. 148, \\ exact function//  |  $$ G = \frac{5}{4} - ln \left( 2 · \frac{p}{c} \right) $$ |  (unitless)  |
-|  //(second correction factor) \\ [3], Table 39, p. 150, \\ approximation by S. Zurek \\ (comparison available [[calculator/toroidal_coil_inductance/Grover Table 39 approximation|here]])//  |  $$ H = 0.33790 - 0.43478 · N^{-0.8} + \frac{0.096876}{N^2} $$ |  (unitless)  | +|  //(second correction factor) \\ [3], Table 39, p. 150, \\ approximation by S. Zurek (comparison available [[calculator/toroidal_coil_inductance/Grover Table 39 approximation|here]])//  |  $$ H = 0.33790 - 0.43478 · N^{-0.8} + \frac{0.096876}{N^2} $$ |  (unitless)  | 
-^ //Source: [3] [[https://isbnsearch.org/isbn/9781118717790|Marian K. Kazimierczuk, High-Frequency Magnetic Components, Second edition, John Wiley & Sons, Chichester, 2014, ISBN 9781118717790]] //  ^^^ +
-|  **(4)** \\ //[3], eq. (1.330), p. 48//  |  $$ L = \frac{μ_r · μ_0 · N^2 · h · (D - d) }{π · (D + d)} = \frac{μ_r · μ_0 · A · N^2 }{ l_c }  $$  |  (H)  |+
  
 <box 100% #efffef>↑</box> <box 100% #efffef>↑</box>
calculator/toroidal_coil_inductance.1740523525.txt.gz · Last modified: 2025/02/25 23:45 by stan_zurek
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