Coil dimensions

of a quarter wave oscillator



This calculates the possible dimensions: lengths, diameters, etc.,
given the desired self resonant frequency (fr) of the coil as input.

Assuming that the coil will be a quarter wave oscillator ( λ = 4 ℓw ),  the total length of the wire (ℓw) can be found by:

  c  =  λ fr    

  c  =  ( 4 w ) fr    

  w   =
    4 fr




    •  c   is the speed of light in air, ( 29,970,254,700 cm/s ) 
    •  λ   is the wavelength,  
    •  fr   is the self resonant frequency and,
    •  w   is the total length of the wire.

The coil used here for this calculation is solenoid shaped, and it is assumed to be made of copper.


The input for this calculation requires these properties:

    • resonant frequency desired, fr,
    • number turns per cm, (ρ turns), a density of turns,
    • the wire's gauge,
    • and then either the coils diameter or length in centimeters.

( units are in:   Hz, turns / cm, AWG, cm   respectively )

The output calculated properties of the coil will be:

    • the total length of wire required for the coil, w,
    • diameter of the coil, d,
    • length of the coil, l,
    • inductance. L,
    • approximate self capacitance, C,
    • mass of the coil, M,
    • and the number of turns, N.

( units are in:   cm, cm, cm, Henries, Farads, kg, and turns,  respectively )


   ρ turns    

The density of turns per centimeter will have a minimum value for each thickness of wire. For example, for a 24 gauge wire the minimum number of turns per centimeter is 18. But here we can allow that value to be less turns per centimeter if desired.



This tool allows us to try different resonant frequencies, diameters, and lengths, and then it solves for a variety of properties of the resulting coil.


   resonant frequency:   


frequency units:

           wire gauge:  


turns per centimeter (currently 18 turns/cm)    
Fill in an entry for either the diameter or the length of the coil.    
     diameter:       cm    
     length:       cm    

( If both entries: diameter and length are empty or zero then the diameter will be set to 4 cm, and the calculations will be made to find the length.

Or if both entries: diameter and length are filled in with a value, then only the diameter's value will be used to then calculate the length. )




The total length of wire for the coil, (ℓw) :

         =  0.0 cm


coil diameter  (d) :

4.000  cm


coil length  (l) :

0.000  cm




self resonant freq ( fr ):  0.00000   
full wavelength ( λ = 4 ℓw ):  0.00000  m  
mass of the coil  (M, kg):  0.00000  kg  
mass of the coil  (M, ounces):  0.00000  oz  
mass of the coil  (M, pounds):  0.00000  lb  
coil diameter  (d, inches) :  1.57  in  
coil length  (l, inches) :  0.00  in  
wire gauge   :    AWG  
density of turns turns):  18.0 turns / cm  
total number of turns (N):  0.0 turns  
wire radius (inches)  :  1.00000  in  
wire radius (meters)  :  1.00000  m  
total wire length (w, inches)  :  0.00  in  
total wire length (w, meters)  :  0.00  m  
inductance (L)   :   0.000    μH
self-capacitance (C)  :   0.000   pF

density of copper (ρ)  =  8940 kg / m3

  The mass (kg) of the coil is computed from:  
M  =  ρ  w   π  r 2


    •  M  = the mass of the coil (kilograms)
    •  ρ   = the density of copper (kg / m3)
    •  w   = total length of the wire for the coil 
    •  r    = radius of the copper wire (meters)

  Then the number of turns (N) is computed from:  
N   =  
     π 2  ρ  d  r 2


    •  N   = the number of turns of the coil 
    •  M   = the mass of the coil (kilograms)
    •  ρ   = the density of copper (kg / m3)
    •  d   = diameter of the coil (meters)
    •  r   = radius of the copper wire (meters)

  The relation above is derived from:  

      total length of the wire for the coil :    w = N C    

       ( length of the wire = turns, times, circumference of the coil )


      total length of the wire for the coil :   w = M / ( ρ A )

       ( length of the wire = mass of the wire, divided by:
                               density of copper times
                                       the cross-sectional area of the wire )


  The inductance, in μH,  is estimated by the following, with the length and diameter measured in inches. **  
                  Inductance ( μH )      
    d 2  N 2
L    =  
    18 d  +  40 ℓ
  ... and the self-capacitance is derived from the resonant frequency:  
f r  =  
    2 π L C
                  Capacitance ( pF )   
C  =  
    4  π 2  L  f 2

** The ARRL Handbook for Radio Communications 2011
section 2.8 Inductance and Inductors

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