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HOW TO CALCULATE LARGE CAPACITIVE
AND INDUCTIVE DEVICES


Sometime a radioamateur is also home-brewer, usually it's a question of antennas, in some case having strange forms, so requiring large loading coils, capacitive loads, as well as bifilar transmission lines and so forth. I think a summary of my considerations and calculuses, carried out to realize these devices, as used in my antennas, may be helpful. The equations used are almost all classic, only a few are written by me on account of my experience on this matter.
All dimensions are in centimeters, capacitance in pF and inductance in µH; consequently e0 is in pF/cm and µ0 in µH/cm.

CALCULUS OF CAPACITANCE

List of the main symbols used:

π = 3.1415927
C,Cs = capacitance in pF
e0 = absolute dielectric constant 8.854 10-2 pF/cm
er = relative dielectric constant (polyethylene = 2.3, paper = 3.5, resin = 4 5, porcelain = 4 7, bakelite = 5.5, = mica 6.5)
S = surface of one plate
n = number of plates
s = separation between plates
d = diameter of the sphere - of the disc - of the cylinder
r = radius of the sphere - of the disc - of the cylinder
l = lenght
h = height of a plate in respect to a conductive plane (soil)

We start wth the general equation for the calculus of a capacitor:

(1)   C =  e0 er S  (n−1)
          s          

This equation, to be valid, must be applied to a capacitor having a very small separation between plates, compared with their surface, otherwise an error appears, having disregarded the capacitance between the external plates and the space. This error in some cases may be noticeable.

To develop this argument we will examine a special capacitor, more theoretical than practical, that avoid the problem. This is the "spherical capacitor", realized with two conductive concentric spheres, the internal one connected to the circuit by means of a small wire that comes out through a small hole of the external sphere. It is evident that, if a voltage is applied to the internal sphere in respect to the external one, which we will keep at potential 0, the electric field is confined exclusively in the space between the spheres, distributed uniformely in each direction. Although it is not a normal capacitor, (1) is still applicable since there is not missing field but, the surface of the spheres being different, S must be considered as geometrical average between them, and s as the difference between their radius; if the space between the spheres is air or vacuum, (1) may be rewritten in this form:

(2)   C =  e0 4 π re ri
      re − ri      
   where     re = radius of the external sphere    ri = radius of the internal sphere

being 4 π re ri = spherical surface, geometrical average between the spheres. As the gap between the spheres rises, the capacitance diminishes slowly and, at last, becomes proportional only to the radius of the internal sphere. This feature appears clear, rewritting (2) this way:

(2 bis)   C =  e0 4 π ri
      1 − r'      
   where    r' = ri
re

Now suppose the radius of the external sphere equal to infinite, that is the same as suppose the internal sphere alone in the free space. In this case, being re = and r = ri from (2 bis) developing, we have:

(3)   Cs =  e0 4 π r  = 1.112 r

The equation (3) demonstrates that a metallic object isolated in the free space (in this case a sphere), shows in any case a capacitance Cs which is simply proportional to the radius. If the object is a disc or a cylinder with heigth equal to its radius, these values are given as a rule: for the disc Cs = 0.7 r, for the cylinder Cs = 1.6 r. Starting from equation (3) and accounting for the different conformation of the surface, it is possible to write two equations that yield approximately the same results; for the disc:

(4)   Cs =  e0 4 π r (2
π
) = 0.708 r

and for the cylinder:

(5)   Cs = e0 4 π r ( 2  
 π 
+  2  
 π 
) = 1.596 r

The equation (4) is applicable also in the case of polygonal surfaces, with approximation increasing according to the number of sides, and with r = (S / π). The espressions seen above remain quite valid even if the surfaces involved are reticular, provided that the meshes have dimensions very small in respect to the wavelength.
From what we have seen, is clear that a simple capacitor with two plates, besides the capacitance C resulting from (1) shows also some capacitance toward the space; this last, if the separation of the plates is large in respect to their surface, may reach the value Cs/2 as result from (4) applied to one plate, while C will be much smaller.
If opposite is the case, that is, if the separation of the plates is very small in respect to their surfaces, C will be much greater than Cs that diminishes, being referred to the external part of the plates only. In the intermediate case, whenever C and Cs may have a comparable value, becomes rather difficult to calculate exatly the real capacitance, being necessary a careful analysis of the field around the plates. In any case it is possible to obtain some approximation using the empirical equation below:

(6)   C =  e0S
π
[er π S
π
 + s(2 + 2
r'
)]1
s
   where     r' = 1 + (S
π
) 1
s

At this point may be useful to take into account the case of two spheres of the same size, located in the free space at a certain distance beetwen them; in this case the capacitance of a so made capacitor may be evaluated using the approximate equation below, being s the distance beetwen the centers of the spheres:

2 π e0 s rs
(7)   C =  
   where     r' =  
(s − r')s
d
 + [(s
d
)2− 1]

In the case of a sphere alone at a certain distance from a conductive plain (soil), being h the distance beetwen the center of the sphere and the conductive plane, obviously will be:

4 π e0 2 h r2 h
(8)   C =  
   where      r' =  
   setting s = 2 h
(2 h − r')2 h
d
 + [(2 h
d
)2− 1]

The case of two discs has been already resolved with (6), in the case of a disc located at some distance from a conductive plane (soil) the expression (6) may be used, providing that s = 2 h and doubling the result.
The calculus of the capacitance C = C+Cs of two cylinders with heigth equal to their radius located in free space at some distance, lined up on the same axis, may be approximated, being s = distance between their nearer bases, using the empirical equation shown below:

(9)   C =  e0 r[er π r + s(7 + 2
r'
)1
s
   where    r' = 1 + r
s

If there is only a cylinder located at some distance from a conductive plane (soil), with a base parallel to the plane, its capacitance C = C+Cs may be approximated, being h = distance from its base and the plane, using (9), providing that s = 2 h, and doubling the result.

CALCULUS OF INDUCTANCE

Here are taken in account only single layer coils, without maqnetic core, cylindrical, flat spiral and toroidal. All dimensions are in centimeters, capacitance in pF and inductance in µH; consequently e0 is in pF/cm and 0 in H/cm.
List of the main symbols used:

π = 3.1415927
C,Cp = capacitance in pF
L,La,Lc,Lo = inductance in µH
e0 = absolute dielectric constant 8.854 10-2 pF/cm
µ0 = absolute permeability 4 π 10-3 µH/cm
µr = relative permeability
rs = resistivity (copper = 1.8 10-6, alum.=2.9 10-6 ohm cm3)
D = diameter of the coil
R = radius of the coil
d = diameter of the wire
rc = radius of the wire
N = number of turns
p = pitch of winding
pe = external pitch of winding in a toroid
pi = internal pitch of winding in a toroid
l = length
cosh-1 = area of hyperbolic cosine, cosh-1 (x) = ln (x (x2 - 1) )
f = frequency in MHz

We start with the well known equation proposed by H.A.Wheeler (Wheeler 1928, p. 1399) for the calculus of a cylindrical, single layer coil:

(1)   L =  R2 N2
  9 R + 10 l  

This equation that requires dimensions in inches, because of its praticability, is fluently used but, being an empirical equation, there are some limits; its accuracy may be acceptable only if the length of the coil is greater than a third of its diameter, and if the pitch of winding is smaller than a sixth of its diameter.
To permit its use with dimensions in centimeters or millimeters, it has been modified as follows:

(1 bis)   L =  R2 N2
  Km ( 9 R + 10 l)  
   where    Km = 2.54 for dimensions in cm and 25.4 for dimensions in mm

The inductance of a single turn coil, if its diameter is greater than then times the diameter of the wire, may be calculated with the simplified equation:

(2)   L =  µ0 R ln1.0827 R
       r        

There is a method, more close to the reality, that avoids the limits of (1); unfortunately it is a bit complex, using some constants calculated some time ago by H.Nagaoka and by E.B.Rosa (Rosa, Grover 1912, p. 119 and 122).

 K =  D
 l 
ratio, from 0.01 to 100
 A =  d
 p 
ratio, from 0.005 to 1
 B =  number of turns from 1 to 1000

they are shown usually as a number of curves what makes tedious to take every time the proper value; nevertheless it's possible, accepting enough accuracy, the calculus of the constants using the approximating polynomials as below

K = 1 − .419 X + .126 X2 − .0171 X3   where    X =  D
 l 
  valid for  D
 l 
 from .01 to 2 

K = .9145 − .271 X + .04536 X2 − .00299 X3    where    X =  D
 l 
  valid for  D
 l 
 from 2 to 5 

K = .534 −.05347 X +.0023876 X2 −.000041287 X3 + 2.226 10−7 X4   where    X =  D
 l 
  valid for  D
 l 
 from 5 to 100 

A = ln(1.7d
 p 
)  valid for  d
 p 
 =.0051 

B =  (ln N).75
     6.5     
  valid for  N =15 

B = .336 (1  −  2.5
 N 
 +  3.8
 N2 
)  valid for  N = 51000 

The process starts from the calculus of an ideal coil, having a great length, made with a tape of infinitesimal thickness, whose turns are separated with an infinitesimal space, thus resembling a tube; in this case may be used the general equation:

(3)   Lo =  µ0 N2 π R2
          l          

being the length of an actual coil practically limited, it is necessary a correction of the inductance using the constant K as below:

(4)   Lc = Lo K

another correction is necessary, using the constants A and B because the coil is made using a round wire, maybe with spaced turns:

(5)   L = Lc − µ0 R N (A + B)

the resolutive expression, developing, therefore appears as follows:

(6)   L = µ0 R N [π R N K
       l       
 − (A + B)]

this process is quite valid also in the case of a single turn, making l = diameter of wire, A and B = 0; however it is preferable to use (2). For a better accuracy the length of the coil must be calculated as: l = (N − 0.5) p, the diameter of the coil must be measured between axes of wire.
Summing up, it is profitable to use (1) every time it is possible, using (6) when a dimension of the coil results out of the limits seen for (1).
There are other kinds of coils, the flat-spiral coil and the toroidal coil; as for the flat-spiral, may be used once more an equation proposed by H.A.Wheeler (Wheeler 1928, p. 1400), having about the same limits as (1 bis):

(7)   L =  R2 N2
  Km ( 8 R + 11 l)  
   where    Km = 2.54 for dimensions in cm and 25.4 for dimensions in mm

or, to avoid limits, the equation proposed by Rayleigh and Niven (Rayleigh 1881, p. 108; Rosa, Grover 1916, p. 116-117), if used for round wires as below:

(8)   Lo = µ0 R N2[ln8 R
   l   
 − 0.5 + l2
96 R2
(ln8 R
   l   
  +  43
12
)]

In both equations     R = mean radius of the spiral     l = radial extension of the winding = (N − 0.5) p

Expression (8) may be further on simplified without a significant loss in accuracy as below:

(9)   Lo = µ0 R N2(ln8 R
   l   
 − 0.5)

At last, for a toroidal coil, a general equation is shown below:

(10)   Lo = µ0 N2(RT − RT2 − R2)   where RT = mean radius of the toroid

The calculuses seen above give the value of low frequency inductance of a coil, but every coil has a self capacitance that must be considered if it operates on high frequency. This capacitance appears difficult to calculate, depending from many factors i.e. its dimensions, pitch of winding/diameter of wire ratio, type of insulation, type of support and so on. An equation proposed by A.J.Palermo (Palermo 1934) is shown below:

π D
(11)   Cp =  
   3.6 cosh-1( p
d
)

this equation may be rewritten as:

e0 π2 D
(12)   Cp =  
   cosh-1( p
d
)

In this form appears clear that the calculus is referred to a coil made with bare wire without support; the result, in many cases, appears quite far from reality. A better approximation may be obtained with (12), empirically modified as below:

e0 π2 D Ki ln(p
d
).34
(13)   Cp =  
   cosh-1( p
d
)

where Ki is a constant with values ranging between 1 and 1.6, as appears in the table below:

Orientation values of Ki
Type of winding Closed turns Spaced turns
Bare wire without support - 1
Bare wire wound on a tube - 1.05
Enamelled wire without support 1.1 1
Enamelled wire wound on a tube 1.2 1.1
Cotton ins. wire wound on a tube 1.4 1.2
Vinyl ins. wire wound on a tube 1.6 1.3


This equation can also be used for flat-spiral coils, taking D = mean diameter of the coil; for toroidal coils, a sufficient approximation may be obtained taking p = pe pi
Being used in the equation cosh-1, it is necessary, in case of closed turns, to calculate p taking in account the tickness of the insulation of the wire, also if it is only enamel, and taking d = diameter of the bare wire.

In the case of a single turn (a loop), the capacitance may be calculated with the equation as below:

(14)   Cp =  1.11 D π
      q'      
   where   q' = 4 π ln(π2 D
  2 d  
)

In any case, if a reliable result is requested, the only way is to measure directly this capacitance using a Grid-Dip or a HF generator and an oscilloscope. The coil may be excited by a loop connected to the generator and located far enough from the coil, to obtain a loose coupling; the oscilloscope must be connected to a turn of a thin insulated wire, wound on the center of the coil; the coil must be far away from metallic structures.
At first the natural resonance is measured and named f1, at this point a small capacitor Cn is connected to the coil, again the resonance is measured and named f2, the capacitance of the coil results:

(15)   Cp =  Cn
    q'    
   where   q' = (f1
 f2 
)2 −1

At last, a very important feature of a coil is its equivalent HF resistance that may be calculated as follows:

we start with the d.c resistance of the wire that is obviously:

(16)   Rcc =  rs l
  π (rc2

because of the skin-effect we have:

(17)   Raf =  rs f 
 5.033 π d
   where   l = π D N    length of the wire used for winding

the proximity-effect, as proposed by R.Mesny, will be calculated only if the p/d ratio is smaller than 9; if this is the case we have:

(18)   Kp = ( 1 + 0.95 d
   p    
)2 − 0.22    where   Kp = correction factor

(19)   Re = Raf  [ 1 + (Kp − 1) 1 − Rcc
   Raf    
]   where   Re = effective resistance of the wire

if the p/d ratio is = 9 or greater Re = Raf.

If the dimensions of the coil are noticeable, the radiation resistance must also be considered:

(20)   Rrad =  N2 D4 f4
  42 1012  
 + 73
q'
   where   q' = (15000
    f l    
)2

The part of the equation after the plus sign refers to the axial radiation of the coil.

The effect of the loss in the insulating elements of a coil may be approximately calculate using the empirical constants Kr1 and Kr2 (see the tables below), with the following equations:

(21)   Rp =  106
2 π f Cp (Kr1 + Kr2)

(22)   Rde =  Rp
   1 + q'    
   where   q' = (Rp
 2 π f L 
)2

Values of Kr1
Type of winding Closed turns Spaced turns Very spaced turns
Bare wire - -00001 .00001
Enamelled wire .008 .0025 .00001
Cotton insul. wire .0013 .0004 .00001
Vinyl insul. wire .0016 .0005 .00001




Values of Kr2
Type of winding Closed turns Spaced turns Very spaced turns
No support .00001 .00001 .00001
PVC sticks .0009 .0013 .0018
PVC tube .0008 .0012 .0016
Ceramic tube .00038 .0005 .00076
Bakelite tube .0017 .0025 .0034


The total equivalent HF resistance Req results therefore = Re + Rrad + Rde

the Q factor of a coil results:

(23)   Q =  2 π f L
   Req    

The behaviour of a coil with self capacitance must be verified if it works in a parallel resonance circuit at a frequency lower than its natural resonant frequency; if the operating frequency of the circuit not exceed 80 % of the natural resonant frequency, the apparent inductance and the apparent resistance of the coil may be calculated using the approximated equations (Terman 1951, p. 52) shown below:

(24)   La =  L
  1 − kn2   
   where   kn = operating frequency / natural resonance frequency

Remember that the coil's own resistance Re also undergoes an apparent increase due to the presence of the parasitic capacitance of the coil. The apparent resistance, in the conditions mentioned above, is given by:

(25)   Ra =  Rp
(1 − kn2)2 
   where   kn = operating frequency / natural resonance frequency

CALCULUS OF BIFILAR AND COAXIAL LINES

Here are taken in account bifilar and coaxial lines; as in the previous parts all dimensions are in centimeters, capacitance in pF and inductance in H; consequently e0 is in pF/cm and µ0 in µH/cm.

List of the main symbols used:

π = 3.1415927
C = capacitance in pF
L = indutctance in µH
Z0 = caracteristic impedance
e0 = absolute dielectric constant 8.854 10-2 pF/cm
er = relative dielectric constant (polyethylene = 2.3, paper = 3.5, resin = 4 5, porcelain = 4 7, bakelite = 5.5, mica = 6.5)
µ0 = absolute permeability 4 π 10-3 µH/cm
s = distance between centers of parallel cylinders
d = diameter of the cylinder (wire)
r = radius of the cylinder (wire)
re = internal radius of the external cylinder
ri = external radius of the internal cylinder
l = length
h = heigth between the center of a cylinder and a conductive plane
cosh-1 = area of hyperbolic cosine, cosh-1 (x) = ln (x (x2 - 1) )

As it is well known, a very important parameter of a line is its caracteristic impedance; for coaxial lines the following equation is used:

(1)   Z0 =  (138 log re
 ri 
) 1
er   
 = (60 ln re
 ri 
) 1
er   

and for bifilar lines

(2)   Z0 =  (276 log s
 r 
) 1
er   
 = (120 ln s
 r 
) 1
er   

the equation (2)appears incorrect when the s/r ratio is very small; in fact, for s / r = 2, this means that the wires are in contact, the result is Z0 = 83 / er, while it would be Z0 = 0. Some authors (Jasik 1961, p. 30-3), to avoid this anomaly, adopt the following equation:

(3)   Z0 =  (120 cosh-1 s
 d 
) 1
er   

This last, for s / d relatively great ratios coincides with (2), but tends to zero for s / d = 1 (that is for conductors in contact), that's why it appears more close to the reality.
Later on, only air-insulated bifilar lines are taken in account, that's why the term / er will be omitted.
In a transmission line there are at the same time self capacitance and self inductance, we start with the calculus of the capacitance; the proposed equations are valid for lines having length very greater in respect to the transversal dimensions, so minimizing the error due to the different conformation of the electric field at the extremities. Here are the equations:

coaxial line

(4)   C =  2 π e0 er l
 ln(r') 
   where   r' = re
 ri 

bifilar line

(5)   C =  π e0 l
 ln(r') 
   where   r' = s
 r 

when the distance between the conductors in a bifilar line becomes of the same order of its length, that is when s > 0.733 l, it is necessary to modify the equation as follows:

(6)   C =  π e0 l
 ln(r') 
   where   r' = 0.733 l
     r     

sometime is necessary to calculate the capacitance of one of these conductors in respect to a conductive plane (soil); the equation in this case is:

(7)   C =  2 π e0 l
 ln(r') 
   where   r' = 2 h
  r  

also in this case, when 2 h > 0.733 l, it is necessary to modify the equation as shown:

(8)   C =  2 π e0 l
 ln(r') 
   where   r' = 0.733 l
     r     

this equation, being not more dependent on the distance, calculates simply the capacitance of a single conductor in the free space.
It is to be noted that (6) and (8) are proposed by some authors in a form sligthly different; in fact ln(0.733 l / r) becomes ln(π l / 4 r). The result is not much different, except in the cases where the l / r ratio results very small.
Expressions (5) and (7) are valid only if the distance or the heigth from the conductive plane are much greater than the radius of the conductor; to get in any case a valid result it is necessary to modify the equations as follows:

bifilar line

(9)   C =  π e0 l
 cosh-1(q')
   where   q' = s
 d 

conductor vs. conductive plane

(10)   C =  2 π e0 l
 cosh-1(q') 
   where   q' = 2 h
  d  

the necessity of these last modifications derives from what has been said about (2) and (3).

Now, as done about the self capacitance, we attempt to calculate the self inductance of a transmission line; also in this case, the equations are valid if the length of the line is much greater than its transversal dimensions:

coaxial line

(11)   L =  (µ0 l lnre
 ri 
)1
 2 π  

bifilar line

(12)   L =  (µ0 l lns
 r 
)1
  π  

single wire (cylinder)

(13)   L =  (µ0 l ln0.736 l
     r     
)1
 2 π  

Also here, to have a valid result whatever may be the distance between the wires of the line, (12) must be modified as follows:

(14)   L =  (µ0 l cosh-1s
 d 
)1
  π  

The use of cosh-1 depends, as stated before, from the fact that we have adopted (3) instead of (2) for the calculus of the caracteristic impedance of a bifilar line; in fact, as is well known, the caracteristic impedance of a transmission line, supposed without losses, is given also from: (L / C), where L e C are the values of inductance and capacitance for unity of length. Developing, from (9) and from (14)

(15)   Z0 =   [µ0(cosh-1s
 d 
)2  ]1
π2 e0

               =  [ (µ0
 e0 
) 1
π
] (cosh-1s
 d 
)

              but    [ (µ0
 e0 
) 1
π
]  = 120    that's why     Z0  =  120 cosh-1 s
 d 

As stated before, the term er has been omitted in the calculus of the capacitance, therefore the term er does not appear in the equation Z0.
In support of the above, we have the equation by A.J. Palermo, that uses cosh-1 to find the capacitance between two parallel wires, in the specific case two adjacent turns of a coil.

REFERENCES

Jasik, Henry, Editor, Antenna Engineering Handbook. New York: Mc Graw-Hill, 1961

Palermo, A. J., "Distributed Capacity of Single-Layer Coils" in Proceedings of the Institute of Radio Engineers, vol. 22, n. 7 (1934), pp. 897-905
<https://worldradiohistory.com/Archive-IRE/30s/IRE-1934-07.pdf>

Rayleigh, John William Strutt, "On the Determination of the Ohm in Absolute Measure" in Proceedings of the Royal Society of London, vol. 32 (1881), pp.104-141
<https://royalsocietypublishing.org/doi/pdf/10.1098/rspl.1881.0015>

Rosa, Edward B., Grover, Frederick W., "Formulas and Tables for the Calculation of Mutual and Self-inductance" in Scientific Papers of the Bureau of Standards, 3. ed, n. 169 (1916)
<https://ia801308.us.archive.org/35/items/formulastablesfo813rosa/formulastablesfo813rosa.pdf>

Terman, Frederick E,, Sc.D, Radio Engineering. 3. ed. New York: Mc Graw-Hill, 1951

Terman, Frederick E., Sc.D, Radio Engineers Handbook. New York: Mc Graw-Hill, 1943
<https://electrooptical.net/static/oldsite/OldBooks/Terman-RadioEngineersHandbook_1943.pdf>

Wheeler, Harold A., "Simple Inductance Formulas for Radio Coils", in Proceedings of the Institute of Radio Engineers, vol. 16, n. 10 (1928), pp. 1398-1400
<https://worldradiohistory.com/Archive-IRE/20s/IRE-1928-10.pdf>