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VK2ZRG > TECH 03.09.08 21:44l 115 Lines 5121 Bytes #999 (0) @ WW
BID : 2598_VK2ZRG
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Subj: Inductance calculations
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Sent: 080829/0428Z @:VK2WI.#SYD.NSW.AUS.OC #:1748 [Sydney] $:2598_VK2ZRG
From: VK2ZRG@VK2WI.#SYD.NSW.AUS.OC
To : TECH@WW
VK2ZRG/TPK 1.83d Msg #:2598 Date:29-08-08 Time:4:27Z
Hello techies,
I've been looking into the various inductance formulae in the past few
weeks and have found some interesting things. The formula most often found in
the manuals (are we still allowed to call them manuals? -personals I suppose
according to the feminists) is Wheeler's (1928) approximation for single
layer air cored solenoids. This is it, where length and diameter are in the
antiquated inch measurements (^2 means N squared and * means multiply):
Microhenrys = (Turns^2 * Diameter^2) / (18 * Diameter + 40 * Length)
Divide the inductance by 25.4 if coil dimensions are in millimetres.
There is also a later (1982) Wheeler's formula which is more accurate,
especially for coils with length/diameter < 0.4.
Another formula is known as the "current sheet" formula. It is based on
the inductance of a very thin, flat metal strip wound into a solenoid, with
a very small space separating the turns. This can be calculated to great
accuracy if the solenoid is very long (tending towards infinity) compared
to its diameter. The fundamental inductance formula is this, where A is coil
area in square centimetres (across the diameter), and the length is also in
centimetres:
Microhenrys = (4*Pi*Turns^2*A) / (Length*1E3)
Because practical coils have length/diameters of, say 0.2 to 10, a correction
factor must be applied according to the L/D ratio. This factor is called
Nagaoka's Constant (K) which is somewhat of a misnomer as the number isn't
constant...it varies with L/D.
The raw Current sheet formula is this (for coil dimensions in mm, and
using diameter instead of radius):
Microhenrys = (Pi^2 * Turns^2 * Diameter^2 / Length * K)*1E-4
Nagaoka's Constant may be calculated to an accuracy of 3 ppm by Lundin's
approximation.
There is a further correction factor (for pitch and number of turns) for
the current sheet formula which is deducted from the raw number.
The Raditron Designer's Handbook says that the current sheet formula is
accurate to one part in a 1000. Wheeler's 1928 formula differs by < 0.6%
compared to raw inductance value from the above formula, for coils with
length/diameter of 0.4 or more.
Most manuals say that the radius or diameter in these formulae is a mean
value measured to the centre of the wire and, as far as Wheeler's formula
is concerned, that you may use any size wire as long as the number of turns
will fit into the designated length.
The first interesting point comes from comparing Wheeler's 1928 formula
with the current sheet formula (with the pitch/turns correction factor).
Comparing inductance of some 10 turn coils (10 mm mean diameter by 10 mm
long), using different wire diameters and pitch. There is a 22% difference
between the coil with 1 mm wire and the one with 0.1 mm wire.
Wire dia mm Pitch Current Wheeler
sheet (1928)
1.00 1.000 0.6269 uH 0.6788 uH
0.50 2.000 0.6704 uH "
0.4375 2.286 0.6788 uH "
0.20 5.000 0.7280 uH "
0.10 10.000 0.7715 uH "
However if the inside diameter of the coil is held constant at 10 mm,
meaning that the mean coil diameter increases with wire diameter, then
there is less variation in inductance from the current sheet formula.
Wire dia mm Pitch Current Wheeler
sheet (1928)
1.00 1.000 0.7391 uH 0.7966 uH
0.50 2.000 0.7279 uH 0.7369 uH
0.20 5.000 0.7519 uH 0.7019 uH
0.10 10.000 0.7839 uH 0.6903 uH
One book that I have, "Principles of Radio" 6th ed by Henney and Richardson,
clearly shows the diameter of a coil using Wheeler's formula as being an
inside diameter measurement. I wonder what measurement Wheeler used in 1928.
Maybe someone who is adept at trawling the internet may find the answer.
The second interesting point comes from comparing inductance of a straight
wire and the inductance of a coil wound from this same length of wire. The
formula for inductance of a straight wire at low frequency, using millimetres,
is this: (-0.75 becomes -1.0 for high frequency)
Microhenrys = 0.0002*Length*(LN((4*Length)/Wire_diameter)-0.75)
For a wire 1 mm dia by 25 mm long the result is 0.0193 or 19.3 nanohenrys.
Wound into a two turn coil of 3.928 mm dia by 4 mm long it is 10.6 nanohenrys
by the current sheet formula and 10.5 nanohenrys by Wheeler's formula. A one
turn coil of 7.93 mm dia by 2 mm long has 12.0 nanohenrys by the current
sheet formula and 11.1 nanohenrys by Wheeler's formula.
There is a formula for single turn loops of round wire. This is it for
low frequency and dimensions mm:
Microhenrys = 0.0002*Pi*Diameter*(LN((8*Diameter)/Wire_diameter)-1.5)
For a 7.93 mm diameter loop of 1 mm wire the inductance is 13.2 nanohenrys
by this formula.
Of course the proper thing to do is to actually measure the inductance,
easier said than done.
73's from Ralph VK2ZRG.
P.S. To G8MNY, you may not use any this SB in your TECH bulls.
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