Single-layer Coils
(Joseph-Henri Lévy)
Determining the inductance of a coil can be necessary for
the restoration of vintage radios or for understanding their design.
There are no analytical formulas giving the exact value of the inductance.
That is why at the beginning of the 20th century, the first tools for
approximate calculations based on tables or empirical formulas appeared
and gave satisfactory results for common needs.
The only formula for calculating the inductance of a coil, in analytical form, applies to a solenoid of « quasi-infinite » length \(l\), with diameter \(D\) and linear turn density \(n\) (see Eq.1). It can take the form: $$L(µH)=\frac{ (\pi n D)^2 }{1000} l \quad \quad \quad (1)$$ The turns can be contiguous or not. \(D\) and \(n\) are in CGS units, therefore in centimeters. Relation (1) gives \(L\) in micro-Henrys (µH).
The only formula for calculating the inductance of a coil, in analytical form, applies to a solenoid of « quasi-infinite » length \(l\), with diameter \(D\) and linear turn density \(n\) (see Eq.1). It can take the form: $$L(µH)=\frac{ (\pi n D)^2 }{1000} l \quad \quad \quad (1)$$ The turns can be contiguous or not. \(D\) and \(n\) are in CGS units, therefore in centimeters. Relation (1) gives \(L\) in micro-Henrys (µH).
a tabular method - Nagaoka's formula
fig.1 Hontaro Nagaoka
In the case of a finite length,
a few millimeters or centimeters for vintage radios, approximate calculation formulas can
take the form:
$$L(µH)=K \frac{(\pi n D)^2 }{1000} l \quad \quad \quad \quad (2)$$
The parameter \(K\) depends on the «geometry» of the coil expressed by the ratio \(D/l\).
The Japanese scientist Hontaro Nagaoka published at the beginning of the 20th century a table allowing
to determine \(K\) as a function of the ratio \(D/l\).
Another more practical formula for use and of the same precision uses a constant \(K1\) derived from \(K\). It depends in the same way on the ratio \(D/l\). $$L(µH)=K_1 \frac{N^2 D}{100} \quad \quad \quad \quad (3)$$ A calculation example here:
Another more practical formula for use and of the same precision uses a constant \(K1\) derived from \(K\). It depends in the same way on the ratio \(D/l\). $$L(µH)=K_1 \frac{N^2 D}{100} \quad \quad \quad \quad (3)$$ A calculation example here:
fig.2 Coil
Example:
Let's calculate the inductance L of a coil of 100 contiguous turns wound on a former of
0.67945 x 2.6 x 2.62/100
i.e., L ≈ 177 µH.
This value and the dimensions of the coil correspond to the tuning coils found in the medium wave (MW) range.
- diameter D = 2.6 cm
- length l = 2.6 cm
0.67945 x 2.6 x 2.62/100
i.e., L ≈ 177 µH.
This value and the dimensions of the coil correspond to the tuning coils found in the medium wave (MW) range.
Single-layer Coils
(Joseph-Henri Lévy)
Determining the inductance of a coil can be necessary for
the restoration of vintage radios or for understanding their design.
There are no analytical formulas giving the exact value of the inductance.
That is why at the beginning of the 20th century, the first tools for
approximate calculations based on tables or empirical formulas appeared
and gave satisfactory results for common needs.
The only formula for calculating the inductance of a coil, in analytical form, applies to a solenoid of « quasi-infinite » length \(l\), with diameter \(D\) and linear turn density \(n\) (see Eq.2). It can take the form: $$L(µH)=\frac{(\pi n D)^2}{1000} l \quad \quad \quad (1)$$ The turns can be contiguous or not; \(D\) and \(n\) are in CGS units, therefore in centimeters. Relation (1) gives \(L\) in micro-Henrys (µH).
The only formula for calculating the inductance of a coil, in analytical form, applies to a solenoid of « quasi-infinite » length \(l\), with diameter \(D\) and linear turn density \(n\) (see Eq.2). It can take the form: $$L(µH)=\frac{(\pi n D)^2}{1000} l \quad \quad \quad (1)$$ The turns can be contiguous or not; \(D\) and \(n\) are in CGS units, therefore in centimeters. Relation (1) gives \(L\) in micro-Henrys (µH).
a tabular method - Nagaoka's formula
fig.3 Hontaro Nagaoka
In the case of a finite length,
a few millimeters or centimeters for vintage radios, approximate calculation formulas can
take the form:
$$L(µH)=K \frac{(\pi n D)^2}{1000} l \quad \quad \quad \quad (2)$$
The parameter \(K\) depends on the «geometry» of the coil expressed by the
ratio \(D/l\). The Japanese scientist Hontaro Nagaoka published at the beginning of the
20th century a table allowing to determine \(K\) as a function of the ratio \(D/l\).
Another more practical formula for use and of the same precision uses a constant \(K1\) derived from \(K\). It depends in the same way on the ratio \(D/l\). $$L(µH)=K_1 \frac{N^2 D}{100} \quad \quad \quad \quad (3)$$ A calculation example here:
Another more practical formula for use and of the same precision uses a constant \(K1\) derived from \(K\). It depends in the same way on the ratio \(D/l\). $$L(µH)=K_1 \frac{N^2 D}{100} \quad \quad \quad \quad (3)$$ A calculation example here:
fig.4 Coil
Example:
Let's calculate the inductance L of a coil of 100 contiguous turns wound on a former of
0.67945 x 2.6 x 2.62/100
i.e., L ≈ 177 µH.
This value and the dimensions of the coil correspond to the tuning coils found in the medium wave (MW) range.
- diameter D = 2.6 cm
- length l = 2.6 cm
0.67945 x 2.6 x 2.62/100
i.e., L ≈ 177 µH.
This value and the dimensions of the coil correspond to the tuning coils found in the medium wave (MW) range.
For information \(K_1\) is deduced from \(K\) by the relation:
\( \displaystyle K_1 = \frac{\pi^2}{10}(\frac{D}{l}) K_{(D/l)}\) with \(D\) and \(l\) in centimeters.
| D/l | K2 |
|---|---|
| 0.00 | 7.11200 |
| 0.01 | 69.00800 |
| 0.02 | 60.29900 |
| 0.03 | 55.20600 |
| 0.04 | 51.59400 |
| 0.05 | 48.79300 |
| 0.06 | 46.50700 |
| 0.07 | 44.57400 |
| 0.08 | 42.90200 |
| 0.09 | 41.42800 |
| 0.10 | 40.11100 |
| 0.11 | 38.92000 |
| 0.12 | 37.83500 |
| 0.13 | 36.83800 |
| 0.14 | 35.91600 |
| 0.15 | 35.05800 |
| 0.16 | 34.25800 |
| 0.17 | 33.50700 |
| 0.18 | 32.80000 |
| 0.19 | 32.13200 |
| 0.20 | 31.50000 |
| 0.21 | 30.90000 |
| 0.22 | 30.32900 |
| 0.23 | 29.78500 |
| 0.24 | 29.26500 |
| D/l | K2 |
|---|---|
| 0.25 | 28.76700 |
| 0.26 | 28.29000 |
| 0.27 | 27.83200 |
| 0.28 | 27.39200 |
| 0.29 | 26.96800 |
| 0.30 | 26.56000 |
| 0.31 | 26.16600 |
| 0.32 | 25.78600 |
| 0.33 | 25.41800 |
| 0.34 | 25.06300 |
| 0.35 | 24.71900 |
| 0.36 | 24.38600 |
| 0.37 | 24.06300 |
| 0.38 | 23.75000 |
| 0.39 | 23.44600 |
| 0.40 | 23.15000 |
| 0.41 | 22.86300 |
| 0.42 | 22.58400 |
| 0.43 | 22.31300 |
| 0.44 | 22.04900 |
| 0.45 | 21.79200 |
| 0.46 | 21.54100 |
| 0.47 | 21.29700 |
| 0.48 | 21.05900 |
| 0.49 | 20.82700 |
| D/l | K2 |
|---|---|
| 0.50 | 20.60100 |
| 0.51 | 20.38100 |
| 0.52 | 20.16500 |
| 0.53 | 19.95500 |
| 0.54 | 19.75000 |
| 0.55 | 19.55000 |
| 0.56 | 19.35400 |
| 0.57 | 19.16200 |
| 0.58 | 18.97600 |
| 0.59 | 18.79300 |
| 0.60 | 18.61400 |
| 0.61 | 18.44000 |
| 0.62 | 18.26900 |
| 0.63 | 18.10200 |
| 0.64 | 17.93900 |
| 0.65 | 17.77900 |
| 0.66 | 17.62300 |
| 0.67 | 17.47000 |
| 0.68 | 17.32100 |
| 0.69 | 17.17400 |
| 0.70 | 17.03100 |
| 0.71 | 16.89100 |
| 0.72 | 16.75400 |
| 0.73 | 16.62000 |
| 0.74 | 16.48900 |
| D/l | K2 |
|---|---|
| 0.75 | 16.36000 |
| 0.76 | 16.23500 |
| 0.77 | 16.11200 |
| 0.78 | 15.99200 |
| 0.79 | 15.87400 |
| 0.80 | 15.75900 |
| 0.81 | 15.64600 |
| 0.82 | 15.53600 |
| 0.83 | 15.42800 |
| 0.84 | 15.32300 |
| 0.85 | 15.22000 |
| 0.86 | 15.12000 |
| 0.87 | 15.02100 |
| 0.88 | 14.92500 |
| 0.89 | 14.83100 |
| 0.90 | 14.74000 |
| 0.91 | 14.65000 |
| 0.92 | 14.56300 |
| 0.93 | 14.47800 |
| 0.94 | 14.39400 |
| 0.95 | 14.31300 |
| 0.96 | 14.23400 |
| 0.97 | 14.15800 |
| 0.98 | 14.08300 |
| 0.99 | 14.01000 |
Abacus
For those who like nomograms, this graph gives \(K\) as a function of the ratio \(D/l\).
For those who don't like tabular methods.
$$K = \displaystyle \frac{100}{\displaystyle \pi^2 (4 \frac {D}{l}+11)} \quad \quad (4)$$
fig.5 - Approximate calculation of K
fig.6 Acceptable error in K calculation
For those who do not like this tabular approach, it is possible to approximate K by the relation in fig.3
provided that an error margin evaluated in fig.4 is accepted.
For example, for a ratio D/l = 1, we find K = 0.68.
For example, for a ratio D/l = 1, we find K = 0.68.
An empirical formula for a compact coil (multilayer)
fig.7 Multilayer Coil
An approximate calculation formula for a compact or multilayer coil.
$$ \left\{\begin{array}{l} L_{µH}\ :\ \displaystyle \frac{10\pi N^2R_1^2 }{6R_1+9l+10(R_2-R_1)} \\ L\ :\ inductance (H)\\ R\ : \ Mandrel radius (m) \\ N\ :\ Total number of turns\\ R_2\ :\ Outer radius of the winding (m)\\ l\ :\ Winding length (m)\\ R_2-R_1\ :\ is the thickness of the winding (m) \end{array}\right. $$
$$ \left\{\begin{array}{l} L_{µH}\ :\ \displaystyle \frac{10\pi N^2R_1^2 }{6R_1+9l+10(R_2-R_1)} \\ L\ :\ inductance (H)\\ R\ : \ Mandrel radius (m) \\ N\ :\ Total number of turns\\ R_2\ :\ Outer radius of the winding (m)\\ l\ :\ Winding length (m)\\ R_2-R_1\ :\ is the thickness of the winding (m) \end{array}\right. $$
An empirical formula for a flat spiral coil
fig.8 Multilayer Coil
$$
\left\{\begin{array}{l}
L(µH)=K2_{(D/l)} \displaystyle \frac {N^2 D}{2000} \\
D =\displaystyle \frac {D_2 + D_1}{2} \\
C=\displaystyle \frac {D_2 - D_1}{2} \\
N \quad turns
\end{array}\right.
$$
Example:
Let's calculate the inductance L of a flat coil of 100 turns with dimensions:
The inductance L will be 25.418 x 1002 x 3/2000 = ≈ 381 µH
- outer diameter D2 = 4 cm
- inner diameter D1 = 2 cm
The inductance L will be 25.418 x 1002 x 3/2000 = ≈ 381 µH
| D/l | K2 |
|---|---|
| 0.00 | 7.11200 |
| 0.01 | 69.00800 |
| 0.02 | 60.29900 |
| 0.03 | 55.20600 |
| 0.04 | 51.59400 |
| 0.05 | 48.79300 |
| 0.06 | 46.50700 |
| 0.07 | 44.57400 |
| 0.08 | 42.90200 |
| 0.09 | 41.42800 |
| 0.10 | 40.11100 |
| 0.11 | 38.92000 |
| 0.12 | 37.83500 |
| 0.13 | 36.83800 |
| 0.14 | 35.91600 |
| 0.15 | 35.05800 |
| 0.16 | 34.25800 |
| 0.17 | 33.50700 |
| 0.18 | 32.80000 |
| 0.19 | 32.13200 |
| 0.20 | 31.50000 |
| 0.21 | 30.90000 |
| 0.22 | 30.32900 |
| 0.23 | 29.78500 |
| 0.24 | 29.26500 |
| D/l | K2 |
|---|---|
| 0.25 | 28.76700 |
| 0.26 | 28.29000 |
| 0.27 | 27.83200 |
| 0.28 | 27.39200 |
| 0.29 | 26.96800 |
| 0.30 | 26.56000 |
| 0.31 | 26.16600 |
| 0.32 | 25.78600 |
| 0.33 | 25.41800 |
| 0.34 | 25.06300 |
| 0.35 | 24.71900 |
| 0.36 | 24.38600 |
| 0.37 | 24.06300 |
| 0.38 | 23.75000 |
| 0.39 | 23.44600 |
| 0.40 | 23.15000 |
| 0.41 | 22.86300 |
| 0.42 | 22.58400 |
| 0.43 | 22.31300 |
| 0.44 | 22.04900 |
| 0.45 | 21.79200 |
| 0.46 | 21.54100 |
| 0.47 | 21.29700 |
| 0.48 | 21.05900 |
| 0.49 | 20.82700 |
| D/l | K2 |
|---|---|
| 0.50 | 20.60100 |
| 0.51 | 20.38100 |
| 0.52 | 20.16500 |
| 0.53 | 19.95500 |
| 0.54 | 19.75000 |
| 0.55 | 19.55000 |
| 0.56 | 19.35400 |
| 0.57 | 19.16200 |
| 0.58 | 18.97600 |
| 0.59 | 18.79300 |
| 0.60 | 18.61400 |
| 0.61 | 18.44000 |
| 0.62 | 18.26900 |
| 0.63 | 18.10200 |
| 0.64 | 17.93900 |
| 0.65 | 17.77900 |
| 0.66 | 17.62300 |
| 0.67 | 17.47000 |
| 0.68 | 17.32100 |
| 0.69 | 17.17400 |
| 0.70 | 17.03100 |
| 0.71 | 16.89100 |
| 0.72 | 16.75400 |
| 0.73 | 16.62000 |
| 0.74 | 16.48900 |
| D/l | K2 |
|---|---|
| 0.75 | 16.36000 |
| 0.76 | 16.23500 |
| 0.77 | 16.11200 |
| 0.78 | 15.99200 |
| 0.79 | 15.87400 |
| 0.80 | 15.75900 |
| 0.81 | 15.64600 |
| 0.82 | 15.53600 |
| 0.83 | 15.42800 |
| 0.84 | 15.32300 |
| 0.85 | 15.22000 |
| 0.86 | 15.12000 |
| 0.87 | 15.02100 |
| 0.88 | 14.92500 |
| 0.89 | 14.83100 |
| 0.90 | 14.74000 |
| 0.91 | 14.65000 |
| 0.92 | 14.56300 |
| 0.93 | 14.47800 |
| 0.94 | 14.39400 |
| 0.95 | 14.31300 |
| 0.96 | 14.23400 |
| 0.97 | 14.15800 |
| 0.98 | 14.08300 |
| 0.99 | 14.01000 |
Sources and references
[1] Frederick W. GROVER, "Inductance calculation", Dover Publications, Inc., New York 1946, 2001
[2] Paul BERCHE, "Pratique et théorie de la TSF", Librairie de la Radio, Paris, 1937, revue par Roger RAFFIN, 1958.
[4] F. E. TERMAN, "Radio Engineer's Handbook", McGraw-Hill, New York, 1943.
