Tuning Coil Calculator

This calculator takes as input:

1. the maximum and minimum frequencies [f_max & f_min] of your desired tuning range,

2. the reactance [X_mid] you want the tuned circuit to have in order for it to match with whatever it is connected to,

3. the coil's diameter [d]. Make this as large as you can comfortably accom­mo­date while leaving adequate 'breathing' space around it. This will ensure that it has the highest possible Q [quality] factor.

Enter new values on the 4 input lines — those with the light brown background. Press the carraige-return key within one of these fields to calculate the parameters on the lines with the blue background. The default values shown initially are for a Long Wave Amateur Band coil.

The calculator displays the following 12 outputs:

1. mid-frequency of the tuning range [f_mid],
2. the coil's inductance [L],
3. the capacitance required for resonance at the mid-frequency [C_mid],
4. number of turns of wire the coil must have [n],
5. coil pitch [the distance between consecutive turns] [p],
6. total length of wire required to wind the coil [w],
7. minimum capacitance of the variable tuning capacitor [C_min],
8. maximum capacitance of the variable tuning capacitor [C_max],
9. reactance of tuning capacitor when tuned to minimum frequency [XC_min],
10. reactance of coil at minimum frequency [XL_min],
11. reactance of tuning capacitor when tuned to maximum frequency [XC_maxn],
12. reactance of coil at maximum frequency [XL_max].

The formulae used by the calculator are:

mid-frequency	fmid	= (fmax + fmin) ÷ 2;
radius of coil	r	= d ÷ 2
coil circumference	c	= d × π
angular frequency	ω	= 2π × fmid
inductance of coil	L	= Xmid ÷ ω
mid-frequency capacitance	Cmid	= 1 ÷ (ω × Xmid)
number of turns on coil	n	= √(736000L ÷ r)
pitch [dist between turns]	p	= √(r³ ÷ 184000L)
minimum capacitance	Cmin	= 1 / (4πfmax² × L)
maximum capacitance	Cmax	= 1 / (4πfmin² × L)
Reactance of Cmin at fmax	XCmax	= 1 / (2πfmax × Cmin)
Reactance of L at fmax	XLmax	= 2πfmax × L
Reactance of Cmax at fmin	XCmin	= 1 / (2πfmin × Cmax)
Reactance of L at fmin	XLmin	= 2πfmin × L

Dialectric Constants
Material	Min	Max
Air	1	1
Styrofoam	1·03	1·03
Dry Wood	1·4	2·9
Dry Paper	1·5	3

For best results, wind coils on cylinders made of a mat­er­ial of low dialectric constant. However, since the cylinders will be, for the most part, hollow, the material of which a coil's cylinder is made shouldn't influ­ence the coil's induc­t­ance significantly. For frequencies up to 1·5 MHz, wind coils with Litz Wire to minimize los­ses due to skin and proximity effects. Above 1.5 MHz, use en­amelled copper wire.

Derivations of The Above Formulae

L = 0·001 × n² × r² ÷ (228r + 254l)    l = length of coil winding.
But for best results, coil length = coil diameter, so
L = 0·001 × n² × r² ÷ (228r + 254 × 2 × r)
L = 0·001 × n² × r² ÷ (736 × r)
L = n² × r ÷ 736000

The pitch, Pich, of a coil is the distance between successive turns.
The number of turns, Trns, of a coil is its length, l, divided by its pitch, p.
But for best results, we decreed that l = 2 × r (its diameter), so
n = 2 × r ÷ p     n² = 4 × r² ÷ p²
Now substitute this in the equation for L above:

L = (4 × r² ÷ p²) × r ÷ 736000
L = 4 × r² × r ÷ (736000 × p²)
L = r³ ÷ (184000 × p²)
r³ = 184000P² × L
r = ³√(184000 × p² × L)

Now knowing the value of r, calculate the number of turns required:
L = n² × r ÷ 736000
n² = L × 736000 ÷ r
n = √(736000 × L ÷ r)

The amount of wire, w, required to wind the coil is the amount of wire, t, needed for one turn times the number of turns, n. If you unwind one turn, while maintaining the pitch, its length is the diagonal of a rectangle whose sides are the circumfer­ence, c, of the coil and the pitch, p, of the coil.

c = π × d     d = 2 × r
t² = c² + p²
t = √(c² + p²)
w = n × t
w = n × √(c² + p²)

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©13 September 2016 Robert John Morton