Inductance Calculation

In these discussions of numerical methods for inductance calculation, it's very easy to lose touch with reality. Every so often, it's a good idea to look at a real world application, and compare the theoretical predictions with the actual numbers.

The Mystery Crystal Set, which was first described in the July 3rd, 1932 Brisbane [Australia] Sunday Mail (page 23), is unique in the way that its main coil is wound. Because of the periodic resurgence of the Mystery Set's popularity (and partly because of its 80th anniversary), it may be interesting to have a look at the coil and develop a way of analyzing its parameters. The Mystery Set circuit is reproduced below:

One of the more interesting characteristics of the Mystery Crystal Set is that there is no apparent signal path between the antenna and ground. The antenna is connected to the primary, while the circuit's only ground point is connected to the secondary. This set makes use of the inter-winding capacitance to complete the signal path. By means of a two position switch, the antenna can be connected to either end of the primary winding, giving either a Selective or Broad tuning characteristic. Presumably, the Broad setting also gives more volume. (I've reproduced the schematic as it originally appeared. However, it should be noted, that if the coil primary and secondary leads are located the way the diagram implies, then the Broad-Selective switch positions are likely backwards.)

The Mystery Set is described by Ray Creighton on the Historical Wireless Society Site, and is an interesting read:

http://www.hws.org.au/.../mystery.html

The coil has the primary and secondary wound side by side (i.e., bifilar). In the original design, a 3 inch (75 mm) diameter coil form was used. The primary was close-wound 12 turns. Then, after the 12th primary turn, the winding continued with the primary and secondary close-wound side by side for the next 25 turns. The secondary was then terminated, and the primary then continued on for another close-wound 13 turns. Thus, the primary has a total of 50 turns, and the secondary has 25 turns.

A very thorough analysis of the mystery crystal set has been given previously by Ben Tongue on his webpage:

Mystery Crystal Radio Circuit Analysis

My own findings essentially corroborate those of Ben Tongue, although I've limited my analysis, as much as possible, to just the coil part of the receiver.

The following diagram shows a cross sectional view of a smaller version of the mystery coil, with a 14 turn primary (blue conductors), and a 6 turn secondary (red conductors). The number of primary and secondary turns are designated by nP and nS respectively. The start position of the secondary is designated by the variable a and is equal to 5 in this example.

Note that in both the original coil and the one shown in the diagram, the secondary is not exactly centred. In order for the secondary to be centred, the total number of primary and secondary turns would have to be an odd number. Also, it can be a bit confusing when counting turns. As in most cases, it's easier to count turns on the side away from the leads, and that is what we will do here.

From the diagram we can also make note of several important characteristics:

- The primary winding has a variable pitch: initially
, then 2**p**, and finally p again;**p** - The secondary winding has a constant pitch of 2
**p** - If the coil has the primary and secondary connected in series then it
effectively becomes a single winding constant pitch coil of pitch
, and total turns**p**=**N****n**_{P}+**n**_{S}turns (20 turns in the diagram example).

(If you're not interested in the derivation of the Mystery coil calculations, and only want to see the results, then go to the Practical Example further down this page.)

In the previous parts of the discussion of inductance calculation, we've demonstrated that it is a simple matter to calculate the inductance of a solenoid with a constant pitch. For this reason it is relatively trivial to calculate the inductance of the secondary winding. Likewise, we can calculate total inductance of the series connected primary + secondary windings.

The formula which relates primary and secondary inductance, to mutual inductance is as follows:

**L**_{T}* *=

where **L**_{T}
is the total inductance when primary and secondary are series
connected, and * M* is the mutual inductance between the
primary and secondary windings. Since we can easily calculate

An efficient method of
calculating either **L**_{P} or * M*
is not immediately apparent. We can, of course, use a brute force
calculation of every combination of pairs of turns. For example, the
following Basic program will calculate the self inductance of the
primary winding:

```
function LpBrute(ByVal np,ns,a,rc,p,dw) As double
Dim Locn(1) as double
Redim Locn(np) as double
'Calculate the absolute position of each turn
' and store in the Locn() array
for i=1 to np
Locn(i)=i
if i>=a+ns-1 then
Locn(i)= Locn(i)+ns
elseif i>=a then
Locn(i)= Locn(i)+(i-a+1)
end if
next
'Step 1 - Calculate sum of self inductances of each turn
M=Mut(rc,rc,dw/2*exp(-.25))*np
'Step 2 - Calculate sum of mutual inductances of each pair of turns
for i=1 to np-1
for j=i+1 to np
M=M+Mut(rc,rc,abs(Locn(j)-Locn(i))*p)*2
next
next
LpBrute = M
end Function
```

The
program input parameters are as indicated in the above diagram. The
actual calculation of the mutual inductances is done by calling the
function * Mut()* which has been described previously in
Part 1b.

In this discussion, the positions of the various turns of the winding will be designated as shown in the diagram just below the upper half of the coil cross section, and numbered from 1 to 20. Hence the primary winding starts at position 1, and the secondary starts at position 5.

It can be seen that the
distance between any two turns is equal to the absolute difference
between their positions multiplied by the pitch * p*. At
the beginning of the program, the position of each primary turn is
calculated and stored in the array

Although this program is not highly efficient it provides a means to cross check more efficient algorithms. The importance of this cross check will soon become apparent.

In the case of a single layer
coil with constant pitch we have seen that the number of unique
spacings between turns is equal to the number of turns * N*
in the coil. Therefore, we only need to call the

As mentioned above, we can
calculate either **L**_{S} or * M*
and then determine the other quite simply. Probably the lesser of two
evils is the calculation of

Now let's consider the turns spaced two positions apart. We see that the red turn at position 5 can be paired with a blue turn at position 3. However, when we look at the turn located at position 7 we see that it's not a blue turn; it's a red turn, so we can't count it as a mutual inductance pair. Only pairs of dissimilar colours can be counted. Next, when we look at the red turn at position 7, we've already seen that it can't be paired with turn 5, nor can it be paired with turn 9. The same thing happens with turns 11 and 13. Finally, turn 15 can be paired only with turn 17. So, for a spacing of 2, the count is only 2.

Next, we look at a spacing of
3. This time, turn 5 can be paired with turns 2 and 8, turn 7 can be
paired with turns 4 and 10, and so on, so that for a spacing of 3,
there are 10 occurrences. We can now see that when the spacing is an
odd number, we don't have the situation of like coloured turns
pairing up, and so the pair count is simply 2**n**_{S}.
Good. We've found a pattern for odd numbered spacings.

Next, we look at a spacing of
4. In this case turn 5 can be paired only with turn 1, and turn 7 can
be paired only with turn 3. Turns 9 and 11 cannot be paired at all.
Turn 13 can be paired with turn 17, and turn 15 can be paired with
turn 19. So, for a spacing of 4 we have 4 occurrences. It appears
that we now also have a pattern for even numbered spacings: for a
spacing of * k*, the number of occurrences is

If the secondary winding were located in an infinitely long primary winding, the pattern which we've determined would be complete. However, we see that as the required spacing increases, we eventually run off the end of the coil. So, we have to account for that situation. When the spacing reaches 5, turn 5 loses one of it's pairings. When the spacing reaches 7, then both turns 5 and 7 lose one of their pairings. In general when the spacing value becomes greater than or equal to a, then the count must be reduced. In terms of a computer formula the reduction is:

`Int((k-a+1)/2)`

However, for small values of * k*, this formula will give a
negative number, and for large values of

`Min(ns,Int(Max(0,(k+2-a)/2)))`

This now gives a proper correction when we run past the left end of the coil, but we also need one for the right end:

`Min(ns,Int(Max(0,(k+1-(np+ns-b))/2)))`

Now that we can precisely count each category of turn pairs, it's simply a matter of coding it into a variation of previously discussed mutual inductance summation functions:

```
function L_Mystery(ByVal np,ns,a,rc,p,dw) As double
'np=no. of pri. turns; ns=no. of sec. turns; rc=coil radius; p=winding pitch;
'dw=wire conductor diameter; a=start position of secondary;
'All dimensions are in centimeters; result is in uH
a1=a-2
b1=np-ns-a+1
n=np+ns
ns2=ns+ns
'Step 1 - Calculate sum of self inductances of each turn
Lt=Mut(rc,rc,dw/2*exp(-.25))
Lp=Lt*np
Ls=Lt*ns
Lt=Lt*n
Mps=0
'Step 2 - Calculate the sum of the mutual inductances
' of all turn pair combinations
for k=1 to n-1
k_even=(k+1) mod 2
M=Mut(rc,rc,k*p)
LtPairs = n-k
LsPairs = Max(0,ns-k/2)*k_even
MPairs = Max(0,Min(ns2,ns2-Max(0,(k_even*(ns2-k)))_
-Min(ns,INT(Max(0,k-a1)/2)) - Min(ns,INT(Max(0,k-b1)/2)))
LpPairs = LtPairs - LsPairs - MPairs
Mps=Mps+M*Mpairs
Lp=Lp+M*LpPairs*2
Ls=Ls+M*LsPairs*2
Lt=Lt+M*LtPairs*2
next
return [?]
end function
```

The function shown above, is a straightforward implementation of the
mutual inductance summation algorithm modified for the mystery coil.
The formulae for the various turn pair counts are as described above.
The variable `k_even`

is used to keep track of even and odd values of
* k*. It has a value of zero when

The * L_Mystery()*
function calculates primary self inductance

Following is a Javascript implementation of the same calculation:

```
function MysteryCalc(np,ns,a,rc,p,dw,sel) {
//All dimensions are in meters
n=np+ns;
//physical constants:
pi=Math.PI;
muo=pi*4e-7;
// set up variables for efficient inner loop calculation
a1=a-2;
b1=np-ns-a+1;
ns2=ns+ns;
rc=dc/2;
// Calculate self inductance component
lt=mut(rc,rc,dw/2*Math.exp(-.25))*1e6;
lp=lt*np;
ls=lt*ns;
lt=lt*n;
m=0;
for (k=1; k<n; k++){
k_even=(k+1) % 2;
mi=mut(rc,rc,k*p);
sc=Math.floor(Math.max(0,Math.min(ns2,k-a1))/2);
ec=Math.floor(Math.max(0,Math.min(ns2,k-b1))/2);
ltPairs = n-k;
lsPairs = Math.max(0,ns-k/2)*k_even;
mPairs = Math.min(ns2,ns2-(k_even*(ns2-k))) - sc-ec;
lpPairs = ltPairs - lsPairs - mPairs;
m+=mi*mPairs*1e6;
lp+=mi*lpPairs*2e6;
ls+=mi*lsPairs*2e6;
lt+=mi*ltPairs*2e6;
}
//coupling coefficient:
kk=m/Math.sqrt(lp*ls);
//return the user selected value:
switch(sel){
case 1:
outval=m;
break;
case 2:
outval=lp;
break;
case 3:
outval=ls;
break;
case 4:
outval=lt;
break;
case 5:
outval=kk;
break;
}
return outval;
}
```

It includes a * Switch()* construct to allow the user to
select which of the calculated parameters is to be returned. The
return parameter is determined by the value of the input variable

```
1 Mutual Inductance
2 Primary Self Inductance
3 Secondary Self Inductance
4 Combined Series Aiding Primary Plus Secondary Inductance
5 Coupling Coefficient
```

This function has been expanded and implemented as an online calculator:

Javascript Mystery Coil Calculator.

Like other inductance calculators, this one lets us see the result of changing various dimensions and number of primary and secondary turns, as well as the effect on mutual inductance by changing the starting position of the secondary winding.

One thing that may be of interest is to see the effect of changing the relative start position of the secondary winding. The following diagrams show how the primary inductance varies depending on the relative position of the secondary winding:

The blue line is the inductance of the mystery coil primary, showing how
it varies as the position of the secondary coil is shifted. The value
* a *= 26 indicates the secondary is exactly
centred, and this gives the minimum inductance in every case. The red
line shows the inductance of an evenly spaced solenoid coil having
the same length and same number of turns as the mystery coil primary.
This is the inductance value that would be calculated by a
traditional inductance formula using average pitch.

The next two diagrams show the same results for coils of different Diameter/Length ratios:

We see that, in all cases, the primary inductance is minimum when the secondary is centred relative to the primary. We also see that, as the coil's Diameter/Length ratio decreases, the mystery coil primary inductance increases, as compared to a winding of the same length with constant spacing.

It should be noted that, because of the vertical scale in the diagrams, the variation in inductance with secondary position is not as great as the curve may seem to indicate at first glance. However, this does give an indication of the amount of error that we may expect, if we use a traditional inductance formula to calculate the primary inductance based on averaging the turn spacing, instead of using the more accurate calculation.

As mentioned earlier, in the
Mystery Crystal Set, there is no apparent signal path between the
antenna and ground, and hence, the inter-winding capacitance is an
important parameter. For that reason, it's useful to know the value
of the inter-winding capacitance. This capacitance calculation is
also included in the online calculator. It's based on the formula for
the capacitance for two long parallel cylinders. In this case, the
cylinders are the adjacent primary and secondary windings. The fact
that these cylinders are wound around a coil form has very little
effect on capacitance, as long as the wire diameter is considerably
less than the coil diameter, which is generally the case.
Complicating matters, the dielectric which separates the 'plates' of
the capacitor consists of three different materials: the wire
insulation, the coil form material, and air. The relative
permittivity values, **ε**_{R},
(aka, dielectric constant) for these are typically as follows:

- Modern magnet wire enamel insulation: 3 to 3.5;
- Coil form materials: 2 to 3.5;
- Air: 1.

On this basis, the effective value of εR will be somewhere between 1 and 3.5. We can do the calculation with the low value and then repeat it with the high value to see the range of capacitance values.

In order to test the Online Calculator, I wound a mystery coil using an empty vitamin bottle as the coil form.

Coil specifications are as follows:

- Number of primary turns: 51;
- Number of secondary turns: 25;
- Secondary start position
: 15;**a** - Coil diameter: 76.76 mm;
- Total coil length: 42.85 mm;
- Coil form material: HDPE,
**ε**_{R}= 2.3; - Wire type: #24 AWG magnet wire, polyamide insulation:
**ε**_{R}= 3.5; - Estimated effective
**ε**_{R}: 2; - Conductor diameter: 0.5106 mm;
- Wire outside diameter including insulation: 0.5588 mm;
- Winding pitch: 0.5638 mm.

Aside from the modern materials, this is very close to the specifications for the coil in the original Mystery Set article.

The output from the calculator is:

```
Wire Length: 18327.35 mm
Primary Inductance LP: 187.1718 µH
Secondary Inductance LS: 58.1545 µH
Mutual Inductance M: 92.3427 µH
Series (Pri. + Sec.) Inductance LT: 430.0117 µH
Coupling Coefficient kC: 0.8851
Inter-winding Capacitance CPS: 1481.4 pF
```

Actual coil measurements:

```
Primary Inductance LP: 188.67 µH (C0=4.61 pF)
Secondary Inductance LS (Q-meter): 58.07 µH (C0=9.34 pF)
Secondary Inductance LS (Meter-3): 59.11 µH
Series (Pri. + Sec.) Inductance LT+: 424.23 µH (C0=345 pF)
Series (Pri. - Sec.) Inductance LT-: 50.68 µH (C0=533 pF)
Mutual Inductance M=(LT+-LT-)/4: 93.39 µH
Coupling Coefficient kC=M/sqrt(LP*LS): 0.892
Inter-winding capacitance (Meter-1): 1380 pF
Inter-winding capacitance (Meter-2): 1590 pF
Inter-winding capacitance (Meter-3): 1560 pF
Inter-winding capacitance mean value: 1510 pF
Primary Q (secondary unconnected): 142 to 184 (see graph below)
Secondary Q (primary unconnected): 80 to 96
```

Inductance and Q measurements were done on an elderly but serviceable Heathkit QM-1 Q-meter by finding the parallel capacitance necessary to
resonate with the coil at a number of different frequencies (exact
frequency measured with an external HP 5381A frequency counter), and
then performing a regression analysis to calculate both the winding
inductance and self-capacitance **C**_{0},
which is also shown for each winding.

The primary self-inductance was
measured over 11 different frequencies from 600 kHz to 1600 kHz.
Standard deviation in the measured * L* and

Secondary self-inductance was
measured over 17 different frequencies from 1000 kHz to 3000 kHz. The
Q values ranged from 95 at the highest frequency to 80 at the lowest
frequency, and peaking at 96 at 2200 kHz. Standard deviation in the
measured * L* and

Secondary self-inductance was
also measured with a simple * L*/C meter, but has no
correction for

Combined Primary plus Secondary
inductance **L**_{T}_{+} was
measured over 8 different frequencies from 240 kHz to 350 kHz. The Q
values ranged from 11 at the highest frequency to 59 at the lowest
frequency. Standard deviation in the measured L and **C**_{0}
values are 3.4 µH and 6.2 pF respectively.

Combined Primary minus
Secondary inductance **L**_{T}_{–}
was measured over 6 different frequencies from 630 kHz to 800 kHz.
The Q values ranged from 4 at the highest frequency to 11 at the
lowest frequency. Standard deviation in the measured L and **C**_{0}
values are 0.63 µH and 14.8 pF respectively.

Combined Primary plus Secondary
inductance **L**_{T}_{+}, and
Primary minus Secondary inductance **L**_{T}_{–},
were also measured with Meter-1 and Meter-3, mentioned below.
However, these measurements did not produce meaningful results, most
likely because of the extremely large values of self-capacitance.

The calculated and measured parameters are summarized in the following table:

Allowing for experimental error, I would consider agreement to within 3% to be quite acceptable. Since the worst deviation here is only 1.34%, I believe this should validate the accuracy of the calculator.

Inter-winding capacitance was measured with three different meters:

- Meter-1: Inexpensive L/C meter; measurement frequency: 8 kHz;
- Meter-2: Medium quality digital multimeter with a capacitance function; measurement frequency estimated to be ~10 kHz;
- Meter-3: Better quality L/C meter; measurement frequency: ~600 kHz.

If we take the inter-winding capacitance mean value of 1510 pF to be
correct, then we can rerun the calculator with different values of **ε**_{R}
in order to find which one gives the same value of 1510 pF, and thus
get a better estimate of the true effective **ε**_{R}
value. Doing this, we get a value of **ε**_{R}=2.04,
which seems reasonable. However, we have to be careful in
interpreting the results, because there is no guarantee that 1510 pF
is the correct value. It's very likely that coil inductance has
affected the capacitance measurement. This is a subject for future
study. Regardless, the measured inter-winding capacitance values are
in the same ballpark as the calculated value, and for something as
non-critical as a Mystery Crystal Set, that's probably good enough.

In any event, the calculator
can be run with the minimum and maximum values of **ε**_{R}
to get capacitance values which will bracket the true value. For this
example, these values of **ε**_{R}
are 1 and 3.5 which give capacitance values of 741 pF and 2593 pF.

One other significant parameter
in the mystery coil is the coupling coefficient between primary
and secondary. The calculated value of 0.8851 shows a high degree of
coupling between primary and secondary, as we might expect with a
bifilar close wound coil. However, it is even more interesting to
look at the coupling between the secondary and just the adjacent
portion of the primary. To do this, we change the number of primary
turns **n**_{P}, to be **n**_{S}+1,
or 26, and the secondary start position, a, to be 2. Doing this with
the example coil, we get a coupling coefficient of 0.9871 which is
close to the value of 0.966 determined by Ben Tongue in his analysis
of the circuit. With such a high coupling coefficient, the coil
behaves more like a transformer than a loosely coupled coil, and as
Ben concluded, the secondary could very likely be omitted and
replaced with taps on the primary at the 25 and 38 turn positions.
Then, a physical capacitor could be added in the antenna lead, to
account for the inter-winding capacitance. This would result in a
simpler receiver, likely with the same performance.

Study of the Mystery Crystal Set coil has provided an opportunity to show how the mutual inductance summation method can be adapted to different special situations. This has also provided the opportunity to compare calculated values with actual measurements, and satisfy ourselves that the numbers provided by theoretical calculations do indeed agree quite well with reality.

Continue to:

Part 2 – Inductance Calculation Refinements Back to:

Numerical Methods 1c

Numerical Methods Introduction

Radio Theory

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This page last updated: November 29, 2022

Copyright 2010, 2022 Robert Weaver