How to build a Tesla coil. Design, theory and compromises!

A live broadcast that I did on Sunday, February 4, 2018 with focus on designing Tesla coils with special focus on the DRSSTC topology. Questions …

WARNING!: Working with electricity is dangerous, all information found on my site is for educational purpose and I accept no responsibility for others actions using the information found on this site.

Introduction

DO NOT REPLICATE THIS PROJECT!

If you do anyway, be aware of large switching transients that may damage nearby electronics, read this entire article before proceeding.

The idea to this coil came with Steve Ward showing off his first QCW DRSSTC that used a buck regulated DC supply to ramp up the supply voltage along with a long on-time to grow straight and very long sparks compared to the secondary coil length.

I thought it could be done simpler, yet with less control, by using the rising edge of 50 Hz mains supply voltage. From start of the sine wave to the top it corresponds to a on-time of 5000 uS and to be able to use large IGBT bricks the frequency would have to be kept down. Sword like behaviour of sparks is however mostly seen at above 300-400 kHz, where as lower than that results in more branched sparks.

Considerations

A high impedance primary circuit is needed to keep peak current at a level that the IGBTs can handle to switch for very long pulses, for a DRSSTC, up to 5000 uS. In order to get enough primary windings, I went for a upside-down U shape primary as a regular helical coil with high enough coupling would quickly get as tall as the secondary coil itself.

To use 3 IGBT bricks in parallel it is important to ensure as even current sharing as possible, this is done by mounting them close to each other on the same heat sink, drive them from the same gate drive transformer with individual gate resistors matched as close as possible.

Steve Wards universal driver 2.1b only has a robust enough 24 VDC section to run up to about 300 uS on-time with large gate capacitance, when trying to run with longer on-times than that, the 24 VDC 1.5 A regulator is now longer enough to supply the needed current. A external 26 VDC 8 A power supply is used instead and output stage will have to be reinforced to conduct higher currents and dissipate more heat.

Specifications

 Bridge 3x  SKM145GB123D IGBT bricks in a parallel half bridge configuration Bridge supply 0 – 260VAC through a variac Primary coil 21 turns of 8 mm copper tubing in a up-side-down U shape MMC 10 strings in parallel of 10 in series Kemet R474N247000A1K capacitors for 0.047 uF at 9000 VDC, 280 A peak and 40 A rms rating. Secondary coil 160 mm diameter, 330 mm long, 1500 windings, 0.2 mm enamelled copper wire. Resonant frequency Around 100 kHz. Topload 100 x 330 mm spun aluminium toroid. Input power 10BPS, 500 cycles, 50A limiter: 750W at 260VAC at 3A. Spark length Up to 500 mm long sparks.

Schematic

Bridge section

Driver section

Same as Steve Wards universal driver version 2.1b. Just made on single sided PCB without SMD components and the 24VDC part has traces reinforced, MOSFETs heat sinked and uses a external power supply. A external 26VDC / 8 Ampere power supply is used to ensure that under voltage will not be a problem, at least not before something starts to smoke.

Construction

31st October 2011
I put the bridge together on a heat sink with 3 phase rectifier used with all inputs in parallel for 1 phase supply and connected all 3 half bridge IGBT bricks in parallel with 3 straight bus bars. All recycled components from a DC link inverter.

Designed staccato PCB as the old layout used in my VTTC I was on a vero board. A optical output was added to use the interrupter with a standard DRSSTC driver.

Started construction of the secondary coil.

2nd November 2011

Etched PCB board for staccato controller.

Finished winding the secondary coil. It was made with a total of 1500 turns of 0.2 mm wire and dimensions 160 x 330 mm. Varnished the secondary coil with polyurethane varnish.

3rd November 2011
The secondary coil was given a second thick layer of varnish, not the most pretty job as I tried to pour as much varnish on as possible and let it rotate and settle it around the coil by itself. As the secondary coil was not completely in level it result in a little running, but overall a fair result of adding a lot of thin varnish.

Finished assembly of the staccato controller and bench tested it.

The project got shelved due to starting on a new job, after a long rest of over 3 years the box with parts was once again brought out in the light and construction could continue.

7th March 2015

Etched driver PCB and started populating the board with all passive components.

6th October 2015

Construction of a very cheap MMC from capacitors that was bought from a 10\$ ebay auction for 100x Kemet R474N247000A1K, rated for 900VDC 28 A peak and 4 A rms. A easy and uniform construction, with current sharing in mind, is to construct it around a round piece of wood or plastic tube.

The resulting MMC has a capacitance of 0.047 uF at 9000 VDC, 280 A peak and 40 A rms rating. Which is spot on for this coil to be running with design goal primary inductance of around 100 uH, 300 A peak, 5000 uS on-time and maximum 10 BPS.

11th October 2015

Construction of acrylic primary supports, that has 21 slots and is formed for a up-side down U shape primary coil. A way of getting a large primary inductance and still maintain a certain distance to the topload as the secondary coil is very short.

The supports are made by hand using a saw, file and drill press.

16th October 2015

Getting the coil winded from the inside and out was no easy task, the whole large roll of copper tubing is heavy, easily bends too sharp and is like a spring. It will lock itself in the wrong slots and it can be a very frustrating piece of work. The complete result is however worth the effort, it looks smooth and even.

As the slots was not made to snap the copper tube in, from shear fear of cracking the acrylic, I made a small hole behind each slot that made it possible to tie each turn at each primary support, with a little piece of copper wire it is secured from deforming the coil.

As water cooling of the primary coil is going to be a must with the long on-times, simple clamps was made from copper sheet and two screws the fastens the 4 braided copper flexible wires to the tubing. The same 4 braided copper wires was soldered to the MMC terminals, as even as possible distributed around the circular copper wire terminal.

Having made the MMC on a wooden stick makes it easy to mount with two wooden blocks with holes in for the extra length of the round rod.

29th December 2015

Driver board populated with all active components. 24 VDC regulator is left out as this will be supplied externally from a 26 VDC, 8 A power supply. All traces related to the 24 VDC is reinforced by soldering a 0.5 mm2 copper wire along them. Four 2200 uF 35 VDC capacitors was added to the underside of the board, one at each N- and P-channel MOSFET. All output stage MOSFETs have heat sinks mounted.

All these precautions are hopefully enough to ensure no under voltage or over heating situation is possible when running at 5000 uS on-time.

25th July 2016

The coil have been put together with power supply, bridge, driver, platforms, secondary and topload. Two fuse holders for large bussman fuses was used at the end of the flexible copper braids for primary tap.

7th October 2016

First test of the coil after all the components have been put together, there are still a few things not in place, but it is good enough for initial testing.

The first test is without power on the DC bus. This is solely to test if the driver and power supply is good enough to drive the 3 IGBT bricks’ gates in parallel with a satisfying gate waveform. The following oscilloscope shots show the coil being driven at 5 ms on-time at 100 BPS, corresponding to every rising edge of the half wave rectified 50 Hz mains supply.

The following months, where I had only little time to do more testing, I could not get the coil to run properly with power on the DC bus. I could measure oscillation at the resonant frequency, but nowhere near enough for the coil to actually produce a output ever so slightly or just so small as to light up a close by fluorescent tube.

I tried running the coil with added 6000 uF capacitance, with normal interrupter, with and without DC bus snubber capacitor, with much fewer primary windings etc. I did a lot of changes to it, so it would be more like a regular DRSSTC, than a QCW, little did it help and I did not get much further before putting it away for Christmas holidays.

26th March 2017

After some online discussions as to what the problem could be, and showing off the coil for the first time, it was suggested that the regular 1:1000 cascaded current transformer for feedback was simply not delivering a strong enough signal as a high impedance coil naturally works with a much lower peak current. From the thread on the forum here https://highvoltageforum.net/index.php?topic=24.0 it is decided that I will try to make a new 1:50 turns ratio current transformer to see if increased feedback will help the coil oscillate.

The only capacitance on the DC bus was a 0.68 uF snubber capacitor, this was also increased to a 10 uF capacitor bank of MKP film capacitors. A small amount of energy storage is needed to make the phase correction driver able to run stable.

First light was achieved with some 20 cm long sparks, remember that the coil is far from tuned for maximum performance and the input voltage was only 200 VAC.

1st April 2017

I made a wide range of secondary and primary circuit measurements to find the resonant frequencies, as the U-shaped primary makes it difficult to simulate with tools like JavaTC.

The measurements was done with secondary in place inside the primary. But secondary ground was unconnected and primary circuit was left open loop by removing the tapping point. I am not sure if this is the correct method, as resonant frequencies are much different when measured with secondary ground or primary loop closed, this is because energy is then transferred between the two resonant systems. The results could however vary with 10-20 kHz compared to the open loop measurements…

Secondary circuit test results
Setup with a 80 cm long wire with 3 bend wires hanging over and pointing down to be “branches”. Signal from signal generator connected to ground terminal on the secondary coil, ground left floating. Signal into oscilloscope captured from open loop probe hanging next to secondary coil.

Unloaded result: 101 kHz, 80 cm wire result: 91 kHz and 80 cm wire with branches result: 88 kHz.

Primary circuit test results
Setup with signal generator and oscilloscope connected across the primary LC circuit and with a jumper across the IGBTs to have a closed loop. Signal generator is connected through a 10K resistor.

Primary resonance with secondary ungrounded – 7th turn from bottom 102kHz, 6th turn from bottom 96kHz, 5th turn from bottom 90kHz and 4th turn from bottom 86kHz.

Rest of the measurement results did not have individually saved oscilloscope shots, so here is a overview of the primary tapping frequencies.

I recorded video from oscilloscope and spark formation (dark dark video, blerg, sorry). There are 4 tests where primary is tapped at 96 kHz, 90 kHz, 86 kHz and last at 65 kHz, where it for reasons I still do not fully understand, performed the best! This is a huge detuning compared to the loaded 88 kHz secondary measurements. I also tried all the taps between 86 kHz and 65 kHz, with only increasing performance until I could detune it no further.

The staccato interrupter is not particular stable and does not really give a good clean 5ms on-time, it starts conducting before the zero crossing, possibly due to non-adjusted phase correction on the driver, this will get looked into next time its running. Waveforms are highly distorted, peak currents are low in the magnitude of 50 A peak. (Blue 100V/div inverter output – Yellow 100A/div current – 5-6 ms on-time)

and also some close up pictures of the beautiful sparks, still much shorter than what I expected, but maybe the very high impedance primary circuit just limits the current way too much, future experiments would be with a step up transformer for a higher primary peak voltage.

4th April 2017

From last nights experiments, I think that this idea might work on a small scale, the peak currents drawn by this large coil simply creates too large switching transients.

I tried tuning the coil at 120 kHz and 130 kHz, way above the estimated loaded secondary resonant frequency of 88 kHz. It performed better than ever before at 120 kHz tap, which properly makes a little sense compared to the better performance at 65 kHz, it certainly does seem to be a harmonics pattern here. But I do not think I can tap it any further down on the inner side of the primary coil right now.

There was however also much higher current draw, loud clunks from the variac and lights dimming! The voltage spikes on the mains supply are at levels where my voltmeter was damaged in my variac. This is also why I call quit on the project as it is, its future will be rebuilding it to a conventional, properly PWM controlled, QCW.

I had sparks fly out to about 50 cm as it can be seen in the video

Conclusion

So far the prototype has worked and shown that the concept works. The spark formation is more straight than first anticipated, as most QCW coils operate above 300-400 kHz to get long sword like sparks. It is however clear that the sparks produced by this coil, resonating below 100 kHz, is swirling a lot.

Tuning is very different from a regular DRSSTC where the sweet spot that produces the longest sparks at the lowest current can be within a few centimetre on the primary coil. Here I could get the same performance over a wide span of 60 kHz, tapping the primary anywhere would give me around 30 cm sparks, but it was easy to recognize when a true sweet spot was found, as the very abrupt current draw could be heard clearly from the variac clunking loudly and lights dimming slightly.

The switchings transients are however a great danger to nearby electronics and is of a magnitude where filtering is properly not enough, certainly it is not a solution to add more passive components to counter a problem that can be completely eliminated by using a different topology and have a control scheme that can control a ramped voltage from a capacitor energy storage, like the class D amplifier, phase shifted or PWM controlled QCW coils demonstrated by other Tesla coil builder.

I wanted to try this method, to see if simplicity could do the same job, it could not.

Demonstration

There is not a overall demonstration video yet, but the 3 videos from research development above.

Kaizer DRSSTC III update #7 – Show and tell

While we had the box open for removing the real-time current control, as you can see in the previous post, I did a walk through …

Busbar and primary circuit design for Tesla coils and inverters

This is chapter 2: Busbar and primary circuit of the DRSSTC design guide

Busbar

A physically large busbar will help components like IGBTs and capacitors dissipate heat through their terminal connections. So it is important that it is the busbar that is cooling the components and not the other way around.

When large currents are switched fast, switching voltage transients will develop. These are very short but very high amplitude voltages that can damage the IGBTs. The switching transients can be lowered by either lowering the inductance or reducing switching speed. Slower switching speed will only result in further losses from the IGBTs spending longer time in the linear region, there is more on this topic in the IGBT chapter.

A primary circuit with a 8 uH inductance and a CM300 IGBT switching 1000 A at 100 kHz would see voltage transients in the order of 3500 V. To understand the rate of change of current, more on this topic in the IGBT chapter.

If there is 100 cm extra AWG14 / 2.5 mm2 wire it would result in the switching voltage transients being higher by

$V_{transient}=L\cdot\frac{dI}{dt}=1.43uH\cdot628\frac{A}{us}=898V$

If there is 100 cm extra AWG8 / 10 mm2 wire it would result in the switching voltage transients being higher by

$V_{transient}=L\cdot\frac{dI}{dt}=1.29uH\cdot628\frac{A}{us}=810V$

If there is 100 cm extra 300 mm2 busbar it would result in the switching voltage transients being higher by

$V_{transient}=L\cdot\frac{dI}{dt}=0.91uH\cdot628\frac{A}{us}=571V$

The following graph shows four different kind of busbar constructions. Busbar side by side, wires side by side, coaxial and laminated busbar.

Busbar side by side needs to be as wide as possible and as close to each other as possible for lowest possible inductance.

Wire side by side needs to be as large a conductor as possible and as close to each other as possible for lowest possible inductance.

Coaxial needs to have both conductors as close to each other in diameter for the lowest possible inductance.

Laminated busbar have to be as close to each other as possible and as wide / large surface as possible to have the lowest possible inductance.

Examples of busbar constructions in DRSSTCs

Primary coils

Most primary coils are made from 10 mm copper tubing for a variety of reasons.

• Easy to find, can be bought everywhere.
• Skin depth at Tesla coil frequencies can not utilize a solid wire, more on this topic further down in this chapter.
• It comes in a coil from the manufacturer, so it is easy to bend into a primary coil.
• It can be cooled by running liquid or compressed air through it.

Copper tubing made for water service and heat exchangers are treated in a way so that the oxygen content in the copper is low, it is to avoid the tubing becoming brittle if it reacts with hydrogen. Normal coiled copper tubing that we all use is phosphorus deoxidised copper (Cu-DHP), the phosphor content reduces the electrical conductivity of the copper which will be around 92% at a phosphorus content of 0.015% down to about 78% at 0.05%. This is actually a help to us at high frequencies as the skin depth lies deeper in materials with higher resistance, it could be comparable with the conductivity of aluminium.

Primary coils made from wire should still have adequate spacing between turns to avoid short circuit if the insulation around the wire melts. It is not uncommon that wire primaries get real hot since its often smaller gauges and the insulation hinders air cooling.

Using flat copper ribbon as primary coil can result in changing resonant frequencies at very high peak currents.

The reason is that the current will be concentrated at the two narrow ends on top and bottom of the copper ribbon, effectively acting as two primary coils in parallel. Care will have to be taken when using copper ribbon for the primary coil.

Primary coil water cooling

Normal tap water can be used for cooling a DRSSTC primary coil. Distilled water is a better solution as it contains less impurities. The resistance of water is however still very high. Distilled water should be changed after each use, as explained below.

Distilled water coming straight from its container has a pH of about 7 and a resistance around 18 MΩ/cm. If it is exposed to open air, the water will take on CO2 which forms carbonic acid. The pH value of the water will fall to ~5.75 and resistance to around ~1 MΩ/cm.

The entry point of the water cooling should be on the inner turn of the primary coil, due to the proximity effect where the inner turns gets induction heated from the outer turns, this is where we will have the highest losses and therefore highest temperature.

A small closed loop with the water in a container or bucket should be enough for shorter runs. There have been no reports on problems with sparks hitting the water where a pump connected to mains was lying in. A strike ring with a wire into the water and connected to the rest of the grounding system could be a protective step to take.

Corrosion of the copper tubing will only occur if there is a DC current flowing, it is however not regarded as a problem for DRSSTCs that genereally have short run times of some hours a day. Some anti-freeze cooling additives does also contain anti-corrosion additives.

Primary coil geometry

Basically there is three different types of primary coil geometries. A flat coil, helical coil and conical coil. The difference between them is the coupling between primary and secondary coil, the closer the primary coil is to the secondary, the higher coupling.

The voltage potential difference between the primary coil and secondary coil, along with varnish on the secondary coil and other barriers like acrylic shields between the coils determines how close the coils can be together without flash over between them occur.

A flash over is a spark directly from secondary to primary coil. It can be very destructive against the power electronics and IGBT switches. It can also burn and melt the secondary coil.

Due to the large currents switched in a DRSSTC, the coupling can not be too tight, the magnetic field is generally strong enough for a flat coil or conical to be the best choice. It also gives the greatest distance from break out point on the topload to the primary coil and strike ring. More about coupling will be discussed in the chapter about the secondary coil.

Examples of primary coil constructions in DRSSTCs

Primary circuit

Leads and connections from MMC / tank capacitor and output terminals from the IGBTs that go to the primary coil itself are often made with too small gauge wires.

It often has to be flexible to allow tuning of the primary resonant frequency by moving the tap point on the primary coil.

A practical solution is to make a Litze (many, from each other isolated conductors and braided / twisted in a way to eliminate each others magnetic field) cable by combining a number of smaller gauge wires in parallel into one cable. This way we can utilize more of the copper and still maintain a good flexibility to allow it to be moved around in all directions.

The primary leads should be kept as short as possible and thus it makes sense to have them come out from the inverter as close to the center and underneath of the primary coil. This way it can have the shortest length possible and still reach all possible tapping points on the primary coil.

If you can not find 90 degree Machine Tool Wire to use for the wiring of the primary circuit, there are cheaper, but also less safe options. A easy source of heavy gauge cable is speaker / power cabling for car audio. The insulation on these cables are however not rated for high voltage or temperature, so it is important to have it in a place where it can never touch any other conducting material.

Primary support and standoffs

It is important to use materials that can withstand high temperatures. Short circuits and failures due to melting insulation and materials is a well-known problem in high current inverters.

Good materials: FR-4 glass fiber material,

Okay materials: ABS, PVC

Bad materials: Zip ties, Acrylic, 3D printed plastics with low melting point

Primary coil and primary circuit proximity to metal

The primary coil and circuit is designed to deliver its energy into a secondary coil that is placed inside the center of the primary coil.

It works like a normal transformer, just with an air core. So the magnetic coupling between the primary coil and secondary coil is solely based on their physical dimensions in regard to each other. More about coupling will be discussed in the chapter about the secondary coil.

Any metal inside the powerful electromagnetic field that we create will heat up from eddy currents induced into the metal from the primary circuit, what is know as induction heating. In high power cases where we switch several thousand Ampere, metal that is just near/below the primary can get heated from induction heating. The primary leads that is even a part of the primary circuit can suffer the same.

It is necessary to construct the primary coil platform in nonmetal materials and also add a safety distance to nearest metal object. A rule of thumb could be 10 cm for each 1000 A switched in the primary circuit.

Primary coil and primary circuit protection against sparks

A strike rail can be installed to protect the primary coil against sparks from the breakout point or topload. The strike rail just needs to be slightly larger in radius and elevated a bit over the primary coil.

It is very important that the strike rail is made of a similar good conductor as the primary coil and that it is gapped, it must under no circumstance form a closed loop. A closed loop would look like a 1 turn coil to the primary coil and would take up a lot of energy and possible pose a fire hazard.

More details on proper grounding techniques will be covered in the chapter about how to set up a DRSSTC for a safe and reliable run.

Decoupling capacitor (more to come)

Skin depth

For alternating current (AC), the interaction of electric and magnetic fields in the conductor distribute the currents to the outside of the wire. This skin effect increases with frequency so that for high frequencies, barely into the tens of kHz, a thin outside layer of the conductor carries essentially all the current.

63% of the current will flow from surface and 1 times the skin depth into the material. 98% of the current will flow from the surface and 4 times the skin depth into the material. This is due to the exponential nature of the drop of current density in conductors. [2]

This illustration show a sinusoidal current in a 2D plane. The amplitude of the current is damped exponentially and is only here to show why we can assume 4 times skin depth and not to explain how skin effect works.

In the table below a range of common Tesla coil frequencies are listed along with the associated skin depth in copper and aluminium conductors.

 Frequency Copper Aluminium 30 kHz 0.38 mm 0.48 mm 40 kHz 0.32 mm 0.41 mm 50 kHz 0.29 mm 0.37 mm 60 kHz 0.27 mm 0.34 mm 70 kHz 0.25 mm 0.31 mm 80 kHz 0.24 mm 0.28 mm 90 kHz 0.22 mm 0.27 mm 100 kHz 0.21 mm 0.26 mm 150 kHz 0.17 mm 0.23 mm 200 kHz 0.16 mm 0.19 mm 300 kHz 0.13 mm 0.16 mm 400 kHz 0.11 mm 0.14 mm 500 kHz 0.095 mm 0.13 mm

As an example I will compare a 10 mm diameter copper tube with a 4 mm diameter solid copper wire of the same cross section area.

Despite the tube is hollow, it can only be considered a single surface as there is only a magnetic field around the entire conductor. Therefore we multiply the skin depth from the above table with 4 for both tube and solid wire.

The 1 mm wall thickness of the copper tubing will utilize atleast 98% of the copper cross section area up until 300 kHz, only after that will there be nonconducting copper at the middle of the wall.

The 4 mm solid wire will have a core of 1 mm diameter that is not conducting current, already at 30 kHz. At 300 kHz, 98% of the current would flow in only 56% of the copper material.

Failure modes of primary coils

Heat is often the root cause of many primary coil failures, like heat melting off insulation of cables that leads to short circuits, heat that melts supports which leads to changed inductance or short circuits. The following pictures shows perfectly how the inner turns have suffered from extensive heating due to the proximity effect described above. It also perfectly demonstrates how acrylic is not a suitable primary support material due to its poor heat resistance.

Conclusion

Busbar conclusion: Laminated busbar is by far the most superior in regard to low inductance and cooling. Busbar side by side is the easiest to build offering good cooling and current capability. Wire is also very easy but have bad cooling abilities and have to be very thick and close to have low inductance. Coaxial is not practical possible.

Primary coil conclusion: Hollow copper tubing has the best electrical and mechanical features in regard to primary coil construction. There is very little electrical difference from 10 mm to 25 mm copper tubing compared to the extra physical size.

Primary circuit conclusion: Keep all leads as short as possible. Flat and wide is better than round or squared in regard to skin depth and stray inductance. Keep all primary circuit wiring away from touching anything it could short circuit against if the insulation melts.

Skin depth conclusion: To find the optimal material size, you can multiply the above numbers by 8 for hollow conductors and by 4 for solid conductors at your resonant frequency and find the diameter or maximum thickness.

It is however advised to use larger conductors for mechanical ruggedness and cooling in term of area and mass. You might find a value that would utilize all the copper but the mechanical and thermal properties would be highly neglected if the material is very thin, so a larger size is highly recommended.

 Previous topic: Rectifiers Next topic: IGBTs

References

[1] Copper Development Association, “High Conductivity Coppers. For Electrical  engineering”, Publication 122, 1998.

Spiral coil calculator

Published July 10, 2014. Updated February 13, 2021.

Here you can calculate the inductance for a given size of a spiral coil wound in one layer. It is optional to add the capacitance for f.ex. a primary tank capacitor or topload capacitance to find the resonant frequency of the LC circuit.

The formulas used to derive the inductance is simplified and correct to within 1%. Source “Harold A. Wheeler, “Simple Inductance Formulas for Radio Coils,” Proceedings of the I.R.E., October 1928, pp. 1398-1400.”

Switch between the input fields to automatically calculate the values.

 Number of turns Turns Inner diameter mm Wire diameter mm Turn spacing mm Outer diameter mm Wire length (estimate) m Inductance uH Optional extra f.ex. tank capacitance size Capacitance nF Resonant frequency kHz

Formulas used

Outer diameter = inner diameter + ( 2 * number of turns * ( wire diameter + wire spacing))

Wire length = ((Pi * number of turns * (outer diameter + inner diameter)) / 2) / 1000
The wire length is calculated as an average of the evenly spaced turns, this results in a estimated spiral length from being calculated as single circles.

Inductance
Width w = ((wire diameter / 25.4) + (wire spacing / 25.4)) * number of turns
Radius r = ((inner diameter / 25.4) + w) / 2
Inductance = (radius^2 * number of turns^2) / (8 * radius + 11 * width)

Resonant frequency = (1 / (2 * pi * sqrt((inductance / 1000000) * (capacitance / 1000000000)))) / 1000