## Power Inductors and Inductors In Power Electronics – Isolation, Day 17

I show a range of different inductors, chokes, reactors and other kinds of spiral wound wires with or without a core. Iron or ferrite core. …

# 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

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 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

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

# Helical coil calculator

Here you can calculate the inductance for a given size of helical 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.

 Coil diameter mm Number of turns Turns Wire diameter mm Turn spacing mm Wire length meters Coil height mm Inductance uH Optional extra f.ex. tank/topload capacitance size Capacitance nF Resonant frequency kHz

Formulas used

Wire length in meters = ((coil diameter * pi) * number of turns) / 1000

Coil length in mm = number of turns * (wire diameter + turn spacing)

Coil inductance in uH = (number of turns * (((coil diameter / 25.4) / 2)*((coil diameter / 25.4) / 2))) / ((9 * ((coil diameter / 25.4) / 2)) + (10 * (coil length / 25.4)))

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