COIL

Current, Magnetic Fields, and Induction

Let’s start with the basics of electromagnetism. One of Maxwell’s equations, Ampere’s law, tells us that current flowing through a wire creates a magnetic field around it.

 If we want to use this magentic field to our adavantage, as we do in an electromagnet, we coil the wire.The magnetic fields from the individual turns add together in the center.

 A constant current makes a static magnetic field. What happens with we put a changing current through the wire? Another of Maxwell’s equations, Faraday’s law of induction, tells us that a magnetic field changing in time induces a voltage across the wire proportional to the rate of change of the magnetic field:

If the current is abruptly shut off, Faraday’s law tellls us that there will be a sharp spike of voltage. If an oscillating current flows through the coil, it induces an oscillating magnetic field inside it. This, in turn, induces a voltage across the coil which tends to oppose the driving current. Intuitively, the magnetic field is “stubborn,” inducing a voltage that opposes any change to the field.

Transformers

A transformer takes advantage of the law of induction to step AC voltages up or down. It consists of two coils of wire around a core. The core is soft iron or ferrite, materials which are easily magnetized and demagnetized.

An oscillating current in the primary winding establishes an oscillating magnetic field in the core.  The core concentrates the field, ensuring that most of it passes through the secondary.  As the magnetic field oscillates, it induces an oscillating current in the secondary coil. The voltage across each turn of wire is the same, so the total voltage across the coils is proportional to the number of turns:

Because energy is conserved, the current on the side of the transformer with the higher voltage is smaller by the same proportion.

 The Tesla coil is a very souped-up transformer. Let’s briefly consider what would happen if it were a perfect transformer. The primary winding has six turns and the secondary has about 1800 turns. The primary is driven with 340 volts, so the secondary will have 340V x 300 = 102kV across it. That’s a lot! But not quite a quarter million. Additionally, because the Tesla coil is air-cored and the coils are positioned relatively far apart, only a small fraction of the magnetic field produced by the primary is actually interlinked with the secondary. To understand more of what’s going on, we need to introduce resonant circuits.

Resonant Circuits

A resonant circuit is like a tuning fork: it has a very strong amplitude response at one particular frequency, called the resonant or natural frequency. In the case of the tuning fork, the tines vibrate strongly when excited at a frequency determined by its dimensions and the material properties. A resonant circuit achieves the highest voltages when driven at its natural frequency, which is determined by the value of its components.

Resonant circuits use capacitors and inductors, and therefore are also known as LC circuits. They are also known as “tank circuits,” because of the energy storage elements present.

Capacitors store energy in the form of an electric field between two plates separated by an insulator, known as a dielectric. The size of the capacitor is dependent upon the size of the plates, the distance between them, and the properties of the dielectric. Interestingly, the topload on the Tesla coil acts like a one-plate capacitor, with the ground plane surrounding the coil acting as the opposing plate. The capacitance of the topload is determined by its dimensions and its proximity to other objects.

Inductors store energy in the form of a magnetic field around a wire, or in the middle of a loop of wire. The primary inductor in the oneTesla 10” coil is six turns of AWG14 wire, and the secondary is approximately 1800 turns of AWG36 wire.

An LC circuit can have an inductor and capacitor in series or parallel. Here, we are using series LC circuits like this:

 Consider what happens when you don’t drive the circuit (assume that the AC source in the above figure is replaced by a wire), but start out with the capacitor charged. The capacitor wants to discharge, so charge flows around the circuit, through the inductor, to the other plate. In the process, a magnetic field builds up inside the inductor. When the charge on each plate of the capacitor is zero, current stops flowing. But at this point, the inductor has energy stored up in a magnetic field - which tends to oppose change. The magnetic field collapses, inducing a continuing current in the same direction, thereby recharging the capacitor and restarting the cycle in the opposite direction.

The resonant frequency of an LC circuit, or the frequency at which the energy cycles between the capacitor and inductor as described above, is:

Driving the circuit at its resonant frequency adds energy during each cycle. By providing a succession of well-timed pushes, we can build up to extremely high voltages! In the Tesla coil, a spark breaks out and discharges the circuit once the voltage is high enough.

DRSSTC

The oneTesla 10” coil employs a double-resonant topology, hence the name double-resonant solid-state Tesla coil, or DRSSTC. In a DRSSTC, the circuit driving the secondary LC circuit is another LC circuit, tuned to the same resonant frequency.  In the following diagram, L_pri and L_sec are the primary and secondary inductors, respectively. They are weakly coupled, linking around one-tenth of their magnetic fields.

There are several reasons why Tesla coils do not employ a magnetic core. First of all, the voltages in the Tesla coil are so high that the core would quickly saturate, meaning it would no longer be magnetizable past a certain point. Also, most materials pose a resistance and heat up in a magnetic field that switches rapidly, as is the case in the coil. The high voltage the coil produces also has the potential to arc to the core. But most importantly, it’s critical that the primary and secondary coils be loosely coupled—so the secondary is not loaded down by the primary.

Half-Bridge

How do we go about exciting the primary? We use a DC voltage source, and apply the voltage in alternating directions across the primary.

 The switches that we use to apply a DC voltage in alternating directions across the primary are IGBTs, short for insulated gate bipolar transistors. An IGBT is a transistor capable of controlling very high voltages and currents. This is its schematic symbol:

Its terminals are labelled collector, gate and emitter as a holdover from vacuum tubes, before the era of transistors. A simplified model of an IGBT is normally open switch that closes when a positive gate voltage (VGE) is applied. In the following diagram of a half-bridge, S1 and S2 represent the IGBTs. They alternately turn on and off, which switches the polarity of V_bus/2 across L_primary and C_primary, the primary inductor and capacitor. The oneTesla 10” coil runs off a bus voltage of 340VDC, which we get from rectified and doubled line voltage.

 On the control board, we get the bus voltage from rectfied and doubled line voltage.We’ll go into detail about that portion of the circuitry later.

Zero-Current Switching

When the IGBTs are fully on (the switches closed), they are nearly perfect conductors. When they are fully off (the switches fully open), they are nearly perfect insulators. When they are in transition between fully open and fully closed, or vice versa, however, they behave like resistors. Recall that the amount of power dissipated in a circuit is P=VI. If we try to switch the IGBT while the current through the circuit is large, then it will heat up a lot! We have to time the switching of the IGBTs to the natural zero-crossings of the primary LC circuit. On the oneTesla board, we achieve zero current switching by sensing the primary current and using control logic to ensure the transistors switch at the correct times.

Gate Driving

The IGBTs are far from ideal switches. We want them to switch fast, to minimize the time during which they are resistive and dissipate power. The issue with switching gates fast is that they have significant internal capacitance, and it takes a lot of charge to fill up that capacitance and achieve the turn-on voltage across the gate (the voltage of a capacitor is given by V=Q/C).

 To charge CGE in as short a time as possible, we want to use a short, high-current pulse. Gate drive ICs are designed to do exactly this. We use UCC3732x ICs, which can source up to 9A for brief pulses. The logic circuitry preceeding the gate drivers isn’t even close to being able to source enough current to turn the gates on fast, so the gate drivers are essential components. Lastly, we need to isolate the gate drivers from the IGBTs using a gate drive transformers (GDT) Each IGBT needs gate voltage to be applied between its gate and emitter to turn on. This is easy on the low-side (bottom) IGBT - its emitter is always at ground, meaning its gate only needs to be brought up to +15V. Things are not so easy with the highside (top) IGBT, because its emitter is referenced to the collector of the low-side IBGT, a node which swings between 0 and V_bus/2 (which, in our case, is 170V). This means we need to bring the gate of the high-side IGBT up to V_bus/2 + 15V to turn it on.

 Fortunately, there is a simple way around this! We can drive the primary of a 1:1:1 transformer with the (bipolar) drive signal derived from a push-pull pair of UCCs. More specifically, we drive the primary of the transformer with the difference of the outputs of an inverting and non-inverting gate driver. This ensures that half the time, this signal is positive, and half the time, this signal is negative. Due to transformer action, the voltage across each secondary of the GDT is guaranteed to replicate the voltage across the primary, no matter where we connect the ends. This means we can simply connect a secondary across the gate and emitter of each IGBT, and guarantee that V_ge will always swing between 0 and 15V (regardless of the emitter’s potential).

Rectifier and Doubler

 The half-bridge in oneTesla is driven by a doubling rectifier as shown in the diagram above. This rectifier alternately charges each capacitor on alternating half-cycles of the AC input, resulting in twice the source voltage across the load. On the positive portion of the cycle, the top diode conducts and charges the top capacitor.

 On the negative portion of the cycle, the bottom diode conducts and charges the bottom capacitor. The voltage across the load is the sum of the voltages on each capacitor.