A few turns of heavy copper around the base of the secondary. It is not connected to the secondary anywhere, and it is not trying to be.
What it is and why
The bridge pushes current back and forth through those turns at the coil's resonant frequency. That current makes a magnetic field, the field passes through the secondary, and because the field is changing it induces a voltage there. The secondary has hundreds of times more turns, so the voltage that appears across it is hundreds of times larger.
That is a transformer, and if it were only a transformer nothing interesting would happen. What makes a Tesla coil is that the secondary is a resonator, so the voltage builds up cycle after cycle instead of being stepped up once.
Why it is thick. The current in it is measured in tens or hundreds of amps. Copper tube or wide strip: at these frequencies the current only travels on the surface, so the middle of a round wire is not doing anything, which is also why several thin wires beat one thick one of the same area.
Why it is short. A handful of turns is enough inductance to be part of the tuned circuit. More turns means more inductance, less current and a lower frequency, and all three of those are levers you use deliberately.
Why it stands beside the secondary rather than around it. Coupling is how much of the primary's field actually reaches the secondary, and it depends on geometry: how tall the primary is relative to the secondary, its diameter, and where the secondary sits inside it. More coupling means more of the field is shared. It does not mean a longer arc, which surprises everybody, and there is an article about why.
The one decision this page is enough to make
Not the inductance. The impedance, which is the number a designer actually sets and which the primary current then falls out of rather than being chosen. That is the primary is an impedance, and it is where turns and tank capacitance get their values.
What will get you
And bringing the two coils too close breaks the secondary across its own winding. That is the ceiling on coupling, and it is set by the winding's geometry rather than by any calculation you can do in advance.
What goes wrong
- The copper gets hot enough to soften. High bangs per second. Air and cross-section, both taken seriously.
- A flashover across the secondary's winding, low down. Racing sparks. Too much coupling, and the ceiling is lower on the lower pole than on the upper.
- The frame draws kilowatts and the bolts melt. The shorted turn above.
- The current is nothing like the calculation. The calculation probably chose turns. The impedance chooses the current.
Where next
- The primary is an impedance, the design number.
- Sizing a tank capacitor by RMS current, the part in series with it.
- An SSTC primary is a different part, where the same component is wound the opposite way for the opposite reason.
The primary on the department diagram stands beside the secondary rather than lying inside the bridge, because the two being on one axis is the only way they are coupled at all.