The Radio Amateurs Handbook has description and chart (In 1968, P-48) showing conditions for tuned-tuned coil coupling. When building a first Tesla Coil (1953) I knew of this relationship but did not grasp full implication nor have a clue how to go about implementing optimum coupling. A coil coupling evaluation technique is now at hand. Needed Components:
The potentiometer is excited with appropriate fixed level of DC voltage. The pot output wiper was connected so the pot output is fed into both the Oscilloscope X-axis drive and also into the Function Generator VCO Input. The way the VCO works is that for each knob selected output frequency range, zero DC volts VCO-input produces lowest frequency and max DC-volts produces highest frequency. So when the (linear taper) potentiometer knob is turned from low to high, the Function Generator output frequency sweeps from low to high as the pot is turned low to high. At the same time the Oscilloscope X-trace moves along the X-axis from left to right. (BTW; If you buy a Function Generator -> be sure it has a VCO input!)
The Function Generator (sine wave) output frequency is connected to the Tesla Coil primary winding through a resistor. The primary tuning capacitor is essentially disconnected by virtue of the spark gap. A 10x Scope Probe is connected to O-Scope Ch-2; the probe tip and ground are connected to the isolated Sense Coil. When the Secondary Coil resonates, an large increase in AC voltage will be induced in the Sense Coil and drive the Y-axis of the Oscilloscope. The Oscilloscope is now performing as a Spectrum Analyzer that will 'bloom' in Y-amplitude when the X-position of the trace is at a frequency near resonance of the secondary coil.
One last chore is to mount the TC secondary into the primary in such a way that the secondary can be raised and lowered concentrically in a systematic way. I used a small nylon rope connected to the top terminal and threaded over an overhead pulley A potentiometer sweep test is implemented between each S-coil height change. This is when the 'way-cool' happens. The coil coupling charts in Radio Amateurs Handbook can be now be replicated for real and in much better detail.
When the S-coil is somewhat removed from the P-coil, the resonant response will be a single sharp peak when the potentiometer is adjusted around one small area; this is when the pot output voltage driving the VCO is near the resonance frequency of the secondary. Note; the secondary breakout electrode should be installed during this test. As the S-coil is lowered, the peak will get larger in amplitude but remain 'sharp'. Continued lowering will eventually cause the peak to decline and broadened in width. At the height where the peak first begins to dip is called critical-coupling. Further lowering of the S-coil causes over-coupling with a very interesting result. If the pot is swept just below to just above the resonant frequency, the full envelope of resonant response is seen. Further S-coil lowering into over-coupling continues to diminish the S-coil response at the center resonant frequency position, but two side bands peaks come up, one on either side of center resonance.
An explanation of what determines secondary coil resonant frequency is needed here. The secondary always has a fundamental resonant frequency. It is measured with the bottom lead connected to ground plane. In addition the HV top capacitance terminal should be installed on the secondary. The secondary coil resonant frequency is controlled by the inductance and distributed capacitance (coil turns & coil geometry). Distributed capacitance includes winding to winding femto-farads plus the coil top terminal capacitance to free space (pF). I might inject here that capacitance to free space goes to square root of the area of the body (terminal capacitor). Also be aware; Walking up along side a secondary will add significant secondary capacitance and lower the resonant frequency that the coil needs to be driven at for an exciting output (more on this topic later). The S-coil resonant frequency can be computed from (f)=1/(2PI)/SQRT(LC). (L) can be computed and (C) can be estimated or measured with a C-meter. Or alternatively, the effective (C) can be found by determining and plugging in the resonant frequency and inductance.
Secondary coils have a very sharp stand alone resonant frequency. The Tesla Coil primary is tuned to match this frequency (Tuned-Tuned). If the secondary is critically coupled, the resonant peak is very narrow (in bandwidth). Anything that changes the S-coil resonant frequency will drive resonance away from the carefully fix-tuned primary frequency; the HV output will decline. Here is the payoff. If the secondary is lowered slightly further into the primary coil, the two coils will be over coupled causing the secondary apparent resonant frequency peak to be MUCH broader. This will minimize inadvertent tuning mismatch during coil operation.
If the spectrum analyzer is used, actual coil output at various couplings can be compared to the shape of the coupled resonance response at that coupling position to determine best operation with minimum experimentation. One gets to see intrinsic physics instead of a chart in a book.