Dissertation on the Quarter-Wavelength Principle as Applied to Tesla Coils

by
G. W. Legel

Judging by questions asked by TCBA members, including myself, there seems to be a need to explore the quarter-wavelength principle used in the design of single-ended (grounded) coils used for Tesla coil secondaries. The principle stated by present day Tesla coil designers and by Tesla is presented following:

Use of this principle as applied to the design of Tesla coil quarter-wave grounded secondaries will be examined to determine the extent of applicability. A design frequency of 100 kHz is chosen. The quarter-wave wire length is therefore:

Winding a secondary coil with this length of wire will usually result in a Tesla coil whose frequency is other than the selected value of 100 kHz. This comes with no surprise to experienced coil builders but newcomers may be frustrated when they find their coil operating on a different frequency than they had intended. This event is really of little consequence since one need merely accept this frequency and tune the Tesla coil primary to suit, by changing the number of turns. This is the method used in the great majority of coils, since the primary capacitor is usually of a fixed value.

There are a number of reasons for the frequency being other than that provided by a coil whose wire length is determined by the quarter-wavelength principle. These follow:

1. The wave propagation velocity of an electromagnetic wave in space is the same as that for the velocity of light, that is 300,000,000 meters per second. This is the figure used in the quarter-wavelength principle. Propagation velocity in a conductor is slower than in space. The actual velocity is dependent mainly on the conductor length to diameter ratio. A factor known as the K factor is one which must be applied to a given size conductor to determine its length as compared to the free space length. A typical example is that for number 12 wire with a length/diameter ratio of 10,000 K is 0.98. Thus a wire of actual quarter-wavelength would resonate at a lower frequency. However, this particular factor has a relatively small effect on a Tesla coil secondary frequency.
2. Of greater consequence in changing the frequency of a coil, wound with a given wire length, is the effect of various capacitances inherent in the construction of a Tesla coil secondary. These include the interwinding capacity, the terminal capacity (sphere or toroid), and the virtual sheet or cylinder capacity of the coil surface to ground. The Tesla coil secondary circuit can be considered to be a parallel tuned circuit utilizing the coil and the capacitances listed. These capacitances are difficult to ascertain by calculation. Terminal capacity, sphere or toroid, is dependent on size and physical location with respect to ground. Empirical data is available for terminal capacitance. Interwinding and cylinder surface capacitances are obtained with difficulty.
3. Of great significance is the variation in inductance of a coil wound with a given length of wire, with variation in coil diameter. Inductance changes and, therefore, resonant frequency changes with a change in coil diameter. Inductance vs. coil diameter for the fixed length of wire chosen is given in Table I.

Table I

Coil Inductance vs. Coil Diameter for a Given Wire Length