## By Antonio Carlos M. de Queiroz

Below are two plots that show the maximum output voltage obtained by varying the elements of the primary circuit (C1 and L1) and the coupling coefficient of a lossless Tesla coil. A normalized ideal Tesla coil with C1 = 1 F, L1 = 1F, C2 = 1F, and L2 = 1F, with vC1(0) = 1 V, was simulated for 70 seconds, and the maximum absolute output voltage obtained in this time recorded. The maximum time was chosen to guarantee that the maximum output is reached even with the lowest k used.

In the first plot, L1 (x) was varied between 0.01 H and 4 H, while the coupling coefficient k (y) was varied between 0.05 and 0.9. The results show that tuning (L1C1 = L2 C2) is more important when the coupling is low, and that the coupling has relatively small influence on the maximum voltage. Sets of maxima correspond to a series of particular values of k, as is well known (k = (b2-a2)/(b2+a2) with a and b being integers with odd difference, b>a, that define the operating mode a:b). The widest maximum being for k = 0.6 (mode 1:2). A series of trenches appear in the plot, the most deep corresponding to k = 0.8 when L1 = 1.

The next plot shows the effect of varying C1, between 0.01 F and 4 F, while k varies between 0.05 and 0.9. It can be observed that larger C1 always result in larger output voltage. With low k, this happens for just a small increase in C1 before detuning reduces the effect, but after a certain value of k, increasing C1 increases the voltage continuously, with it tending to a value proportional to k for large C1. The output voltage is always greater than the normal with C1 increased after k = 0.5, tending to twice the normal value for large C1 and k = 1. This is, however, of little practical interest, since the system loses efficiency, with just a fraction of the initial energy in C1 being transferred to C2. Note the trenches again, with depths that vary periodically.

The two plots below illustrate the effect of losses. In this case a resistance of 20 Ohms in parallel with the output capacitance, corresponding to a quality factor Q = 20 for the secondary circuit. The variation of L1 just produces more clearly defined peaks, practically at the same places.

The variation of C1 shows maxima skewed in the direction of larger C1, and reduction in the output voltage for large C1. (further increase in C1 results again in the maximum output voltage proportional to the coupling). The trenches are smoother in these lossy cases.

For a regular Tesla coil, only the range with low k is of interest, and there the effects of changing L1 or C1 are similar. But for energy conversion devices operating with high coupling, the increase in the output voltage caused by increasing C1 may be of interest.

Thanks to Terry Fritz for the idea of making this kind of plot. These were made with a program that I wrote, but the same thing can be done with Terry's Scantesla program (see the archives of the Tesla list) and a suitable plotting program.

Created: 21 July 2005
Last update: 28 July 2005
Developed and Maintained by Antonio Carlos M. de Queiroz