Recently, the author has received many questions regarding the correct calculation of the Tesla transformer [1]. Indeed, there are many methods and calculators for such calculations, in which you can quickly get lost. And we are talking only about fairly accurate methods, for example [2]. And if we consider this transformer as a long line, it becomes completely confusing, because the quarter wavelength frequency can differ by 40 percent or more from the actually measured resonant frequency. With such introductory information, any engineer can quit this thankless task without even starting it :)

In this work we will try to eliminate the contradictions in methods and develop a unified algorithm for calculating not only the resonant frequency in the quarter-wave mode, but also higher modes, which were either not previously calculated at all, or were simply taken as a multiple of the fundamental frequency, which, in relation to the Tesla transformer, is a gross mistake. To do this, we will focus here on the secondary winding L2 of this transformer, and we will further call it Tesla coil (TC). And we will apparently devote a separate work to the features of calculating the primary winding L1, which is also called an inductor.

Also, in this note we will focus on one non-classical effect that occurs when distributed and concentrated resonance are combined. It will reveal another facet of this amazing transformer, will help our readers master its quick calculation, which means that as a result we will get another direction for the study of free energy.

If we approach TC from a classical point of view, then it should be considered as a long line [3]. But if you calculate the length of a quarter wave and convert it into frequency, and then measure the resonant frequency of a real TC scan, then the discrepancies can sometimes be 40 percent or more. Taking into account the wave slowdown in a real waveguide will not improve the situation much. Let us further consider real examples in which the TC winding diameter is 50 mm, and the resonant frequency is determined by the quarter-wave mode of its operation.

Example TC No. 1
• Number of turns: 200
• Winding height, mm: 200
• Calculated frequency along the wire length, MHz: 2.38
• Real frequency, MHz: 3.19 (calculator).

The actual operating frequency of the TC is quite accurately calculated by this calculator, taking into account the grounding capacitance, which for small coils is usually several picofarads. As you can see, the difference between the frequency calculated along the length of the wire and the actual operating frequency of the TC differs by 34%. It is this parameter that is indicated in the presented calculator as the “wave propagation speed coefficient”. Using the wave resonance effect, some researchers build their own free energy generators.

Example TC No. 2
• Number of turns: 1000
• Winding height, mm: 400
• Calculated frequency along the wire length, MHz: 0.477
• Real frequency, MHz: 0.807 (calculator).

In this example, the difference between the frequency calculated along the length of the wire and the actual operating frequency of the TC differs by 69%. The general relationship is obvious: the greater the ratio of the winding height to its diameter, the higher the wave propagation speed coefficient. This directly leads to the conclusion that it is better not to make this ratio too large; 1.3-1.8 is considered optimal. The residual mismatch with wave resonance can be eliminated by adding an isolated capacitance to the hot end of the TC. For example No. 2, this can be done like this: attach a metal sphere with a diameter of 200 mm to the TC (calculator). As we can see, despite the rather large ratio of the winding height to its diameter, the wave propagation speed coefficient will become equal to unity.

From all of the above, we can conclude that for TC, calculation along the length of the conductor is important only for wave resonance. Quarter-wave resonance is calculated using an expression close to Thomson’s formula for lumped resonance [4], which is included in the calculator. Detailse research on this issue was carried out by the author here.

TC can operate at higher frequencies, where the same winding will “ring” at 1/2, 3/4, 4/4 wavelengths, but the calculation of these frequencies looks even more interesting. Researchers usually try to avoid this issue, because here the differences between theory and practice can be even greater. The differences are clearly visible, for example, in two graphs from this work, in which the author studied this issue. To find wave modes in TC, calculation along the length of the conductor is again not needed, no matter how contradictory it may sound. Why? We will try to answer this question at the end of the note.

So, how to count wave modes for TC? To do this, you will need to know only two frequencies: \(f_g\) and \(f_s\). In real measurements they are obtained very simply: \(f_g\) is the resonant frequency of the TC with connected grounding and, if any, with a solitary capacitance (sphere, toroid), and \(f_s\) is the resonant frequency of the TC in its pure form, without grounding and isolated capacitance. They are obtained in the classic quarter-wave mode for TC (first mode), when there is a maximum voltage at the hot end, and a maximum current at the cold end.

These two frequencies can be approximately obtained using a calculator. For example No. 2, with a connected sphere, they will be located like this: \(f_g\) = 0.493 MHz, \(f_s\) = 1.09 MHz.

Then the wave modes will be calculated according to the following rule.
Frequencies for odd modes: \[ f_1 = f_g \\ f_i = \left({i+1 \over 2}\right) f_s + {f_g \over 4}, \quad i \in 3, 5, 7, 9 ... \tag{ 1} \] Frequencies for even modes: \[f_i = \left({i+2 \over 2}\right) f_s, \quad i \in 2, 4, 6, 8 ... \tag{2} \] where: \(f_i\) is the resonant frequency of the coil on the i-th mode, \(i\) is the number of the wave mode.

For example No. 2, with a connected sphere, the first mode is 0.493 MHz, the second is 2.18 MHz, the third is 2.3 MHz, the fourth is 3.27 MHz, etc. From the resulting series it is clear that to obtain the mode frequency of a TC, there can be no talk of any simple multiplication of any fundamental frequency by integers!

The algorithm for calculating TC is as follows. First you need to decide on the mode number and operating frequency. Then select the frame for the coil and the wire with which the TC will be wound. Using calculator find the number of turns in the TC and the winding height. Then, if required, select the parameters of the solitary capacitance in the calculator to adjust the TC to the wave resonance.

Let's look at this algorithm using an example. We need the first mode (quarter-wave mode of the classic Tesla transformer) and a resonant frequency of 1 MHz. We have a 50 mm frame and a 0.3 mm thick wire (the thickness is taken taking into account the thickness of the varnish coating). In the calculator we need to get the “Resonant Frequency” at 1 MHz and “Winding step” 0.3 mm. In fact, it is impossible to wind the turns without a gap, so, usually, they take a margin of 5-7%. Thus, we will adjust the “Winding pitch” to 0.32 mm. Let's enter the diameter of the frame into the calculator, plus half the thickness of the wire, which in total will be equal to 50.15 mm. We will also add a grounding capacitance, which for such a coil can be about 3 pF. And then, by selecting the number of turns and winding height, we achieve a resonant frequency of 1 MHz and a winding pitch of 0.32 mm. In this case, the author got such a TC.

As we can see from the example, we have a wave propagation coefficient equal to 1.3. If we need to obtain wave resonance from this TC, then we need to reduce this coefficient to unity. To do this, let's do one more action - connect the “External Capacity”, for which we will select, for example, a metal ball. In the calculator you need to enter its diameter, constantly increasing this parameter, until you obtain a wave propagation coefficient equal to unity. For our example, it will look like like this.

AfterConnecting a solitary container (ball), the resonant frequency of the TC dropped to 711 kHz. If we fundamentally need the initially set frequency of 1 MHz, then it must be selected by manipulating three parameters, without forgetting about the wave propagation coefficient, if we need to receive wave resonance from the TC. The author, for the example under consideration, came up with this version of TC.

Obviously, instead of a ball, you can connect a toroid, also calculating its parameters in this calculator.

Also, the author can recommend a wonderful approach to calculating TC for wave resonance, performed by Sergey Stalker in this video [5]. There, the author also offers a calculation of the inductor to obtain a full-fledged Tesla transformer.

By developing this algorithm, we can calculate higher modes for the operation of the resulting TC at 1/2, 3/4, 4/4 wavelengths (2nd, 3rd and 4th modes). To do this, it is enough to use formulas (1, 2) from this work. In the above example, without connecting a separate container, the frequency values for the modes will be as follows: the first mode is 1 MHz, the second is 2.94 MHz, the third is 3.19 MHz, the fourth is 4.41 MHz. By applying these frequencies to the TC using an inductor, we will be able to observe the corresponding resonant distributions of current and voltage on the coil.

When working with the Tesla transformer, many researchers note its non-classical behavior in some modes. The same difficulties arise when compiling a mathematical apparatus that describes its properties. The author managed to assemble in this work a mathematical model of the primary coil of this transformer (TC), and present it in the form of formulas, a calculator and the corresponding algorithm. All this makes it possible to calculate TC both for the quarter-wave mode and for higher modes.

As a result of the research, results were obtained that clearly indicate that TC cannot be calculated using classical formulas for a long line, although they can be used to achieve wave resonance. Apparently, two waves propagate in TC: longitudinal and transverse. The first one has a speed that sometimes exceeds light speed [2], and can move not only along the conductor. It carries most of the total energy of the wave and is calculated using the lumped resonance formulas. The second wave, transverse, propagates in a classical manner and is calculated using the distributed resonance formulas.

Combining the frequencies of these two types of waves leads to the appearance of a new non-classical effect - wave resonance, manifested in an increase in the amplitude of current and voltage in the TC.