11 Tesla coil theoretical model and experimental verification J. Voitkans (Researcher, RTU), A. Voitkans (Researcher, LU) Abstract – In this paper a theoretical model of a Tesla coil ending. This construction allows to slightly increase output operation is proposed. Tesla coil is described as a long line with voltage. scattered parameters in a single-wired format, where the line voltage is measured against electrically neutral space. By using a principle of equivalence of one-wired and two-wired schemes it is C C shown that equivalent two-wired scheme can be found for a single-wired scheme and the already known theory of long lines can be successfully applied to a Tesla coil. A new method of multiple reflections is developed to characterize a signal in a long line. Formulas for calculation of voltage in a Tesla coil by coordinate and calculation of resonance frequencies are proposed. Theoretical calculations are verified experimentally. Resonance frequencies of a Tesla coil are measured and voltage L2 L2 standing wave characteristics are obtained for different output capacities in a single-wired mode. Wave resistance and phase coefficient of a Tesla coil is obtained. Experimental measurements show good compliance with proposed theory. Formulas obtained in this paper are also usable for a regular two-wired long line with scattered parameters. e(t) L1 e(t) Keywords – Tesla coil, single-wire scheme, long line, resonance frequencies. L1 Introduction Tesla transformer is a device which is used for obtaining a b high voltage. It is invented by genial Serbian scientist Nikola Figure 1: Tesla transformer Tesla. Importance of his works in science and technology is hard to overestimate. Many of N. Tesla’s discoveries so There are other Tesla transformer construction variations predated their time, that only now we can fully assess their where is winded in cone or like flat spiral. Transformer size essence, but some of them are still waiting for their time, for may vary from few centimetres to few tens of meters. Output example, wireless transfer of energy in great distances. voltage may be from few tens to millions of volts. In Currently Tesla transformer operating principles are not experiments in Colorado Springs N. Tesla gained voltage of sufficiently explained, model of operation that would describe approximately 50 million volts. Electrical voltage spatial physical processes in Tesla coil is not created. It is very distribution by a length of coil coordinate is shown in difficult to design Tesla coil and predict its necessary Figure 2. properties and parameters. Tesla transformer design is shown in Figure 1 (a). It consists of primary winding with small number of winds which is operated by generator . Operating currents of this winding can be relatively big; it is winded with increased diameter wire because of that. Secondary winding (Tesla coil) consists of much more winds which are winded with small diameter wire usually in one layer. Upper ending of the winding is connected to load, in this case spatial capacity (conducting sphere, hemisphere, ellipsoid, it may also be a load with two connectors). Tesla coil lower ending is connected ether to ground or generator casing. In Figure 1 (b) Tesla autotransformer variant is shown. In this variant upper ending is connected with lower Figure 2: Voltage U distribution by coil length 12 In Figure 2 can be observed that voltage by Tesla coil Cv Cv length coordinate is gradually increasing with a maximum in Cs Cs its output. General transformer output voltage is proportional to ratio of secondary and primary winding number of winds. Tesla transformer output voltage does not correspond to this regularity, it is many times higher. To get high voltage in output it is necessary to tune it to some particular frequency Cs Cs which is called working or resonance frequency. Anomalous Cv Cv voltage increase in output could be explained by occurring of series electrical resonance in secondary winding because this Figure 4: Tesla coil winding scheme resonance is characterized by voltage increase on reactive elements times comparing with input voltage, where is In Tesla coil theory processes occurring in Tesla coil can be contour goodness [1]. Nevertheless attempts to describe described knowing its inductivity by length unit and processes occurring in Tesla transformer in bounds of the winding slef-capacity by length unit , resonance frequencies voltage resonance theory have not yielded results [2], [3]. can be calculated, phase coefficient , characteristic In this work Tesla coil theoretical model is given and impedance can be determined. comparison of theoretically acquired parameters with Co(F/m) Co(F/m) Cizo(F) experimental measurements is performed. Lo(H/m) Csl(F) Tesla coil theoretical model e(t) In the proposed theoretical model it is assumed that Tesla Co(F/m) Co(F/m) Co(F/m) transformer high voltage winding is operating in a long line with distributed parameters mode with multiple reflections Figure 5: Tesla coil model electrical scheme from coil endings. Direct and reflected voltage and current waves are forming which mutually summing create different Tesla coil model electrical scheme (Figure 5) consists of standing wave sights by corresponding resonance frequencies. sinusoidal electrical voltage generator , where one end is In this work Tesla transformer secondary coil operation is connected with ground or other big spatial capacity. Its output described as long line in single wire electrical system format voltage is and frequency . To the output of a generator [4], [5], [6], where circuit currency is measured against beginning of a Tesla coil winding with a length and diameter infinitely distant sphere (i.e. against neutral surrounding is connected. Other end of the winding is connected to a space). In free space positioned electric wire of which spatial load capacity which together with the coil idle secondary winding consists equivalent scheme is shown in running capacity (Figure 5) is forming combined output Figure 3. capacity . Any electrically conductive body with self- C capacity can be a load capacity. Spherical body has Lv self-capacity , where sphere radius. It is possible to measure a self-capacity of a conductive body [8]. Lv Lv Main characteristics of Tesla winding are inductivity by C C length unit (H/m) and capacity by length unit (F/m). Figure 3: Single wire equivalent scheme Virtual ground of single-wire scheme In scheme (Figure 3) is straight wire inductivity by Any single-wired electrical scheme can be substituted by an length unit (several microhenries on meter), is wire self- equivalent two-wired scheme where as a common connection capacity (capacity against infinitely distant sphere) by length serves virtual ground. Its theoretical description and unit (approximately 5pF/m). and values are dependant experimental verification is shown in work [7]. Existence of from length and diameter of wire. Tesla coil that is winded virtual ground is discovered theoretically by describing one- from such a wire is shown in Figure 4. In this case , wired schemes with Kirchhoff equations, where electrical because there are several winds in length unit, and capacity and electrical potential of a conductive body is because additional inter-wind capacity is introduced. is determined against infinitely distant sphere. Illustration of dependant from by the side positioned wind distance and virtual ground is shown in Figure 6.a. diameter. In Figure 6.a. a spherical two layer condenser is shown As we are viewing Tesla coil as single-wire long line with where one electrode is in free space placed spherical distributed parameters, by applying alternating input voltage, electrically conductive body with diameter , which is voltage and current travelling wave is introduced with centrically encased by spherical hollow electrode with multiple reflections from winding ends. At certain frequencies inner radius . Electrical capacity of such two layer condenser with different wave lengths resonance occurs. is 13 distance to VZ boundary for coil is smaller than for sphere because the diameter of the coil is smaller than the diameter of the sphere. Virtual ground is connected with a If radius tends toward infinity, which corresponds to casing of the generator , because displacement currents infinitely remote sphere, as defined in electro-physics, with which are generated by single-wire scheme capacities connect zero electrical potential , then sphere with radius to it. gains capacity Physically free, neutral space with a zero electrical potential , (2) serves as a virtual ground. Displacement currents created by which is also called self-capacity of a conductive body self-capacity of bodies which are proportional to a speed of towards infinitely remote sphere. Electrical potential of a change of electric field and capacity value flow in it. body is determined against a zero potential of this sphere, in electrical resistance for displacement currents is close to this case zero. It is ensured by huge cross-section area of a virtual . electric wire. Joule – Lance losses of are close to zero, too. φ0=0 r2 Tesla transformer equivalent two-wire scheme As one-wire connection electrical scheme (Figure 5) is r1 completely equivalent to two-wire scheme (Figure 7) and it C corresponds to traditional long line, a known theoretical description of long line with distributed parameters can be applied to one-wire system [9]. V Z Lo Lo Lo Lo a Ri Co Co Co Ciz0 Csl e(t) V Z Virtuālā zeme r 0 l x L rL C e(t) C Figure 7: Tesla coil equivalent two-wire scheme and frame of reference φ0=0 Zero point of coordinate (Tesla coil beginning), direction of change and length of line (Tesla coil ending) is shown in b Figure 7. Figure 6: Conductive body To simplify calculations, capacity , as shown in Figure and its virtual ground (VZ) 7, which is created by last winds of the coil and output, and load capacity are unified into one output capacity .