Tesla Coil Theoretical Model and its Experimental Verification

Transcription

1 Electrical, ontrol and ounication Engineering 4/7 Tesla oil Theoretical Model and its Experiental Verification doi: 55/ecce-4-8 Janis Voitkans (Researcher, Riga Technical niversity, Arnis Voitkans (Researcher, niversity of Latvia Abstract In this paper a theoretical odel of Tesla coil operation is proposed Tesla coil is described as a long line with distributed paraeters in a single-wire for, where the line voltage is easured across electrically neutral space By applying the principle of equivalence of single-wire and two-wire schees an equivalent two-wire schee can be found for a single-wire schee and the already known long line theory can be applied to the Tesla coil A new ethod of ultiple reflections is developed to charactere a signal in a long line Forulas for calculation of voltage in Tesla coil by coordinate and calculation of resonance frequencies are proposed The theoretical calculations are verified experientally Resonance frequencies of Tesla coil are easured and voltage standing wave characteristics are obtained for different output capacities in the single-wire ode Wave resistance and phase coefficient of Tesla coil is obtained Experiental easureents show good copliance with the proposed theory The forulas obtained in this paper are also usable for a regular two-wire long line with distributed paraeters e(t L L e(t L Keywords Transforers; Resonance; Transission lines; Matheatical odel; Electroagnetic odeling I INTRODTION Tesla transforer is a device which is used for obtaining high voltage It was invented by genial Serbian scientist Nikola Tesla The iportance of his works in science and technology can scarcely be overestiated Many of N Tesla s discoveries so far passed ahead their tie, that only today we can fully assess their essence, but soe of the are still waiting for their tie, for exaple, wireless transfer of energy along large distances However, the operating principles of Tesla transforer have not been sufficiently explained and the odel of operation that would describe physical processes in Tesla coil has not been created It is very difficult to design Tesla coil and to predict its necessary properties and paraeters The design of Tesla transforer is shown in Fig (a It consists of a priary winding L with a sall nuber of turns which is operated by a generator The operating currents of this winding can be relatively high as it is wound with increased diaeter wire The secondary winding (Tesla coil L consists of uch ore turns which are wound with a sall diaeter wire usually in one layer The upper terinal of the winding L is connected to the load, in this case spatial capacitance (conducting sphere, heisphere, ellipsoid, it ay also be a load with two connectors The Tesla coil L lower terinal is connected either to ground or to the generator casing In Fig (b Tesla autotransforer variant is shown In this variant the upper terinal L is connected with the lower one L This construction allows slight increase of the output voltage L (a (b Fig Tesla transforer There are other Tesla transforer construction variations where L is wound in a cone or like a flat spiral The transforer se ay vary fro a few centieters to a few tens of eters The output voltage ay be fro a few tens to illions of volts In experients in olorado Springs N Tesla gained voltage of about 5 7 volts The electrical voltage spatial distribution along the length of coil l coordinate x is shown in Fig In Fig we can observe that the voltage along the Tesla coil L length coordinate is gradually increasing to axiu at its output Fig Voltage distribution by coil length Download Date /8/8 9: AM

2 Electrical, ontrol and ounication Engineering 4/7 The general transforer output voltage is proportional to the ratio of the nuber of turns of the secondary and priary winding The output voltage of Tesla transforer does not correspond to this regularity, it is any ties higher To get high voltage at the output requires tuning it to a particular frequency which is called a working or resonance frequency The anoalous voltage increase in output could be explained by occurring of series of electrical resonance in the secondary winding because this resonance is charactered by the voltage increasing across reactive eleents by Q ties as copared to the input voltage, where Q is contour Q-factor [] Nevertheless the attepts to describe the processes occurring in Tesla transforer in bounds of the voltage resonance theory have not yielded results [], [3] In this work the theoretical odel of Tesla coil is presented and the coparison of theoretically acquired paraeters with experiental easureents is perfored II THEORETIAL MODEL OF TESLA OIL The proposed theoretical odel assues that the high voltage winding of Tesla transforer is operating in the ode of long line with distributed paraeters with ultiple reflections fro coil terinals The direct and reflected voltage and current waves are foring the utual su, which creates different standing wave sights by corresponding resonance frequencies v s L v L v L v Fig 3 Single wire equivalent schee s v In this work Tesla transforer secondary coil operation is described as a long line in single wire electrical syste forat [4], [5], [6], where the circuit current is easured in accordance to infinitely distanced sphere (ie, against neutral surrounding space The equivalent schee of the secondary winding consisting of an electric wire positioned in free space is shown in Fig 3 In the schee (Fig 3 L V is the straight wire inductivity per length unit (several icro-henries per eter, V is wire selfcapacitance (capacitance against infinitely distant sphere per length unit (approxiately 5 pf/ L V and V values depend on the length and diaeter of the wire Tesla coil wound of such a wire is shown in Fig 4 In this case L > L V, because there are several turns per length unit, and > V because additional inter-turn capacitance S is introduced S depends on the distance and diaeter of the turn positioned by the side As we are viewing Tesla coil as a single-wire long line with distributed paraeters applying alternating input voltage, voltage and current travelling wave is introduced with ultiple reflections fro winding ends At certain frequencies with different wave length resonance occurs In the theory of Tesla coil the processes occurring in Tesla coil can be described knowing its inductivity per length unit L and winding self-capacitance per length unit, resonance frequencies can be calculated, phase coefficient, characteristic ipedance Z can be deterined The electrical schee of Tesla coil odel (Fig 5 consists of sinusoidal electrical voltage generator, where one terinal is connected to ground or other large spatial capacitance Its output voltage is and frequency is f To the output of a generator beginning of Tesla coil a winding with length l and diaeter D is connected The other terinal of the winding is connected to a spatial load capacitance sl which together with the coil idle running capacitance (Fig 5 is foring a cobined output capacitance Any electrically conductive body with self-capacitance p = sl can be a load capacitance Spherical body has self-capacitance S = 4 r, where r is sphere radius It is possible to easure self-capacitance of a conductive body [8] r φ = s s r v v VG Fig 4 Tesla coil winding schee (a V G L(H/ (F/ (F/ (F sl(f e(t L r L r e(t (F/ (F/ (F/ Fig 5 Electrical schee of Tesla coil odel φ = (b Fig 6 onductive body (a and its virtual ground VG (b Download Date /8/8 9: AM

3 Electrical, ontrol and ounication Engineering 4/7 Main characteristics of Tesla winding are inductivity per length unit L (H/ and capacitance per length unit (F/ III VIRTAL GROND OF SINGLE-WIRE SHEME Any single-wire electrical schee can be substituted with an equivalent two-wire schee where virtual ground serves as a coon connection Its theoretical description and experiental verification is shown in work [7] The existence of virtual ground is discovered theoretically by describing onewire schees with Kirchhoff equations, where electrical capacitance and electrical potential of a conductive body is deterined regarding to infinitely distant sphere Illustration of virtual ground is shown in Fig 6(a In Fig 6(a a spherical double-layer capacitor is shown where one electrode is in a free space placed spherical electrically conductive body with diaeter r, which is centrically encased by spherical hollow electrode VG with inner radius r Electrical capacitance of such double-layer capacitor is expressed as 4 r ( r / r If radius r tends toward infinity, which corresponds to infinitely reote sphere, as defined in electro-physics, with zero electrical potential ( =, then the sphere with radius r gains capacitance 4 r, ( which is also called self-capacitance p of a conductive body towards infinitely reote sphere Electrical potential c of a body is deterined regarding to a zero potential of this sphere, in this case Nevertheless, current understanding of infinitely reote sphere in electro-physics creates probles for its practical use as it is difficult to understand how electrical force field lines could be copleted as it is required by principle of continuity of electric current after passing infinite distance if electrical field propagates with liited speed (speed of light Is it possible to change soething in our understanding of infinitely reote sphere? The answer is in forula ( It turns out that the conductive body acquires a value close to its selfcapacitance p if the radius of outer sphere r gets only several tens of ties larger than r Five percent boundary is achieved when r r, but one percent difference of p fro selfcapacitance value is when r r It can be concluded that the infinitely reote sphere is close to conductive body and it would be ore understandable to call it a virtual ground VG (Fig 6 For a coplex configuration body like generator e(t, shown in Fig 6(b, series connection with coil L and capacitance area of virtual ground VG depends on linear diensions of each separate coponent As we can see in Fig 6(b, the distance to the VG boundary r L for the coil L is saller than r for the sphere because the diaeter of the coil is saller than that of the sphere The virtual ground is connected with a casing of the generator, because the displaceent currents generated by the single-wire schee capacities are connected to it Physically free, neutral space with a zero electrical potential serves as a virtual ground The displaceent currents created by self-capacitance of bodies which are proportional to the speed of change of electric field and capacitance value flow in it The VG electrical resistance for displaceent currents is close to zero It is ensured by huge cross-section area of a virtual electric wire Joule-Lance losses of VG are close to zero too IV TESLA OIL EQIVALENT TWO-WIRE SHEME As the one-wire connection electrical schee (Fig 5 is copletely equivalent to the two-wire schee (Fig 7 and it corresponds to the traditional long line, the known theoretical description of long line with distributed paraeters can be applied to the one-wire syste [9] Zero point of coordinate x (Tesla coil beginning, direction of change and length of line l (Tesla coil ending is shown in Fig 7 To siplify calculations, capacitance as shown in Fig 7, which is created by last turns of the coil and output, and load capacitance sl are unified into one output capacitance These capacities are connected in parallel, due to that = + sl The long line is considered hoogenous and without losses It is assued that loss resistance per length unit R = and loss conductivity per length unit G = In the first approxiation it ay be done because Tesla coil Q-factor is large: Q >> V DIRET AND REFLETED VOLTAGE WAVE IN TESLA OIL When describing physical processes in Tesla coil (Fig 5 and its equivalent schee (Fig 7 it is assued that ultiple voltage and current wave reflections are occurring In addition to that coil ending atches idle running ode with a nature of capacitive load and beginning atches short circuit because generator e(t is connected to ground and its inner resistance R i is close to zero Generator e(t is a source of sinusoidal voltage with a frequency f and aplitude e( t sin( t, (3 where f It is assued that e(t is an ideal source of voltage, in this case inner resistance R i = as for Tesla coil R i << Z c e(t L R L R L G Virtual ground (VG l x Fig 7 Tesla coil equivalent two-wire schee and coordinate reference syste G sl Download Date /8/8 9: AM 3

4 Electrical, ontrol and ounication Engineering 4/7 The generator output voltage (3 in the long line induces a direct running-wave which is describable with two-arguent function u( x, t sin( t x, (4 t where is the phase coefficient The direct running-wave of the voltage propagates in axis x direction until reaches the end of line u( l, t t sin( t l (5 Phase offset angle value and sign is deterined by the load In case of capacitive load is negative and in case of inductive load is positive While idle running, when =, = too nder the influence of an unbalanced load (Z Z wave is reflected and propagates in a direction of generator oving the sae distance as the direct wave which depends on and hanging coordinate has to be counted off the end of the line, due to that x has to be substituted by (l x As a result we get a reflected wave u( x, t a sin( t ( x l (6 Reflected voltage wave at the beginning of the coil where x = is u(, t a sin( t l (7 At this point wave is reflected again in the direction of the direct wave due to short circuit Let us label it as u(x,t at VI ALLATION OF TESLA OIL LOAD INFLENE Angle that characteres the influence of a load to a Tesla coil can be calculated using the coplex reflection coefficient at the end of the line ~ ~ a ( l Z Z N ~, (8 ( l Z Z t (l a (l t where: coplex direct voltage at the output of a line; coplex reflected wave at the output of a line; Z characteristic ipedance; coplex output ipedance Inserting into the expression (8 the value of output ~ ipedance Z j /( we get Z ~ N ~ j j Module of reflection coefficient is ~ N Z Z (9 Its phase can be expressed using sinus or tangents of N ~ but ~ preference is given to arg( N function due to the fact that this function also respects the position of this angle in a quadrant and using (8 it can be expressed jz arg ( j Z j a ( l t ( l e ( ~ N There is a short circuit at the beginning of a line, due to that At this point there is no significance to a value of input capacitance because it is shunted by short-circuit which is created by the inner resistance of a generator which is close to zero Wave resistance Z value has a sall significance, either Hence at the x = coordinate a wave reflected in the direct direction is in an opposite phase with the falling wave ( ( at( a VII TESLA OIL AMPLITDE-FREQENY HARATERISTI Tesla coil is a resonance syste with a high Q-factor Q This coefficient shows how any ties the output voltage is higher than the input voltage Q To ake the deterination of an aplitude-frequency characteristic of Tesla coil easier it is assued that a voltage wave propagates in a line without losses, it is reflected Q ties without changing aplitude and then iediately disappears As a result the output voltage will be increased Q ties which confors to the reality When Q-factor of the coil Q would change the resonance frequencies should also change slightly, but this approxiation does not anticipate it It should be considered a shortcoing of this ethod To get a Tesla coil aplitude-frequency characteristic a transition fro two arguent function to a single arguent function is desirable It is possible to get rid of the tie diension by transferring direct (4 and reflected (6 wave functions to the coplex plane: e t t ie jx e( x, t e, (3 ( xl ( x, t a a e (4 This transforation allows writing direct and reflected signals in a coplex for as shown in the expressions (3 and (4 Direct wave in line jx t e Direct wave at the end of a line x = l together with the influence of a load l t ( l e Reflected wave at the end of a line x = l, using ( l a ( l e Reflected wave in a line x l x ( x l e a Reflected wave at the beginning of a line l a ( e It is reflected fro the beginning of a line again as shown in ( and becoes a reflected direct wave 4 Download Date /8/8 9: AM

5 Electrical, ontrol and ounication Engineering 4/7 l at( e Reflected direct wave at a current coordinate of a line ( xl at e Siilarly further direct and reflected waves can be found, ( x4l 4 ta e, ( x4l 4 at e, ( x6l 6 t3a e, ( x6l 6 3 a3t e and siilarly further Q ties If Q = 64 the last reflected wave in a line is ( x64l 64 3 t3a e It is beneficial to choose Q-factor of a coil equal to 4n where n is a natural nuber starting with, because in this case an analytical function of su of different waves can be easily found As an exaple a total voltage in a coil with n = 6 which corresponds to Q = 64 is shown x ( (5 64( t a at 3t 3a x Inserting into expression (5 corresponding direct and reflected waves and suing the by pairs, after siplification, an expression is obtained 64 4 csc(( l sin( l sin(3( l cos( ( l x je Multiplier je 3 l 3 l corresponds to the tie function (6 cos( t 3( l (7 It can be observed that cosine (7 arguent t 3( l does not anyore depend on coordinate x That eans that a standing wave has fored in a line, but the rest part of the expression (6 64 ( x 4 csc(( l sin( l sin(3( l cos( ( l x (8 shows an aplitude of the standing wave in a dependence on coordinate x Expression (6 in a general way for different values of Q-factor can be written as Q 4 csc( ( l sin( l l Q ( l Q sin( cos( ( l x je, (9 at Q/4 = n, where n is a natural nuber,, 3, sing (9 voltage for a line with other Q = 4n can be written For exaple, if n = 5 then Q-factor of the coil is Q = and the following expression is obtained: 4 csc(( l sin( l sin(5( l cos( ( l x je 5 l ( It is not allowed to use Q-factor of a coil Q = 98 in forula (9 because in this case Q/4 = 45 and it is not a natural nuber If n = is inserted in the expression (9 and it is siplified, a four wave suary voltage which corresponds to Q-factor Q = 4 is obtained: 4 4 sin( l cos( ( l x je l 4 ( Q = 4 is the sallest allowed value that is usable in forula (9 There is no upper liit for the natural nuber n Inserting into the equation (8 x = l, phase coefficient f L, substituting angle by the forula (, taking Z L / and applying odule to 64(x, an expression for the Tesla coil aplitude-frequency characteristic at the end of a line when Q = 64 is obtained where 64( f csc(( f l L sin( f l L cos( sin(3( f l L, ( j f L / arg (3 j f L / In Fig 8 an exaple of aplitude-frequency characteristic 64( f with values = V, l = 8, L = H/, = pf/, = pf is shown In Fig 8 it can be observed that at increasing frequency resonance output voltage decreases, it happens due to increasing phase shift which is caused by output capacitance At frequencies between base resonances icro resonances with sall aplitude are being fored, these can be considered as a background voltage in a coil Fig 8 Tesla coil aplitude-frequency characteristic Download Date /8/8 9: AM 5

6 Electrical, ontrol and ounication Engineering 4/7 VIII DETERMINATION OF TESLA OIL RESONANE FREQENIES Differentiating voltage aplitude ultiplier in the expression (9 by variable and equaling this derivative with zero its frequencies of voltage axiu in a line can be calculated To avoid an influence of icro axius it is advisable to use the ultiplier of the four wave variation ( of the expression (9 4 sin( l cos( ( l x 4 4 ( x by substituting x with line end coordinate x = l Deriving by variable and equaling it with zero, an expression is obtained cos( l and l / n (4 where n =,,, 3 Inserting into (4 expression L and solving it in respect to, resonance frequencies are obtained (n n (5 l L For exaple, for a quarter wave resonance n = : n (6 l L For wave resonance n = and its angular frequency is 3 n (7 l L Siilarly resonance angular frequencies for other waves can be found IX TESLA OIL STANDING WAVE ALLATION To acquire standing wave sight aplitude ultiplier of the expression (9, which depends on coordinate x, can be used and axiu voltage in a line can be noraled As a result noraled voltage is obtained, it is necessary to take the odule of it: N ( x cos( ( l x (8 sing the expression (8 and noraling it to the length unit l a voltage distribution in a hoogenous line without losses can be obtained The value of the phase coefficient has to be such that a corresponding wave resonance would set in Tesla coil Standing wave sight for a quarter wave resonance is shown in Fig In Fig 9 voltage distribution exaples for 3/4 (Fig 9(a and 5/4 (Fig 9(b wave resonances by coordinate x are shown Output of the coil is loaded by electrically conductive sphere with a diaeter D = 394 and self-capacitance p = sl = 9 pf Output idle-load capacitance which is created by last windings of the coil and coil output terinal is approxiately = pf Suary output capacitance = + sl = 39 pf (a (b Fig 9 Voltage distribution in Tesla coil for (a 3/4 and (b 5/4 wave resonances In idle load ode when = axial voltage of a line ax is equal to both all half waves, and output quarter wave If there is a reactive load at the output of a coil < ax In the case of a capacitive load voltage iniu point coordinate x oves closer to the load and output quarter wave gets shorter, ie, gets saller than /4, but in the case of inductive load coordinate x oves away fro the load and output quarter wave gets longer than /4 X DETERMINATION OF LOAD INDED ANGLE IZ IF STANDING WAVE DISTRIBTION IS KNOWN If the closest to a load iniu coordinate x and nuber of whole half waves in a coil n are known value can be calculated equaling the expression (8 to zero: cos( ( l x Physically a corresponding solution of this equation is ( l x / (9 By substitution of = / and = x /n into the equation (9 it can be obtained l n ( n(, (3 x which is usable both with capacitive and inductive loads in the case when n =,, 3 is a natural nuber of full half waves in Tesla coil If n =, the equation (3 gives = at x = l/3 that corresponds to idle-load of a long line If n =, idle-load ode corresponds to coordinate Forula (3 is not applicable to quarter wave resonance because it gives undeterined state (n = and x = For this resonance has to be deterined by other eans 6 Download Date /8/8 9: AM

7 Electrical, ontrol and ounication Engineering 4/7 XI DETERMINATION OF LOAD INDED PHASE SHIFT FOR QARTER WAVE RESONANE To deterine angle in a quarter wave resonance it is necessary to know the angle frequency of this resonance, the three quarters resonance frequency and its Deterining cos( fro the expression (9 and writing it for two first resonances an equation syste is obtained ( Z cos( ( Z ( Z cos(, (3 ( Z which is solved in respect to unknowns and Z As a result the product of output capacitance and wave resistance tan( Z (3 and the load induced phase shift for quarter wave resonance are obtained ( cos( arccos (33 ( cos( If approxiated angle ay be calculated easier As resistance of capacitive output is inversely proportional to frequency x = /(, knowing the voltage shift angle introduced by coil output reactivity at one frequency it can be proportionally calculated for other frequency, therefore ( / (34 Equation (34 gives a result which is approxiately by 5 % different coparing with forula (33 Knowing the quarter wave resonance, its wave length can be calculated that is not obtainable fro standing wave sight In order to do it, it is necessary to find at what coordinate x voltage reaches axiu, equation (8 has to be derived by x, substituted by = / and x = / 4, as well as equaled to zero Expressing the quarter wave resonance wave length fro the obtained equation we get: (x 4l (35 Forula (35 shows that in the case of negative the quarter wave / 4 is higher than a coil length l XII TESLA OIL AS A LONG LINE EXPERIMENTAL RESEARH Experiental schee for Tesla coil research is shown in Fig In a one layer wound as one-wire connection Tesla coil is attached to the voltage generator Γ-A output (Fig 5 Diaeter of the coil is D S = 8 c, length l = 8c, diaeter of enaeled wire including isolation D V = 3, nuber of turns n 6 Tesla coil inductivity if easured with alternating current bridge E7-8 at frequency f = khz is L = 76 H that eans L = H e(t x Fig Schee of the experient In the experients when load capacitance is required different diaeter etallic balls or cylindrical bars that are attached to coil output with an approxiately 5c long wire are used In easureents without load connection output wire reains and together with the last turn for an idle-load output capacitance of the coil Electrical field intensity distribution close to Tesla coil is registered either with a neon lap by observing the intensity of its glow or by voltage induced in an oscilloscope probe, which has high input resistance Measureents are perfored both when load is not connected, and with electrically conductive spheres with diaeter D = 394 and D = 65 connected to coil output Self-capacities of these spheres are sl = 9 pf and sl = 36 pf The length of the connection wire is approxiately 5 c During the experients different resonance frequencies f n are easured and standing wave characteristics for each of the are registered The results of the easureents are shown in Table I The voltage knot point coordinates are labelled with x n, where n is resonance sequence nuber, is knot sequence nuber starting fro the closest to the load Resonances also occur at higher frequencies but their voltage level is low and qualitative easureents are difficult to perfor Fro the easureent results it can be observed that under the influence of load capacitance sl resonance frequencies slightly decrease and voltage knot points ove closer to the end of the line sing standing wave knot coordinates it is possible to verify if the line is hoogenous by coparing nuber of half waves of one standing wave sight with the axiu nuber of half waves When the load is not connected for a 7/4 resonance nuber of whole half waves in the coil n = 3 and their lengths are show in Table I, where / x 45 c, 3 3 / x x c, / x x 6 c oparing different half wave lengths in different Tesla coil regions it can be observed that they are different That eans phase coefficient and characteristic ipedance Z depend on the coordinate of the long line It can be concluded that the basic paraeters of the line L and are also changing At the ends of the coil half wave lengths increase, therefore L, and decrease Magnetic field at the centre of the coil is hoogenous Therefore inductivity by length unit L is the highest By getting closer to the ends of the coil agnetic field starts to scatter and L is decreasing l L sl Download Date /8/8 9: AM 7

8 Electrical, ontrol and ounication Engineering 4/7 Sphere D sl pf pf pf TABLE I RESLTS OF MEASREMENTS Wave type /4 3/4 5/4 7/4 Frequency, MHz f = 635 f = 455 f = 736 f 3 = 966 Knot point coordinates, c x = 57 x = 675 x = 34 x 3 = 75 x 3 = 465 x 3 = 45 Frequency, MHz f = 473 f = 456 f = 6633 f 3 = 8836 Knot point x = 63 x = 735 x 3 = 775 coordinates, c x = 366 x 3 = 495 x 3 = 63 Frequency, MHz f = 36 f = 3977 f = 6478 f 3 = 877 Knot point x = 665 x = 765 coordinates, c x = 38 x 3 = 785 x 3 = 55 x 3 = 65 Fig Experiental Tesla coil resonance frequencies In Fig experientally deterined resonance frequency placeent of a Tesla coil is shown The aplitude is given in relative units because the accurate easureent of the voltage in the line and its terinals without changing its value is a proble It is difficult because there is no such volteter that would have input capacitance ie equal with Experients show that Tesla coil working conditions are being influenced even by connection of pf capacitance Output voltage value can be indirectly estiated by the length of electric spark at the output of the coil By coparing experiental AFR (Fig with theoretical (Fig 8 it can be observed that they are close and are atching each other A resonance network fors in Tesla coil Theoretical distances between resonances f = f n+ f n should be equal (Fig 8, but experiental easures indicate that with increasing frequency the distance f slightly decreases (Fig It ay be due to the fact that real line is not hoogenous, and losses of electroagnetic energy while frequency is rising are increasing because the radiated wave length EM is closing to the diensions of the coil By increasing the losses the resonance frequency decreases, therefore decreases f sing the closest to the output of a coil voltage iniu coordinate x for different resonances (x, x, x 3, average wave length n = x n /n and angle values for different resonances with not connected output capacitance sl can be deterined: = 57 / = 4 for a 3/4 wave resonance; = 675 / = 675 for a 5/4 wave resonance; 3 = 75 / 3= 483 for a 7/4 wave resonance Forula (3 allows calculation of the angle: = 74 for a 3/4 wave resonance; = 33 for a 5/4 wave resonance; 3 = 34 for a 7/4 wave resonance Knowing and using forulas (33 and (35 quarter wave resonance paraeters can be calculated: = 63 and = 4 86 = 344 It ust be aditted that the coordinates for higher resonances x 3 and x 3 are located close to the end of the coil, therefore their deterination is associated with high easureent error Measureent offset in bounds of one centieter causes the change of value by ore than degrees Knowing average resonance wave lengths and applying forula = / allow calculation the corresponding average phase coefficients n: = 83 rad/; = 55 rad/; = 93 rad/; 3 = 3 rad/ Knowing average the coefficients n allow calculating of product L for each resonance: n / n L 36, L, L, L Then average capacitance per length unit is =L /L : 44 pf/, 7 pf/, pf/, 7 pf/ Average characteristic ipedance Z L / Z 4 k, Z 3 k, Z 4 k, Z 985 k 8 Download Date /8/8 9: AM

9 Electrical, ontrol and ounication Engineering 4/7 By viewing the obtained data it can be noticed that the average capacitance per length unit by rising frequency is increasing by approxiately 35 % and Z is decreasing by approxiately %, although these values should be constant It ight be explained with heterogeneity of the line To obtain the idle-load output capacitance and local characteristic ipedance in reflection zone Z it is necessary to obtain together with idle-load resonance frequencies and output angles the sae paraeters for the known load capacitance The obtained results have to be processed using forula (3 and the equation syste for idle-load and loaded cases have to be solved tan( Z tan( ( sl Z (36 By solving equation (36 characteristic ipedance at the reflection zone Z tan( tan( (37 and idle-load output capacitance are obtained sl sl tan( (38 tan( tan( By inserting into the equations (37 and (38 respective, fro Table I, and, fro (3, values the following results are obtained For resonance 3/4 and for 394 sphere: = 74, = 44, Z = 4 k, = 6 pf For resonance 3/4 and for 65 sphere: = 74, = 535, Z = 9 k, = 93 pf For resonance 5/4 and for 394 sphere: = 33, = 58, Z = 3 k, = 74 pf For resonance 5/4 and for 65 sphere: = 33, = 735, Z = 3 k, = 48 pf Differences between the calculated results Z and, which are obtained for both resonances, can be observed The resonance data should be considered ore correct because of the increasing easureent errors for higher order waves XIII ONLSION The following is concluded: Tesla coil can be considered as one-wire electrical syste; Tesla coil is a long line with distributed paraeters that as a coon wire has a virtual ground; 3 At particular frequencies electrical resonances with different spatial configurations for in a coil; 4 Resonance processes in Tesla coil can be theoretically described using traditional theory of a long line; 5 Existence of resonance networks in Tesla coils is approved experientally; 6 The obtained experiental results coply with theoretically calculated values; 7 The obtained atheatical single wire line results can also be applied to a traditional two wire long line; 8 Paraeters Tesla coil L,, and Z depend on the length of coordinate x; 9 It is better to calculate Tesla coil base paraeters fro lower resonances due to lower easureent errors REFERENES [] I Dūiņš, K Tabaks, J Briedis u c Elektrotehnikas teorētiskie paati Stacionāri procesi lineārās ķēdēs, I Dūiņa redakcija Rīga: Zvaigzne AB, lpp [] M Tilbury, The ltiate Tesla oil Design and onstruction Guide, McGraw-Hill, 8 [Online] Available: docs/-np--the-ultiate-tesla-coil-desig_9_6 [3] M Denicolai, Tesla Transforer for Experientation and Research [4] J Voitkāns, J Greivulis, Elektriskās enerģijas pārvadīšanas iespējas pa vienvada līniju II Pasaules latviešu zinātnieku kongress Tēžu krājus, Rīga,, 63 lpp [5] J Voitkāns, J Greivulis, Vienvada elektropārvades līnijas eksperientālās iekārtas tehniskie raksturojui Zinātniskā konference Elektroenerģētika tehnoloģijas Tēžu krājus, Kauņa, 3 [6] J Voitkans, J Greivulis and A Locelis, Single wire transission line of electrical energy, EPE PEM scientific conference, Riga, 4 [7] J Voitkāns, J Greivulis and A Voitkāns, Single Wire and Respective Double Wire Schee Equivalence Principle 7th International Scientific onference Engineering for Rural developent, Jelgava, 8 [8] J Voitkāns, S Voitkāns, A Voitkāns LV patents Nr 3785 Elektriski vadoša ķereņa paškapacitātes ērītājs, Patenti un preču zīes, 8 Nr [9] I Dūiņš Elektrotehnikas teorētiskie paati Pārejas procesi, garās līnijas, nelineārās ķēdes, Zvaigzne AB, Rīgā, 6 [] J Voitkāns, A Voitkāns and I Osanis, Investigations on Electrical Fields and urrent Flow through Electrode Syste within Electrode 8th International Scientific onference Engineering for Rural developent, Jelgava, 9 [] J Voitkāns, J Greivulis LV patents Nr 343 Sinusoidāla spriegua pārvades vienvada līnija Patenti un preču zīes, 6 Nr 6 [] J Voitkans, J Greivulis Loading aspects of a single wire electric energy transission line, International scientific conference Advanced Technologies for Energy Production and Effective tilation Jelgava, 4 [3] J Voitkāns, J Greivulis LV patents Nr 93 Vienvada regulējaā elektropārvades līnija Patenti un preču zīes, Nr [4] J Voitkāns, J Greivulis LV patents Nr 33 Sietrēta vienvada elektropārvades līnija Patenti un preču zīes, 3 Nr 7 [5] B B Anderson, The lasic Tesla oil, [Online] Available: [6] G L Johnson, Solid State Tesla oil [Online] Available: on/tcchappdf [7] G F Haller and E T unningha, The Tesla High Frequency oil New York, 9 Janis Voitkans graduated fro Riga Polytechnical Institute in 976 as a radio-engineer He received Dr sc ing in electrical engineering at Riga Technical niversity, Riga, Latvia, in 7 His research interests are connected with electro physics and power electronics He is presently a Senior Researcher with the Institute of Industrial Electronics and Electrical Engineering, Riga Technical niversity Address: Āzenes, Riga, Latvia; E-ail: janisvoitkans@rtulv Arnis Voitkans received B Sc and M Sc in physics at the niversity of Latvia, Riga, Latvia in 3 and 6, respectively urrently he is a leading systes analyst with the IT Departent of the niversity of Latvia Address: Aspazijas Boulevard 5, Riga, Latvia; E-ail: arnisvoitkans@lulv Download Date /8/8 9: AM 9