Design of Tesla s Two-Phase Inductor

Transcription

1 X International Symposium on Industrial Electronics INDEL 14, Banja Luka, November 68, 14 Design o Tesla s Two-Phase Inductor Đorđe M. Lekić, Petar R. Matić University o Banja Luka Faculty o Electrical Engineering Banja Luka, Republic o Srpska, Bosnia and Herzegovina djordje.lekic@etbl.net, pero@etbl.net Abstract This paper describes a new method or designing Tesla s two-phase inductor or demonstration o rotating magnetic ield, based on the amous Tesla s Egg o Columbus experiment. The design is based on electromagnetic and thermal analytical models o the two-phase inductor and veriied by computer simulation and FEM analysis. Keywords-FEM analysis; rotating magnetic ield; single-phase induction motor; Tesla s Egg o Columbus; NOMENCLATURE D, d Outer and inner iron core diameter [mm], a, h Iron core width and height [mm], μ = 4π 1-7 Permeability o ree space [H/m], ω, Angular requency [rad/s] and requency [Hz], α, β Phase index, N α, N β, N Number o turns per phase, m = N β / N α Transer coeicient, N 1/ = N/ Number o turns per hal-winding, R g Equivalent reluctance o air gap [H -1 ], X α, X β, X Reactance o phase winding [Ω], X 1/ = X/ Reactance o one hal-winding [Ω], R α, R β, R Resistance o phase winding [Ω], Z α, Z β, Z Impedance o phase winding [Ω], X C, R C Capacitor reactance and resistance [Ω], U C, C Capacitor voltage [V] and capacitance [μf], ρ = 1/57 l, L = N l Speciic resistance o copper at C [Ωmm /m], Average length o turn and phase winding [m], S α, S β, S Copper wire cross section [mm ], d α, d β, d Copper wire diameter [mm], n α, n β, n Number o conductor layers, I α, I β, I RMS value o phase and total current [A], U α, U β, U RMS values o phase and supply voltage [V], F α, F β, F Amplitude o magneto-motive orce [Aturns], Φ gα, Φ gβ, Φ g Amplitude o air gap magnetic lux [Wb], B gα, B gβ, B g Amplitude o air gap magnetic lux density [T], Δ, P rrent density [A/mm ] and copper losses [W], m, V Total mass [kg] and volume o copper [m 3 ], γ = 89 Speciic mass o copper [kg/m 3 ], θ, θ a Operating and ambient temperature [ C], θ = (θ + θ a )/ Average (ilm) temperature [ C] θ = θ θ a Temperature increase [ C] α c, α r, α t Heat transer coeicients [W/ C/m ], S hc, S hr Heat transer surace areas [mm ], σ = Stean-Boltzmann constant [W/m /K 4 ], ε Emissivity coeicient, Nu, Gr, Pr Nusselt, Grasho and Prandtl number, λ Thermal conductivity o air [W/ C/m], ν Kinematic viscosity o air [m /s], g = 9.81 Acceleration o ree all [m/s ], β = (35 + θ ) -1 Thermal expansion coeicient [ C -1 ], γ = 1.98 Speciic mass o air [kg/m 3 ], c = 15 Speciic thermal capacity o air [J/kg/ C]. I. INTRODUCTION In February 188, the Serbian inventor and scientist Nikola Tesla came up with the design o the rotating magnetic ield and in 1888 he constructed the irst induction motor. Tesla s irst induction motor had a toroidal stator with a two-phase inductor, unlike a cylindrical stator with a three phase inductor which is nowadays a standard design. In order to demonstrate the eects o the rotating magnetic ield at the World air exhibition in Chicago in 1893, Tesla used his two-phase inductor to lit and spin a metal egg, thus perorming the eat o Columbus without cracking the egg [1]. Tesla s original design o the two-phase inductor is shown in Fig. 1. Figure 1. The original design o the two-phase Tesla inductor [] 115

2 Two coils are wound on an iron core o toroidal shape. The two phase coils are displaced by 9 (geometrical) degrees and are supplied with two alternating currents phase shited by 9 (electrical) degrees. In this way a rotating magnetic ield is generated in the air gap. I a metal object, e.g. a copper or aluminum egg, is placed on a platorm in the centre, it will start to rotate, and eventually stand on its main axis, due to gyroscopic eect. Tesla s two-phase inductor is in act the stator o a two-phase induction motor, with the metal egg being the rotor. Replicas o Tesla s Egg o Columbus can be ound in museums dedicated to Tesla and almost every Faculty o electrical engineering has its own unique copy. There are even companies that create authentic replicas or museums and private collectors [3]. These replicas are not diicult to reproduce, however in most cases they deviate rom Tesla s original design. For example in [4] using the laws o electrical machine resemblance, a stator o a conventional our-pole, three-phase induction motor was redesigned to demonstrate the Egg o Columbus experiment, and in [5] an electronically controlled rotating magnetic ield inductor, using power switches, oscillator and counter circuits is presented. All these replicas don t properly illustrate the brilliance and simplicity o the rotating magnetic ield concept and they are not a practical educational tool which, in a simple way, conveys basic electro-technical engineering principles to young students. This paper gives the design and construction principles o Tesla s two phase inductor. The paper is organized as ollows. In Section II the analytical model or electromagnetic design is presented and proper level o magnetic lux density in air gap is selected. Equations or calculating the phase reactance and resistance, required capacitance in one phase and number o turns in each phase are derived. Section III gives a method or determining adequate current density, based on analytical thermal analysis. Using the derived equations, all the necessary parameters are calculated in Section IV and the results o the design are veriied via computer simulation and Finite Element Method (FEM) analysis. Conclusion is given in Section V. II. ELECTROMAGNETIC DESIGN The design o electrical machines is an iterative procedure based on calculations and assumptions on the irst step, and on necessary corrections and experimental veriication on the second step. Assuming ixed dimensions o used iron core, the electromagnetic design starts with the selection o magnetic lux and current density and ends by calculating number o turns in each phase. A. Magnetic Circuit The magnetic circuit or one phase winding o the Tesla inductor is shown in Fig.. The phase winding is divided in two halves which are connected in series, but wound in opposite directions and placed on opposing sides o the iron core. The resulting magnetic lux Φ g passes through the air gap, while only hal o the resulting lux Φ Fe = Φ g / passes through each side o the iron core. For design analysis, the magnetic circuit in Fig. can be represented with an equivalent circuit, shown in Fig. 3. The magnetic circuit equivalent takes only the equivalent magnetic resistance (reluctance) o the air gap into account, while the reluctance o the iron core is neglected due to high values o iron core permeability. F m F Ф Fe Ф g Figure. Magnetic circuit ФFe g Ф g Ф Fe Figure 3. Magnetic circuit equivalent Ф Fe B. Phase Impedance Assuming a homogeneous magnetic ield in the air gap, the phase reactance can be calculated by using similar approach to slot leakage reactance calculation or standard induction motors [6]. Later in the text, by using computer simulation and FEM analysis, it will be shown that this is a very good approximation. The reactance o one hal-winding o each phase, is: N1/ 1 1/ 1/ g 3 X h N where the equivalent air gap reluctance is: F F m (1) 3 g h () Since two hal-windings are connected in series to orm one phase winding, ater introducing requency and number o turns per phase in (1), the ollowing equation or reactance o each phase winding is obtained: 1 X X1/ h N (3) 3 116

3 The phase resistance at θ = C can be calculated as: R L N l (4) S S I U α The cross section o copper wire in (4) is: R α S The average length o copper wire is: d (5) 4 I α X α C U C 1/ L N l N a h n d (6) The coil resistance at operating temperature θ is: X β I β R β R R (7) U β The impedance o each phase winding is then: Figure 4. Electric circuit equivalent Z R X (8) In equations (1)-(8) phase indexes have been let out, meaning that the equations are valid or both phases. In order to calculate the parameters or a particular phase, one must use the number o turns, number o conductor layers and cross section o that particular phase winding. C. Permenantly Connected Capacitor In order to get a symmetrical rotating magnetic ield, it is needed to have equal amplitudes o Magneto-Motive Forces (MMFs) or both phases, and an electrical phase delay o 9º between the phase curents. In standard single-phase induction motors, this is achieved by using permenantly connected capacitor in one o the phases, e.g. phase β in Fig. 4. For calculating the capacitance in phase β a standard procedure described in [7] will be used. The transer ratio is calculated using [7]: D. Number o Turns per Phase The magnetic lux density in the centre o the air gap due to excitation rom one phase can be calculated rom Fig. 3. For that purpose, the Kapp-Hopkins law or magnetic circuits is used: F (1) g The Magneto-Motive Force (MMF) generated rom both phases is: g F N I N I (13) With the assumption o a homogeneous magnetic ield in the air gap, the amplitude o the magnetic lux is given by: g Bg Sg Bg h d (14) 4 C X X R R m R (9) Substituting (), (13) and (14) in (1) the ollowing equations or calculating the number o turns in each phase are derived: while capacitor reactance and capacitance are [7]: C X m X m R (1) 1 1 C X X C C (11) While calculating the transer ratio, one must assume the value o capacitor resistance, but or rough calculations, the capacitor resistance can be neglected. N N 3 Bg d I (15) 3 Bg d I (16) From (15) and (16) it can be seen that required level o magnetic ield density in the air gap can be achieved by increasing the number o turns, or by increasing the current. 117

4 The required level o magnetic ield density is determined rom the condition that it only has to spin the egg, and thereore can be much lower than as in conventional electric machines. This level is determined by analyzing a prototype. The results o the experiments show that or rotating magnetic ield demonstration, a magnetic lux density o 5-1 mt is large enough to lit and spin a smaller metal egg. Heaving this range o values in mind, the design problem is to select the appropriate number o turns or each phase winding in order to obtain required magnetic lux density without overheating the coils due to large currents needed to generate this lux density in the large air gap. I a standard value o copper wire diameter or phase α is selected, with a proper value or current density without overheating, the phase current and the number o turns in phase α, using (15) can be calculated. Ater the number o turns is ound, the reactance and resistance o phase α using (3) and (4) and the required transer ratio using (9) can be calculated. In that case, phase current and number o turns in phase β is urther calculated as: N I I m (17) m N (18) From (17) the diameter and cross section o copper wire in phase β should be adopted to the closest standard values. Since the core has ixed dimensions, it must be must ensured that there is enough space to place both phase windings on the core. The windings will it on the core i they are wounded in n layers, where n is calculated or each phase independently using: n Nd d (19) where the irst larger integer value should be adopted. At the end the number o turns or phase β is calculated by using the same equations as or phase α. III. THERMAL DESIGN rrent density o conventional induction motors is typically in the range 3-8 A/mm [6]. rrent density determines the cross section o copper wire, or in other words, the total mass o copper used to wound the inductor. Since the losses depend on the value o current density, the steady state temperature o the windings also depends on current density. In order to prevent overheating, a thermal model is used to select the required value o current density. A. Coper Losses The design is based on requirement that the steady state temperature o windings does not exceed 8ºC. Since the magnetic lux density is in the range o 5-1 mt, the iron losses can be neglected, and only the copper losses are considered. The total copper losses in both phases at operating temperature θ are: P R I R I R I () I it is assumed that operating temperature should be θ = 8ºC, the resistance should be increased using (7) and the ollowing equation or copper losses in common electrical machines is obtained [6]: m (1).4 m P where Δ is given in A/mm and total mass o copper is: m V L S L S () From (1) the losses per cubic meter o copper depend only on the current density: P V (3) I the copper losses rom (1) are combined with (4) and (15) or (16), the ollowing equation is derived: 6 dl P B k B (4) g g in which k is a constructive constant which depends on characteristics o used materials and core dimensions. From (4) the characteristic product o magnetic lux and current density can be ound which will provide that copper losses don t lead to overheating. B. rrent Density The allowed copper loss can be calculated using Newton s law o cooling [6,8]: P S (5) max t hc where convection and radiation heat transer surace areas and total heat transer coeicient are: Shc Shr hd 4hD d D d 4 hc (6) Shr t c r (7) S Using (5) the copper losses which will lead to steady state temperature rise o θ = 8ºC can be calculated, and ater that, using (4) the maximum allowed current density or a given value o magnetic lux density is calculated. 118

5 The heat transer coeicient due to radiation is calculated using Stean Boltzmann s law [8]: r a a (8) where the emissivity is taken to be. or new, and close to 1 or old copper conductors. The heat transer coeicient due to natural convection is calculated using dimensional thermal analysis, involving Prandtl, Grasho and Nusselt numbers, and is given by ollowing equation [8]: c Nu (9) h In (9) Nusselt number is a unction o the product o Grasho and Prandtl number and is calculated using [8]: n Nu A Gr Pr (3) where values or coeicients A and n depend on product o Grasho and Prandtl number and are given in Table I [8]. In (9), (3), (31) and (3) thermal conductivity o air, Prandtl number and viscosity o air are unctions o average (ilm) temperature and are calculated as [8]: (33) 4 Pr (34) (35) Assuming a value o steady state and ambient temperature and using equations (8)-(35), the total heat transer coeicient is calculated, and value or allowed copper losses is obtained rom (5). The allowed current density is then calculated rom (4) and the inal result o this calculation is showed on diagram in Fig. 5. The diagram shows allowed current density versus operating temperature, or several values o ambient temperature. TABLE I. NUSSELT NUMBER COEFFICIENTS Gr Pr A n A/mm Grasho number takes into account orces due to dierences in density o heated air layers and is calculated using ollowing equation [8]: 3 g h Gr (31) Prandtl number depends on physical properties o air and can be calculated as [8]: a1 = 15 o C.5 a = o C a3 = 5 o C a4 = 3 o C o C Figure 5. rrent density versus operating temperature c Pr (3) but is oten taken to have a constant value o.7. It can be seen rom diagram on Fig. 5 that or operating temperature o θ = 8 C and ambient temperatures o 15 C to 3 C, allowed current density is in the range o 3-4 A/mm. For most likely ambient temperatures o C to 5 C current density is approximately 3.5 A/mm. 119

6 IV. RESULTS OF DESIGN TABLE III. RESULTS OF CALCULATION A. Calculation For demonstration o the proposed calculation o twophase Tesla inductor a toroidal iron core rom an instrument transormer or voltage 58V and current 5A is selected. The dimensions o the iron core are given in Fig. 6 with the values in Table II. Parameter Unit Value d α mm 1. d β mm.8 S α mm.7854 S β mm.57 B g mt 1 A/mm 3.5 I α A.75 I β A 1.84 N α turns 74 N β turns 15 m / 1.5 n α layers 3 n β layers 4 R α Ω 4.11 R β Ω 9.49 X α Ω 6.15 X β Ω Z α Ω 7.4 Z β Ω X C Ω R C Ω / Figure 6. Dimensions o the used iron core C µf 16 U α V 1 TABLE II. CORE DIMENSIONS U β V 31 Parameter Unit Value D mm 6 d mm 18 h mm 6 a mm 4 For selected standard copper wire with diameter o 1 mm or phase α, using equations derived in Section II and Section III, with the selected values o 1 mt or magnetic lux density and 3.5 A/mm or current density, all the parameters or constructing a two-phase Tesla inductor are calculated and their values are given in Table III. U C V 37 L α m 15 L β m m kg V m P W 63.3 P /V W/m It should be noted that total wire lengths in each phase should be increased or approximately 1% in order to take into account side connections and terminal connections. In this way, the total mass and volume o copper will be slightly increased. 1

7 B. Electromagnetic Simulation For electromagnetic and thermal simulation, FEMM sotware is used. Fig. 7 shows magnetic lux density plot rom phase α, at the moment when the current in phase α is at its maximum, while current in phase β is equal to zero. This justiies the assumption o homogeneous magnetic iled in the air gap. Fig. 8 shows the similar results or phase β. From Fig. 7 and Fig. 8 it can be concluded that the magnetic ields are orthogonal, thus creating a rotating magnetic ield in the air gap B, Tesla 5.8e- : >6.15e e- : 5.8e- 5.16e- : 5.519e e- : 5.16e- 4.61e- : 4.913e- 4.38e- : 4.61e- 4.5e- : 4.38e- 3.7e- : 4.5e e- : 3.7e- 3.96e- : 3.399e-.793e- : 3.96e-.49e- :.793e-.188e- :.49e e- :.188e- 1.58e- : 1.885e- 1.79e- : 1.58e e-3 : 1.79e e-3 : 9.761e e-3 : 6.733e-3 <6.754e-4 : 3.74e-3 Density Plot: B, Tesla Figure 7. Magnetic lux density plot or phase α 5.388e- : >5.667e- 5.18e- : 5.388e- 4.89e- : 5.18e e- : 4.89e- 4.69e- : 4.549e- 3.99e- : 4.69e- 3.71e- : 3.99e- 3.43e- : 3.71e e- : 3.43e-.871e- : 3.151e-.59e- :.871e-.31e- :.59e-.3e- :.31e e- :.3e e- : 1.753e e- : 1.473e e-3 : 1.193e- 6.34e-3 : 9.138e e-3 : 6.34e-3 <7.495e-4 : 3.546e-3 Density Plot: B, Tesla Figure 8. Magnetic lux density plot or phase β Fig. 9 shows value o magnetic lux density along the air gap. It is clear rom Fig. 9 that the magnetic lux density changes rom 11 mt to 1.5 mt along the diameter o the air gap, which is more than anticipated 1 mt. The coil parameters are also calculated by FEM simulation (Table III), and compared with analytically calculated values shown in Table II. It can be seen that the calculation error is less than 3% in all cases Length, mm Figure 9. Magnetic lux density distribution along air gap TABLE IV. SIMULATION RESULTS Parameter Unit Value Figure 1. Temperature distribution Calculation error % R α Ω R β Ω X α Ω X β Ω C. Thermal Simulation The electromagnetic FEM simulation considered a planar problem, while the thermal analysis should consider a axisymmetric problem. The copper winding is modeled with a hollow torus, with the same volume as calculated in Table II, while copper losses per cubic meter are calculated rom (3). The calculated temperature distribution is showed in Fig. 1 rom which it can be seen that the highest temperature is 354 K or 81 C, which is almost equal to the designed temperature o 8 C. During simulation an ambient temperature o C was assumed. 3.54e+ : >3.575e+ 3.51e+ : 3.54e e+ : 3.51e e+ : 3.478e e+ : 3.446e e+ : 3.413e e+ : 3.381e e+ : 3.349e+ 3.85e+ : 3.317e+ 3.5e+ : 3.85e+ 3.e+ : 3.5e e+ : 3.e e+ : 3.188e+ 3.13e+ : 3.156e+ 3.91e+ : 3.13e+ 3.59e+ : 3.91e+ 3.7e+ : 3.59e+.994e+ : 3.7e+.96e+ :.994e+ <.93e+ :.96e+ Density Plot: Temperature (K) 11

8 It may seem that the model shown in Fig. 1 is oversimpliied, but with current level o design, without experimental veriication, it would be unreliable to assume special insulation types or conductors and special wooden or metal platorms or the metal egg to spin on them which may inluence the basic calculation. o current densities can be allowed without overheating the winding, the FEM simulation showed that with only 6 W o losses the temperature rise is close to given 8 C limit. The next step would be to build the prototype based on the proposed calculation. Only by experimental veriication, the the design process will be completed. V. CONCLUSIONS In this paper a ull design procedure or calculating the parameters o Tesla s two-phase inductor is explained. Starting rom basic concepts o electrical machine design, a new method dedicated to special machines with large air gaps and toroidal inductors is developed. The main problem o the proposed design procedure is ound to be the selection o magnetic lux density and current density values. While magnetic lux density, required to lit a small metal egg, is determined experimentally, the value or current density is selected as a result o detailed thermal analysis. Even such analysis implies complicated set o equations and is parameter sensitive, it is shown that or reasonable values o operating and ambient temperature, the current density is ound to be in a relatively small range o values. It is shown that the analytical model gives very similar results as the FEM computer simulation. It should also be noted that the total copper losses o such a device are around 6 W which is equivalent to a light bulb. Even it seems that the larger values REFERENCES [1] Tesla universe, Tesla s Egg o Columbus [Online]. Available: [] N. Tesla, System o Electrical Distribution, U.S. Patent May 1, [3] RT17, Tesla s Egg o Columbus Production o working replicas [Online]. Available: [4] P. Matić, M. Milanković, G. Kondić, M. Radivojević, An laboratory model or induction machine practice (in Serbian), in Proc. Symposium Industrial Electronics INDEL 4, Banja Luka, Republic o Srpska Bosnia and Herzegovina, Sep. 11-1, 4, Vol. V, pp [5] D. Jagodić, A. Ilišković, Forming o rotary magnetic ield (in Serbian), in Proc. Symposium Industrial Electronics INDEL 1997, Banja Luka, Republic o Srpska Bosnia and Herzegovina, Sep. 4-6, 1997, Vol. 1, pp [6] V. Petrović, Uput u proračun asinhronog motora, vol. I. Beograd Naučna knjiga, 1974, pp [7] B. Mitraković, Asinhrone mašine, vol. IV. Beograd Naučna knjiga, 1989, pp. 15 [8] V. Morgan, Thermal behaviour o electrical conductors, vol. I. Taunton Research Studies Press,