International Journal of Pure and Applied Mathematics Volume 114 No. 7 2017, 531-546 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu
Serial Step Up Resonant Frequency Static Discharge System - Tesla Gun
R. Ramya1, Abhinash Kumar Patra2, Saurodeep Adhikary3, Rishav Ranjan Paul4 SRM University, Kattankulathur [email protected] and [email protected]
Abstract Currently weapons research and development takes up the greatest share of any defense budget. In this aspect, India is lagging, mostly due to economical and institutional constraints. It is the largest importer of arms and ammunitions in the world. However, there is still a need for a failsafe defense system. This paper is a step towards addressing this shortcoming of the Indian military. However, this is not the first prototypal weapons system in the world. The U.S. Defense Strategic Defense Initiative put into development the technology of a similar type using a particle beam to be used as a weapon in outer space as part of the Beam Experiments Aboard Rocket (BEAR) project. This is the next step to build a weapon system that rises above ammunition constraint and environmental hazard. The basic premise of a Tesla Gun involves static discharge at a very high voltage. There are three main elements of the system. The first is the voltage step up. The next is the resonant circuit and the final element is the targeting system. Key Words and Phrases: Tesla Coil, Static Discharge, Resonant Frequency, Bounces.
1. Introduction
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A Tesla coil is a device producing a high frequency current, at a very high voltage but of relatively small intensity. Basically, it works as a transformer and as a radio antenna, even if it differs radically from both. A Tesla coil is a radio frequency oscillator that drives an air-core double-tuned resonant transformer to produce high voltages at low currents. Tesla's original circuits as well as most modern coils use a simple spark gap to excite oscillations in the tuned transformer. More sophisticated designs use transistor or thyristor switches or vacuum tube electronic oscillators to drive the resonant transformer [1]. Tesla coils can produce output voltages from 50 kilovolts to several million volts for large coils. The alternating current output is in the low radio frequency range, usually between 50 KHz and 1 MHz. Although some oscillator-driven coils generate a continuous alternating current, most Tesla coils have a pulsed output; the high voltage consists of a rapid string of pulses of radio frequency alternating current.
The common spark-excited Tesla coil circuit consists of these components:
A high voltage supply transformer (T), to step the AC mains voltage up to a high enough voltage to jump the spark gap. Typical voltages are between 5 and 30 kilovolts (kV).
A capacitor (C1) that forms a tuned circuit with the primary winding L1 of the Tesla transformer.
A spark gap (SG) that acts as a switch in the primary circuit.
The Tesla coil (L1, L2), an air-core double-tuned resonant transformer, which generates the high output voltage.
Optionally, a capacitive electrode (top load) (E) in the form of a smooth metal sphere or torus attached to the secondary terminal of the coil. Its large surface area suppresses premature corona discharge and streamer arcs, increasing the Q factor and output voltage.
Figure 1: Basic Tesla Coil Circuit
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2. Simulations for system parameter
2.1 Software used
JAVATC by Barton B. Anderson TESLAMAP
2.2 Spark size
The spark size is directly dependent on the input power from the NST. An approximate value of the spark can be derived using
(1)
(2)
where L is the spark length and P is the input power.
Input Arc Length P Voltage Inches Cm 10 13.04 6.13 15.60 20 26.09 8.68 22.06 30 10.63 10.63 27.02 40 52.17 12.28 31.20 50 65.22 13.73 34.89 60 78.26 15.04 38.22 70 91.30 16.24 41.28 80 104.35 17.37 44.13 90 117.39 18.42 46.81 100 130.43 19.42 49.34 230 300 29.44 74.82
Table 1: Correlation of input power and arc length
2.3 Primary inductance
The inductance is necessary for the calculation of the resonant frequency. The primary inductance is determined from the primary coil as the other components have negligible inductances. The system has been designed to provide tuning options used taps in the primary coil. It is found using
(3)
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(4)
Where L is the inductance, N is the number of turns, A is the area of conductor, D1 is the internal diameter, W is the wire diameter and S is the wire spacing. S = 7.7917 mm, W = 12 mm, D = 165 mm, D0 max = 640 mm Turns Inductance 11 44.434 11.25 46.801 11.5 49.247 11.75 51.772 12 54.378
Table 2: Variation in inductance
2.4 Resonant frequency
The resonant frequency dictates the capacitor charging rate and thus the capacitor used. It is also instrumental in the operation of the coil as a gun. The resonant frequency is calculated from the primary circuit inductance and capacitance.
(5)
where L is the inductance, C is the capacitance and F is the resonant frequency. L C F 44.43 11.11 226.519 46.801 11.11 220.717 49.247 11.11 215.165 54.378 11.11 204.763
Table 3: Resonant frequency computation 3. Prototype of Tesla Gun
3.1 Charging system
The first stage is the voltage step. The supply is from a DC battery source or from the AC mains (230 V-1ph, 415 V-3ph). In the case of DC source, a fly-back transformer is used to convert low voltage of 18 V to high voltage of more than 20 kV. In case of AC source, a High Voltage Transformer used. The components of the charging system are:
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A. Neon Sign Transformer
The high voltage transformer is the most important part of the system. It is simply an induction transformer which acts as the power supply of the system. Its role is to charge the primary capacitor at the beginning of each cycle. Apart from its power, its ruggedness is very important as it must withstand terrific operation conditions.
The widely-used type of transformer is the neon sign transformer (NST), which is, as its name suggests, generally used to power neon signs. They generally supply between 6 and 15 kV and are current-limited often at 30 or 60 mA. They are safer and easier to find but are more fragile. But newer NST should be avoided, as they are provided with a built-in differential circuit breaker, which will prevent any Tesla gun operation as it provokes repeated spikes of current and voltages that will trigger the breaker. Beside from these two common types, one can also use a fly back transformer or a microwave oven transformer (MOT).
In the case of this system the NST used has the following specifications:
Output Voltage (V) = 10 kV
Maximum Current (I) = 30 mA
Also, important to computations is the power and impedance.
Power (P) = V × I = 300 W
Impedance (Z) = V/I = 333.33 kΩ
B. Capacitor bank
Energy stored in the capacitor is given by:
The output from the transformer is used to charge the capacitor bank. The benefit of using a capacitor bank is instantaneous current release for the spark gap. This discharge goes to the primary coil. Its capacitance must be such as there is resonant amplification in the primary circuit (LC circuit in series with an alternating voltage generator). The capacitance is Cres. If the capacitance is lower than Cres the energy available for the rest of the cycle will be lower. The same thing would happen if the capacitance is larger than Cres, but the larger capacitance allows more charge to be stored, which
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compensates the first problem. But due to resonance that it might be judicious to make a "bigger" capacitor in order to prevent the amplification from becoming too powerful, which could easily destroy the transformer as well as the capacitor.
By definition, Cres could be found with the resonant frequency formula for 50 Hz, but the total inductance of the primary circuit should be known. It is important to note that the inductance of the transformer is much greater than the inductance of the primary coil, and the same is true for their impedances. Therefore, the primary inductor's contribution is neglected. The aforementioned formula is the following:
-1 Cres = (2 × π × Z × f) = 9.54 nF
The specifications of the Alcon FF – 06 are as follows:
Capacitance 100 nF DC Voltage Tolerance 2000 V Maximum pulse rise time (dV/ dt) 1500 V/µs Dissipation factor (Tan δ) 0.01 kHz at 25 C Temperature Range -25 C to 85 C
Table 4: Capacitor specifications
The notion of dV/dt represents the speed at which a capacitor is charged or discharges (its units are V/s). In this context, it is not really the time-derivative of the potential but rather the maximal value it can take.
The Alcon FF-06 boasts a dV/dt of 1500 V/μs. The amplitude of the 1/2 NST's voltage is equal to Vmax = 2 × Vrms = 14140 V. But with 9 capacitors in series, each one is exposed to only one ninth of this value, i.e. 1571.11 V. For the angular speed, ω = 2πf where the frequency f is the resonant frequency. It attains a maximum value of 226.52 kHz in this case. Thus, the obtained dV/dt is 3162V/µs.
3.2 Targeting system The targeting system includes the following components:
A. Primary Coil
The primary inductance can be computed by the formula in the simulation or the formula given below. However both give approximate values.
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(6)
where N is the number of turns, R is the mean radius and W is the coil width.
B. Main Spark gap
The function of the spark gap is to close the primary LC circuit when the capacitor is sufficiently charged, thus allowing free oscillations inside the primary circuit. This is a component of prime importance in a tesla coil because its closing/opening frequency will have a considerable influence on the final output.
C. Secondary Coil
In the Tesla gun the secondary coil is made of enameled copper "magnet wire" with a diameter of 0.559 mm (SWG 24) which is wound around a PVC pipe of 110 mm external diameter. The winding height is 40.3 cm. The number of turns that have been wound can be calculated by
N = H/d (7)
where H is the height of the coil and d the diameter of the wire used [3].
In this case, the number of turns is 720 turns (more or less a few turns) which takes almost 249m of magnet wire. Length is given by
l = (2×π×N×r) (8)
where r is the radius of the coil. The self-inductance L in henrys of adjacent-turns solenoid is given by (9)
where μ represents the magnetic permeability of the medium (≈ 1.257 · 10-6 N/A2 for air, very close to the value of the void), N the number of turns of the solenoid, H its total height, and A the area of a turn.
Injecting Tesla Gun secondary coil values:
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Ltheoretical = 15.4 mH and Lexperimental = (16.4 ± 0.01) mH
which is not too far from the value measured from an LCR meter:
Resistance of Secondary Coil: The resistance of the secondary coil cannot be measured using a multimeter as the current through it is a high frequency AC current. Another method proposed by Fraga, Prados and Chen takes the corona and skin effects into account and has been proven to be precise for solenoids. The simulator yields the following value:
Rac = 94.856 Ω
Inductance and capacitance at resonance: The inductance calculated and measured previously are again valid for low frequencies only. The self-inductance of the secondary coil driven at resonance is slightly different, because of the non-uniform repartition of current and because the length of the coil is comparable to the wavelength of the signal it will carry. Once again, a more precise formula must be used.
The simulator gave the following value:
Lres = 13.471 mH and Cres = 6.74 pF
D. Top load
The top load acts like the upper "plate" of the capacitor formed by the top load and the ground. It adds capacity to the secondary LC circuit and offers a surface from which arcs can form. Computing its capacitance is difficult in the general case. However, simulation provides with an approximate value. The formula is:
(10)
where D1 is the outer diameter, D2 is the width, and C is the capacitance.
C = 11.198 pF
3.3 Protection system
The protection system consists of the following components:
A. NST protection Equipment
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Tesla coils give a harsh treatment to the main transformer. It must withstand powerful transient currents and voltages as well as high- frequency alternating currents
B. Security Spark gap
It is made of two electrodes, hence the name two element static spark gap. This is an extra spark gap which is placed across the capacitor bank.
C. Auto-transformer
In order to protect the electronic devices connected to the mains from dangerous currents spikes and voltages fluctuation from main supply, it is recommended to connect at least a simple auto transformer between the mains and the transformer. There's an important empirical rule about the ratio between the height H and the diameter D of the coil. It has been observed that the best performances are attained with an H/D ratio between 3 and 5. Tesla coil is thus conforming to a ratio of 4.
The auto-transformer also provide definitive measurement of the NST output. This is instrumental in keeping the dV/dt of the capacitors within limit. 4. Construction Constraints of Tesla Gun
4.1 Description of a cycle
Considering that the primary and secondary circuits are RLC circuits with low resistance, which accords with reality. For this reason, internal resistance of the components is not represented. The current- limited transformer is also replaced. This has no impact regarding pure theory.
Some parts of the secondary circuit are drawn in dotted lines. This is because they are not directly visible on the apparatus. Regarding the secondary capacitor, it is seen that its capacity is actually distributed, the top load only being "one plate" of this capacitor. Regarding the secondary spark gap, it is shown in the schematic as a way to represent where the arcs will take place.
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Figure 2: Schematic of the basic features of a Tesla coil
4.2 Charging
The first step of the cycle is the charging of the primary capacitor by the generator. The input has a frequency is 50 Hz. Because the generator is current-limited, the capacity of the capacitor must be carefully chosen so it will be fully charged in exactly 1/100 seconds.
Figure 3: The generator (NST) charges the primary capacitor
4.3 Oscillations
The coupling constant k between the two circuits is kept low, generally between 0.05 and 0.2. Several oscillations will therefore be required to transfer the totality of the energy.
Figure 4: Oscillations in primary and secondary coils
The oscillations in the primary will thus act a bit like an AC voltage generator placed in series on the secondary circuit. To maximize the voltage in the secondary, it is intuitively clear that both circuits must share exactly the same resonant frequency, as it will be demonstrated
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below.
(11)
This will allow the voltage in the secondary to increase dramatically, as in equations describing a harmonically driven LC circuit. This is called the resonant rise. The voltage becoming rapidly enormous, generally several hundreds of thousands of volts, sparks will form at the top load.
Figure 5: Oscillations of diminishing amplitude in the primary (ring- down)
Figure 6: Oscillations of increasing amplitude in the primary (ring- up)
4.4 Bounces
All the energy is now in the secondary circuit. The ring-up is very fast and the path of ionized air in the spark gap subsists a few moments even when the intensity of the field has fallen below the critical value. The energy of the secondary can therefore be transmitted to the primary in a similar fashion. Current and voltage in the secondary will then diminish while those in the primary will increase. Such bounces can occur 3, 4, 5 times or even more. At each rebound, a fraction of the energy is definitively lost. This is the reason why the envelop of the waveform falls exponentially. After a certain number of bounces, voltage will have decreased significantly and the spark gap finally opens at the next primary notch [5].
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These rebounds are actually important for the creation of long arcs, because arcs grow on the ionized air path created during the previous rebounds. At each bounce, the spark gets longer. The complete process happens several hundred times per second.
Figure 7: General waveform of the oscillation in the primary circuit with 3 rebound
Figure 8: General waveform of the oscillation in the secondary circuit with 3 rebounds
Higher number of rebounds is very helpful for a classic tesla coil operation. However, for the coil to act as a gun it should discharge instantaneously. Hence the coupling coefficient is slightly greater to ensure less number of bounces and full discharge of energy in minimum cycles.
4.5 Decay
Once the main spark gap has stopped firing, the primary circuit is open and all the remaining energy is trapped in the secondary. This situation is thus the same as in a free RLC circuit. Oscillations will decay exponentially as the charge dissipates through the sparks.
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Figure 9: When the spark gap has opened, secondary oscillation decays exponentially
4.6 Voltage Gain
The following derivation is based on conservation of energy, we'll suppose there's no ohmic losses (R = 0) and that the primary and secondary circuits are perfectly at resonance. From [6], the formula giving the energy stored in a capacitor as a function of the voltage E = 2 CV /2. The energy of the primary Ep circuit is thus given by
(12)
where Cp is the capacity of the primary circuit. The energy Es stored in the secondary circuit (capacity is Cs) is
(13)
It is more convenient to express this gain in terms of the inductance of the circuits.
(14)
The gain can therefore be rewritten this way: (15)
So, this is how the Tesla coil can reach such tremendous voltages: the secondary coil, which has around 800 turns, has an inductance considerably higher than the primary coil, which generally has about 10 turns. Using typical values, this formula will yield 5 6 a Vout of the order of 10 V or even 10 V for the largest coils.
4.7 Energy Losses
In most cases, primary and secondary circuits are rarely at perfect resonance, which contributes to a further reduction of the secondary voltage. And there are certain losses like copper loss, corono loss and Electromagnetic loss.
4.8 Distribution of Capacitance within the secondary circuit
The capacitance of secondary coil consists of many contributions and is difficult to compute, but its major components are top load, secondary circuit.
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Figure 10: The major contributions to the secondary capacitance C
The total capacitance — as all these "capacitors" are in parallel, so the total capacity of the secondary circuit will be given by:
(16) Cs = Ct + Cb + Ce
where, Cs is the stray capacitance, Ct is the capacitance of the top load, Cb is the capacitance of the secondary circuit.
The reason why secondary capacitance is so hard to define is because it is truly small (a few dozens of Pico farads). That is why all the factors must be taken in account. Indeed, they would also apply for the capacity of the primary circuit, but this one has a much greater capacity, which is totally concentrated in the primary capacitor, so the aforementioned factors are truly negligible.
Figure 11: Full tesla coil setup
5. Conclusion
The aim of this paper is to develop a static discharge weapon system, more popularly known as a Tesla Gun. Although there are some experimental prototypes but such system has not been given proper
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scientific attention yet due to difficulties in the development cycle and material constraints.
This paper addresses these issues in the design and development phases. The Tesla Gun is a low cost, relatively stable static discharge prototype. The firing system is a High-Tension Tesla Coil undergoing a two-phase step up. The study begins with a thorough analysis of the working of LC, RLC circuits with a focus on resonance condition and quality factor as these are instrumental in the design of the primary and secondary coils. The next step is simulating the design to identify the range of the system.
This prototype acts as an active defense mechanism within the ascertained range which is around 30 inches. The static discharge length can be varied by adjusting the input, capacitor bank, or the tap in the primary to obtain fine control. References
[1] N. Tesla, "Apparatus for transmitting electrical energy", US Patent no1119732, 1914.
[2] Maxwell, J. Clerk "A Dynamical Theory of the Electromagnetic Field". Philosophical Transactions of the Royal Society of London .1 January, 1865.
[3] Hayt, William H. “Engineering Electromagnetics”, 5th edition, McGraw-Hill, 1989.
[4] Marco Denicolai “Tesla Transformer for Experimentation and Research”, Helsinki University of Technology, 2001.
[5] C. Gerekos, "The Zeus Tesla Coil - Introduction and overview | Hazardous Physics"
[6] Katsuhiko Ogata “System Dynamics”, 4th edition. University of Minnesota, 2005.
[7] Cartwright, K. V.; Joseph, E.; Kaminsky, E. J. "Finding the exact maximum impedance resonant frequency of a practical parallel resonant circuit without calculus”. The Technology Interface International Journal, 2010.
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