TESLA COIL

                                                TESLA COIL

The world of wireless technology is increasing the comfort levels of every single person. Innumerable wireless applications like wireless powered lighting, wireless smart homes, wireless chargers, and so on are developed due to wireless technology. The initiator of this wireless technology was a Serbian American inventor, engineer as well as one of the biggest futurists, the great Nikola Tesla (10th of July 1856 – 7th of January 1943), best known for his contributions towards the design of modern AC electricity supply system. Of his many futuristic inventions, one of the most interesting one till date stays to be the famous ‘Tesla Coil’ invented around 1891, which he then patented in 1914 at the US Patent and Trademark Office as the “Apparatus for transmitting Electrical Energy”.

What is Tesla coil?

            A Tesla coil is a radio frequency oscillator that drives the air-core double-tuned resonant transformer to produce high voltages with low currents.

What is a Radio Frequency Oscillator?                                

             An electronic oscillator is a device that produces electrical signals of either a sine wave or a square wave. This electronic oscillator produces signals in the radio frequency range of 20 kHz to 100 GHz, known as a radio frequency oscillator.

                   

Tesla Coil Working Principle

          This coil can produce output voltages up to several million volts based upon the size of the coil. The Tesla coil works on a principle to achieve a condition called resonance. Here, the primary coil emits huge amounts of current into the secondary coil to drive the secondary circuit with maximum energy. The fine-tuned circuit helps to shoot the current from primary to secondary circuit at a tuned resonant frequency.

Tesla Coil Circuit Diagram

          This coil has two main parts – a primary coil and a secondary coil, with each coil having its own capacitor. A spark gap connects the coils and capacitors. The functionality of the spark gap is to generate the spark to excite the system.

It consists of two parts:

1. Primary coil
2. Secondary coil

Each coil has a capacitor. Spark gap is used to close the primary circuit. HV transformer is used to charge the capacitor. Sparks generated on the toroid.

 

Tesla Coil Working Mechanism

         This coil uses a specialized transformer called a resonant transformer, a radio-frequency transformer, or an oscillation transformer.

         The primary coil is connected to the power source and the secondary coil of a transformer is coupled loosely to ensure that it resonates. The capacitor connected in parallel with the transformer circuit acts as a tuning circuit or an LC circuit to generate signals at a specific frequency.

         The primary of the resonant transformer steps up to generate very high levels of voltage ranging between 2kv to 30 kV, which in turn charges the capacitor. With the accumulation of massive amounts of charge in the capacitor, eventually, breaks down the air of the spark gap. A simple ‘spark gap’ i.e., a separation between two conducting electrodes to allow an electric spark to pass between them, it is used to excite the oscillations in the tuned transformers. The capacitor emits a huge amount of current through the Tesla Coil, which in turn generates a high voltage at the output.

         The functioning of a Tesla Coil is based on numerous practical applications of concepts of resonance, capacitance, inductance, impedance, RLC circuits and many more.

Resonance in RLC Circuit:

         When the frequency of input or applied signal is equal to the natural frequency of the circuit then the net amplitude of the signal in circuit is very high. This phenomenon used in a tesla coil for a very high voltage across the top mounted circular capacitor.

        The angular frequency of a circuit with no resistance in it is,

                                               ω =1/√(LC)

         For non-zero resistance the frequency is same but the oscillation becomes damped. So, the frequency must be                           

                                                f = 1/ (2π√(LC))

  At this frequency reactive part of the circuit becomes zero.

Quality factor:

      It is the ratio of energy stored to the energy dissipation in an oscillation in a circuit. It defines the Current(I) vs angular frequency (ω) curve of a particular RLC circuit.

                                               Q = E_stored/E_{dissipated (per cycle)}

Replacing power in place of energy in above definition,

                                              Q = Reactive power/ Active power

Where, Reactive power = X*I²

             Active Power = R*I²

                                           ⇒ Q = (X*I²)/(R*I²)

                                            ∴ Q = X/R

At resonant frequency quality factor becomes,

                                            Q = (1/R) (√(L/C))

If quality factor is high than power stored is higher than the power dissipated.

Voltage Gain:

       If power is stored in a circuit, then it will constantly be transferred from capacitor to inductor and vice versa so we can say that net power is always constant.

       Initially all reactive power is at the capacitor. So, the input power will be

                                            P_in = (C_p*V_in²)/2

 

 Similarly, for output power,

                                            P_out = (C_s*V_out²)/2

Here it is considered that there is no resistive property in the circuit then there is no loss in the circuit.

Then,

             (C_p*V_in²)/2 = (C_s*V_out²)/2

           ⇒   V_in/V_out = √(C_s/C_p)

           ⇒   V_in/V_out = √ (L_p/L_s) [ ∵ for resonance C_p*L_p=C_s*L_s]

 L is proportional to n², so

                  V_in/V_out = n_p/n_s.

         For secondary the inductance is 100 times larger than that of primary so from the above formula V_out will be very large in comparison with primary.

Arc Length:

           After many experiments it was found that the maximum length of arc produced is approximately equal to 4.32 * √ P_in i.e.,

                                           L ≈ 4.32 √P_in                                        

Capacitors in Tesla Coil:

           Capacitors are one of the most important components in tesla coil operation. For parallel plates the value of capacitance is:

                                           C = ε *(A/d)

For many such capacitors in parallel, capacitance will be sum of all capacitances:

                                           C=n* ε *(A/d)              (for n capacitors of same capacitance each)

            Practical capacitor also has internal resistance which is in series with capacitor. And we can calculate all quantities like net impedance, phase angle etc. by simple ac circuit calculations.

Inductors in Tesla Coil:

           The primary inductor is used to generate a magnetic field to be injected into the secondary circuit as well as forming an LC circuit with the primary capacitor. It can transport heavy current without excessive losses.

           The inductance of secondary inductor is larger than the primary inductor due to this high voltage is established at output.

 

The Coil:

          The primary coil is used with the primary capacitor to create the primary LC circuit and it is also responsible for transferring power to the secondary coil.

            The main purpose of the secondary coil is to bring an inductive component to the secondary LC circuit and to collect the energy of the primary coil.

HV transformer:

            It is simply an induction transformer and it is the most important part of a Tesla coil. It is used to charge the primary capacitor at the beginning of each cycle.

Toroid:

            It acts as a capacitor to hold the HV charge and also serves the purpose of offering some electrostatic shielding to the secondary by producing sparks.

            In tesla coil there is a spark gap in secondary circuit. A parasitic capacitor developed at the top of coil, to understand that phenomenon we need to understand the breakdown voltage in capacitor.

Parasitic capacitance:

             It is a phenomenon that occurs when there exists an unavoidable capacitive action even though there is no capacitor present in the electrical circuit. This occurs when two conductors of the same circuit are in proximity to each other with a dielectric in the middle. This creates a virtual capacitor between the two conductors.

Breakdown voltage with PASCHEN'S LAW

          This law gives the equation for the breakdown voltage with respect to gap pressure and gap length. Taking a parallel plate capacitor and any gas as dielectric medium between them, we can make some observations by changing gap pressure and gap length between plates.

The observations are as follows:

1. If we keep constant gap length and then increase the pressure of the gas then breakdown voltage will decrease gradually but after a certain pressure it will increase with pressure (the starting pressure is 0 torr cm, and at 0 torr cm V tends to infinite). The plot can vary with gas.

2. If we keep pressure constant and increase the gap length then breakdown voltage will increase.

 Hence, we can conclude that the breakdown voltage is a function of both gap length and gap pressure i.e., V= f (p, l).

According to above equation, graphs of voltage (V) against pressure (p) and gap (d) have been plotted using MATLAB. 

PISLER Method:

         This is one of the method to find the capacitance between two spheres of same radius (let's say r) and kept at a distance of d (center to center). Pisler also uses method of charge images and values are calculated with the help of computer.

Method of charge images:

        This method is also known as method of mirror charges. In this method certain elements in original layout are replaced with imaginary charges, which replicates the boundary condition of the problem. If we know the potential at each boundary and charge density in the volume inside the boundaries then we will be definitely able to determine the potential at any point inside the boundary, Gauss law is application of this Corollary.

The equations computed by Pisler for x, beta and S as follows:

                     x = 1 + (d − 2r)/2r               (here d is distance and r is radius)

                     β = ln (x + √ (x²-1))                (x is from above equation)

                     S = sinh(β)*[ ∑_∞ (sinh(nβ))^-1 ]            (β is from above equation).

 According to these equations, graph of capacitance v/s gap is plotted using MATLAB.

APPLICATIONS:

● Spark gap radio transmitters: It generates radio waves by means of electric sparks from a Tesla Coil.

● Induction and Dielectric heating: In large scale Induction heating, Vacuum tube and Spark gap types of Tesla Coils are used to pass a high frequency alternating current (AC) through the electromagnet. The rapidly alternating magnetic field penetrates the object, generating electric currents inside the conductor, called eddy currents that produce the heating effect.

● Induction Coils: It is a type of electrical transformer used to produce high-voltage output pulses from a low-voltage DC supply. Induction coils were used to provide the high voltage for electrical discharges in gases at low pressure.

● Medical Devices: X-ray devices, Violet ray devices etc.

● Electrotherapy: Electrotherapy is the use of electrical energy for medical treatments.

● Ozone Generators: The sparks, brush discharges and corona produced by Tesla Coils can generate Ozone from atmospheric molecular Oxygen.

● Particle Accelerators: It is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for basic research in particle physics.

● Leak Detectors: Small Tesla Coils are used to detect leaks in scientific high vacuum systems and igniters in arc welders.

● Electrical stage shows and entertainment purposes: Tesla coils are widely used for production of Artificial Lightning as well as Musical sounds. These are also used for Educational Purposes.

● Power source: Tesla coils are used to generate extremely high voltage with relatively high-power levels.

 

Advantages and Disadvantages of Tesla Coil

The advantages are

·       Uniform voltage distribution throughout the winding coils.

·       Builds up the voltage at a slow pace and hence no damage.

·       Using 3-phase rectifiers at higher powers offer tremendous load sharing.

·       Wireless non-radiative energy transfer.

·       Wastage of power is less.

·       Highly resonant strong coupling provides high efficiency over distance.

The disadvantages are

·       Wireless power transmission can be possible only for few meters.

·       Buying large DC smoothing capacitor involves high costs.

·       Complex circuit and construction of circuit consumes much time as it needs to be perfect to resonate.

·       Electronic equipment damage occurs due to tesla coils.

·       Tesla coil usage causes several health hazards due to high voltage radio frequency emission that includes skin burn, damage to the nervous system and heart and it produces ozone which is poisonous.

·       There is a possibility of fires and explosions in tesla coil.

 

Future of tesla coil

·       Wireless transmission & charging.

·       Transmission of electrical energy to electrical load from power source without using wires.

·        World will be free of wires.

 

Conclusion

        Tesla coil can produce high voltages at the output with low current intensity. It can transmit electrical energy to load without using wires and power loss will be less as it is wireless. But it cannot be used to transmit electrical energy for long distances. It's applications where in different sectors.

(By EE Team, AIM, NIT Rourkela)

Shor's Algorithm and RSA Encryption

 

INTRODUCTION

Every positive number can be factorized to the product of primes. But factoring a large number is computationally expensive and it takes more time to factorize as the number of digits increase. Finding whether a number is prime or composite can be easily done on classical computers. The problem arises when we have to factorize the composite number. Even the most efficient classical factoring algorithm (number-theoretic sieve algorithm) couldn’t factor a composite number in polynomial time. RSA encryption leverages this fact in cryptography. Peter Shor followed the path started by Benioff, Bennett, Deutsch, Feynman, Simon, and others great scientists in the field of quantum computing created an algorithm based on Quantum Fourier Transform and modular exponentiation that can factorize number leveraging one of the properties of factorization and that is its ability to reduce to a period finding the problem.

 

PERIOD-FINDING PROBLEM

A factoring problem can be converted to a period finding problem. A function that repeats itself at a regular interval is called periodic functions and that interval is called the period of a function (e.g. trigonometric functions). Finding the period of modular function plays an important role in factoring a number.

MODULAR ARITHMETIC (f(x) = a mod N)

If X and Y are two integers, then we know X/Y = P, where P = XQ+R. 
Here, X is dividend; Y is divisor; Q is quotient; R is remainder.
R can be represented as R = X mod Y (X modulo Y is equal to R)

Example: 7 mod 5 = 2

In terms of the clock, to find the result of X mod Y we can follow these steps ;

1.  Construct a clock for size Y.
2.  Start at 0 and move around the clock X steps. 
3.  Wherever we land, is remainder.

            If X is negative, we move anticlockwise and if X is positive, we move clockwise.

MODULAR EXPONENTIATION (f(x) = a^x mod N)

Let us take the sequence S(n) = 2ⁿ, where n is a natural number.

           S(n) = 2, 4, 8, 16, 32, 64, 128,...                                  --------------> (1)

So, if we take modulo 15 of S(n), we get :

           S(n) mod 15 =  2ⁿ mod 15 = 2, 4, 8, 1, 2, 4, 8,...        ---------------> (2)

One can notice that the function (2) repeats itself with a period 4. For smaller values of a, we can find period of f(x) = a^x mod N just by looking at it. For bigger values of a, we use something called Quantum Fourier Transform (QFT). These modular exponentiations and QFT consist the heart of Shor's algorithm.

SHOR’S ALGORITHM

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer or Shor’s Algorithm is a period finding algorithm that is used to factorize numbers. Basically, it involves 5 steps to factorize a number and the one that involves quantum computers is step 2.

CONVERTING PRIME FACTORIZATION PROBLEM TO PERIOD-FINDING PROBLEM 

Let N = p*q where p and q are two prime numbers (for simplicity let’s take only 2 prime numbers). Factors (or divisors) of N are 1, p, q and N. Now 1 and N are the trivial solution (i.e.) they are too obvious. We have to find non-trivial solutions. If N is even, then 2 is one of the factors since it is the only even prime number. The other factor will be. So let’s focus on odd numbers, factoring using Shor’s algorithm involves 5 steps.

1. Choose a random positive integer 1 < m < N. Find the greatest common divisor form of N i.e. gcd(m, N) using the polynomial-time Euclidian algorithm. There are two possible outcomes.

Outcome (I):  gcd(m, N) != 1

Here, m<N so the gcd(m, N) cannot be equal to N and according to outcome (I), gcd(m, N) != 1. So if m and N had a gcd it has to be either p or q. If one of them is known (say p), other (say q) can be obtained by dividing N with it (i.e. p).

Outcome (II):  gcd(m, N) = 1

GCD of two numbers will be 1 only if they both don’t have any common divisor (or factor). Proceed to step 2. 

2. Now that we have a number m, we define a function f(x) = MX mod N. Using QFT we find the period P of this function on a quantum computer. P is the period of the function if f(x) = f(x + P). If P is odd, move back to step 1. If even proceed to step 3.

3. If P is odd, move back to step 1. If even, proceed to step 3.

4. P is the period,

        =>  m^P = 1 mod N                                                    -------------------> (3)

        =>  m^P-1 = 0 mod N                                                 -------------------> (4)

Therefore, m^P-1 is a multiple of N. Again, m^P-1 = (m^P/2 - 1)(m^P/2 + 1).

If (m^P/2 + 1) = 0 mod N, then m^P/2  = (-1) mod N           -------------------> (5)

Then return back to step 1, else proceed to step 5.     (see Table 1 for reference)

5.  Therefore, d = gcd(m^P/2-1, N). Since equation (5) is not true i.e. (m^P/2 + 1) != 0 mod N (or m^P/2  != -1 mod N), it can be easily shown that d is non-trivial (i.e. neither 1 nor N) factor (or divisor) of N.     

                Table 1: finding factors of 15 using Shor’s algorithm

m	period P	gcd(15, ar/2-1)	gcd(15, ar/2+1)
2	4	3	5
4	2	3	5
7	4	3	5
8	4	3	5
11	2	5	3
13	4	3	5
14	2	1	15

Step 4 explanation: if m^p/2 = -1 mod N, then gcd(m^p/2-1,N)  would give either 1 or N which are trivial solutions. For example, look at m = 14 in Table 1 which has a period P = 2 and 14^2/2 = -1 mod 15. The factors obtained using m = 14 are 1 and 15 which are out of our scope. So we go back to step 1 and start the whole procedure with different m (< N).

 

APPLICATIONS OF SHOR’S ALGORITHM 

One of the popular applications of Shor’s algorithm is breaking RSA encryption but this can also be used to solve any problem involving period-finding. Though we have quantum computers that can run this algorithm and find factors of small numbers, we need more powerful quantum computers to actually break RSA encryption. Scientists are working to counter this application of Shor’s algorithm before it’s too late. Quantum Computing is an upcoming field and it’s not going to replace classical computers. Quantum computers are not efficient at doing every-day task that we do with our phones or laptops. They are built for a specific purpose like factorization, optimization, model simulations, etc.

(by Naveen V.)