CHAPTER 5. DESIGN OF kj PULSED POWER SUPPLY USING COMPUTER SIMULATION

Transcription

1 150 CHAPTER 5 DESIGN OF kj PULSED POWER SUPPLY USING COMPUTER SIMULATION 5.1 INTRODUCTION The electromagnetic rail guns require compact pulsed power supplies that are to be installed in combat vehicles. The components with high degrees of energy and power density as well as complex system designs need to be realized in such vehicle in order to meet the requirement for the integration of pulsed power supply (Wisken et al 2003). The pulsed power supply is related to choice of energy storage system. An energy storage device that has an acceptably small weight and volume yet can provide appropriately conditioned power is an essential component for the successful implementation of a portable electrical energy system (Gully 1991). Today the energy storage systems which feed the rail launcher are still large. Extensive work and research are being conducted all over the world to minimize the volume occupied by the power supply and its weight. Capacitor based systems are the most familiar type of pulsed power supplies system used in a rail gun research for the past several years. They consist of a number of individual capacitors grouped together called a module and these modules can be grouped together called segments. The capacitors in a module are discharged simultaneously and then the segments are discharged sequentially. The purpose of modules and segments is to give a more constant current pulse to the rail gun load. It is possible to maintain a near constant output current, over a substantial part of the current pulse by varying the inductance and

2 151 capacitance values of segments and then sequentially discharging the segments (Dwight warnock 2003). The design of 500-kJ capacitor module pulsed power system to be applied for the electromagnetic rail gun is currently under investigation by the Government of India. The Anna University, Chennai has initiated a program to develop 500-kJ pulsed power supply using computer simulation techniques. In order to design a 500-kJ pulsed power supply using computer simulation in this thesis, the basic electric requirements of electromagnetic guns to be fulfilled by the appropriate PPS systems are studied. From these requirements, design criteria for high energy discharge modules and their auxiliary systems are derived. Finally, an attempt is made to design a 500-kJ PPS using computer simulation to accelerate the projectile with a velocity of 1 to 1.5 km/s. In order to achieve this task in this work, computer simulation packages called PSPICE software and MATLAB software are used. In PSPICE simulation, trade off study is made to find the optimum number of capacitor stages to get desired current pulse shape and rail parameters. MATLAB is used to obtain the optimum value of rail parameters such as muzzle velocity, current at exit, and effective barrel length. By applying a systematic approach to optimizing the power supply this study has shown that by distributing the capacitance into five equal sized banks, the 75 grams of projectile can be accelerated with a velocity of 1.25km/s. The design is achieved by trial and error methods by changing rail parameter and pulsed power supply parameter. Finally, this work gives some basic considerations on volume and weight requirements of 500-kJ capacitive PPS systems to be applied for rapid fire of electromagnetic rail guns. 5.2 INTRODUCTION TO PULSE FORMING NETWORK Pulsed power supply (PPS) system is important to an electromagnetic gun system( Yongxia Han 2009). Usually, the PPS is made

3 152 up of modules called pulse forming unit (PFUs), which are capacitor based and switched by semiconductors. These PFUs are grouped into several segments and then the modules in the same segment are switched simultaneously as shown in Figure 5.1. RACK -1 CAPACITOR BANK MODULE 1 MODULE 2 MODULE 3 PULSE SHAPING INDUCTOR CROWBAR SWITCH RACK -2 MODULE 1 MODULE 2 MODULE 3 RACK -3 PULSE SHAPING INDUCTOR CROWBAR SWITCH RAIL GUN PROJECTILE MODULE 1 MODULE 2 MODULE 3 PULSE SHAPING INDUCTOR CROWBAR SWITCH Figure 5.1 Simple schematic diagram of capacitor based PPS (Tatake et al 1994)

4 153 For the past several years, a lot of work has been done on research and development of PPS based on a pulse forming network. Many researchers have reported their work on simulation model and real model previously. Lehmann et al (1995) have presented an overview of what can be done to enhance the efficiency of a rail launcher fed by capacitor banks by adjusting the rail design in view to accelerate payloads of several kilograms up to velocities of few km/s with minimum of charged electrical energy. Wey et al (1995) have developed the capacitor bank, consisting of two stages is used for experiments with a 50mm round bore rail gun at a maximum current of 2MA. The rail gun is fed at two different points in order to test distributed energy storage system rail gun concept. Deadrick et al (1982) have developed and validated simulation code at the Lawrence Livermore National Laboratory (LLNL) to predict the performance of a rail gun. The code is called MAGRAC (MAGnetic Rail gun ACCcelerator ), models the performance of a rail gun, driven by a magnetic flux compression current generator (MFCG).The MAGRAC code employs a time-step solution of the non-linear time varying element rail gun circuit to determine rail currents. Emelin et al (1995) have described the pulse formation of a programmable multi-stage pulsed power supply for electric rail guns. This power supply allows optimal operation of the rail gun by maintaining a near constant acceleration force over the launch period. A 450-kJ pulsed-power supply (PPS) for electromagnetic launcher has been built up and tested with a short-circuit load and a rail gun load (Yi Liu 2010). Tatake et al (1994) have developed simulation code using Pascal language to predict the performance of the rail gun. Timothy et al (2004) have described the preliminary design assessments associated with the 200 MJ Naval PoC Facilities. The circuit model was created using MicroCap software to validate the PFN performance at each of the installation increments. Hildenbrand (1993) has described the simulator developed for the Army Research, Development and Engineering Center in The code was in FORTRAN. Jack Bernardes et al (2003) have

5 154 proposed a generic design for the capacitor based PPS. They simulated the PPS and rail gun model using Micro-Cap VI, which is an electrical circuit analysis software package. From the above literature survey, it is found that worldwide research is on to develop the PPS, with respect to their requirements, in order to predetermine the performance of rail using various simulation techniques. In this work, the PSPICE and MATLAB simulation packages have been used to predict the performance of PPS system. 5.3 KEY COMPONENTS OF PULSED POWER SUPPLY SYSTEM The block diagram in Figure 5.2 shows the key components necessary to design a capacitor based PPS system for an electromagnetic rail gun pulsed power system in a combat vehicle. HV - charger Charge Dump Panel Resistor network protection diode PPS-Module Load (Rail gun) Crowbar power diode Pulse forming inductor Energy Storage Power Switch Figure 5.2 Key components of pulsed power supply system The PPS module is connected with charging device through charge dump resistors and relays. The discharging process of the capacitor inside the PPS module will be initiated by a high power switch which should have a

6 155 blocking voltage of several kilovolts and a current capability of several kilo amps. During the discharge, the energy stored in a capacitor is transferred to rail gun load through a pulse shaping inductor. During the crow bar operation the inductor is charged and stores the energy Capacitor Capacitors are used as an energy storage device to store the energy and then to discharge the energy over time. The accumulated charges on the electrodes of a capacitor induce charges in the dielectric. These induced charges determine the permittivity of the dielectric medium. Every dielectric material has its own value of permittivity. Permittivity introduces another term called dielectric constant K of the dielectric medium. The dielectric constant K is defined as the ratio of the permittivity of the dielectric to the permittivity of free space or a vacuum (Cletus Kaiser 1998). In general the stored energy density (MJ/m 3 ) of a capacitor can be calculated using (MacDonald et al 2009) D PFKE (5.1) where K, is the dielectric constant, E is breakdown strength of the dielectric medium and PF is packing fraction (fraction of total volume which stores energy). So the energy density of a dielectric material is proportional to its dielectric constant ( K ) and square of its dielectric breakdown strength ( E ) values. Therefore, high K and E values are significant to achieve high energy density in capacitors. Most capacitors used in pulsed power applications are oil-filled, film and/or paper dielectric. Most polymers used in capacitors have dielectric constant between 2 and 4; impregnated paper has a dielectric constant between 3 and 6, and the polar polymer polyvinylidene fluoride has a dielectric constant of 11 ( Ennis et al 1990). Typical values for electric field in pulsed power capacitors ranges from 500 V/µm 800 V/µm.

7 156 The electric field is limited by the breakdown strength of the polymer dielectric, which is the maximum field that can be applied before a breakdown event short-circuits the electrode. There have been many improvements over the last 10 to 20 years in energy storage capacitor fabrication and the energy densities of 1.5 to 3.0 J/cm 3 are now available (MacDonald et al 2009) Switches One of the most critical components in pulsed power systems is the main output switch due to the electrical and operational safety requirements. The switching device has to pass high currents and withstand high voltages and with this it should be reliable, durable and provide high technology efficiency. Requirements to switching devices depend also on the type of energy storage system. In capacitor based PPS, a switching device should switch on the capacitor bank to the load quickly and reliably and provide multiple passing currents up to several hundreds of kilo amperes with duration from several microseconds to several milliseconds (Dmitrii Alferov et al 1999). On one hand, it has to block the charging voltage of the capacitive storage, without any pre-triggering or pre-firing, on the other hand it has to perform reliable switching when triggered with a minimum of self impedance due to the switching losses to be avoided ( Wisken et al 2004). The switching devices are mainly classified as 1. Non solid state switches 2. Solid State Switches Non solid state switches are 1. Spark gap 2. Triggered vacuum switch

8 157 Solid State Switches Solid state power devices consist of a gate, cathode, and anode. The gate is used to trigger the device by closing the connection between the anode and cathode. The device can only be turned off, once all the current has been dumped to zero. The solid state devices are limited by its break over voltage and di/dt ratings. The solid state devices will fail if either of these two parameters are exceeded.the different types of solid state switches are given below 1. Silicon controlled rectifies 2. Silicon carbide thyristors. 3. Gate turn off SCR. 4. MOS Controlled thyristors. 5. Light activated thyristors. Singh et al (1995) have made a comparison between the above switching devices for their ability to conduct high peak current at high coulomb levels while operating in a mobile tactical system having severe volumetric constraints. He has concluded that for series-parallel arrangement of silicon controlled rectifiers was considered to offer the best approach for the tactical system Diode In pulsed power supply systems, the crowbar diodes conveniently perform important functions. They automatically close upon when the capacitor charges in reverse direction. The diode string will be placed parallel with the capacitor bank, as the voltage on the bank attempts to reverse the parallel diodes will be forward biased. Thus the crow bar diode, preventing voltage reversal on capacitors and trapping energy in pulse shaping coils. The

9 158 advantages of crow bar operation are: it increases the life of the energy storage capacitors, increases energy transfer efficiency, and additional options for pulse shaping (Rolf Dethlefsen 1993). The diode will immediately begin to conduct at the peak bank current, therefore it must be rated at the same standard as the bank switches Cable The cables for the rail gun system must be capable of handling large currents with minimum power loss. The cable will be self-supporting, co axial cable as far as electromechanical force is concerned, so conductor bracing will be eliminated. Another great advantage of the coaxial bus is there is minimal leakage of coaxial cable (Jiannian Dong et al 2005).This will be very helpful in minimizing interaction with the data acquisition and control system. 5.4 COMPONENTS USED IN DESIGN OF 500-kJ PULSED POWER SUPPLY of India as follows: The requirement of pulsed power system given by the Government Energy rating = 500 kj/shot and 8 rounds/minute (Total of 4MJ) Muzzle velocity = 1 to 1.5km/s, Pulse width= 4-5ms and Volume occupied by pulsed power supply and its weight. According to the electric requirements of the pulse power supply, the key components of PPS have to be chosen. In order to carry out this work, the required components that are essential to build 500-kJ power supply design are chosen from the manufacture data. There are many other smaller

10 159 circuits and parts such as timing and control circuits for the bank isolation switches, to protect the capacitors that will be required, in order to make the overall system fully operational. These items are essential to the design and its safe operation, but will not be considered in this work, as they are not the main focus of this thesis Capacitor The capacitors used in the 500- kj PPS system are self healing metalized electrode capable of storing 50-kJ of energy, designed and manufactured by General Atomics Electronic Systems Inc. The pulse power supply (PPS) has to store 500-kJ of energy, hence totally 10 capacitors have been chosen to design a pulsed power supply using computer simulation. These capacitors are divided into 5 independently triggerable capacitor bank modules and each module has two capacitors and they are connected in parallel. The energy density of capacitor used in the PPS is about 3MJ/m 3. The life duration of capacitor reaches up to 1000 times. The technical data of 3MJ/m 3 capacitor is given in Table 5.1. Table 5.1 Technical data of the capacitor ( Capacitance 23685µF Voltage 6.6kV Stored energy 50kJ Pulse discharge current 30kA Size mm Internal inductance <500nH Weight 30kg All the five banks are identically configured therefore the current pulse from each bank will also be identical. Each bank consists of total of two

11 160 capacitors and connected in parallel, resulting in a peak current per capacitor of 30 ka. So the predicted maximum peak discharge current from each capacitor bank is 60kA Inductor As the volume of the capacitors and the switching assemblies are reduced, the volume of the pulse shaping inductor begins to dominate system size. This is particularly true for conventional solenoid inductors, where their external magnetic field must be taken into account. In some cases, other system components must be kept in a low field region, effectively expanding the volume of the inductor. In general, it is necessary to keep other components approximately one radius and one half-length away from the inductor. The effective volume of the inductor is thus four to eight times the volume of the inductor itself. The pulse-forming inductance is decreased on selected modules in an effort to generate an increase in peak launcher current and launch velocity. The inductor used in this pulsed power supply has toroid design to reduce the external magnetic field during the discharge. This is necessary to prevent electromagnetic inference to other electronic equipment Fuse Fuses are used to enable the capacitor to handle large amount of current while providing isolation in the event of short circuit at high voltages (Miguel Del Güercio 2003). Fuses are designed and manufactured by GEAP. Each Maxwell high voltage fuse is rated at 6 ka and 22 kv. Five fuses are connected in parallel and placed in series with the output of each capacitor providing a maximum rating of 30 ka per assembly. Table 5.2 gives the technical data of the fuses used in 500-kJ PPS system.

12 161 Table 5.2 Technical data of the Fuse ( Rated Pulsed Current amps 6000A at 1.5 m.sec Cold Resistance 13 milli-ohms at 20 C Hot Resistance 25 milli-ohms at 250 C Rated Fuse Carrying Action 54,000 A 2 sec Weight 2kg Size 6.1(D)cm 30.4cm (L) Crowbar Switch (diode) The GA diodes selected for this design have a reverse voltage limit of 10% of the rated voltage or 6500 V. As the current delivered by each capacitor bank is 60kA under normal conditions and 120kA under fault conditions, the diodes have to withstand this current. A total of six diodes in two stacks parallel, with three diodes in series in each stack, form a crow bar diode. Each diode is rated at 6 kv and 60 ka for a 8.3-ms half-sinusoid surge current. The technical data of diode is given in Table 5.3. Table 5.3 Technical data of the diode ( Rep. Peak Reverse Voltage and Current VRRM = 6000 V ; IRRM=100 ma Non Repetitive Peak Surge Current IFSM(8.3 ms, VR=0) Maximum Average Current IF(AV) Size Weight IFSM(8.3 ms, VR=0) = 60,000 A 3500A at T case =100 o C cm 5kg Spark Gap Switch As the expected maximum current from each capacitor module is 60kA under normal condition and 120 ka under fault condition, the spark gap

13 162 switch has to withstand the maximum peak current discharged by the each capacitor bank. The spark gap used in this design is ST-150 spark gaps rated at 300 ka with a standoff voltage of 22 kv at zero pressure. The spark gap switch is manufactured by L3 communication. The switch shows ratings of at least a factor of three over the required duty in peak currents to ensure long life and reliability. The spark gaps are connected in series with each capacitor module. The technical data of the spark gap switch is given in Table 5.4. Table 5.4 Technical data of the spark gap switch ( Operating Voltage Peak Current Charge Transfer Size Weight kv 300 ka 120 coulomb per shot 8 inch diameter,11inch height 10kg 5.5 DESIGN OF 500-kJ PULSED POWER SUPPLY USING PSPICE SIMULATION Capacitor Based Pulsed Power Supply Capacitor based pulsed power systems can be described using a simple RLC circuit as shown in Figure 5.3. In which, C is the system capacitance, L is pulse shaping inductance, R is the internal resistance of the inductor, L is the variable inductance of the rails and R is the variable resistance of the rails. As the projectile moves along the barrel, the L and R values will be changed.

14 163 S L R C R L Rail impedance L variable inductance R variable resistance Figure 5.3 An ideal rail gun circuit represented by using RLC circuit When the switch S is closed, the capacitor will start to discharge the stored energy into the system. During the discharging time of the capacitor, the energy stored in a capacitor is transferred to the inductor and inductor gets charged. The inductor will be fully charged, when the capacitor is completely discharged and current flowing through circuit will reach to maximum value at this point. After this point, the current value will exponentially decrease, until the projectile exits the barrel resulting in an open circuit Problem Statement The rail gun is essentially a current operated device. For maximum piezometric launch efficiency, it requires a current pulse which rises rapidly and is then maintained at a constant level and drops to zero, just before the time of projectile exit (Fred Charles Beach 1996).In capacitor based pulsed power supply, capacitor is used only to store the energy and initiate the current pulse. The capacitor discharge phase does not provide majority of the accelerating force to the projectile. But the inductor discharge phase gives the majority of accelerating force to the projectile. That s why it is important to maintain the peak current level high in a rail gun system as long as possible. Obtaining the constant level pulse depends on the values of L, C, and the

15 164 delay between pulses. If the inductance value increases in PPS, the peak current value delivered to the load decreases and the decaying time of the pulse current increases. Also when increasing the inductance of the PPS, the current density in the rails decreases, as it increases the barrel length, but results in higher barrel current when the projectile leaves the rails. If the delay times between the banks discharge increases, the peak current value reduces which result lower acceleration force on the projectile, subsequently the muzzle velocity also being lower. In order to balance for the loss in velocity, it is necessary to increase the amount of capacitance in the power supply, when the voltage is constant. Increasing the capacitance is in disagreement with the stated objective to minimize the capacitance, and also results in increased peak current and a current density that may be higher than in the lower inductance case. From the above discussion, it is clear that designing a power supply for a rail gun is an exercise in system optimization (Dwight warnock 2003). In the multiple-bank model, it is possible to shape the pulse by implementing a non-uniform capacitance and inductance distribution among banks. The capacitance and inductance values are varied on a per bank basis. In case of capacitor, it is necessary to keep large percentage of system capacitor in first bank to deliver a high current to the load and then lower percentage of system capacitor placed for the following banks to boost the current. In the case of inductance, initially higher value of system inductance is placed to limit the peak current and lower value of system inductance is placed to following bank to reduce the decaying time current, resulting in a lower exit current (Alexey Alexeev et al 1992). In order to identify optimum solution for 500-kJ PPS in this work, different types of capacitor banks connection are studied. Capacitors are grouped into single, double, triple and five stages. For each capacitor connection pulse shaping inductance value, peak current at exit, muzzle velocity, peak acceleration, pulse width, and effective barrel length of the rails are calculated. These values are then used to estimate the size of the capacitor banks and system inductances.

16 IMPORTANT MODELS USED IN PSPICE DESIGN The following PSPICE models are used in design of PPS for electromagnetic rail gun (PSPICE Manual 1996) Current Controlled Current Source (CCCS) Figure 5.4 shows, the current controlled device used in simulation. As name implies, the output current can be controlled by its input current and gain. In a 500-kJ PPS design by using this model, the rail current can be separated from the circuit without affecting rail gun circuit parameter. This current can then be used to find the acceleration, velocity and distance traveled by the projectile. As the output of this model depends on its input and gain, in PPS design the gain of CCCS chosen as 1. So that input and output current will be same. Output function F1 F Input function Figure 5.4 Current controlled current source model Power Model Figure 5.5 shows the power model used in the simulation. This model is used to get the powered value of input and it depends on the gain specified by the user. This model can be used to get the squared value of

17 166 current in a simulation. When PWR value is 2, this block will square the input current and gives as an output current. PWR 2 Power Figure 5.5 Power model Gain Model Figure 5.6 shows the gain model used in the simulation. This model is used to multiply the given input with the gain specified by the user. This model can be used to get the acceleration value of the projectile in simulation. In a simulation, the squared rail current is multiplied by a gain of x10-6 using this block and then output is obtained that will be equal to the acceleration of the projectile. Gain Figure 5.6 Gain model Integrator with Gain Figure 5.7 shows the integrator model used in simulation. This model is used to integrate the given input. This model can be used to get the velocity and distance traveled by the projectile in simulation. The 500kJ PPS design has two integrator models and they are connected in series. The first integrator which is connected between gain model and 2 nd integrator model (see in Figure 5.10) gives the velocity of projectile as an output. The second

18 167 integrator which is connected between first integrator and ZX model (see in Figure 5.11) gives the displacement of projectile as an output V Figure 5.7 Integrator with gain Variable Inductance and Resistance In the operation of the rail gun, resistance and inductance of the rails are varied with respect to distance travelled by the projectile. So rail inductance and resistance are calculated with respect to distance travelled by the projectile while simulating a rail gun model, for accurate performance prediction. The implementation of rail impedance in simulation is difficult. In order to implement variable impedance of rail using simulation, a suitable model has to be developed. To incorporate above mentioned varying parameter in simulation ZX model can be used. Figure 5.8 shows the ZX element which can be used to represent the variable rail resistance and inductance in simulation. 1 2 X 1 + ZX _ Reference X 2 + ZX _ Reference R1 Figure 5.8 PSPICE model of ZX (Wey et al 1997)

19 168 The basic operation of the model is explained below The resistance or inductance across the terminals 4 and 5 of ZX element is the multiplication of voltage across the terminals 1and 2, and resistance or inductance that is connected across the terminal 3 and ground. In the rail gun simulation, output of the second integrator which is proportional to distance travelled by the projectile is connected to terminal 1 and terminal 2 is grounded. This indicates that the voltage across the terminal 1 and 2 is proportional to distance travelled by the projectile. The resistance or inductance per meter of the rail is connected across terminal 3 and ground in ZX model. The multiplication of voltage across the terminal 1 and 2, and the resistor or inductor across the terminal 3 and ground, will give the live resistance or inductance of the rails across the terminal 4 and 5 (Wey et al 1997). 5.7 THE ELECTRICAL EQUIVALENT CIRCUIT OF CAPACITOR BASED PULSED POWER SUPPLY Figure 5.9 shows, the electrical equivalent circuit of capacitor bank pulsed power supply with electromagnetic rail gun load. The system has been divided in to an energy storage module, rail module and projectile module. The energy storage module consists of a capacitor followed by a spark gap switch. The spark gap switch is modeled by the series combination of a switch, a diode, and an on-state resistance. The crowbar diode is represented by a diode and an effective forward resistance. The pulse-shaping inductor is modeled as an inductance and a related conductor resistance. The output cable is represented by its inductance and resistance. The inductance and resistance of the rails are varies linearly with respect to projectile position. The rail inductance value depends on the rail dimension and designs. The calculation of rail inductance values with respect to rail dimensions and design have been explained in chapter 3 and 4.

20 169 Cable Rail 1 Rail 2 Pulse shaping inductance Projectile Spark gap Energy Module + _ HV Figure 5.9 Electrical equivalent circuit of capacitor bank PPS Calculation of Acceleration, Velocity and Distance Travelled by the Projectile The equations of motion for the projectile in the barrel can be solved using the rail gun acceleration equation (Jack Bernardes et al 2003). a 2 ' I L 2m (5.1)

21 170 where I is the current delivered to load in Amps. m is the mass of the projectile in kg. ' L is inductance gradient of the rail in µh/m. The inductance gradient of the rail is assumed as L = 1.096µH/m Mass of the projectile is assumed as m = 75 grams Then the acceleration of the projectile is given as a I I 3 m/s 2 (5.2) From the above equation 5.2, it is observed that the squared value of current has to be multiplied by to get the acceleration of projectile in simulation. Velocity of projectile can be obtained by integrating the acceleration of projectile and given as v adt m/s (5.3) The distance traveled by the projectile can be obtained by integrating the velocity of projectile and given as d vdt m (5.4) In order to calculate acceleration and velocity of the projectile, and distance traveled by the projectile using simulation, the total current supplied to the rail has to be separated from the circuit without affecting the rail parameter values.

22 SINGLE STAGE CAPACITOR (SSC) BANK PULSED POWER SUPPLY Figure 5.10 shows the electrical equivalent circuit of single stage capacitor bank of 500-kJ pulsed power system designed using PSPICE simulation. Energy storage device used in simulation is capacitor of mf and manufactured by GEAP. RC Lp L C X 1 + _ X 2 + ZX _ Reference R p 0.101m 1.096µH Spark Gap T G FPOLY PWR x10-6 C D 1.0 0V 1.0 0V + _ 6500V RD C = mf Rp = 6µ Lp = 5µH L C = 70nH R C = 60µ TG = 0.1ms a = acceleration of the projectile v = velocity of the projectile d = distance travelled by the projectile a v d Figure 5.10 Single stage capacitor bank design using PSPICE simulation The energy stored by one capacitor is 50 - kj. Totally 10 capacitors have been chosen and they are connected in parallel to form a single stage capacitor bank. Each capacitor is rated for 30 ka current, so the predicted maximum current delivered by single stage capacitor bank to the rail gun load is 300kA. The capacitor is initially charged to 6500V. To get the rated current

23 172 of single stage capacitor bank module, the pulse shaping inductance value varied from 1µH onwards and the output current is checked. It is observed that for the inductance value of 5µH, the peak value of the load current is 270kA, which is less than the rated value of SSC bank module. So this value has been chosen as a pulse shaping inductance value for single stage capacitor bank module. The other components of pulsed power supply are selected as per the design requirement and given in Table 5.5. In order to calculate the rail gun parameters such as acceleration, velocity, and distance traveled by the projectile using simulation, the total current supplied to the rails has to be separated from the circuit without affecting the rail impedance values. The portion FPOLY block, shown in Figure 5.10, is used to separate the load current from the circuit to calculate the rail gun parameters without affecting the rail impedance values. The current obtained from the FPOLY block is squared and then multiplied by the gain to calculate the acceleration of the projectile based on equation (5.1) and then uses two integration ABM blocks to calculate the velocity of the projectile and distance traveled by the projectile. Table.5.5 Key components of pulsed power supply for SSCB S.No Components Value 1 Capacitor C= mf U= 50kJ 2. Inductor 5µH 1 3 Spark gap Operating Voltage = kv Peak Current = 300 ka (ST- 150) 4 Number of component required 10 capacitors and connected in parallel 4 Diode Rep. Peak Reverse Voltage & Current VRRM = 6000 V ; IFSM =60.000A 30 (Ten stacks connected in parallel, each stack consist of three diode

24 Performance of 500-kJ SSCB Pulsed Power Supply System Capacitor Discharge Characteristics Figure 5.11 shows the predicted capacitor discharge characteristics of the SSCB PPS obtained from simulation. It is observed that all the capacitors are simultaneously discharged at the same time. It is also observed that the time taken to complete the discharge process is less than 1ms. Capacitor Voltage (kv) Time (ms) Figure 5.11 Capacitor Discharge characteristics Total load current delivered to the load Figure.5.12 shows the predicted total current delivered to rail gun load obtained using simulation. In SSCB all the capacitors are connected in parallel and have the same rating, hence all capacitors will deliver an equal amount of current to the load. As the current delivered by individual capacitor is 30 ka and then the expected maximum current delivered by SSCB will be 300kA. From the Figure 5.12, it is observed that the maximum current delivered to the load is 270 ka, which is less than that it s rated value. From the figure, it is also observed that the peak current value is 270kA at 0.5ms.

25 174 Total load current (ka) Time (ms) Figure 5.12 Total current delivered to the load Acceleration of the projectile Figure 5.13 shows the predicted acceleration of the projectile obtained from simulation. It is observed that the peak acceleration of the projectile at 0.5ms is 520 km/sec 2. It is also observed that the total current and acceleration pulse has same shape. This is because according to equation (5.1), the acceleration force is directly propositional to the current which is supplied by capacitor bank. Acceleration (km/s 2) Time (ms) Figure 5.13 Acceleration of the projectile

26 Velocity of the projectile Figure 5.14 shows the predicted velocity of the projectile obtained from simulation. It is observed that the velocity of the projectile is constant after the time period 1.5ms.Therefore, the projectile under frictionless condition would exist the rails at 1.5ms. This graph can be used to get muzzle velocity of the projectile. The point after which the projectile velocity is constant, called muzzle velocity of the projectile. It is observed that the muzzle velocity of the projectile is 275 m/sec at 1.5 msec. From the Figure 5.12, it is observed that the current at exit is 10kA. Velocity (m/s) Time (ms) Figure 5.14 Velocity of the projectile Displacement of the projectile Figure 5.15 shows the predicted distance traveled by the projectile obtained from simulation. The distance travelled by the projectile graph can be used to calculate the effective barrel length of the rail. The point at which acceleration stops is called the effective barrel length, since the projectile does

27 176 not gain velocity after this point. It is observed that the effective barrel length of the rail is 0.3m at 1.5ms. Distance (m) Time (ms) Figure 5.15 Distance travelled by the projectile 5.9 FIVE STAGE CAPACITOR BANK (FSCB) OF 500kJ PULSED POWER SUPPLY Figure 5.16 shows the electrical equivalent circuit of FSCB of 500- kj pulsed power system for electromagnetic rail gun designed using PSPICE simulation package. In this PPS, the capacitors are divided into five different groups in which two capacitors are connected in parallel. Usually capacitor groups are switched on sequentially in order to get nearby constant current. Normally the capacitor banks will be fired at the peak current level of the previous bank. As per the requirement given by the Government of India, in FSCB connection, the duration of current that is supplied to rail is assumed as 5ms. So in FSCB connection the values of acceleration, velocity and effective barrel length of the rails are calculated at 5ms.

28 Figure 5.16 Five Stage Capacitor Bank (FSCB) pulsed power supply system 177

29 178 The components that are used in the five stage capacitor modules have already been given in section Performance of 500-kJ FSCB Pulsed Power Supply Capacitor discharge characteristics Figure 5.17 shows the predicted capacitor discharge characteristics of the FSCB pulsed power supply obtained using simulation. Once all the capacitors are charged to 6500 V, the first capacitor bank is fired at a time t= 0.1ms. The remaining capacitor banks are fired at peak current level of the previous bank. From the figure, it is observed that remaining capacitor banks are fired at 0.8ms, 1.45ms, 2.25ms and 3.0ms respectively, because these are the time for which current will be maximum in each capacitor bank. Capacitor Voltage (kv) C 1 Voltage C 2 Voltage C 3 Voltage C 4 Voltage C 5 Voltage Time (ms) Figure 5.17 Capacitor discharge characteristics of FSCB Individual capacitor current delivered to the load Figure 5.18 shows the predicted individual capacitor current delivered to the rail gun load obtained from simulation. As the predicted maximum current from each capacitor bank is 60kA from the figure, it is observed that the current which is delivered by each bank is less than the

30 179 predicted maximum current of capacitor bank. It is also observed that the peak current time period (time taken to reach peak current value) for each capacitor bank is 0.8ms, 1.45ms, 2.25ms and 3.0ms respectively. The first capacitor is fired at 0.1ms and the remaining capacitor banks can then be fired sequentially at these time periods. It is also observed that each segment current has the same pulse shape, this because all capacitor banks are essentially identical. Individual load current (ka) C 1 Current C 2 Current C 3 Current C 4 Current C 5 Current Time (ms) Figure 5.18 Individual Capacitor bank current delivered to the load Total current delivered to the load Figure 5.19 shows the predicted total current delivered to the rail gun load obtained using simulation. From the figure, it is observed that the total current which is delivered by PPS is a composite of the five segment currents, which are introduced into the breach end of the rail gun at different times by switching on the segments one by one. It is also observed that the peak current value delivered by capacitor bank is less than the rated value.(i.e 300kA) From the figure, it is observed that the peak current value of the load current is 214kA at 3.4ms. It is also observed that current in rail is 142kA at 5ms. This current is called exit current in which projectile gets into the space from rail.

31 180 Total load current (ka) Time (ms) Figure 5.19 Total current delivered to load Acceleration of the projectile Figure 5.20 shows the predicted acceleration of the projectile obtained using simulation. It is observed that the total current and acceleration pulse has same shape. This is because according to equation 5.1, the acceleration force is directly propositional to the current which is supplied by capacitor bank. It is also observed that the peak acceleration value of projectile at 3.4 ms is 334km/sec 2. Acceleration (km/s 2 ) Time (ms) Figure 5.20 Acceleration of the projectile

32 Velocity of the projectile Figure 5.21 shows the predicted velocity of the projectile obtained from simulation. It is observed that the muzzle velocity of the projectile is 0.9 km/ sec at 5 ms. Velocity (m/s) Time (ms) Figure 5.21 Velocity of projectile Displacement of the projectile Figure 5.22 shows the distance travelled by the projectile obtained using simulation. It is observed that the effective barrel length of rail is 1.8m at 5ms. Distance (m) Time (ms) Figure 5.22 Distance travelled by the projectile

33 RESULTS AND DISCUSSION In order to get optimum number of capacitor stages for 500-kJ pulsed power supply, in this work the capacitors are grouped in to a different module. Capacitors are grouped into single, double, triple, and five stages. For each capacitor bank connection, pulse shaping inductance value, peak current value, current at exit, muzzle velocity, peak acceleration, pulse width and effective barrel length of the rails are calculated and the values are given in Table 5.6. From the Table 5.6, it is observed that increasing the number of stages in a capacitor bank module causes an increase in value of pulse width time, muzzle velocity of projectile, the time at peak current value, effective barrel length of the rails, current at exist and decrease in value of peak current and peak acceleration. Table 5.6 Comparison of capacitor bank configurations S. No. Components and rail gun parameters 1 Capacitor (mf) Energy stored per bank(kj) 2 Inductor (µh) No. of stages 1stage 2 stage 3 stage 5 stage 10x x x per stage x x x Each stage has 2 x per stage 3 Current pulse 1.5ms 3ms 4ms 5ms width 4 Peak current 270 ka 240 ka 220kA 214 ka value 5 Time at Peak 0.5 ms ms 1.4 ms 3.4 ms current value 6 Peak acceleration km/sec km/sec 2 334km/sec 2 km/sec 2 7 Muzzle velocity 275 m/sec 370m/sec 700m/s 900m/s 8 Length of rail 0.3m 0.8m 1.7m 1.8m 9 Exit current 10 ka 77 ka 75KA 142 ka 45

34 183 The aim of this thesis is to design a 500-kJ pulsed power supply system which can be used to accelerate the projectile at a velocity of 1000 m/s to 1500m/s. But from the Table 5.6, it is observed that by dividing the capacitor banks into five stages, it can be able to accelerate the projectile at a velocity of only 900m/s. In order to achieve the goal, the work is extended, through which the firing time of capacitor banks have been changed. Two options have been taken to fire the capacitor banks sequentially. Case 1. Case 2. The capacitor banks are fired before the peak level of previous bank current. The capacitor banks are fired after the peak level of previous bank current. (a) Performance of 500kJ PPS for case 1 study In order to improve the performance of the 500-kJ PPS, the firing time of each capacitor banks have been changed. In case study 1 the capacitors are fired before the peak level of previous bank current. The firing times are randomly selected to fire the capacitor bank sequentially. Individual load current (ka) C 1 Current C 2 Current C 3 Current C 4 Current C 5 Current Time (ms) Figure 5.23 Individual load current for case study 1

35 184 Figure 5.23 shows the predicted individual capacitor current delivered to the rail gun load obtained from simulation. It is observed that the peak current time period (time taken to reach peak current value) for each capacitor bank is 0.8ms, 1.45ms, 2.45ms and 3.0 respectively. The first capacitor is fired at 0.1ms and the remaining capacitor banks can then be fired sequentially at earlier to this time period. In this case, the capacitor banks are fired at 0.4ms, 0.6ms, 1ms, and 1.2ms. Total load current (ka) Time (ms) Figure 5.24 Total load current for case study 1 Figure 5.24 shows the predicted total current delivered to the rail gun load obtained using simulation. From the figure, it is observed that the peak value of load current is 263kA at 1.68ms which is less than the rated value of PPS. It is also observed that current in rails is 112kA at 5ms. This current is called exit current in which projectile getting into space from rail.

36 185 Acceleration (km/s 2 ) Time (ms) Figure 5.25 Acceleration of the projectile for case study 1 Figure 5.25 shows the predicted acceleration of projectile obtained using simulation. It is observed that the peak acceleration value of projectile is 492km/ sec 2 at 1.68 ms. Velocity (m/s) Time (ms) Figure 5.26 Velocity of the projectile for case study 1 Figure 5.26 shows the velocity of projectile obtained from the simulation. It is observed that the muzzle velocity value of the projectile is 1.2 km /sec at 5 ms.

37 186 Distance (m) Time (ms) Figure 5.27 Distance travelled by the projectile for case study 1 Figure 5.27 shows the distance travelled by the projectile obtained using simulation. It is observed that the effective barrel length of rail is 2m at 5ms. (b) Performance of 500kJ PPS for case 2 study In case study 2, the capacitor banks are fired after the peak level of previous bank current. The firing time of capacitor banks are randomly selected to fire the capacitor banks sequentially. In this case, the capacitor banks are fired at 0.1ms, 1ms, 1.65ms, 2.45 and 3.2ms. Total load current (ka) Time (ms) Figure 5.28 Total load current for case study 2

38 187 Distance (m) Time (ms) Figure 5.29 Projectile positions for case study 2 The total load current and projectile position for case 2, obtained from simulation is shown in figures, Figure 5.28 and Figure The rail gun parameters such as peak current, current at exit, muzzle velocity, peak acceleration, pulse width, and effective barrel length are calculated for the two case study and the values are given in Table 5.7. Table 5.7 Comparison of five stage capacitor bank configuration with respect to firing time of capacitor bank module S. No. Rail gun parameters values Peak level time of previous bank) 1 Firing time of Bank1= 0.1 capacitor bank Bank2= 0.8 (ms) Bank3=1.45 Bank4=2.25 Bank5 =3.0 2 Current pulse width Firing time of capacitor banks Before the peak level time of previous bank Bank1=0.1 Bank2=0.4 Bank3=0.6 Bank4=1 Bank5 =1.2 After the Peak level time of previous bank Bank1=0.1 Bank 2=1 Bank3=1.65 Bank4=2.45 Bank5=3.2 5 ms 5 ms 5 ms

39 188 Table 5.7 (Continued) S. No. Rail gun parameters values 3 Peak current value 4 Time at Peak current value 5 Peak acceleration 6 Muzzle velocity 7. Effective barrel length Peak level time of previous bank) Firing time of capacitor banks Before the peak level time of previous bank After the Peak level time of previous bank 214 ka 263 ka 204 ka 3.2 ms 1.68 ms 3.7ms 348 km/sec km/sec km/sec m/s 1200 m/s 850 m/s 1.8 m 2 m 1.5 m 8. Exit current 142 ka 112 ka 150 ka From the Table 5.7, it is observed that as the capacitor banks are fired before the peak level of previous bank 1. The peak current value delivered to the load, peak acceleration value of projectile, velocity of projectile and the distance travelled by the projectile are increased. 2. The current at exit and the time at peak current values are decreased. of previous bank It is also observed that as the capacitor are fired after the peak level 1. The peak current value delivered to the load, peak acceleration value of the projectile, velocity of the projectile and distance travelled by the projectile are decreased.

40 The current at exit and time at peak current values are increased. It is also observed that the 500kJ pulsed power supply can be able to accelerate a projectile at a velocity of 1.2km/s, when the capacitor banks are fired before reaching peak level current of previous bank. These parameters can be chosen as optimized parameters of 500kJ pulsed power supply OPTIMIZATOIN OF 500 kj PULSED POWER SUPPLY DESIGN USING MATLAB SIMULINK PSPICE has large library of electrical linear and non linear components. It has a user friendly interface. It is only necessary to draw an electrical circuit. Current and voltage can be obtained at every node or mesh in the circuit. However there are disadvantages. Details of the solver are not known to the user. The limit on the circuit complexity is not known and it takes a long time, to become acquainted with the code, especially with the large library of component (Wey et al 1997). To overcome the drawback of PSPICE simulator package, in this work MATLAB software is also used to optimize the pulsed power supply for electromagnetic launcher application. MATLAB is a high-level technical computing language and interactive environment for algorithm development, data visualization, data analysis, and numerical computation. Using MATLAB, technical computing problems can be solved faster than with traditional programming languages, such as C, C++, and FORTRAN. In this work, the 500-kJ pulsed power supply system is designed using MATLAB software to optimize the rail gun design parameters. The electrical equivalent circuit of 500-kJ PPS system and rail gun model is drawn

41 190 in MATLAB and value of rail parameters such as the velocity, acceleration of the projectile and effective barrel length of the rails have been calculated DESIGN OF 500-kJ CAPACITOR BASED PULSED POWER SUPPLY Figure 5.30 shows, the capacitor based pulsed power supply and rail gun model designed using MATLAB simulation package. The system has been divided into energy storage module, rail module and projectile module. The energy source used in this design is capacitors. An energy storage module consists of capacitors, spark gap, and a diode. The metalized self healing capacitors, which are used in a 500-kJ PPS, have a capacitance of mF and 50kJ of energy, and a maximum peak voltage of 6500V D.C. Each capacitor is capable of delivering a peak current of 30kA under normal condition. The capacitors are divided into five groups each will have two capacitors and they are connected in parallel. So the total capacitance in each bank is 4.767mF and the total energy stored per bank is 100-kJ. In the simulation, triggering circuit of a spark gap is modeled as a switch. The opening condition and closing condition of the switches are specified by using voltage level. The voltage level is specified with respect to distance traveled by the projectile. The crowbar diode is represented by a diode and an effective forward resistance. The pulse shaping inductor is modeled as an inductance and a related conductor resistance and the output cable is represented by its inductance and resistance. The variable inductance of rails is represented by variable inductance element. In MATLAB simulation the variable resistance of rails cannot be taken into consideration as its value is low, when compared to the rail inductance.

42 Figure 5.30 Capacitor based pulsed power supply with rail gun load 191

43 Implementation of Variable Inductance in MATLAB Simulation Figure 5.31 shows the equivalent circuit of variable inductance which can be used to represent a variable inductance of rails in MATLAB simulation. D L' Voltage measurement + - Discrete time integrator 1/C X v/l L is variable D = Distance travelled by the projectile L = inductance gradient of the rail = 1.096µH/m 2 Injected inductance current Figure 5.31 Variable inductance used in MATLAB This block allows the modeling of the variable inductance of the rail, which is dependent on projectile location in the barrel. The variable inductor block represents a linear time varying inductor. This block provides two options for the relationship between the voltage v across the device and the current through the device is i, when the inductance at port L is L, the equation parameter determines which of the following equations the block uses: di dl v L i dt dt (5.5) where v is the voltage across the inductor and the rails. L ' is inductance gradient of

44 193 the formula The current flowing through the inductor can be calculated using 1 i vdt L (5.6) In the variable inductance block shown in Figure 5.35, the voltage across the inductor is measured between the terminal 1 and 2. The measured voltage which appears across the inductor is integrated and then divided by inductance gradient of the rails to measure the current through the inductor. The inductance gradient of the rail is multiplied by the distance traveled by the projectile, so the inductance gradient of the rail value changes with respect to projectile position. Once the L value changes, automatically the current flowing through inductor and the voltage across the terminal will change with respect to time. Once the voltage across the terminals changes, automatically the impedance across the terminals will change with respect to time Realization of Projectile Motion The equations of motion for the projectile in the barrel can be solved by defining a separate series circuit, using current source and two electronic integrators as shown in the Figure The total current delivered by the energy source is squared and then multiplied by the inductance gradient value of the rail. The resultant value is then divided by 1/2m to get the projectile acceleration (a) and then the acceleration is integrated using discrete time integrator to get velocity of the projectile (v). The obtained velocity is then integrated once again using the discrete time integrator to get the projectile position (d). The distance would be used, as if it was a B-dot signal to make the switches turn on. In simulation instead of B dot probe, relays are used to sense the distance traveled by projectile.

45 194 Acceleration Velocity a v I Current I 2 L' X Divide Discrete time integrator Discrete time integrator D Distance 2m m = Mass of the projectile L = Inductance gradient Figure 5.32 Equivalent circuit of projectile motion Implementation of B Dot Probe in MATLAB Simulation A B-dot probe is a closed wire that will generate a signal from an induced voltage, based on Faraday s Law v ind d N (5.7) dt The induced voltage will come from change in magnetic flux density generated when the magnetic field passing through the wire. The B- dot probe is placed over the path of armature and the current will be flowing parallel to the B-dot probe through the armature, causing a magnetic field to circulate through the closed wire. The signal induced by a B-dot looks like a negative sinusoid. As nearer to armature, the induced voltage will go negative. When the armature is directly over the B-dot, the induced voltage goes to zero because the magnetic field that goes through the loop twice, canceling itself.

46 195 Finally, when the armature passes the B-dot probe, the induced voltage will be positive (Bryan Mcdaniel 2006). In simulation instead of B dot probe, relays are used to sense the distance traveled by projectile. The relays send the signal to spark gap switch once the projectile reaches specified distance. Under normal condition the output of the relay will be zero volts, once the projectile reaches specified distance the output of relay become one volt (High level). If the relay output becomes one, the switch which is connected to the corresponding relay will be closed, so that the spark gap switch which is connected to that corresponding switch will be triggered. Charging Timer Bank 1 Timer Bank 2 Distance Relay 1 Bank 3 Relay 2 Bank 4 Relay 3 Bank 4 Relay 4 Figure 5.33 Electrical equivalent circuit of controller circuit Figure 5.33 shows, the electrical equivalent of control circuit designed using relays in MATLAB simulation. The charging block is used to

47 196 protect the power supply. Once the capacitors are charged to its rated voltage, it will send the signal to all switches which are connected between capacitor banks and source voltage, which results open the switches. Once the switches from the source side opens, the relay will send signal to the spark gap switch which is connected to first capacitor bank (see in Figure 5.33 bank1) and it gets triggered, once the spark gap gets triggered the current will be flowing through rails. Remaining capacitor banks spark gap switches are then triggered by other relays with respect to projectile position. In this work, the blocks, charging and Bank 1 shown in Figure 5.33 are used to send the signal to the switches which are connected to them by using the timer block Optimization of 500kJ Pulsed Power Supply and Rail Gun Design The aim of this work is to design a 500-kJ pulsed power supply which can be used to accelerate the projectile with a velocity of 1.0 to 1.5km/s. In order to get the desired velocity of projectile, First, proper value of inductance gradient of the rails has to be selected, since the force acting on the projectile is directly proportional to mass of the projectile and inductance gradient of rail. The inductance gradient value depends on rail geometry. The calculations of inductance gradient of rails for different rail geometry have been discussed in chapter 3. In this work, the inductance gradient value is chosen as 1.096µH/m. For this L of the rails, the rail dimensions are as follows Rail height = 1cm Rail separation = 4cm Rail width = 4mm

48 197 The electromagnetic rail gun requires flat shape of current pulse to utilize the entire barrel length of the rails effectively. To utilize the entire barrel length effectively the capacitor banks are fired at different time period. Usually capacitor bank are triggered at peak current level of previous bank. In this work, an attempt is made to fire the capacitor banks with respect distance traveled by projectile in order to utilize the entire length of barrel effectively. In order to get desired velocity of the projectile in this work, two parameters such as B dot probe positions of the rails and mass of the projectile have been considered in MATLAB. For various B dot positions such as 1. 5cm,10cm, 40cm, and 70cm 2. 5cm, 20cm, 30cm, and 60cm 3. 5 mm, 10cm, 15cm, and 25cm 4. 1mm, 5mm, 1cm, and 5cm, the rail gun parameters such as peak current value, peak value of acceleration, muzzle velocity and effective barrel length of the rails are calculated for various mass of the projectile using MATLAB simulation. In order optimize the entire system, first the pulsed power supply is considered to get desired output current pulse. Initially the B dot positions and mass of the projectile are assigned as 5cm, 10cm, 40cm, 70cm, and 25grams. For these rail gun parameters the desired current pulse can be obtained by varying the C and L of pulse forming network. Since the capacitance is already selected, in this work, the pulse shaping inductance value is varied. The pulse shaping inductance value is varied from 10µH onwards and the output current is checked. Figure 5.34 shows the individual capacitor current and the total load current delivered to the load, for the inductance value of 10µH, obtained from simulation.

49 198 Individual load current and total current (ka) Time (ms) Total Current C 1 Current C 2 Current C 3 Current C 4 Current C 5 Current Figure 5.34Capacitor individual and total load current From the figure, it is observed that the current delivered by each capacitor bank is 120kA. It is more than that it s rated value. In order to get the rated value of capacitor bank, the pulse shaping inductance values are increased gradually until the desired output current is obtained. Figure 5.35 shows the individual and total current delivered to load by PPS for the pulse shaping inductance value of 45µH. Individual load current and total current (ka) Total Current C 1 Current C 2 Current C 3 Current C 4 Current C 5 Current Time (ms) Figure 5.35 Capacitor individual and total load current

50 199 It is observed that each capacitor bank delivered 60kA current to the load and the total current delivered to the load is 240kA. It is less than its rated value. So this value can be considered as a pulse shaping inductance in pulsed power design. Relay 1 Relay 2 Voltage (V) Relay 3 Relay 4 Time (ms) Figure 5.36 Output pulse from the control circuit Figure 5.36 shows the output pulse from the control circuit for the B dot probe positions 5cm, 10cm, 40cm, and 70cm. It is observed that before the projectile reaching projectile to the specified distance of B dot probe the output of the relay is zero. Once the projectile reaches specified distance the output of the relay becomes one. Figure 5.37 shows the acceleration, velocity and distance traveled by the projectile graphs obtained from simulation, for the B dot positions 5cm, 10cm, 40cm, 70cm, and 25 grams of projectile.