AUTOMATIC PROJECTILE PROPULSION SYSTEM

Abstract:

Nowadays, the requirement for natural fuels is going up in an

increased scale, but the available amount of natural fuel resources is going

down in a steep rate. Also, the amount of pollution in the atmosphere is

becoming a great threat to mankind. It is the duty of engineers to overcome

these difficulties by effectively cultivating alternative energy for the essential

needs. This project undoubtedly, is one of the kinds by manipulating the most

effective electrical energy, which has proved its importance since Edison

invented Electrical bulb.

The prevailing scenario, the Projectiles widely referred as bullets

use only the very ancient source for propulsion i.e., the gun powder is used. The

gun powder used in these propulsion systems is very costly. Then the design of

projectile has to be precise, or else it might end up in an accident for the user.

Also there are added difficulties in handling and storage of those weapons.

In our project, we use the magnetic energy in the place of gun

powder for projectile propulsion. Here, electromagnetic coils are used, that

could be energized and their electromagnetic force generated is utilized to make

the projectile propel in the direction of the target at great speeds. As the speed

of the projectile can be easily varied, variable ranging can be achieved easily by

varying magnetic flux created at small time. This automatic propulsion system

allows manual control too. The automatic control is handled by a

microcomputer programmed efficiently, so that it could detect objects in

vicinity and launch the projectile automatically or by command. It has an option

of alarm to alert the personnel in the environment when an obstacle in tracking

zone is detected. This gadget is a wall mounting type that can be mounted on

walls of potentially high security zones. Economically this propulsion system is

very much considerable, as our nation imports costly weapons to secure our

peoples peaceful lives, and can be manufactured in home country with our own

natural and human resources.

AUTOMATIC PROJECTILE PROPULSION SYSTEM:

There are three important stages required to construct and make the

automatic projectile propulsion system working. The stages are

Coil and trigger section

Drive system and controller

Infra red detector

WORK OF COIL AND TRIGGER SECTION

It has to accelerate a piece of iron or steel through a tube. The tube

is mounted with an electromagnetic coil. There are no sparks or noise or parts to

wear out. The principle of magnetic attraction draws the projectile along at

rapidly increasing speed. This effect is timed so that the field lasts only up to

few milliseconds. Therefore the magnetic field intensity reaches maximum level

when the projectile is halfway through the electromagnet.

Hardware components required to make coil and trigger section

are,

Electromagnetic coil

Low internal resistance capacitor

Capacitor charger

Switching device

ELECTROMAGNETIC COIL

Number of Turns

The number of turns will have a direct effect on the coil's D.C. resistance. A

large resistance will decrease the current, if the power supply is not changed.

The coil's magnetic field is directly proportional to the number of turns which

is actually turns/inch, and to the coil current. Performance can be maximized

piling on turns and increasing the current.

Steps To Get Maximum Efficiency

1. Storing maximum energy in the form of magnetic field around a coil.

2. Using a projectile that can couple with the field as much as possible, and

has the minimum mass possible.

When you put the projectile near the coil, the system seeks the minimum energy

state. That occurs when the projectile is in the center of the coil. So the system

dumps a bunch of mechanical energy into the projectile so that it can finally

reach that minimum energy state.

Limitations in Designing the Coil

1) Current density

Current density inside the wire gets too high; the coil cannot

dissipate the heat fast enough. You can get around it a little bit by reducing the

duty cycle, but eventually the wire melts during a single shot. Thicker wire is

better to handle high currents.

2) Instantaneous Current

The economical way to supply huge current is by using big

capacitors. Their physical dimensions play an important role in defining the

portability of the system. These physical dimensions are a trade-off between the

capacitance and the WVDC (Working Voltage DC) rating.

3) Output Current

There are limits to managing high currents. The 2N2955 are rated

for 15A continuous current, or 150W total power dissipation. You could use

another device with higher ratings. For example, the IGBT is intended for

electric motor control and can handle a lot more current.

Magnet Wire

This is copper or aluminum wire with a thin insulation to prevent

short circuits. It is single-strand wire insulated with enamel, varnish, cotton,

glass, asbestos or a combination.

The Institute of Electrical and Electronics Engineers (IEEE)

thermal classes of insulation, defined by upper temperature limits at which the

untreated insulation will have a life expectancy of at least 20,000 hours, are

O for 176F (80C)

A for 221F (105C)

B for 266F (130C)

F for 311F (155C)

H for 356F (180C)

In general, materials such as cotton, paper, and silk are class O.

Organic materials, such as oleo resinous and Formvar enamels, varnish-treated

cotton, paper, and silk are class A. Asbestos, mica, silicone varnishes, and

polyamide are class H, while various synthetic enamels fall in the B and F

classes. The polyesterimide enamels, however, are capable of withstanding

temperatures of 356-392F (180-200C).

Almost all magnet wire is insulated soft-drawn electrolytic

copper, but aluminum is being used more, due to scarcity and high cost of

copper. Round aluminum wire, being soft flattens under pressure giving a

higher space factor in coils. At temperatures above 392F (200C) copper

oxidizes rapidly. It also becomes brittle when under stress at such temperatures.

Current ratings

Wire size has to be chosen that will greatly exceed the current

rating for its size. The most relevant would be its fuse rating or short-circuit

current.

A 0.5 mm^2 wire is rated at 3A in some applications but will carry

over 8A in free air without overheating.

Some typical current ratings are:

Maximum current ratings for copper wire

A/mm2 R/mOhm/m I/A

6 3.0 55

10 1.8 76

16 1.1 105

25 0.73 140

35 0.52 173

50 0.38 205

Wire Sizes used in Fuses

The Standard Handbook for Electrical Engineers lists the following formula:

33 * (I/A) ^2 * S = log ((Tm - Ta) / (234 + Ta) + 1)

I = current in Amperes

A = area of wire in circ. mils

S = time the current flows in seconds

Tm = melting point, C (copper's melting point is 1083 C)

Ta = ambient temp, C

Measuring the Coil Strength

First length of the screw is to be measured. Then these steps are followed to

obtain coil strength

1. Position the coil above the projectile.

2. Measure the height from the table to the coil.

3. The coil is held steady.

4. The supply voltage is gradually increased, and the exact voltage at which

the screw is barely lifted from the table is noted. This force is by

definition precisely one G.

5. This measurement is repeated by varying the height and voltage.

Adding External Iron

Air coil is taken and an iron is placed in the flux return path around

the coil. Adding iron increases the total magnetic flux, by adding its

magnetization to the field. The iron also guides the flux toward the firing tube,

where it is much closer to the projectile.

Adding external iron is cheap and easy to do. Just add a flat washer

or two onto each end, and enclose the coil in an iron pipe. An exploded view is

shown here, cut in half to show the interior.

However, this is not very good for heat dissipation. It is going to

take longer for the assembly to cool off after each shot. This iron will saturate at

very large values of magnetic flux.

Magnetic Field Lines

The above figure illustrates how the external field is very small

because the iron carries most of the magnetization. The total magnetic flux is

large because it is primarily guided through the iron parts.

This is a cross-sectional view of the coil surrounded by iron. The

object in the middle is the projectile. The thin strips at the corners are the iron.

The thin black lines indicate the flux lines.

The washers are with a slightly larger outside diameter than the

pipe. The washers are left and right ends of the coil. They are larger than the

pipe because, in an effort to get a very tight fit and squeeze sections of pipe

between washers. An external clamp will ensure they remain tight. Any air gap

should be minimized.

Overview of Current Scenario

Recent advances in energy storage, switching and magnetic

technology make electromagnetic acceleration a viable alternative to chemical

propulsion for certain tasks, and a means to perform other tasks. Launchers of

interest include the dc rail-gun driven by energy stored as inertial in a homo-

polar generator and transferred through a switching inductor, and the opposite

extreme, the synchronous mass driver energized by a high voltage alternator

through an oscillating coil-capacitor circuit. A novel system described here is

the momentum transformer which transfers momentum from a massive

chemically driven armature to a much lighter, higher velocity projectile by

magnetic flux compression. Potential applications include the acceleration of

gram-size particles for hypervelocity research and can be use as reaction

engines in space transport; high velocity artillery; stretcher-size tactical supply

and medical evacuation vehicles; the launching of space cargo or nuclear waste

in one-ton packets using off-peak electric power.

Background

Magnetic guns and launchers have received periodic attention for

many years. The fact that none of these evolved into a practical device reflects

largely the immaturity of required support technology and lack of coordinated

follow-up programs.

Since 1972 considerable attention has been devoted to linear

electric motors in the context of air cushion and magnetically levitated high

speed trains; an extensive review published in 1975 contains over 140

references. Most early efforts utilized linear induction motors (LIMs) which do

not lend them to high acceleration.

High Power Coil Driver

The general idea is to use a high-current PNP transistor (Q2 = 2N2955),

to bring the bottom end of the coil to near ground. The other end of the coil is

attached to the high-power driver, VDD = +15 VDC. When Q2 is turned on, it

goes into saturation. That means there is only 0.2v between its collector and

emitter.

One end is at +15 VDC, and the other is at 0.2 volts. So the current

through a 5-ohm coil will be the voltage v divided by resistance r, or (15 -

0.2)/5 or almost 3 amps. For a comfortable design margin, Q2 has to be driven

fully on with at least 5 amps output current capability.

The base current required to turn on a 2N2955 is the final current

divided by the transistor gain B (beta). With minimum beta B of 20, the

necessary Q2 base current is 5/20 = 0.25 amps base current. It is higher than a

typical JK flip-flop can provide. So another transistor, Q1, can be used to drive

the 2N2955. Both transistors have to be driven into saturation for maximum

current handling and minimum collector-emitter voltage drop.

The base current required to turn on the 2N2222 will be its output

current divided by its gain B (beta). Its gain is at a minimum B = 25, then its

base current must be at least 250mA/25 = 10 mA. This is easily provided by a

TTL chip such as a JK flip flop. The low-power LS family 74LS73 cannot

source high currents.

The transistors should be protected from the coil's kickback voltage. When its

magnetic field collapses as the output transistor turns off, a high voltage spike is

generated. This can easily overcome a transistor's maximum breakdown

voltage. So a protection diode is provided across the coil, which passes current

to keep its forward voltage drop down to about 0.7v.

The 2N2222 configuration also provides another piece of protection for the

sensitive TTL logic. A typical failure mode of output transistors is to short the

collector to the base. With this 2N2222, the fragile TTL output is protected by

the collector-base junction of Q1. The typical reverse breakdown voltage of a

2N2222 is 60 volts or more, so it provides ample protection for this application.

However, if you plan to raise VDD above 60 volts, you should replace both Q1

and Q2 with transistors that have a higher c-e reverse breakdown voltage.

Power Supply Design

It depends on coil resistance and your chosen coil current, and

the number of coils energized at a time. You will usually build your coils first,

and choose your power transistors next, and these will dictate your power

supply requirements.

Generally speaking, a bigger supply is always better. A larger

supply with extra capacity will maintain the voltage better than a wimpy supply.

And if the voltage droops, the current will slump, causing the magnetization

inside the coil to be reduced.

The power supply is basically constant voltage. This assumption

includes practically all modern sources of electrical power, whether or not it

contains a "voltage regulator" component. It includes the most basic

"transformer and rectifier" supply, because the voltage is determined by the

transformer itself. It includes battery and capacitor-driven supplies, because

they both try to provide whatever current is needed to maintain the same

voltage.

CAPACITOR CHARGE AND DISCHARGE

Discharge

Suppose your capacitor is charged to 9 volts, and at time t = 0 the switch is connected to a one ohm resistor. The discharge time is regulated by the resistance.

The initial current (t = 0) is I = V/R = (9 volts)/ (1 ohm) = 9 amps.

For a moment, let us assume the rate of discharge is constant. That is, it will follow a linear discharge curve over time. At this rate it would discharge in time: t = C * V / I = (0.022)*(9 volts)/ (9 amps) = 0.022 sec = 22 milliseconds.

The rate is not actually linear, because the current drops as the voltage drains away. This means discharge is at a progressively slower rate over time. When the capacitor voltage reaches 6 volts, there will only be 6 amps. When it is 3 volts, the current is 3 amps. An ideal capacitor will never completely discharge. It will gradually approach zero volts but never quite reach it.

Exponential Decay

Math for a capacitor discharge is an exponential decay curve: V (t) = V0 e-t / RC

This curve starts at the initial capacitor voltage (V0), and diminishes quickly at first. As time elapses, the slope becomes lesser while the voltage approaches zero. In practical, the capacitor might as well be empty by the time 99% of the initial charge has escaped.

This graph shows that an exponential decay curve at 22 ms is only 64% discharged. It has one-third of its charge left. For this circuit at 40 ms, the exponential decay curve still has 16% of the original charge remaining.

(A) Charge

With the switch at A, the capacitor is charging. Current flows from the battery through the capacitor. The electrons move to one plate, but they do not jump the insulating gap inside the capacitor. They collect on the surface of the plate.

Meanwhile, electrons are removed from the other plate from the abundance that is always there in metals. That gives the plate a net positive charge. And removing the charge completes the path around which current flows.

The current is always the same on both terminals of a capacitor. You cannot move charge into one terminal without removing it from the other.

As the current flows from the battery to the capacitor, it travels through the LED. This emits light during the charging cycle, and then dims and finally turns dark when the capacitor is fully charged.

(B) Disconnected

With the switch at B, the capacitor is disconnected. What happens? There is no current on one terminal of the capacitor. There must be no current on the other terminal.

With no current flowing, the capacitor will keep its 9-volt charge nearly forever. It is stored in the electric field between the two places. It cannot move due to the insulator -- the charges cannot jump the gap.

In practice, no insulator is perfect and the charge will eventually leak away. But this may take months in a high-quality capacitor. Additionally, a significant charge can remain forever, stored in the chemical reaction that ionizes the plate surfaces of an electrolytic capacitor.

(C) Discharge

With the switch at C, the capacitor is connected to the 1-ohm resistor. The charges stored in the capacitor's electric field now have an escape route. They can finally flow from one plate to the other, by travelling through the resistor.

The rate of discharge current depends on the circuit resistance, and the strength

of the internal electric field (voltage).

Launch Position Effect

The launch position can be defined as the distance between the

leading face of the projectile and the rear face of the coil. The sign convention

for delta is such that when the leading face of the projectile is outside the rear of

the coil, delta is negative.

COILGUN BASICS

The Reluctance Coil gun

A reluctance coil gun is basically a solenoid which can launch iron or steel

projectiles by careful timing of the coil current. The cutaway diagram below

shows the very simplest of coil gun designs.

A coil is wound over a non-conducting flyway tube and the

projectile is positioned at the breech end of the tube. If a short current pulse is

passed through the coil the projectile will accelerate into the coil, and if this

pulse is terminated just as the projectile gets to the middle of the coil it will

leave with a gain in velocity. This is how a reluctance coil gun works. One of

the most important facets of coil gun design is the correct timing and shaping of

the current pulse. There are many refinements which can be implemented to

improve the performance and this site explores several avenues of investigation.

Research on the reluctance coil gun is not as widespread in the literature as its

cousin the induction coil gun; however there are some papers that provide a

basic theoretical framework.

While coil guns do not have any industrial application at present,

there have been suggestions that these systems could be used to launch payloads

into orbit. A more realistic application may be launch boosting where a vehicle

is given an initial speed from a long coil gun accelerator. After leaving the coil

gun the vehicle would fire its rockets to achieve orbit. Launch boosting could

result in significant savings in fuel costs. From a military point of view coil gun

technology may have a place in future combat vehicle where, for example, it

could form part of a so-called active electromagnetic armor system. Hyper-

velocity launching still remains the domain of the rail gun, where there is great

deal of ongoing research.

There are two distinct types of coil gun. The first is the reluctance

coil gun which uses the attractive ferromagnetic properties of the projectile to

generate acceleration. The second type is the induction coil gun in which the

accelerating force is repulsive and comes from the eddy currents induced in the

projectile when the coil is fired.  

(1) The use of a non-conducting tube is preferred because with a

conducting tube there is a large electromagnetic braking effect as the

magnetized projectile moves through it. Slotting of the tube can help with

reducing eddy current braking if a conducting tube must be used.

(2) It is termed a reluctance coil gun because the force acts to move the

projectile in the direction of decreasing magnetic reluctance.

The Induction Coil gun

The induction coil gun is identical to the simple reluctance coil gun

in terms of its general construction. The difference is that the projectile is

repelled out of the coil through the action of eddy currents induced in the

projectile. The projectile must be non-ferromagnetic (e.g. copper or aluminium)

and the starting position needs to be slightly off-centre in the coil otherwise it

would not experience a net force when the coil is fired. The impulse

experienced by the projectile depends on mutual inductance and magnetic

diffusion processes, which must be taken into account to affect a good design.

When a multistage launcher is designed, the individual drive coils can be made

short compared to the projectile length, allowing a smoother acceleration

profile.

The projectile need not be a solid conductor - a projectile

comprising a coil shorted can also be used and offers efficiency advantages. The

projectile can also be tubular where it rides a cavity formed by the drive coils

and an inner, rifled mandrel. The mandrel provides support to the projectile

against the radial component of the driving force, while also imparting spin.

 The Reconnection Coil gun

The plate or disc launcher consists of two coils stationed either side

of the projectile. When a current is pulsed into the coils eddy currents form in

the projectile and the interaction of the coil current and eddy currents propels

the projectile. As the projectile leaves the coils the flux lines from the coils

reconnects. A multi-stage cylindrical induction coil gun can also be referred to

as a reconnection coil gun.

The Thompson Coil gun

This is another style of induction coil gun and it works on the same

principle as the classical induction coil gun. Again, the projectile is made from a

non-ferromagnetic material like copper or aluminium. The diagram below

shows one possible design for a single stage Thompson coil gun.

When the coil is fired the ferromagnetic core becomes magnetized,

and as the magnetic flux increases in magnitude it causes a circumferential eddy

current in the ring projectile. This induced current is repelled by the coil current

and the projectile shoots off the core. The faster the flux is increased, the greater

is the induced current and the resulting force on the projectile. For best results

the core should be constructed from either laminations, bundled rods, or

powdered material. This is necessary to minimize eddy currents in the core and

therefore permit rapid flux swings.

 The Helical Coil Launcher

In this type of launcher there are two coils; the drive coil (stator)

and the launch coil (armature). The armature coil is connected electrically in

series with the stator coil via brush gear that contacts the armature's trailing

arms. The projectile is orientated such that the currents in the stator and

armature travel in opposite directions, producing a repulsive force similar to that

in the tubular induction launcher. The difference in this case is that the duration

of the repulsion is not limited to magnetic diffusion timescales.

 

It is possible to construct a helical coil launcher in which the

armature/projectile rides on the outside of an elongated stator. Rails are placed

either side of the projectile and brushes channel current into the stator and

projectile. The brush gear is arranged such that there is a section of energized

stator immediately behind the armature regardless of the position of the

armature on the stator.

 The SPEAR coil gun

This coil gun topology employs passive commutation of the drive

coils eliminating the need for active sensing. The armature is charged with an

initial current that persists during the launch (i.e. the L/R time constant of the

current decay is longer than the launch time). At the start of the launch the drive

coils are fired and, as the projectile is drawn into the coils, each successive coil

has its current driven to zero. The SCR switches that commutate current to the

drive coils automatically turn off when the current goes through zero so no suck

back, or braking force, is produced.

Biot-Savart Law:

It is possible to determine the magnetic field generated by a current element

using the Biot-Savart Law.

 

  Eqn 2.1

 

Where H is the field component at a distance r generated by the current I

flowing in the elemental length l . u is a unit vector directed radically from

l.

Consider an infinitely long wire carrying a current i. The Biot-

Savart Law can be used to derive a general solution for the field at any distance

from the wire.

The general solution is:

  Eqn 2.2

  

 

The field is circular and concentric with the current.

Another configuration which has an analytical solution is the axial

field of a current loop. An analytical solution can be developed for the axial

field, it is not possible to do this for the field in general. In order to find the field

at some arbitrary point complex integral equations should be solved, this best

done with numerical techniques.

 3. Ampere's Law -

This is an alternative method of determining the magnetic field due to a group

of current carrying conductors. The law can be stated as:

 

  Eqn 3.1

 

Where N is the number of conductors carrying a current i and l is a

line vector. The integration must form a closed line around the current. Looking

at the infinite wire again Ampere's Law can be applied as follows:

 

 

The field is circular and concentric with the current so H can be

integrated around the current at a distance r to give:

  

 

 

 

Eqn 3.2

  The integration is very straightforward and shows how Ampere's

Law can be applied to provide quick solutions in some types of geometry. A

knowledge of the field pattern necessary before this Law can be applied. 

4. Field of a Solenoid

When charge flows in a coil, it generates a magnetic field whose direction is

given by the right hand convention - Take your right hand and curl your fingers

in the direction of the current while extending your thumb, the direction of your

thumb now points to the magnetic north end of the coil. The convention for the

direction of flux has the flux emerging from a north pole and terminating on a

south pole. The field and flux lines form closed loops around the coil.

Remember that these lines do not actually exist as such; they simply connect

points of equal value. It is a bit like the contours on a map where the lines

represent points of equal height. The ground height varies continuously between

these contours, in the same way the field and flux from a coil are continuous

(the continuum is not necessarily smooth - a discrete change in permeability

will cause field values to change sharply, a bit like a cliff face in the map

analogy).

 

Fig 4.1

  If the solenoid is long and thin then the field inside the solenoid

can be considered almost uniform.

10. Force on Charged Particle -

 

Eqn

10.1

This force is determined by the vector cross product between the

charges velocity, v, the magnetic induction, B, and is proportional to the value

of the charge. Consider a charge, q = -1.6x10-19 C, moving at 500m/s in a

magnetic field of induction 0.1T, as shown below.

Charge in motion experiences a force

 

The force experienced can be calculated as follows:

The velocity vector is 500 i m/s and the induction is 0.1 k T so:

 

  Obviously, if there is nothing resisting this force then the particle

will be deflected. It would describe a circle in the x-y plane. There are plenty of

interesting things which can be achieved with free charged particles and

magnetic fields.

11. Force on Current Carrying Conductor

Relating this to the force experienced on a current carrying

conductor. There are a couple of different ways of deriving the relationship.

The conventional current as the rate of flow of charge

  Eqn 11.1

 Now we can differentiate the force equation above to give  

  Eqn 11.2

 Combining these equations results in -

  Eqn 11.3

The dl vector points in the direction of the conventional current.

This expression can be used to analyze physical arrangements such as the DC

motor. If the conductor is straight then this can be simplified to  

 Eqn

11.4

  The direction of the force is always at right angles to the flux and

the current direction. When using the simplified equation, the direction of the

force is given by the right hand rule.

12. Induced Voltage, Faraday's Law and Lenz's Law

This is just an extension of the analysis of the force on a charged

particle. If we take a conductor with mobile charge and move it at some speed,

V, relative to a magnetic field, the free charges will experience a force which

will push them to one end of the conductor. In a metal bar there will be a charge

separation where some electrons have been forced to one end of the bar. The

diagram below shows the basic idea.

Voltage induced across a moving conductive bar

  The result of any relative motion between a conductor and a

magnetic induction will be an induced voltage generated by charge motion.

However, if the conductor moves parallel to the flux (the Z direction in fig X)

then no voltage will be induced.

Another situation is where an open planar surface is penetrated by

a magnetic flux. If we set up a closed contour, C, then any change in the flux

linking C will induce a voltage around C.

Flux bounded by contour 

Now if we introduce a conductive loop in place of C then the

changing flux will induce a voltage in this conductor driving a current around

the loop. The direction of the current can be found by applying Lenz's Law

which basically states that the effect acts in opposition to the cause. In this case

the induced voltage will drive a current which opposes the change in flux - if

the flux decreases then the current will try to maintain the flux (anticlockwise

current), if the flux is increasing then the current will try and oppose the

increase (clockwise current). Faraday's Law states the relationship between

induced voltage, changing flux and time -

  Eqn

12.1

 The negative symbol is a consequence of including Lenz's Law.

 13. Inductance

Inductance can be described as the ratio of flux linkage to the current producing

the flux. For example, consider a wire loop of cross-sectional area, A, carrying a

current I.

 

The self inductance can be defined as

  Eqn

13.1

 If the loop is composed of more than one turn then the expression becomes

  Eqn

13.2

 Where, N is the number of turns.

It is important to realize that the inductance is only a constant if the

loop has an air core, i.e. is surrounded by air. When a ferromagnetic material is

part of the magnetic circuit, it introduces a non-linear behavior into the system

which results in a variable self inductance.

14. Electromechanical Energy Conversion -

The core principles of electromechanical energy conversion apply

to all electrical machines and the coil gun is no exception. Imagine a simple

linear electric motor consisting of a stator field and an armature immersed in the

field. This is illustrated in figure. Note that in this simplified analysis the

voltage source and armature circuit have no inductance associated with them.

This means that the only induced voltage in the system is due to the motion of

the armature with respect to the magnetic induction.

Primitive linear motor

When a voltage is applied across the ends of the armature a current

will be developed according to its resistance. This current will experience a

force (I x B) causing the armature to accelerate. This induced voltage acts in

opposition to the applied voltage (Lenz's Law). Below figure shows the

equivalent circuit in which electrical power is converted into thermal power, PT,

and mechanical power, PM.

Motor equivalent circuit

Since the armature is positioned at right angles to the field

induction, the force is given by a simplified version of Eqn 11.4

  Eqn

14.1

 So the instantaneous mechanical power is the product of the force and velocity,

  Eqn

14.2

 Where, v is the velocity of the armature. If we apply Kirchhoff's voltage law

around the circuit we get the following expression for the current, I.

  

 

 

 Eqn 14.3

Now the induced voltage can be expressed as a function of the armature

velocity

  Eqn

14.4

 Substituting Eqn 14.4 into 14.3 yields

 Eqn

14.5

Substituting Eqn 14.5 into 14.2 gives

 Eqn

14.6

 Thermal power developed in the armature is given by Eqn 14.7 

 

 

Eqn

14.7

 And finally we can express the power supplied to the armature as

  

 

 

Eqn

14.8

 Notice also that the mechanical power (Eqn 14.2) is equivalent to

the current, I, multiplied by the induced voltage (Eqn 14.4).

We can plot these curves to show how the power supplied to the

armature is distributed over a range of speeds. In order for this analysis to have

some bearing on coil guns, give variables values that are in keeping with the

coil gun pistol accelerator conditions. Starting point is the current density in the

wire, which helps in determining values for the rest of the parameters. The

maximum current density during the coil testing was 90A/mm2 so if we fix the

wire length and diameter as

l = 10 m

D = 1.5x10-3 m

Then the wire resistance and current become -

R = 0.1

I = 160 A

Now that we have values for resistance and current, we can specify the voltage

needed to drive the current -

V = 16 V

These are all the parameters required to plot the steady-state characteristics of

the motor.

Characteristic curves for a frictionless motor model.

 

Model can be made a bit more realistic by adding a constant friction force, of

say, 2 N, such that the mechanical power loss is proportional to the armature

velocity. This friction value is deliberately large to show its effects more

clearly. The new sets of curves are shown in fig 14.4.

Characteristic curves with constant friction.

 

The presence of friction slightly modifies the power curves such

that the maximum armature speed is slightly less the zero friction case. The

most striking difference is the change in the efficiency curve which now peaks

and then rapidly drops off as the armature approaches its no-load speed. This

form of efficiency curve is typical of permanent magnet dc motors.

If we substitute Eqn 14.5 into Eqn 14.1 we get an expression for F

in terms of v. 

 Eqn

14.9

Plotting this expression we get the following graph -

Armature force vs. velocity

 

Clearly the armature starts off with a maximum acceleration force

that begins to decrease as soon as the armature starts moving. Although these

characteristics give a snapshot of the various operational parameters at any

particular speed, it would be useful to see how the motor behaves in time, i.e.,

dynamically.

14. Electromechanical Energy Conversion -

The dynamic response of the motor can be determined by solving

the differential equation which governs its behavior. Fig 14.6 shows the free

body diagram of the armature from which we can determine the net force and

then write the differential equation.

Armature free body diagram 

Fm and Fd are the magnetic and drag forces respectively. Since the voltage is a

constant we can use Eqn 14.1 and the net force, Fa, on the armature is

  

 

Eqn

14.10

 We can now write an expression for the acceleration of the armature

 

 

 

 

 

 

Eqn

14.11

 

If we write the acceleration and velocity terms as derivatives of displacement, x,

with respect time, and rearrange the expression we get the differential equation

for the motion of the armature

  Eqn

14.12

This is a second order non homogeneous equation with constant coefficients,

and it can be solved by determining the complementary function and the

particular integral. The solution method is straight forward. One point to note is

that this particular solution uses the initial conditions: x = 0, dx/dt = 0.

  

Eqn

14.13

 

 

Eqn

14.14

  Assign values to the friction, magnetic induction, and armature

mass. Determining a value for the induction, that will produce a similar

accelerating force in the model as it does in the test coils for a given current

density, requires that we look at the radial component of the flux density

distribution coming out of a magnetized coil gun projectile. This is integrated

over the volume occupied by the coil and a force expression is generated by

multiplying this by the current density, J, and filling factor, F. The expression is

then equated to BIL for our model and Bmodel is obtained by solving equation

14.15, where is the wire diameter.

 

 

 

 

 

Eqn

14.15

We can look at the magnetic flux from a magnetized projectile (without a coil

current) as shown in fig 14.7.

Determining volume integral of radial flux density using FEMM

 

The projectile is magnetized by giving it a B-H curve and an Hc

value in the FEMM material properties dialog. Values were chosen to resemble

strongly magnetized iron. FEMM gives a value of 6.74x10-7 Tm3 for the flux

density volume integral Bcoil, so using F = /4 we arrive at Bmodel = 3.0x10-2 T.

This flux density value may seem very small considering the flux density inside

the projectile is around 1.2T, however, realize that the flux expands into the

much greater volume around the projectile with only a fraction of the flux

resolves to a radial component. The essence of the system is the paired co-linear

forces acting on the stator and armature, so we can fix the copper part and allow

the stator field generator to move instead. Since the stator field generator is

acting as our projectile it will have a mass of 12 g assigned to it.

We can now plot the displacement and velocity as functions of time as shown in

fig 14.8 

Dynamic behavior of the linear motor

We can also combine the velocity and displacement equations to give a velocity

vs. displacement function as shown in fig 14.9.

Velocity vs. displacement characteristics

  A relatively long accelerator is needed before the armature begins

to reach its maximum speed. This has implications for the maximum efficiency

of a practical accelerator.

Close up of velocity vs. displacement curves

  There are several significant differences between this model and an

actual coil gun - e.g. in the coil gun the force is a function of the velocity and

displacement coordinates whereas, in the present model, the force is only a

function of the velocity coordinate.

Fig 14.11 is a plot of the cumulative efficiency of the motor as the projectile

accelerates.

Cumulative efficiency as a function of displacement with no friction losses

Cumulative efficiency as a function of displacement with constant friction

losses

The cumulative efficiency illustrates a fundamental property of this

type of electrical machine model. The maximum possible efficiency of an ideal

accelerator fired by a step voltage is 50%. If friction is present then the

cumulative efficiency exhibits a maxima turning point caused by the machine

doing work against the friction.

Effect of B on velocity-displacement gradient

Small displacement region where increasing induction yields a greater velocity

 

These curve sets show an interesting property of this model in

which a larger field induction initially yields a higher velocity over a given

displacement, but as the velocity increases the lower induction curves overtake

those representing stronger induction. This makes sense when you consider that

the stronger induction will yield a greater initial acceleration, however, the

correspondingly larger induced voltage causes the acceleration to decrease more

rapidly allowing the lower induction curves to catch up.

  The instantaneous efficiency increases as the projectile gains speed

due to the induced voltage reducing the current. This increases the efficiency

because the resistive power loss is dropping while the mechanical power is

increasing; however, since the acceleration is also dropping it takes

progressively more displacement to make use of the improving efficiency. In

short, a linear motor subjected to a step voltage forcing function is going to be

quite an inefficient machine unless it is very long.

This model of a primitive motor is instructive in that it points to the cause of the

typically poor efficiency of coil guns namely a low motion-induced voltage.

The model is oversimplified as it takes no account of the nonlinearities and

inductance elements of a practical system, so to refine the model we need to

incorporate these elements into our electrical circuit model. The next section

will develop a generalized differential equation for a single stage coil gun.

Coil gun Fundamentals

The Coil gun

A coil gun consists of two interacting parts, the coil and the

projectile. Attraction occurs because the coil magnetizes the rod, effectively

creating two separate magnets. The rod is magnetized in the same sense as the

coil so the end of the rod which faces the coil sees an opposing pole. Regardless

of which end of the coil the rod is placed, it will experience an attraction since

the coil will always magnetize the rod in the same sense as its own magnetic

field. It would be a different story if the rod was an independent magnet. If this

were the case the direction of the current and the orientation of the rod could

result in either an attraction or repulsion. Little more detail can be added by

considering the interaction of the flux from the rod and the current in the coil.

The diagram below shows a coil and rod in close proximity. The rod is

magnetized such that it sees the opposite pole when it faces the coil.

 

It is almost impossible to calculate a value for the attractive force

by applying the force equation from the previous page, the complexities

involved would likely result in ball park values at best. There would be far too

many simplifications required to get an accurate value. Integrate the force value

obtained from each elemental part of the coil. This would require some

estimation of the flux distribution which is not possible using analytical math.

This can be done using numerical field solution programs such as Quick field or

FEMM. These help to determine the flux distribution and forces in a static

magnetic system.

Solving for a static situation ignores a very important mechanism

of electromagnetic, namely induced voltage and current. As the projectile

accelerates into the coil, the flux linkage increases, generating an induced

voltage in the coil which opposes the supply voltage. This tries to reduce the

coil current and the magnetic field which, in turn, induces a voltage that tries to

maintain the coil current. In many instances, the magneto static solution may

not be a good indicator of the dynamic performance. An exception to this is the

situation in which the induced voltage is small compared to the supply voltage,

such as a slowly moving projectile. In this instance the current will only be

affected slightly. This means that a series of magneto static simulations could be

used to produce a rough estimation of the muzzle velocity. An example of this

comparison can be found in the results section.

 2. Finite Element Force Simulation -

In order to get the best out of a coil gun understanding how the

force varies with the position of the projectile in the coil is necessary. The graph

below illustrates the typical force variation on a rounded nose projectile

measuring 20mm long x 10mm diameter. The force curve is almost symmetrical

but the asymmetry of the projectile means that the force drops to zero just

beyond the midpoint. 

The force curve is plotted from a series of 21 simulations with the

projectile incremented at evenly spaced intervals, with the coil current density

held constant at a conservative 50Amm-2. This is the general form of the force-

displacement curve, although there will be variations of the exact shape due to

differences between coil and projectile geometries. The force curve generated

from a series of magneto static simulations is of course a simplification since

the induced voltage in the coil will affect the current. The important thing is that

it shows quite clearly that the maximum force occurs approximately halfway

into or out of the coil. It should also be noted that the force gradient is quite

steep as the projectile crosses the midpoint. This suggests that the current pulse

should be extinguished promptly; otherwise the projectile will start to be

decelerated at a rapidly increasing rate - something which we want to avoid.

This type of simulation can be used to give a reasonable estimation of the force

curve if the induced voltage is small compared to the supply voltage. 

3. Estimating Muzzle Energy and Velocity from a Force Curve -

There a coil gun system which satisfies the small induced voltage assumption,

to determine the approximate muzzle energy from the force curve is actually

quite straight forward; all that is needed is a simple integration. 

A 3rd order polynomial fits this part of the curve very well so

integrating this equation gives us

The energy units are in mJ because the distances are expressed in

mm. With the mass of the projectile known, the below formula can be used to

determine its velocity,

In this example the current was set to 50Amm-2, this is small

compared to what is needed for a really fast projectile. Current densities of

around 1000Amm-2 will produce respectable velocities.

4. Simple Force Formula -

There is a very neat little formula which describes the force on the projectile of

a solenoid, it goes like this:

  Eqn

4.1

Where N is the number of turns, I is the current, and df/dx is the rate of change of

flux linkage with plunger displacement. N and I are straightforward, but the flux

linkage is a quite difficult to determine since it is dependent on the geometry of

the coil and the plunger material. Perhaps the best thing to take away from

examining this formula is that the force can be increased by increasing the

number of turns, increasing the current, or increasing the change in flux linkage.

5. Enhanced Flux Linkage -

In the force equation above, the flux linkage is one of the

parameters which affect the force on the projectile. Increasing the flux linkage,

for a given coil current, will increase the force on the projectile. The flux

linkage can be enhanced by two means; either by using a projectile with a

higher saturation flux density (such as iron-cobalt) or by adding external iron to

the flux path around the coil.

 

The increased flux linkage produces an increased inductance which increases

the time constant of the circuit. The cure for this is to run the system at a higher

voltage with an external resistor to achieve similar dynamic and peak current.

This is a technique which can be used to improve the performance of stepper

motors.

6. Projectile Saturation

Saturation is a state in which the specimen has reached maximum

magnetization. Now magnetization can be thought of as the amount of

microscopic atomic dipoles which are aligned with one another. Application of

an external field to dipoles will tend to align themselves with this field. The

stronger the external field becomes, the more dipoles become aligned. When all

the dipoles are aligned the specimen is said to be saturated. This is a simplified

description but it conveys the general idea. Ultimately the forces are due to

charge motion; there is motion of charge in the coil - the coil current, and have

orbital electron motion in the specimen - the dipoles. The force between these

wires will be attractive if the currents are parallel and repulsive if the currents

are anti-parallel. Now imagine two loops of wire sitting side by side as shown in

figure below.

Current carrying loops are mutually attracted

 

If the wire currents are in the same direction the force is attractive.

Now the force between these loops will depend on their diameter, their

respective current, and their separation, as well as the medium in which they are

placed (e.g. free space). Increasing the current in either loop will increase the

force. The coil is basically a large number of wire loops, each carrying the same

current. The projectile is composed of many tiny current loops - the orbital

electrons. In an unmagnified ferromagnetic material the current loops are

organized into small groups called domains. These domains are orientated in

random directions so that macroscopically, the material exhibits no

magnetization. Fig illustrates the randomly aligned loops. When an external

field is applied, the loops within the domains experience a torque force which

tries to align them with the field. This means that domains which are originally

more aligned with the field tend to grow at the expense of the less well aligned

domains. The stronger the external field becomes, the loops become more fully

aligned.

An unmagnified material contains randomly ordered magnetic dipoles

 

Fig shows some alignment of the loops as the coil field begins to influence

them. As the coil field is increased, more and more of the loops align until we

reach a point where, for our purposes, all the loops become aligned i.e. the

material is saturated. Saturation of the projectile is by the right hand loop

reaching a maximum current, but the force depends on the current in both loops.

Since the current can be increased in the left hand loop coil the attractive force

will increase even though the projectile is saturated. Right hand loop current

fixed.

An external field causes alignment of the dipoles and an attractive force

develops

 

IMHO saturation of the projectile is of limited consequence, in so far as it does

not place a limit on the attractive force. There is another way to think about this

using the pole concept of magnetics. Basically a projectile can be thought of as

one big dipole which has maximum pole strengths determined by its saturation

magnetization. A magnetic pole experiences a force when placed in a magnetic

field. Now since the projectile is a dipole it has two poles of opposite

magnitude, so if it is placed in a magnetic field each pole will experience a force

depending on the field strength around it and the sign of the pole (+north / -

south). Since the sign of the pole determines the direction of the force we find

that the front of the projectile is attracted towards the centre of the coil and the

end of the projectile is repelled from the coil. The field is stronger towards the

centre of the coil so the front pole experiences an attractive force which is

stronger than the repulsive force generated by the rear pole, resulting in a net

attraction. However, since the coil field can still be strengthened by increasing

its current the attractive force can still be increased. The strengthening of the

coil field increases the difference in the force on the projectile poles and so

increases the net attractive force. As fig 4 shows, any dipole which is placed in

a field gradient will experience a net force, the larger the gradient becomes the

stronger the resulting force. The forces on the poles can be expressed

mathematically as:

  Eqn 6.1

Where, P is the pole strength. Since our projectile consists of two opposing

poles in different field strengths, we can write

 Eqn

6.2

Now if we assume that the poles are of equal magnitude this reduces to

  Eqn 6.3

A dipole experiences a net force in a field gradient.

 

If the field is increased around a saturated material the flux density

will continue to increase with a dB/dH equal to vacuum. The reason for this is

that the material is full of space. The space still contributes to the flux with a

relative permeability of 1. This is important when using B-H curves for finite

element analyses since the correct dB/dH must be used for large values of field

beyond the materials saturation point.

Optimized Coil Geometry

Thickness of the coil affects the field strength at its centre. The basic coil

parameters are illustrated below.

Firstly let's define some basic geometrical relationships

It can be shown that the field, Ho, at the centre of the solenoid is

Where N is the number of turns, I is the coil current.

F (a,b) is the field factor defined as

We can use these equations in conjunction with the coil resistance equation to

plot Ho as a function of Ro for any combination of L, R i and wire diameter. Fig

2 shows the field strength for three different wire diameters using a coil with

L=26mm and Ri=7mm.

Field strength vs. outer radius for various wire diameters

  The field strength plots are based on a source of 1V. The particular

voltage level is not important; it varies in the relative effect of wire diameter

and outer radius. It is obvious that for any given wire diameter there is an

optimal outer radius which maximizes the field strength. The central field

strength is probably the most important factor governing the muzzle energy of

the projectile.

Fig 3 shows the results of a simple experiment which involved measuring the

muzzle speed from a coil with varying values of outer radius. Coil used was

1mm wire. As each speed measurement was completed a layer of was removed

until the coil was 2 layers thick. Suck back is reduced by using the multi-diode

commutation arrangement. The projectile energy is plotted along with the

calculated central field strength.

Correlation between central field and projectile energy

  There is a good degree of correlation between the energy and the

central field. The most notable aspect of these curves is that the peak energy and

peak field occur at the same coil outer radius. The correlation between these

peaks is very striking and reflects the likelihood that central field strength will

play a key role in determining the best coil geometry.

The first thing that comes to mind is the source resistance value.

This is based on the battery source internal resistance, the MOSFET module

resistance, the current sensor resistance and the wiring resistance. Apart from

the current sensor none of these resistances are known to a high degree of

accuracy so the absolute magnitudes of Ho could be off by some amount.

Another reason could simply be that the field strengths involved in this

experiment is insufficient to induce saturation magnetization in the projectile so

it is still subject to the nonlinear portion of its B-H curve. There's also suck back

acting as an uncontrolled factor which will no doubt vary with the changes in

coil radius. I suspect that running a similar experiment with higher currents, and

hence field strengths, will produce a tighter correlation.

This experiment is by no means definitive, additional experiments need to be

run with a selection of base geometries - length and inner radius combinations -

in order to determine if the outer radius can be confidently specified using the

peak Ho. The Ho analysis is based on the pulse width being at least 6 time

constants - steady state approximation. The Coil B configurations satisfied this

limitation but obviously not all coils will. Some type of dynamic factor is

required to refine the Ho analysis.

AWG Dia nom Diaturns /

inturns / sq in

Ohms / 1000'

Operating

Current

Fusing Current

lb/1000'breaking force,

lb

170.04526

00.0468 19.23 369.8 5.063 2.076 98.6 6.200 62.1

180.04030

00.0418 21.53 463.6 6.384 1.646 82.4 4.917 49.1

190.03589

00.0373 24.13 582.2 8.051 1.306 69.7 3.899 39.0

200.03196

00.0334 26.95 726.1 10.15 1.035 58.6 3.092 31.0

210.02846

00.0298 30.20 912.1 12.80 0.821 49.3 2.452 24.6

220.02535

00.0266 33.83 1145 16.14 0.6511 41.2 1.945 19.4

230.02257

00.0238 37.82 1430 20.36 0.5164 34.8 1.542 15.4

240.02010

00.0213 42.25 1785 25.67 0.4095 29.2 1.223 12.7

250.01790

00.0190 47.37 2244 32.36 0.3247 24.5 0.9699 10.1