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The main difficulty in constructing high-efficiency multistage accelerators is a huge laboriousness of this process. One of solutions could be to make species are examples of such approach.thisBut a probem arises here. In conventional coilgun constructing, a tricky process of adjustment of geometrical parameters of each coil is conducted (with a help of situation. The latter coilgun shows that we can get approximately the same peak current by tunning the caps for each stage - this allows the same power switches to be utilized, but the geometry variation is unavoidable.EM-3 ("Electric bow")Trying to save a time and using identical coils, we get substantially lower efficiency. If a common capacitor is used, the situation is even worse because the cap would try to keep the same forms of current in all stages which is far from optimal conditions for acceleration. However, the common-capacitor sytem with identical coils looks so attractive that I tried to assess the limits of its application without any complicated calculations. The resuls are below.
1. General suggestions. At first, let us remind the waveforms for current in RLC-circuit:
As we see, the current rises and reaches its maximum at the moment t), we have to switch the current off to avoid suck-back effect. Then, it decreases according to different laws in dependence on a damping scheme applied in the circuit._{0}
Suggesting the length and inside diameter (caliber) of the coil to be fixed, and the moments of the current inset ( outside diameter of the coil and wire gauge. Like shown , these geometrical parameters fully determines the waveforms (provided the capacitance and its initial voltage, i.e.total energy, is fixed). hereSecondly, an accelerating froce acting on the projectile is proportional to the magnetic field i.e. ampere-turns: Thus, the optimization of the system can be described as
2. Problem formulation. The discourse articulated above is, in general, obvious and concerns a common case of multistage accelerator where it is possible to vary geometry of the accelerating coil and wire gauge (like gauss-builders do in many situations). Getting down to our "unified" system, we can conclude that its optimization reduces to the choice of geometry of (unitary) coil which provides the best efficiency throughout all the distance of acceleration. How can we choose it? Let's look at fig.1 again. If all the parameters of RLC-circuit are set, then pulse shape (including t changes only, which is connected with continuous acceleration of a projectile. This situation is illustrated on fig. 2, where oscillograms of current are shown for all stages basing on assumption of a very large common capacitor (which is close to our case). It is described _{0} that current decays monotonically in such systems (AKA "overdamped").here
The question is how long the pulse can be until a substantial efficiency of acceleration preserves? Two limiting situations may be distinguised (fig. 3): 1) The projectile moves so fastly that current hasn't time to reach its maximum ( U/L (U - initial cap voltage, L - inductance);2) The projectile moves so slowly that current has overcome its maximum long before RC constant when the moment of t happens._{0 }
It is obvious that the acceleration In the first situation it follows from the simple fact that further current increase could supply much higher velocity to the projectile, but we just "don't give it a chance" to do it. Thus, the coil is used ineffectively. In the second situation we can imagine a thicker coil containing more turns of wire of higher caliber, in such a way that the moment of current maximum is closer to switch-off (this hypothetical case is shown with a dashed line on the right of fig. 3). Charge consumed from a cap is nearly the same for both coils (it is clear because a square under the curves is close to identical), but the force would be stronger for the hicker one (because it contains more ampere-turns at the same current). Thus, the thicker coil would be more effective. So, we can expect that the most optimal coil must allow the current to achieve its peak but not to decrease substantially until the moment when a switch is closed. These speculations are validated by FEMM modelling, which demonstrates that the optimal range for
So, out task is to choose a form of a coil ensuring relation (1) in a range of velocities Provided coil form is fixed, an inverse problem may be stated: we want to assess maximum projectile speed which keeps a satisfactory efficiency of acceleration (as (1) says it makes l is length of the coil). That's what we are most interested in.
3. Solution. For the beginning, we can express , in a strongly overdamped circuithere(2) where capacitive constant inductive constant damping factor
The inductive constant (as shown (3) where To express This can be done as following. The capacitance through the energy conserved in a capacitor and initial voltage is U3/4_{f} ≈ U - about half of the energy is spent in this case. So, the capacitance would be _{0}C≈4E (where _{w}/U_{0}^{2}E is the energy spent per one shot), and the capacitive constant is _{w}RC/2 ≈ 2E_{w}(R/U_{0}^{2}). It's easy to note that the relation in the right part of this formula is an inverse ohmic power. In our situation current in gauss-gun during nearly all process of acceleration is limited by ohmic resistance, and acceleration efficiency is rather small, so we can say with good precision that ohmic power equals to total power (4) where Let us note this interesting fact: Expression (4) can be further modified if we remember that for a unformly accelerated motion (which is close to truth in our constant current-mode coilgun) mean velocity is half of the output one, so (5) where At last, differ in no more than 4 times. Hence, assuming unformly accelerated motion again, we get that maximum number of stages is 4. If more, either the first of them would have a weak influence on projectile (especially in comparison to hypothetical more optimal coils, see a right part of fig. 3), or the last ones would accelerate the shell only slightly (because the current couldn't reach a considerable value in them). So, ^{2}=16l≈_{s}16·l, and we can write finally:(6) Thus, we expressed the capacitive constant through the individual coil's length and maximum speed.
Combining (1), (2), (3) and (6), we get for (7) Having this, we can assess maximum speed on a basis of geometry of a coil. As we see, the value subject to assessment stands both in the left and right and part of the equation under the logarythm. Such relations in mathematics are called "transcendent" - they don't have exact analytical solution and are usually resolved graphically. Fig. 4 shows an example of such a solution for a system with following parameters:
Tables of solution of (7) for _{ }for different calibers d are resolved in fig. 5. The range of lengths of a coil is in the limits from l = d to l = 4·d (longer coils are ineffective according to FEMM modelling and lot of experimental data), and outer diameters are from D = 1,1·d to D = 2·d.
Coloration of some cells will be explained further. We can see that theoretical values of A critical clause must be made here: the values above are assessed theoretically basing only on geometrical parameters of a system. If an accelerator has a geometry shown in fig. 5 - speed can be close to those theoretical limits - more realistic estimations may be obtained by modelling of the specific system in special programs like FEMM.
4. Limitations. Viewing the results in fig. 5, one can naturally ask: why such high speeds are not obtained in practice? Indeed, v more than 130 m/s till this moment (while majority of them utilize coils attenuated for different stages). This question can be answered taking into account natural limitations layed by physical laws and modern level of development of electronic components. _{max}Let us consider two of them only. 1) Coil heating. As an ohmic energy deposition occurs in a very short time duration, we can assume thermoexchange to be negligible (such processes are called "adiabatic" in physics). Thus, a final temperature is proportional to mass of a coil, and the shortest and thinnest coils of the smallest calibers are expected to be mostly heated. A calculation of the peak temperature was conducted for the coils with parameters depicted in fig. 5, accounting for the heat capacity of copper 0,4 J/(g·°C) and acceleration efficiency to be ≈ 8% for 8 mm dia and proportional to the caliber (which coincides to results of FEMM modelling). Then, a limit was suggested to be exceeded if the temperature rose higher than 300 °C - this is a point when heat-resistant electrical varnish loses its isolating abilities. As a strong tension occurs in a winding during a shot, the short circuit is expected in such situation. Moreover, in worst cases (according to right higher corners of the tables), melting point of copper is surpassed. As result, a range of geometrical characteristics of the coils for each caliber shrinks to band colored by green in fig. 5.
2) Weight of the power capacitor. State-of-art electrolitic capacitors have some boundary energy accumulated per unit of mass, which varies depending on a specific type of a cap, and makes about 3 grams per Joule. Thus the maximum energy spent per shot is about 1000 J (this corresponds to total accumulated energy 2 kJ and capacitor mass of 6 kg - clearly close to a limiting value for a portable system). Comparing this number to the projectile muzzle energy corresponding to These "exorbitant" values are marked red in fig. 5. They correspond to situations when Generally, we can see that both limits under concern work in one direction, constraining
Other limiting factors like wight of the coils can also be taken into account by a similar way.
5. Conclusions and discussion. Following conclusions can be made from the results of fig. 5.
It is clear that small calibers are the most attractive for breakthrough to high velocities. They are even more tempting if we note that mass is not the only reason which limits us in a portable system - the length acts, too. 16 stages of 12 mm gaussgun with The first table of fig. 5 says that supersonic velocities can be obtained theoretically. Сooling of windings with liquid nitrogen or Peltier modules can be used prior to shot to prevent overheat (such a method was proposed by many authors for induction launchers). In this case, the efficiency would increase not only because of active resistance reduction, but also because of the speed growth (see As for large caliber coilguns (more 10 mm) with identical windings, the results above make us claim that velocities of projectile of more than 100 m/s cannot be obtained with state-of-art electronics (unless some tricky technologies are utilized like
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