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Lately, an accumulator-driven construction of the gauss gun has been widely discussed. A main feature which differs this design from the “traditional” constructions is no need of charging of a power capacitor before each shot. Consequently, at first, building of voltage converter can be avoided. Secondly, the acceleration of the next bullet can be stated at once after the previous projectile leaves the barrel, and a theoretical fire rate is limited by a feeding system only. It’s obvious that the main parameter of the accelerator – its output velocity – depends on pulse power output of the accumulators. So, the creation of this sort of coilgun became possible only after compact high-current Lithium-Polymer (LiPo) accumulators appeared. In this article I will try to assess a maximum velocity reached in accumulator-driven gauss-gun (ADG).
For the beginning, let us imagine the electric circuit of a single-stage ADG (fig. 1). One can see that a typical component of any coilgun – the capacitor – is absent, but an internal resistance of the accu (R
The accumulator is illustrated as a series connection of DC voltage source and its internal resistance, which is a simplified but close to true performance for LiPo accus in a pulse mode. Thus, there is a simpler
«Time constant» of the current rise is usually included which is
Current builds up to approximately 63% of its maximum value R parameters (moreover, that is true for J.Murray’s construction). Besides, they are usually switched in sequence, and each next coil is activated approximately on the moment when the previous one is deactivated. Thus, the current can be suggested something about DC with the value of approximately _{L}Imax (fig. 3).
The role of the accumulator is to ensure that current.
Let us remind that the question we try to answer sounds like follows: what is the maximal output velocity of a projectile can be obtained in ADG using actual for today LiPo accumulators? m to speed v with efficiency of η along a path l is determined by the next equation:
Note that we are talking about the power which dissipates
Substituting here the equation (1) for current, we have finally for the power:
Dependence of P (i.e. maximum current at fixed supply voltage) is obtained when coil resistance equals the sum of all other resistances of the chain. Consequently, the projectile will be accelerated the best when a gauge and length of the wire are chosen in such a way that its resistance R _{L} = R_{int}+R_{par}.
At this point
v.The parasitic chain resistance is treated rather simply – it is determined by on –resistance of the switch, and interconnections. On-resistance of the modern low-voltage MOSFETs is limited not by their chips (which are very low-ohmic), but their leads and makes no more than 5 milliohms (mOhms) as a rule. The same can be suggested for the resistance of the connections (provided they are made of copper buses of enough cross-sections and well-soldered everywhere). In total we get about 10 mOhms ( i. e. one-hundred part of Ohm) for R. Perhaps it is too “overoptimistic” assessment and _{par}R in a real construction (especially with long bore demanding prolonged buses) may be higher, but it is OK for our rough calculation._{par}The situation is much more interesting for the internal resistance of the accu. It seems impossible to fix some unified value of R for a huge variety of all types of LiPos. Indeed, their capacity may be in a range from hundreds to thousands mA·h, and voltage ranges from 3,7 V (for 1S-type, i.e. single bank) up to 22,2 V for 6S which means six sequentially connected accus. It is strightforward to suggest that _{int}R will significantly differ in all that cases._{int}The way to overcome this difficulty is to introduce some coefficient r which can be called “relative internal resistance” of the accu. Its dimension is Ohm·(mA·h)/V, and a resistance of an arbitrary accumulator will be deduced from r by the following equation:
where U,C – battery voltage and capacity, accordingly.This relation demonstrates that the internal resistance of the battery increases with voltage builds-up and capacity decreases. C = 4000 mA·h has the resistance of about 2,5 mOhm for the individual 3,7 V bank, but, including the interconnections between the elements, it totally gives R = 17,9 mOhm for 4-bank battery (4S), so _{int}r = 4·10^{-3} · 4000 / 3,7 ≈ 4 Ohm·(mA·h)/V. That is what we will use for the following calculations.Considering the results obtained, the eq. (5) may be simplified if we suggest the internal accu resistance to dominate (i.e. R). Indeed, in the example above the parasitic resistance (which is fixed to 10 mOhms as we said) will be nearly twice lower than _{int}>>R_{par}Rwhich is for 14,8 Volts-battery, while much more voltage (and consequently higher _{int }R) is used in real constructions. So, _{int}R can be suggested negligible for the simplification, and then (5) (accounting for (6)) will transform to_{par}
Fig.5 shows the results obtained from (7) for
It is interesting to compare the results to the parameters of the really built ADGs. Fig. 6 gives such comparison for two models; form ArcFlashLabs (in fact, that is the same coilgun with the capacitor added between the accu and the coils for the second model). The parameters of the accelerators taken for the calculation (according to the author’s information) are in table 1.EMG-01
What can we say looking at the results? without any extraordinary engineering tricks). overcome 100 m\s speed limitThirdly, the parasitic resistance has a substantial influence on the parameters of ADG at voltages low than 20 V. So, for the analyzed-above case of 3200 mA·h capacity the speed lowers doubly for the single-bank accu, and to about 10 % for 6S – type (which corresponds to η ≈ 20 % reducing efficiency) when considering R. This says that the lowest voltage limit of the accumulators for ADG lays somewhere about 20 V and corresponds to 6S-accus._{par}Finally, the parameters of EMG-01 coilgun show that a capacitor added between power supply and accelerating coils increases the efficiency dramatically by deleting such factor as internal resistance of the accumulator. RL-circuit of ADG turns here to a “traditional” RLC-type with only difference in cap charging directly from the accu (not from a voltage booster, I examined this type of the coilguns ). This fact reveals that conventional capacitor-driven coilguns are still more effective due to lower internal resistance of the capacitors (also called ESR) in comparison to hereR of the LiPo accumulators._{int }
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