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Heat Generation during the Firing of a Capacitor Based Railgun System Heat Generation during the Firing of a
Capacitor Based Railgun System

Andrew N. Smith
Mechanical Engineering Department
U.S. Naval Academy

Benjamin T. McGlasson and Jack S. Bernardes
Naval Surface Warfare Center
Dahlgren Division

Abstract Railguns use a high-current, high-energy electrical
pulse to accelerate projectiles to hypersonic velocities. Pulse
forming networks that employ capacitors as the energy store are
typically used to shape the required electrical pulse. A
significant fraction of the stored energy (25 40% in large
caliber railguns) is converted to projectile kinetic energy during
launch. After the projectile exits the launcher, the balance of the
energy has either been dissipated as heat in the circuit
components or is stored in system inductance. If an energy
recovery scheme is not employed, the inductor energy will also
be dissipated in the resistance of the active circuit components.
A circuit analysis has been performed in order to calculate the
current profile from the PFN. A higher fidelity solution was
achieved by accounting for the temperature dependent resistance
of the rails. This information along with individual component
resistance and inductance was used to calculate the distribution
of energy subsequent to a single pulse. Detailed component
heating information is important when considering the overall
thermal management of the system. Once this information has
been obtained, the components that require external cooling can
be identified, and an appropriate thermal management system
can in turn be designed.
I. I
NTRODUCTION

The electromagnetic railgun has been proposed as a weapon that could enable the
US Navy to conduct long-range surface fire support missions [1]. Several investigations have been published to date that examine the capacitor based Pulse
Forming Network (PFN) that would be required [2]. These investigations have
typically focused on determining the total energy required in order to achieve the
desired launch velocity. In many cases the individual resistance of several components
have been combined into a total effective resistance. Bernardes et al. examined a
capacitor based system that would be used to launch a 20 kg launch mass at 2500 m/s
velocity [2]. Pitman et al. [3] used equations developed by Parker et al. [4] to simulate
a capacitor based system by solving the governing equations using a Runge Kutta
method. This investigation will use a similar approach however the system of equations
has been solved using a second order finite difference method.
The most complicated component in the system is the launcher itself. The
resistance of the launcher changes with time as the projectile physically moves down
the barrel and the magnetic field diffuses into the rail. Several investigations have been
performed using an analytical approach [5], segmented rail with an effective height [6]-
[7], segmented rails where the magnetic field diffuses into multiple surfaces [8], and
full three dimensional simulations of the launcher [9]. In this paper, an effective height
model is used to simulate the transient rail resistance based on the method presented by
Parker et al. [7]. These calculations were performed concurrently with the solution of
the PFN in order to account for mutual effects.
II. C
IRCUIT
M
ODEL

In order to determine the heat generation in each of the components of the PFN,
the entire circuit had to be modeled. The specific circuit that was modeled in this
investigation is shown in Figure 1 and is based on the requirements for the notional
naval railgun. Each module is triggered by an array of thyristors. An array of crowbar
diodes prevents the circuit current from ringing. The following rules were implemented
in the circuit code: (1) the diode is not included in the circuit until after the capacitor
has discharged; (2) once the capacitor has fully discharged the diode prevents that
capacitor from recharging and thereby removes the thyristor, bus bar and capacitor from
the circuit. As shown in Figure 1, the individual resistances of the following
components were included in the circuit: thyristors, diodes, bus bars, capacitors,
inductors, launcher and the shunt. At the completion of each shot, all of the energy not
transferred to the launch mass ultimately represents a resistive heat load in one of these
components. It was assumed that the railgun muzzle shunt is purely resistive, and that
no attempt was made at energy recovery.
The governing differential equation for the charge in the circuit is given by the
following equation [4]:

2
2
2
2
1
1
0
n
n
i
b
i
b
b
s
b
s
b
i
i
b
q
q
q
q
q
L
L
R
R
t
t
t
t
C
=
=
+
+
+
+
=
(1)
where q
b
is the charge of the capacitor for that bank and the summations are taken over
the banks that are currently active. The subscript b denotes a particular bank and is
used to identify the resistance, R; capacitance, C; and the inductance, L, of that bank.
The subscript s refers to the system components that experience the total current
summed over the individual banks. The resistance of the system, R
s
, includes the
resistance of the launcher, the resistance of the armature and the electromotive force on
the armature.
Equation (1) is then be used to express the time varying charge of each active
bank. Using a finite difference approach for the first and second order derivations, the
system of differential equations can be solved simultaneously. Equation (2) shows the
system of equations when there are three active banks, the size of the matrix grows as
more banks are triggered.
(
)
(
)
,1
,1
1,1
,2
,2
1,2
,3
,3
1,2
3
3
s
,
1,
b,1
,1
1,1
,
,2
,1
,1
,1
1
1
s
,
2
2

2
s
b
s
b
s
s
s
s
i
s
s
s
b
s
b
s
s
i
s
s
s
s
s
b
s
b
i
i m
i
m
i
i
s i m
b
i
b
i
m
m
i
q
q
q
q
q
q
q
q
q
q
q












+
+
+
=
= +
+ +
+
+ +
+
+ +
+ =
+
+
+
+ +
+ +
+ (
)
(
)
(
)
(
)
3
3
1,
b,2
,2
1,2
,
,2
,2
,2
,2
1
1
3
3
s
,
1,
b,3
,3
1,3
,
,3
,3
,3
,3
1
1
2
2
2
m
i
m
i
i
s i m
b
i
b
i
m
m
i m
i
m
i
i
s i m
b
i
b
i
m
m
q
q
q
q
q
q
q
q
q
q
q
q
q =
=
=
= + +
+
+ +
+ (2)
R
shunt
Launcher
R
inductor
Diode
Thyristor
Inductor
R
bus
R
capacitor
Capacitor
R
launcher
R
armature

Figure 1. Capacitor-based PFN circuit that was used in the simulation. Although this figure
only shows three banks, there are actually 35 individually triggered banks in the simulation. where,
,
,
2
s b
s b
L
t = (2a)

,
,
s b
s b
R
t = (2b)

1
b
b
C =
(2c)
The subscript i represents the time increment. The solution of this system of
equations yields the charge in each capacitor banks at the future timestep (i+1).

Figure 2 shows both the total current and the current of each individual bank
during a simulated shot of a notional naval railgun. This simulation is based on
launching a 20 kg mass to 2500 m/s using a 135mm square bore and a barrel length of
12 m. Table 1 provides the values of the key parameters used in this simulation. These
values were typically based on the components to be used in the pulsed power system
designed for installation at NSWC Dahlgren Division. It was assumed in this
simulation that only the launcher resistance varied with time.

0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0
1
2
3
4
5
6 x 10
6
Time, (s)
C
u
r
r
ent
,
(
A
)
Figure 2. Total system current shown along with the current of each individual bank.
The resistance of the launcher was determined using an effective height model
based on a 135mm square bore [6]-[8]. In order to calculate the effective height, the
non-uniform surface current profile around the perimeter of the rail cross-section must
be calculated. The effective height is then determined by equating the power dissipated
in a surface layer on the perimeter of the rail, with a uniform current distribution on the
front surface of a rail with an effective height [7]. Since the assumption was made that
the current is uniform on the inside surface of the rail, a one dimensional finite
difference approach was used to account for the diffusion of the magnetic field into the
conductor. The tem