Greater voltage gain for TeslaMtransformer accelerators Jay L.
Reed Science Applications/nternational Corporation, 1247-B North
Eglin Parkway, Shalimar, F'lorida 32579 (Received 8 February 1988;
accepted for publication 16 June 1988)
The governing equation of secondary potential is reformatted and
analytically optimized. A circuit adjustment is predicted that
yields an 18 % increase in voltage gain.
INTRODUCTION The Tesla transformer finds convenient application
as a high-voltage pulsed power supply for intense relativistic
electron beam machinery. 1-4 Figure 1 contains a circuit dia-gram
of the transformer. Drude5 and subsequent investiga-tors6 .7
indicate a circuit adjustment to obtain maximum vol-tage gain V1/Vj
with the device. The previous analyses demonstrate that ifLIC1 = L
2C2 and the mutual inductance Mis 3/5 of JI~~, the voltage maxima
of the two coexisting oscillations will occur concurrently, at a
particular instant in time, and cooperate to produce maximum
voltage across the secondary circuit. The voltage gain of the
transformer under this adjustment is
( 1 ) The present analysis is an extension of the work
ofOber-
beck,8 with an added degree of freedom obtained by allowing LICI
L 2Cz, and predicts an adjustment that yields an 18% increase in
voltage gain.
I. THEORY Upon commutation at the spark-gap switch, the
voltage
across C2 , at any insiant in time t, for high-Q circuits, can
be written9 as
(2) where WI and (ilz are the radian frequencies of the two
coex-isting oscillations within the coupled system. In writing Eq.
(2), W Z >f1J) is assumed.
Define the coupling coefficient k as
k = M I{Lll; (3) and the tuning ratio T as
(4) In this discussion the inductance of each winding is fixed
and k is varied by altering the proximity of the circuits. The
dis-tributed capacity of the secondary circuit C2 is also
consid-ered to be constant.
Equation (2) is now rewritten in terms of the coefficient of
coupling and the tuning ratio as
Vz(t)/V[ = [k /lO-=--:fF+4kT:tlJL;:I1.~ (cos WIt -- cos (U2t).
(5)
The cooperative interference in time of the two cosine terms has
been discussed and it has been shown that align-ment of the voltage
maxima occurs in the least time if the individual frequencies of
oscillation differ by a factor of 2. 10
"Drude's adjustment" produces this desirable frequen-cy ratio.
The old and well-known result shown in Eq. (1) is found upon
substituting T = I, k = 3/5, UJ 2/ OJ i = 2, and t = rr/{t)[ into
Eq. (5).
A circuit adjustment that yields a higher voltage gain is
predicted by optimizing Eq. (5) in the following manner. The ratio
of W2 to (I}I can be written in terms of the tuning ratio and the
coefficient of coupling by slight rearrangement of Oberbeck's
equations for magnetically coupled circuits [l as
By setting (U2/WI = 2 in Eq. (6) and solving for k, one
obtains
(7) Utilizing Eq. (7), one can rewrite the first factor
orEq.
(5) as a function of the tuning ratio alone, giving k / ~Tf =
-T)T +" 4P1' = r~4-T'-+-f7Y' =4T j[3T( T + I)}. (8)
M
FIG.!' Tesia-transformer circuit consisting of two high-Q LC
circuits cou-pled through the mutual inductance J{. The primary
capacitor C, is charged to the voltage V, by a source of emflabeled
S. Oscillations are initi-ated by commutation at the spark-gap
switch SG. The distributed capacity oflhe secondary circuit is C2 .
The largest voltage occurring across C2 during the oscillation is
V2
2300 Rev. Sci. Instrum. 59 (10), October 1988
0034-6748/88/102300-02$01.30 (c) 1988 American Institute of Physics
2300
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Let expression (8) be equal to z. Then z assumes an extreme
value when
dz/dT=O. Equation (9) is satisfied by
21'3 - 191'2 + 6T + 2 = 0
(9)
(10) The three roots ofEq. (10) are 9.160594, - 0.201730,
and 0.541136. Only the last of these has physical signifi-cance,
namely,
T=O.54!I36. (II) With the tuning ratio now known, the
corresponding
coefficient of coupling can be calculated from Eq. (7). We
find
k = 0.545659. (12) The increase in voltage gain is seen by
substituting the
new circuit adjustment with T= 0.541136, k = 0.545659, w2/Wj =
2, and t = 1T/(v, into Eg. (5). We find
Vz/Vj = - 1.1802jt-;n:,-. (13)
II. IMPLEMENTATION OF THE ADJUSTMENT All interconnection of the
transformer secondary cir-
cuit to capacitive structures and gun assemblies must be
complete before the coefficient of coupling can be set, since it is
sensitive to the distribution of charge within the windings. It is
suggested that the coefficient of coupling be determined through
the change in the resonant frequency of the second-ary circuit when
the primary winding leads are short-circuit-ed, Wsc ' and when
open-circuited, (Voc' The coefficient of coupling in terms of these
two frequencies '2 is
..---'---'-2' k='./l- (wojwsc ) . (14) The proximity of the
windings is adjusted until a coefficient of coupling near 0.545659
is obtained.
The primary winding leads are then connected to a suit-able
capacitance decade box and the oscillation frequencies may be
determined by simple resonance or grid-dip tech-niques. The proper
capacity wiIl produce two distinct reso-nant frequencies differing
by a factor of 2.
llio PRACTICAL CONSIDERATIONS The circuit adjustment defined by
Eqs. (11) and (12)
increases secondary potential at the expense of energy-trans-fer
efficiency. Energy considerations show that, unlike Drude's
adjustment, the larger primary capacitor is not
2301 Rev. SCi.lnstrum., Vol. 59, No. 10, October 1988
completely discharged at the instant of maximum secondary
potential. The capacitor will still contain 24.6% of its charge.
The designer may wish to consider the impact of this imperfect
energy transfer on the supply power requirements in the
construction of a high-repetitive firing rate system. The
inefficiency can be destroyed by preserving the untrans-ferred
energy from "pulse to pulse" by employing special-ized commutation
schemes. 13.14
A pulse repetition rate of 120/s can be obtained by uti-lizing
the line frequency (j)L of the commercial mains. In this
configuration, a high-voltage transformer is chosen as the source
of emf and must be impedance matched into the pri-mary capacitor at
60 Hz, that is, the rms rating of the trans-former voltage v and
current i must substantially satisfy
v/i= l/({{)LCt ), (15) This matching for maximum power transfer
will generally be difficult unless the high-voltage transformer is
custom wound to obtain the desired impedance. A technique to avoid
this complication is to construct a parallel-wired array of n
transformers, with identical ratio and impedance char-acteristics,
and match into C] through the relation
V/(il-t-iZ-+-'" +inhd/(MLC,). (16)
ACKNOWLEDGMENT The author wishes to acknowledge discussions
with
Fred B. Hagedorn concerning this work.
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Tesla transformer 2301
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