Below I describe what I have obtained working on the ideas that I found in the papers [1] and [2]. A set of formulas that allow, at least ideally (losses ignored, lumped model assumed as valid), the design of a perfect Tesla magnifier circuit, or triple resonance transformer, are presented.

A Tesla magnifier is a variation of the Tesla transformer where the resonator is separated from the secondary coil. The main objective of the modification is to obtain fast energy transfer without excessive voltage stress in the transformer. The magnifier is here modeled by the lumped circuit shown below:

The capacitor C_{1} is initially charged to a
moderately high voltage, and when this voltage is high enough it
discharges through the spark gap to the primary of the
transformer L_{1}L_{2}, that has a coupling
coefficient k_{12}. The system enters then in an
oscillatory transient, that eventually produces a very high
voltage at the top terminal of the resonator coil, or third coil,
L_{3.}. C_{3} is usually a distributed
capacitance, that represents the sum of the "self-capacitance"
of L_{3} and the capacitance of a terminal placed above
it. C_{2} represents the output capacitance of the
transformer, added to other distributed and lumped capacitances
at that point.

When C_{2} is negligible, this system behaves exactly
as a normal Tesla coil [5], or double resonance transformer, with
two oscillatory modes, as discussed below.

If C_{2} is significant, however, a third oscillatory
mode appears, and the transient is more complex. Note that C_{2}
may be very significant if the driver transformer has its
windings at short distance. The ideal design would eventually put
all the energy that was stored in C_{1} in C_{3}
only, with zero voltages at C_{1} and C_{2}, and
zero currents at L_{1}, L_{2}, and L_{3}
at that instant.

The problem is at first sight very complex, but it has a
closed form solution, that is described (with some errors in the
formulas that I had to figure out) in [1][2]. The nulling of the
currents when the energy transfer is complete leads to sets of
ideal relations among the three oscillation frequencies that
makes them to be ratios of integer numbers k: l: m, so that l =
k+1, k+3, k+5, ..., and m = l+1, l+3, l+5, ...

Ex: k:l:m = 1:2:3; 1:2:5; 2:3:4; 2:3:6; etc.

These three numbers define the "mode" of the operation.
Some extra conditions must be added to force the condition of
total energy transfer to C_{3}. The details are in the
papers.

The design formulas, as functions of these three integers, are:

1) w^{2}L_{1}C_{1}= (2m^{2}k^{2}+(m^{2}-l^{2})(l^{2}-k^{2}))/(2k^{2}l^{2}m^{2}) 2) w^{2}L_{2}C_{2}= l^{2}/(k^{2}m^{2}) 3) w^{2}L_{3}C_{3}= 1/l^{2}4) L_{2}/L_{3}= ((l^{2}-m^{2})(k^{2}-l^{2}))/(2*k^{2}m^{2}) 5) k_{12}^{2}= ((k^{2}-l^{2})(l^{2}-m^{2}))/(k^{2}(l^{2}+m^{2})-l^{2}(l^{2}-m^{2}))

The circuit oscillates simultaneously at the frequencies wk, wl, and wm rad/s. The complete energy transfer
occurs after p/w
seconds, at the "l"th peak (semicycle) of V_{C3}.

As for every k:l:m there are 8 unknowns to be computed from 5
relations, 3 values are arbitrary. For example, w, C_{1} and C_{3}, fixing
the voltage amplification factor, that is (C_{1}/C_{3})^{1/2},
and the energy transfer time. The formulas allow the computation
of the three inductances, C_{2}, and k_{12}. Once
defined the geometry of the system, the self-capacitance of the
third coil would be computed (by Medhurst's empirical formula
[3], for example) and subtracted from the terminal capacitance,
and an extra capacitor, distributed if possible (a toroidal
terminal in L_{2}) would be added in parallel with the
transformer output capacitance and the input capacitance of L_{3}
(approximately identical to its self-capacitance if the base of L_{3}
is sufficiently far from the ground) to make C_{2}.

Additional useful relations that can be obtained from the equations above are:

6) L_{1}C_{1}= (L_{2}+L_{3})C_{3}7) k_{12}^{2}= L_{2}/(L_{2}+L_{3}) 8) C_{2}/C_{3}= 2l^{4}/((l^{2}-m^{2})(k^{2}-l^{2})) 9) L_{1}C_{1}(1-k_{12}^{2}) = L_{3}C_{3}

A design can then be made without direct use of w:

L_{3} and C_{3} can be obtained from an
adequate resonator coil with a top terminal.

L_{2} can be obtained from the equation for L_{2}/L_{3}
(4) and the mode k:l:m.

L_{1} comes from a given C_{1} and the first
additional relation (6) above.

C_{2} comes from the relation for C_{2}/C_{3}
(8) and the mode.

k_{12} comes from the second (7) or from the fourth (9)
additional relation.

With these formulas, C_{2} can be left to be
determined experimentally, as the distributed capacitances
associated with the transformer and the base of L_{3} may
be difficult to predict.

Observe that the fourth relation above says that the resonator
L_{3}C_{3} resonates at the same frequency of the
primary system when C_{2} is short-circuited. (3) then
says that both circuits, alone, resonate at the central frequency
of the three where the whole circuit resonates.

When m is much greater than k and l, the circuit behaves as
when C_{2}=0, and the formulas still work correctly, with
the special values of the coupling coefficient for optimal
transfer obtained from the several combinations of k and l.

Optimum designs for a conventional Tesla coil can also be
obtained, by adding L_{3} to L_{2} (forming L_{3}'),
and adjusting the coupling coefficient to k_{123} = k_{12}(L_{2}/(L_{2}+L_{3}))^{1/2}.
The resulting formulas are:

w^{2}L_{1}C_{1}= w^{2}C_{3}L_{3}' = (k^{2}+l^{2})/(2k^{2}l^{2}) k_{123}= (l^{2}-k^{2})/(l^{2}+k^{2})

Note that in this case the first formula serves only for the
determination of "w". The
oscillation frequencies of the resulting system are then wk and wl rad/s.
The important relation is L_{1}C_{1} = L_{3}'C_{3}.

Example of a magnifier (nothing specially optimized or realistic chosen):

Let k:l:m = 2:3:4, and let's fix L_{1}, L_{3},
and w:

w=2p200000
(total energy transfer in 2.5 µs)

L_{1 }= 0.1 mH

L_{3 }= 20 mH

The remaining elements are then:

L_{2 }= 5.47 mH

C_{1} = 896 pF

C_{2} = 16.28 pF

C_{3 }= 3.52 pF

k_{12 }= 0.463

A simulation of this circuit shows perfect transfer in 2.5
µs, and 159.5 kV in C_{3} for initial 10 kV in C_{1},
corresponding exactly to the ideal factor of (896/3.52)^{1/2}=15.95.
All the other voltages and all the currents (not shown below) are
zero at the peak of V_{C3}.

Voltage waveforms for the modes with klm=1:2:3,
1:2:5, 1:2:7, and 1:2:9, and for the modes with k:l:m=1:2:3, 2:3:4, 3:4:5, and 4:5:6. All the
examples with C_{1}=1 nF, C_{3}=10 pF (for
voltage amplification of 10 only), and w=2p200 krad/s.

Take also a look in my double resonance simulator TeslaSim, that allows experiments with Tesla coil circuits, and also inversion of Laplace transforms. The program MagSim, that designs systems based on the formulas presented here, and can generate Laplace transforms to be used in the TeslaSim program, is also available. See also the program Mrn6.

The theory initially developed in [1][2] can be generalized and extended to more complex configurations, with the help of some classical linear circuit theory. The papers [4][6] show the generalization to the case of "multiple resonance networks". See also [7], for a variant design technique, and [8], where it's shown that the initial energy can be in the first inductor instead of in the first capacitor. See also my pages about a transformerless Tesla coil, a capacitive transformer Tesla coil, a transformerless magnifier, a classical Tesla coil, and a Tesla magnifier, designed with help of the theory.

The paper [5] is an old reference about the theory of Tesla transformers. A partial translation of it to English is available.

References:

[1] F. M. Bieniosek, Review of Scientific Instruments 61 (6)
p. 1717, June 1990.

[2] F. M. Bieniosek, Proc. 6th IEEE Pulsed Power Conference, p.
700, 1987.

[3] R. G. Medhurst, "H. F. resistance and self-capacitance
of single-layer solenoids," Wireless Engineer, February
1947, pp. 35-43, continued in March
1947, pp. 80-92 (^{1}).

[4] A. C. M. de Queiroz, "Synthesis of multiple resonance
networks," 2000 IEEE ISCAS, Geneva, Switzerland, May 2000,
Vol. V, pp. 413-416. Text (^{2}).

[5] Drude, P. "Uber induktive Erregung zweier elektrischer
Schwingungskreise mit Anwendung auf Perioden und
Dampfungsmessung, Tesla transformatoren und drahtlose
Telegraphie," Annalen der Physik, pp. 512-561, vol. 13,
1904. Text (^{1}).

[6] A. C. M. de Queiroz, "Multiple resonance networks,"
IEEE Transactions on Circuits and Systems I, Vol. 49, No. 2,
February 2002, pp. 240-244.

[7] A. C. M. de Queiroz, "A simple design technique for
multiple resonance networks," ICECS’2001, St. Julians,
Malta, September 2001, Vol. I, pp. 169-172. Text (^{2}).

[8] A. C. M. de Queiroz, "Generalized Multiple Resonance
Networks, " 2002 IEEE ISCAS, Scottsdale, USA, May 2002, Vol
III, pp. 519-522. Text (^{2}).

^{1}) Djvu format.

^{2}) PDF format. Copyrighted material by the IEEE.

Created: 6 May 2000

Last update: 8 October 2003

Developed and Maintained by Antonio Carlos M. de
Queiroz