In some cases the two circuits of the Tesla coil, the primary and secondary, are sections of one single coil. In this form the arrangement is called a resonator or auto transformer, and is much used for producing high frequency discharges for medical purposes. The construction of a resonator is as follows: A bare copper wire is wound upon an ebonite or wooden cylinder or frame, and one end of it is connected to the outside of a Leyden jar or battery of Leyden jars, the inner coating of which is connected to one spark ball of the ordinary induction coil. The other spark ball is connected to a point on the above-named copper wire not very far from the lower end. By adjusting this contact, which is movable, the electric oscillations created in the short section of the resonator coil produce by resonance oscillations in the longer free section, and a powerful high frequency electric brush or discharge is produced at the free end of the resonator spiral. An electrode or wire connected with this free end therefore furnishes a high frequency glow discharge which has been found to have valuable therapeutic powers.
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| Fig. 3. | |
C1, Condenser in primary circuit. C2, Condenser in secondary circuit. | L1, Inductance in primary circuit. L2, Inductance in secondary circuit. |
The general theory of an oscillation transformer containing capacity and inductance in each circuit has been given by Oberbeck, Bjerknes and Drude.[2] Suppose there are two circuits, each consisting of a coil of wire, the two being superimposed Theory of Oscillation Transformers. or adjacent, and let each circuit contain a condenser or Leyden jar in series with the circuit, and let one of these circuits contain a spark gap, the other being closed (fig. 3). If to the spark balls the secondary terminals of an ordinary induction coil are connected, and these spark balls are adjusted near one another, then when the ordinary coil is set in operation, sparks pass between the balls and oscillatory discharges take place in the circuit containing the spark gap. These oscillations induce other oscillations in the second circuit. The two circuits have a certain mutual inductance M, and each circuit has self inductance L1 and L2. If then the capacities in the two circuits are denoted by C1 and C2 the following simultaneous equations express the relation of the currents, i1 and i2, and potentials, v1, and v2, in the primary and secondary circuits respectively at any instant:—
| L1 | di1 | + M | di2 | + R1i1 + v1 = 0, |
| dt | dt |
| L2 | di2 | + M | di1 | + R2i2 + v2 = 0, |
| dt | dt |
R1 and R2 being the resistances of the two circuits. If for the moment we neglect the resistances of the two circuits, and consider that the oscillations in each circuit follow a simple harmonic law i = I sin pt we can transform the above equations into a biquadratic
| p4 + p2 | L1C1 + L2C2 | + | 1 | = 0. |
| C1C2 (L1L2 − M2) | C1C2(L1L2 − M2) |
The capacity and inductance in each circuit can be so adjusted that their products are the same number, that is C1L1 = C2L2 = CL. The two circuits are then said to be in resonance or to be tuned together. In this particular and unique case the above biquadratic reduces to
| p2 = | 1 | · | 1 ± k | , |
| CL | 1 − k2 |
where k is written for M √ (L1L2) and is called the coefficient of coupling. In this case of resonant circuits it can also be shown that the maximum potential differences at the primary and secondary condenser terminals are determined by the rule V1/V2 = 2√C2/√C1. Hence the transformation ratio is not determined by the relative number of turns on the primary and secondary circuits, as in the case of an ordinary alternating current transformer (see [Transformers]), but by the ratio of the capacity in the two oscillation circuits. For full proofs of the above the reader is referred to the original papers.