Nikola Tesla Books
Books written by or about Nikola Tesla
Now if there are n turns there will be (n - 1) such systems as illustrated in Diagram 2. Taking air as insulation and neglecting the effect of the small thickness of other dielectric, we would then have the total capacity: $! {C_{1} = {{S_{1} \over {4 \pi r \log {d \over r}}} \times {n - 1} \times {r \over d}}} $!. Now S1 = 2πrl, l being the length of one turn. The values calculated out in this particular case of a secondary of 37 turns (36 1/2 + connecting wires = approx. 37 turns) are as follows:
n = 37
S1 = 3600 sq. cm.
r = 0.125 cm.
d = 4.2 cm.
K = 1
l = 143 feet = 4360 cm.
$! {{d \over r} = 33.5} $!, $! {\log {d \over r} = 1.525045} $! Substituting these values we have:
$! C_{1} = {{3600 \times 36} \over {4 \pi \times 1.525 \times 4.2}} = {{9 \times 3600} \over {\pi \times 1.525 \times 4.2}} $!
$! {= {32,400 \over {3.1416 \times 1.525 \times 4.2}} =} $!
$! { = {32,400 \over {4.791 \times 4.2}} = {32,400 \over 20.12}} $!
C1 = approximately = $! {32,400 \over 20} $! = 1620 cm.
This is not far from the value found. It shows that an approximate estimate may be made in this manner.
Colorado Springs
July 10, 1899
The following consideration conveys an idea of the drawbacks of distributed capacity in the secondary. Suppose the total capacity of 12 such large turns would be 1200 cm. as may be frequently the case, or 100 cm. per one turn and now let us ask to what potential this condenser may be charged - assuming further the energy to be carried away for the performance of work - by expending 1 H.P. In this case we would have $! {{1000 P^{2} \over {9 \times 10^{11} \times 2}} 2n = 750} $! watts. Here P is the potential, n the number of vibrations per second.
In our case n may be 20,000 and then we may put
$! {750 = 40,000 \times {1000 P^{2} \over {9 \times 10^{11} \times 2}} = {2 P^{2} \over {9 \times 10^{4}}}} $! or
$! {P^{2} = {{9 \times 10^{4} \times 750} \over 2} = 9 \times 10^{4} \times 375 = 3375 \times 10^{4}} $! or
$! {P = 100 \sqrt{3375}} = $! nearly 5800 volts.
This shows that by only charging the internal capacity to the insignificant pressure of 5800 volts we would have to expend 1 H.P. Of course, normally the power is small
74
July 9
In order to check the secondary distributed capacitance influence, Tesla prepares new experiments with secondary coils where considerable reduction of undesirable capacitance could be expected. He performs the operating frequency calculation by a previously established method using inductance and capacitance in the primary circuits. A new secondary is wound by thinner wire (0.23mm in diameter instead of 2.6mm), and parameters of this coil he determines according to normal coil parameters on the same core (for equations explanations, please see June 28) with factor D as parameter. When calculating the factor D (which represents the ratio between turns separation of old and new secondary), the period is exchanged with the number of periods per second by mistake, and instead of having D = 2.45, D = 83. Other numerical errors crept in on other equations from which the relationship between D and C is obtained (the number 38 is omitted under square root) and consequently C obtained was 10,000cm instead of 227 cm. Because these results were not used, it is obvious that Tesla did not want to draw attention to this mistake.
The method by which Tesla measured the oscillator frequency by means of additional (auxiliary) coil is interesting. This coil with its distributed capacitance represented actually an absorbing resonant circuit, and the spark at the coil terminals served as indicator (to a certain extent this circuit is similar to a Hertz resonator). Tesla was adjusting the coil by varying the number of its turns until he obtained the biggest spark at coil terminals.
Then he calculated the oscillations wavelength on the basis of wire length on the coil at an achieved resonance (considering that the wire length in the coil is then equal to one quarter of wavelength. The wire length he finds on the basis of measured wire resistance value and known resistance per unit of length. This method, which in itself, hides the systematic error due to neglecting the speed propagation reduction through the coil(45), is applicable for oscillators of larger power. In Tesla's present experiments this is the most reliable method for determining the oscillations' frequencies.
In the calculations method of determining the oscillation period, Tesla applies two equations: one in which he neglects the secondary influence (as e.g., the beginning of note July 9) and the other, when he takes into account the secondary influence. In the latter case he considers that the primary inductance is reduced by a ratio 1 - M2/NL, which corresponds to a primary inductance reduction when the secondary is short circuited. How much of this is correct, it is difficult to estimate, because the oscillator with intensive discharge does not follow the simple theory of an oscillator with a resonant transformer. Secondary circuit is then considerably damped, and free oscillations in secondary quickly disappear, and therefore it would be necessary to apply a theory which takes into account oscillations of a high damped level.
The same day Tesla returns to a problem of approximate distributed capacitance calculations. This time he applies the Kelvin equation for determining the capacitance of a co-axial conductor. He considers that the outside conductor consists of two adjacent turns so that the capacitance calculated for the co-axial conductor reduces the ratio of wire radius and distance between two coil turns. Although not proving the applicability of such approach (e.g., how this equation could be applied in the case when two turns are around a middle turn shown in Fig. 2 at different potential) he calculates the coil total distributed capacitance and obtains a value which he says matches the measured one. The capacitance so determined is not equal to the ideal coil capacitance, which is shown in an equivalent coil schematic in parallel with a pure inductance.
July 10-11
In order to try and increase the secondary voltage of the HF transformer by keeping down the distributed capacity of the secondary Tesla added a third oscillatory circuit, thus obtaining an oscillator with three resonant circuits of which two are tightly coupled*. The third circuit will not necessarily be most strongly excited when its resonant frequency coincides with that of the primary and secondary (assuming these are the same) and the primary and secondary are tightly coupled. If the spark in the primary circuit lasts long, then the tightly coupled primary-secondary system will produce two distinct oscillations, and the third circuit will be most strongly excited if it is tuned to one (strictly speaking to near one) of these two frequencies. On the other hand, if the spark is of short duration the tightly coupled system may oscillate strongest at the resonant frequency of the secondary, and then the third circuit will be excited the strongest when all three have the same resonant frequency. Tesla believed that his system of coupled circuits was producing a single vibration, which under certain conditions is in fact feasible.
July 10
In distributed capacitance, Tesla sees the main obstacle which limits the voltage in the secondary. His assumption regarding the influence of such capacitance he checks now by calculation of a coil model with evenly distributed voltage and capacitances between windings. In order to calculate the active power (for purpose of simplification) he assumes that all energy received by "capacitors" (they represent distributed capacitance) during one quarter period is used up in the second quarter of a period, and so on. In the second half-period, the event is similar, but only the charging is in opposite direction. He thinks that the number of charges (and discharges) is equal to double the number of high frequency current periods.
After mentioning the damaging effect of distributed capacitance, he continues the description of his thoughts on a secondary with thin wire on the same day. He estimates that the induced secondary voltage will be approximately 40% smaller than the ratio between primary and secondary windings, which corresponds to the induced voltage in an open secondary. In our case, however, a current in the secondary circuit will appear which influences the current in the primary, and therefore the induced voltage will have another value.
By adjusting the capacitance in the primary and secondary circuits, he achieves the maximum secondary voltage. How well Tesla knows the physics of the process could be based on the fact that he doesn't lose sight of the influence of the secondary validity factor, and the magnitude of the primary capacitance which determines the amount of power which the source could provide to the oscillator for conversion into high frequency current.
The voltage increase in the secondary, he achieves by addition of a coil with 260 turns wound with wire 2.6mm in diameter. By addition of this coil the oscillator becomes a system with three oscillating circuits, of which two are well linked*. The third circuit does not have to be the most excited, through a strong link with the primary and secondary circuits, when the resonant frequency of the third circuit is matched with the resonant frequencies of primary and secondary circuits (assuming that both of them are the same). If a spark in the primary circuit lasts for a long time (please see app. Fig., 2P-a) then strongly linked system primary-secondary will produce two emphasized oscillations, and the third circuit will be the most excited if it is adjusted on one of two oscillations (strictly taken in vicinity of one of them). If however, the spark in the primary is of short duration, it could happen that with a strong link, the strongly linked system oscillates the most on its own secondary resonant frequency, and consequently then the third circuit would be the most excited when all three circuits are on the same resonant frequency. Tesla assumes that his linked circuits system produces one vibration, and that is under certain conditions achievable if under the term "one" the most emphasized vibration is understood.
For determination of additional coil inductance, he applies the equation for an indefinitely long coil. The coil's self-capacitance he estimates on the basis of known coil resonant frequency, by assuming that it is equal to the reciprocal period value of the primary circuit which he finds from inductance and capacitance of the primary circuit. When with all the calculation of the additional coil validity factor he makes approximations, because he neglects the coil resistance increase with frequency and consequently he gets an unproportionally high validity factor.
* Similar systems were analyzed in 1906 and 1907 by M. Wien, 1907 by C. Fischer and in 1909 by J. Kaiser(46). From these works it can be seen that the effective value of current in a poorly linked circuit will be at maximum if the resonant frequency of this circuit is the same as the other two better linked circuits only when their link is poor.
