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On Electrostatic Transformers and Coupling Coefficients.1 On Electrostatic Transformers and Coupling Coefficients.1 BY F. C. BLAKE Physical Laboratory, The Ohio State University, Columbus, Ohio. Starting from the experimentally determined fact that the capacity of an air condenser is independent of the fre­ quency of electrical oscillation, it is shown by means of Lord Rayleigh's equations for the mutual reaction between two circuits each having inductance, resistance, and capacity, that for high-frequency conditions when the resistance is negligible compared to the reactance, the capacity reaction between the two circuits can be expressed best in terms of elast- ances. Definitions are given for self and mutual elastances as well as for self and mutual capacitances and the definitions are tested by our knowledge of spherical condensers. The coefficient of elastic coupling is shown to be the ratio between the mutual elastance and the square root of the product of the two self elastances, the analogy with the coefficient of inductive coupling being exact. The coefficient of capacitive coupling between two circuits each having capacity with a capacity in the branch common to both is shown to be a limiting case of the coefficient of elastic coupling, and thereby a condenser of the ordinary or close form is shown to be an electrostatic transformer with a coupling coefficient of unity. The true relationship between Maxwell's coefficients of capacity and the elastances or capacitances is pointed out in the case of the spherical condenser. The ideas developed are applied to the thermionic tube and thereby the behavior of the ultraudion and the experiments of Van der Pol are readily explained. Attention is called to the alternative view of the behavior of condensers toward alternating currents, viz., instead of being paths of low impedance, they are paths of ready yielding or low stiffness or elastance, as suggested by Heaviside and by Karapetoff. N a paper presented before the American Physical wave-length of the harmonic under consideration, I Society in November 1919, the writer was able K the capacity of the condenser and k the capacity experimentally to verify the equations developed per unit length of the Lecher wires. In such systems by Lord Rayleigh1 for the reaction of one circuit upon equation (3) takes the place of Lord Kelvin's equation another, each circuit having both kinetic and potential for the discharge of a condenser, viz., energy as well as dissipation, i. e., inductance, capacity p2 L K - 1 = 0. (4) and resistance. If L L and L are the self and lh 22 12 In equations (1) and (2) all the self coefficients were mutual inductances, E , E and E the self and n 22 12 determined from geometrical considerations, the mutual mutual elastances, (see below) R , R and R the n 22 u coefficients being determined from the experimental self and mutual resistances, and p the generalized observations. For instance, En is the reciprocal of frequency, then the effective elastance, E and the e the capacity of the plate attached to the Lecher cir­ effective resistance, Re are given by the equations: cuit when it exists alone in space, E22 being the re­ 2 2 (E12-p L12) ciprocal of the capacity of the plate attached to the Ee = En-p'L, Ei P2 U receiver circuit under the same conditions. E12 on the other hand was determined by noting the difference in p2 [Ru (E - P2 L ) - R (E - p2 L ) ]2 (1) 22 22 22 12 12 frequency of the two fundamental tones as the coupling + (E - p2 L ) [ (E - p2 L )2 + p2 R 2} 22 22 22 22 22 between Lecher and receiver circuits was tightened. Similarly, Ln is the self inductance of half of the R = R ~ Rl22/R22 e n Lecher circuit and L22 is the self inductance of the 2 2 receiver circuit. Substituting these values of L and [Ru (E22- P2L ) - R (E V L ) } 22 22 12 12 (2) + R 2[(E - p2L )2 + p2R 2} E in equation (3) one can find the natural frequency n 2 22 22 22 of each circuit when alone in space. It is, of course, The verification was accomplished by working with this natural frequency n that must be used in equations Lecher and receiver systems, thus using short electric (1) and (2) for either primary (Lecher) or secondary waves. For such systems the distributed capacity (receiver) circuit when existing separately. Hence, of the Lecher wires has to be taken into account as equations (1) and (2) become well as the capacities of the condensers. Thus the generalized formula for the discharge of a condenser (E - P2L )2 E = E - n2 L 12 l2 e n n 2 was found to hold, viz., E22 — n L22 2wl . 2irl kl p2 [R (E - n2 L ) - R (E V2L ) ]2 (3) 12 22 22 22 l2 -Y-tan-x- K + (E - n2 L ) [ (E 2 2 2 2 22 22 22 n L22) + p R22 ] ' (5) where I is the length of the Lecher circuit measured from the exploring bridge up to the condenser, X the Re = Rll — R2 *Read before the American Physical Society, Feb. 28, 1920. 2 [ Rn (E - n L ) - g (Ei, - p2 Lu) ]2 1. Phil. Mag. XXI. pp. 369-381, 1886; Scientific Papers, M 22 22 + R22 [ (E - n* L )2 + v2 R™2\ Vol. II p. 484. 2i 22 (6) 23 24 BLAKE: ELECTROSTATIC TRANSFORMERS Journal A. I. E. E. By causing the Lecher and receiver circuits to approach frequency currents. It is to be noted that equation each other both Eu and R12 were made to vary, the (11) is the exact analog for reacting systems having system being so disposed that L12 was practically zero stiffness or elastance, of the well-known equation for always. Naturally E12 would be expected to increase inductively reacting systems, viz., and Ru to decrease as the circuits approached each other. This was borne out by observation. Calcu­ T - T Ll2* lation really showed that for the frequencies used Lie — Li\ 1 — j • L (12) (n > 108) the last term in (5) was negligible compared 22 Now inertia (inductance) is a measure of the re­ to the other terms so long as R was small. Because u sistance a body offers to a change in its condition of the currents are of such high frequency p2 jR 2 is 22 motion, and stiffness (elastance) is a measure of the entirely negligible compared to (E — n2 L )2 and 22 22 resistance an elastic body offers to its instantaneous con­ (5) and (6) reduce to dition of vibratory motion. If in (12) the coefficient of 2 inductive coupling k is equal to —^12 as is well- E.-En-n IN- E22-n L22 y/Ln L22 2 2 2 2 2 known, then (12) may be written L = L (1 — k ). (13) V [R12 (E22 - n L22) - R22 (E12 - p L12) ] e n 2 So also the coefficient of elastic coupling k is equal to + (E22 - n L22f (7) E 12 and (11) may be written J? - I? #2i2 /in ~ J5 JXe — s/Eu E22 JX22 2 Ee = En (1-k ) (14) 2 2 2 [Ru (E22 - n L22) - R22 (E12 - p L12) ] From the definitions of self and mutual capacitances 2 2 ^ R22 (E22 - n L22) * (g) given it is manifest that if in (10) we take Ln and L22 For distances between the circuits less than 15 per each zero the effective capacity of the primary cir­ cent of the diameter of the capacity plates it was found cuit becomes that Ru was practically equal to zero and (7) reduced r< _ n C122 w — On -p p , in this case to r 2 O12 — on o ^25) 22 C C and if we substitute k2 for —^ 22 (15) reduces to For electrostatic (elastic) coupling between the c — di circuits L12 is zero and we have e~ ^ ' (16) E. = E - n2 L - El2* . n n T which obviously may be made as large as we please &22— Wr L/22 ^ Thus for radio frequency currents the effective by making k approach unity more and more nearly. elastance of a circuit is apparently a function not This accounts for the capacity of a condenser being only of its own elastance, the elastance of a neighbor­ so enormous compared to the capacities of its com­ ing circuit and their mutual elastance but also of their ponent conductors when far removed from each other. self and mutual inductances. If now the self Just as in (13) the maximum value of k is unity so the maximum value of k in (14) is also unity. Thus (di and C22) and mutual (G\2) capacitances of the conductors forming the condensers between the Lecher just as the effective inductance of a coil is decreased and receiver circuits be defined as the reciprocals by the presence of a second coil so also the effective of the self and mutual elastances of these conductors elastance of a conductor is decreased by the presence of a second conductor. and if Ce be the effective capacity of the primary cir­ cuit, then In discussing two mutually reacting inductive cir­ cuits Lord Rayleigh derives the equation Ce — Cu 1 2 T — T — -4- (£12 R22 — L22 R12) , r n \ C22 1 e ~ 11 L22 ^ L22 (P2 L 2 + R222) 9 a 22 n L C ) ^5- • !_ 2 n u W LmCm (17) (10) If in (9) the inductances L and L are negligible or in which the second fraction is positive and with n 22 increasing p2 continually diminishes.
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