Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1950"

Theoretical limitations on the broadband matching of arbitrary impedances

Abstract: This paper deals with the general problem of matching an arbitrary load impedance to a pure resistance by means of a reactive network. It consists primarily of a systematic study of the origin and nature of the theoretical limitations on the tolerance and bandwidth of match and of their dependence on the characteristics of the given load impedance. Necessary and sufficient conditions are derived for the physical realizability of a function of frequency representing the input reflection coefficient of a matching network terminated in a prescribed load impedance. These conditions of physical realizability are then transformed into a set of integral relations involving the logarithm of the magnitude of the reflection coefficient. Such relations are particularly suitable for the study of the limitations on the bandwidth and tolerance of match. Definite expressions for these quantities are obtained in special cases. The practical problem of approaching the optimum theoretical tolerance by means of a network with a finite number of elements is also considered. Design curves are provided for a particularly simple but very important type of load impedance. In addition, a very convenient method is presented for computing the values of the elements of the resulting matching network.

The distribution of particle sizes

Abstract: In this paper the older literature on distribution of particle sizes is reviewed critically. The distribution law should be connected with the law of growth of crystals, for which is chosen the exponential law as Galton indicated long ago in an analogous case. The law which then follows is the so-called Logarithmico-Normal law. In the application of that law to experimental data, graphical analysis by means of special charts is generally used. It is shown why this purely graphical method should be replaced by an algebraic one, such as is used, for instance, in psychophysics with a similar problem.

A note on the solution of certain approximation problems in network synthesis

Abstract: This paper presents a method of determining appropriate functions to be used as amplitude characteristics of networks when certain ideal behaviors are to be approximated within specified tolerances. This method is based on the use of suitable transformations in the complex plane which, by simplifying the geometry of the problem, permit the direct determination by inspection of the desired functions. In the simplest cases of practical interest, this method has the only advantage of presenting results already known in a more intuitive form. On the other hand, it yields also alternative solutions satisfying additional and less conventional requirements and, at the same time, it indicates an approach likely to be fruitful in the solution of more complex approximation problems.

On certain matters pertaining to electrets

Abstract: The object of the paper is to trace the mathematical consequences of the assumption that an electret consists of (1) a distribution of polarization which decays with time, (2) a distribution of surface and volume charge which disappears according to ohmic conductivity having no relation to the decay of the polarization. Theelectret is supposed provides with metal plares permanently attached to its surface and to have a thickness of 1 cm. and a radius of 3 cm. Starting with one plate at zero potential and the other at a negative potential V0, symbolizing in the ordinary terminology the condition in which the electret shows a “heterocharge,” the absolute value of V decays with time to zero.It then rises to a positive maximum, after which it again decays with time, ultimately to zero.It turns out that the functional relation between potential difference and time is absolutely independent of the charge distribution within the electret and of its decay.The only parameters involved are the resistance R connecting the plates and the parameter determining the rate of decay of the polarization. However, while the initially produced polarization plays a part in determining the maximum positive potential Vm attained and while V0 plays a part in determining the time for the potential to change from V0 to zero, neither of these quantities plays any part in determining the subsequent time for rise to the maximum Vm. If the resistance is such as would be determined by ordinary ohmic conductivity in a substance of the normal specific resistance of a wax, if the polarization decays according to a factor e−αt, and if α is 10−8 sec−1, corresponding to a decay of.the polarization to 1 e of its initial value in about three years, then, starting with V0 = − 2000 volts and with the initially produced polarization such as to make Vm = 2000 volts, there results the following: V will change from its original value − 2000 volts to zero in about one and a half days, then it will rise to the positive maximum Vm in about two weeks, after which it will decay to one third of that maximum value in about three years. If the electret be exposed to the open air, the conductivity of the air would produce a smaller resistance than the wax itself, and the maximum would be attained in about 42,000 seconds after the value V = 0 was reached. If, with the conductivity of the wax alone operating in determining R, we desire a maximum value of V equal to 2100 volts, it would only be necessary to have an initial polarization less than 1 per cent of that which would be produced by a complete orientation of the dipole moments of the molecules of the wax in one direction. Expressions are also developed for the case where the charge within the electret disappears by self-repulsion to the plates, in addition to its decay by ohmic conductivity. Such a phenomenon can, of itself, simulate a decaying polarization. A discussion of experiments pertaining to the field outside of the electret is also given.

Application of near-equilibrium criteria during adiabatic flow to representative propellant systems☆

Abstract: Near-equilibrium criteria for chemical reactions during adiabatic flow through a nozzle are summarized.These criteria are then applied to two representative chemical systems. It is shown that the adiabatic expansion of the combustion products produced in the hydrogen-fluorine motor using the reactants in stoichiometric proportions should occur under conditions very closely approximating thermodynamic equilibrium. One of the near-equilibrium criteria is modified and used to demonstrate the departure from thermodynamic equilibrium for the decomposition of nitric oxide. This study of the equilibrium involving nitric oxide, nitrogen, and oxygen suggests the occurrence of practically frozen flow below a certain temperature, a result which is in agreement with the conclusions reached by a detailed numerical calculation.

Variations of grid contact potential and associated grid currents

Abstract: This survey is intended to serve as a guide to the literature on grid contact potentials and the associated grid currents.The difference of potential between grid and cathode of an electron tube is defined and separated into its component parts. The Volta potential, or “true contact potential,” is shown to be the difference between work functions of grid and cathode. When the grid is negative the current in the external grid circuit consists of several small currents due to electron initial velocities, gas ions, interelectrode leakage, grid emission, and other miscellaneous factors.The grid voltage at which the resultant grid current is zero is a composite quantity V τ = − ( V gk + V ξ ) where Vgk is the grid-cathode Volta potential and Vχ is the voltage equivalent of the various components of grid current. The quantity Vχ is a spurious “contact potential.” The significance and use of the calculated correction potential ϵ for space-current equations is discussed briefly. In addition, the relationship between ϵ and tube characteristics is shown. The stabilization of dynamic characteristics of low-bias tubes having oxide cathodes has been described in recent literature. O'Neill, Champeix, and Gorelik have shown how a control grid can be coated with active material in order to stabilize (φk - φg), and, in addition, Gorelik has also described a procedure for reducing grid emission from such a coated grid. Darbyshire suggests methods for controlling initial velocity currents by suitable processing. The appendices list thermionic work functions for 44 pure metals, for contaminated metals, and for oxides. Also is listed the manner in which changing physical factors within the tube will tend to alter grid potential.

Carrier compensation for servomechanisms

Abstract: A carrier servo system is a servo system which employs carrier-frequency techniques for data transmission. The provision of compensation to improve the stability and the performance of such a system presents several unique problems. Theoretically, the compensation transfer characteristic should have an amplitude response which is symmetrical about the carrier frequency and a phase response which is skew-symmetrical about the carrier frequency, both considered relative to an arithmetical frequency scale. Transfer characteristics normally obtained from physically realizable structures possess the required symmetry if considered on a logarithmic frequency scale; therefore, they introduce distortion into the system. Accuracy of physically realizable transfer characteristics is limited by tolerances required on components of the compensation device and by the dependence of many of the devices on the absolute stability of the carrier frequency. Carrier-compensation transfer characteristics may be realized through the use of passive electrical networks, electrical or electromechanical demodulation, compensation, and remodulation systems, feedback devices, and other means. Two main types of passive electrical networks have been employed, namely, those involving resistances, inductances, and capacitances, and those involving only resistances and capacitances. The former type of networks is the most general but is limited by the non-linearities introduced by non-ideal inductances. Parallel-T and bridge-T networks of the latter type of networks have found wide usage in carrier lead compensation. Electrical demodulation, compensation, and remodulation systems consist of the application of readily available electronic and non-carrier compensation techniques; however, electromechanical systems for obtaining similar results have been developed for carrier servo systems. The electrical systems do make available for carrier compensation a wide variety of compensation transfer characteristics, whereas electromechanical systems are more limited. Feedback devices are useful for obtaining time derivatives of the output. At least one method for minimizing the effect of carrier-frequency instability has been developed. Passive electrical networks present perhaps the best method of obtaining carrier-compensation transfer characteristics; however, characteristics other than simple lead are normally difficult to obtain, and the use of non-carrier compensation techniques appears advisable. Carrier servo systems do have inherent advantages as compared with non-carrier systems and consequently require less compensation for acceptable response. Great advancement may be expected in the field of carrier compensation during the coming years.

Radiation transfer considerations in the heating of a cathode sleeve

Abstract: The theory of radiation heat transfer is applied to the problem of heating a cathode sleeve to high temperatures. Experiments have been made in which metal and ceramic sleeves are heated by an internal cylindrical radiator. The results are discussed quantitatively in terms of the total emissivities of the several surfaces.