Statistical Analysis of Chopper-Modulated Circuits

This paper describes an approach to the statistical analysis of chopper-modulated circuits excited by random inputs. In general the output response of those networks containing time variable elements such as a periodically operated switch, to say, the chopper, becomes the nonstationary random function, even though the input signal of the random function to networks is stationary, therefore the analytical method has to be considered for the nonstationary random process. In this paper we introduce the available technique for such process according to Zadeh's methods which are probably the powerful tools for the analysis of linear variable networks. The two types of practical interest are treated, that is, the transformer coupled chopper-modulated circuit and the resistance-capacitance coupled chopper-modulated circuit which are frequently used to conventional chopper amplifiers. The results obtained in this paper make clear commonly the fundamental statistical characteristics of these circuits

The Steady-State and Transient Characteristics of the Chopper-Modulated Circuit Having Four Circuit Modes

The problem considered in this paper is one, in general, of making clear the characteristic of chopper-modulated circuits from the theoretical point of view. The chopper-modulated circuit under consideration is assumed to have four circuit modes. In a previous investigation, we have already discussed a chopper-modulated circuit having common two circuit modes, and also have developed the valuable analytical method for a periodically interrupted electric circuit having two circuit modes excited by a complex sinusoidal input. Here we shall generally extend the foregoing analytical method for a periodically interrupted electric circuit having m circuit modes driven by a complex sine wave input and basing on this method, we shall discuss a chopper-modulated circuit having four circuit modes. Only a type of practical interest is treated, that is, the transformer coupled chopper-modulated circuit which is frequently used with conventional chopper amplifiers, and other types which are not considered here. The method considered in this paper is an available technique explained to make clear the steady-state and transient performance of these types

A Study of the Transfer Function of Chopper Amplifiers

The chopper amplifier is frequently used in several types of d.c and a.c amplifiers as well as in d.c servomechanisms employing a.c control elements. In this paper the transfer function of the chopper amplifier is derived for a sinusoidal input signal. The chopper amplifier has a chopper, or, a contact modulator at the input element of an a.c coupled amplifier to provide a means of converting d.c and low frequency signals to signals which lie within the pass band of the amplifier. Therefore, we first consider on the general analysis of the circuits containing a periodically operated switch such as a chopper. This analytical method is based on the theories of the Periodically Interrupted Electric Circuits and the complex Fourier series. The demodulation of the amplified signal is obtained by employing a synchronizing rectifier circuit, or a 2-phase induction motor at the output of the amplifier. In these systems we can easily derive the transfer function of some types of chopper amplifiers and of which the steady-state performance can be clearly stated in detail

Approximate Solution of Mathieu's Differential Equation

This paper presents a method for the approximate solution of the differential equations of the Mathieu-Hill type. This method is based on the analytical method of the Periodically Interrupted Electric Circuits. The second-order differential equations with periodic coefficients, considered in this paper, are represented by the general form : d²y/dz²+f(z)y=0, where f(z) is a single-valued periodic function of fundamental period Zᵀ, when f(z)=a+16q cos 2z, it is known as Mathieu's differential equation. Based on the procedure in this paper, the periodic function f(z) is subdivided into m functions, f₁(z), ··· , fᵣ(z), ··· , fₘ(z), each of which has a different interval zᵣ, (r=l, 2, ···, m) for one period zᴛ of f(z). Namely the function fᵣ(z) represents the linear approximation of f(z) in each interval, that is, fᵣ(z)=2cz+d, 0≦z≦zᵣ, (r=1, 2, ···, m) where the values of c and d are constant. From this practical linear approximation, the present method is adequate for the determination of the approximate solution of the differential equations of the Mathieu-Hill type and this method has certain advantages, especially for the stability of the solution and also the transient solution. The stability chart for Mathieu's differential equation is obtained and plotted for the ranges of -3≦a≦34 and 0≦q≦2. This result is very well coincident with Ince's numerical one computed for the range of q=0 to 5.0. The obtained solutions and their numerical results may be extensively accurate. And the procedure considered in this paper is useful for the mathematical analysis of a large class of physical problems

On the Detonation Pressure Produced at the Inner Surface of the Charge Hole

In these experimental studies, the magnitude of the detonation pressure produced at the inner surface of the charge hole by the detonation, of a confined explosive and the change in that pressure with time were chiefly investigated by means of measuring the stress wave produced within a steel rod by this explosion, utilizing a specially designed piezo-electric pressure gauge. The main element of the above pressure gauge was a transducing unit which contains a barium titanate ceramic transducer enclosed tightly in the gauge body. The strong stress caused by the explosion must be properly reduced before being imposed upon the transducing unit. In this gauge such reduction of stress was accomplished by placing an ebonite directly on top of the upper surface of the ceramic transducer. The results obtained are briefly shown below. In the case of the detonation of a confined 45g- or 90g-cartridge charged in a bore hole with a charging density of 0.7g/cm³, the rise time to the peak pressure was less than about 2μs, and this peak pressure decreased rapidly to the state of so-called static pressure about 15μs later, decreasing very slowly thereafter. The duration time of such detonation pressure in the charge hole was about half a millisecond in these experiments. The value of about 3×10⁴kg/cm² was obtained as the steady detonation pressure (generally known as Pᴄ₋ᴊ) of a 45g-cartridge of No. 3 Take dynamite under our experimental conditions, corresponding to a velocity of detonation of 3, 3000 m/s

An Analysis of Non-Linear Sampled-Data Feedback Control Systems

Higher order sampled-data feedback systems which contain a saturating element or a backlash element are investigated in this paper. This study introduces a new approach to the analysis of non-linear sampled-data control systems. At first the authors describe a new analytical method for such systems using the theorem of Periodically Interrupted Electric Circuits and how to apply the Digital Computer (KDC-1) to this theorem. The method presented here can be applied to any higher order systems with any non-linear elements by making use of the digtal computer simulation of the above theorem and the non-linear element. Some illustrative examples are given to clarify the method involved. One example of the third order sampled-data feedback system with a saturating element is investigated in case of initial conditions being given without external forces, a unit step function and sine wave inputs being given. The examples show that in the case of a step input as well as initial conditions existing, slight variations of initial values result in difierent modes of periodic oscillations, while in the case of a sinusoidal input, a slight modification of non-linear characteristics results in forced oscillations in one case and in sub-harmynic oscillations in another. Two illustrative examples of the second order system with a backlash element are considered in the case where the linear system is followed by the backlash or follows the backlash. Some results obtained by numerical computations are presented to show the performance of the system dynamics on the basis of the new analytical method presented here

Analysis of Sampled-Data Feedback Control Systems with Finite Pulse Width

A method for analysis of sampled-data feedback control systems with finite pulse width is presented in this work. The analysis is based on introducing a new technique with some approximations by the theories of the Periodically Interrupted Electric Circuits. The results make it possible to obtain the transient and steady-state response of such systems containing a sampler with any pulse width. Furthermore, by utilizing this method, the stability criterion is readily developed and its application to systems under consideration make clear the effects of the sampler with finite pulse width on the performance of sampled-data feedback control systems, which are of considerable importance in view of its engineering applications. Some illustrative examples are given to clarify the method involved and its numerical results are presented to show the performance of the system dynamics. Moreover the results in this investigation are examined by the simulation of the control system in question by means of the analog computer

A Field Determination of Permeability

This paper was published in Japanese on Feb. 15, 1952 in the Journal of the Japan Soc. of Civil Engrs.In order to determine the permeability of the undisturbed field ground, a new method using a simple pool is proposed instead of the conventional unreasonable assumption or the troublesome method proposed hitherto. The theoretical formulae on which this method depends are derived, and a discussion is presented on the result obtained by this method and its verification