Logarithmic Frequency Research Articles

An apparatus for measuring the dynamic shear elasticity and viscosity of solids at high audio frequencies

An apparatus has been designed to measure the dynamic properties of organic glasses at high audio frequencies. The principle of the method employed was that the material, in the form of a rod, was used as the suspension of a moving coil galvanometer. Freely decaying torsional oscillations were detected and displayed as a repeating trace on a cathode-ray oscilloscope. This enabled the logarithmic decrement and frequency of the oscillations to be measured directly. Some typical results are presented.

A graphical contribution to the analysis and synthesis of electrical networks

The contribution of the poles and zeros of a network function to the steady-state characteristics at real frequencies is considered by means of vectors, and the connection between attenuation and phase is thereby accentuated. Bode's relation between attenuation and phase in minimum-phase networks is derived by considering the contributions from individual singularities and using a known definite integral in a real variable. Details are derived for the construction of a series of templates, useful for finding pole-zero positions which yield certain attenuation, phase and delay characteristics, and vice versa. These are attenuation and phase templates for individual singularities on a linear frequency scale, and delay and phase templates on a logarithmic frequency scale. Apart from uses in the usual analysis and synthesis problems, the templates provide a particularly easy means, for simple networks, of obtaining the phase and delay characteristics corresponding to a given attenuation characteristic of a minimum-phase network, and vice versa, and of obtaining the transient response corresponding to given steady-state characteristics. The paper concludes with illustrations of the procedure for estimating the phase-angle characteristic from a given impedance-magnitude characteristic of a driving-point impedance; for estimating the phase and delay characteristics corresponding to the given attenuation characteristic of a minimum-phase wide-band amplifier; for estimating the response of the amplifier to a step-function drive; and for the design of a first-degree delay equalizer.

Dielectric Dispersion in Materials Having a Distribution of Relaxation Times

On the basis of considerable experimental evidence, a particular form for the dielectric loss factor, ε″, is assumed to hold for many materials exhibiting a distribution of relaxation times. The dielectric constant, ε′, corresponding to the assumed form of ε″ is then computed by means of the Kronig-Kramers relations and tabulated as a function of log frequency and of the distribution function half-width parameter, α. Using these results, experimental ε′ and ε″ curves can be easily tested for mutual consistency and can be fitted individually. In conclusion, a brief discussion of other methods of dealing with the many-relaxation-time dispersion problem is given.

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.

Carrier compensation for servomechanisms

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.

A Note on Frequency Transformations for Use with the Electrolytic Tank

Two frequency transformations are described which transform the circular electrolytic tank described by Hansen and Lundstrom into rectangular and elliptical tanks, respectively. The rectangular tank is particularly suited for simulating highly damped circuits where response extending over many octaves in frequency is desired. Furthermore, the co-ordinates of the tank may be calibrated directly against "logarithmic frequency" and dissipation factor, respectively. The elliptical tank is particularly adapted to representation of band-pass circuits having characteristics which are geometrically symmetric about a center frequency. Not only does this transformation expand the scale of the tank and increase its accuracy, but the geometrical relationships between the pole electrodes for simple resonant circuits become substantially independent of the relative bandwidth.

Permeability as a Function of the Size Parameters of Unconsolidated Sand

Abstract The relation between permeability and the size parameters of unconsolidatedsand is approached by considering sands as logarithmic frequency distributionshaving the basic parameters mean size and standard deviation. Sieve separateswere mixed in accordance with the normal logarithmic probability law, so thatthe mean and standard deviation could be varied at will. Experiments wereconducted in which one parameter was kept fixed and the other varied; in thismanner the dependence of permeability on the individual parameters wasevaluated. It was found that permeability can be expressed as the product of apower function of the mean size and an exponential function of the standarddeviation. Introduction Earlier studies of the effect of size of particles on the permeability ofunconsolidated sand have shown that the permeability varies with the square ofsome average diameter. This was first pointed out by Hazen in 1892 and verifiedby Slichter in 1898. Hazen's mean size was defined as the 10 percentile of thecumulative curve (i.e., size that is smaller than 90 per cent of thedistribution and larger than 10 per cent). Slichter's effective size, on theother hand, was found by Wilsey to agree with the 31 percentile. Fair and Hatchfound that the geometric mean size of sieve separates yielded the same powerfunction; Hulbert and Feben found that the permeability of sieve separatesvaried as the 1.89 power of the median diameter (i.e., the size associated withthe 50 percentile of the cumulative curve). Relatively few studies included the effect of the "sorting" of thesediment on the permeability. It was recognized that the nature of the sizedistribution, in terms of the spread of the particles about the mean, affectsthe permeability, but no systematic investigations of this effect are known tothe writers. Cloud demonstrated that the effect of the grain-size distributionon permeability is definitely measurable. In the present paper the authorsinvestigate this relation analytically, and show that the effects of the sizeparameters on permeability may be predicted for at least a restricted set ofconditions. T.P. 1492

Improved Techniques in the Study of Violins

First Page

Terminology for Logarithmic Frequency Units

Fletcher has proposed the use of a logarithmic frequency scale such that the frequency level equals the number of octaves, tones, or semitones that a given frequency lies above a reference frequency of 16.35 cycles/sec., a frequency which is in the neighborhood of that producing the lowest pitch audible to the average ear. The merits of such a scale are here briefly discussed, and arguments are presented in favor of this choice of reference frequency. Using frequency level as a count of octaves or semitones from the reference C0, a rational system of subscript notation follows logically for the designation of musical tones without the aid of staff notation. In addition to certain conveniences such as uniformity of characters and simplicity of subscripts (the eight C's of the piano, for example, are represented by C1 to C8) this method shows by a glance at the subscript the frequency level of a given tone counted in octaves from the reference C0 = 16.352 cycles/sec. From middle C4, frequency 261.63 cycles/sec., the interval is four octaves to the reference frequency, so that below C4 there are roughly four octaves of audible sound. Various subdivisions of the octave are considered in the light of their ease of calculation and significance, and the semitone, including its hundredth part, the cent, is shown to be particularly suitable. Consequently, for general use in which a unit smaller than the octave is necessary it is recommended that frequency level counted in semitones from the reference frequency be employed.

Terminology for Logarithmic Frequency Units

Fletcher has proposed the use of a logarithmic frequency scale such that the frequency level equals the number of octaves, tones, or semitones that a given frequency lies above a reference frequency of 16.35 cycles/sec., a frequency which is in the neighborhood of that producing the lowest pitch audible to the average ear. The merits of such a scale are here briefly discussed, and arguments are presented in favor of this choice of reference frequency. Using frequency level as a count of octaves or semitones from the reference C0, a rational system of subscript notation follows logically for the designation of musical tones without the aid of staff notation. In addition to certain conveniences such as uniformity of characters and simplicity of subscripts (the eight C's of the piano, for example, are represented by C1 to C8) this method shows by a glance at the subscript the frequency level of a given tone counted in octaves from the reference C0 = 16.352 cycles/sec. From middle C4, frequency 261.63 cycles/sec., the interval is four octaves to the reference frequency, so that below C4 there are roughly four octaves of audible sound. Various subdivisions of the octave are considered in light of their ease of calculation and significance, and the semitone, including its hundredth part, the cent, is shown to be suitable. Consequently, for general use in which a unit smaller than the octave is necessary it is recommended that frequency level counted in semitones from the reference frequency be employed.

Rapid Graphical Analysis of Circuit Performance by the Use of Logarithmic Charts

IN connexion with some recent work, use has been made of charts with log (| impedance |) and log (frequency) as axes, and the properties of this plane have been found to lend themselves to graphical use. The chief properties are:

On the Logarithmic Frequency Distribution and the Semi-Logarithmic Correlation Surface

I NTRODUCTION* * 'T'he metho(d of treatinig frequency curves as developed chiefly l)y Fdgeworth, Kapteyn. Vanl U veni and( NVicksell occupies an iml)ortant place in both tlheoretical anid applied statistics. The essenice of this metlhod maxv he briefly sunmimarized as follows: Suppose a fulnctioni of the variable z is distributed according to the normal law of error. 'rlTen. z certainily canniiot be also normallv (listributed, ivunkss the funiction is a liniear functioni of z . Withlout losinig genierality. we shall write the niormiially distributed functioni in stanidard units as = f6z). Thus the origin of z is its mean ani(l the unit of x is its stand(lar( (leviation. Trhe relative frequency of values of x betw-een x andt x + dx is, therefore

Automatic Logarithmic Recorder for Frequency Response Measurements

This instrument was designed to facilitate obtaining records of frequency-response in cases, such as the investigation of electro-phones, where the response varies very irregularly with frequency and the point-by-point method becomes prohibitively tedious. The automatic registration is photographic and both the frequency and response coordinates are logarithmic. The logarithmic frequency scale (50–10,000 cycles) is obtained by the rotation of a special condenser connected to the platten carrying the photographic paper, the plates of the condenser being so shaped as to provide a logarithmic variation of frequency of the associated beat frequency audio oscillator. A logarithmic scale of ordinates is obtained by means of a new “logarithmic voltmeter” employing variable-mu exponential tubes which are automatically controlled. The response or sound-pressure range covered by the present instrument is 100x, although 1000x is obtainable if necessary. Specimen records will be shown and the application of this recorder to studies of the response of loud speakers in the actual reverberant surroundings of the average home will be described.

Logarithmic Scale for Beat-Frequency Oscillator

The equation governing the shape of a condenser's plates, which if placed in a beat-frequency oscillator would give a logarithmic frequency scale, is developed.