Phase-shifted Rectified Sine Waves

ILL INOI SUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

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UNIVERSITY OF ILLINOIS

BU LETINVoL 40 December 1, 1942 No. 15

ENGINEERING EXPERIMENT STATIONBULLETIN SERIES No. 839

PROPERTIES AND APPLICATIONSOF PHASE-SHIFTED RECTIFIED

SINE WAVESBY

J. TYKOCINSKI TYKOCINER

LOUIS R. BLOOM

PRICE SIXTY CENTSPtBLSH lPD BY THE UNIVERSITY OF ILLINOIS

URBANAid weekly. Egered as aecond-clasa matter at the post office at Urbaa, Illinois, under the Act o

Agu 2, 191. %Oce I Publicatioun S8 Adminiotranto Building, Urban, Il Aceptance form a thepeca rtpW of poatage provided for n ection 1108, Act of Octobr 8, 1017, authorized

* -1 * t

HE Engineering Experiment Station was established by actof the Board of Trustees of the University of Illinois on De-Scember 8, 1903. It is the purpose of the Station to conduct

investigations and make studies of importance to the engineering,manufacturing, railway, mining, and other industrial interests of theState.

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/',

UNIVERSITY OF ILLINOISENGINEERING EXPERIMENT STATION

BULLETIN SERIES No. 339

PROPERTIES AND APPLICATIONSOF PHASE-SHIFTED RECTIFIED

SINE WAVES

BY

J. TYKOCINSKI TYKOCINERRESEARCH PROFESSOR OF ELECTRICAL ENGINEERING

AND

LOUIS R. BLOOMRESEARCH ASSISTANT IN ELECTRICAL ENGINEERING

PUBLISHED BY THE UNIVERSITY OF ILLINOIS

PRICE: SIXTY CENTS

3000-11-42-24425 * ILLIN

CONTENTSPAGE

I. INTRODUCTION . . . . . . . . . . . . . 7

1. Origin and Purpose of Investigation . . . . . 72. Acknowledgments .... . . . . .. 7

II. PROPERTIES OF BIPHASE RECTIFIED SINE WAVES . . . 93. Graphical Presentation of Properties of Wave Forms

Obtained by Subtraction . . . . . . . 94. Graphical Presentation of Properties of Wave Forms

Obtained by Addition . . . . . . . . 105. Properties of Sine Waves Combined With Rectified

Sine Pulses . . . . . . . . . . . 14

III. APPARATUS FOR PRODUCING PHASE-SHIFTED RECTIFIEDSINE WAVES.. . . . . . . . . . . . .20

6. Methods of Phase-Shifting . . . . . . . . 207. Phase-Shifting by Addition of Vectors in

Quadrature Relation. . . ... . . . 218. Sound-Radiation Method of Phase-Shifting . . . 229. Acoustic Phase-Shifting Apparatus . . . . . 23

10. Phase-Shifting by Adjusting Circuit Constants . . 26

IV. EXPERIMENTAL VERIFICATION OF NEW WAVE FORMS ANDOF THEIR PROPERTIES . . . . . . . . . .27

11. Scope of Oscillographic Investigation . . . . . 2712. Experiments With Wave Forms Obtained by Vectors

in Quadrature Relation at 60 Cycles . . . .2713. Experiments With Wave Forms Obtained by Sound-

Radiation Method at 1500 Cycles . . . . . 2914. Experiments With Wave Forms Obtained by Sound-

Radiation Method at 6000 Cycles . . . . . 3315. Experiments With Wave Forms Obtained by Adjust-

ing Circuit Constants at 20 000 to 100 000 Cycles 33

CONTENTS (CONCLUDED)

PAGE

V. MATHEMATICAL DISCUSSION OF WAVE FORMS . 36

16. General Relations . . . . . . . . . . 3617. Wave Forms Obtained by Subtraction . . . . 38

18. Wave Forms Obtained by Addition . . . . . 4019. Wave Forms Obtained by Combination of Rectified

Pulses With Full Sine Waves . . . . . . 4320. Relations Between Wave Forms . . . . . . 46

VI. APPLICATIONS . . . . . . . . . . . . . 4921. Review of Applications . . . . . . . . . 4922. Oscillographic Method of Phase Angle Measure-

ment Using Difference Curve D . . . . . 4923. Oscillographic Method of Phase Angle Measure-

ment Using Sum Curve S . . . . . . . 5024. Oscillographic Method of Phase Angle Measure-

ment Using Composite Curve M . . . . . 5025. Phase Angle Measurement by Means of Direct-

Reading Instruments. . . . . . . . . 51

VII. RiSUME OF INVESTIGATION . . . . . . . . . 51

26. Summary . .. .. . . . . . . . 5127. Results . . . . . . . . . 52

LIST OF FIGURESNO. PAGE

1. Graphical Subtraction of Two Rectified Phase-Shifted Sine Waves . . 82. Amplitude as Function of Phase Angle for Difference Curve D . . . . 93. Graphical Addition of Two Rectified Phase-Shifted Sine Waves . . . . 114. Amplitude as Function of Phase Angle for Sum Curve S . . . . . . 125. Graphical Subtraction of a Phase-Shifted Rectified Sine Wave From a

Full Sine Wave ........ ...... .156. Amplitude as Function of Phase Angle for Difference Curve M . . . 197. Diagram of Phase-Shifting Circuit ... . . . . . . . . . 208. Component Values of Vectors for Adjusting Phase Angles . . . .. 219. The Acoustic Phase-Shifting Apparatus . . . . . . . . . . 23

10. Diagram for Phase-Shifting by Varying Circuit Constants . . . . . 2511. Oscillograms Showing Subtraction of Two Rectified Phase-Shifted Sine Waves 2812. Oscillograms Showing Addition of Two Rectified Phase-Shifted Sine Waves 3013. Oscillograms Showing Subtraction of a Phase-Shifted Sine Wave From a

Full Sine Wave ................ .3214. Oscillograms Showing Subtraction of Two Rectified Phase-Shifted Sine Waves 3415. Subtraction and Addition of Two Full Sine Waves . . . . ... .47

LIST OF TABLESNO. PAGE

1. Coordinates of Critical Points and Relative Periods of Difference Curve D 102. Coordinates of Critical Points and Relative Periods of Sum Curve S . . 133. Coordinates of Critical Points and Relative Periods of Composite Curve M 184. Comparison of D, S, and M Curves . . . . . . . . . . . . 48

PROPERTIES AND APPLICATIONS OF PHASE-SHIFTEDRECTIFIED SINE WAVES*

I. INTRODUCTION

1. Origin and Purpose of Investigation.-The development of anautomatic recorder of spectral sensitivity of photoelectric surfacestrequired means for so controlling a pair of modulated light sourcesthat their relative intensities be maintained in definite phase relations.In this connection a theoretical study of rectified phase-shifted sinewaves was made. Enhanced by an urgent demand for a practicalmethod of precise determination of small phase shifts, this work hasbeen further developed experimentally.

The purpose of this investigation is(a) To analyze by graphical methods the properties of wave forms

obtained by subtraction or addition of two phase-shifted rectified sinepulses.

(b) To show that new wave forms are also obtainable by subtrac-tion or addition of a full sine wave and a phase-shifted rectified sinepulse.

(c) To verify experimentally the predicted properties and to illus-trate them by oscillograms.

(d) To develop means of producing phase-shifted rectified sinewaves whose phase angles can be precisely adjusted.

(e) To substantiate the graphical method by a mathematical dis-cussion of the new wave forms.

(f) To devise means for the applications of the properties of thenew wave forms.

2. Acknowledgments.-This investigation has been carried on aspart of the work of the Engineering Experiment Station under thegeneral administrative direction of DEAN M. L. ENGER, Director ofthe Engineering Experiment Station, and PROFESSOR ELLERY B. PAINE,Head of the Department of Electrical Engineering. Acknowledgmentis due to DR. HENRY J. MILES of the Mathematics Department forchecking the mathematical discussion and for making suggestionswhich aided a clearer presentation of Chapter V.

*A brief account of this investigation was given in a paper presented before the annualmeeting of. the Illinois State Academy of Science on May 8, 1942.

tJ. T. Tykociner and L. R. Bloom, Journal of the Optical Society of America; Vol. 31,pp. 689-692; 1941.

ILLINOIS ENGINEERING EXPERIMENT STATION

vs~

m

I

co

a:

- I

qHN (

::

0

PI

Cv

a

S6

PHASE-SHIFTED RECTIFIED SINE WAVES

Phase Angle P# n 6 Degrees

FIG. 2. AMPLITUDE AS FUNCTION OF PHASE ANGLE FORDIFFERENCE CURVE D

II. PROPERTIES OF BIPHASE RECTIFIED SINE WAVES

3. Graphical Presentation of Properties of Wave Forms Obtainedby Subtraction.-Figure 1 shows the results of subtraction of tworectified sine waves, A and B, of equal amplitudes Ao = Bo which areshifted in phase by 0, 15, 30, 45, 60, 90, and 120 deg. respectively.Curve D in each of the figures represents wave forms produced by thedifference of the ordinates, A minus B. The critical points are markeda, b, c, d, and al. A study of such difference curves has revealed thefollowing properties:

(a) The original rectified sine wave with a period To = 2 r is trans-formed into alternating pulses of nearly triangular wave shapes.

(b) Each pair of superimposed phase-shifted pulses is transformedinto an alternating cycle. Thereby the frequency of the resultingwave D is doubled.

(c) For phase angles corresponding to odd quadrants 0 = 0 to 7r/ 2or 4 = 7 to 3/2 7r, etc., the part of the curve a, b, c, which faces the


TABLE 1*COORDINATES OF CRITICAL POINTS AND RELATIVE PERIODS OF

DIFFERENCE CURVE D

Critical Abscissas Ordinates PeriodsPoints x D * TD

a . = 0 (8a)* Da= -sin 4 (min) (8b)b xb = 4/2 (9a) D= 0 (9a)c Xe = 4 (7a) D, = sin 4 (max) (7b) TD = To/2 (10)d x = ,/2+ 4/2 (9b) Dd= 0 (9b)at x.=T Di = sin(r+ 4) (min)

*Numbers in brackets refer to mathematical derivations in Chapter V.

origin and connects the minimum point a, with the maximum point c,is very close to a straight line. The other part of the curve c, d, a,, fol-lows more noticeably a curved line with the zero point d as its inflec-tion point.

For phase angles corresponding to even quadrants where 4 = 7/2to r or 4 = 3/2 r to 2r, etc., these characteristics of the two parts ofthe curves are reversed.

The deviation from linearity of the c, d, a, part of the curve firstincreases with increasing angles 4, reaches a maximum at = r/6,then decreases and becomes smallest at 0 = r/2.

. At the latter phase angle (0 = 90 deg.) the difference curve D iscompletely symmetrical and its wave form resembles an isoscelestriangle.

(d) The amplitude of the difference curve is proportional to thesine of the phase differences; this is shown graphically in Fig. 2. Asis evident from Fig. 2, the greatest amplitude of the difference curvesoccurs at a phase angle q = 7r/2.

(e) In Table I are given the equations for the abscissas and ordi-nates of the critical points. They show the dependence of the coordi-nates upon the phase angle.

The numbers in brackets refer to mathematical proofs given inChapter V.

4. Graphical Presentation of Properties of Wave Forms Obtainedby Addition.-If instantaneous values of two biphase rectified sinewaves of equal amplitude, Ao, Bo, are added, wave forms are pro-duced which are different from those obtained by subtraction. Figure 3represents a series of such wave forms as they evolve by varying step-wise the phase angle 0 from 0 to 15, 30, 45, 60, 90, and 120 deg.

*Throughout this bulletin "critical points" is to be taken to mean "points of minimum,maximum, and zero values."


03

w

g

o

hI


A A

SA(cos-cos s),(sinf-si

Ol if i==^=^S= ==0

0 30 60 90 1e0 160 /80Phase Angle 0 in De/rees

FIG. 4. AMPLITUDE AS FUNCTION OF PHASE ANGLE FOR SUM CURVE S

The properties of these wave forms are as follows:(a) The resultant curve S has the form of a unidirectional double

pulse with conspicuous critical points a, b, c, d, and e. The ordinatesmay be regarded as being composed of three components.

(b) The first component a, b, c is sinusoidal in character. Itsamplitude at b, denoted by So', depends upon the phase angle 0, asshown graphically in Fig. 4. This amplitude becomes zero for < = 0, 27r,etc., and: reaches a maximum 2Ao, or twice the amplitude of the

, original rectified sine pulse, for = zn, 3r, etc.'*: '(c) The second component c, d, e is also sinusoidal in character.

Its amplitude at d, denoted by So", also depends upon the angle 0.This amplitude is a maximum, 2Ao, at < = 0, 27r, etc., and reduces tozero for r = rr, 37r, etc.

(d) The third component a, c, e, represented by So'", may becalled the constant component, since it remains constant throughoutthe period of the pulse. Its magnitude is proportional to the phaseangle 0, as shown in Fig. 4. This component becomes zero for < = 0,7r, 27r, etc., while for p = 7r/2 it reaches a maximum value Ao = Bo.

(e) Tabulated in Table 2 are the equations for the abscissas X,the ordinates S, including the component amplitudes for each of thecritical points as dependent upon the phase angle 0.

PHASE-SHIFTED RECTIFIED SINE WAVES 13

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(f) Figure 4 shows three characteristic intersection points P, Q, andR. Point P signifies that the component amplitudes, So' = So'" atS= 120 deg., (P = 0.866). Point R indicates that So" = So'" at

= 60 deg., (R = 0.866). Finally, point Q marks the point at whichSo' = So" at 0 = 90 deg., (Q = 0.414).

(g) By assigning to the component pulses (S', S", and S'") periodsT', T", and T"', relative to the values To of the original full sine wave,relations can be expressed for each component period as enumeratedin the last column of Table 2.

5. Properties of Sine Waves Combined With Rectified Sine Pulses.-A further series of wave forms, M, is obtained by combining sinewaves with phase-shifted sine pulses of equal amplitudes Ao = Bo,either by addition or subtraction of their ordinates. Figure 5 representsthe wave shapes for phase angles, 0 = 0, 30, 60, 90, 120 deg., respec-tively. They are obtained by subtraction of the rectified wave B fromthe sine wave A, so that the resultant wave form is M = A - B.

The following properties of the curve M are described below:(a) The zero axis divides the curve into a positive and negative

part whose shapes and amplitudes vary depending upon the phaseangle. The critical points of the curve are marked by the pointsa, b, c, g, d, e, f, hi, and a1, where a and c are the points where thecurve crosses the axis. The part of the curve marked by points b, c, g,d, and e is most noticeably influenced by a change in the phase angle.Starting with p = 0 deg., when this part of the curve is a straight linecoinciding with the abscissa, this line with increasing phase angle isgradually formed into a convex-concave curve with an inflection pointat the point c on the abscissa axis. At angles approaching 4 = 90 deg.,the point d marking the change in direction becomes more and moreprominent until at 0 = 90 deg. the point e is at the terminus of twosymmetrical concave curves.

(b) A similar curve M, but reversed in phase 180 deg., is obtainedif the ordinates of the rectified sine wave B are added to the ordinatesof the full sine wave.

(c) The abscissas X of the critical points depend on the phaseangle 0. The corresponding functions are enumerated in Table 3.

(d) The ordinates of the critical points which differ from zero arealso functions of the phase angle p. They are all included in Table 3.The ordinates Mb, Md, Me, and Mf represent maxima and minima ofthe complex curve M. Their dependence on the phase angle 0 is showngraphically in Fig. 6. The intersection point L indicates that forS= 90 deg., the amplitudes Md Mf = 1.41.


0< c54z3 Is

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0

z0r/1l

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0

0

[SC7d* 0

dx2

(minimum).

By inserting x = 0/2 + 7 in (6b) we obtain for the minimumpoint, d, the value of the ordinate

Mmin'= Md = sin( - + 7r) - sin ( + 7 -

Md = -2 sin -.2


Next we explore for a minimum or maximum region three, xbetween (7 + 4) and (2r + 4). In this region the rectified phase-shifted wave B undergoes a change of sign. Equation (6) musttherefore be written

M = sin x + sin (x - 4). (6c)

By differentiating we find

dM-- = cos z + cos (x - 4)

dxcos x = -cos (x - 0)

3x =Xf= -7r + (29)

2 2d2M

S= -sin x - sin (x - ) > 0dx2 (minimum).

3By inserting x = -- + - in Equation (6c) we obtain for the

2 2second minimum the value of the ordinate

/3 /3 \Mmin"= Mf = sin -r +- + sin( -r +

\2 2/ \2 2

/3Mf = - 2 sin^ + -(2 2)

= -2 cos-. (30)2

Finally, we investigate the critical point

x = x, = 7 + . (31)

At this point the second and the third ranges meet, and the valuesof M given by Equations (6b) and (6c) are equal. From (6b) and(6c) it follows that the derivative of M changes sign from + to -as x increases through x = (r + 4). Hence the continuous function

46 ILLINOIS ENGINEERING EXPERIMENT STATION

M has a maximum value at x = (7r + 4). Inserting x = (r + 4)in either (6b) or (6c) we obtain

M = Me = sin (7 + 4) - sin (7 + 4 - 4)= sin (7 + 4) = -sin 4. (32)

By applying the procedure used in Section 18 for the calculationof the periods of three component pulses TM', TM", and TM"' rela-tive to the period, To = 2 7, of the original sine wave, we obtainfor TM'

T 7rTM" = X - -xa = -- +

2

TM' 2

To 2vTo

TM' = (33)4

for TM" r w 71 4TM = x, -x= = --

2 2 2 27r +

TM'" 2

To 27r

TM"' = To (34)4r

for TM"' d = - xTM'0 - Xa, - Xe= 2Fr+- e- - - = tr ---

2 2

TM"1' 2

To 2r

27 - )42

20. Relations Between Wave Forms.-The question of the charac-ter of the component parts of the D, S, and M curves can now beanswered.


Ang/es /7 Degrees

FIo. 15. SUBTRACTION AND ADDITION OF TWO FULL SINE WAVES

In Table 4 are listed the Equations (4a) to (6c) for the componentparts of the D, S, and M curves and the respective ranges of abscissaefor which they are valid. For better identification each componentpart is marked by letters referring to the graphical presentation inFigs. 1, 3, and 5. Comparison of the equations and the ranges for thefour component parts of the M curve with those for the D and S curvesleads to the following conclusions:

Wave form Mhab, Equation (6a), is identical with curve Dabe, Equation (4a)(in phase)

Wave form Mbeg, Equation (6b), is identical with curve Dcda, Equation (4b)(in phase)

Wave form Mgde, Equation (6b), is identicalwith curve Sabe, Equation (5a)(phase shifted 180 deg.)

Wave form Mefh, Equation (6c), is identical with curve Scde, Equation (5b)(phase shifted 180 deg.)

(6) Addifion I(F +F) for q= 60o

I I I I I

48 ILLINOIS ENGINEERING EXPERIMENT STATION

TABLE 4COMPARISON OF D, S, AND M CURVES

WaveForm

D

D

S

S

M

MM

PointsMarking

Comp.Parts

a, b, c

c, d, ai

a, b, c

c, d, e

h, a, b

b, c,g

g, d, e

e,f, hi

Fig.

1

1

3

3

5

5

5

5

Rangeof

Abscissa

0 x < 4,

0:5 x 5

o__. x& 4r

(< X< (5 2O0 x-< 2

Equations forComponent

Parts

sin x + sin(x - 4,)sin x- sin(x - 4)sin x - sin(x - 4)sin x + sin(x - 4)sin x + sin(x - 4)sin x - sin(x - 4,)sin - sin(x - 4,)sin x + sin(x - 4)

Equa-tion

(4a)(4b)(5a)(5b)(6a)(6b)(6b)(6c)

Iden-tity

Marks

0

Ax

0

AX

WaveForm

FS

FD

FD

FS

FS

FD

FD

FS

PointsMarking

Comp.Parts

i, i, k

s, t, u

9, V, Ug, r, a

k, 1, m

i, j, ks, t, u

u, v, w

m, n, p

Fig.

15b

15a

15a

15b

15b

15a

15a

15b

Further proof of the identity is derived by comparing the expres-sions for the amplitudes Mb, Md, and Mf in Table 3 with the corre-sponding expressions for Dc in Table 1, and with the expressions forSb and So in Table 2. Thus the equality of the respective absolutevalues of the amplitudes are obtained, namely

Mb = Dc; Md = Sb; and M, = Sd.

By the way of synthesis all the component parts enumerated on theleft side of Table 4 may be combined to form a difference curve FDand a sum curve FS of two full sine waves (FA and FB respectively)shifted in phase by an angle 0b. The corresponding expressions for thesewave forms are

FD = sin x - sin (x -


VI. APPLICATIONS

21. Review of Applications.-The main field of applications of thewave forms described in the preceding chapters relates to the meas-urement of the phase angle between two sine waves throughout thelower, the audio, and the radio frequency spectrums. Other applica-tions include the use of these new wave forms in square wave oscilla-tors, harmonic oscillators, frequency multipliers, and in sweep circuitsfor use in oscillographs, television, and electronic time-control devices.

The various applications and the forms in which they are carriedout, will be.described elsewhere. Only the principles and some of theparticular methods of phase measurements are included in the fol-lowing discussion.

22. Oscillographic Method of Phase Angle Measurement UsingDifference Curve D.-A simple method of phase angle measurementsusing the difference curve D consists in observing and evaluating thecoordinates of the critical points of the wave forms traced on the screenof the oscillograph, making use of the relations (8b) and (7b) inTable 1 (see also Fig. 1). It can be seen that the amplitudes of thedifference curve are proportional to the sine of the phase angle P. Bymeasuring on the oscillograph screen the length cc, where c, is theintersection of a perpendicular dropped from the maximum point C tothe X axis and measuring the amplitude of the rectified sine wave,A, the phase angle in radians is given by the relation

eel eel= sin 4 or 4 = sin - 1

A A

A more direct and accurate method makes use of the relation (9a)in Table 1 (see Fig. le). A perpendicular line is projected from themaximum point c to the intersection ci on the X axis. By measuringthe lengths bc and bd, the phase angle in radians is given by therelation

bc17r = v -=.bd

Since the points b, c,, and d are well defined, these distances can befairly accurately determined and the angles computed from the fore-


going relation. The limitations of accuracy are set by the sharpnessof the focus of the electronic beam and the linearity of the time axisof the sweep circuit.

23. Oscillographic Method of Phase Angle Measurement Using SumCurve S.-The oscillographic method becomes even more simplified ifthe sum curve S (Fig. 3) is used in place of the difference curve D.Making use of the relations (17) and (15) in Table 2 it follows thatthe length ac is related to the phase angle as the length ae is relatedto r-. Therefore

ac0 = 7r --.ae

The points a, c, and e are very well defined, and the respectivedistances between them can be easily measured. Therefore, the phaseangle q can be determined accurately.

Any of the other relations valid for the critical points, as enumer-ated in Table 2, may be used. For example, from relation (18) itfollows that since the d.c. component So'" is proportional to sin q, thephase angle can be determined by obtaining the distance of any ofpoints a, c, or e to the X axis, and using the relation

SO,,,4 = sin-'

Ao

24. Oscillographic Method of Phase Angle Measurement UsingComposite Curve M.-In this case the critical points b and e are themost practical points for accurate determination of the phase angle.The relation (25) in Table 3 gives the maximum ordinate bb 1, whichis proportional to the sin 0. This property is similar to that mentionedin Section 21 for the maximum ordinate of the difference curve D.Therefore, knowing the lengths A and bb6, the phase angle is com-puted from

bbl4 = sin-1 .A

Another method consists in measuring the relative lengths ab, andac and using the relation

abi


The critical points e and g may also be applied for phase anglesbetween 30 and 90 deg. In this case a line parallel to the X axis isdrawn between the points e and h. The distance ge is related to thephase angle 0 as the length hg is related to 7. Therefore

gehg

25. Phase Angle Measurement by Means of Direct-Reading In-struments.-A direct-reading method has been developed in order toavoid the necessity for measuring lengths on the oscillographic screenand of making computations to obtain the phase angles. In this con-nection use is made of any of the simple relations enumerated inTables 1, 2, and 3, so that known commercial indicating meters can beadapted to the mixing circuits shown in Figs. 7, 9, and 10. Indi-cating instruments such as thermionic voltmeters for measuring crestvoltages, or a.c. instruments for measuring the integrated values of theD curve, or galvanometers for measuring the d.c. component of the Scurve are all equally applicable.

In all cases the condition must be fulfilled that the amplitudes,A and B, of the rectified pulses be of equal magnitudes. This makes itnecessary that current-limiting devices be installed for keeping theseamplitudes constant and equal to each other.

A detailed account of these methods and devices will be publishedelsewhere.

VII. RiSUME OF INVESTIGATION

26. Summary.-The researches described in this bulletin werepartly theoretical and partly experimental. The task of the theoreticalinvestigation was to find and to formulate composite wave forms madeup of component sine waves, having sharply marked critical points,and also to derive relations which could be used to determine the phaseangles of the component waves.

The experimental part consisted of developing methods of produc-ing such composite wave forms and utilizing them for the determina-tion of phase angles of any two given phase-shifted sine waves of likefrequency. Further, the investigation necessitated the development ofaccurate methods of producing arbitrarily adjustable phase shifts.

These researches finally led to a number of applications which mayprove useful in the field of electronics.


27. Results.-The results of the investigation may be stated asfollows:

(1) By combining two phase-shifted rectified sine waves, newwave forms have been obtained. It was found that the critical pointsof these wave forms depend upon the phase angle between the com-ponent rectified sine waves.

(2) Three types of such wave forms have been investigated:(a) those obtained by the subtraction of two rectified phase-

shifted sine waves,(b) those obtained by the addition of two rectified phase-

shifted sine waves,(c) those obtained by a rectified phase-shifted sine wave

added to or subtracted from a full sine wave.(3) Expressions for the coordinates of the critical points have been

formulated for all three types of wave forms.(4) The variation of the wave shapes as a function of the phase

angle has been investigated graphically and analytically.(5) Various methods have been experimentally studied for the

production of phase-shifted rectified sine waves, each adaptable fora definite frequency range.

(6) For the low frequency range, from 30 to 250 cycles per second,the method of vector addition in quadrature relation has been adoptedfor producing phase-shifted sine waves.

(7) A method based on sound radiation has been developed forphase-shifting in the audio frequency range from 250 to 20 000 cyclesper second.

(8) For the radio frequency range between the limits of 20 000and 100 000 cycles per second the method of phase-shifting by vary-ing the constants of the circuits coupled to the oscillator has beenapplied.

(9) Circuits for the observation of the new wave forms have beendevised and applied.

(10) The properties of the wave shapes have been experimentallyverified by means of oscillograms.

(11) By utilizing the properties of the critical points, oscillographicmethods of determining phase differences have been developed for allthree types of wave forms.

(12) Direct-reading methods of measuring phase differences havebeen devised.

RECENT PUBLICATIONS OFTHE ENGINEERING EXPERIMENT STATIONt

Reprint No. 12. Fourth Progress Report of the Joint Investigation of Fissuresin Railroad Rails, by H. F. Moore. 1938. None available.

Bulletin No. 307. An Investigation of Rigid Frame Bridges: Part I, Tests ofReinforced Concrete Knee Frames and Bakelite Models, by Frank E. Richart,Thomas J. Dolan, and Tilford A. Olsen. 1938. Fifty cents.

Bulletin No. 308. An Investigation of Rigid Frame Bridges: Part II, Labora-tory Tests of Reinforced Concrete Rigid Frame Bridges, by W. M. Wilson, R. W.Kluge, and J. V. Coombe. 1938. Eighty-five cents.

Bulletin No. 309. The Effects of Errors or Variations in the Arbitrary Con-stants of Simultaneous Equations, by George H. Dell. 1938. Sixty cents.

Bulletin No. 310. Fatigue Tests of Butt Welds in Structural Steel Plates, byW. M. Wilson and A. B. Wilder. 1939. Sixty-five cents.

Bulletin No. 311. The Surface Tensions of Molten Glass, by Cullen W.Parmelee, Kenneth C. Lyon, and Cameron G. Harman. 1939. Fifty-five cents.

Bulletin No. 312. An Investigation of Wrought Steel Railway Car Wheels:Part I, Tests of Strength Properties of Wrought Steel Car Wheels, by Thomas J.Dolan and Rex L. Brown. 1939. Seventy cents.

Circular No. 36. A Survey of Sulphur Dioxide Pollution in Chicago andVicinity, by Alamjit D. Singh. 1939. Forty cents.

Circular No. 37. Papers Presented at the Second Conference on Air Condition-ing, Held at the University of Illinois, March 8-9, 1939. 1939. Fifty cents.

Circular No. 38. Papers Presented at the Twenty-sixth Annual Conference onHighway Engineering, Held at the University of Illinois, March 1-3, 1939. 1939.Fifty cents.

Bulletin No. 313. Tests of Plaster-Model Slabs Subjected to ConcentratedLoads, by Nathan M. Newmark and Henry A. Lepper, Jr. 1939. Sixty cents.

Bulletin No. 314. Tests of Reinforced Concrete Slabs Subjected to Concen-trated Loads, by Frank E. Richart and Ralph W. Kluge. 1939. Eighty cents.

Bulletin No. 315. Moments in Simple Span Bridge Slabs with Stiffened Edges,by Vernon P. Jensen. 1939. One dollar.

Bulletin No. 316. The Effect of Range of Stress on the Torsional FatigueStrength of Steel, by James O. Smith. 1939. Forty-five cents.

Bulletin No. 317. Fatigue Tests of Connection Angles, by Wilbur M. Wilsonand John V. Coombe. 1939. Thirty-five cents.

Reprint No. 13. First Progress Report of the Joint Investigation of ContinuousWelded Rail, by H. F. Moore. 1939. Fifteen cents.

Reprint No. 14. Fifth Progress Report of the Joint Investigation of Fissures inRailroad Rails, by H. F. Moore. 1939. Fifteen cents.

Circular No. 39. Papers Presented at the Fifth Short Course in Coal Utiliza-tion, Held at the University of Illinois, May 23-25, 1939. 1939. Fifty cents.

Reprint No. 15. Stress, Strain, and Structural Damage, by H. F. Moore.1940. None available.

Bulletin No. 318. Investigation of Oil-fired Forced-Air Furnace Systems inthe Research Residence, by A. P. Kratz and S. Konzo. 1939. Ninety cents.

Bulletin No. 319. Laminar Flow of Sludges in Pipes with Special Reference toSewage Sludge, by Harold E. Babbitt and David H. Caldwell. 1939. Sixty-five cents.

Bulletin No. 320. The Hardenability of Carburizing Steels, by Walter H.Bruckner. 1939. Seventy cents.

Bulletin No. 321. Summer Cooling in the Research Residence with a Con-densing Unit Operated at Two Capacities, by A. P. Kratz, S. Konzo, M. K. Fahne-stock, and E. L. Broderick. 1940. Seventy cents.

Circular No. 40. German-English Glossary for Civil Engineering, by A. A.Brielmaier. 1940. Fifty cents.

Bulletin No. 322. An Investigation of Rigid Frame Bridges: Part III, Testsof Structural Hinges of Reinforced Concrete, by Ralph W. Kluge. 1940. Forty cents.

Circular No. 41. Papers Presented at the Twenty-seventh Annual Conferenceon Highway Engineering, Held at the University of Illinois, March 6-8, 1940. 1940.Fifty cents.

tCopies of the complete list of publications can be obtained without charge by addressing theEngineering Experiment Station, Urbana, Ill.


Reprint No. 16. Sixth Progress Report of the Joint Investigation of Fissures inRailroad Rails, by H. F. Moore. 1940. Fifteen cents.

Reprint No. 17. Second Progress Report of the Joint Investigation of Con-tinuous Welded Rail, by H. F. Moore, H. R. Thomas, and R. E. Cramer. 1940.Fifteen cents.

Reprint No. 18. English Engineering Units' and Their Dimensions, by E. W.Comings. 1940. Fifteen cents.

Reprint No. 19. Electro-organic Chemical Preparations, Part II, by SherlockSwann, Jr. 1940. Thirty cents.

Reprint No. 20. New Trends in Boiler Feed Water Treatment, by F. G. Straub.1940. Fifteen cents.

Bulletin No. 323. Turbulent Flow of Sludges in Pipes, by H. E. Babbitt andD. H. Caldwell. 1940. Forty-five cents.

Bulletin No. 324. The Recovery of Sulphur Dioxide from Dilute Waste Gasesby Chemical Regeneration of the Absorbent, by H. F. Johnstone and A. D. Singh.1940. One dollar.

Bulletin No. 325. Photoelectric Sensitization of Alkali Surfaces by Means ofElectric Discharges in Water Vapor, by J. T. Tykociner, Jacob Kunz, and L. P.Garner. 1940. Forty cents.

Bulletin No. 326. An Analytical and Experimental Study of the HydraulicRam, by W. M. Lansford and W. G. Dugan. 1940. Seventy cents.

Bulletin No. 327. Fatigue Tests of Welded Joints in Structural Steel Plates, byW. M. Wilson, W. H. Bruckner, J. V. Coombe, and R. A. Wilde. 1941. One dollar.

Bulletin No. 328. A Study of the Plate Factors in the Fractional Distilla-tion of the Ethyl Alcohol-Water System, by D. B. Keyes and L. Byman. 1941.Seventy cents.

*Bulletin No. 329. A Study of the Collapsing Pressure of Thin-Walled Cylinders,by R. G. Sturm. 1941. Eighty cents.

*Bulletin No. 330. Heat Transfer to Clouds of Falling Particles, by H. F. John-stone, R. L. Pigford, and J. H. Chapin. 1941. Sixty-five cents.

*Bulletin No. 331. Tests of Cylindrical Shells, by W. M. Wilson and E. D. Olson.1941. One dollar.

*Reprint No. 21. Seventh Progress Report of the Joint Investigation of Fissuresin Railroad Rails, by H. F. Moore. 1941. Fifteen cents.

*Bulletin No. 332. Analyses of Skew Slabs, by Vernon P. Jensen. 1941. Onedollar.

*Bulletin No. 333. The Suitability of Stabilized Soil for Building Construction,by E. L. Hansen. 1941. Forty-five cents.

*Circular No. 42. Papers Presented at the Twenty-eighth Annual Conference onHighway Engineering, Held at the University of Illinois, March 5-7, 1941. 1941.Fifty cents.

*Bulletin No. 334. The Effect of Range of Stress on the Fatigue Strength ofMetals, by James O. Smith. 1942. Fifty-five cents.

*Bulletin No. 335. A Photoelastic Study of Stresses in Gear Tooth Fillets, byThomas J. Dolan and Edward L. Broghamer. 1942. Forty-five cents.

*Circular No. 43. Papers Presented at the Sixth Short Course in Coal Utiliza-tion, Held at the University of Illinois, May 21-23, 1941. 1942. Fifty cents.

*Circular No. 44. Combustion Efficiencies as Related to Performance of DomesticHeating Plants, by Alonzo P. Kratz, Seichi Konzo, and Daniel W. Thomson. 1942.Forty cents.

*Bulletin No. 336. Moments in I-Beam Bridges, by Nathan M. Newmark andChester P. Siess. 1942. One dollar.

*Bulletin No. 337. Tests of Riveted and Welded Joints in Low-Alloy StructuralSteels, by Wilbur M. Wilson, Walter H. Bruckner, and Thomas H. McCrackin.1942. Eighty cents.

*Bulletin No. 338. Influence Charts for Computation of Stresses in ElasticFoundations, by Nathan M. Newmark. 1942. Thirty-five cents.

*Bulletin No. 339. Properties and Applications of Phase-Shifted Rectified SineWaves, by J. Tykocinski Tykociner and Louis R. Bloom. 1942. Sixty cents.

*A limited number of copies of bulletins starred are available for free distribution.

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Phase-shifted Rectified Sine Waves

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