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 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAIGN 
 
 5io.e>^ 
 
 r\o.2a>7-22>2 
 
 cop. 2 
 
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 in 2013 
 
 http://archive.org/details/calibrationofdop292lend 
 
' «- ' V s 
 
 J/& r Report No. 292 
 
 /f/a~'C*A- 
 
 T 
 
 CALIBRATION OF DOPPLER RADAR USING TUNING FORKS 
 
 by 
 
 Lester Martin Lendrum 
 
 September 1, 1968 
 
 NOV 9 1972 
 
 UNIVERSITY Or ILLINOIS 
 AT URBANA-CHAMPAIGN 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
Report No. 292 
 
 CALIBRATION OF DOPPLER RADAR USING TUNING FORKS* 
 
 by 
 Lester Martin Lendrum 
 
 September 1, 1968 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 Submitted in partial fulfillment of the requirements for the 
 degree of Master of Science in Electrical Engineering in the 
 Graduate College of the University of Illinois, 1968. 
 
Ill 
 
 ACKNOWLEDGMENT 
 
 The author wishes to thank Professor T. A. Murrell for his 
 patience and guidance in the preparation of this paper. The generosity 
 of Muni Quip Corporation in supplying the Radar Timer was greatly 
 appreciated. The author also wishes to express appreciation to the 
 University Research Board for providing funds for computer computations. 
 Special thanks go to Miss Marian Buch whose help in typing and proof- 
 reading the preliminary drafts was invaluable. 
 
IV 
 
 TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION 1 
 
 2. TARGET SPEED MEASUREMENT BY DOPPLER SHIFT 3 
 
 3- ANALYSIS OF MIXING AND DETECTING PROCESSES 7 
 
 k. THEORETICAL DETECTOR OUTPUTS Ik 
 
 5- EXPERIMENTAL VERIFICATION 17 
 
 6. SUMMARY 29 
 
 7. CONCLUSIONS 31 
 
 BIBLIOGRAPHY 32 
 
 APPENDICES 33 
 
1. INTRODUCTION 
 
 There are in use today many radar units which depend on the 
 Doppler shift in frequency to measure the velocity of a moving object. 
 In most instances, such as measurement of speeds of vehicles on a 
 highway, the difference frequency falls in the audio range (Appendix A) 
 When this is the case, a convenient and commonly used method of calibra- 
 tion consists of placing a vibrating tuning fork close to the antenna 
 and in the direct path of the radiation, and adjusting the conversion 
 circuits to bring the indicated speed into agreement with the reading 
 corresponding to the tuning fork frequency. In spite of the advantages 
 afforded by this method, no investigations in this area are available. 
 
 The purpose of this study is to evaluate the reliability of 
 tuning fork calibration. In this respect, the operation of the radar 
 unit will be analyzed for normal use in regard to reading speeds and for 
 calibrating conditions. Therefore, the objectives of this study are to: 
 
 1. Analyze the process of speed measurement by Doppler radar. 
 
 2. Analyze the mixing and detecting processes in the particular 
 radar unit under study. 
 
 3. Predict the theoretical detector output wave forms for the 
 following cases: 
 
 a) where the reflecting object is moving at a constant rate 
 of speed. 
 
 b) where the reflecting object is a vibrating tuning fork. 
 h. Correlate the theoretical detector outputs with actual 
 
 detector outputs for the case of a vibrating tuning fork. 
 
2 
 5 . Show that the signal generated by a vibrating tuning fork 
 
 is a satisfactory calibration signal for Doppler radar units. 
 
2. TARGET SPEED MEASUREMENT BY DOPPLER SHIFT 
 
 In normal usage, the radar beam, usually shaped to a width of a 
 few degrees by the antenna assembly, is aimed at the target and 
 radiation from the target is received by the same or another antenna. 
 
 As shown in the block diagram of Figure 1, a small portion of 
 the transmitted frequency is fed to a mixer, where it is added to the 
 signal received from the moving object. For electromagnetic radar units, 
 operating at higher 10,525 mHz. or 2,^50 mHz., the radio frequency energy 
 is fed directly to the mixer, which consists of a length of waveguide or 
 transmission line. The detector is usually a semiconductor crystal 
 diode designed to respond to these high frequencies. To a good 
 approximation, the output of the detector is proportional to the square 
 of the envelope of the mixed wave (square-law detector). 
 
 The various waveforms of interest are shown in Figure 2. Mixing 
 the two sinusoidal R.F. waves produces an interference pattern with 
 envelope variations at the difference frequency, Af . Although the 
 envelope variation is not sinusoidal, (it is the square root of a 
 sinusoidal), the square law detector will generate a sinusoidal output. 
 The output of the detector is amplified and then converted into a square 
 wave which is at the same frequency. The square wave is then clipped 
 and differentiated to form a pulse train of period t = l/Af . The pulse 
 train is then fed into the conversion circuit which produces a direct 
 current proportional to the number of pulses per second and hence to the 
 difference frequency, Af. The square-wave convertor, the differentiator, 
 and the pulse to direct current convertor are used to be certain that 
 

 
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 TIME (X10 1 ) 
 
 80,00 
 
 Figure 2. Waveforms of Doppler Radar Unit. 
 
6 
 the speed indication is a function only of the difference frequency 
 and not of the amplitude of the received signal. 
 
 Thus, using this method, the Doppler difference frequency Af 
 controls the current through a D.C. meter calibrated in miles per hour. 
 By adjusting the circuit values, the meter can be calibrated to read 
 10 mph for a difference frequency of 31^ Hz., 20 mph for a difference 
 frequency of 628 Hz., etc. . Thus, the meter indicates the velocity of 
 the moving object by measuring the frequency of the Doppler shift it 
 causes . 
 
 Since it is the signal output from the amplifier section which 
 contains the information of interest, the remaining circuitry will not 
 be discussed. The relationship under consideration is that existing 
 between the movement of the target and the output of the amplifier 
 section. With this in mind, the mixing and detecting processes will be 
 analyzed in detail. 
 
3. ANALYSIS OF MIXING AND DETECTING PROCESSES 
 
 The particular radar unit under study operates at 10,525 mHz. 
 The mixer in this case is a length of RG 52/w waveguide. The detector is 
 a 1N23C semiconductor crystal diode. 
 
 3.1 Mixing 
 
 The dominant mode in rectangular guides such as RG 52/w is TE . 
 
 i, u 
 
 The transverse electric field is the quantity of interest since diode 
 
 detectors respond to this field only. The electric field of a TE mode 
 
 2 
 
 in a waveguide is given by: 
 
 e j(o>t + e) e -j/(jr/a) - O) [i.£ z (l) 
 
 v.here K is a constant depending on the strength of the signal, 9 is the 
 time phase angle with respect to some reference, co is the radian 
 frequency of the signal, p. is the permeability of air, e is the 
 permittivity of air, a is the width of the waveguide, and z is the 
 distance along the axis of the guide. This wave is propagating in the 
 positive z-direction. 
 
 The mixer consists of an antenna coupled to a shorted section of 
 waveguide as shown in Figure 3. 
 
 1. E. C. Jordan, Electromagnetic Waves and Radiating Systems , p. 289- 
 
 2. Ibid., p. 267. 
 
Figure 3« Mixer and Detector 
 
 If an R. F. signal is received by the antenna, a TE.. n wave propagating 
 
 1, 
 
 to the right is set up in the guide. Let the electric field of this 
 wave be E . Since the waveguide is shorted at z = 0, the incident wave 
 E T is reflected totally; the reflected wave is E . The incident 
 electric field may be represented by 
 
 / 2 2 
 
 j(co t + 9 ) -j/(jt/a) - oo n ue z 
 
 E (t,z) = K co Q e ° ° e * 
 
 u & 
 
 (2) 
 
 where K depends on the strength of the signal, and 9 is the time phase 
 angle referenced to the R.F. source. 
 
 The reflected wave, E_.> is propagating in the negative 
 
 n 
 
 z-direction and its magnitude is the negative of that of E , since 
 
 E (t,0) + E (t,0) = (a conducting surface). Thus 
 -i- R 
 
 2 2 
 j(co t + e ) j/(n/a) - co ^ 
 
 E R (t,z) „ -K co e ° ° e" 
 
 (3) 
 
9 
 The incident and reflected electric field form a standing wave 
 pattern in the guide. The resultant electric field in the guide is the 
 sum of the incident and reflected fields. 
 
 ^esultant^' 2 ) = E I (t ' z) + \ (t ' z) 
 
 3(<D.t+eJ 
 
 = -2jK^o> e slnj{-n/a)^ - ay-ie z (k) 
 
 The diode detector is mounted in the guide at a fixed positon z_, 
 as shown in Figure 3. This position is chosen in the design of the -unit 
 such that sin is fairly close to unity. Since the electric field of 
 interest is that at the position of the detector, the above equation may 
 be simplified to: 
 
 j(o> n t+e ) 
 ^esultant^ " K ' e (5) 
 
 This is the electric field at the detector for a single incident 
 
 wave. If more than one incident wave is present, superposition is used 
 
 to find the total electric field in the guide. 
 
 In the following calculations, all frequency shifts will be 
 
 JOXj_t 
 treated as continuous phase shifts; that is, a signal having e 
 
 j(a) t + e(t)) 
 
 time dependence will be written as having e time dependence, 
 
 where 9(t) = (a*, ~<^ n H i- n this case (Appendix B) . 
 
 In Doppler type radar units using a waveguide mixer-detector as 
 described above, energy from the source is fed to the mixer -detector by 
 two paths (Figure h) . A small reflector, R, is build into the antenna 
 assembly to send a small portion of the source energy into the mixer- 
 detector (path A). This causes a standing wave to be set up in the 
 
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11 
 
 mixing guide and the resulting electric field E is of the same frequency 
 ou, as the source. 
 
 El (t,z ) = K ± e ° (6) 
 
 where 9 n is the fixed phase angle of E with respect to the source which 
 is due to the propagation delay over the fixed path A. 
 
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 reflected into the mixer-detector along path B. This causes a second 
 standing wave in the guide, 
 
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 where varies if the length of path B varies . In fact, 6=9 + — — - 
 where 9 is the phase angle between E and the source when d = 0, X is the 
 wavelength corresponding to angular frequency co n , and d is the distance 
 measured along the axis of the antenna system. Thus, the total electric 
 field in the guide is E + E by superposition. 
 
 j(a) n t+9 ) «i(o) o t40 + — ) 
 E totel = V + V C X W 
 
12 
 
 3.2 Detecting 
 
 The semiconductor crystal diode is positioned in the guide so that 
 
 the electric field, E^ .. is along the axis of the diode holder. The 
 
 ' total D 
 
 electric field, E, and the diode current i, are related by the following 
 formula: i = A(e - 1), where A and OL depend on the particular diode 
 
 used. Diodes of this type are referred to as square-law since the filtered 
 
 3 
 output signal is proportional to the square of the input signal. Thus, 
 
 in the case discussed here, the filtered output signal varies as the 
 
 square of the amplitude of E , . 
 
 xo"Ca_L 
 
 A convenient method of representing the filtered detector output 
 
 k 
 
 is vector representation. Let the signal reaching the receiver directly 
 
 from the source be taken as a reference. This signal will be represented 
 by a vector e , of length |E | and referenced horizontally to the right 
 (Figure 5). 
 
 Figure 5. 
 
 3. R. V. Pound, Microwave Mixers , p. 19. 
 
 h. D. Luck, Frequency Modulated Radar , p. 12. 
 
13 
 The signal returned from the target is represented by vector e , where 
 |E | is the magnitude of the returned signal and is the phase angle 
 with respect to e . The vector e is the sum of e and e . Let 
 
 J_ \> J- C. 
 
 e l = VW- S 2 " VW an<J 8 t " E totar Then l\l = h> l E 2 l " K 2> 
 and 9=9 - 9 + btd/X; see Equations 6, 7, and 8. Thus, the magnitude 
 
 of vector e , |E corresponds to the magnitude of E . By the law of 
 
 "G X XOX3J- 
 
 ^l 2 = |E 1 | 2 + |E 2 | - 2|E 1 | |E 2 | cos * (9) 
 
 Since 
 
 <J> = « - 9, cos <t> - -cos 9: 
 yielding 
 
 l^l 2 = lEj 2 + |E 2 | 2 + 2| El | |E 2 | cos (9 c - 9 Q + ^) (10) 
 
 Since the detector output is the square of the envelope of the total 
 electric field, the final result is 
 
 DETECTOR OUTPUT = IeJ 2 + IeJ'" + 2 1 E_ I |E_| cos(9 -0_ + -7% (ll) 
 
 '1' '2' ' 1' ' 2' c X 
 
11+ 
 
 k. THEORETICAL DETECTOR OUTPUTS ' 
 
 Using the procedures developed in the last section, the detector 
 outputs have been related to d, the distance between target and antenna. 
 If this distance varies with time, then the detector output will also 
 vary with time. In this section, the detector output is determined for 
 two cases: first, the distance decreasing or increasing at a constant 
 rate with respect to time (corresponding to the Doppler shift caused by 
 a constant velocity target either closing or moving away); second, the 
 distance varying sinusoidally about a fixed position (corresponding to 
 reflection from a vibrating tuning fork). 
 
 k.l Doppler Shift 
 
 Without loss of generality, only the case in which the target is 
 moving directly away from the antenna assembly at a fixed velocity, 
 v, is discussed. Thus, d = d + vt where d is the distance between the 
 antenna and target at t = 0. In Equation 11, the argument of the cosine 
 becomes 9' + 1+itvt/X where 1 = 9 - 9 + kitd.^\. 
 
 DETECTOR OUTPUT = C + C C0S(9' + 1+jrvt/X) (12) 
 
 This corresponds to the square law detector output as shown in Figure 2. 
 The time dependence of the cosinusoid is 4rcvt/X, which, since ^— = f , is 
 
 A. (J 
 
 2f v 
 equivalent to , the Doppler difference frequency if the direction of 
 
 c 
 
 travel of the target is neglected (Appendix A). This result agrees with 
 
 the discussion in Section II. 
 
15 
 
 k.2 Vibrating Tuning Fork . 
 
 If the reflecting surface is a vibrating tuning fork, the 
 distance, d, varies about a fixed point d . This oscillation is very 
 nearly sinusoidal and therefore d = d + 8 cos w t, where 8 is the 
 maximum distance the tines of the tuning fork move from d and to is 
 the angular frequency of the tuning fork oscillation. In Equation 11, 
 the argument of the cosine becomes ©" + hnd^'k + (hit&/\) cos CD„t, where 
 9" = 9 - 9 . Thus, the detector output is given by, 
 
 DETECTOR OUTPUT = C* + C" cos (9" + kxdJ\ + (kiib/\) cos to t) 
 
 (13) 
 
 If — =r— is small enough, a very close approximation of the expressions 
 can be realized by expanding the cosine and sine functions in a power 
 
 series. 
 
 [(*W5/X) cos to t] 2 
 cos [ ( i +Jt8/X)cos to f t] = 1 - ^ 
 
 [(kn&/\) cos to t] 
 
 + m - • • • 
 
 [(^&A cos 03x.t] 
 
 sin [(JfrtsA) cos co«t] = -^— cos to^t - 3J 
 
 I At 
 
 [(iwoA)cos at] 
 + - . . . 
 
 (1*0 
 
 3 
 
 5-' 
 
 (15) 
 
 Using only the first two terms of Equation 14, and only the first term 
 of Equation 15, the expression for the detector is approximated 
 
 by, 
 
16 
 DETECTOR OUTPUT - C* + C" (l - kx 2 £/l?) cos (9'* + ka<lJ\) 
 
 - C" -~ sin (0" + Wq/X) cos ait 
 
 . c" -2LL- C os (e" + W^x) cos 2co f t (16) 
 
 A 
 
 The true expression for the detector output involving Bessel functions 
 is given here for completeness: 
 
 TRUE DETECTOR OUTPUT = C f 
 
 00 
 
 + C" cos (0" + ta^X) [J Q (Ujto/X) + 2 Z (_i) k j {k«b/\) cos (2kto f t] 
 
 k=l 
 
 - C" sin (0" + kmJX) [2Z (-1) J 2k+1 (^«5/X) cos [ (2k+l)<D t] ] (17) 
 
 k=0 
 
 Actually, the coefficients of the first and second harmonic terms are 
 very close in the two expressions. The only difference between 
 Equations 16 and 17 resides in the higher harmonics, and for all 
 practical purposes, the harmonics higher than the second are so small 
 as to be unobservable for any reasonable value of 5. Thus, the formula 
 of Equation 17 is sufficient for the purpose, of this discussion. 
 
 Comparing the detector output for a target moving at a constant 
 velocity (Equation 13) with that for a vibrating tuning fork (Equation 
 16), it is evident that the only significant difference is the second 
 harmonic component of Equation 16. Thus, if it can be shown that the 
 second harmonic component caused by the vibrating tuning fork does not 
 cause incorrect indications on the output meter, then the tuning fork 
 method of calibration must be considered valid. The following section 
 describes experiments performed to show this. 
 
17 
 5. EXPERIMENTAL VERIFICATION 
 
 The radar unit used for experimental work was manufactured by- 
 Muni Quip Corporation of Decatur, Illinois, (Radar Timer Series 6h). 
 The circuit diagrams of this unit are shown in Figures 6 and 7* 
 
 Two separate experiments were performed: first, measurement of 
 the actual detector output when a vibrating tuning fork is placed in 
 front of the antenna to verify the results of Sections II and III, 
 especially Equation 16; second, determination of the maximum amount of 
 second harmonic distortion which can be tolerated in the detector output 
 without affecting the accuracy of the conversion circuitry. A secondary 
 observation of the maximum displacement of the tines of the vibrating 
 tuning forks was also made; however, this observation was made only 
 to determine an upper bound on 8, and therefore no attempt was made 
 to measure it precisely. 
 
 5-1 leasurement of Detector Output . 
 
 Although measurements at single points were made with actual 
 tuning forks, it was necessary to measure the first and second harmonics 
 of the detector output for many values of d , the distance between the 
 target and antenna. It is desirable that the value of 8 be constant 
 throughout each set of distances d . This was impossible to achieve 
 with a tuning fork since the oscillations are damped by the air 
 surrounding the fork and by internal friction. To provide a vibrating 
 surface which oscillated at a constant frequency and amplitude, a small 
 metal disk was attached to the cone of a permanent magnet speaker and 
 the speaker was excited by an oscillator-amplifier system. An optical 
 
18 
 
 bench and micrometer positioning drive were used to maintain alignment 
 and to determine the distance d precisely (Figure 8). 
 
 The signal observed was that present at point A, Figure 6. This 
 point was chosen because the signal there is unaffected by any external 
 adjustment of the RANGE or CALIBRATE controls. The previous stages, 
 TR1 and TR2, are frequency compensated amplifiers. 
 
 The signal was monitored with frequency selective VTVM's 
 (Hewlett-Packard Model 302A Wave Analyzers), one adjusted to measure 
 the first harmonic, V , of the speaker frequency, the other adjusted 
 to measure the second harmonic, V p . The magnitude of the first and 
 second harmonics were recorded for 66 values of d , each d differing 
 from the previous one by \/k millimeter. This gave a total change in 
 d of 1.625 centimeters. A typical set of measured values is given in 
 Table 1. A series of data sets like that of Table 1, were averaged 
 and plotted by computor (Appendix C) along with the same quantities 
 calculated from Equation 16. This plot is shown in Figure 9« 
 
 Difficulties in obtaining data resulted from radio frequency 
 noise caused by electrical equipment in the laboratory and from stray 
 reflections from the optical bench and speaker frame. To reduce the 
 effect of stray reflections, the optical bench and the speaker assembly 
 were masked with a microwave absorbing material. A small channel was 
 made through the speaker assembly mask directly in front of the 
 vibrating disk. As a result of these precautions, the only moving 
 reflector in the radiation beam was the vibrating disk. 
 
19 
 It will be noted that no measured voltage was less than 2 mV 
 RMS. This residual voltage might be caused by noise voltage present 
 in the system due to crystal noise and stray pick-up. However, the 
 most likely cause of the failure of the second harmonic voltage 
 to reach zero is a second harmonic component generated by the square- 
 law detector. An ideal square-law detector will generate an output 
 proportional to the square of the input, but if the detector is not 
 perfectly square-law (most crystal detectors are not), harmonics 
 of the input signal may also be generated. Only 2. 5 percent second 
 harmonic generation would be required to give the results found 
 experimentally. 
 
 After finding similar results for other frequencies and for 
 other values of 5, it was apparent that Equation 16 accurately 
 describes the physical situation. 
 
 1. H. C Torrey and C. A. Whitman, Crystal Rectifiers, p. 3- 
 
20 
 
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21 
 
 HOLD METER DIAGRAMS 
 
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 FOR NT MODELS 
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 SERIES 64 S 63 
 
 ANTENNA DIAGRAM 
 
 
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22 
 
 
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24 
 
 TABLE 1 
 First and Second Harmonic Voltages 
 
 £d (cm) 
 
 V 1 (mV) 
 
 V 2 (mV) 
 
 Ad (cm) 
 
 V 1 (mV) 
 
 V 2 (mV) 
 
 0.000 
 
 75-0 
 
 3-7 
 
 0.825 
 
 67.O 
 
 4.5 
 
 0.025 
 
 75-0 
 
 3-9 
 
 0.850 
 
 63.0 
 
 5-5 
 
 0.050 
 
 73.0 
 
 4.4 
 
 0.875 
 
 57-0 
 
 6.5 
 
 0.075 
 
 70.0 
 
 5.2 
 
 0.900 
 
 51.0 
 
 7-h 
 
 0.100 
 
 66.0 
 
 5-9 
 
 0.925 
 
 45.0 
 
 8.2 
 
 0.125 
 
 61.0 
 
 6.7 
 
 0.950 
 
 38.0 
 
 8.7 
 
 0.150 
 
 57-0 
 
 7-1 
 
 0.975 
 
 30.0 
 
 9.3 
 
 0.175 
 
 51.0 
 
 8.2 
 
 1.000 
 
 27.5 
 
 9-7 
 
 0.200 
 
 44. 
 
 8.8 
 
 1.025 
 
 14.5 
 
 10.0 
 
 0.225 
 
 38.0 
 
 9-3 
 
 1.050 
 
 5.6 
 
 10.0 
 
 0.250 
 
 31.0 
 
 9-9 
 
 1.075 
 
 3.8 
 
 10.0 
 
 0.275 
 
 23.0 
 
 10.0 
 
 1.100 
 
 12.5 
 
 10.5 
 
 0.300 
 
 15-5 
 
 10.2 
 
 1.125 
 
 20.5 
 
 10.0 
 
 0.325 
 
 7-9 
 
 10.3 
 
 1.150 
 
 29.0 
 
 9.7 
 
 0.350 
 
 2.8 
 
 10.5 
 
 1.175 
 
 37-0 
 
 9.1 
 
 0.375 
 
 9-5 
 
 10.2 
 
 1.200 
 
 45.0 
 
 8.5 
 
 0.400 
 
 16.5 
 
 10.2 
 
 1.225 
 
 52.0 
 
 8.0 
 
 0.1+25 
 
 24. 5 
 
 10.0 
 
 1.250 
 
 58.O 
 
 7.2 
 
 0.450 
 
 31.0 
 
 9.7 
 
 1.275 
 
 64.0 
 
 6.5 
 
 0.475 
 
 38.0 
 
 9-1 
 
 1.300 
 
 68.0 
 
 5.8 
 
 0.500 
 
 45.0 
 
 8.4 
 
 1.325 
 
 72.0 
 
 5.1 
 
 0.525 
 
 51.0 
 
 8.0 
 
 1.350 
 
 74.0 
 
 4.3 
 
 0.550 
 
 57-0 
 
 7.5 
 
 1.375 
 
 76.0 
 
 4.2 
 
 0.575 
 
 61.0 
 
 6.5 
 
 i.4oo 
 
 77.0 
 
 3.8 
 
 0.600 
 
 66.0 
 
 5.7 
 
 1.425 
 
 77-0 
 
 4.3 
 
 0.625 
 
 69.0 
 
 4.8 
 
 1.450 
 
 76.0 
 
 4.4 
 
 0.650 
 
 72.0 
 
 3-8 
 
 1.475 
 
 74.0 
 
 5-4 
 
 0.675 
 
 74.0 
 
 2.9 
 
 1.500 
 
 71.0 
 
 5.9 
 
 0.700 
 
 75-0 
 
 2.4 
 
 1.525 
 
 67.0 
 
 6.8 
 
 0.725 
 
 75-0 
 
 2.1 
 
 1-550 
 
 63.0 
 
 7.3 
 
 0.750 
 
 75.0 
 
 2.3 
 
 1-575 
 
 57-0 
 
 8.4 
 
 0.775 
 
 73.0 
 
 2.9 
 
 1.600 
 
 50.0 
 
 8.8 
 
 0.800 
 
 70.0 
 
 3.6 
 
 1.625 
 
 44.0 
 
 9-5 
 
25 
 
 5«2 Measurement of Allowable Distortion 
 
 If the amplifier output is sufficiently distorted, it is possible 
 that the square-wave convertor may produce a waveform which is not at the 
 fundamental frequency of the input signal. This might possibly lead to 
 miscalibration of the radar unit. Let it be noted that not all forms 
 of distortion can cause miscalibration. Only distortion which causes 
 a change in the sign of the slope of the signal may cause miscalibration; 
 for example, the waveform in Figure 10A. However, distortion caused 
 by saturation or cut-off of an amplifier (clipping) cannot cause 
 miscalibration; Figure 10B. The particular type of distortion of 
 concern here is that generated by a vibrating tuning fork. Since 
 the first and second harmonics are the only ones of significance for 
 vibrating tuning forks (Section IV"), measurements involving only second 
 harmonic distortion were taken. 
 
 A B 
 
 Figure 10 
 
 A Hewlett-Packard Model 203A Variable Phase Function Generator 
 and a General Radio Model 1319A Oscillator were used to provide a signal 
 in which the magnitudes of the first and second harmonics could be varied 
 
26 
 independently. This signal was connected through a 5000 to 1 divider to 
 W-X on Figure 6. W-X is the normal mixer input point. For these 
 measurements, the mixer diode was removed and the klystron power supply 
 disabled. Measurements were again taken at point A, Figure 6, using two 
 frequency selective VTVM's. 
 
 With the RANGE control in the extreme clockwise position 
 (maximum sensitivity), 13*5 mV RMS at point A was required to activate 
 the meter. The frequency of the fundamental was approximately 1500 
 hertz and the output meter of the radar unit was indicating approximately 
 50 miles per hour as was expected. The second harmonic component of 
 the signal was increased until the output meter began to move upscale. 
 This was called the break point and the magnitude of the second harmonic 
 at this point was recorded. If the amount of second harmonic is 
 increased beyond the break point value, the indicated speed increased 
 until it reached twice the original reading or approximately 100 miles 
 per hour. The second harmonic voltage was also recorded at this point, 
 which was called the leveling point. Table 2 shows the values obtained 
 for various first harmonic voltages. 
 
 TABLE 2 
 
 
 
 
 
 
 
 V 2 at 
 
 v l 
 
 RMS 
 
 V 2 
 
 at break 
 
 point 
 
 leveling point 
 
 13-5 mV 
 
 10.0 
 
 mV 
 
 RMS 
 
 16.0 mV RMS 
 
 30.0 
 
 
 
 19.5 
 
 
 
 31.0 
 
 100.0 
 
 
 
 62.0 
 
 
 
 110.0 
 
 300.0* 
 
 
 
 2*K).0 
 
 
 
 300.0 
 
 * Distortion caused by clipping becomes evident 
 at 210 mV RMS. 
 
27 
 It is obvious from Table 2 that V must be greater than l/2 V, in order 
 to cause an improper speed indication. It should be mentioned that as 
 V increases beyond 210 mV RMS, the amount of second harmonic distortion 
 necessary to cause an improper speed indication becomes close to 100 
 percent; that is V. — V . 
 
 5. 3 Approximation of 8 for Tuning Forks 
 
 Three aluminum-alloy tuning forks are supplied by Muni Quip for 
 the purpose of calibration. These forks vibrate at frequencies 
 corresponding to 30, h5, and 65 miles per hour and these values are 
 stamped on the respective forks. Normally, these forks are struck 
 lightly and then placed directly in the path of the radiation. However, 
 the second harmonic component generated by the mixing of these signals 
 in the radar unit strongly depends upon the value of 8, the tine 
 displacement. Therefore, to obtain the worst case conditions the forks 
 were struck as hard as possible and the tine movement observed by means 
 of a stroby lamp. It was found that none of the forks showed more than 
 one millimeter total tine displacement, thus giving a maximum 8 of 0. 5 
 millimeters or 0.05 centimeters. It was also noted that the higher 
 frequency forks had smaller displacements than the lower frequency forks. 
 Table 3 shows the maximum first harmonic voltages obtainable at point A, 
 Figure 6, due to reflections from the tuning forks. 
 
TABLE 3 
 
 
 
 Maximum First 
 
 Fork 
 
 Serial No. 
 
 Harmonic Voltage 
 
 30 mph 
 
 ho6k9 
 
 1.5v 
 
 U5 mph 
 
 k0650 
 
 1.3v 
 
 65 mph 
 
 40651 
 
 o.8v 
 
 28 
 
29 
 6. SUMMARY 
 
 When the operator wishes to calibrate the radar unit, he strikes 
 the tuning fork, places it in front of the antenna, and looks for a 
 steady reading on the output meter. When he obtains a steady reading, 
 he adjusts the CALIBRATE control to bring the indicated speed into 
 agreement with that stamped on the fork. Using the 30 mile per hour 
 fork, it is possible to generate approximately 1. 5 V RMS of first 
 harmonic signal at point A, Figure 6. Since 5 = 0.05, Equation 16 
 indicates that V (max) = r — V.. (max) or 83. 3 niV RMS of second harmonic 
 
 d A _L 
 
 signal can be produced if the fork is held in the proper position. If 
 the RANGE control is set further clockwise than the midpoint, this is 
 a sufficient second harmonic signal to cause the output meter to read 
 double the true reading. However, to achieve this double reading, the 
 operator must hold the tuning fork within l/k millimeter of the proper 
 position. 
 
 Throughout the course of the experiment, the author has tried 
 repeatedly to obtain a double reading with a hand-held tuning fork. 
 Although one can observe the reduction of first harmonic component which 
 indicates the fork is near the proper position, even an unsteady reading 
 greater than that stamped on the tuning fork has never been achieved. 
 This can be readily understood since it can be shown that the fork must 
 be held in the plane perpendicular to the path of the radiation in addi- 
 tion to the fact it must be held exactly at the proper distance from 
 the antenna. 
 
 Furthermore, as the operator seeks a more stable reading by 
 moving the fork, he seeks a position where the voltage produced is 
 
30 
 greater. This is equivalent to moving to a first harmonic maximum 
 and thus away from the position which may cause a double reading. 
 
31 
 7- CONCLUSIONS 
 
 On the basis of the measurements which have been made, it has 
 been shown that the vibrating tuning fork approximates the signal 
 caused by a moving target well enough to be used for calibration, and 
 that it is practically impossible to miscalibrate a radar speed meter 
 using the tuning fork method regardless of the position of the RANGE 
 control. Furthermore, if someone with an extremely steady hand did 
 manage to obtain a steady second harmonic reading when calibrating a 
 radar unit, the miscalibration would be by a factor of two. This 
 would cause a vehicle traveling at 60 miles per hour to register on 
 the meter as 30 miles per hour. A miscalibration of this magnitude 
 would be noticed immediately by the operator and result in recalibration. 
 
32 
 
 BIBLIOGRAPHY 
 
 Fink, D. G. , Radar Engineering , New York, McGraw-Hill, 19^7- 
 
 Jordan, E. C. , El e c t r omagn e t i c Wave s and Radiating Systems , Englewood 
 Cliffs, Prentice-Hall, 1950. 
 
 Luck, D. G. C, Frequency Modulated Radar , New York, McGraw-Hill, 1949. 
 
 Pound, R. V. , Microwave Mixers , New York, McGraw-Hill, 19^8. 
 
 Ridenour, L. N. , Radar Systems Engineering , New York, McGraw-Hill, 
 I9U7. 
 
 Terman, F. E. , Radio Engineering , New York, McGraw-Hill, 1932. 
 
 Torrey, H. C, and Whitman, C A., Crystal Rectifiers , New York, 
 McGraw-Hill, 19^7- 
 
33 
 
 APPEND DC A 
 
 DOPPLEE SHIFT 
 
 1,2 
 
 Given incident electromagnetic radiation of frequency f in a 
 stationary medium, propagating to the left with velocity c, the speed 
 of light. A reflecting surface S, is moving to the right with velocity 
 v. All wavelengths, frequencies, and velocities are those observed by 
 a stationary observer. 
 
 Figure 11 
 
 The wavelength of the radiation, X, is given by c/f . Points A 
 and B are successive maximum points in the electric field. Thus, points 
 A and B are moving to the left at the velocity of light and are 
 separated by X, the wavelength. Point A strikes the reflecting surface 
 at time t = 0, and at position x • At time t , point B strikes the 
 surface. Since the reflecting surface and point B are approaching each 
 other at velocity c + v, and the distance between them at t = is X, 
 
 1. D. Fink, Radar Engineering , p. 287. 
 
 2. D. Luck, Frequency Modulated Radar, p. 7« 
 
X 
 
 34 
 
 C + V 
 
 At time t , the reflecting surface S has moved to position 
 : = x + v(t ) and point B of the wave is at S. Also at time t , point 
 A has traveled from the point of reflection x to x + ct since the 
 reflected wave is now traveling to the right at the velocity of light. 
 The distance between points A and B is the wavelength of the reflected 
 wave X ' . 
 
 X' = et - vt 1 (19) 
 
 Substituting for t from Equation 18. 
 
 (C ^ V)X (20) 
 
 c + v v ' 
 
 Since f _ = — and the frequency of the reflected wave f ' equals X'/ c 
 
 c - v v ' 
 
 The difference frequency, Af = f ' - f 
 
 Af = f 2v (22) 
 
 c - v 
 
 If the incident electromagnetic radiation and the reflecting 
 surface are moving in the same direction, the formula for the difference 
 frequency is: 
 
Af o f 
 
 35 
 
 2y 
 
 c + v 
 
 The two expressions are for all practical purposes equal to 2f v/c for 
 all velocities yet attained of man-made vehicles. The following graph 
 shows the Doppler shift (difference frequency) for various target speeds 
 and electromagnetic wavelengths. 
 
 In the experimental work described in this paper, the incident 
 frequency was 10, 525 mHz. This corresponds to a difference frequency 
 of 31. 367^+23^ hertz per mile per hour. The error in disregarding the 
 direction of travel is about .000015 per cent at 100 miles per hour. 
 
36 
 
 10,000 r 
 
 5000- 
 
 1000- 
 
 - 500- 
 
 X 
 c/) 
 
 ac 
 
 UJ 
 
 _j 
 
 o 
 o 
 
 100 
 
 100 
 
 500 1000 5000 10,000 
 
 CARRIER FREQUENCY IN MHe. 
 
 50,000 100,000 
 
 Figure 12 
 
37 
 APPENDIX B 
 ANGLE MODULATION AS APPLIED TO DOPPLER EFFECT 
 
 Angle modulation is defined to be modulation of the argument of 
 the trigonometric function. That is, the signal S = A cos <t> is angle 
 modulated if <t> is modulated. Both frequency and phase modulations are 
 special cases of angle modulation. In the case of an unmodulated signal, 
 <t> = to t + , where co is the angular carrier frequency and 6 is some 
 phase angle. The instantaneous angular frequency ft is defined as 
 3©/<3t; for an unmodulated signal ft = to . 
 
 Now, frequency modulate the signal by causing the instantaneous 
 angular frequency ft to vary. 
 
 Let ft = o> + Aof(t) (24) 
 
 where o> is the carrier frequency, Ao is the maximum frequency deviation, 
 and f(t) is a function of time with continuous first derivative. It 
 
 follows that, 
 
 <J> = /ftdt = /[flj + Aof(t)] dt (25) 
 
 c 
 
 = co t + Ao / f (t) dt (26) 
 
 c 
 
 S = A cos [co t + Ao / f(t) dt] (27) 
 
 Take the unmodulated signal, S = A cos (co t + ) and cause to vary 
 as Ao / f(t) dt. Then the resulting signal is the same as expressed 
 by Equation 27. 
 
38 
 Now, consider the Doppler shift as a frequency modulation of 
 the carrier. In Equation 27, £*o is the angular difference frequency, 
 and f(t) = 1. Applying the results from above, a difference frequency 
 of £*a is equivalent to a continuous change of phase of £w radians 
 per second. 
 
 S = cos (a) t + Ao>t) (28) 
 
APPENDIX C 39 
 
 COMPUTER PROGRAM 
 
 THE PROGRAM REPRODUCED HERE IS THAT USED TO PRODUCE THE 
 GRAPH OF THE FIRST AND SECOND HARMONICS SHOWN IN FIGURE 9. 
 FOLLOWING THE PROGRAM IS THE PRINT OUT WHICH RESULTS. 
 THE AVERAGED DATA SHOWN HERE IS THE SAME AS IS PLOTTED IN 
 FIGURE 9. 
 
 MAIN PROGRAM 
 
 CALL CPB1 
 DIMENSION 
 RIT 7,1,N 
 
 LALL LHB1 
 
 DIMENSION DATA! 10,2,66), AV<2,66) ,DD(66) 
 
 RIT 7,1,N 
 
 FORMAT( 12) 
 
 RIT 7,2, ( ( (DATA(I,J,K),J=1,2),K=1,66),I=1,N) 
 
 FORMAT(2F10.2) 
 
 CALL AVER(N,DATA,AV) 
 
 CALL MATCH ( AV , FMAX , SMAX , DD, THET A, THETA 1 , THETA2* 
 
 Z = ABSF(THETAl) -ABSF ( THETA2 ) 
 
 IF(ABSF(Z)-.2) 67,100,100 
 67 CALL GRAPH(THETA,DD,AV,FMAX,SMAX) 
 
 GO TO 200 
 100 CONTINUE 
 
 WOT 6,5 
 5 F0RMAT(46H BAD MATCH IN THETA *** PROGRAM TERMINATED ***) 
 200 CONTINUE 
 
 END 
 
 PROGRAM FOR SUBROUTINE AVER 
 
 THIS SUBPROGRAM AVERAGES UP TO 10 DATA SETS AND CORRECTS FOR 
 ERRORS IN METER CALIBRATION. 
 
 SUBROUTINE A VER ( N, DATA, AV ) 
 
 DIMENSION DATA( 10 ,2 ,66 ) , AV ( 2 ,66 ) 
 
 FN = N 
 
 DO 1 L = 1,N 
 
 DO 1 M = 1,66 
 1 DATA(L,2,M) = ,919*DATA ( L , 2 , M ) 
 
 DO 10 I = 1,2 
 
 DO 10 J = 1,66 
 
 DO 15 K = 1,N 
 15 AV(I,J) = AV(I,J) + DATA(K,I,J) 
 10 AV( I, J) = AV( I,J)/FN 
 
 RETURN 
 
END ^0 
 
 PROGRAM FOR SUBROUTINE MAT*~H 
 
 THIS SUBPROGRAM FINDS THE EXPERIMENTAL VALUE OF .DELTA AND ALSO 
 EQUATION 17, 
 
 FINDS THE VALUE OF THETA TO BE SUBSTITUTED INTO THE THEORETICAL 
 
 12 
 13 
 14 
 15 
 16 
 
 17 
 18 
 
 10 
 
 100 
 
 4 
 
 37 
 
 38 
 39 
 
 40 
 41 
 
 50 
 71 
 59 
 60 
 
 SUBROU 
 
 DIMENS 
 
 PI = 3 
 
 FMAX = 
 
 SMAX = 
 
 FMIN = 
 
 SMIN = 
 
 WOT 6, 
 
 FORMAT 
 
 DO 10 
 
 FK = K- 
 
 DD(K) 
 
 WOT 6, 
 
 FORMAT 
 
 IF( FMA 
 
 FMAX=D 
 
 IF(SMA 
 
 SMAX=D 
 
 IF(FMI 
 
 FMIND= 
 
 FMIN=D 
 
 IF( SMI 
 
 SMIND= 
 
 SMIN=D 
 
 CONTIN 
 
 WOT 6, 
 
 FORMAT 
 
 WOT 6, 
 
 FORMAT 
 
 EXDELT 
 
 WOT 6, 
 
 FORMAT 
 
 B =4.0 
 
 THETA1 
 
 THETA2 
 
 DO 65 
 
 IF( 1-2 
 
 THETA= 
 
 GO TO 
 
 THETA= 
 
 DO 59 
 
 IF(THE 
 
 I F( P I / 
 
 THETA= 
 
 GO TO 
 
 IF(THE 
 
 THETA= 
 
 CONTIN 
 
 IF( 1-2 
 
 TINE M 
 ION D 
 .14159 
 
 0.0 
 
 0.0 
 
 100. 
 
 100. 
 7 
 
 ( 1H1,2 
 K=l,66 
 1 
 
 = 0.02 
 6,DD(K 
 (3F20. 
 X-DATA 
 ATA( 1, 
 X-DATA 
 ATA(2, 
 N-DATA 
 DD(K) 
 ATA( 1, 
 N-DATA 
 DD(K) 
 ATA( 2, 
 UE 
 100 
 
 (///// 
 4, FMAX 
 ( / 
 
 = (2.8 
 2,EXDE 
 ( 1H ,/ 
 *PI/2. 
 
 = -B* 
 
 ATCH( DATA, FMAX , SMAX , DD,THET A , THETA1 , THETA2 ) 
 ATA(2,66) ,DD(66) 
 
 OX, 13HAVERAGED DATA) 
 
 5*FK 
 
 ) ,DATA( 1,K) ,DATA(2,K) 
 
 5) 
 
 ( 1,K) ) 12,13,13 
 
 K) 
 
 (2,K)) 14,15,15 
 
 K) 
 
 ( 1,K) ) 17,16,16 
 
 K) 
 
 (2,K) ) 10,18,18 
 
 K) 
 
 /, 10X,7HMAXIMUM, 1 4X , 7HM I N I MUM, 14X , 8HD I STANCE ) 
 , FMIN, FMIND, SMAX, SMIN, SMIND 
 3F20.5,//,3F20.5) 
 05*SMAX)/(FMAX*PI ) 
 LT 
 
 //,31HEXPERIMENTAL VALUE OF DELTA IS ,F10.5,//) 
 805 
 
 FMIND +PI/2. • 
 
 -B*SMIND 
 
 ,38 
 
 1 = 1,2 
 
 )37,38 
 
 THETA1 
 
 39 
 
 THETA2 
 
 J =1,5 
 
 TA) 40 
 
 2. +TH 
 
 THETA 
 
 59 
 
 TA -PI 
 
 THETA 
 
 UE 
 
 ) 61,62,62 
 
 ,50,50 
 
 ETA)41,60,60 
 + PI 
 
 /2.) 60,71,71 
 -PI 
 
kl 
 
 61 THETA1=THETA 
 GO TO 65 
 
 62 THETA2=THETA 
 65 CONTINUE 
 
 THETA=(THETAl+THETA2)/2.0 
 WUT 6,3,THETA1,THETA2,THETA 
 3 F0RMAT(9H THETA1 = , F 10 . 5 , / / , 9H THETA2 = , F 10 . 5 , / / , 8H THETA = 
 1 ,F10.5) 
 RETURN 
 END 
 
 PROGRAM FOR SUBROUTINE GRAPH 
 
 THIS SUBPROGRAM GENERATES THE THEORETICAL CURVES FROM EQUATION 17 
 AND PLOTS THEM ALONG WITH THE AVERAGED DATA. 
 
 SUBROUTINE GRAPH ( THETA , DD, DATA , FMAX , SM AX ) 
 
 DIMENSION DATA ( 2,66) , FHARM(660) ,S HARM (660) , DD ( 66 ) , D ( 660 ) , 
 1 TX(2) ,TY1 (2) 
 
 PI = 3.14159 
 
 B =4.0*PI/2.805 
 
 DO 20 1=1,660 
 
 F 1= 1-1 
 
 D( I )=.0025*FI 
 
 ANG = THETA +B*D( I ) 
 
 FHARM( I )=ABSF( FMAX*COS ( ANG ) ) 
 20 SHARM(I)= ABSF( SMAX*S I N ( ANG ) ) 
 
 CALL CCP4SC(D, 10.0, 660, 1,TX) 
 
 CALL CCP4SC(FHARM,8.0,660,1,TY1 ) 
 
 CALL CCP1PL(2. 0,2. 0,-3) 
 
 CALL CCP5AX(0.0,0.0,20HFIRST HARMONIC ( BOX ) , 20, 6 . 5 , 90. , TY 1 ) 
 
 CALL CCP5AX(0.0,0.0,8HDISTANCE,-8,08.5,0.0,TX) 
 
 CALL CCP5AX( 8.5,0.0,26HSEC0ND HARMONIC ( TR I ANGLE ) , -26, 6 . 5 , 
 1 90.0,TY1) 
 
 CALL CCP6LN(D f F HARM, 660, 1,TX,TY1 ) 
 
 CALL CCP6LM(D,SHARM, 660,1 ,TX,TY1 ) 
 
 DO 30 I =1,66 
 
 Y1=(DATA( 1, I )-TYl ( 1 ) )/TYl (2) 
 
 Y2=(DATA(2, I)-TY1(1))/TY1(2) 
 
 X = (DD( I )-TX( 1) )/TX(2) 
 
 CALL CCP2SY(X,Y1, 0.075, 0,0. 0,-1 ) 
 
 CALL CCP2SY( X,Y2, 0.075, 2, 0.0,-1) 
 30 CONTINUE 
 
 CALL CCP2SY(2.0,6.5,0.15,29HDETECT0R OUTPUT IN MILLIVOLTS, 
 1 0.0,29) 
 
 CALL CCP2SYI 2.0,6.25,0.15, 16H VS,0.0,16) 
 
 CALL CCP2SY(2.0,6.0,0.15,26H DISTANCE IN CENT I METERS, 0.0, 
 1 26) 
 
 CALL CCP1PL{ 12.0,0.0,-3) 
 
 RETURN 
 
 END 
 
U2 
 
 AVERAGED DATA 
 
 DISTANCE 
 
 FIRST HARMONIC 
 
 SECUND HARMONIC 
 
 .OOOOO 
 .02500 
 .05000 
 .07500 
 . 10000 
 .12500 
 . 15000 
 .17500 
 .20000 
 .22500 
 .25000 
 .27500 
 .30000 
 .32500 
 .35000 
 .37500 
 .40000 
 .42500 
 .45000 
 .47500 
 .50000 
 .52500 
 .55000 
 .57500 
 .60000 
 .62500 
 .65000 
 .67500 
 .70000 
 .72500 
 .75000 
 .77500 
 .80000 
 .82500 
 .85000 
 .87500 
 .90000 
 .92500 
 .95000 
 .97500 
 1.00000 
 1.02500 
 1.05000 
 1.07500 
 1.10000 
 
 75.00000 
 74.50000 
 72.50000 
 70.00000 
 66.00000 
 61.00000 
 56.50000 
 50.50000 
 44.00000 
 37.75000 
 30.50000 
 23.00000 
 15.50000 
 7.70000 
 2.65000 
 9.10000 
 16.50000 
 24.50000 
 31.00000 
 38.00000 
 44.50000 
 50.50000 
 56.50000 
 61.00000 
 65.50000 
 69.00000 
 72.00000 
 73.50000 
 74.50000 
 75.00000 
 74.50000 
 72.50000 
 70.00000 
 64.00000 
 63.00000 
 57.00000 
 51.00000 
 45.00000 
 38.00000 
 30.00000 
 25.25000 
 14.25000 
 5.55000 
 3.90000 
 02.50000 
 
 3.30840 
 3.63005 
 3.99765 
 4.73285 
 5.42210 
 6.06540 
 6.61680 
 7.53580 
 8.04125 
 8.59265 
 9.05215 
 9.19000 
 9.28190 
 9.46570 
 9.64950 
 9.41975 
 9.37380 
 9.19000 
 8.96025 
 8.54670 
 7.85745 
 7.44390 
 6.98440 
 6.06540 
 5.33020 
 4.54905 
 3.58410 
 2.84890 
 2.13667 
 1.83800 
 2.02180 
 2.59617 
 3.17055 
 3.99765 
 4.96260 
 5.83565 
 6.70870 
 7.35200 
 7.94935 
 8.50075 
 8.91430 
 9.19000 
 9.32785 
 9.32785 
 9.51165 
 
^3 
 
 1.12500 
 
 21.00000 
 
 1.15000 
 
 29.00000 
 
 1.17500 
 
 37.00000 
 
 1.20000 
 
 44.50000 
 
 1.22500 
 
 51.50000 
 
 1.25000 
 
 58.00000 
 
 1.27500 
 
 63.50000 
 
 1.30000 
 
 68.00000 
 
 1.32500 
 
 71 .50000 
 
 1.35000 
 
 74.00000 
 
 1.37500 
 
 76.00000 
 
 1.40000 
 
 77.00000 
 
 1.42500 
 
 77.00000 
 
 L. 45000 
 
 76.00000 
 
 1.47500 
 
 73.50000 
 
 1.50000 
 
 71.00000 
 
 1.52500 
 
 67.00000 
 
 1.55000 
 
 62.50000 
 
 1.57500 
 
 57.00000 
 
 1.62500 
 
 50.50000 
 
 1.62500 
 
 44.00000 
 
 9.23595 
 9.05215 
 8.59265 
 8.04125 
 7.58175 
 6.93845 
 6.11135 
 5.46805 
 4.68690 
 4.04360 
 3.67600 
 3.40030 
 3.53815 
 3.85980 
 4.54905 
 5.37615 
 6.11135 
 6.66275 
 7.53580 
 8.04125 
 8.73050 
 
 MAXIMUM (-IKST HARMONIC VOLTAGE IS 77.00000 MILLIVOLTS 
 
 MINIMUM FIRST HARMONIC VOLTAGE IS 2.65000 MILLIVOLTS 
 
 POSITION OF FIRST HARMONIC MINIMUM IS 0.35000 CENTIMETERS 
 
 MAXIMUM SECOND HARMONIC VOTAGE IS 
 MINIMUM SECOND HARMONIC VOTAGE IS 
 
 9.64950 MILLIVOLTS 
 1.83800 MILLIVOLTS 
 
 POSITION OF SECOND HARMONIC MINIMUM IS 0.72500 CENTIMETERS 
 
 EXPERIMENTAL VALUE OF DELTA IS 
 
 0.11189 CENTIMETERS 
 
 THETA1 = 
 THETA2 = 
 THETA = 
 
 0.00280 
 -0.10640 
 -0.05180 
 
*tfj 
 
 a 
 
 \$1* 
 

 
■ 
 
 I I