CHAPTER 10 SONAR RESOLVING EQUIPMENT Model OKA-1 Sonar Resolving Equipment The model OKA-1 sonar resolving equipment is designed for use with echo-ranging and depth-determining equipments. Its primary function is to calculate and record the horizontal range of a sonar target. This calculation is based on sonar range and target depth-factors that are determined graphically by separately recording the acoustic outputs of the azimuth echo-ranging and depth-determining equipments. The range recorder is a unit of the OKA-1 equipment, and the depth recorder is a unit of the depth-determining equipment. Calculating circuits of the OKA-1, however, provide all the electric controls to the depth recorder. This design ensures that the indications of the depth recorder accurately represent target depth. One function of the OKA-1, derived from the requirement of the calculation of horizontal range, is prediction of the angle of depression to the sonar target. This prediction provides a form of aided tracking to the depression controls of the depth-determining equipment. It must be emphasized that the OKA-1 equipment is not concerned with the primary task of sound transmission or reception, and the accuracy of its performance depends directly on the validity of the information obtainable from the primary equipments. See figure 10-1 for a pictorial diagram of the OKA-1 equipment. THEORY OF SOUND AS APPLIED TO THE OKA-1 In order to understand the operation of the OKA-1 equipment, certain elements of the physical theory of sound transmission must be thoroughly understood. Because rigorous adherence to theory would make the calculations excessively complex, certain simplifying assumptions are made in the solving circuits. The validity of these assumptions under all circumstances is beyond the scope of this text. The velocity of sound in sea water is affected by salinity, pressure, and water temperature. If a sound ray passes at a small angle through layers of water in which these factors are altered (figure 10-2), bending or refraction of the sound ray occurs, in accordance with a relationship known as Snell's law- (V-ΔV)/V cos Eq= cos Eqr,  (10-1) where V is the velocity of sound in the first layer, ΔV is the velocity change in adjacent layer, Eq is the angle of ray in first layer with respect to boundary between layers (which may be considered a plane parallel to the surface of the ocean), and Eqr is the angle of ray in second layer with respect to the plane of the layer. The principal variable affecting the velocity of sound is the temperature of the water. Temperature may be determined to great depth by use of the bathythermograph. If certain assumptions concerning salinity are made, the velocity of sound at any depth may be calculated from the bathythermograph record. Under most conditions the bathythermograph indicates a layer of water of nearly constant temperature from the surface down to an appreciable depth. This layer is known as the mixed layer. Below this mixed layer the temperature falls rapidly through a region known as the thermocline and then again changes relatively slowly. A sound beam that is narrow in the vertical dimension if directed at a small downward angle with respect to the surface of the ocean passes through the mixed layer without bending, because the velocity is constant. In passing through the thermocline, it undergoes marked refraction and, 196
 Figure 10-1 -OKA-1 equipment. 197 Figure 10-2 -Sound-ray refraction. because sound velocity decreases with decreasing temperature, the ray is directed downward at a greater angle to the surface. If a sonar target lies below the thermocline, echoes can be obtained at long range-when the depression angle is small-only from sound rays which would pass well above the target if there were no refraction. Positive temperature gradients result in upward deflection of the sound ray, but they are rare in most ocean areas. It is also somewhat common to find almost continuously decreasing temperature from the surface down. Thus, the ray is continuously passing from a layer of higher velocity to a layer of lower velocity. Figure 10-2 illustrates the various angles and factors in sound-ray refraction affecting the OKA-1. Not only the mixed layer but also the water below the thermocline are assumed to be isothermal. The sound-ray path is curvilinear in the thermocline, where sound does not travel at constant velocity. The first of the fundamental assumptions made in the equipment is concerned with this fact. Range is measured by a recording stylus, which keys the sound transmitter at its zero position and traverses a recording chart at constant speed proportional to the average velocity of sound. If this speed is proportional to the velocity of sound, the distance of an echo trace on the chart from the zero point may be defined as range. A suitable adjustment in the recording equipment provides for setting the excursion rate of this stylus to a particular sound velocity. If the velocity selected is proportional to the velocity of the ray, and if the ray path is a straight line, the recorded range is correct. Because the ray path is not a straight line, some compromise on how to use the indication of the bathythermograph must be made. This stylus speed adjustment is a matter of basic doctrine formulated and disseminated to the fleet to ensure best results. Figure 10-2 shows that the distance from the echo-ranging transducer to the target may be appreciably less than the distance along the sound-ray path. Because no compensation is provided for this error, the recorded range is incorrect in an amount dependent on the magnitude of the temperature gradient. Figure 10-2 shows the error that refraction can introduce in determining target depth. The circuits of the OKA-1 equipment deliver to the recorder stylus motor of the depth-determining equipment a frequency proportional to the velocity of sound times the sine of Eq. This expression may be termed Vz, the vertical component of the velocity of sound. The angle Eq is determined by the depth-determining equipment and may be defined as the angle of depression of the center of the acoustic beam with respect to the surface, measured in a vertical plane through the line of sight to the sonar target. The recorder stylus speed is therefore proportional to Vz, and the position of an echo trace across the depth-recorder chart depends on the stylus speed and the time for the echo to return. If refraction were neglected, the stylus speed and time of arrival of the echo would lead to the conclusion that the target was at position A and that the target depth was therefore Rq sin Eq, where Rq is the sound range. If the value of Eqr is appreciably different from that of Eq, it is evident that a notable error will be made from this assumption. The circuits of the OKA-1 equipment and the depth recorder of the depth-determining equipment provide a measure of compensation for such errors. A solving circuit continuously computes the refracted depression angle, Eqr, by means of an adjustment made on the basis of the velocity change indicated by the bathythermograph. This relation is governed by Snell's law, the velocity change being V1-V2. To make the speed of the depth-recorder stylus proportional to the vertical component of the velocity of sound after refraction, it is necessary only to substitute the angle Eqr for the angle Eq in the frequency-determination circuit. This change must occur, however, when the sound ray has reached a depth equal to that of the mixed layer. Regardless of the magnitude of Eq, the change in speed should occur when the recorder stylus reaches a point on the chart that is equal to the layer depth. Change-over of control between angles Eq and Eqr is accomplished by 198 placing a contact device in the path of the depth-recorder stylus. The contact device is adjustable along its traverse. These contacts actuate a relay in the OKA-1 equipment, which selects the angle, Eqr, for the rest of the recorder excursion. The accuracy of correction provided by the circuit described varies appreciably, depending on water conditions. This situation creates a complex problem, and the adjustment of the refraction controls cannot be set forth on a purely theoretical basis. The refraction adjustments are important features of the fleet doctrine that relates the bathythermograph to the sonar installations. After several echoes have been obtained on the depth recorder, it may be assumed that a valid indication of target depth is available, if the refraction compensations are reasonably correct. Target depth below the depth-determining transducer, H'q, and target range-which is based on the assumption that sound range, Rq, measured by the range-recording device is correct-are therefore available for other calculations required of the OKA-1 equipment. The angle, the sine of which is the ratio of H'q to Rq, is determined by the equipment and is defined as the computed target depression angle (cEtq). At a late stage in a sonar attack the target-depression angle may change very rapidly, and this calculated angle is transmitted by synchro order to the depth-determining equipment to provide aided tracking to the depression control of the depth-determining equipment. As may be seen from figure 10-2 and from the assumptions concerning Rq, this angle has no physical reality until Eq becomes large and refraction is negligible. Fortunately, this is the condition that exists when Eq must change rapidly. The angle cEtq is employed further in the circuits of the equipment to calculate the horizontal target range, Rhq. The assumption is made that the calculated depression angle is the true target-depression angle and that Rq is the target distance along a straight line. The exact calculation is Rhq=Rq cos cEtq.  (10-2) Figure 10-2 shows that, if refraction is appreciable, the value calculated is not the distance across the surface of the ocean to a point directly over the target. With a temperature gradient, the calculated horizontal range always exceeds the actual horizontal range. SERVO SYSTEMS The various angles and components of range with which the OKA-1 equipment is concerned are transmitted between its units and between other equipments by synchro order. Consequently, a large number of servo amplifiers are involved in accepting these various orders. Additionally, the solving circuits of the equipment are electro-mechanical systems positioned by servo action. Finally, various motor drives such as those for the range-recording stylus and depth-recording stylus require adjustable constant speed. These drives are effected by servo combinations which may be called rate servos, because they accomplish mechanical rotation at a fixed rate, in contrast to the customary servo which accomplishes mechanical rotation to a fixed position. In all, there are 13 servo systems, which constitute the major part of the OKA-1 equipment. Because all of the servo systems can be grouped into four classes, the basic theory of operation will be explained for one example of each class. The four classes of servo systems are: (1) Comparison servo; (2) single-speed positioning servo; (3) dual-speed positioning servo; and (4) rate servo or integrator. Servomotors Two types of servomotors are employed in the equipment. For systems in which the speed or torque requirements are low, a 2-phase 60-cps induction cup motor is employed. For all high-speed or high-torque applications, a 2-phase 60-cps squirrel-cage induction motor is employed. This motor is also equipped with an induction cup generator element mounted on the same shaft. Figure 10-3 -Servo-amplifier power stage. 199 The generator is wound as a 2-phase device and has the property of producing from one phase-if the other phase be excited by a 60-cps voltage-a signal proportional to the speed of the rotor and to the degree of excitation of the other phase. All servomotors in the equipment are driven by a power stage which consists of a type 6V6 beam power tube, triode-connected and arranged as a cathode follower with one phase of the motor completing the circuit from cathode to ground. A simplified diagram of a typical stage is shown in figure 10-3. This phase of the motor is tuned by a parallel capacitor, 2 μf, for the induction cup motor and 5.7 μf for the squirrel-cage motor. The other phase is excited by a reference voltage. Comparison Servo System The comparison servo system converts a signal voltage into a mechanical rotation proportional to the signal. An example of the comparison servo is the cEtq servo system in the range computer. Functional and schematic diagrams of the complete circuit are shown respectively in figure 10-4, A and B. The mechanical part of the system includes a miniature synchro or resolver, B-316, which is driven by an induction cup motor, B-314. The synchro rotor is excited by a fixed voltage from a 60-volt transformer through a series potentiometer, R-528, which is used for adjusting the phase of the synchro output voltage. The resolver, B-316, has the usual properties of synchros-namely, that the voltage of a particular stator coil varies trigonometrically as the rotor is turned. The resolver is positioned in the mechanism in such a manner that the voltage of the stator coil employed is zero when cEtq is zero and increases directly with the sine of cEtq. This voltage is applied through a series calibrating potentiometer, R-305, to a 10-revolution helipot, R-301, which is located in the Rq gear train. The helipot is a wound variable resistor with its resistive element constructed in the form of a helix and requires one or more turns to cover its range. The arm of this helipot is positioned by Rq. Because R-301 is at a position representing Rq, and is excited by sin cEtq, the voltage available at the arm is therefore Rq sin cEtq, which, by definition, is H'q. In solving for cEtq, therefore, it is necessary only to compare Rq sin cEtq voltage with a voltage derived from H'q. The H'q helipot, R-524, is located in the H'q gear train. The arm of R-524 is positioned by H'q and the excitation of R-524 is derived from a locked rotor synchro similar to B-316. Across the helipot, R-524, is a divider, consisting of R-506 and R-507nA, which provides a zero reference point which is a short distance above the end of the helipot instead of being at the extreme end of travel. All helipots in the OKA-1 equipments are used with a similar divider to provide a more precise zero, and as a safety precaution. The junction of the two divider resistors is connected to ground, and is the ground reference for the cEtq calculating circuit. The excitation for R-524 is a fixed voltage, and the arm of R-524 is positioned by H'q. Therefore, the voltage available from the arm to the zero reference point (ground) is, by definition, H'q. The H'q voltage is combined with the Rq sin cEtq voltage to obtain a difference signal for the cEtq servo amplifier. A basic requirement of the system is that the voltages being compared be precisely in phase because a small phase difference can result in a large phase-angle error in the difference signal to the amplifier. This phase-angle error would seriously alter the normal quadrature phase relationship of the motor voltage and the result would be a loss of motor torque and overload of the motor amplifier. To ensure that the voltages being compared are in phase, the helipot excitation voltages are derived from similar sources, and a phasing adjustment is provided for precise phasing. The fixed phase of the servomotor is connected to the 117-volt source through a 1.0 capacitor, providing a fixed-phase potential which leads the excitation by approximately 90°. The phase of the input signal to the amplifier is such that if this signal is applied directly to the amplifier, the motor voltages will be in quadrature. The servomotor receives a control signal from the output of a three stage resistance coupled amplifier. The input stage is a pentode with conventional screen bypass. The output of this stage is coupled to the grid of a triode intermediate driver which in turn supplies the control grid of the final cathode follower. The over-all gain is such that the voltage developed across the motor is approximately 200 times that of the input signal. The polarity of signals is determined by the fact 200
 Figure 10-4 -Comparison servo system. A, Functional block diagram; B, schematic diagram. that the Rq sin cEtq, and the H'q signal must tend to run the motor in a direction to increase cEtq. When the two signals are equal, the difference signal is zero and the motor remains at rest. Thus the position of the cEtq gear train is determined by the values of Rq and H'q. Direct-current power supply for the cEtq servo amplifier is obtained from a filter network which decouples the amplifier from all other systems. Single-Speed Positioning Servo System The single-speed positioning servo system consists of a motor-driven synchro control transformer with synchro orders from a remote transmitter connected to the transformer stator. The stator voltage produces in the rotor of the control transformer a voltage which varies in trigonometric relation to the rotor position. The rotor voltage 201
 Figure 10-5 -Positioning-servo schematic. therefore passes through zero at two points separate d by 180° and has a phase for one-half of a revolution opposite to that for the other half of a revolution. Because the rotor voltage reverses in phase upon passing through zero, the rotor signal may be employed as the input to an amplifier which drives the servomotor in a direction which reduces the control-transformer rotor signal to zero. At this point the system remains at rest until a change in the synchro order to the stator is received. A suitable example of the single-speed positioning servo system is the H'q servo in the range computer. The complete circuit is shown in figure 10-5. This system has the function of causing the H'q gear train in the range computer to assume a position determined by the H'q transmitter in the depth recorder of the depth-determining equipment. The order from the transmitter is connected to the stator of the 1CT control transformer B-309. The system is driven by an induction cup motor, B-310, the fixed phase of which is excited from the 117-volt a-c source through a 1.0 μf capacitor. This series capacitor causes the voltage applied to the reference winding to lead the excitation voltage by 90°. The phase of the control-transformer rotor signal is such that it can be applied directly to the amplifier input tube with the result that quadrature voltages are obtained at the servomotor. The amplifier is a conventional resistance-coupled amplifier with a triode input tube coupled to a type 6V6 power tube which drives the variable phase of the servomotor. Dual-Speed Positioning Servo System Where maximum precision is necessary in positioning a servomechanism, the dual-speed system is employed. In such a system the controlling synchro orders originate from a transmission system consisting of one synchro transmitter geared to the system at a 1-to-1 or 2-to-1 ratio and a second synchro transmitter geared to the system at a 36-to-1 ratio. The controlled servomechanism is correspondingly arranged with two control transformers, one geared at 1 to 1 or 2 to 1 and the other at 36 to 1. It is a requirement of the associated amplifier that the 1-speed control transformer govern the approximate position of the system to within a few degrees and that the 36-speed control transformer govern the exact position. Because the gearing provides 10° of system motion for one complete revolution of the 36-speed control transformer, extreme accuracy of position is obtainable. As an example of the dual-speed positioning servo system, the system controlling the position of the Rq gear train in the range computer will be described. The complete circuit is shown in figure 10-6. The control transformers B-302 and 202
 Figure 10-6 -Dual-speed positioning servo schematic. B-303 are excited by Rq derived from transmitters in the range recorder of the OKA-1 equipment. The system is driven by an induction cup motor, B-304. The maximum rotor voltage from either control transformer is 55 volts rms, and if the mechanism is displaced from the synchro order by an angle θ, the voltage from B-302, the 1-speed control transformer, is 55 sin θ, whereas the signal from B-303, the 36-speed control transformer, is 55 sin 360. To provide the accuracy attainable by means of the 36-speed order, it is a requirement of the circuit that the 36-speed order take control as θ approaches zero. As θ increases from zero, the first maximum voltage from the 36-speed control transformer occurs at 0 equals 2.5°, the signal then being 55 volts rms. The next maximum occurs at θ equals 7.5° but the polarity is reversed. For values of θ less than 5° the polarity of the connections to the servo amplifier and motor causes the motor to drive θ to zero. If θ were greater than 5° the reverse polarity of the signal would drive the system to θ equals 10° where the 36-speed signal is again zero. The system would then be, and remain, 10° out of position. To prevent this condition from occurring the 1-speed control transformer signal must take control when θ is greater than 5°. The polarity of the signal from the 1-speed control transformer does not reverse until θ exceeds 180°. It is therefore evident that control by the 1-speed order cannot result in a spurious position of the servo system. To summarize, when the values of θ are less than 5°, the 36-speed order must control the servo-mechanism; when these values are more than 5° the 1-speed order must control. The method of accomplishing this requirement is stated in the following paragraph. The 1-speed signal is injected in series with the cathode circuit of the pentode, V-308, self-biased by resistor R-352. The tube current passes through the rotor winding of the control transformer to ground. This current produces a d-c voltage at the R2 terminal of the control transformer of +1 volt to ground. The instantaneous voltage at R2 to ground is (1+55 squareroot(2)) sin θ sin ωt volts. This voltage is applied to twin diode V-340 through current-limiting resistor R-350. The twin diode limits the voltage at the junction of R-350 and R-351 to +4.4 volts and -2.4 volts. The signal reaching the grid of V-308 through the current-limiting resistor R-351 is, therefore, limited to values between +4.4 volts and -2.4 volts to ground. The 1-volt positive bias of the control 203 transformer signal, therefore, means that limiting will occur if the peak amplitude of the a-c signal is greater than 3.4 volts. The signal attains this value when θ reaches 2.5° and hence limiting does not occur for values of θ from zero to ±2.5°. For values of θ less than 2.5°, the signals injected into the cathode and applied to the grid are identical and there is no output at the plate of the pentode except for a very small component which is the equivalent of the effect caused by a power-supply ripple equal to the cathode-injected signal. This plate is coupled to the grid of the triode, one section of V-310, through current-limiting resistor R-357 and, for values of θ less than 2.5°, negligible signal is present at this grid. Three-fourths of the signal from the 36-speed control transformer B-303 is injected into the cathode circuit of the triode by means of R-358 and R-360, the values of which serve to degenerate the amplification of the triode to a gain of 10. Therefore, a signal 10 times the 36-speed signal appears at the plate of the triode, and this voltage is coupled to the grid of the power tube, V-303, which drives the servo-motor, B-304. To provide correct phase relationship, the fixed phase of the motor is energized through capacitor C-301 from the same line exciting the synchro transmitters. The result is that the system is driven until the error signal is eliminated and θ equals zero. When θ exceeds 2.5° the signal amplitude at the Figure 10-7 -Servo-amplifier waveforms at 5°. A, Waveform of signal at grid of V-310; B, waveform at grid of V-303. cathode of the pentode, V-308, can increase to the maximum of 55 volts. The grid, however, is constrained to positive swings of 4.4 volts and negative swings of 2.4 volts. There is consequently established a voltage difference between grid and cathode during those portions of the wave when signal amplitude exceeds ±3.4 volts. During this part of the wave, that portion of the signal in excess of ±4.5 volts is amplified at the full gain of the pentode and appears at the grid of the triode, V-310, with a gain of 30. When θ equals 5°, the 36-speed signal becomes zero and the only signal at the output-tube grid is that produced by the amplified 1-speed order. The relative polarity of the control transformers must be such that the amplifier delivers to the grid of the output tube, a signal of the same polarity as that obtained at the same point from the 36-speed control transformer for values of θ between zero and 5°. These conditions must exist if the 1-speed signal is to drive the motor in a direction to reduce θ to zero. The waveform of the signal at the grid of the triode, V-310, when θ equals 5°, is indicated in figure 10-7, A. The peak-to-peak amplitude is approximately 70 volts, but the signal is zero for definite portions of the wave, namely, those portion s during which the 1-speed signal is less than ±3.4 volts. This level, as delivered to the triode grid is obviously excessive and the amplifier limits, delivering to the grid of the power tube a waveform as shown in figure 10-7, B. The load in the cathode of the power tube is a parallel-tuned circuit and the waveform of the grid signal produces an entirely satisfactory voltage across the servomotor, the harmonic content of the grid signal having negligible effect on the performance of the motor. It should be clear that for values of θ between 2.5° and 5°, when the 1-speed signal is beginning to appear at the 36-speed amplifier grid, this signal merely aids whatever 36-speed signal is present in the cathode, because both signals must tend to drive the motor in the same direction for values of θ between zero and 5°, as previously explained. When θ increases to 7.5°, a more complex waveform results in the output of the triode. In this condition, maximum 36-speed signal of opposite polarity is present in the cathode circuit of the triode and proper action of the servo depends on the fact that the amplitude of the 1-speed signal is 204 Figure 10-8 -Servo-amplifier waveforms at 7.5°. A, Waveform of signal from the 1-speed amplifier; B, waveform of signal from the 36-speed amplifier. 120 volts, whereas the amplitude of the 36-speed signal is approximately 40 volts (three-fourths maximum rotor voltage). The 1-speed signal is zero for appreciable portions of a cycle, however, and the 36-speed amplifier triode during this time must receive a net signal of proper polarity for rotation of the motor to reduce θ to zero. When θ equals 7.5°, the signal from the 1-speed amplifier is zero for an electrical angle of from zero to approximately 20° and increases rapidly to a limiting value of approximately 130 volts because the net grid-to-cathode signal reaches a maximum of 15 volts causing the pentode to limit. The waveform of this signal is shown in figure 10-8, A. For the first 20° of a cycle, the cathode of the triode, V-310, goes positive, and because the grid signal is zero, the plate potential increases. At 20° the grid begins to go positive as the 1-speed signal becomes amplified and the a-c component of the anode voltage rapidly reduces to zero, goes negative, and limits. Toward the end of the half cycle, as the 1-speed signal diminishes, the plate again goes positive, reaching the maximum at an electrical angle of 160°. This process inverted repeats in the next half cycle. The waveform of the a-c voltage at the anode of the 36-speed amplifier is shown in figure 10-8, B. 239276°-53-14 So far as the motor is concerned, this peculiar waveform is of no consequence because of the tuned circuit and the filtering action of the motor itself. It is necessary only that the voltage wave for a half cycle have correct polarity for the desired direct ion of motor rotation. For most suitable action of the servo system, however, it is desirable that the voltage at the motor does not dip too low when going through the 7.5° position of θ. For this reason a definite relationship must be maintained between the amplification of the 1-speed signal and the proportion of the 36-speed signal employed. As θ continues to increase, the "notch" in the output signal diminishes, and vanishes when the value of θ is between 10° and 15° because in this interval the results of the two control-transformer signals have the same polarity. At 17.5° the "notch" is again at maximum, but the magnitude is reduced as compared with that existing when 0 equals 7.5° because the magnitude of the 1-speed signal has more than doubled and the electrical angle during which the 1-speed signal is zero, has diminished. As θ is further increased, the "notch" finally becomes nearly imperceptible. For small values of θ less than 2.5°, one factor which controls the accuracy of the positioning, as well as the rapidity with which the servomotor restores the system to balance, is called the "stiffness" of the system. The degree of stiffness is determined essentially by the gain of the 36-speed order to the variable phase of the servomotor. Greater stiffness results in more sensitive and more rapid positioning for small values of θ but it increases the tendency of the system to oscillate if the damping factor is not adequate for a given system stiffness and moment of inertia. The stiffness of this system and the inherent damping of the induction cup motor are such that no additional electrical damping is required. Rate Servo System The rate servo, or integrator, is an electromechanical system designed to rotate at a speed proportional to an adjustable signal. This servo is one of the most important devices employed in the OKA-1 equipments. In its most exacting application, the rate servo controls the speed of the range recorder stylus mechanism. It is a requirement of this range recorder that the stylus speeds be at either of two basic rates which are 205
 Figure 10-9 -Rate-servo schematic. in the ratio of 2.5 to 1, and that both basic rates be adjustable over a range equivalent to a sound velocity of from 4,600 to 5,100 feet per second. The mechanism must operate at sufficient power level so that the variation in load during the excursion of the stylus does not cause error in speed, and that power is available for the operation of the recorder chart drive. The complete circuit is shown in figure 10-9. The mechanical portion of the system consists of a motor generator, MG-202, driving the mechanical load through suitable gearing. The motor variable phase is driven by an amplifier which will be described in detail, the input signal being the difference between an arbitrary adjustable signal and the output of the induction generator. Polarity of the induction cup generator signal is such that it attempts to reverse the voltage applied to the motor, but complete cancellation is not possible because the generator output must approach zero as the motor speed approaches zero, and the fixed signal amplified at the full gain of the amplifier is available to drive the motor. In all cases the fixed signal exceeds the generator voltage and if the motor attempts to slow down for any reason whatever, more net signal is applied to the power-tube grid, because the signal difference increases and the motor tends to speed up. Correspondingly, if the motor speed is excessive, the amplifier signal is diminished, the motor voltage decreases, and the motor slows down. This method of speed control is analogous to a conventional feedback amplifier and the feedback ratio is the ratio of the generator voltage to the net signal at the input amplifier under normal operation. If this feedback is large enough, it is evident that the motor must run at a speed demanded by the fixed signal, regardless of variation in power supply voltage, mechanical load, or system frequency. Either of two fixed signals is selected by a transfer contact of a relay, K-220. One signal may be adjusted from zero to one-half the available voltage, Ev and the other from one-half to the full voltage of Ev, by two potentiometers-R-212 and R-213. In calibration, the larger signal is first set so that the recorder operates at the proper speed on the 1,500-yard scale excursion rate, and the other adjustment is then set for correct operation at the 3,750-yard rate, the result being that the signals have a ratio of precisely 2 ½ to 1. The voltage, Ev, is externally adjustable to be proportional to the velocity of sound between the limits of 4,600 and 5,100 feet per second. Therefore, the "asking" signal is proportional to the velocity of sound, and the motor speeds are 206 proportional to the velocity of sound at the two basic rates required. It is essential that the asking signal and the generator induced voltage be of identical time phase in order to avoid an excessive out-of-phase component of signal which would overload the amplifier and produce no torque in the motor. The generator is excited from the 60-volt winding of a transformer through a series resistance consisting of R-211 in parallel with the series combination of R-208, R-209, and R-210, all three of which are resistors with a negative temperature coefficient. The magnitude of these resistances and the temperature coefficient of the temperature-sensitive resistors are such that the magnitude and phase angle of the generator output voltage are independent of temperature. The difference signal is amplified in a pentode, V-217, the gain of which is reduced by omission of the bypass capacitor in the screen circuit. The output of this pentode is delivered to a phase-shifting frequency selective network which eventually furnishes the signal to the biased grid of a triode, V-215. This network performs two important functions. First, it serves to prevent regeneration of high frequencies through the system and more specifically, to prevent regeneration of harmonics of the 60-cycle system voltage. Secondly, this capacitor causes a large lagging phase shift in the signal across it. This angle of lag is further increased by R-269 and the other section of C-236. The phase angle change is excessive because of the amount of harmonic regeneration suppression required. Because of this change, approximately 20° of leading phase shift is provided in the coupling to the triode amplifiers V-215. The output of the triode is coupled to the power-tube grid which is biased at -27 volts. The lagging phase shift of the amplifier output necessitated by the harmonic suppression in the output of the pentode amplifier accomplishes quadrature relation of the motor voltages. The fixed phase is connected to the line through a series resistor, R-222, to aid in obtaining the required phase shift. Relation of the OKA-1 to Sonar Systems With the background of the OKA-1 equipment computations and the fundamental theory involved in its operation set forth previously, it is possible to appreciate the exact relationship of the OKA-1 equipment to the azimuth echo-ranging and depth-determining equipments. The complexities of the synchro systems involved and of the interrelationships between OKA-1 and the depth-determining equipment make very desirable a simplified block diagram of that portion of the complete sonar installation with which the OKA-1 equipment is directly concerned. Such a diagram is shown in figure 10-10. The meaning of the various symbols used is explained in the legend accompanying the diagram. This diagram does not show any details of primary equipments other than those involved directly in the calculating requirements of OKA-1. Only two sources of primary information need be considered in describing the circuit relationships depicted-sonar depression, Eq, and sonar range, Rq. The quantity, Eq, originates in the Eq mechanical system, which is controlled by a 1DG differential generator in the depth-determining equipment depression control. Regardless of the input synchro order to this generator, the operator is required to position the 1DG rotor by means of mechanical correction, jEq, so that on-target indications of the DDI cathode-ray tube are obtained. This synchro order is accepted by a 1CT and associated servo amplifier in the OKA-1 equipment, supplying the angle Eq as a mechanical position, and positioning three 1G synchro transmitters. One of these 1G synchros transmits Eq as a 2-speed order, to a 1F synchro motor in the depth-determining equipment to indicate Eq to the depression-control operator. The other two 1G synchros, geared at 2-speed and 36-speed transmit Eq to the stabilization computer. Sonar range, Rq, is made available at 10,000 yards per revolution at 1-speed and 36-speed. These synchro orders are accepted by the Rq servo amplifier, making Rq mechanically available in the calculating circuits. This mechanical system has the primary task of driving two voltage-variable devices, having an output proportional to sonar range. The Eq mechanical system drives a resolver which provides two voltages-one proportional to sin Eq and the other proportional to cos Eq. The 207
 Figure 10-10 -OKA-1 simplified functional diagram. 208 latter voltage is further altered by the ratio of sound velocity below the thermocline to sound velocity in the isothermal layer. The device effecting this alteration is the circuit associated with the B adjustment dial of the range computer unit. The B dial is set to ΔV obtained from BT readings and computed according to fleet doctrine. In the circuit associated with the B adjustment the equation (10-1) is solved to provide the sin Eqr output. This altered signal is supplied to an amplifier driving a servo system to provide the sine of an angle defined as the refracted depression angle, Eqr. Cos Eqr is converted into sin Eqr by means of a servo system, because the sine of Eqr is required by the Vz frequency generator. The two signals, sin Eq and sin Eqr, form the basic speed controls for a rate servo and power amplifier combination delivering a variable frequency to the recorder stylus motor of the depth-determining equipment. It has been established previously that Vz, the vertical component of the velocity of sound, is defined as V sin Eq before refraction and (V-ΔV) sin Eqr after refraction. The maximum velocity difference provided in the refraction adjustments is 200 feet per second, and because V is of the order of 5,000 feet per second, (V-ΔV) cannot differ from V by more than 4 percent. The calculation of Eqr cannot be considered valid to an accuracy of greater than 4 percent and therefore it is assumed that the sound velocity itself, before and after refraction, is the same, and in actual magnitude may be some average value. This average value of V is designated Va and is set on the A adjustment dial in the Vz frequency generator. The output frequency at the Vz frequency generator is proportional to Vz and equals Va sin Eq and Va sin Eqr before and after refraction, respectively. The layer depth contacts of the depth recorder govern selection of the depth at which Eqr is substituted for Eq. The entire combination is shown for simplicity in the diagram as the Vz frequency generator. The Vz frequency generator supplies its output to the stylus motor of the depth recorder, the stylus of which is to be driven at a speed proportional to the Vz frequency generator output. The stylus then provides target depth, Hq, indication. An equipment function not depicted on the diagram is the keying and clutch control of the depth-determining equipment with the related function of synchronization of keying of both sonar equipments. It is essential that these equipments be synchronized in order that transmission pulses of one equipment do not blank out incoming echoes on the other equipment. Synchronization is accomplished simply by supplying synchronized keying pulses to the sonar equipments. An inspection of the recorder chart and the positioning of a suitable index to the average indications of the chart enables an observer to determine target depth. Target depth could be considered a third item of primary importance injected into the calculating circuits, but target depth is a calculation modified by observer opinion in respect to the exact setting of the index, and hence is not primary externally derived information. A 1G transmitter geared to the depth recorder index makes available an order of relative target-depth- H'q at 2,000 feet per revolution. This target-depth calculation is accepted in the calculating circuits by a 1CT and associated servo amplifier, making depth available as a mechanical function. A 5G transmitter displaced with respect to the 1CT by an angle equivalent to the projector depth below the surface, Pvq, makes true target depth, Hq, available for remote indication. A voltage variable device driven by this system provides a voltage proportional to target relative depth, H'q. For simplicity, these circuits are shown as a single block marked "target-depth servo system" in the diagram. The voltage H'q is combined with Rq from the range servo system to produce sin cEtq- H'q / Rq = sin cEtq.  (10-3) This signal then drives two resolvers, one of which gives outputs equal to cos cEtq and sin cEtq. In the other, range rate, dRq, is multiplied by sin cEtq, producing dRq sin cEtq. This value in turn is combined with horizontal range, Rhq-the derivation of which will be explained later-in a comparison servo system to produce ΔcEq which is supplied to the depth-determining equipment as the aiding tracking signal. Horizontal range is computed by taking the cos cEtq output previously mentioned and combining it in a voltage-variable device in accordance with the equation- Rq cos cEtq=Rhq.  (10-4) 209 This output is then fed (1) to the horizontal range indicators which utilize 1-speed and 36-speed to position its stylus, and (2) to the ΔcEtq computing circuits explained in the previous paragraph. The ΔcEq rate servo system performs the function of transmitting the incremental computed depression angle, ΔcEtq, to the depression control in the depth-determining equipment. The advantage of providing the rate of change of cEtq rather than cEtq directly as aided tracking is that cEtq may be changed abruptly and cause loss of contact with the target. When for any reason the depth recorder operator moves the cursor to a different position, the H'q order changes and therefore angle cEtq suddenly has a different value. If the order cEtq were supplied directly to the depression control, the system would tend to move off the target and it would be difficult to regain contact. When the rate of change of cEtq, is supplied to the depression control, however, Eq is not immediately affected by changes in H'q, but the rate at which cEtq changes provides the aided tracking to facilitate maintenance of contact. The relationship which the ΔcEq rate servo system must satisfy may be demonstrated best by reference to the simplified diagram in figure 10-11, in which point O represents the ship's transducer, and points A and C two respective positions of the target. As the target moves from point A to point C, the change in slant range, ΔRq, is equal to the distance BC and the change in depression angle, ΔcEtq, is equal to angle AOB. Because small angles expressed in radians are equal to their sines, ΔcEtq=AB/AO,  (10-5) Figure 10-11 -Depression-angle aided tracking diagram. Angle ACB is essentially equal to cEtq, and distance OA is equal to Rq. Therefore, omitting second-order differences, equation (10-5) may be restated as follows: ΔcEtq = -(ΔRq tan cEtq) / Rq = (ΔRq sin cEtq) / (Rq cos cEtq) = (ΔRq sin cEtq) / Rhq. If dRq is substituted for ΔRq and direction is neglected, dcEtq = (dRq sin cEtq) / Rhq.  (10-6) The manner in which the ΔcEtq rate servo system accomplishes the function expressed in equation (10-6) is described as follows. The second voltage variable device in the cEtq servo system is excited by a voltage proportional to range rate dRq and produces as its output a voltage which is proportional to the range rate, dRq, times sin cEtq. The voltage constitutes the input signal to the ΔcEq rate servo system. The rate servo previously described in this chapter was provided with fixed excitation to the induction generator. The ΔcEq rate servo, however, is provided with excitation which is proportional to horizontal range, Rhq. This excitation voltage is derived from a power amplifier the input signal of which is voltage Rhq, previously described as being the output of a voltage variable device in the Rq servo system. The output of the induction generator is dependent upon its speed, and upon the magnitude of its excitation voltage. If the input signal to the rate servo is constant, the speed of the motor generator must be equal to some constant times the input signal in order to provide a generator output voltage which is equal to the input signal. Actually, the output voltage must be slightly less than the input signal to furnish a difference signal which is amplified to drive the motor. The magnitude of this difference signal is dependent upon the gain of the amplifier, and in this system the gain is sufficient to make the difference signal required small enough relative to the input signal to be disregarded in the required calculations. The generator output voltage, therefore, equals the rate servo input signal. 210
 Figure 10-12. -OKA-1 function in the sonar system. 211 The induction generator output may be expressed as an equation- E= K X S X Rhq,  (10-7) where E is the generator output voltage, K is a constant, S is the speed of the motor generator, and Rhq is the excitation voltage. But the generator output voltage equals the input signal to the rate servo system, and therefore, K X S X Rhq = dRq sin cEtq or, K X S = (dRq sin cEtq) / Rhq From equation (10-6)- K X S = dcEtq.  (10-8) Thus the speed of the ΔcEq rate servo motor is proportional to the rate of change of cEtq. A 1G synchro transmitter is geared to the motor, making the incremental apparent depression angle, ΔcEq, available as a synchro order. A control circuit is provided in the OKA-1 equipment whereby the full scale value of the depth-recorder chart can be changed from 1,500 to 150 feet. This change is effected by causing the Vz rate servo to operate at a speed proportional to 10 sin Eq instead of sin Eq, when excitation is applied to the expanded scale relay in the range computer. It is imperative that Eq be at least 2° when operating with the expanded scale on, because a limit switch in the Eq system stops the Vz generator when Eq is 2° or less. For reference, figure 10-12 shows a complete functional diagram of the OKA-1, including the range recorder. 212
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