Matched Transformers for Synchro and Resolver Applications

Electrocomponent Science and Technology (C) Gordon and Breach Science Publishers Ltd. 1975, Vol. 2, pp. 121-134 Printed in Great Britain MATCHED TRANSFORMERS FOR SYNCHRO AND RESOLVER APPLICATIONS M. PRATT Professional Components Division, Ferranti Ltd., Dundee, UK (Received December 17; 1974; in final form March 3, 1975) Transformer pairs in Scott Tee and similar transformer arrangements have been used for some considerable time in precision synchro/resolver angular measuring gear and in synchro to digital converters. Literature on the subject of transformer requirements is, however, scant or non-existent.I, 2 This paper describes the basic principle of transformer operation and develops a design approach which, although aimed primarily at the minimisation of transformer hardware size, still maintains the required level of angular accuracy. The method applies to transformers utilised in mobile systems, particularly transformer arrangements working under loaded conditions. INTRODUCTION manufacturing approach leading to minimised weight and volume, especially in transformers looking Synchro shaft angle to digital conversion techniques towards and positioning the synchro. (Digital to are used extensively in airborne and shipborne synchro mode application.) systems to give direct read-out positional data on synchro and resolver elements with an ability to 2 THREE TO FOUR WIRE CONVERSION reposition synchro devices from a central control computer. The use of transformers in three to four wire Shaft angle data, to and from the control point, is conversion is readily understood by first considering by a three wire system, usually with the synchro Figure 1. which is descriptive of a synchro element elements energised at a frequency of 400 Hz. connected to a pair of transformers in an arrangement For accurate translation of the positional data, it is known as the Scott Tee connection. This transformer convenient to perform a three wire to four wire connection is used basically in power applications to conversion to produce two output signals propor- give three phase to two phase conversion. Used as tional to the sine and cosine of the synchro angular shown, the line-to-line voltage outputs from the position. By suitable process methods, a single synchro will be in the form: can obtained which is analogue signal be readily gs sin 0 converted to the equivalent digital angle. Vl While various conversion methods can be used, it V2 3 Es sin(0 + 120) can be demonstrated that the incorporation of Vl 2 gs sin(0 + 240) specially matched transformers in the synchro to where Es is the maximum voltage across any two lines. digital loop will, in addition to a general simplifica- and 0 is the synchro electrical angle. tion of the converter design, give added benefits in the way of: It should be noted that the value Es can be written in the instantaneous form es sin cot and is related to the (a) Complete isolation, input to output. voltage induced in the three stator coils from the (b) A high degree of stability against extreme energised rotor coil. For simplicity, however, only the climatic and temperature variation. peak voltage values need be considered. An to withstand sustained overloading (c) ability l/r2 3 and V12 can be re-arranged to give: with general overall rugged characteristics unsur- passed by other components. -1/2 sin 0 + cos 0 Mobile applications demand a proper design and V: 3 Es 121 -- 122 M. PRATT R 1 V 2 3% V, cos 0 Vo cos 0 2 R2 0 Synchro Angle E MAX L-L voltage V3 s Vi sin/9 $3 Vo sin/9 FIGURE 1 Synchro-to-transformer connections. 3 % transformer will indicate this in the form: V12 =Es sin O cos 0 (_1A E. The volts across the individual synchro stator coils Vcr Cos 0 + Cos 0 will be: 2- 3 % Es Cos 0 V2 3,---S cos 0 2 3 w V1 s 1/2 c s O sin 0 (1) Also 7E ( ) Es 3v2 and VCT2 T sin 0 Es v3 (2) sin 0 ( -cos0+Ts0 2 Writing down the volts across---each half of the centre and tapped transformer as E 3 v2 s sin 0 c ( + 73) VCTx r and substituting for d from Eqs. (1) and (2) V V sin 0 gives 2 Es Cos 0 From this it will be seen that the signal voltage across Vcr 2 3v the whole of the tapped tramformer behaves according to the sine of the synchro-- angle, while that across Further consideration of the volts across the untapped the untapped transformer behaves according to the MATCHED TRANSFORMERS 123 cosine of the synchro angle. The volts amplitude of Consideration of transformation errors, related to the the latter is, however, 31/2/2 times that of the tapped mechanism of four wire coversion, will indicate that transformer. the voltage excitation across the tramformers in the With an equal turns per volt relationship on each Scott pair, as already indicated, is in a differential transformer, output windings of equal turns will give mode, giving a coincidence of maximum and output voltages varying with the respective inputs and minimum core flux change in each unit. Only at the equal to each other in maximum amplitude. mid points (45 degrees, 135 degrees etc) in each quadrant of synchro travel will the excitation be equal. This is illustrated graphically in Figure 2. 3 ERRORS AND ERROR CAUSES The effect of this variable excitation will be to vary the apparent voltage transformation ratio The two main sources of system error are through the transformers, the voltage transformation being the actual measured input to output voltage (a) Errors through the tramforrners ratio as against the physical turns ratio value of each (b) Errors due to the tramformers loading the transformer. Related to the mid qua&ant points, the synchro element. tendency is to give an apparent decrease in the ratio decreases an increase in the ratio Investigation of error mechanism, allied with a as the excitation and the excitation points. knowledge of synchro action, will indicate that towards high If errors the transformers are minimisation of most loading errors, both through angular through two of synchro travel, the transformers and back to the synchro element, plotted over the first qua&ants the ratio of the two outputs as the are directly related to the degree of balance obtain- by extracting of the angle over the first and third octants able in the series and shunt impedance values. tangent and the co-tangent of the angle over the second and fourth octants, the error factor under classic con- 3.1 Transbrmation Errors ditions will be as illustrated in Figure 3. will indicate that, Considering errors through the transformers, the two Examination of this error form the 45 and 135 degree main contributory parameters are considered from degree balanced points, the resultant errors will bear a direct (a) Transformation voltage ratio relationship to the variation of the effective transfor- (b) Series winding impedance mation ratio through both tramformers in the pair. Output- max cosine zero sine 0...... Output- max sine zero cosine Equal outputs 180 FIGURE 2 Differential voltage distribution. 124 M. PRATT 0 20 40 60 80 100 120 140 160 180 L _.__1 1 l .I _[ 1. J Synchro Rotation Degrees l-z FIGURE 3 Idealised error form. Matched winding resistance. R2 Roc Vo cos 0 $3 R3 R 1 Ros Vo sin O S 1 FIGURE 4 Equivalent circuit showing winding and load resistances. 3.2 Loading Errors though the Transformers Roe are the output winding resistances of the sine and cosine transformers, and the input The variation of output terminal voltage on each R1 R3 resistance of both halves of the sine transformed and transformer and the variation of the voltage ratio the input winding resistance of the cosine between with the of R transformers, application loads transformer. across the output, will be dependant on the value of Referring all values to the appropriate output transformer series winding impedance and the value windings will give" of the applied loads. Consider Figure 4. which represents the equivalent R(sin) R' +R3 circuit of a Scott Tee transformer pair loaded on both +Ros outputs with matched resistors R];. Resistors Ros and Rs + Ros MATCHED TRANSFORMERS 125 1R3 care that the output voltage balance can be obtained R2 + +Ro = +R3 under two distinct conditions, i.e. _4 Rs (a) Completely matched winding and load con- -3 R2 +Roc ditions. +- (b) Winding resistance values tending towards where R's R'I + R'2 Total input sine zero. resistance referred to the output winding. The latter condition implies a unit tending to be infinitely large and, therefore, unacceptable in com- R Input cosine resistance pact systems under consideration. referred to the output With imposed requirements of matched winding winding. resistance, it is clear that a method must be derived to The effective voltage at each output will be: lay down limits of allowable winding resistance variation between the transformers in the pair and Fo(sin) KE's sin 0 Io sin O(R's + Ros) between the windings on each transformer. Writing down the output volts in terms of winding and resistance and load resistance will give :- + Vo(cOs) KE cos O lo cos O [4 , + __R's) + Roc s'n 0 0 The first term in the expression is the voltage due to and transformation through the transformers; the second term is the voltage drop caused by the load current Io Vo(cOs) E' K cos O cos O through the total series winding resistance on each s R+ +-L transtbrmer. At the mid quadrant points, both outputs will be The output voltage ratio at 45 degrees can be written equal as will each term in the equation.

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