VERNIER RESOLVER 1489 



number of laminations (one hundred in the present units) the effect of 

 individual imperfections in laminations is greatly reduced. 



In preparation for a close study of the vernier resolver we shall describe 

 the performance of an ideal unit, and also introduce some technical 

 terms. The output of the vernier resolver consists of two amplitude 

 modulated voltages one of which is called the sine-voltage and the other 

 the cosine voltage. The amplitudes of these voltages are proportional to 

 the sine and cosine of "n"-times the rotor orientation. The factor "n" 

 which, of course, is a function of the rotor configuration will be called 

 the order of the resolver. We shall call the arctangent of the ratio of the 

 secondary voltages — sine-voltage over cosine-voltage — the "signal- 

 angle". Furthermore, to define a positive sense of rotation and to make 

 the signal-angle definition unambiguous, we shall assume that, with 

 continuous positive shaft rotation, the signal-angle runs through a se- 

 quence of cycles, each going from zero to 360°. Thus, one signal-angle 

 cycle corresponds to a shaft rotation of (l/w)th; of one revolution. This 

 angular interval is called the "vernier" interval. 



II. DESIGN PRINCIPLES 



2.1 .1 Simplified Description 



Fig. 1 represents a simplified model of a third order vernier resolver. 

 The unit consists of a laminated rotor with three equally spaced teeth 

 and a laminated 4-pole-shoe stator. Each pole-shoe bears one exciting 

 coil (not shown in the figure) and one output coil. Successive exciting 

 coils are wound in opposite directions, connected in series, and energized 

 from an ac source. Thus, successive pole-shoe fields alternate in phase. 

 The two output windings each consist of two diametrical output coils 

 connected in phase opposition. 



If the rotor were a circular cylinder, the net voltage in either output 

 winding would be zero. However, because of the three rotor teeth, the 

 induced voltage of either output winding goes through three identical 

 cycles per rotor revolution. Consequently, the amplitude Ec of the in- 

 duced cosine-voltage ec can be represented as a Fourier series of three 

 times the shaft angle dm , 



Ea = E, cos (30,„) + E, cos [3(30„,)] + • • • , (1) 



where Ei , E3 are the Fourier components of Ec with respect to (3^m). 

 The series is free of even harmonic terms because of the s\'mmetry be- 

 tween positive and negative half-cycles. The expression for the ampli- 

 tude Es of the sine-voltage Cs is obtained by substituting [6^ — (tt/G)] 



