As a numerical example let a = 10°, VFS = 10.24 V, M = 12, and Vp = 1 V; this leads to 4e2(max) = 0.072°. In order to reduce this error we should increase M and Vp, and decrease a. Table II illustrates the change of a/sin a with et. It is obvious that we have to choose a reasonably high sampling frequency which is at least, say, four times the signal frequency.
IV. OVERALL ERRORThe phase error due to mislocating a single zero-crossing is +± 4eI ± 0ke2 However, the overall error ke is due to mislocating the two zero-crossings of the signals x(t) and y(t), and is given by =e = ± Oelx ± Oe2x ± + ely ± ¢e2y (16) From the previous analysis it is obvious that kei of Table I has a double hump shape in the range 0 < 0 c a, whereas Oe2 is maximal when 0 = 0 or a. Consequently, the overall worst-case Oe is not the sum of the worst-case 4)el and ke2, but depends on the various parameters involved.V. CONCLUSIONS Calculating the phase difference between two signals, using synchronous real-time sampling, has been proposed. The method is based on linear interpolation, which gives the least measurement accuracy. It was shown that this accuracy can be improved by increasing the sampling frequency, the signal amplitudes, and the number of A/D quantization levels. However, more accurate results are expected if we resort to computer-based nonlinear interpolation, since the sampled values can be fitted into a sinusoidal function.ACKNOWLEDGMENT As a first author, I would like to present the research of my life to the memory of my mother, Hekmat Mahmoud Fawzy, who passed away in Cairo, Egypt, on December 13, 1984, during the preparation of this paper. Exceptional signal stability and rapid response when changing operating parameters can be achieved by using a method of digital waveform generation for the analog output [1]. With this method two sets of calculated values, representing sample points at equal time intervals along two waveforms, are converted to voltages using dual-channel analog-to-digital converters. The resulting stepped sine waves (see Fig. 1 Even though converter characteristics depart from the ideal, the spectral purity of the output sine wave can be made relatively high by using a reasonable large number of sample points per waveform. The lower limit of the harmonic content, assuming that the sampling harmonics have been filtered out, is given by the quantization noise which for an ideal 16-bit converter is -97.8 dB, and -73.8 dB for a 12-bit converter [4]. Fig. 3 shows measured spectra for 50/60 kHz using 16-bit conversion. The quantization noise is more evident for the 12-bit conversion at 50 kHz and appears in the form of larger harmonics (Fig. 4). The measured harmonic components, which are above the theoretical limit, are attributed to imperfect converter and amplifier characteristics. pling rate, resolution is traded for conversion speed with faster 12-bit converters in place of the 16-bit converters used at lower frequencies. Active low-pass filters with a corner frequency of approximately 0.35 MHz reduce...