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Sine-wave crossing technique as a basis for a sinusoidalphase shifterR. S. FYATH a & A. K. HASSAN aa Department of Electrical Engineering , College of Engineering, University of Basrah ,Basrah, IraqPublished online: 24 Feb 2007.

To cite this article: R. S. FYATH & A. K. HASSAN (1993) Sine-wave crossing technique as a basis for a sinusoidal phase shifter,International Journal of Electronics, 75:4, 655-664, DOI: 10.1080/00207219308907141

To link to this article: http://dx.doi.org/10.1080/00207219308907141

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INT. J. ELECTRONICS, 1993, VOL. 75, NO. 4, 655-664

Sine-wave crossing technique as a basis for a sinusoidal phase shifter

R. S. FYATHt and A. K. HASSANt

A versatile phase-shifting scheme for processing sinusoidal signals has been implemented using a sine-wave crossing (SWC) technique. The circuit is capable of producing a continuous phase shift over the range 0-360" using one potentiometer. Experimental results from a laboratory set-up support the analysis and demon- strate the feasibility of this approach.

1. Introduction Circuits wich produce a fixed phase shift across a band of frequencies are

required in certain areas of electronic instrumentation, measurements and communi- cations. The conventional techniques for providing a phase shift for a sinusoidal signal usually use R C networks (National Semiconductor 1977).

The drawbacks of these circuits arise from the frequency dependence of both the amount of phase shift, and the amplitude of the output signal. Recently, Wilson (1990) proposed precision phase-shifting networks which exhibit very low phase deviations over relatively wide bandwidths. Unfortunately, these circuits should be designed only for fixed phase shifts. On the other hand, Barker (1978) has described another scheme that avoids the use of reactive components and this scheme is capable of generating frequency independent phase shifts up to 90'. The complexity of this circuit is due to the use of three analogue multipliers which require perfect balance otherwise undesirable signals may be generated. An alternative phase- shifting approach is based upon generating the required shift for square waves then using square-to-sine wave converters (Christiansen et al. 1988). These converters are usually implemented with a square-to-triangular wave stage and are then followed by a smoothing filter (Middebrook and Richer 1965) or a sine function shaping circuit (Graeme et al. 1981) that transfers the triangular waves into sine waves. These converters should be carefully designed to reduce the harmonic distortion at the output.

In this paper a new phase shifting scheme based on the sine-wave crossing (SWC) technique is proposed and implemented for sinusoidal signals. The essential features of this method are the possibility of using comparators rather than analogue multipliers and the ability of changing the phase shift from 0 to 360" using a single potentiometer.

2. The proposed scheme The basic idea of our scheme is illustrated in Fig. I (a) which is basically a

quadrature amplitude modulator using two D C signals a and b to modulate the sinusoidal carrier ei(t)cosw,t. The output signal e, is given by:

e,(t) = J(a2 + b2)'I2 cos (wt, - 0) (1) where O = tan-'alb is the required phase shift. By varying the D C signals a and b, a continuous phase shift covering the range 0-360" can be obtained.

Received 4 July 1992; accepted 22 July 1992. tDepartment of Electrical Engineering, College of Engineering, University of Basrah,

Basrah, Iraq.

002C-7217193 $10.00 0 1993 Taylor & Francis Ltd.

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R. S. Fyarh and A . K. Hassan

b cos w,t COO

I A, cos w , l

network hr rectangular pulses

Figure 1. Illustrative block diagrams: (a) conventional quadrature amplitude modulator; (b) quadrature amplitude modulator using SWC technique; (c) single oscillator SWC modulator.

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Sine-wave crossing technique 657

Expressions relating both a and b to the control voltage V,.

The main limitations of this system come from (i) the use of multipliers adds complexity to the circuit; (ii) the cosine-to-sine converter requires a precise phase shifting scheme; and (iii) the phase shift can be controlled by two signals a and b rather than one.

To overcome the first limitation we use a quadrature amplitude modulator based on the SWC technique (Piwinicki 1983). This method is simple since it employs

710 : Cornoarator I I (-17 1

Figure 2. A proposed circuit for generating the DC voltages a and b using one potentiometer p.

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R. S . Fyarh and A . K. Hassan

Figure 3. Circuit diagram of the proposed full wave rectifier.

comparators and an Ex-OR gate instead of multipliers (see Fig. I (b)). The theoretical background behind this modulator is outlined briefly in the Appendix. The output of the bandpass filter, centred a t the input frequency o o , can be expressed as:

e,(r) = k[a, sin wot + b, cos w,~] = k,/(ai -+ b:)'l2 cos (wOt- 0 ) (2)

where a,=a/A, and b,=b/A, with A, the amplitude of the input signal. Further B=tanW1 alh and k is constant. The DC voltages a and b should be bounded to AJJ2 which is an essential requirement for such a modulator (see the Appendix).

To use a single oscillator instead of the quadrature one, the DC voltage a is also compared with cos o,t instead of sin o0r and then the resultant pulses are delayed by a quarter of the period while keeping the same pulse width a t the output (Fig. 1 (c) ) . This is the main part of our work and will be described in detail in 54.

Figure 4. Waveforms at different points of the circuit shown in Fig. 3. Horizontal scale: 0.2 msdiv-'. Vertical scale: input and output 50 mVdiv-I, square wave 5 Vdiv- '.

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Sine-wave crossing technique

/ -Predicted

. . Measured

I I I 0.2 0.4 0.6 0.8 I .O

Peok input voltage ( V )

Figure 5. The relation between output and input peak voltages for the full-wave rectifier.

The next step is to set the required phase shift using a single control voltage. This will be explained in the following section.

3. Phase shift adjustment We have adopted an efficient approach to control the phase shift using a single

control voltage V,. This voltage is related to the peak amplitude of the input signal A, and can be varied from -2V0 to 2V0 to cover the range from 0" to 360". Here, V, is a positive DC voltage which is bounded to A0/2J2. The D C levels a and b are generated from V, according to the relation summarized in the Table and illustrated schematically in Fig. 2.

The performance of this part of the phase shifter is governed by the accuracy in measuring the peak amplitude A,. This has been carried out using a precision full-wave rectifier based on the zero-crossing technique (see Fig. 3) followed by a low pass filter with 1.57 (=n/2) D C gain. This circuit avoids the use of diodes and hence can be used to rectify low-level signals. Figure 4 shows waveforms at different points of the rectifier circuit while Fig. 5 illustrates the relation between the input and output peak voltages versus the peak input voltage, which indicates a good linearity and accuracy for a peak input voltage greater than 30mV. This threshold level can be extended to a lower value if a perfect zero-crossing detector is used. The same approach has been used to realize the 'absolute value' circuit in Fig. 2.

4. The delay network This section describes a simple and versatile circuit for producing a 90"

frequency-independent phase shift for rectangular pulses while maintaining the same input pulse duration a t the output. Figure 6 shows a block diagram of the proposed circuit along with its timing waveforms. The input waveform e , having a period T

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660 R. S. Fyath and A. K. Hassan

and width T is first converted to very narrow pulses e , a t the rising edges using a monostable multivibrator. The pulses are used to trigger the sawtooth generator to yield the waveform e , having a peak amplitude E which is detected by the peak detector. The sawtooth waveform is compared with two DC voltages e,= E / 4 and e,= E / 4 + E T I T to generate the required output pulses e,. The DC voltage E T / T can easily be obtained by sampling the peak value E by the input pulse train and filtering the resultant waveform e , . A detailed diagram for a practical implementation of this circuit is given in Fig. 7. The circuit has been tested and its performance is displayed in Figs 8 and 9. These results demonstrate conclusively the ability of such a system to delay the coming pulses by a quarter period while maintaining the same pulse

Time

Figure 6. (a) Block diagram of 774 delay network; (b) related timing waveforms

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Sine-wave crossing technique 66 1

Figure 7. Circuit diagram of the delay network.

width at the output over one decade of frequency. To extend the frequency response of the circuit the capacitor in the sawtooth generator should be altered for different decades o r by changing the charging current through this capacitor automatically according to the input frequency using a frequency-to-voltage converter.

5. Results A phase shifter of this type has been fabricated and tested. A simple LC tuned

circuit is used to select the required signal in the SWC modulator and the amount of phase shift is controlled by a precision potentiometer. Figure 10 depicts the variation of the amount of phase shift measured for 10 kHz, 0.5 V peak sinusoidal signal as a

Figure 8.

Input pulse width (nsl

Measured output pulse width and introduced phase shift versus input for the delay network.

pulse width

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R. S. Fyath and A. K. Hassan

Figure 9. Frequency response of the delay network.

- P 8- 0

6

L

2

0

f = I 0 kHz V i = 0.5 V peak

- Theoretical r . s r Experimental

r = 2 0 ns <,:+I5 V

-

-

- I I 1

Figure 10. Variation of the phase shift as a function of the attenuation factor a of the potentiometer.

5 10 20 30 Frequency (kHz1

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Sine-wave crossing technique 663

function of the attenuation factor u ( O S a S I) of the potentiometer. The dots in Fig. 10 refer to the experimental data points while the solid line indicates the theoretical performance, which is related to tan- a according to the Table. One can clearly observe a good correspondence between practical and theoretical character- istics. The variation of the phase shift as a function of the frequency of the input signal has been experimentally investigated. In this measurement the resonant frequency of the tuned circuit is adjusted manually and set equal to the input frequency. The phase error is reported to be less than k0.5" over the frequency range I-IOkHz. This error comes mainly from the performance of the simple sawtooth generator used in this circuit.

A better performance is expected when the passive bandpass filter is replaced by an active one whose resonant frequency can adapt itself automatically by sensing the input frequency. Care should be taken in designing this scheme since the phase shift is strongly affected by the time delay introduced by the filter. This is the subject of future work.

6. Conclusions The feasibility of using a sine-wave crossing technique to realize a versatile phase

shifter for sinusoidal signals has been demonstrated. The circuit comprises a quadrature amplitude modulator, implemented without analogue multipliers, plus a T/4 delay network for processing rectangular pulses. The required phase shift can be controlled from 0 to 360" using one potentiometer.

Appendix A quadrature amplitude modulator based on the SWC technique

A bounded band-limited signal can be described by a set of points on the time- axis at which it crosses a reference sine-wave whose magnitude and frequency are greater than the peak amplitude and the bandwidth of the signal (David 1974). Based on this representation Piwinicki (1983) proposed a double-edge position square-wave modulator (see Fig. 1 (b)) where the spectrum of modulated pulses p(t) may be expressed as sums of an infinite number of sinusoidal carrier amplitudes modulated by Chebyshev polynomials of the modulating signals a(t) and b(t) (in general a and b are time-varying signals).

2A ~ ( 1 ) = dA + C

n = l (A 1)

where A is the amplitude of the modulated pulses, d is the normalized value of the D C component (with respect to A ) and

Where T, represents the Chebyshev polynomials of the first kind of degree n. In (A I) the two modulating signals a([) and b(t) are taken to be band-limited such that the carrier frequency w, is greater than the highest frequency component of both signals and their peak amplitudes are bounded to AJJ2 so the intersection of both modulating waveforms with the carrier will be interlaced (Piwinicki 1983). Here a(t) modulates the positive edges of the square wave while b(t) modulates the negative ones.

A suitable bandpass filter with centre frequency o, can be used to select the

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664 Sine-wave crossing technique

spectrum of n = I and t o reject all other terms. T h e filter ou tput is given by

e, = k[a(r) sin o0r + b(t) cos wet] ( A 3)

where k is a constant related t o the gain of the filter. Equation ( A 3 ) shows a quadrature modulated signal o n a carrier frequency o, modulated by both inputs.

REFERENCES BARKER, R. W., 1978, Phase-shifting technique without reactive components. lnrernarional

Journal of Elecfronics, 44, 573-576. CHRISTIANSEN, C. F., HERRADA, J. L., VALLA, M. I., and MARTINEZ, N. H., 1988, Further

improvement in a three-phase sinewave generator. I.E.E.E. Transacfions on Industrial Elecfronics, 35, 338-339.

DAVID, B., 1974, An implicit sampling theorem for bounded band-limited functions. hter- national Journal of Confrol, 24, 36-44.

GRAEME, J. G., TOBEY, G. E. and HUELSMAN, L. P., 1981, Operafional Amplifiers Design and Applications (Tokyo: McGraw-Hill).

MIDDI~BROOK, R. D., and RICHER, I . , 1965, Non-reactive filters converts triangular waves to sines. Electronics, 8, 995- 100 1.

NATIONAL SEMICONDUCTOR, 1977, FET Data Book. 0-360" Phase Shifter (Santa Clara, CA). PIWINICKI, K., 1983, Modulation methods related to sinewave crossings. I.E.E.E. Transacrions

on Conmiunications, 31, 503-508. WILSON, G., 1990, Precision RC-active frequency compensated phase shifters. I.E.E. Proceed-

ings on Circuits. Devices and Systems, 137, 44-48.

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