Generation of a sine wave from two in-quadrature triangular waves
3
284 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-31, NO. 4, DECEMBER 1982 . B0l· ■ 50 (0 <- 30 fu > ^20 (ü -10 h O : t h e BVRg des i gn 1 T-1.9BK,Q-5.4E7,CF-16.4 O :the conventional RT c u t design Γ T-1.9K,Q«9.2E6,CF-1.8 o h O 1E-9 fractional O L _ frequency shift - °°°Oo 0 o 0 <> ° o o o o o o o ° - -70 -60 -50 Dissipated Pouuer -40 -30 i n dBm Fig. 2. Frequency versus dissipated power in crystal resonator near 2 K for the same samples as shown in Fig. 1. The zero on the vertical axis is chosen arbitarily. at 4.2 K for the same crystals, where the temperature coefficient is almost ten times that at 2 K, did not show any noticeable change in the results shown in Fig. 2. DISCUSSION The very small temperature coefficient near 2 K and increased Q imply that the use of cryogenic temperature can be a promising way for a highly stable crystal-controlled oscillator, along with the fact that the temperature regulation of 10 microdegrees is possible within a helium dewar [6] and the inherent \/f frequency fluctuation of crystal resonators decreases following a l/g 2 -dependent law [10]. Although the drive level of a crystal unit at cryogenic temperature must be held at a low value, the increased Q will make it possible to maintain enough signal-to-noise ratio when it is applied to a frequency control system. To obtain better performance, crystal resonators with optimum cut for cryogenic temperature operation should be used [4], [11]. Frequency stability of 4 X 10~ 14 has been achieved by a low- temperature crystal oscillator based on a tunnel junction diode [4]. The application of the cryogenic-temperature crystal resonator to the passive dual crystal system has also been studied [12]. ACKNOWLEDGMENT The author wishes to thank S. R. Stein for providing him the op- portunity to perform this work and for many helpful discussions. He acknowledges the many beneficial discussions and suggestions from C. M. Manney and F. L. Walls. Thanks are also due to Prof. R. J. Besson for the offers of several quartz crystals. REFERENCES [ 1 ] A. W. Warner, "Ultra-precise quartz crystal frequency standards," IRE Trans. Instrum.,\o\. 1-17, pp. 185-188, Dec. 1958. [2] G. Mossuz and J. J. Gagnepain, "Quartz crystal oscillator at very low température," Cryogenics, vol. 16, no. 11, pp. 652-656, Nov. 1972. [3] A. G. Smagin, "A 1-MHz quartz resonator with a Q factor of 4.2· 10 9 at a temperature of 2°K," lustrum. Exp. Tech., vol. 17, no. 6, pp. 1721-1723, Nov.-Dec. 1974. [4] , "A low-temperature oscillator with a frequency stability 4 \0- ]4 :'Instrum. Exp. Tech., vol. 18, No. 6, pp. 1853-1855, Nov.-Dec. 1975. [5] R. J. Besson, "A new electrodeless resonator design," in Proc. 31st Ann. Symp. on Frequency Control, pp. 3-7, 1977. [6] S. R. Stein, "Application of superconductivity to precision oscillators," in Proc. 29th Ann. Symp. on Frequency Control, pp. 321-327, 1976. [7] J. J. Gagnepain, J. C. Poncot, and C. Pegeot, "Amplitude-frequency behavior of doubly rotated quartz resonators," in Proc. 31st Ann. Symp. on Frequency Control, pp. 17-22, 1977. [8] V. J. Johnson, "Compendium of the properties of materials at low temperature," WADD Tech. Rep. 60/56 (reprinted by Pergamon Press, 1961). [9] G. K. White, Experimental Techniques in Low-Temperature Physics. Oxford, England: Oxford Univ. Press, 1968. [10] J. J. Gagnepain, J. Uebersfeld, G. Goujon, and P. Handel, "Relation between 1/f noise and Q factor in quartz resonators at room and low temperatures. First theoretical interpretation," in Proc. 35th Ann. Symp. on Frequency Control, pp. 476-483, 1981. [11] F. P. Phelps, "Stability of quartz resonators at very low temperatures," in Proc. llth Ann. Symp. on Frequency Control, pp. 256-276, 1957. [12] B. Komiyama, "Quartz crystal oscillator at cryogenic temperature," in Proc. 35th Ann. Symp. on Frequency Control, pp. 335-339, 1981. Generation of a Sine Wave from Two In-Quadrature Triangular Waves ROBERT GENIN Abstract—A simple and effective method is proposed for transforming two in-quadrature triangular waves into a low-distorsion sine wave. A circuit per- forming this operation is described that includes two multipliers and an absolute value device. Several recent papers [l]-[3] described circuits which can generate simultaneous, triangular periodic waves, 90° out of phase. Useful complement to these circuits as function generators would be to shape the triangular waves into a sine wave with low distortion. In this short paper we report a method and we describe a circuit to solve this problem. Product P{t) of two in-quadrature triangular waves e\(t) and ^ ( 0 of period T is a periodic function of period T/2 (Fig. 1). The graph of function P(t) is formed by a sequence of parabolic arcs. It has the same properties of symmetry as a sine curve and, considered as an approximation to a sine curve, it may be characterized by a total harmonic distortion equal to 3.8 percent. Note also that the derivative of function P(t) is a continuous function of /. Let us introduce the following notations: B amplitude of P(t), u = 4t/T. A positive parabolic arc is represented by the function P = 4Bu(\ - «), 0 < w < 1. (1) The sine wave of same amplitude and period is then S = Bsinwu. (2) Functions P and S are equal for u = 0, u = 1 (P = S = 0), and for u = ^(P = S = 1 ), for other values we have P > S. To obtain a signal with low distortion we may attempt to correct signal P(t) generating signals of the form S = Pf(\P\) (3) S = Pf(P 2 ) where function/is chosen so that signal S has a total harmonic dis- tortion as low as possible. We adopt the First expression using absolute value of P which leads to a simpler realization. Ideally, the function f(\P\) is defined b y / ( | P | ) = S/P. Putting Manuscript received May 27, 1982; revised July 9, 1982. The author is with Faculté des Sciences et Techniques, 29283 Brest Cedex, France. 0018-9456/82/1200-0284$00.75 © 1982 IEEE
Transcript of Generation of a sine wave from two in-quadrature triangular waves
284 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-31, NO. 4, DECEMBER 1982
. B0l·
■ 50
(0
<- 3 0 fu > ^ 2 0
(ü
-10 h
O : t h e BVRg des i gn 1 T-1.9BK,Q-5.4E7,CF-16.4
O : t h e c o n v e n t i o n a l RT c u t d e s i g n
Γ T-1.9K,Q«9.2E6,CF-1.8 o
h O
1E-9 f rac t iona l O L _ frequency s h i f t
- ° ° ° O o 0 o 0 <> ° o o o o o o o °
-
-70 -60 -50 D i s s i p a t e d Pouuer
-40 -30 i n dBm
Fig. 2. Frequency versus dissipated power in crystal resonator near 2 K for the same samples as shown in Fig. 1. The zero on the vertical axis is chosen arbitarily.
at 4.2 K for the same crystals, where the temperature coefficient is almost ten times that at 2 K, did not show any noticeable change in the results shown in Fig. 2.
DISCUSSION
The very small temperature coefficient near 2 K and increased Q imply that the use of cryogenic temperature can be a promising way for a highly stable crystal-controlled oscillator, along with the fact that the temperature regulation of 10 microdegrees is possible within a helium dewar [6] and the inherent \/f frequency fluctuation of crystal resonators decreases following a l/g2-dependent law [10]. Although the drive level of a crystal unit at cryogenic temperature must be held at a low value, the increased Q will make it possible to maintain enough signal-to-noise ratio when it is applied to a frequency control system. To obtain better performance, crystal resonators with optimum cut for cryogenic temperature operation should be used [4], [11].
Frequency stability of 4 X 10~14 has been achieved by a low-temperature crystal oscillator based on a tunnel junction diode [4]. The application of the cryogenic-temperature crystal resonator to the passive dual crystal system has also been studied [12].
ACKNOWLEDGMENT
The author wishes to thank S. R. Stein for providing him the opportunity to perform this work and for many helpful discussions. He acknowledges the many beneficial discussions and suggestions from C. M. Manney and F. L. Walls. Thanks are also due to Prof. R. J. Besson for the offers of several quartz crystals.
REFERENCES
[ 1 ] A. W. Warner, "Ultra-precise quartz crystal frequency standards," IRE Trans. Instrum.,\o\. 1-17, pp. 185-188, Dec. 1958.
[2] G. Mossuz and J. J. Gagnepain, "Quartz crystal oscillator at very low température," Cryogenics, vol. 16, no. 11, pp. 652-656, Nov. 1972.
[3] A. G. Smagin, "A 1-MHz quartz resonator with a Q factor of 4.2· 109
at a temperature of 2°K," lustrum. Exp. Tech., vol. 17, no. 6, pp. 1721-1723, Nov.-Dec. 1974.
[4] , "A low-temperature oscillator with a frequency stability 4 \0-]4:'Instrum. Exp. Tech., vol. 18, No. 6, pp. 1853-1855, Nov.-Dec. 1975.
[5] R. J. Besson, "A new electrodeless resonator design," in Proc. 31st Ann. Symp. on Frequency Control, pp. 3-7, 1977.
[6] S. R. Stein, "Application of superconductivity to precision oscillators," in Proc. 29th Ann. Symp. on Frequency Control, pp. 321-327, 1976.
[7] J. J. Gagnepain, J. C. Poncot, and C. Pegeot, "Amplitude-frequency behavior of doubly rotated quartz resonators," in Proc. 31st Ann. Symp. on Frequency Control, pp. 17-22, 1977.
[8] V. J. Johnson, "Compendium of the properties of materials at low temperature," WADD Tech. Rep. 60/56 (reprinted by Pergamon Press, 1961).
[9] G. K. White, Experimental Techniques in Low-Temperature Physics. Oxford, England: Oxford Univ. Press, 1968.
[10] J. J. Gagnepain, J. Uebersfeld, G. Goujon, and P. Handel, "Relation between 1/f noise and Q factor in quartz resonators at room and low temperatures. First theoretical interpretation," in Proc. 35th Ann. Symp. on Frequency Control, pp. 476-483, 1981.
[11] F. P. Phelps, "Stability of quartz resonators at very low temperatures," in Proc. llth Ann. Symp. on Frequency Control, pp. 256-276, 1957.
[12] B. Komiyama, "Quartz crystal oscillator at cryogenic temperature," in Proc. 35th Ann. Symp. on Frequency Control, pp. 335-339, 1981.
Generation of a Sine Wave from Two In-Quadrature Triangular Waves
ROBERT GENIN
Abstract—A simple and effective method is proposed for transforming two in-quadrature triangular waves into a low-distorsion sine wave. A circuit performing this operation is described that includes two multipliers and an absolute value device.
Several recent papers [l]-[3] described circuits which can generate simultaneous, triangular periodic waves, 90° out of phase. Useful complement to these circuits as function generators would be to shape the triangular waves into a sine wave with low distortion. In this short paper we report a method and we describe a circuit to solve this problem.
Product P{t) of two in-quadrature triangular waves e\(t) and ^ ( 0 of period T is a periodic function of period T/2 (Fig. 1). The graph of function P(t) is formed by a sequence of parabolic arcs. It has the same properties of symmetry as a sine curve and, considered as an approximation to a sine curve, it may be characterized by a total harmonic distortion equal to 3.8 percent. Note also that the derivative of function P(t) is a continuous function of /.
Let us introduce the following notations: B amplitude of P(t), u = 4t/T. A positive parabolic arc is represented by the function
P = 4Bu(\ - «), 0 < w < 1. (1)
The sine wave of same amplitude and period is then
S = Bsinwu. (2)
Functions P and S are equal for u = 0, u = 1 (P = S = 0), and for u = ^(P = S = 1 ), for other values we have P > S.
To obtain a signal with low distortion we may attempt to correct signal P(t) generating signals of the form
S = Pf(\P\) (3)
S = Pf(P2)
where function/is chosen so that signal S has a total harmonic distortion as low as possible. We adopt the First expression using absolute value of P which leads to a simpler realization.
Ideally, the function f(\P\) is defined by/ ( |P | ) = S/P. Putting
Manuscript received May 27, 1982; revised July 9, 1982. The author is with Faculté des Sciences et Techniques, 29283 Brest Cedex,
France.
0018-9456/82/1200-0284$00.75 © 1982 IEEE
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-31, NO. 4, DECEMBER 1982 285
Fig. 1. Quadrature triangular waves (e\, ei) and the parabolic curve resulting from product e\ei.
0.78 0.80 070 0.72 0.74 0.76 Fig. 2. Theoretical total harmonic distorsion as a function of parameter
a.
x = P/B and assuming P ^ 0, we obtain from (1) and (2)
f{x) = x~x sin |TT/2[1 - (1 - x)l/2]\.
The graph of this function shows that, in the range 0 ^ x ^ 1, it is remarkably approximated by a line, the equation of which may be written in the form
f(x) = a + (1 - a)x. (4)
In first approximation a may be determined by the simple condition /(0) =/(0) . It results in a = x /4 = 0.785 and the relative error \f(x) — f(x)]/f(x) is less than 0.55 percent. A better value of a may be found by minimizing the total harmonic distortion of signal S. The determination of the total harmonic distortion in function of a is tedious but elementary, the result is shown in Fig. 2. The curve passes through a minimum for a = a0 - 0.777 and for this value the total harmonic distortion is equal to 0.071 percent.
286 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-31, NO. 4, DECEMBER 1982
AD 534 1 ΑΌ 534 z H
M2.87K
-1
Πιοκ
Fig. 3. Schematic diagram of proposed implementation.
Signal S, defined by (3) and (4) is easily generated by a circuit associating an absolute value circuit and two multipliers, as shown in Fig. 3.
The multipliers (Analog Devices AD534) perform the operation [4]
(*i - X2)(Yi - Yi) = V{Zx - Z2), V = 10.00 V. (5)
If A is the amplitude of the input triangular waves, the amplitude of signal P at the first multiplier output is B = A2/\0. Equation (5) applied to the second multiplier gives
P · (R/R\)\P\ = 10 ' [eo - PRs/iRi + *3)1
i.e.,
e0 = P[R3/(R2 + Ä3) + (R/Rx)(\P\/\0)] = §. (6)
The comparison of (3), (4), and (6) yields the relations
R3/(R2 + R3) = a0 = 0.777
(R/Ri)(B/l0) = 1 - a0 = 0.223.
The first relation is independent of B and for R3 = 10 kfì it results in /?2 = 2.87 kQ (2.872 ki2); on the contrary, the second relation is dependent on B. In consideration of this point, a variable resistor, as a part of R \, is provided to adjust ratio R\/Rasa function of the exact value of B which depends on the exact value of A.
REFERENCES
[ 1 ] B. Z. Kaplan and Y. Tatrash, "New method for generating precise triangular waves and square waves," Electron. Lett., vol. 13, no. 3, pp. 71-73, Feb. 1977.
[2] R. Genin, C. Jézéquel, and J. Genin, "Simplified method for generating precise triangular waves," Electron. Lett., vol. 14, no. 6, pp. 162-163, Mar. 1978.
[3] J. R. Pimentel, "One chip oscillator generates in-quadrature waveforms," Electronics, vol. 51, no. 24, p. 147, Nov. 1978.
[4] B. Gilbert, "New analogue multiplier opens way to powerful function-synthesis," Microelectronics, vol. 8, no. 1, pp. 26-36, Jan. 1976.
