Multiphase sinusoidal oscillator using the CFOA
pole
D . 4 . WU S.4. Liu Y.-S. Hwang Y.-P. w uIndexing ferns: Multiphase sinusoidal oscillator, Amplifiers
Abstract: A multiphase sinusoidal oscillator using the parasitic poles of current feedback operational amplifiers (CFOAs) is presented. The oscillator can generate n signals which are equal in ampli- tude and equally spaced in phase. The oscillator using the parasitic poles of CFOAs is suitable for high frequency applications and monolithic IC fabrication. The effects caused by the nonlinearity of the CFOA on the oscillation frequency and condition have been analysed. Experimental results show that a three-phase sinusoidal oscil- lator has a wide operating range from 10 kHz to 10 MHz and a large output voltage swing up to 7 V peak to peak while using f 5 V power supply. The total harmonic distortion (THD) of each oscillation frequency is dominated by its third harmonic component which is at least 33dB down from its fundamental.
1 Introduction
Multiphase sinusoidal osciilators (MSOs) play an impor- tant role in communications, signal processing and power electronics. A number of MSOs based on different tech- niques have been developed in literature [l-71. MSOs implemented by the methods of References 1-4 exhibit good performance, but those circuits suffer complex cir- cuitry from using a large number of components.
O n
the other hand, both of the MSOs realised by active-R tech- nique [5] and operational transconductance amplifiers (OTAs) [SI feature simple circuit structure, but they are restrained by the limited operating range and output voltage swing.Because CFOAs possess more extended bandwidths and higher slew rates than conventional operational amplifiers (OAs), they have been widely used in place of OAs as active devices in analog circuit applications [S, 91. However, little attention has been paid on realisation of CFOA based MSOs.
In this paper, we propose a simple scheme for realising MSOs which features the following merits: using only the parasitic poles of CFOAs that makes it suitable for high frequency oscillation and monolithic IC fabrication, pos- sessing large output voltage swing and moderately low
0 IEE, 1995
Papa 1682G (ElO), first received 17th March and in revised form 8th November 1994
The authors are with the Department of Electrical Engineetink Nation-
al Taiwan University, Taipei, Taiwan, 10664, Republic of China IEE Proc.-Cirnrits Devices Syst., Vol. 142, No. I , February 1995
THD, and having simple circuit structure that only two resistors and one CFOA are required for each phase. 2 Circuit description
The circuit symbol and equivalent circuit for a CFOA including a compensation terminal are shown in Fig. 1
a
Fig. 1 CFOA
a Circuit symbol
b Equivalent circuit
[lo]. The transfer function between the output and input terminals of the CFOA can be given by
where k is the current transfer ratio, R, is the input resistance of the inverting input terminal, (Ro//Co) is the equivalent parasitic impedance at the compensation ter- minal, and upl is the CFOA dominant pole. According to eqn. 1, a CFOA can be used as a first-order low-pass filter.
Based on Fig. 1, the generalised circuit for realising an MSO is shown in Fig. 2.
This
oscillator is composed ofFig. 2 Generdised circuit for realising t n u l t i p h e oscillator two resistors and one CFOA for each phase. Assuming that all CFOA in Fig. 2 are identical and k = 1, the loop gain can be expressed as
Table 1 : Frequency and condition of oacillation of multi-
smo - mo -
phase oscillator
Number Frequency Condition of symmetrical
of phase of oscillation multiphase oscillation
suo - ~ n Wn R .
+
R:- derived as & -s,
-s;;
= - 1 -RI 3 5 wo = 0 . 7 2 7 ~ ~ R , 2 1.236R, 7 wo = 0 . 4 8 2 ~ ~ R , 3 1.1 1 OR, w, = 1 . 7 3 2 ~ ~ R, 2 2R, I . I“ whereNote that eqns. 10 are valid for each n-phase oscillation circuit. All of the absolute values of the sensitivities are no more than one. The frequency stability factor S , is defined as Reference 12: (3) 1 R b = ( R 2 / / R O ) - _ _
-
1 0, = ( R 2 / / R 0 ) C 0 R b Go = Rb/R, R, = R I+
Rim (4)According to the Barkhausen criterion Ell], the condi- SF -
d::)l”=,
(11)tion for the proposed circuit to provide sinusoidal oscil-
lations of frequency wo is that where U = w/wo is the normalised frequency, and
4(u)
represents the phase function of the open loop transfer function in Fig. 2. And the frequency stability factor S , ( I + $ r - ( - G o ) ” = O ( 5 ) can be derived as
By expanding eqn. 5, it is easy to show that eqn. 5 would have a solution only if the value of n is odd (n 3 3). By equating the imaginary and real parts to zero, respec- tively, the frequency and condition of oscillation can be shown as R n wo = ob tan - (6) and Rb 3 R, sec n (7)
From eqns. 6 and 7, the oscillation frequency and condi- tion of a three-phase sinusoidal oscillator (n = 3) can be expressed as
= J(3)wb (8)
and
Rb 3 2R, (94
or
According to eqns. 8 and 9, this three-phase sinusoidal oscillator has the maximum and minimum oscillation fre- quencies when R I = 0 and R , = (R0/2) - R,, respec- tively.
Finally, the oscillation frequency and condition for realising an n-phase sinusoidal oscillator, equal in ampli- tude and equally spaced in phase, are summarised in Table 1.
3 Sensitivity analysis and frequency stability
From the above analysis, the sensitivities of the passive and active components on oscillation frequency can be 38
-n 2x
S, =
-
sin -2 n
The value of IS,I is proportional to that of n, and is equal to 1.30 and x while n = 3 and CO, respectively.
4
Considering the effects of the CFOA nonlinearity, we can use a two-pole model to describe the performance of a nonideal CFOA. Thus, it is suitable to modify the current transfer ratio of a nonideal CFOA as
Effects of the CFOA nonlinearity on oscillation frequency and condition
(13) 1 1 + - U P 2 k = - S
where wp2 is the CFOA second pole. Substituting eqn. 13 into eqn. 1, we can carry out the characteristic equation of the three-phase oscillation circuit in Fig. 2 as
a.
+
a,s+ a2sz+
a3s3+
a4s4+
ass5+
a6s6 = 0 (14) where a, = 1+
(2y
3 3 a , = - + - O b O p 2 3 9 3+ z
a , = - + - a , = - + - + - 0: w b w p 2 O p 2 1 9 9 1 0: w:wpz Wbwi2+%
9 3+-+-
a4 =-
3 wbJwp2 O b W : 2I E E Prcx-Circuits Devices Syst., Vol. 142, No. I , February 1995
For simplicity, the fifth and the sixth order terms in eqn. 14 can be neglected because their coefficients are much smaller than that of the fourth and lower terms.
Based
onL I I I I I I I I I I Fig. 3 Output waveform of three phase oscillator
fo = 10.3 MHZ
Horizontal scale is about U) ns/div; vertical scale is 1 V/div
MKR: 10.30MHz -37.68dBrn W A
RL: -31.2dBm 5 d 0 l ATOdB ST 50rns D:PK
r
r-iI 1
Fig. 4
Horizontal d eis 5 MHZIdiv; vertical scale is 5 dB/div
Frequency spectrum of one output signal of Fig. 3
1031 I I 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 , , ,, ,,,,
,
, , , ,,,,,,
, , ,,,, 102 103 1 ob 105 106 1o7
R ~ .n
Fig. 5 ~ Linear . ___. . - nonlin. x x x cxp.Variation of oscillationjiequency f, with resistor R ,
this condition, we would derive the approximate expres- sions of the oscillation frequency and condition as
0 0 = Am, (16)
and R b 2 pRa
IEE Proc.-Circuits Devices Syst., Vol. 142, No. 1, February 1995
or
Fig. 6
~ b e a r
___.___
nob 000 exp.Relationship of oscillation-condition between R, and R,
-10r .a where 1/2 3 A = (1
+
8R+
R') 9(1+
8R+
17RZ+
8R3
+
R4)
- 1)113A detailed analysis indicates that the values of A and p are equal to J(3) and 2, respectively, while R - 0 . This consequence shows that the oscillation frequency and condition derived from the linear and nonlinear model of CFOA are almost equal when on 4 w P 2 .
5 Experimental results and discussion
To verify the theoretical analysis, the circuit of a three- phase sinusoidal oscillator has been realised by a com- mercial CFOA (AD844AN). The typical values of
R,"
andRo of a CFOA are 50 R and 3 MR [lo], respectively. The total parasitic capacitance caused by the AD844AN and 39 (19) =
(
(1+
8R+
R2)2the circuit bfeadboard is about 8.3 pF measured at each compensation terminal of the CFOAs. The second pole of the AD844AN is about 70 MHz [13].
The output waveform of the three-phase sinusoidal oscillator at 10.3 MHz is shown in Fig. 3, with Ri = 1
k Q
R, = 1.44 KZ, ande5
V
power supply. Fig. 4 shows the frequency spectrum of one output signal of Fig.3.
It shows that the THD is dominated by the third harmof& component which is 33 dB down from the fun- damental. The relationship between R, and fo(fo
=
wo/2+) is shown in Fig. 5 which indicates good agreement between the linear model prediction and experimental results while fo is under 1.5 MHz, but, for higher fre- quencies, the larger error between the linear model pre- diction and the measured results occurs. On the contrary, the nonlinear model prediction in high frequency range matches much better with the measured results than the linear model does.
The oscillation conditions between R, and RI are plotted in Fig. 6 which shows that the experimental results are well matched with the nonlinear model predic- tion. The THD and the output voltage swing of various oscillation frequencies are shown in Fig. 7. It shows that the output voltage swing remains almost constant when fo is under
3
MHz, and that the THD of each frequency is at least 33 d S down from its fundamental.Again, we restudy both Fig. 5 and Fig. 7. It turns out that, due to the effects of the higher order poles and the slew rate, the output voltage swings decline rapidly; meanwhile, the measured oscillation frequencies deviate from the nonlinear model prediction when the oscillation frequeacy is above 3 MHz.
6 Conclusions
A simple scheme, which exploits the parasitic poles of the CFOAs, has been presented for realising an MSO. The three-phase sinusoidal oscillator has been built and tested. Experimental results which agtee very well with the theoretical result are obtained. Because the frequency of oscillation (i.e. eqn. 6 ) is inversely proportional to the
total parasitic capacitance at the compensation tetminal
of the CFQA, it means that a voltage controlled multi- phase oscillator can be realised if each phase has a voltage-controlled grounded capacitor (VCGC) con- nected to the compensation terminal of the CFOA. A VCGC can be realised by the active-R technique [14]. The capacitor [14] is a fiinction of the gain bandwidth product (GBP) of the OAs. The GBP, however, is a biasing supply dependent parameter, so it is feasible to realise a VCGC by controlling the biasing supply of the circuit. The results are expected to be useful in analogue signal processing applications.
7 References
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