Active-R Oscillators Based On Cascaded First-Order Building Blocks
Dr. Mohamed M. Heima
Mesratha University, Mesratha, Libya,
ABSTRACT
In recent years a number of papers have been published dealing with the design of active-R oscillators. The design strategy adopted in the research is based on a survey made of the popular Wien bridge active-RC oscillator. The open-loop transfer function of this oscillator can be recognized as a second-order bandpass function with negative real-axis poles. Hence the transfer function can be realized by cascading first-order lowpass, and highpass building blocks. The aim of this paper is to present analytical approach to the systematic generation of active-R oscillators, based on two cascaded first-order single time constant building blocks. There are four ways of first-order building blocks which may be cascaded to form a sinusoidal oscillator are presented in this paper. In each case, expressions of threshold gain required for oscillation and the oscillation frequency were obtained. Furthermore, the relative merits of the four oscillators were considered by comparing their frequency stability factors and their harmonic rejection factors. Experimental results are given which matched with simulation results.
Index Terms— Active-R oscillator, single time constant building blocks, stability factor..
1. INTRODUCTION
Over a number of years, various authors have described designs for active-R sinusoidal oscillators. The absence of capacitors makes the designs attractive for monolithic IC fabrication, or for realization using commercially available operational amplifiers where compact designs are required. In many papers, designs are presented without giving much indication of the strategy employed in their design [1], [2], [3]. The design strategy adopted in the research leading to this paper is based on a survey made of the popular Wien bridge active-RC oscillator. The open-loop transfer function (TF) of this oscillator can be recognized as a second-order bandpass function with negative real-axis poles. Hence the TF can be realized by cascading first-order lowpass and highpass building blocks. This approach has already been used in the design of a multi-op amp active-RC oscillator [4],[5]. Methods of realizing inverting and non-inverting first-order active-R lowpass, highpass and allpass TF building blocks were described [6]. The inverting circuit was found to be more useful than the non-inverting circuit, since its time constant could be adjusted without significantly altering the dc gain. Thus giving a total of three first-order single time constant (STC) building blocks which could be used in oscillator design. There are four ways in which two of these three first-order building blocks may be cascaded to
form a sinusoidal oscillator. The possible combinations are:
1. Lowpass-Highpass;2. Lowpass-Allpass;
3. Highpass-Allpass;4. Allpass-Allpass.
One form of allpass-allpass oscillator has already been reported by Vosper [7]. This paper will deal with all four cascades in order to find, in each case, expressions for the threshold gain required for oscillation and the oscillation frequency. Furthermore, the relative merits of the four oscillators will be considered by comparing their frequency stability factors SFand their harmonic rejection factors [8].
2. OSCILLATORS INCORPORATING TWO FIRST-ORDER BUILDING BLOCKS
The transfer function (TF) of the first-order STC lowpass, highpass and allpass building blocks are given respectively by the following expressions:
(1)
(2)
(3)
Consider the general oscillator of Fig.1 in which the building blocks with TF Ha(s)and Hb(s)are first-order STC building blocks. The overall open-loop TF will be
H(s) = Ha(s)Hb(s)(4)
The characteristic equation of the closed-loop system formed by closing switch S in Fig. 1 is
1 - H(s) = 0(5)
When this equation has conjugate imaginary-axis roots s1,2 = ± jω, the system will oscillate continuously at a frequency fo= ωo/2π. The four structures mentioned in the introduction will now be considered.
A. Lowpass-Highpass Cascade
The open-loop TF of lowpass-highpass is given by
(6)
where K=K1K2. Substitute equation 6 into 5, yield roots of imaginary axis conjugates, and when K = 2 the system will oscillate at foof
(7)
B. Lowpass-Allpass Cascade
The open-loop TF of lowpass-allpass is given by
(8)
where K=KlK3. Substitute equation 8 into 5, yield roots of imaginary axis conjugates, and when K =-2the system will oscillate at f0of
(9)
C. Highpass-Allpass Cascade
The open-loop TF of highpass-allpass is given by
(10)
where K = K2K3. Substitute equation 10 into 5, yield roots of imaginary axis conjugates, and when K = 2the system will oscillate at f0of
(11)
D.Allpass-Allpass Cascade
The open-loop TF of allpass-allpass is given by
(12)
where K = K3K3. Substitute equation 12 into 5, yield roots of imaginary axis conjugates, and when K = -1 the system will oscillate at f0of
(13)
3. HARMONIC REJECTION
Strauss [8] has proposed a general model of the sinusoidal oscillator incorporating a linear active network of transfer function H„ followed by a non-linear amplitude control network. The amplitude control network injects harmonics into the oscillator output. Each harmonic k„ will be acted upon by the feedback present at the harmonic frequency, which is given by the following equation
(14)
Where Hn = H(jf) evaluated at f= nf0. The factor k„ is sometimes designated the harmonic rejection factor.
In a well designed oscillator, the feedback will change from positive at the oscillation frequency, to negative at the harmonic frequencies which requires that |kn| will be < 1 at the harmonic frequencies. When saturated operation is allowed, the resulting clipped output waveform will contain mainly odd order harmonics, the amplitudes of the third and fifth being the largest. Hence, it is informative to evaluate k3and k5for the four types of oscillators [9], [10].
A. Lowpass-Highpass Cascade
Substituting H(s)which is given by equation 6 with K = 2 (the threshold value required for oscillation) to the corresponding equation 14. Hence, the resultant frequency response function is
(15)
Evaluating this at the nth harmonic frequency nωo= nlτand taking the modulus yields
(16)
The values of Since | kn| for the 3rd , 5th harmonics
and the limiting value (as n→∞) are respectively
| k3| =1.25, | k5| =1.083, and | k∞| =1.0(17)
These results show that there is positive feedback at harmonic frequencies which will accentuate the distortion.
B. Lowpass-Allpass Cascade
Using an approach similar to thatadopted for the lowpass-highpass cascade it may be shown that the value of | kn| for the lowpass-allpass oscillator can be calculated by substituting Hn = H(jnωo)into equation 14, where H(s)is given by equation 8, with K=-2. Then, taking the modulus of the resultant frequency response function evaluated at the nth harmonic frequency yields
(18)
Similar to the previous section the values of | kn| for the 3rd,
5th harmonics and the limiting value are respectively
| k3| =1.167, | k5| =1.056, and | k∞| =1.0 (19)
However, these values are slightly better than those obtained for the lowpass-highpass cascade.
C. Highpass-Allpass Cascade
Similar to the previous two oscillators, the value of | kn| for highpass-allpass oscillator can be calculated by substituting Hn = H(jnωo)into equation 14, where H(s)is given by equation 10, with K = 2. Hence, evaluating the nth harmonic frequency yields
(20)
The values of | kn| for the 3rd, 5th harmonics and the limiting value are respectively
| k3| =0.50, | k5| =0.389, and | k∞| =0.33 (19)
Since | k3| , and | k5| are both < 1, this indicates negative feedback atthe harmonic frequencies which will reduce the distortion.
D. Allpass-Allpass Cascade
Using an approach similar to that adopted for the previous sections, the value of | kn| for highpass-allpass can be calculated by substituting Hn = H(jnωo)into equation 14, where H(s)is given by equation 12, with K = -1. Hence, evaluating the «th harmonic frequency yields
(22)
The values of | kn| for the 3rd, 5th harmonics and the limiting value are respectively
| k3| =0.625, | k5| =0.542, and | k∞| =0.50 (23)
These values are higher than those for the highpass-allpass cascade.
4. FREQUENCY STABILITY
The frequency stability factor, SF, has been used as a design pointer towards the likely frequency stability of an oscillator [7]. It is defined by
(24)
The value of SFdetermines the change in frequency which results if some change occurs in the open-loop phase shift. When such a change in phase occurs, the oscillator responds by changing its frequency to re-establish the open-loop phase shift to zero. The larger the value of SP, the smaller the fractional change in frequency f/fo= /o.However, it is worth evaluating SFfor the four oscillators described in this paper in order to establish their relative merits.
A. Lowpass-Highpass Cascade
The open-loop transfer function H(jω)of the lowpass-highpass cascade is given by equation 6 with K = 2. Hence
(25)
and the open-loop phase shift
= 90°- 2arctanω(26)
Using radian oscillation frequency of ωo = 1 / , then the
frequency stability factor for this system is given by
(27)
This shows that this oscillator has a better stability factor than that of the popular Wien bridge oscillator, for which SF = -2/3. Active-RC and active-R oscillators have very poor frequency stability when compared with active-LC or crystal-controlled oscillators. For crystal-controlled oscillators, the expected values of SFare between 1,000 and 50,000 [7].
B. Lowpass-Allpass Cascade
The open-loop transfer function H(jω)of the lowpass-allpass cascade is given by equation 8 withK= -2. Hence
(28)
and the open-loop phase shift
= 180°- 3arctanω(29)
Using radian oscillation frequency of , then
the frequency stability factor for this system is given by
(30)
This is superior to the SFvalue for the lowpass-highpass cascade.
C. Highpass-Allpass Cascade
The open-loop transfer function H(jω)of the highpass-allpass cascade is given by equation 10 with K = 2. Hence
(31)
and the open-loop phase shift
= 90°- 3arctanω(32)
Using radian oscillation frequency of then the frequency stability factor for this system is given by
(33)
This is superior to the SFvalue for the lowpass-highpass cascade and exactly the same asSFvalue for the lowpass-allpass cascade.
D. Allpass-Allpass Cascade
The open-loop transfer function H(jω)of the lowpass-allpass cascade is given by equation 12 withK= -1. Hence
(34)
and the open-loop phase shift
= 180°- 4arctanω(35)
Using radian oscillation frequency of ωo = 1 / , then the
frequency stability factor for this system is given by
(36)
Therefore, it is been found that the best stability factor for these four types of oscillators presented is the allpass-allpass combination.
5. EXPERIMENTAL OSCILLATORS
The experimental oscillators are based on the systems given in section 2. The technique employed is to design the oscillators using simple hand calculations, then to simulate them using PSPICE, finally building the circuits if the simulation results appeared satisfactory.
A. Lowpass-Highpass Cascade
Fig.2 shows an active-R oscillator consisting of cascaded inverting lowpass and highpass stages. Since the threshold gain required for oscillation is K = 2, the design was implemented with K1 = -1 and K2 = -2, although provision was made to make K > 2 to allow the oscillation amplitude to build up.
A dual op amp package, the Texas Instruments TL072 was used for op amp 1 and 2. The use of a single-chip dual op amp ensures that the transition frequenciesft, will be matched to within 1%providing close adherence to the theory which assumes equal time constants in both lowpass and highpass blocks. The value offtused for the TL072 in calculations was 3.6 MHz. Hence in order to minimize the amplitude and phase errors caused by op amp 3, a wideband op amp, the OP-64 (ft = 60 MHz) was used. The slew rate of the TL072 of 13 V/s allows operation up to approximately 200kHz without slew-rate distortion occurring. All resistors used were of ±1% tolerance with R1=R2=R4=R5=100 k. The value of R6is 200 k. This was increased to 220 kto allow the oscillation amplitude to build up. Various values were chosen, based on (τ=1/ωtβR), and equation 9 to provide different frequencies of oscillation.
Experimental results are given in Table 1. The frequency of oscillation, the stability factor, and the amplitudes at nodes A, B, and C are recorded. Stability factor for this oscillator should be SF = -1. Phase angles acand bcare given, in theory these should be 135° and -90°. Since, measured values of bc are close to their theoretical value at all frequencies, therefore, circuit functions as an excellent quadrature oscillator.
Table 1 Experimental results for the oscillator of Fig. 2
R3[] / 330 / 1 k / 3.3 k / 6.8 kDesign fo[kHz] / 11.80 / 35.29 / 111.4 / 215.5
Actual fo[kHz] / 12.75 / 35.94 / 104.1 / 151.9
SF / -1.00 / -1.04 / -1.02 / -1.10
THD at node A [%] / 4.50 / 4.75 / 5.40 / 6.10
VA[Vpp] / 21.80 / 22.05 / 23.75 / 24.3
THD at node B [%] / 2.20 / 2.50 / 2.70 / 3.10
VB[Vpp] / 15.63 / 15.66 / 17.25 / 18.66
THD at node C [%] / 9.50 / 9.90 / 11.0 / 13.0
VC[Vpp] / 28.50 / 28.52 / 28.9 / 28.95
ac[deg.] / 135 / 135 / 137 / 137
bc[deg.] / -90 / -90 / -91 / -90
Fig.3 Output voltage waveforms at nodes A, B, and Cof the active-Rlowpass-highpass oscillator of Fig. 2
Fig.3 shows the output voltagewaveforms at nodes A, B, and C at measured frequency of 12.75 kHz. However by taking the output at node B the THD is moderately low, even though op amp 3 is allowed to saturate.The differences between the measured and designfrequencies are caused by passive component tolerances,and increasingly as the oscillation frequency is raised by the finite gain-bandwidth product of the op amp 3.
B. Lowpass-Allpass Cascade
Fig.4 shows an active-R oscillator consisting of cascaded inverting lowpass and non-inverting allpass stages. Since
K=-2is required for oscillation, the design was implemented with K= -1 and K3 = 2. Resistors used were of R1=R2=R4=100 kandR5=2R4=200 k. The value of R6 =2KR4=400k. This was increased to 470 kto allow the oscillation amplitude to build up. Various values were chosen for R3based on equation 11 to provide different frequencies of oscillation.
Experimental results are given in Table 2. Stability factor for this oscillator should be SF-1.29. The phase angles ac and bc are given, in theory these should be 120° and -120° respectively.The THD figures show an increase in distortion with increasing fo.
Table 2 Experimental results for the oscillator of Fig. 4
R3[] / 180 / 560 / 1.8 k / 3.9 kDesign fo[kHz] / 11.18 / 34.53 / 108.3 / 225.6
Actual fo[kHz] / 12.22 / 36.46 / 97.62 / 165.8
SF / -1.30 / -1.37 / -1.47 / -1.61
THD at node A [%] / 3.80 / 4.40 / 6.00 / 7.40
VA[Vpp] / 16.87 / 16.55 / 17.39 / 18.81
THD at node B [%] / 1.62 / 1.80 / 2.55 / 3.60
VB[Vpp] / 8.75 / 8.34 / 9.47 / 11.17
THD at node C [%] / 10.0 / 10.50 / 13.50 / 16.0
VC[Vpp] / 28.50 / 28.52 / 28.95 / 28.95
ac[deg.] / 122 / 123 / 124 / 125
bc[deg.] / -116 / -116 / -115 / -114
Fig.5 Output voltage waveforms at nodes A, B, and Cof the active-Rlowpass-allpass oscillator of Fig. 4
Fig. 5 shows the output voltage waveforms atnodes A, B, and C for an oscillation frequency of 12.22 kHz. By taking the output at node B the THD is moderately low, even though op amp 3 is allowed to saturate.
C. Highpass-Allpass Cascade
Fig. 6 shows an active-R oscillator consisting of cascaded inverting highpass and inverting allpass stages. Since the threshold gain required for oscillation is K = 2, the design was implemented with K2 = -2 and K3 = -1.
Dual op amp package, TL072 was used for op amp 1, 3and wideband op amps, AD OP 64 were used for op amp 2 and 4. Resistors used were R1 = R2 = R4 = R5 = R7 = R10 = 100 k, R6 = R8 = 2R1 = 200 k. The value of R10 = 2R9 =200 k. This was increased to 220 k to allow the oscillation amplitude to build up. In order to ensure equal time constants in the highpass and allpass stages, resistors R3were made equal. Various values were chosen, based on equation 11 to provide different frequencies of oscillation.
Experimental results are given in Table 3. Thephase angles AD, BD and CD, are given, since in theory these should be 150°, -120° and 30° respectively.
Fig. 7 shows output voltage waveforms at nodes A, B, C, and D for an oscillation frequency of 12.60 kHz. By taking the output at node C the THD is moderately low, even though op amp 4is allowed to saturate.
Table 3 Experimental results for the oscillator of Fig. 6
R3[] / 560 / 1.8 k / 5.6 k / 12 kDesign fo[kHz] / 11.51 / 36.11 / 104.7 / 201.1
Actual fo[kHz] / 12.60 / 37.82 / 93.30 / 156.5
SF / -1.30 / -1.44 / -1.48 / -1.54
THD at node A [%] / 1.75 / 2.25 / 2.94 / 4.26
VA[Vpp] / 23.75 / 23.50 / 23.47 / 23.47
THD at node B [%] / 4.30 / 5.05 / 5.40 / 6.80
VB[Vpp] / 26.63 / 26.20 / 26.25 / 26.25
THD at node C [%] / 1.60 / 1.75 / 2.41 / 3.80
VC[Vpp] / 20.94 / 21.90 / 22.95 / 22.95
THD at node D [%] / 7.40 / 7.55 / 8.50 / 8.56
VD[Vpp] / 28.50 / 28.52 / 28.95 / 28.95
AD[deg.] / 151 / 153 / 153 / 155
BD[deg] / -123 / -123 / -124 / -128
CD[deg] / 30 / 31 / 31 / 32
D. Allpass-Allpass Cascade
Fig.8 shows an active-R oscillator consisting of cascaded non-inverting and inverting allpass stages. Since, the threshold gain required for oscillation is K = -1, the design was implemented with K3=± 1. Values of resistors used
Fig. 7 Output voltage waveforms of the active-Rhighpass-allpass cascade oscillatorofFig. 6
were R1=R2=R4=R7=R9=100 k, R5=R6=R8=200k. The value of R9required for threshold oscillation conditions is R10 =2KR9=200k. This was increased to 220 k to allow the oscillation amplitude to build up. Various values of R3were chosen, based on equations 13 to provide different frequencies ofoscillation.
Experimental results are given in Table 4.
Table 4 Experimental results for the oscillator of Fig. 8
R3[] / 330 / 1 k / 3.3 k / 6.8 kDesign fo[kHz] / 11.80 / 35.29 / 111.4 / 215.5
Actual fo[kHz] / 12.75 / 36.15 / 95.45 / 162.0
SF / -1.99 / -2.01 / -2.03 / -2.15
THD at node A [%] / 1.70 / 1.90 / 2.55 / 4.05
VA[Vpp] / 20.31 / 20.50 / 20.77 / 21.16
THD at node B [%] / 4.40 / 4.65 / 5.70 / 6.81
VB[Vpp] / 27.50 / 27.50 / 27.35 / 27.27
THD at node C [%] / 1.60 / 1.75 / 2.25 / 3.85
VC[Vpp] / 20.47 / 20.50 / 20.75 / 20.95
THD at node D [%] / 5.45 / 6.35 / 7.50 / 8.60
VD[Vpp] / 28.13 / 28.15 / 28.25 / 28.15
AD [deg.] / 133 / 133 / 132 / 131
BD[deg] / -90 / -90 / -91 / -91
CD [deg] / 44 / 44 / 43 / 43
Fig.9 shows output voltage waveforms at nodes A, B, C, and D for an oscillation frequency of 12.75 kHz. The outputs at nodes A and C are with THD moderately low, despite the fact that an op amp 2 and 4 reaches saturation.
Fig. 9 Output voltage waveforms at nodes A, B, C and D of the active-R allpass-allpass cascade oscillatorofFig. 8
6. CONCLUSIONS AND COMPARISONS
It has been found possible to design oscillators based on two cascaded first-order STC building blocks. Table 5 summarizes the main differences between the four types of oscillator. From this table it may be observed that the lowpass-highpass cascade and lowpass-allpass cascade have the lowest number of resistors and op amps. Furthermore, the lowpass-highpass cascade circuit could be used as a quadrature oscillator since the phase angle BCis very close to -90° at all recorded frequencies.
The lowpass-allpass cascade has a better frequency stability factor than the lowpass-highpass cascade and |SF|=1.29 compared with unity for the lowpass-highpass cascade and the harmonic rejection was found to be superior. However, this did not always result in lower distortion, since it was not easy to produce the same amplitudes at various points in both circuits. The most important distinctive feature of this oscillator is that the outputs at nodes A, B and C are the outputs of a three-phase oscillator, although the voltage amplitudes at these three nodes do not have the same values. This could be achieved by means of resistive potential dividers or by the addition of another op amp-based amplifier. Frequency adjustment for this oscillator requires the simultaneous adjustment of two equal grounded resistors.
The harmonic rejection factor knwas found the lowest for the highpass-allpass cascade, which employ two cascaded first-order STC building blocks. The results of measurements given did not agree with theory. This can be attributed to two factors. Firstly, the amplitudes at the points of lowest distortion were considerably lower. Secondly, Lowpass-highpass oscillator, the node at which the distortion is lowest is fed from the point of highest distortion via two directly cascaded lowpass building blocks.