# All-Pass Filter Based Linear Voltage Controlled Quadrature Oscillator

Authors: Koushick Mathur; Palaniandavar Venkateswaran; Rabindranath Nandi
Journal: Active and Passive Electronic Components
 (2017)

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## Abstract

A linear voltage controlled quadrature oscillator implemented from a first-order electronically tunable all-pass filter (ETAF) is presented. The active element is commercially available current feedback amplifier (AD844) in conjunction with the relatively new Multiplication Mode Current Conveyor (MMCC) device. Electronic tunability is obtained by the control node voltage (V) of the MMCC. Effects of the device nonidealities, namely, the parasitic capacitors and the roll-off poles of the port-transfer ratios of the device, are shown to be negligible, even though the usable high-frequency ranges are constrained by these imperfections. Subsequently the filter is looped with an electronically tunable integrator (ETI) to implement the quadrature oscillator (QO). Experimental responses on the voltage tunable phase of the filter and the linear-tuning law of the quadrature oscillator up to 9.9 MHz at low THD are verified by simulation and hardware tests.


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## Body

## 1. Introduction

Realization of first-order all-pass filters had been reported earlier using various types of active building blocks (ABBs), namely, VOA [1], Current Conveyor and its variants [2–5], DDA [6], DDCC [7], DVCC [8, 9], VDIBA [10], and OTA, with differential amplifier [11] as listed in Table 1; some of these building blocks are implemented with a basic device (OTA, CC, or DVCC) combined with some signal (current or voltage) differential unit. Recent literature suggests that electronic function circuits fabricated by commercially available IC modules are capable of providing desired results [12–16].Table 1
Comprehensive summary of recent APFs.
Ref.
ABB
ABB implementation
f
p reported (Hz)
Tunability
[1]
VOA
HA 2544C opamp
125 K
R
C
[2]
CCII
LF 356N opamp with current mirror
1.6 K
R
C
[3]
DO-CCCII
Dual-output current controlled conveyors
1.63 M
I
b
[4]
CCC II with OA
Current controlled conveyor
12.15 K
I
b
[5]
DO-CCII
CC II with additional z-copy node
3.89 M
R
C
[6]
DDA
Differential amplifier (DA)
16 K
R
C
[7]
DDCC
Current conveyor with DA
318 K
R
C
[8]
DVCC
VDU followed by OTA
400 K
I
b
[9]
DVCC
VDU followed by CC
1.59 M
R
C
[10]
VDIBA
OTA with differential input buffer
9.4 M
I
b
[11]
OTA
OTA with DA
375 K
I
b
Proposed
CFA-MMCC
CFA844 and multiplier AD835
9.91 M
V
Notes. (a) Designs in [1, 2, 10], based on commercially available IC modules; (b) comprehensive listing of recent APFs using various ABBs presented in [5].A varied genre of ABBs emerged during the recent past for realizing diverse functions towards signal processing/wave forming/filtering applications. Fabrication of internal design of such ABBs in CMOS technology leads to easy verification. However, recent studies address the issues of such approach concerning involvement of fabrication cost and complexity in their nodal relations with such wide variety of ABBs [12, 14, 16].Use of readily available off-the-shelf devices thus may be an alternate approach to obtaining satisfactory results. Some recent composite blocks yield quite useful results on signal processing applications wherein two types of basic commercially available chips are conjoined (e.g., DDCC, DVCC, and VDIBA). It has been suggested that composite blocks may provide better results in comparison to the constituent elements [17]. So for such ABBs requiring more than one commercially available IC, it is still economical and more convenient compared to chip fabrication [12, 16, 17].The relatively new MMCC element [18] used in this work is thus configured using readily available CFA (AD844) and multiplier (AD835) [19] elements. The useful feature of the MMCC is that it has an in-built control voltage terminal (V) which is conveniently utilized for electronic tuning of a circuit parameter. Hence the conventional method of transconductance (gm) to bias current (Ib) conversion is avoided; such conversion needs additional hardware complexity and involvement of thermal voltage (VT) [3, 10, 16]. We now present here an electronically tunable all-pass filter (ETAF) using the composite CFA-MMCC block.Next a linear voltage controlled quadrature oscillator (LVCQO) is implemented with the ETAF being looped in feedback with an electronically tunable integrator (ETI). Detailed analysis is carried out taking into account the device imperfections, namely, parasitic shunt-rzCz components and the roll-off poles of the port-transfer ratios. Effects of these nonidealities are negligible but the higher range of usable frequencies is constrained. The oscillation frequency (ωo) is active-insensitive relative to port-mismatch error (ε). Experimental results on ETAF and quadrature oscillator responses are verified by PSPICE simulation and hardware test.


## 2. Analysis

The proposed ETAF is shown in Figure1(a) where the nodal relations are as follows:CFA:Iz=αIx, Vx=βVy, and Vo=δVzMMCC:Iz=aIx, Vx=b(kVy1Vy2), and Vw=γVz.The port tracking ratios are ideally unity but may be postulated by error coefficients (ε≪1) as α=(1-εi)=a, β=(1-εv)=b, and δ=(1-εz)=γ, where k≡(0.1/volt) is the multiplication constant [20].Figure 1
(a) First-order ETAF. (b) Implementation of MMCC with commercially available ICs.
(a)
(b)Analysis of Figure1(a) yields the transfer function G≡Vo/Vi as(1)G=δ1δ2m+δ1δ2-1α2β2snτ-α1α2δ1δ2abγsnτ+α1δ1abγparasitic elements, yielding a modified transfer where m=r3/r4, n=r2/r1, and τ=RC/kV.In ideal devices port tracking ratios are all unity; assuming the realizability conditions asm=1=n(r1,2,3,4=r) for simplicity, we get(2)G=sτ-1sτ+1which is the nonminimum phase all-pass function, where transmission gain G=1 and phase (θ) is tunable in a range 180≤θ°≤0, being variable by control voltage (V) as(3)θ=π-2arctan⁡ωRCkV.


## 3. Effects of Nonidealities

### 3.1. Parasitic Components

The parasitic components appear as shunt-rzCz arms at current source nodes of the device; as per data-book [21], the typical values are 2≤rz(MΩ)≤5 and 3≤Cz(pF)≤5.5; ratio of circuit resistors (KΩ) relative to rz is assumed to be negligible. Reanalysis now with finite parasitic elements yields modified transfer elements in Figure 1(a) exhibiting a single-pole roll-off model [22], given by(4)G^=n2s2+n1s+no-1d3s3+d2s2+d1s+do+1,where(5)n2=ττz11+σn1=τ1+σ1+μ1+μ3τz1no=μ31+μ1d3=ττz1τz21+σd2=ττz11+σ1+μ2+ττz21+σ1+μ1+μ3τz1τz2d1=τ1+σ1+μ1+μ3τz11+μ2+τz2μ31+μ1do=μ2σ=Cz3C≪1,μ1,2=rrz1,2≪1,μ3=RkVrz3≪1τz1,2=rCz1,2,τz1,2τ≪1.Then (4) simplifies to(6)G^=s2ττz1+sτ-1s3ττz1τz2+s2τz1+τz2+sτ+1.The frequency-domain behavior may be estimated by writings=jω in (6), given by(7)G^ω=jωτ-1+ζ1jωτ1-ζ2+1–ζ3,where(8)ζ1=ω2ωpωz1≪1;ωpfilter-pole  frequency=1τ,ωz1=1τz1;ζ2=ω2ωz2ωz1≪1ζ3=ω2ττz1+τz2τz1=τz2=τz≈2ω2ττz≈2ω2ωpωz≪1.WithCz≈3.3pF (measured) and r=1KΩ, we estimated value of the parasitic pole frequency fz≈54 MHz. Hence for usable ranges of <fz, effects of parasitic capacitors are negligible and G^=G; therefore the nominal APF function is preserved with the limit f≤fz.


### 3.2. Roll-Off Pole of Device-Port Transfer Ratios

At relatively high-frequency ranges, the port-transfer ratiosα, β, δ and a, b, γ of the active components are as follows:(9)α1,2=αo1,2sp1,2+1δ1,2=δo1,2sp3,4+1β2=βo2sp5+1a=aospi+1b=bospv+1γ=γospz+1.The dc values of all the products of port coefficients in (1) may be taken as unity; for example, αo1,2≈{1-(εi1+εi2)}≈1 since εi1,2≪1.The modified APF function (G′) is now written as(10)G′=Δ1+Δ2Δ1–1sτ-Δ3Msτ+Δ4Ms,where(11)Δ1=δ1δ2≈1s2p3p4+sp3+p4+1(12)Δ2=α2β2≈1s2p1p5+sp1+p5+1(13)Δ3=α1α2≈1s2p1p2+sp1+p2+1(14)Δ4=α1δ1≈1s2p1p3+sp1+p3+1.The function M(s) relates to the MMCC block; it is decomposed as Ms=Mos·Me(s), where Mo=1/sτo and(15)Mes=1spi+1spv+1spz+1.It has been observed that the 3 dB corner frequencies for all the port-transfer ratios appear at close proximity, and hence these may be assumed equal without loss of generality [22]. Therefore calculations for examining the effects of the roll-off poles become simpler if we write p1~5≡1/ωp. Writing u=(ω/ωp)≪1 and solving for (11) we get(16)Δ1jω=11-u2+2juwhich implies Δ1=√(1+4u2)≈1 and ∟Δ1=arctan⁡(2u)≈0. Similar results are checked for (12) to (14). For the effects of the MMCC roll-off, we put ψ=(ω/ωm)≪1, where pi,v,z≡1/ωm; solving (15), we get(17)Mejω=11-3ψ2+j3ψ1-ψ2/3which reduces to Me=1 and ∟Me=-arctan⁡(3ψ)≈0. This analysis indicates that effects of roll-off poles of the active building blocks are quite negligible in the range of operating frequencies (f<fp,m).


## 3.1. Parasitic Components

The parasitic components appear as shunt-rzCz arms at current source nodes of the device; as per data-book [21], the typical values are 2≤rz(MΩ)≤5 and 3≤Cz(pF)≤5.5; ratio of circuit resistors (KΩ) relative to rz is assumed to be negligible. Reanalysis now with finite parasitic elements yields modified transfer elements in Figure 1(a) exhibiting a single-pole roll-off model [22], given by(4)G^=n2s2+n1s+no-1d3s3+d2s2+d1s+do+1,where(5)n2=ττz11+σn1=τ1+σ1+μ1+μ3τz1no=μ31+μ1d3=ττz1τz21+σd2=ττz11+σ1+μ2+ττz21+σ1+μ1+μ3τz1τz2d1=τ1+σ1+μ1+μ3τz11+μ2+τz2μ31+μ1do=μ2σ=Cz3C≪1,μ1,2=rrz1,2≪1,μ3=RkVrz3≪1τz1,2=rCz1,2,τz1,2τ≪1.Then (4) simplifies to(6)G^=s2ττz1+sτ-1s3ττz1τz2+s2τz1+τz2+sτ+1.The frequency-domain behavior may be estimated by writings=jω in (6), given by(7)G^ω=jωτ-1+ζ1jωτ1-ζ2+1–ζ3,where(8)ζ1=ω2ωpωz1≪1;ωpfilter-pole  frequency=1τ,ωz1=1τz1;ζ2=ω2ωz2ωz1≪1ζ3=ω2ττz1+τz2τz1=τz2=τz≈2ω2ττz≈2ω2ωpωz≪1.WithCz≈3.3pF (measured) and r=1KΩ, we estimated value of the parasitic pole frequency fz≈54 MHz. Hence for usable ranges of <fz, effects of parasitic capacitors are negligible and G^=G; therefore the nominal APF function is preserved with the limit f≤fz.


## 3.2. Roll-Off Pole of Device-Port Transfer Ratios

At relatively high-frequency ranges, the port-transfer ratiosα, β, δ and a, b, γ of the active components are as follows:(9)α1,2=αo1,2sp1,2+1δ1,2=δo1,2sp3,4+1β2=βo2sp5+1a=aospi+1b=bospv+1γ=γospz+1.The dc values of all the products of port coefficients in (1) may be taken as unity; for example, αo1,2≈{1-(εi1+εi2)}≈1 since εi1,2≪1.The modified APF function (G′) is now written as(10)G′=Δ1+Δ2Δ1–1sτ-Δ3Msτ+Δ4Ms,where(11)Δ1=δ1δ2≈1s2p3p4+sp3+p4+1(12)Δ2=α2β2≈1s2p1p5+sp1+p5+1(13)Δ3=α1α2≈1s2p1p2+sp1+p2+1(14)Δ4=α1δ1≈1s2p1p3+sp1+p3+1.The function M(s) relates to the MMCC block; it is decomposed as Ms=Mos·Me(s), where Mo=1/sτo and(15)Mes=1spi+1spv+1spz+1.It has been observed that the 3 dB corner frequencies for all the port-transfer ratios appear at close proximity, and hence these may be assumed equal without loss of generality [22]. Therefore calculations for examining the effects of the roll-off poles become simpler if we write p1~5≡1/ωp. Writing u=(ω/ωp)≪1 and solving for (11) we get(16)Δ1jω=11-u2+2juwhich implies Δ1=√(1+4u2)≈1 and ∟Δ1=arctan⁡(2u)≈0. Similar results are checked for (12) to (14). For the effects of the MMCC roll-off, we put ψ=(ω/ωm)≪1, where pi,v,z≡1/ωm; solving (15), we get(17)Mejω=11-3ψ2+j3ψ1-ψ2/3which reduces to Me=1 and ∟Me=-arctan⁡(3ψ)≈0. This analysis indicates that effects of roll-off poles of the active building blocks are quite negligible in the range of operating frequencies (f<fp,m).


## 4. LVCQO Implementation

Quadrature oscillators find diverse application in measurement systems [30–33] and in communication circuits, quadrature mixers, SSB-modulator, vector generator, and data conversion. A comprehensive summary of recent QO designs is listed in Table 2. It may be seen that even though a number of such oscillators were reported, very few are linearly tunable.Table 2
Comparative description of recent QO designs.
Ref.
Device used
Electronic tunability
Linear tuning law
f
o reported (Hz)
Tuning by
THD (%)
[5]
DO-CCII
No
No
2.33 M
R
C
1.17~2.82
[10]
VDIBA
Yes
No
8.5 M
√
I
b
2.25
[12]
CFA
No
No
909 K
R
C
—
[14]
CFA
No
No
398 K
R
C
1.42
[23]
OTA
Yes
Yes
64 K
I
b
—
[24]
VOA
No
No
160 K
R
C
—
[25]
CFOA
No
No
159 K
R
C
3.16
[26]
CDTA
Yes
No
1.73 M
√
I
b
3.0
[27]
CCCDBA
Yes
No
419 K
√
I
b
2~1.2
[28]
CCCCTA
Yes
No
1.1 M
√
g
m
1~2.9
[29]
OTA
Yes
No
1.59 M
√
g
m
1~2.2
Proposed
CFA-MMCC
Yes
Yes
~10.2 M
V
0.9 ~2.8We now present the linear VCO as implemented by forming a regenerative feedback loop formed with the ETAF in Figure1(a) and the ETI in Figure 2; transfer of ETI is(18)Hs=1sτo1+ξ+Ro/rz,where τo=RoCo/kV, ξ=Co/Cz≪1, and Ro/rz≪1; hence H(s)=1/sτo. Grounding of the capacitor (C) facilitates absorbing Cz in C-values at predesign level. Device imperfection relative to port-transfer roll-off pole frequencies (ωa,b,γ) is estimated by writing(19)Hs=abγsτo≡1-εTsτoĦ,where εT=(εi+εv+εz)≪1.Figure 2
Electronically tunable integrator (ETI).The deviation due to roll-off is(20)Ħ=sωa+1sωb+1sωγ+1.As in previous section if we assume ωa,b,γ≈ωm and put ψ=(ω/ωm), then Ħ reduces similar to (17) as(21)Ħ=1,∟Ħ=-arctan⁡3ψ≈0.Now implementation of oscillator is derived by equating {1-GH(s)}≡0 which gives the characteristic equation (CE):(22)CE:s2τoτ+sτo-τ+1=0Realizability  condition:τo=τR=Ro,C=CoOscillation  frequency:ωo=kVRC.So tuned frequency (fo in Hz) is linearly and directly variable by control voltage (V)(23)fo=V20πRC.The proposed method provides direct electronic tunability of oscillation frequency by the control voltage (V). No additional current processing circuitry for gm to bias current (Ib) conversion is needed, as seen in the previous designs shown in Table 1, and thermal voltage (VT) is not involved. It may be seen that ωo is practically active-insensitive, Sεωo≈0.5ε≪1.


## 5. Experimental Results

The response ofV-tunability of the proposed ETAF and LVCQO had been verified by PSPICE simulation [34] and hardware tests; these are shown in Figure 3. ETAF responses, along with its phase error (θe) and THD (measured to 7th harmonic), are shown in Figures 3(a) and 3(b) which are relatively low compared to [10, 24, 25]. Some small phase error (θe) on the experimentally generated sinusoid quadrature signals deviated around 90° was measured as shown in Figure 3(c). The oscillator spectral response in Figure 3(d) shows the LVCQO spectral response wherein phase noise measured is −107 dBc/Hz at an offset of 100 KHz from the tuned frequency of 10 MHz; this response is measured by Tektronix spectrum analyzer (RSA306B) [35].Figure 3
Measured response of ETAF and QO. (a) AP phase response: (A) = 9 V.d.c.; (B) = 6 V.d.c. (b) AP-THD (%) to 7th harmonic. (c) QO wave response tuned at 10 MHz quadrature signals atVo and Eo nodes. (d) QO spectral response of 10 MHz for phase-noise evaluation. (e) QO linear-tuning response. (f) Measured tuning error of VCO.
(a)
(b)
(c)
(d)
(e)
(f)This result is better as compared to that in [14]. A new method of examining the VCO tuning error (β%) is adopted by measuring the shift (Δf) in fo at phase cross-over frequency. The tested value of β% in the overall tuning range of 1 MHz~10 MHz was observed to be less than 4.5%.


## 6. Conclusion

A new design of ETAF function and its subsequent application to the implementation of a VCQO with linear-tuning law are presented. The active blocks are commercially available CFA (AD844) and four-quadrant multiplier (AD835) which are readily available IC modules. Responses of the function circuits are verified experimentally with PSPICE simulation and hardware tests. The quadrature oscillator had been tuned by control voltage (V) in a linear range (band-spread) of 0.5≤fo(MHZ)≤9.9 at low THD. The design does not need any additional current processing circuitry; hence there is no extra quiescent dissipation. Power dissipation of the proposed oscillator circuit is about 33 mW. The circuit is practically active-insensitive relative to device-port track-error (ε). Analysis on the effects of device parasitic components indicates insignificant effect on design equations, even though parasitic capacitors tend to limit the usable high-frequency range. In this work, a relatively new building block, namely, the MMCC, had been utilized for the voltage (V) controlled linearly tunable quadrature oscilator design with 107 dBc/Hz phase noise at 100 KHz offset as shown in Figure 3(d). Thus a comparatively better quality result had been obtained as is shown in Table 2.It may be seen in Tables1 and 2 that in the cited designs active devices used are certain primary blocks with auxiliary voltage/current differencing unit (VDU/CDU) or differential amplifiers with their associated parasitics which tend to limit the desired range of frequency (fp or fo) [6]. Also, tuning by gm or Ib needs additional current processing circuits with parasitic capacitors. The authors believe that, owing to such topological distinction, the frequency tuning ranges are somewhat lower. However in the case of design by VDIBA [10], the range improves owing to the fact that the active component OPA-860 has itself a bandwidth (BW) of 470 MHz. In the proposed design, no such VDU/CDU and gm-to-Ib conversion circuitry are used; this circuit is structured with commercially available AD844 (BW = 60 MHz) and AD835 (250 MHz) IC modules to achieve relatively higher frequency, wherein effects of port-mismatch roll-off poles and parasitics had been included in support.



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*Source: 3454165-2017-08-29.xml*