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All-Pass Filter Based Linear Voltage Controlled Quadrature Oscillator.

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].

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 ([g.sub.m]) to bias current ([I.sub.b]) conversion is avoided; such conversion needs additional hardware complexity and involvement of thermal voltage ([V.sub.T]) [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-[r.sub.z][C.sub.z] 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 ([[omega].sub.0]) is active-insensitive relative to port-mismatch error ([epsilon]). Experimental results on ETAF and quadrature oscillator responses are verified by PSPICE simulation and hardware test.

2. Analysis

The proposed ETAF is shown in Figure 1(a) where the nodal relations are as follows:

CFA: [I.sub.z] = [alpha][I.sub.x], [V.sub.x] = [beta][V.sub.y], and [V.sub.o] = [delta][V.sub.Z] MMCC: [I.sub.z] = a[l.sub.x], [V.sub.x] = b(k[V.sub.y1][V.sub.y2]), and [V.sub.w] = [gamma][V.sub.z].

The port tracking ratios are ideally unity but may be postulated by error coefficients (|[epsilon]| [much less than]: 1) as [alpha] = (1 - [[epsilon].sub.i]) = a, [beta] = (1 - [[epsilon].sub.v]) = b, and [delta] = (1 - [[epsilon].sub.z]) = [gamma], where k = (0.1/volt) is the multiplication constant [20].

Analysis of Figure 1(a) yields the transfer function G [equivalent to] [V.sub.o]/[V.sub.i] as

[mathematical expression not reproducible] (1)

parasitic elements, yielding a modified transfer where m = [r.sub.3]/[r.sub.4], n = [r.sub.2]/[r.sub.1] and [tau] = RC/kV.

In ideal devices port tracking ratios are all unity; assuming the realizability conditions as m = 1 = n ([r.sub.1, 2, 3, 4] = r) for simplicity, we get

[mathematical expression not reproducible] (2)

which is the nonminimum phase all-pass function, where transmission gain |G| = 1 and phase ([theta]) is tunable in a range 180 [less than or equal to] [theta][degrees] [less than or equal to] 0, being variable by control voltage (V) as

[mathematical expression not reproducible] (3)

3. Effects of Nonidealities

3.1. Parasitic Components. The parasitic components appear as shunt-[r.sub.z][C.sub.z] arms at current source nodes of the device; as per data-book [21], the typical values are 2 [less than or equal to] [r.sub.z] (M[OMEGA]) [less than or equal to] 5 and 3 [less than or equal to] [C.sub.z] (pF) [less than or equal to] 5.5; ratio of circuit resistors (K[OMEGA]) relative to [r.sub.z] is assumed to be negligible. Preanalysis now with finite parasitic elements yields modified transfer elements in Figure 1(a) exhibiting a single-pole roll-off model [22], given by

[mathematical expression not reproducible] (4)

where

[mathematical expression not reproducible] (5)

Then (4) simplifies to

[mathematical expression not reproducible] (6)

The frequency-domain behavior may be estimated by writing s = j[omega] in (6), given by

[mathematical expression not reproducible] (7)

where

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible]

With [C.sub.z] [approximately equal to] 3.3 pF (measured) and r = 1 K[OMEGA], we estimated value of the parasitic pole frequency [f.sub.z] [approximately equal to] 54 MHz. Hence for usable ranges of < [f.sub.z], effects of parasitic capacitors are negligible and G = G; therefore the nominal APF function is preserved with the limit f [less than or equal to] [f.sub.z].

3.2. Roll-Off Pole of Device-Port Transfer Ratios. At relatively high-frequency ranges, the port-transfer ratios [alpha], [beta], [delta] and a, b, [gamma] of the active components are as follows:

[mathematical expression not reproducible] (9)

The dc values of all the products of port coefficients in (1) maybe taken as unity; for example, [[alpha].sub.o1, 2] [approximately equal to] {1 - ([[epsilon].sub.i1] + [[epsilon].sub.i2])} [approximately equal to] 1 since [[epsilon].sub.i1, 2] [much less than] 1.

The modified APF function (G ) is now written as

[mathematical expression not reproducible] (10)

where

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

The function M(s) relates to the MMCC block; it is decomposed as M(s) = [M.sub.o](s) * [M.sub.e](s), where [M.sub.o] = 1/S[[TAU].sub.O] and

[mathematical expression not reproducible] (15)

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 [p.sub.1~5] = l/[[omega].sub.p]. Writing u = ([omega]/[[omega].sub.p]) [much less than] 1 and solving for (11) we get

[mathematical expression not reproducible] (16)

which implies [mathematical expression not reproducible] and [mathematical expression not reproducible] Similar results are checked for (12) to (14). For the effects of the MMCC roll-off, we put [psi] = ([omega]/[[omega].sub.m]) [much less than] 1, where [p.sub.i, v, z] [equivalent to] 1/[[omega].sub.m]; solving (15), we get

[mathematical expression not reproducible] (17)

which reduces to [mathematical expression not reproducible]. This analysis indicates that effects of roll-off poles of the active building blocks are quite negligible in the range of operating frequencies (/ < [f.sub.p,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.

We now present the linear VCO as implemented by forming a regenerative feedback loop formed with the ETAF in Figure 1(a) and the ETI in Figure 2; transfer of ETI is

[mathematical expression not reproducible] (18)

where [[tau].sub.o] = [R.sub.o][C.sub.o]/kV, [xi] = [C.sub.o]/[C.sub.Z] [much less than] 1, and [R.sub.o]/[r.sub.z] [much less than] 1; hence H(s) = 1/S[[tau].sub.O]. Grounding of the capacitor (C) facilitates absorbing [C.sub.z] in C-values at predesign level. Device imperfection relative to port-transfer roll-off pole frequencies ([[omega].sub.ab[gamma]]) is estimated by writing

[mathematical expression not reproducible] (19)

where [[epsilon].sub.T] = ([[epsilon].sub.i] + [[epsilon].sub.v] + [[epsilon].sub.z]) [much less than]: 1.

The deviation due to roll-off is

As in previous section if we assume [[epsilon].sub.a,b,y] [approximately equal to] [[omega].sub.m] and put [psi] = ([omega]/[[omega].sub.m]), then H reduces similar to (17) as

[mathematical expression not reproducible] (21)

Now implementation of oscillator is derived by equating {1 - GH(s)} = 0 which gives the characteristic equation (CE):

CE: [a.sup.2] [[TAU].sub.O][TAU] + s ([[TAU].sub.O] - [TAU]) + 1 = 0

Realizability condition: [[TAU].sub.O] = [tau] R = [R.sub.o], (22) C = [C.sub.o]

Oscillation frequency: [[omega].sub.o] = [kV/RC].

So tuned frequency ([f.sub.o] in Hz) is linearly and directly variable by control voltage (V)

[f.sub.o] = [V/(20[pi]RC)]. (23)

The proposed method provides direct electronic tunability of oscillation frequency by the control voltage (V). No additional current processing circuitry for [g.sub.m] to bias current ([I.sub.b]) conversion is needed, as seen in the previous designs shown in Table 1, and thermal voltage ([V.sub.T]) is not involved. It may be seen that [[omega].sub.o] is practically active-insensitive, [mathematical expression not reproducible].

5. Experimental Results

The response of V-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 ([[theta].sub.e]) 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 ([[theta].sub.e]) on the experimentally generated sinusoid quadrature signals deviated around 90[degrees] 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].

This result is better as compared to that in [14]. A new method of examining the VCO tuning error ([beta]%) is adopted by measuring the shift ([DELTA]f) in [f.sub.o] at phase cross-over frequency. The tested value of [beta]% 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 [less than or equal to] [f.sub.o] (MHZ) [less than or equal to] 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 ([epsilon]). 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 Tables 1 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 ([f.sub.p] or [f.sub.o]) [6]. Also, tuning by [g.sub.m] or [I.sub.b] 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 [g.sub.m]-to-[I.sub.b] 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Koushick Mathur, (1) Palaniandavar Venkateswaran, (2) and Rabindranath Nandi (2)

(1) Department of Electronics & Communication Engineering, UIT, Burdwan University, West Bengal 713104, India

(2) Department of Electronics & Telecommunication Engineering Jadavpur University, Kolkata 700032, India

Correspondence should be addressed to Rabindranath Nandi; robjutel@yahoo.co.in

Received 14 June 2017; Accepted 27 July 2017; Published 29 August 2017

Academic Editor: Jiun-Wei Horng

Copyright [c] 2017 Koushick Mathur et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

https://doi.org/10.1155/2017/3454165
Table 1: Comprehensive summary of recent APFs.

Ref.      ABB             ABB implementation

[1]       VOA             HA 2544C opamp
[2]       ecu             LF 356N opamp with current mirror
[3]       DO-CCCII        Dual-output current controlled conveyors
[4]       CCC II with OA  Current controlled conveyor
[5]       DO-CCII         CC II with additional z-copy node
[6]       DDA             Differential amplifier (DA)
[7]       DDCC            Current conveyor with DA
[8]       DVCC            VDU followed by OTA
[9]       DVCC            VDU followed by CC
[10]      VDIBA           OTA with differential input buffer
[11]      OTA             OTA with DA
Proposed  CFA-MMCC        CFA844 and multiplier AD835

Ref.      [f.sub.p] reported (Hz)  Tunability

[1]       125 K                        RC
[2]         1.6 K                      RC
[3]         1.63 M                 [I.sub.b]
[4]        12.15 K                 [I.sub.b]
[5]         3.89 M                     RC
[6]        16 K                        RC
[7]       318 K                        RC
[8]       400 K                    [I.sub.b]
[9]         1.59 M                     RC
[10]        9.4 M                  [I.sub.b]
[11]      375 K                    [I.sub.b]
Proposed    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].

Table 2: Comparative description of recent QO designs.

Ref.      Device used  Electronic tunability  Linear tuning law

[5]       DO-CCII      No                     No
[10]      VDIBA        Yes                    No
[12]      CFA          No                     No
[14]      CFA          No                     No
[23]      OTA          Yes                    Yes
[24]      VOA          No                     No
[25]      CFOA         No                     No
[26]      CDTA         Yes                    No
[27]      CCCDBA       Yes                    No
[28]      CCCCTA       Yes                    No
[29]      OTA          Yes                    No
Proposed  CFA-MMCC     Yes                    Yes

Ref.      [f.sub.0] reported (Hz)  Tuning by

[5]         2.33 M                 RC
[10]        8.5 M                  [square root of [I.sub.b]]
[12]      909 K                    RC
[14]      398 K                    RC
[23]       64 K                    [I.sub.b]
[24]      160 K                    RC
[25]      159 K                    RC
[26]        1.73 M                 [square root of [I.sub.b]]
[27]      419 K                    [square root of [I.sub.b]]
[28]        1.1 M                  [square root of [g.sub.m]]
[29]        1.59 M                 [square root of [g.sub.m]]
Proposed  ~10.2 M                  V

Ref.      THD (%)

[5]       1.17~2.82
[10]      2.25
[12]      -
[14]      1.42
[23]      -
[24]      -
[25]      3.16
[26]      3.0
[27]      2~1.2
[28]      1~2.9
[29]      1~2.2
Proposed  0.9~2.8
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Title Annotation:Research Article
Author:Mathur, Koushick; Venkateswaran, Palaniandavar; Nandi, Rabindranath
Publication:Active and Passive Electronic Components
Article Type:Report
Date:Jan 1, 2017
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