Single VDGA-Based Dual-Mode Multifunction Biquadratic Filter and Quadrature Sinusoidal Oscillator

Abstract: This article relates to the realization of voltage-mode and/or current-mode multifunction biquadratic filter and quadrature oscillator circuits each using one voltage differencing gain amplifier (VDGA), two resistors and two grounded capacitors. The proposed dual-mode filter having one output and three inputs can provide the three standard biquadratic transfer functions with both voltage and current output filter responses simultaneously. It also has the independent tuning of the angular resonance frequency and the quality factor. With a slight modification of the proposed filter, a new dual-mode quadrature sinusoidal oscillator can be obtained. The proposed quadrature oscillator provides orthogonal resistive/electronic control of both oscillation condition and oscillation frequency. Non-ideal and parasitic conditions are also examined and their effects on the circuit performance are discussed. To confirm the theory, several computer simulation results with PSPICE program are given.


Introduction
Multifunction filters, which can simultaneously realize low-pass (LP), band-pass (BP) and high-pass (HP) filter responses with the same configuration, are fundamental circuit elements widely used in the design of several electronic systems, such as phase-locked loop frequency modulators, crossover networks, stereo demodulators, etc.
This contribution led us to the design of the dual-mode multifunction biquad filter and sinusoidal quadrature oscillator which can provide both voltage and current output signals simultaneously. Each of the proposed configurations includes only a single voltage differencing gain amplifier (VDGA) [50], two resistors, and two grounded capacitors. The proposed dual-mode multifunction biquad configuration can orthogonally control of ω o and Q. With a slight modification of the proposed biquad, a compact dual-mode QO circuit with non-interactive adjustment of CO and ω osc is also obtained. Non-ideal characteristics parasitic element effects on the behavior of the proposed circuits are considered. The working of the circuits is evaluated by simulation results.

VDGA Description
The VDGA device is a versatile six-terminal active building block, which is recently introduced in [50]. As a consequence, a variety of VDGA-based analog circuit applications have been developed in technical literature, such as active filters [50]- [51], quadrature sinusoidal oscillators [52]- [53], and capacitance multiplier circuit [54]. Its schematic symbol is shown in Fig.1, where p and n are high-impedance voltage input ports, z+, z-and x are high-impedance current output ports, and w is low-impedance voltage output port. The ideal property of the VDGA device can be characterized as in the following matrix equation: where g mA and g mB are the transconductance gains, and β is the transfer voltage gain of the VDGA. Fig.2 shows a CMOS implementation of the VDGA used in this work, and it is derived from the one in [50]. According to [55], it can be realized from Fig.2 where for i = 1, 2, 3, 4 and the parameter K is the transconductance of the transistor M ik . From equations (2) and (3), the value of g mk can be scaled electronically, since each transconductance g ik is proportional to the square root of the bias current I Bk . Furthermore, a pair of transconductors M 1B -M 9B and M 1C -M 9C performs a current-controlled voltage amplifier with the voltage transfer gain b = v w /v z+ = g mB /g mC .

Proposed dual-mode multifunction biquadratic filter
The proposed dual-mode multifunction biquadratic filter topology is shown in Fig.3. It essentially comprises a single VDGA, two grounded capacitors and two resistors (one of them is grounded). A straightforward analysis with i in = 0 provides the three voltage transfer functions as follows: and where Therefore, the proposed circuit of Fig.3 provides a noninverting BP, a non-inverting LP and an inverting HP filter voltage response at the output voltages v o1 , v o2 , and v o3 , respectively. Equations (4)- (7) suggest that the characteristics ω o and Q of the filter are obtained as : (9) It is important to note that the parameter ω o is now tunable electronically by adjusting g mA and/or g mB , while the value of the Q-factor is adjustable through the value of R 1 without affecting ω o . In other words, the parameters ω o and Q-factor of the filter are orthogonally controllable. Considering again the proposed dual-mode frequency filter given in Fig.3, its routine algebraic analysis with v in = 0 also reveals the three following current transfer functions: and where the denominator D(s) is the same as in equation (7). It is clear from equations (10)-(12) that, by the same structure, current-mode HP, BP, and inverting LP responses are simultaneously obtained at i o1 , i o2 , and i o3 , respectively. For this version, the ω o and the Q-factor are the same as those given in equations (8) and (9).

Proposed dual-mode quadrature sinusoidal oscillator
From Fig.3, it is further noted that the dual-mode quadrature sinusoidal oscillator is obtainable by setting i in = 0, and connecting the p-terminal to the w-terminal of the VDGA. The resulting QO circuit is depicted in Fig.4, and its characteristic equation is found as: From the above equation, the CO and ω osc of the realized QO are respectively given by: and It is clear from equations (14) and (15) that the CO of the oscillator can be varied independently of ω osc by R 1 . This QO provides two voltage outputs and two current outputs, which relate as follows: (17) Figure 4: Proposed dual-mode quadrature sinusoidal oscillator.
Equations (16) and (17) show that the two voltage and current outputs are each other shifted in phase by 90°, thereby exhibiting quadrature property to the proposed QO circuit. Moreover, the proposed oscillator also offers versatility by simultaneously providing both quadrature voltage as well as current outputs.
The proposed dual-mode multifunction biquadratic filter of Fig.3 is re-analyzed using the non-ideal performance relation (18) of VDGA, and the non-ideal ω o and Q become as follows: and 2 1 1 Equations (19) and (20) indicate that the transfer errors directly affect the parameters ω o and Q of the filter. However, since α A and α B are typically close to unity, these small deviations can be compensated by slightly re-adjusting the values of g mA and g mB via the bias currents I BA and I BB , respectively.
Similarly, in the non-ideal case, the characteristic equation of the proposed dual-mode QO circuit in Fig.4 can be found as: where the parameters CO and ω osc of the oscillator for this case are modified as: The CO and ω osc of the oscillator are deviated from the ideal case by the non-ideal transfer gains. In the same manner, because the voltage transfer gain β is proportional to g mB /g mC , the transconductance g mC is adjustable to minimize the influence of α A and δ on the CO. Also, note from (23) that the slight ω osc deviation can be overcome by re-tuning g mA and g mB .

Effect of parasitic elements
In practice, a parasitic problem should be taken into count to determine the effects of the VDGA parasitic impedances on the proposed circuits. Fig.5 displays a sophisticated equivalent model behavior that represents the practical VDGA. In Fig.5, the dashed line encircles the practical VDGA, while the continuous line denotes the ideal VDGA with various parasitic elements at its terminals. As can be seen, there are parasitic parallel resistances and capacitances from terminals p, n, z+ and z-to ground [(R p //C p ), (R n //C n ), (R z+ //C z+ ) and R z-// C z-)], and a parasitic serial resistance (R w ) at the terminal w. Therefore, by applying the practical model of the VDGA to the proposed dual-mode multifunction filter in Fig.3 and assuming the condition R 2 >> R w , the modified ω o and Q can be expressed as: and In a similar way, accounting for the VDGA parasitic elements given in Fig.5, the CO and ω osc of the proposed dual-mode sinusoidal QO in Fig.4 can be derived as: and From inspection of equations (24)- (27), it is possible to reduce the VDGA parasitic influences on the proposed circuits by taking the following conditions in the design: R 1 << R z+ , R x << R n , C 1 >> C z+ and C 2 >> (C x + C n ).

Simulation results and discussions
The correct operation of the proposed dual-mode multifunction filter and QO topologies in Fig. 3 and 4 is assessed through PSPICE simulation results. For this purpose, the transistor models of the TSMC 0.25-µm CMOS process parameters have been used. The CMOS VDGA in Fig.2 is simulated under DC supply voltages of ±1V. The values of transistor aspect ratios (W/L) are provided in Table 1.

Simulation results of the proposed dual-mode multifunction biquadratic filter
For the proposed dual-mode multifunction biquadratic filter in Fig.3, the active and passive components are taken as: g mk ≅ 1 mA/V (I Bk = 100 µA), R 1 = R 2 = 1 kΩ and C 1 = C 2 = 100 pF which result in f o ≅ 1.59 MHz and Q ≅ 1. The simulated and ideal frequency responses for the voltage and current gains are represented in Fig.6. In Fig.7, the input and output waveforms for the proposed BP filters at 1.59-MHz sinusoidal input signals are also shown. Total harmonic distortions (THDs) are less than 4% and 1.6% for the voltage-mode and currentmode BP filters, respectively. In addition to the simulation result, this filter has the total power consumption of about 1.49 mW. Next, the large-signal behavior of the proposed filter is also evaluated by applying a sinusoidal input signal of 1.59 MHz. The BP output responses are found to demonstrate the THD variations of input signal amplitude as shown in Fig.8. To test the adjustability of Q-factor without changing the f o -value, the following circuit components are selected as: g mk = 1 mA/V, C 1 = C 2 = 100 pF, with four different values of R 1 namely 0.5 kΩ, 1 kΩ, 2 kΩ, and 10 kΩ. The four BP voltage responses are illustrated in Fig.9, which accordingly shows the orthogonal variability of Q-factor. Furthermore, by keeping the product of g m1 R 1 constant, and tuning the value of g m1 /R 1 only, an independent tunability of f o can be obtained as shown in To evaluate the mismatch and process variation effects, Monte Carlo (MC) analysis has been performed for 100 iterations. MC simulation results showing deviations in filter responses for 5% changes in all passive elements (R 1 , R 2 , C 1 , and C 2 ) and transconductances (g mk ) are illustrated in Fig.11(a) and 11(b), respectively. Temperature variation impacts on the filter responses are also studied for observed range 0°C to 100°C. The simulation results of the gain-frequency characteristics with a variation in operating temperature are drawn in Fig.12

Simulation results of the proposed dual-mode quadrature sinusoidal oscillator
With the same above designed values, the proposed dual-mode QO circuit of Fig.4 has been simulated for a frequency of oscillation of f osc = 1.59 MHz. Simulation results of the voltage and current quadrature output responses are shown in Fig.13 and 14, respectively. Fig.13(a) indicates the two quadrature voltage outputs difference in phase by 88°, while Fig.14(a) shows 91° phase-shifted quadrature current outputs. The observed THD for both voltage and current outputs are around 3.76% and 3.73%, respectively. The Lissajous patterns for the two voltage and two current outputs are further depicted through Fig.15(a) and 15(b), respectively. It is observed that the resultants produce circles around the origin with no tilt in axis illustrating the quadrature property of the proposed oscillator circuit.

Conclusions
The paper proposes compact circuit configurations for realizing dual-mode (i.e. both voltage-mode and current-mode) multifunction biquadratic filter and quadrature sinusoidal oscillator. Each of the proposed circuit configurations requires a single VDGA, two grounded capacitors, and two resistors (one of them is grounded). The proposed filter can realize simultaneously the three standard biquadratic filtering functions, i.e. LP, BP, and HP ones, with both voltage and current output responses. It has also capable of independent control of ω o and Q-factor. Another notable advantage of the proposed circuit is that it can also be used as a sinusoidal quadrature oscillator to provide two quadrature voltage outputs and two quadrature current outputs simultaneously. The oscillation condition and the oscillation frequency of the proposed dual-mode QO are orthogonally controllable by separate bias currents. The effect of non-idealities influences of the VDGA on the circuit functionality has been studied, and simulation results using PSPICE with TSMC 0.25-µm CMOS technology have also been included to validate the functionality of the proposed circuits in both frequency filter and oscillator mode.