Tunable Mixed-Mode Voltage Differencing Buffered Ampliﬁer-Based Universal Filter with Independently High- Q Factor Controllability

: This paper proposes the design of a mixed-mode universal biquad conﬁguration, which realizes generic ﬁlter functions in all four possible modes, namely voltage mode (VM), current mode (CM), transadmittance mode (TAM), and transimpedance mode (TIM). The ﬁlter architecture employs two voltage differencing buffered ampliﬁers (VDBAs), two resistors and two capacitors, and can provide lowpass (LP), bandpass (BP), highpass (HP), bandstop (BS), and allpass (AP) biquadratic ﬁltering responses without any circuit alteration. All passive elements used are grounded, except VM. The circuit not only allows for the electronic tuning of the natural angular frequency ( ω o ), but also achieves orthogonal tunability of the quality factor ( Q ). It also provides the feature of availability of output voltage at the low-output impedance terminal in VM and TIM, and does not require inverting-type or double-type input signals to realize all the responses. Moreover, in all modes of operation, the high- Q ﬁlter can be easily obtained by adjusting a single resistance value. Inﬂuences of the VDBA nonidealities and parasitic elements are also discussed in detail. PSPICE simulations with TSMC 0.18- µ m CMOS process parameters and experimental testing results with commercially available IC LT1228s have been used to validate the theoretical predictions.


Introduction
The design and synthesis of active frequency-selective filters have a very significant role in the areas of continuous-time signal processing, instrumentation and measurement applications, and wireless communication. In recent years, the design of general mixed-mode universal biquadratic filters with input voltages and/or currents and output voltages and/or currents has received a lot of attention from researchers. Considering the nature of input and output signals, the filters can be classified into four possible modes, i.e., voltage mode (VM), current mode (CM), transadmittance mode (TAM), and transimpedance mode (TIM). The VM and CM operations perform frequency filtering behavior on voltage and current signals, respectively. The TAM and TIM operations can be used as bridges for connecting a VM filter to any of the CM circuits and vice versa. Accordingly, the mixed-mode universal filters that provide all generic filtering responses in all four modes increase the versatility and flexibility of practical filtering applications and requirements. Consequently, these filters are worthy of investigation and research. Therefore, in the recent past, several structures realizing mixed-mode universal biquadratic filters with a variety of high-performance active elements have appeared in the literature . Table 1 presents a comparative study of earlier-reported mixed-mode universal filters based on various types of high-performance active components.
Due to its simple structure, versatility, and CMOS integrability, the voltage differencing buffered amplifier (VDBA) is an alternative and suitable active building block for biquad filter solutions [47]. Interestingly, the internal circuit architecture of the VDBA block consists of an operational transconductance amplifier (OTA) and a voltage follower (VF) [48,49]. This simple circuitry implementation leads to low power consumption and small chip area requirements. Therefore, in this work we adopt the advantages provided by the VDBA device to design a mixed-mode universal biquad filter. The designed filter topology possesses the following salient properties: (i) use of a reasonable number of active and passive elements (i.e., two VDBAs, two resistors, and two capacitors); (ii) capability of realizing universal biquadratic filter functions in all four modes; (iii) employment of all grounded passive elements, except for VM; (iv) exhibits inbuilt tuning capability; (v) noninteractive control of Q; (vi) low-output impedance for VM and TIM operations. The functionality of the circuit has been evaluated through simulation results based upon TSMC 0.18-µm 1P6M CMOS technology parameters, and furthermore through the experimental measurements of the commercially available integrated circuit (IC), LT1228. Additionally, all properties of the proposed mixed-mode filter are mentioned and compared with the previous related works in Table 1. Furthermore, Table 2 presents a comprehensive comparison illustrating the superiority of the proposed mixed-mode universal filter over the earlier reported relevant VDBA-based biquad configurations [50][51][52][53][54][55][56][57][58]. As can be observed, no earlier VDBA-based filter realization can be operated in all four modes of operation. The passive components used for their realizations are all floating. Although the works proposed in [54,56] use a single VDBA as an active element, they suffer from operating in only a single mode and using at least four floating passive components.   The paper is organized as follows, Section 2 describes the VDBA. The proposed mixed-mode universal filter is proposed in Section 3. The non-ideal gain effect, sensitivity performance, and parasitic impedance effect are investigated in Sections 4 and 5, respectively. The simulation results are given in Section 6, while the practical circuit implementation and the experimental results are presented in Section 7. Finally, the paper is concluded in Section 8.

VDBA Description
The electrical symbol of the VDBA is shown in Figure 1. The defining characteristic of the VDBA can be described by the following matrix equation: where g m is the transconductance gain of the VDBA. The transconductance g m , as usual, can be tuned by a bias current or voltage, thereby imparting tunability to the structure. Further, α and β are the non-ideal transconductance gain and nonideal voltage gain, respectively. These non-ideal gains can be defined as α = (1 + ε α ) and β = (1 + ε β ), in which the tracking errors are identified as |ε α | << 1 and |ε β | << 1. Accordingly, the values of α and β are ideally equal to unity. As mentioned above, the VDBA block comprises two essential circuit blocks: an OTA and a VF [47,48]. The simple CMOS implementation of the VDBA used in this work is shown in Figure 2, in which the OTA consists of transistors M 1 -M 6 ; and it is followed by a VF formed by transistors M 7 -M 14 . A pair of diode-connected PMOS active load (M 3 -M 4 ) is driven by a source couple pair (M 1 -M 2 ). The transconductance gain (g m ) of the OTA stage can be externally tuned by the bias current (I B ), as described by the following expression: where K n = µ n C ox is the transconductance parameter, and (W/L) is the ratio of the widthto-length of the transistors M 1 and M 2 . Note from Equation (2) that the transconductance g m is electronically adjustable utilizing I B . Further, the voltage drop across the grounded impedance at terminal z (v z ) is then conveyed to the w terminal with a unity voltage gain by the VF. Thus, the negative-feedback loop established by M 7 -M 11 provides a very low output impedance at the w terminal. For the simulation purpose, the TSMC 0.18-µm level 7 CMOS model parameter has been employed, where the transistor aspect ratios are given in Table 3.

Proposed Mixed-Mode Universal Biquad Filter
The proposed configuration, which is realized by two VDBAs, two resistors, and two capacitors, is shown in Figure 3. It is important to note that, in this realization, the resistors R 1 and R 2 are permanently grounded. From the proposed circuit in Figure 3, the universal biquadratic filter operated in all four possible modes is available as follows. For VM operation: Assuming ideal VDBA (i.e., α =β = 1) and setting i in = 0, the general voltage biquadratic transfer functions of this MISO filter can be obtained as follows.

•
With v in = v 1 (input voltage) and v 2 = v 3 = v 4 = 0 (grounded), then the LP response is realized as: • With v in = v 2 , v 1 = v 3 = v 4 = 0, and g m2 = 1/R 2 , then the BP response is realized as: • With v in = v 4 , and v 1 = v 2 = v 3 = 0, then the HP response is realized as: • With v in = v 1 = v 4 , and v 2 = v 3 = 0, then the BS response is realized as: , v 2 = 0, and g m2 = 1/R 2 , then the AP response is realized as: where Under appropriate conditions, the proposed circuit realizes all five generic biquadratic filter responses at v out , which are taken from the w-terminal of VDBA2. Thus, the voltage output of the circuit has a very low output impedance, which is suitable for VM cascadability. Moreover, in this MISO configuration, there is no requirement for negative and double input voltage signals to realize the desired filter responses.
For CM operation: If v 1 = v 2 = v 3 = v 4 = 0 (grounded), the CM biquad transfer functions for this SIMO filter, attained from the circuit analysis of Figure 3, are given by and where H 0 is the passband gain equal to 1/g m2 R 1 . Additionally, for R 1 = 1/g m2 , the BS current response can be realized by connecting the appropriate output currents as i BS = i HP -i LP . In the same way, the AP response can also be obtained by the interconnection of LP, BP, and HP responses as i AP = i HP -i BP -i LP . For TAM operation: With v in = v 3 and v 1 = v 2 = v 4 = 0, then we obtain the TAM filter functions as follows: and Equations (12) and where H 3 = R 2 .
In all the above working modes, the important filter characteristics ω o and Q according to Equation (8) are found as: and Inspection of Equations (19) and (20) reveals that the characteristic frequency ω o can be tuned electronically through the transconductance g mi (i = 1, 2) of the corresponding VDBA. Moreover, the filter parameter Q is independently controllable by the R 2 . Hence, the high-Q filter could be conveniently obtained by simply adjusting a single resistance R 2 .

Analysis of the Non-Ideal Gain Effect and Sensitivity Performance
Considering only the influence of the non-ideal gains (α =β = 1), the characteristics ω o and Q of the proposed filter will be modified as: and where α i and β i (i = 1, 2) are the parameters α and β of the i-th VDBA, respectively.
The sensitivity analysis of ω o and Q with respect to active and passive components is also carried out, and the calculation results are obtained as: and It can be easily deduced that all the sensitivity coefficients of ω o and Q are not greater than one in all four modes of operation.

Analysis of the Parasitic Impedance Effect
In this section, the effect of various parasitic impedances of the employed VDBA on the performance of the proposed mixed-mode universal filter in Figure 3 is to be analyzed. In practice, the non-ideal VDBA model with its various terminal parasitics is represented in Figure 4. It appears that the finite parasitic resistances and capacitances at the p, n, and z terminals are in the form [R p //(1/sC p )], [R n //(1/sC n )], and [R z //(1/sC z )], respectively, while the low-value serial resistance (R w ) appears at the w terminal. Ideally, these parasitic values are assumed to be R p = R n = R z = ∞, R w = 0, and C p = C n = C z = 0. Under the effect of these parasitics, the non-ideal denominator of all transfer functions in all four working modes becomes: where R 2 = R 2 // R z2 // R n1 , C 1 = C 1 + C z1 , and C 2 = C 2 + C z2 + C n1 . Equation (27) illustrates that the order of the filter function is modified due to the parasitic pole ω parasite , (i.e., ω parasite = 1/R w1 C p2 ). However, this effect can be diminished if the proposed circuit is designed to operate at a useful frequency much less than ω parasite or under the following condition: ω << 0.1 ω parasite . As the term (sR w1 C p2 + 1) is made close to unity, Equation (27) can be further simplified to From D n (s), the expressions for ω o and Q in the presence of parasitic impedances are thus obtained as: and Therefore, it may be concluded that the parasitic effects on the ω o and Q would be alleviated if the following designs must be satisfied: and minimum (C 1 , C 2 ) >> parasitic capacitances (C n1 , C z1 , C z2 ).

Simulation Results
The functionality of the proposed mixed-mode universal filter in Figure 3 was validated by the PSPICE circuit simulation program. The VDBA was modeled using the CMOS structure mentioned in Figure 2 with ±V = 0.75 V and I A = 15 µA. In all simulations, the capacitor values were chosen with C 1 = C 2 = 50 pF. The circuit was designed for f o = ω o /2π = 1.52 MHz and Q = 1; the active and passive components were chosen as: g m1 = g m2 = 0.48 mA/V (I B1 = I B2 = 50 µA), and R 1 = R 2 = 2 kΩ. Figures 5 and 6 illustrate the ideal and simulated LP, BP, HP, BS, and AP frequency responses for VM and TAM (i.e., when the input is voltage), respectively. Figure 7 shows the ideal and simulated LP, BP, and HP gain responses for CM and LP and BP in TIM (i.e., when the input is current). The simulated f o of the BP filter was measured as 1.44 MHz, which is an error of 5.26% concerning its theoretical value. The simulation results of Figure 6 also show that the passband gain H 1 of the LP response for TAM is obtained as −66 dBS, which depends on H 1 = 20 log 10 (1/R 1 ). Similarly, the passband gains H 2 for BP, HP, BS, and AP responses are the same as the gain H 1 of the LP filter because of H 2 = 20 log 10 (g m2 ) due to g m2 = 1/R 1 .
To examine the transient behavior of the proposed filter, the LP, BP, and HP responses were carried out for the VM operation. The sinusoidal input voltage of 50 mV (peak) at a frequency of 1.52 MHz was applied and the corresponding output current waveforms are given in Figure 8. As can be monitored, the phase differences between the input and LP, BP, and HP outputs are found to be −92.73 • , 5.45 • , and 87.29 • , which are consistent with ideal values equal to −90 • 0 • , and 90 • , respectively. The percentages of the total harmonic distortion (THD) for the three filter outputs are 0.22% for LP, 1.12% for BP, and 0.64% for HP. In addition, the THD variations of the LP, BP, and HP output voltages on the input signal amplitudes are also shown in Figure 9. It is shown that when the applied input signal amplitude increases by 100 mV (peak), the THD values are within 2.2%. Through the simulation results, the circuit has a total power consumption of 0.373 mW.
As indicated in Equations (19) and (20), the parameters ω o and Q of the proposed filter can be set orthogonally. Figure 10 shows the Q-factor adjustability of the BP responses in VM for various values of R 2 . In this case, the Q-factors are set as 0.5, 2.4, 9.5, and 95.5 with the R 2 value of 1 kΩ, 5 kΩ, 20 kΩ, and 200 kΩ, respectively. The results demonstrate that the high-Q tuning can be achieved by adjusting R 2 without influencing f o . Figure 11 represents the VM gain responses of the BP filter for three different values of I B and R 2 . The BP filter is designed for f o = 1.12 MHz, 2.15 MHz, and 3.72 MHz, while keeping Q fixed at 9.5. Table 4 gives the component values used in Figure 11 and the corresponding calculated and simulated f o .        To study the effect of temperature variations, the proposed filter was analyzed under various ambient temperatures. Figure 12 demonstrates the simulated frequency responses of the AP filter in VM for different temperatures (0 • C, 20 • C, 50 • C, 75 • C, and 100 • C). At the natural angular frequency f o = 1.52 MHz, the simulation results show that the gain and phase responses lie within the range of -1.3 dBV to -2.7 dBV, and -184 • to -228 • , respectively. This variation does not have a strong effect on the gain and phase responses of the circuit. The noise behavior of the proposed filter versus the frequency has also been evaluated, as shown in Figure 13. The output voltage noises of the BP filter at the frequency of 1.52 MHz were found to be 20.50 nV/Hz 1/2 for VM and TIM operations, while the output current noises for CM and TAM were 8.45 pA/Hz 1/2 .   Figure 14. Additionally, the corresponding histogram demonstrating the f o variations in BP output is shown in Figure 15. According to the statistical results, the mean, median, and standard deviation were, respectively 1.50134 MHz, 1.49936 MHz, and 53.2151 kHz, which implies that the proposed filter exhibits a reasonable sensitivity figure to the passive component tolerances. This further validates the robustness of the design.

Experimental Results
The features of the proposed mixed-mode universal filter configuration in Figure 3 were also verified by laboratory experiments using a commercially available IC LT1228 from Linear Technology [59]. Figure 16 shows the PCB realized for measurement purposes. The supply voltage used was ±5 V. The experimental setup of the proposed mixed-mode universal filter utilizing the PCB board in Figure 16 is also shown in Figure 17. In CM and TIM measurements, an additional AD844 and a conversion resistor R C were employed to perform the voltage-to-current conversion (V-to-I), where R C = 1 kΩ. On the other hand, to obtain CM and TAM filter results, two AD844s and a resistor R C were employed as a current-to-voltage converter (I-to-V). The passive and active components were selected as R 1 = R 2 = 1 kΩ, C 1 = C 2 = 100 pF, and g m1 = g m2 = 1 mA/V (I B1 = I B2 = 100 µA, where g mi = 10 I Bi ). As a consequence, the theoretical filter parameters for this design were f o = 1.59 MHz and Q = 1. Figure 18 shows the experimental measurements in the time domain of the input and output responses in VM operation, for a 50 mV (peak) sinusoidal input voltage (v in ) at 1.59 MHz. The corresponding spectral analyses of the v out were also measured, and the results are provided in Figure 19. The measured results indicate that the THD figures for the LP, BP, HP, BS, and AP output responses were found to be 1.23%, 2.05%, 1.78%, 0.87%, and 2.04%, respectively. Hence, they have no significant distortion that can be observed in our frequency range of interest. The experimental results of Figure 19 also show that the spurious-free dynamic range (SFDR) for the cases of LP, BP, HP, BS, and AP were determined to be 40.70 dBc, 34.60 dBc, 38.07 dBc, 44.82 dBc, and 34.66 dBc, respectively.

Conclusions
In this work, a mixed-mode universal filter configuration has been proposed based on only two VDBAs, two grounded resistors, and two capacitors. The proposed circuit is capable of realizing all five biquadratic filtering functions in VM, CM, and TAM operation. In TIM operation, the circuit can realize LP and BP responses. The circuit satisfies the major advantages simultaneously such as (i) employment of grounded passive components, except for VM operation; (ii) having electronic tunability for ω o ; (iii) independent controllability of its quality factor; (iv) unemploying inverting-type or double-type input signals; (v) having low output impedance for VM and TIM operations; (vi) low active and passive sensitivity features. The high-Q filter can be easily achieved through a single resistance adjustment. The mathematical analyses such as non-ideal gains, sensitivity performance and parasitic analysis along with the numerical simulation results and experimental measurement results are shown, in order to strengthen the design idea.

Conflicts of Interest:
The authors declare no conflict of interest.

Symbols
The following symbols are used in this manuscript: α non-ideal transconductance gain β non-ideal voltage gain ε α tracking error of transconductance gain ε β tracking error of voltage gain K n transconductance parameter of the transistor µ n mobility of the carriers C ox gate-oxide capacitance per unit area W effective channel width L effective channel length Ω Ohm dBV voltage decibel dBA ampere decibel dBS siemens decibel dBΩ Ohm decibel dBc decibels relative to the carrier V/Hz 1/2 the unit of a noise voltage A/Hz 1/2 the unit of a noise current