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CM-Biquad Filter Using Single DO-VDBA ()

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*Circuits and Systems*,

**9**, 133-139. doi: 10.4236/cs.2018.99014.

1. Introduction

Biquads are the major components in the area of electronics. It is extensively used in numerous electronic applications, which include analog and digital signal processing, communication etc. [1] . Looking at the vast literature available, we are well aware that innumerable filters have been designed till date using several active building blocks and each of them has some advantages and disadvantages too. Current mode filters have proven its vitality in several aspects, some of them are high performance, lower power consumption, miniaturization of circuit, higher frequency range, increased slew rate and linearity. So in this paper the current mode is exploited to get such benefits. Filters available in literature have also reportedly suffered few problems such as smaller frequency range, lack of tunability and excessive use of passive components.

The filter proposed in this paper uses current mode VDBA, it is single input and dual output and it is electronically tunable as well [2] .

FB-VDBA is well known for voltage-mode analog signal processing. It is a fully-differential structure. It is used to realize floating inductor [3] .

Using voltage mode configuration, biquad filter has also been realized using VDBA. Those filters use 2 active elements and 2/3 passive components. They were having low passive sensitivity and acceptable value of THD [4] .

An extremely convenient multiphase oscillator with reduced complexity as an easy non-tunable replacement to classical conceptions employing lossy integrators in phase-shifted loop has been designed using VDBA/VDIBA. Linearly tunable quadrature oscillator and square wave generator were also reported in the literature [5] .

Single VDIBA and a capacitor, can be used to realize a novel voltage-mode (VM) resistorless, 1st-order all-pass filter (APF) [6] .

Several analog signal-processing filters are also available using current mode circuit; CDBA is one such ABB [7] .

CM filters have proven its vitality in several aspects, some of them are high performance, lower power consumption, miniaturization of circuit, higher frequency range, increased slew rate and linearity, so in this manuscript the CM is exploited to get such benefits. Filters available in literature have also reportedly suffered few problems such as smaller frequency range, lack of tunability and excessive use of passive components. FB-VDBA is very useful building block for analog circuit design. It is also helpful in the designing of circuit with least number of passive components [8] .

Voltage mode configuration has also been proved helpful in design of a current-mode and voltage-mode electronically tunable quadrature oscillator that consists of both voltage and current output [9] .

To the best awareness of the authors no same type of current mode biquad is available in the open literature. So, in this manuscript we have proposed a biquad using single DO-VDBA whose frequency is electronically controllable. Offering very low active and passive sensitivities are also important characteristics of this circuit.

2. The Proposed New Structure

The DO-VDBA is represented in its symbolic form as shown in Figure 1, where the input terminals are given by P and N while the output terminals are denoted by z, w^{+} and w^{−}. Current flowing through “z” terminal is given by the difference of V_{P} and V_{N} by transconductance (g_{m}). The V_{W} is same as the V_{Z}. The electronic controllability of DO-VDBA has advantages over the traditional VDBA by possibility of controlling of g_{m} through the bias current I_{B}.

The DO-VDBA can be given by the following matrix:

$\left[\begin{array}{c}{I}_{p}\\ {I}_{n}\\ {I}_{z}\\ {V}_{{w}^{+}}\\ {V}_{{w}^{-}}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ {g}_{m}& -{g}_{m}& 0\\ 0& 0& 1\\ 0& 0& -1\end{array}\right]\left[\begin{array}{c}{V}_{p}\\ {V}_{n}\\ {V}_{z}\end{array}\right]$ (1)

Figure 1. Symbolic notation of the DO-VDBA.

After applying KCL at different nodes of Figure 2 the proposed structure yields the following current transfer functions:

${T}_{1}\left(s\right)={\frac{{I}_{C1}}{{I}_{in}}|}_{HPF}=\frac{{s}^{2}}{{s}^{2}+\frac{s}{{R}_{1}{C}_{1}}+\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}$ (2)

${T}_{2}\left(s\right)={\frac{{I}_{C2}}{{I}_{in}}|}_{BPF}=\frac{-\left(\frac{s{g}_{m}}{{C}_{1}}\right)}{{s}^{2}+\frac{s}{{R}_{1}{C}_{1}}+\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}$ (3)

${T}_{3}\left(s\right)={\frac{{I}_{R2}}{{I}_{in}}|}_{LPF}=\frac{\left(\frac{{g}_{m}}{{R}_{2}{C}_{1}{C}_{2}}\right)}{{s}^{2}+\frac{s}{{R}_{1}{C}_{1}}+\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}$ (4)

${T}_{4}\left(s\right)={\frac{{I}_{4}}{{I}_{in}}|}_{APF}=\frac{{s}^{2}-\frac{s{g}_{m}}{{C}_{1}}+\frac{{g}_{m}}{{R}_{2}{C}_{1}{C}_{2}}}{{s}^{2}+\frac{s}{{R}_{1}{C}_{1}}+\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}$ (5)

where ${I}_{4}={I}_{c1}+{I}_{c2}+{I}_{r2}$

${T}_{5}\left(s\right)={\frac{{I}_{5}}{{I}_{in}}|}_{BRF}=\frac{{s}^{2}+\frac{{g}_{m}}{{R}_{2}{C}_{1}{C}_{2}}}{{s}^{2}+\frac{s}{{R}_{1}{C}_{1}}+\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}$ (6)

where ${I}_{5}={I}_{c1}+{I}_{r2}$ .

The cutoff frequency
${\omega}_{0}$ , bandwidth (BW) and quality factor Q_{0} are given by:

${\omega}_{0}=\sqrt{\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}$ (7)

$BW=\frac{{\omega}_{0}}{{Q}_{0}}=\frac{1}{{R}_{1}{C}_{1}}$ (8)

${Q}_{0}=\sqrt{\frac{{g}_{m}{R}_{1}{C}_{1}}{{C}_{2}}}$ (9)

Figure 2. The proposed configuration.

3. Sensitivity Analysis

The various sensitivities of ${\omega}_{0}$ and ${Q}_{0}$ w.r.t. each passive and active element are:

Sensitivity of y with respect to x is symbolized as

${S}_{x}^{y}=\frac{\partial y/y}{\partial x/x}=\frac{x}{y}\frac{\partial y}{\partial x}$

It’s active and passive sensitivity of circuit analysis parameter are expressed as:

1) $\begin{array}{c}{S}_{{g}_{m}}^{{\omega}_{0}}=\frac{\partial {\omega}_{0}/{\omega}_{0}}{\partial {g}_{m}/{g}_{m}}=\frac{{g}_{m}}{{\omega}_{0}}\frac{\partial {\omega}_{0}}{\partial {g}_{m}}=\frac{{g}_{m}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{\partial}{\partial {g}_{m}}\sqrt{\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}\\ =\frac{1}{2}\frac{{g}_{m}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{1}{\sqrt{{R}_{1}{C}_{1}{C}_{2}}}\frac{1}{\sqrt{{g}_{m}}}=\frac{1}{2}\end{array}$

2) $\begin{array}{c}{S}_{{C}_{2}}^{{Q}_{0}}=\frac{\partial {Q}_{0}/{Q}_{0}}{\partial {C}_{2}/{C}_{2}}=\frac{{C}_{2}}{{Q}_{0}}\frac{\partial {Q}_{0}}{\partial {C}_{2}}=\frac{{C}_{2}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\frac{\partial}{\partial {C}_{2}}\sqrt{\frac{{g}_{m}{R}_{1}{C}_{1}}{{C}_{2}}}\\ =\frac{-1}{2}\frac{{C}_{2}\sqrt{{C}_{2}}\sqrt{{g}_{m}{R}_{1}{C}_{1}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\frac{1}{\sqrt[3]{{C}_{2}}}=\frac{-1}{2}\end{array}$

3) $\begin{array}{c}{S}_{{R}_{1}}^{{\omega}_{0}}=\frac{\partial {\omega}_{0}/{\omega}_{0}}{\partial {R}_{1}/{R}_{1}}=\frac{{R}_{1}}{{\omega}_{0}}\frac{\partial {\omega}_{0}}{\partial {R}_{1}}=\frac{{R}_{1}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{\partial}{\partial {R}_{1}}\sqrt{\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}\\ =\frac{-1}{2}\frac{{R}_{1}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{\sqrt{{g}_{m}}}{\sqrt{{C}_{1}{C}_{2}}}\frac{1}{\sqrt[3]{{R}_{1}}}=\frac{-1}{2}\end{array}$

4) $\begin{array}{c}{S}_{{C}_{1}}^{{\omega}_{0}}=\frac{\partial {\omega}_{0}/{\omega}_{0}}{\partial {C}_{1}/{C}_{1}}=\frac{{C}_{1}}{{\omega}_{0}}\frac{\partial {\omega}_{0}}{\partial {C}_{1}}=\frac{{C}_{1}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{\partial}{\partial {C}_{1}}\sqrt{\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}\\ =\frac{-1}{2}\frac{{C}_{1}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{\sqrt{{g}_{m}}}{\sqrt{{R}_{1}{C}_{2}}}\frac{1}{\sqrt[3]{{C}_{1}}}=\frac{-1}{2}\end{array}$

5) $\begin{array}{l}{S}_{{C}_{2}}^{{\omega}_{0}}=\frac{\partial {\omega}_{0}/{\omega}_{0}}{\partial {C}_{2}/{C}_{2}}=\frac{{C}_{2}}{{\omega}_{0}}\frac{\partial {\omega}_{0}}{\partial {C}_{2}}=\frac{{C}_{2}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\frac{\partial}{\partial {C}_{2}}\sqrt{\frac{{g}_{m}}{{R}_{1}{C}_{1}{C}_{2}}}\\ =\frac{-1}{2}\frac{{C}_{2}\sqrt{{R}_{1}{C}_{1}{C}_{2}}}{\sqrt{{g}_{m}}}\sqrt{\frac{{g}_{m}}{{R}_{1}{C}_{1}}}\frac{1}{\sqrt[3]{{C}_{2}}}=\frac{-1}{2}\end{array}$

Figure 3. Frequency response of the proposed biquad.

6) $\begin{array}{l}{S}_{{g}_{m}}^{{Q}_{0}}=\frac{\partial {Q}_{0}/{Q}_{0}}{\partial {g}_{m}/{g}_{m}}=\frac{{g}_{m}}{{Q}_{0}}\frac{\partial {Q}_{0}}{\partial {g}_{m}}=\frac{{g}_{m}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\frac{\partial}{\partial {g}_{m}}\sqrt{\frac{{g}_{m}{R}_{1}{C}_{1}}{{C}_{2}}}\\ =\frac{1}{2}\frac{{g}_{m}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\sqrt{\frac{{R}_{1}{C}_{1}}{{C}_{2}}}\frac{1}{\sqrt{{g}_{m}}}=\frac{1}{2}\end{array}$

7) $\begin{array}{c}{S}_{{R}_{1}}^{{Q}_{0}}=\frac{\partial {Q}_{0}/{Q}_{0}}{\partial {R}_{1}/{R}_{1}}=\frac{{R}_{1}}{{Q}_{0}}\frac{\partial {Q}_{0}}{\partial {R}_{1}}=\frac{{R}_{1}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\frac{\partial}{\partial {R}_{1}}\sqrt{\frac{{g}_{m}{R}_{1}{C}_{1}}{{C}_{2}}}\\ =\frac{1}{2}\frac{{R}_{1}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\sqrt{\frac{{g}_{m}{C}_{1}}{{C}_{2}}}\frac{1}{\sqrt{{R}_{1}}}=\frac{1}{2}\end{array}$

8) $\begin{array}{c}{S}_{{C}_{1}}^{{Q}_{0}}=\frac{\partial {Q}_{0}/{Q}_{0}}{\partial {C}_{1}/{C}_{1}}=\frac{{C}_{1}}{{Q}_{0}}\frac{\partial {Q}_{0}}{\partial {C}_{1}}=\frac{{C}_{1}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\frac{\partial}{\partial {C}_{1}}\sqrt{\frac{{g}_{m}{R}_{1}{C}_{1}}{{C}_{2}}}\\ =\frac{1}{2}\frac{{C}_{1}\sqrt{{C}_{2}}}{\sqrt{{g}_{m}{R}_{1}{C}_{1}}}\sqrt{\frac{{g}_{m}{R}_{1}}{{C}_{2}}}\frac{1}{\sqrt{{C}_{1}}}=\frac{1}{2}\end{array}$

From above values of sensitivity, concluding the values here:

${S}_{{g}_{m}}^{{\omega}_{0}}=\frac{1}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{{R}_{1},{C}_{1},{C}_{2}}^{{\omega}_{0}}=-\frac{1}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{{g}_{m},{R}_{1},{C}_{1}}^{{Q}_{0}}=\frac{1}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{{C}_{2}}^{{Q}_{0}}=-\frac{1}{2}$ (10)

From Equation (10) it is clear that the proposed circuit offers small active and passive sensitivities.

4. Simulation Result

To show the functionality the presented biquad is tested using SPICE simulations. For this purpose we use CMOS DO-VDBA [5] with power supply voltage taken as V_{DD} = −V_{SS} = 1.2 V, I_{b} = 50 μA biasing current and passive elements with C_{1} = 0.1 nF, C_{2} = 0.2 nF, R_{1} = 2 kΩ and R_{2} = 2 kΩ. The frequency response of the circuit is depicted in Figure 3 with the cutoff frequency f_{0} = 0.78 MHz.

5. Conclusion

In this manuscript we present an application of dual output voltage differencing buffered amplifier as current mode biquad filter using single DOVDBA and 4 passive components (2 resistors and 2 grounded capacitors as required for IC fabrication). The presented circuit offers low active and passive sensitivity. The FO can be tuned electronically by varying transconductance (g_{m}) of circuit. The active and passive sensitivity of the proposed work are not more than one. It means that it has low sensitivity. The practicability of reported circuit is tested using pSPICE simulation with 180 nm process parameters of TSMC CMOS.

Acknowledgements

This research is supported by the “Young Faculty Research Fellowship and Research/Contingency Grant”, under the “Visvesvaraya PhD Scheme for Electronics and IT”, Ministry of Electronics and Information Technology, Govt. of India.

Conflicts of Interest

The authors declare no conflicts of interest.

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