ABSTRACT

The paper presents a new universal biquadratic filter using single universal voltage conveyor (UVC), two resistors and two capacitors. The offered structure has three inputs and one output and can realise all the five basic biquadratic filters: high-pass (HP), low-pass (LP), band-reject (BR), band-pass (BP) and all-pass (AP) from the same circuit topology. The proposed universal filter also provides following advantageous features, not available simultaneously in any UVC based universal biquadratic filter so far: (i) low active and passive sensitivities, (ii) independent control of natural frequency (ω₀) and bandwidth (BW) and (iii) no requirement of any component matching condition and inversion of input signal(s) (as needed in most of the earlier reported structures). The workability of proposed structure has been presented by SPICE (Version 16.5) simulation using 0.18 µm TSMC technology.

Keywords:

Analog Circuit, Universal Voltage Conveyor, Voltage-Mode, Filters

1. Introduction

Interest in the design of multi-input single-output (MISO) or single-input multi- output (SIMO), current-mode (CM) or voltage-mode (VM) universal filter configurations have been emerging, due to their flexibility and versatility for practical applications [1] . In literature [2] - [7] , filter structures using different active building blocks/devices (such as current differencing transconductance amplifier, current differencing buffered amplifier, modified current feedback operational amplifier and current feedback operational amplifier) have been presented over the past few years. In ref [8] , Herenscar, Koton and Vrba have presented multifunction filter structures, employing three UVCs could realize only three basic filters (i.e. LP, HP, and BP). In ref [9] , Minarcik and Vrba have proposed three multi-function filter structures: i) realize three basic filters i.e. LP, HP, and BP using three UVCs ii) realize four basic filters i.e. LP, HP, BP and, AP using two UVCs which can iii) four basic filters i.e. LP, HP, BP and, AP, employing single UVC. Filter structures presented in references [8] [9] are unable to realize all the basic biquadratic filter functions from single topology (i.e. LP, HP, BP, BR and AP).

In ref. [10] , Pushkar and Gupta presented another universal filter in which all basic filter functions (LP, HP, BP, BR and AP) are realised however, the normalized gain of bandpass filter is approximately half which limits it’s applications, component matching and scaling of input signal is also required. Therefore, the purpose of this paper is to propose a new universal MISO-type VM biquad with three inputs and one output, which can realize all basic filter functions, namely: LP, HP, BP, BR and AP with no component matching and scaling of input signal is required. Comparison with other previously known MISO-type biquads using different active building blocks is shown in Table 2. The proposed filter is verified by SPICE (Version 16.5) simulations using 0.18 µm TSMC technology.

2. Proposed Filter Configuration

The Universal Voltage Conveyor is a 6-port active element with one voltage input x, two difference current inputs (y⁺, y⁻), two mutually inverse voltage outputs (z⁺, z⁻), and one auxiliary port w. Figure 1(a) and Figure 1(b) show the

schematic symbol and ideal circuit model of UVC respectively. The relationship between port currents and voltages of a six port UVC is given in the following matrix.

$\left(\begin{array}{c}{I}_x\\ I_w\\ V_{y^{+}}\\ V_{y^{-}}\\ V_{z^{+}}\\ V_{z^{-}}\end{array}\right)=\left(\begin{array}{cccccc}0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ -1& 0& 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{V}_x\\ V_w\\ I_{y^{+}}\\ I_{y^{-}}\\ I_{z^{+}}\\ I_{z^{-}}\end{array}\right)$ (1)

A routine circuit analysis (assuming ideal UVC) of Figure 2 yields the following expression for the voltage transfer function of the biquad:

$V_o=\frac{V_2\left(s^2\right)-V_1\left(\frac{s}{C_2{R}_2}\right)+V_3\left(\frac{1}{C_1{C}_2{R}_1{R}_2}\right)}{s^2+\frac{s}{C_2{R}_2}+\frac{1}{R_1{R}_2{C}_1{C}_2}}$ (2)

From Equation (2), second order filter functions (LP, HP, BP, BR and AP) can be realized.

1) If $V_3=V_{in}$ , $V_2=V_1=0$ (grounded), a low pass filter can be realized.

${T\left(s\right)|}_{LP}=\frac{\frac{1}{C_1{C}_2{R}_1{R}_2}}{D(\; s\; )}$

2) If $V_2=V_{in}$ and $V_3=V_1=0$ (grounded), a high pass filter can be realized.

${T\left(s\right)|}_{HP}=\frac{s^2}{D(\; s\; )}$

3) If $V_1=V_{in}$ and $V_2=V_3=0$ (grounded), a band pass filter can be realized.

${T\left(s\right)|}_{BP}=\frac{-\frac{s}{C_2{R}_2}}{D(\; s\; )}$

4) If $V_3=V_2=V_{in}$ and $V_1=0$ , a band reject filter can be realized.

${T\left(s\right)|}_{BR}=\frac{s^2+\frac{1}{R_1{R}_2{C}_1{C}_2}}{D(\; s\; )}$

5) If $V_1=V_2=V_3=V_{in}$ , a all pass filter can be realized.

${T\left(s\right)|}_{AP}=\frac{s^2-\frac{s}{C_2{R}_2}+\frac{1}{R_1{R}_2{C}_1{C}_2}}{D(\; s\; )}$

where: $D\left(s\right)=s^2+\frac{s}{C_2{R}_2}+\frac{1}{R_1{R}_2{C}_1{C}_2}$

The expressions for quality factor (Q₀), bandwidth (BW) and natural frequency ( ${\omega }_0$ ), respectively are given by:

$Q_0=\sqrt{\frac{R_2{C}_2}{R_1{C}_1}},BW=\frac{1}{C_2{R}_2},{\omega }_0=\sqrt{\frac{1}{R_1{R}_2{C}_1{C}_2}}$ (3)

From (3), it can be observed that after adjusting BW by R₂, ${\omega }_0$ can independently be controlled through R₁. Furthermore, it is seen that no inversion of input signal(s) and no component matching condition is required while realizing any of the five filter functions.

In the ideal case, the various sensitivities of ${\omega }_0$ and BW with respect to R₁, R₂, C₁, and C₂ are found to be:

$S_{C_1}^{{\omega }_0}=S_{C_2}^{{\omega }_0}=S_{R_1}^{{\omega }_0}=S_{R_2}^{{\omega }_0}=-\frac{1}{2}$ (4)

$S_{C_1}^{BW}=0,S_{C_2}^{BW}=-1,S_{R_1}^{BW}=0,S_{R_2}^{BW}=-1$ (5)

3. Non-Ideal Analysis and Sensitivity Performance

Taking into account the non-idealities of UVC, the relationship between the port voltages and currents is shown by the hybrid matrix:

$\left(\begin{array}{c}{I}_x\\ I_w\\ V_{y^{+}}\\ V_{y^{-}}\\ V_{z^{+}}\\ V_{z^{-}}\end{array}\right)=\left(\begin{array}{cccccc}0& 0& {\alpha }_1& -{\alpha }_2& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& {\delta }_1& 0& 0& 0& 0\\ 0& {\delta }_2& 0& 0& 0& 0\\ {\gamma }_1& 0& 0& 0& 0& 0\\ -{\gamma }_2& 0& 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{V}_x\\ V_w\\ I_{y^{+}}\\ I_{y^{-}}\\ I_{z^{+}}\\ I_{z^{-}}\end{array}\right)$ (6)

where ${\alpha }_j=1-{\epsilon }_{ij}$ and ${\delta }_j=1-{\epsilon }_{v1j}$ , ${\gamma }_j=1-{\epsilon }_{v2j}$ for $j=1,2$ . Here ${\epsilon }_{ij}\left(\left|{\epsilon }_{ij}\right|\ll 1\right)$ and ${\epsilon }_{v1j}$ , ${\epsilon }_{v2j}$ ${\epsilon }_{ij}\left(\left|{\epsilon }_{v1j}\right|,\left|{\epsilon }_{v2j}\right|\ll 1\right)$ represent the current and voltage tracking errors of the UVC respectively. The parasitic present on the low impedance ports ( y⁺, y⁻, z⁺, z⁻) are quite low as compared to the resistances on the other ports (wand x) [12] . After considering the non-idealities of the UVC, given by the hybrid matrix of Equation (6), transfer function presented in Equation (2) can be modified as follows:

$V_0=\frac{{\alpha }_1{\gamma }_2{V}_2\left(s^2{C}_1{C}_2\right)-{\gamma }_2{V}_1\left(s\frac{C_1}{R_2}\right)+\frac{V_3}{R_1}\left(s{C}_x+\frac{1}{R_2}+\frac{1}{R_x}\right)}{s^2\left({\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2+C_1{C}_x+C_w{C}_x\right)+s\left(\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)\right)+\left(\frac{1}{R_2}+\frac{1}{R_x}\right)\left(\frac{1}{R_1}+\frac{1}{R_w}\right)}$ (7)

${\omega }_0=\sqrt{\frac{\left(\frac{1}{R_2}+\frac{1}{R_x}\right)\left(\frac{1}{R_1}+\frac{1}{R_w}\right)}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}}$ (8)

$BW=\frac{\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}$ (9)

The sensitivity is an important performance criterion of any circuit structure. The sensitivity of Natural Frequency (ω₀) with respect to circuit parameters, say A is given as:

$S_A^{{\omega }_0}=\frac{A}{{\omega }_0}\frac{\partial {\omega }_0}{\partial A}$ (10)

Using above definition, the various active and passive sensitivities of Natural Frequency (ω₀) and Bandwidth (BW), for the biquadratic filter, with respect to $C_1,C_2,C_x,C_w,R_1,R_2,R_x,R_w,{\alpha }_1,{\delta }_1$ and ${\gamma }_2$ are found to be:

$S_{C_1}^{{\omega }_0}=-\frac{1}{2}\left(\frac{C_1{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}\right),S_{C_x}^{{\omega }_0}=-\frac{1}{2}\left(\frac{C_1{C}_x+C_w{C}_x}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}\right)$ (11)

$S_{C_2}^{{\omega }_0}=-\frac{1}{2}\left(\frac{{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}\right)=S_{{\alpha }_1}^{{\omega }_0}=S_{{\delta }_1}^{{\omega }_0}=S_{{\gamma }_2}^{{\omega }_0}$ (12)

$S_{C_w}^{{\omega }_0}=-\frac{1}{2}\left(\frac{C_w{C}_x}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}\right),S_{R_2}^{{\omega }_0}=-\frac{1}{2}\left(\frac{1}{1+\frac{R_2}{R_x}}\right),S_{R_x}^{{\omega }_0}=-\frac{1}{2}\left(\frac{1}{1+\frac{R_x}{R_2}}\right)$ (13)

$S_{R_w}^{{\omega }_0}=-\frac{1}{2}\left(\frac{1}{1+\frac{R_w}{R_1}}\right),S_{R_1}^{{\omega }_0}=-\frac{1}{2}\left(\frac{1}{1+\frac{R_1}{R_w}}\right)$ (14)

$S_{C_1}^{BW}=-\frac{C_1\left(C_x^2\left(\frac{1}{R_1}+\frac{1}{R_w}\right)+{\alpha }_1{\delta }_1{\gamma }_2{C}_2\left(C_1\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)\right)\right)}{\left(\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)\right)\left(C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2\right)}$ (15)

$S_{C_w}^{BW}=-\frac{C_w\left(C_x^2\left(\frac{1}{R_1}+\frac{1}{R_w}\right)+{\alpha }_1{\delta }_1{\gamma }_2{C}_2{C}_1\left(\frac{1}{R_2}+\frac{1}{R_x}\right)\right)}{\left(\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)\right)\left(C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2\right)}$ (16)

$S_{C_x}^{BW}=-\frac{C_x\left({\left(C_1+C_w\right)}^2\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+{\alpha }_1{\delta }_1{\gamma }_2{C}_2{C}_1\left(\frac{1}{R_1}+\frac{1}{R_w}\right)\right)}{\left(\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)\right)\left(C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2\right)}$ (17)

$S_{C_2}^{BW}=-\frac{{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}{C_1{C}_x+C_w{C}_x+{\alpha }_1{\delta }_1{\gamma }_2{C}_1{C}_2}=S_{{\alpha }_1}^{BW}=S_{{\delta }_1}^{BW}=S_{{\gamma }_2}^{BW}$ (18)

$\begin{array}{l}{S}_{R_1}^{BW}=\frac{-\frac{C_x}{R_1}}{\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)},\\ S_{R_w}^{BW}=\frac{-\frac{C_x}{R_w}}{\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)}\end{array}$ (19)

$\begin{array}{l}{S}_{R_2}^{BW}=\frac{-\frac{\left(C_1+C_w\right)}{R_2}}{\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)},\\ S_{R_x}^{BW}=\frac{-\frac{\left(C_1+C_w\right)}{R_x}}{\left(C_1+C_w\right)\left(\frac{1}{R_2}+\frac{1}{R_x}\right)+C_x\left(\frac{1}{R_1}+\frac{1}{R_w}\right)}\end{array}$ (20)

Considering the typical values of various parasitic [13] e.g. $C_x=17.41\text{pF}$ , $R_x=378.73\text{k}\Omega$ , $C_w=4.19\text{pF}$ , $R_w=88.19\text{M}\Omega$ , ${\alpha }_1={\delta }_1={\gamma }_2=1$ along with $C_1=C_2=10\text{pF}$ , $R_1=R_1=100\text{k}\Omega$ , the various sensitivities are found to be $S_{C_1}^{{\omega }_0}=-0.253$ , $S_{C_2}^{{\omega }_0}=S_{{\alpha }_1}^{{\omega }_0}=S_{{\delta }_1}^{{\omega }_0}=S_{{\gamma }_2}^{{\omega }_0}=-0.143$ , $S_{C_x}^{{\omega }_0}=-0.355$ , $S_{C_w}^{{\omega }_0}=-0.104$ , $S_{R_w}^{{\omega }_0}=-0.0056$ , $S_{R_1}^{{\omega }_0}=-0.495$ , $S_{R_2}^{{\omega }_0}=-0.395$ , $S_{R_x}^{{\omega }_0}=-0.104$ and $S_{C_1}^{BW}=$ $-0.5001$ , $S_{C_2}^{BW}=S_{{\alpha }_1}^{BW}=S_{{\delta }_1}^{BW}=S_{{\gamma }_2}^{BW}=-0.2861$ , $S_{C_x}^{BW}=-0.4981$ , $S_{C_w}^{BW}=-0.1484$ , $S_{R_w}^{BW}=-0.0053$ , $S_{R_1}^{BW}=-0.5006$ , $S_{R_2}^{BW}=-0.4125$ , $S_{R_x}^{BW}=-0.1089$ . It is clearly observed that none of the active and passive sensitivities are more than one half in magnitudes for the proposed VM biquad filter.

4. Simulation Results

To confirm the feasibility of the presented biquadratic filter, the circuit was simulated using SPICE. The voltage and current values selected for CMOS implementation of UVC are ±1.9 V and 100 µA, respectively. The passive elements were chosen as $C_1=C_2=10\text{pF}$ , $R_1=R_2=100\text{k}\Omega$ . The natural frequency $\left({\omega }_0\right)$ and BW of the proposed filter for the selected passive elements are 87.498 kHz and 151.266 kHz respectively. CMOS implementation of universal voltage conveyor is shown in Figure 3 and the aspect ratios of MOSFETs used in Figure 3 are given in Table 1 [14] . CMOS UVC was implemented 0.18 μm TSMC CMOS model parameters [15] .

A comparison with other previously known MISO-type biquads using different active building blocks has been shown in Table 2, the proposed structure gives all basic filter functions and free from any matching conditions. The SPICE simulated frequency response of the biquad is shown in Figures 4-6 represent the magnitude and phase plot of APF, respectively. These plots are nearly similar to the standard second order filters: LPF, HPF, BPF, APF and BRF. Therefore, these responses confirm the validity of the proposed biquadratic filter.

5. Conclusions

A new second order voltage-mode MISO-type universal biquadratic filter using single UVC has been presented. The proposed filter offers following advantages: i) employment of single active component; ii) ability to realize all the basic second order filters without altering the circuit configuration; iii) low active and passive sensitivities; iv) free from component matching conditions; v) independent control of natural frequency and bandwidth. On the basis of above mentioned features of the proposed biquadratic filter, it can be used for various low voltage applications. The workability of the presented circuit configuration has been established by SPICE simulation using 0.18 μm TSMC CMOS technology.

Cite this paper

Pushkar, K.L., Gupta, K. and Vivek, P. (2017) A New Voltage-Mode Universal Biquadratic Filter Using Single UVC. Circuits and Systems, 8, 275-284. https://doi.org/10.4236/cs.2017.812020

References