Cascadable First-Order and Second-Order Inverse Filters Based On Second-Generation Voltage Conveyors

1. Introduction

In control and instrumentation, and speech applications, inverse filters are crucial circuit components used to correct distorted output signals resulting from linear or nonlinear transformations caused by signal processors or transmission systems. This filter has a frequency response that is inversely proportional to the frequency response of the system, leading to distortion. The design of inverse filters is also necessary in several communication applications, such as monitoring the relative cleanness of a signal and de-emphasizing FM systems. In control systems, it is necessary to employ inverse lowpass, highpass, and bandpass filter functions for the implementation of proportional-integral (PI), proportional-derivative (PD), and proportional-integral-derivative (PID) controllers. For these reasons, the literature in [1-31] explicitly discusses the design and implementation of inverse filters utilizing different modern active components. The previous studies aim to operate in various modes such as voltage mode (VM) [1,6,8-11,13-15,17-31], current mode (CM) [2-5,7,12,14,15,17,30], and trans-admittance mode (TAM) [16]. In addition, the works of [5-7,23,29,31] are developed specifically as first-order inverse filters, whereas the works referenced as [1-4,8-24,26-30] deal with second-order inverse filters.

In [1], a comprehensive approach utilizing nullors as fundamental components was introduced to derive the inverse transfer function for linear dynamic systems and the inverse transfer characteristic for non-linear resistive circuits. An example of a Sallen-Key highpass filter and the inverse of a half-wave rectifier has also been used to illustrate this approach. The previous study described in [2] provided a procedure for transforming a VM op-amp-based RC filter into CM, four-terminal floating nullor (FTFN)-based inverse filter. The realization approach employs a network representation that utilizes nullors and RC:CR dual transformation. The work in [3] demonstrates a general method for deriving FTFN-based CM inverse filters from well-known VM filters, as well as generating a CM inverse filter from the Friend-Delyannis biquad. Since FTFN is not available commercially, one can realize it using two second-generation current conveyors (CCIIs) or two current feedback operational amplifiers (CFOAs). A number of CM inverse filter implementations using a single FTFN has been described in [4], where the simulation results reported in this study were obtained by synthesizing the FTFN using two CCIIs. Numerous additional inverse filters were given in [5-8,16,20-25]. Each configuration provides a single inverse filter function. Multifunction second-order VM and TAM inverse filter realizations employing more than two active components are suggested in [8-11,14-17,19, 26,27,29]. Some of them also necessitate a significant number of passive components for their realizations, at least seven passive components [17,20[21,24,25,29,30]. In [12], it is observed that the output current of the CM inverse filter is sensed through the passive components, requiring an additional buffer stage to connect to the next stage. The output voltage terminals of the second-order VM inverse filter designs in [13,14,18-21,23,24,28] do not explicitly have a low impedance level. Thus, these filters are not appropriate for cascadable connections.

This study emphasizes the use of second-generation voltage conveyors (VCIIs) as active components in designing multifunction inverse filters. Four configurations implementing first-order and second-order inverse filters in both VM and CM are presented. The suggested first-order VM inverse filter employs only one VCII and three passive components to realize both inverse LP and HP filter functions by appropriately selecting the external impedance type. With its very low output impedance, it is suitable for cascading in VM operation. By cascading the suggested first-order VM inverse filter, the second-order VM inverse filter with all the generic inverse filter realizations, i.e., LP, BP and HP, can be obtained. The third proposed circuit features a first-order CM configuration with two VCIIs and three passive elements. On the other hand, the fourth configuration is the second-order CM type that provides inverse LP, HP, and AP filter responses within the same circuit design. Both of the proposed CM inverse filters exhibit a low-input impedance and a high-output impedance, which permits full cascadability. All of the proposed circuits possess the characteristic of orthogonal control of their pole frequency and minimum gain, and do not require any criteria for component matching. The theoretical predictions are confirmed by PSPICE simulation results using TSMC 0.18-μm CMOS model parameters. Furthermore, the laboratory testing involved the utilization of commercially available IC-type AD844 CFOAs for measurement purposes. Table 1 provides a comprehensive comparison between previously available inverse filters and the proposed circuits in terms of operation mode, filter order, available response, active and passive component count, technology required, supply voltage levels, and power dissipation.

2. Second-Generation Voltage Conveyor (VCII)

The proposed designs are based on the use of the VCII as an active element. The VCII is symbolically represented in Figure 1, which also provides the following description of its terminal relationship:

v_y = 0, i_x = ±β i_y, and v_z = α v_x,

(1)

where β is the current gain between the y and x terminals, and α is the voltage gain between the x and z terminals. In ideal case, the gain values β and α are both equal to unity. The “+” and “–” signs in Equation (1) indicate the positive VCII (VCII+) and negative VCII (VCII-), respectively. Equation (1) also reveals that the VCII has a low-input impedance y terminal, a high-output impedance x terminal, and a low-output impedance z terminal.

3. Proposed Cascadable Voltage-Mode Inverse Filters

Using the VCII in Figure 1 as an active component, the implementations of the first-order and second-order VM inverse filters are demonstrated in Figure 2.

3.1. First-Order VM Inverse Filter Realization

Figure 2a depicts the realization of the first-order VM inverse filter. The circuit requires one VCII-, one resistor (R₁), one capacitor (C₁), and one impedance (Z_A). It is worth noting that the output voltage terminal of the circuit is directly obtained from the z terminal of VCII-. This configuration offers the benefit of a cascadable terminal with a low output impedance. A straightforward circuit analysis assuming ideal VCII yields the following inverse filter transfer function:

$$G_{v1}(s)={\displaystyle \frac{V_{out1}(s)}{V_{in}(s)}}={\displaystyle \frac{s{R}_1{C}_1+1}{s{C}_1{Z}_A}}$$

(2)

Equation (2) clearly indicates that, by appropriately configuring the impedance Z_A, the first-order ILP and IHP filter responses could be realized in a single configuration, as demonstrated in Table 2. As can also be seen from the table, the gain (G₀) values of ILP and IHP filters can be tuned independently through C_A and R_A, respectively. An additional advantage of this realization is that no element-matching realizability conditions are necessary.

3.2. Second-Order VM Inverse Filter Realization

Figure 2b illustrates the proposed realization of the second-order VM inverse filter, which is constructed using the cascade connection of the first-order VM inverse filter in Figure 2a. This topology still exhibits a low output impedance, making it appropriate for cascading in VM operation. The voltage transfer function of the circuit is given by:

$$G_{v2}(s)={\displaystyle \frac{v_{out2}}{v_{in}}}={\displaystyle \frac{N(s)}{s^2{C}_1{C}_2{Z}_A{Z}_B}},$$

(3)

$$\mathrm{where}N(s)=s^2{R}_1{R}_2{C}_1{C}_2+s\left(R_1{C}_1+R_2{C}_2\right)+1.$$

(4)

Similarly, the suitable impedance choices for Z_A and Z_B as specified in Table 3 would lead to the required second-order ILP, IHP, and IBP filter responses.

4. Proposed Cascadable Current-Mode Inverse Filters

4.1. First-Order CM Inverse Filter Realization

By slightly modifying the configuration depicted in Figure 2a, the circuit can perform a CM inverse filter as shown in Figure 3a. It consists of a primary circuit from Figure 2a and an additional VCII- #2. Due to its low input impedance and high output impedance, the scheme supports cascading inputs and outputs for CM operation. By selecting of the appropriate impedance Z_A, the proposed circuit in Figure 3a generates the following filter-order inverse frequency response for CM at the terminal i_out1, which possess the critical characteristics detailed in Table 4.

$$G_{i1}(s)={\displaystyle \frac{I_{out1}(s)}{I_{in}(s)}}={\displaystyle \frac{s{R}_1{C}_1+1}{s{C}_1{Z}_A}}.$$

(5)

4.2. Second-Order CM Inverse Filter Realization

The construction of the second-order CM inverse filter in Figure 3b involves cascading the first-order section of VCII- #2, R₂, C₂, and Z_B from Figure 3a. The resulting circuit realizes the following CM transfer function at the current output terminal, i_out2.

$$G_{i2}(s)={\displaystyle \frac{i_{out2}}{i_{in}}}={\displaystyle \frac{N(s)}{s^2{C}_1{C}_2{Z}_A{Z}_B}}.$$

(6)

By choosing the impedances Z_A and Z_B as shown in Table 5, one can then realize three different second-order CM inverse filter functions.

5. Tracking Error Analysis

In non-ideal operation, current and voltage transfer errors may cause deviations in the actual values of the proposed filters. To evaluate the effect of the VCII tracking errors on the circuit performance, the gain values β and α of the k-th VCII-, where k = 1, 2, and 3, are defined as β_k and α_k, respectively. We also denote that β_k = (1 - ε_i) and α_k = (1 – ε_v), where ε_i (| ε_i | << 1) and ε_v (| ε_v | << 1) are the current and voltage tracking errors, respectively.

Consequently, we perform non-ideal analyses of the proposed circuits in Figure 2 and report their non-ideal parameters in Table 6 and Table 7. Following a similar manner, we also analyze the circuits depicted in Figure 3 for non-ideal transfer gains and present the non-ideal parameters in Table 8 and Table 9.

From Table 6, Table 7, Table 8 and Table 9, it is obvious that the non-ideal values of ω₀ and Q are not influenced by the current and voltage tracking error coefficients. The VCII tracking errors may result in slight changes in the gain values of the realized inversed filters. Nevertheless, the pre-distortion values of R_A, R_B, C_A, or C_B can compensate for this impact without affecting the ω₀ value.

6. Parasitic Element Analysis

To conduct a further non-ideality analysis, it is necessary to consider the effect of the VCII parasitics on the filter characteristics. For this purpose, we use the non-ideal behavior model of the VCII, which includes a finite non-zero input resistance R_y at port y, a parasitic resistance R_x in parallel with a parasitic capacitance C_x at port x, and a low z-port resistance R_z. Figure 4 illustrates the equivalent circuit for the non-ideal VCII with its parasitic elements.

6.1. Parasitic Effects on VM Inverse Filter Realization

A straightforward analysis of the proposed first-order VM inverse filter in Figure 2a using the VCII’s non-ideal model in Figure 4 yields the non-ideal transfer functions of the ILP and IHP filters given by the following expressions:

$$1^{\mathrm{st}}-\mathrm{order}\mathrm{ILP}:G_{v1}(s)={\displaystyle \frac{G_0\left(s{R}_1{C}_1+1\right)}{\left(s{R}_1{C}_x+1\right)\left(s{R}_y{C}_A+1\right)}},$$

(7)

$$1^{\mathrm{st}}-\mathrm{order}\mathrm{IHP}:G_{v1}(s)={\displaystyle \frac{G_0\left(s{R}_1{C}_1+1\right)}{\left(s{R}_1{C}_1\right)\left(s{R}_1{C}_x+1\right)}}.$$

(8)

Typically, the value of R_x is extremely high, while the values of R_y and C_x are extremely small. As a consequence, R_x >> R₁, R_A >> R_y, and C_x << C₁. Equations (7) and (8) demonstrate that the VCII parasitics introduce extra poles at frequencies f_v1 = (1/2πR₁C_x) and f_v2 = (1/2πR_yC_A) for ILP filter response, and at frequency f_v1 in IHP. Therefore, the useful frequencies of the first-order VM ILP and IHP filters are approximated as:

1^st-order ILP: f << min [ f_v1, f_v2 ],

(9)

1^st-order IHP: f << min [ f_v1 ].

(10)

For instance, if the parasitic elements of the VCII- are R_y = 38.88 Ω, R_x = 109.24 kΩ, R_z = 38.23 Ω, and C_x = 3 pF [32], and the filter components are chosen as: R₁ = 1 kΩ and C_A = 100 pF, then the pole frequencies will be located at frequencies f_v1 = 53.08 MHz and f_v2 = 40.93 MHz. This implies that the first-order VM ILP and IHP filters proposed here have high frequency limitations of approximately 40.93 MHz and 53.05 MHz, respectively.

By taking into account the VCII parasitic effect on the second-order VM inverse filter realization of Figure 2b, the non-ideal transfer functions of ILP, IHP, and IBP filters are as follows:

$$2^{\mathrm{nd}}-\mathrm{order}\mathrm{ILP}:G_{v2}(s)={\displaystyle \frac{G_{0}N(s)}{\left(s{R}_1{C}_{x1}+1\right)\left(s{R}_2{C}_{x2}+1\right)\left(s{R}_{y1}{C}_A+1\right)\left(s{R}_{y2}{C}_B+1\right)}}$$

(11)

$$2^{\mathrm{nd}}-\mathrm{order}\mathrm{IHP}:G_{v2}(s)={\displaystyle \frac{G_{0}N(s)}{\left(s^2{R}_1{R}_2{C}_1{C}_2\right)\left(s{R}_1{C}_{x1}+1\right)\left(s{R}_2{C}_{x2}+1\right)}}$$

(12)

$$2^{\mathrm{nd}}-\mathrm{order}\mathrm{IBP}:G_{v2}(s)={\displaystyle \frac{G_{0}N(s)}{s\left(R_1{C}_1+R_2{C}_2\right)\left(s{R}_1{C}_{x1}+1\right)\left(s{R}_2{C}_{x2}+1\right)\left(s{R}_{y1}{C}_A+1\right)}}$$

(13)

where C_xk and R_yk are the parasitic elements C_x and R_y of the k-th VCII-. Suppose that ${f^{\prime }}_{v1}={\displaystyle \frac{1}{2\pi R_1{C}_{x1}}}$, ${f^{″}}_{v1}={\displaystyle \frac{1}{2\pi R_2{C}_{x2}}}$, ${f^{\prime }}_{v2}={\displaystyle \frac{1}{2\pi R_{y1}{C}_A}}$, and ${f^{″}}_{v1}={\displaystyle \frac{1}{2\pi R_{y2}{C}_B}}$, then the operating frequency ranges of the proposed second-order VM ILP, IHP, and IBP filters can be defined as:

2^nd-order ILP: f << min [ f′_v1, f″_v1, f′_v2, f″_v2 ],

(14)

2^nd-order IHP: f << min [ f′_v1, f″_v1 ],

(15)

2^nd-order IBP: f << min [ f′_v1, f″_v1, f′_v2 ].

(16)

These effects on the realized filter performance can be considerably minimized if the operating frequencies are chosen sufficiently lower than the parasitic pole frequencies of VCII-.

6.2. Parasitic Effects on CM Inverse Filter Realization

The non-ideal transfer functions of the proposed CM first-order inverse filter in Figure 3a are derived by accounting for the parasitic nature of the VCII-:

$$1^{\mathrm{st}}-\mathrm{order}\mathrm{ILP}:G_{i1}(s)={\displaystyle \frac{G_0\left(s{R}_1{C}_1+1\right)}{\left(s{R}_1{C}_{x1}+1\right)\left[s\left(R_{z1}+R_{y2}\right)C_A\right]}}$$

(17)

$$1^{\mathrm{st}}-\mathrm{order}\mathrm{IHP}:G_{i1}(s)={\displaystyle \frac{G_0\left(s{R}_1{C}_1+1\right)}{\left(s{R}_1{C}_1\right)\left(s{R}_1{C}_{x1}+1\right)}}$$

(18)

Let $f_{i1}={\displaystyle \frac{1}{2\pi R_1{C}_{x1}}}$ and $f_{i2}={\displaystyle \frac{1}{2\pi \left(R_{z1}+R_{y2}\right)C_A}}$, then the frequency limitations of the proposed first-order CM ILP and IHP filters should be lied in the following ranges:

1^st-order ILP: f << min [ f_i1, f_i2 ],

(19)

1^st-order IHP: f << min [ f_i1 ].

(20)

Furthermore, we have also considered the parasitic effect on the proposed CM second-order inverse filter structure in Figure 3b. The various non-ideal expressions of ILP, IHP, and IBP filter responses are given by:

2^nd-order ILP:

$$G_{i2}(s)={\displaystyle \frac{G_{0}N(s)}{\left(s{R}_1{C}_{x1}+1\right)\left(s{R}_2{C}_{x2}+1\right)\left[s\left(R_{z1}+R_{y2}\right)C_A+1\right]\left[s\left(R_{z2}+R_{y3}\right)C_B+1\right]}},$$

(21)

2^nd-order IHP:

$$G_{i2}(s)={\displaystyle \frac{G_{0}N(s)}{s^2{R}_1{R}_2{C}_1{C}_2\left(s{R}_1{C}_{x1}+1\right)\left(s{R}_2{C}_{x2}+1\right)}},$$

(22)

^2nd-order IBP:

$$G_{i2}(s)={\displaystyle \frac{G_{0}N(s)}{s\left(R_1{C}_1+R_2{C}_2\right)\left(s{R}_1{C}_{x1}+1\right)\left(s{R}_2{C}_{x2}+1\right)\left[s\left(R_{z1}+R_{y2}\right)C_A+1\right]}},$$

(23)

which represent the additional pole frequencies expressed by: ${f^{\prime }}_{i1}={\displaystyle \frac{1}{2\pi R_1{C}_{x1}}}$, ${f^{″}}_{i1}={\displaystyle \frac{1}{2\pi R_2{C}_{x2}}}$, ${f^{\prime }}_{i2}={\displaystyle \frac{1}{2\pi \left(R_{z1}+R_{y2}\right)C_A}}$, and ${f^{″}}_{i2}={\displaystyle \frac{1}{2\pi \left(R_{z2}+R_{y3}\right)C_B}}$.

Therefore, the frequency ranges of operation of the proposed CM second-order inverse filter in Figure 3b can be approximated from the above relations as:

2^nd-order ILP: f << min [ f′_i1, f″_i1, f′_i2, f″_i2 ],

(24)

2^nd-order IHP: f << min [ f′_i1, f″_i1 ],

(25)

2^nd-order IBP: f << min [ f′_i1, f″_i1, f′_i2 ].

(26)

7. Simulation Verification of Filter Responses

The validity of the cascadable inverse filters proposed in Figure 2 and Figure 3 has been evaluated through a simulation conducted by PSPICE program. In the simulation analysis, the CMOS structure of the VCII- as shown in Figure 5 has been implemented [33]-[34]. The ratio of width (W) to channel length (L) of the CMOS transistors employed is listed in Table 10. Supply voltages of +V = -V = 0.75 V and biasing currents of I_B1 = I_B2 = 25 μA are used. All the proposed inverse filter structures are tested with identical resistors and capacitors: R₁ = R₂ = R_A = R_B = 1 kΩ, and C₁ = C₂ = C_A = C_B = 100 pF for the theoretical pole frequency of f₀ = ω₀/2π = 1.59 MHz.

The simulated time and frequency responses of the VM and CM inverse filters in Figure 2 and Figure 3 compared with the theoretical responses are given in Figure 6, Figure 7, Figure 8 and Figure 9. The proposed circuits are tested at a signal frequency of 1.59 MHz with an input signal amplitude of 50 mV (peak) for VM and 50 μA (peak) for CM. The total power consumptions of the first- and second-order VM inverse filters are 0.255 mW and 0.511 mW, respectively, while the first- and second-order CM inverse filters are 0.511 mW and 0.766 mW.

The simulated cut-off frequency f_o for the first-order VM ILP and IHP responses was measured at 1.56 MHz and 1.54 MHz, respectively, and the corresponding f₀ values for the second-order ILP, IHP, and IBP responses were 1.57 MHz, 1.52 MHz, and 1.55 MHz. For the CM inverse filters, the simulated f_o values of the first-order ILP and IHP responses were 1.57 MHz and 1.50 MHz, while the second-order ILP, IHP, and IBP filters provided the simulated f₀ at 1.57 MHz, 1.52 MHz, and 1.56 MHz, respectively.

The f_o-tuning for the IBP filter in Figure 2b was simulated with a constant value of Q. The capacitor and resistor values are C₁ = C₂ = C_A = 100 pF, R_B = 1 kΩ, and R₁ = R₂ = 0.5 kΩ, 2 kΩ, and 5 kΩ. With a fixed Q-value of 0.5, the corresponding values for f₀ are 3.18 MHz, 795.77 MHz, and 318.31 MHz, respectively. Figure 10 illustrates the ideal and simulated gain-frequency responses, demonstrating the ability to independently adjust the f₀-value of the proposed filter. The proposed VM IBP filter in Figure 2b was also simulated at different temperatures including 0^°C, 25^°C, 50^°C, 75^°C, and 100^°C. The simulations were conducted with C₁ = C₂ = C_A = 100 pF and R₁ = R₂ = R_B = 1 kΩ, and the obtained results are shown in Figure 11. At f₀ = 1.59 MHz, the deviations from its nominal value in voltage gain due to temperature variations are approximately -0.88 dBV at 0^°C and +7.10 dBV at 100^°C, which can be considered insignificant.

Furthermore, a Monte Carlo (MC) statistical analysis was performed on the VM IBP filter at the pole frequency of 1.59 MHz, considering nominal 5% tolerances in the values of resistors and capacitors. Figure 12 depicts the MC simulation results with a Gaussian distribution for 200 runs. The standard deviation in f_o resulting from the changes in resistor and capacitor tolerances was obtained at 47.92 kHz, even though it is not considerable.

8. Experimental Verification

The experimental laboratory measurement has been carried out to further validate the practicability of the proposed design. For this purpose, the practical VCII- has been realized using commercially available IC-type AD844s, as shown in Figure 13 [34]. The DC bias voltages were set to +V = -V = 5 V. The resistor and capacitor have values of R₁ = 1 kΩ and C₁ = C_A = 1 nF, respectively, to obtain an ideal pole frequency of f_o = 159 kHz. Figure 14 shows the experimentally measured time and frequency responses for the proposed first-order VM inverse filter in Figure 2a along with the corresponding ideal responses. The input voltage was set at 100 mV peak-to-peak with a frequency of 159 kHz. The testing results, thus, verify the feasibility of the proposed concept.

9. Conclusions

Four circuit configurations for realizing first- and second-order inverse filters using second-generation voltage conveyors (VCIIs) have been presented in this paper. Both proposed first-order inverse filters in VM and CM can realize inverse LP and HP filter functions using the same circuit topology by the appropriate selection of the branch impedance. Using the same circuit concept, both of the proposed second-order VM and CM inverse filters can provide inverse LP, BP, and HP filter functions from a single design. All the suggested circuits are appropriate for cascade connections. By modifying the passive element values, it is possible to independently control the pole frequency and gain of the filters. Analyses of non-ideal performance, such as transfer errors and parasitic effects, have been examined in relation to theoretical performance. The feasibility of the circuits was confirmed by PSPICE simulations in conjunction with laboratory testing measurements.