CDBA-Based Universal Biquad Filter and Quadrature Oscillator

Abstract

The voltage-mode universal biquadratic filter and sinusoidal quadrature oscillator based on the use of current differencing buffered amplifiers (CDBAs) as active components have been proposed in this paper. All the proposed configurations employ only two CDBAs and six passive components. The first proposed CDBA-based biquad configuration can realize all the standard types of the biquadratic functions, that is, lowpass, bandpass, highpass, bandstop, and allpass, from the same topology, and can also provide orthogonal tuning of the natural angular frequency (𝜔_𝑜) and the bandwidth (BW) through separate virtually grounded passive components. By slight modification of the first proposed configuration, the new CDBA-based sinusoidal quadrature oscillator is easily obtained. The oscillation condition and the oscillation frequency are independently adjustable by different virtually grounded resistors. The sensitivity analysis of all proposed circuit configurations is shown to be low. PSPICE simulations and experimental results based upon commercially available AD844-type CFAs are included, which confirm the workability of the proposed circuits.

1. Introduction

Current-mode universal active components have two distinct advantages: they provide wide bandwidths and high slew rate. On the other hand, many of today’s analog signal processing applications require voltage-mode operation. Therefore, it is advantageous to implement current-mode active elements in voltage-mode circuits [1].

Recently, a current differencing buffered amplifier (CDBA) that is one of the current-mode components has been introduced [2]. It can offer advantageously features such as high slew rate, free from parasitic capacitance, wide bandwidth, and simple implementation [3]. Since the CDBA consists of a unity-gain current differential amplifier and a unity-gain voltage amplifier, this element would be suitable for the implementation of voltage and current-mode signal processing applications. As far as the applications of the CDBA are concerned, various voltage-mode filters [2, 4–8] and oscillators [1, 9, 10] have been reported in literature. Only few can perform both biquadratic filter and oscillator in the same circuit configuration [1]. However, it is restricted for only bandpass filter function design and cannot be realized by another type of the standard biquadratic function characteristics. Moreover, it generates a single-phase output voltage when it performs as a sinusoidal oscillator. There is no CDBA-based circuit configuration, which can realize the universal biquad filter and quadrature oscillator both in the same circuit configuration.

Thus, the voltage-mode universal biquadratic filter and sinusoidal quadrature oscillator using two CDBAs, four virtually grounded resistors, and two capacitors are described in this paper. The first proposed CDBA-based biquad filter can realize lowpass (LP), bandpass (BP), highpass (HP), bandstop (BS), and allpass (AP) biquadratic functions without changing the circuit topology. The filter also provides an orthogonal control of the natural angular frequency (𝜔_𝑜) and the bandwidth (BW). Moreover, a CDBA-based sinusoidal quadrature oscillator circuit can be realized by slightly modifying the first proposed CDBAs-based circuit configuration. All passive components used in the circuit realization are really and virtually grounded, which is insensitive to the effects of the stray capacitance. The oscillation condition and the frequency of oscillation can be controlled independently through different virtually grounded resistors. In addition, the sensitivity analysis shows that all of the proposed circuits have low-sensitivity performance. PSPICE simulation and experimental results using commercially available active component AD844 are also included to verify the theoretical analysis.

2. Current Differencing Buffered Amplifier (CDBA)

The circuit symbol of the CDBA is shown in Figure 1, where p and n are the positive and negative current input terminals, respectively, z is the current output terminal, and w is the voltage output terminal. Its current and voltage characteristics can be described by the following matrix equations [1, 2]:𝑖_𝑧𝑣_𝑤𝑣_𝑝𝑣_𝑛=.001−1𝑣100000000000^𝑧𝑖_𝑤𝑖_𝑝𝑖_𝑛. (1) According to the above set of describing equations, the terminal z behaves as a current source that takes the difference of currents at the inputs, and the terminal w behaves as a voltage source that copies the output voltage at the z terminal. Thus, the CDBA can be considered as a collection of a noninverting and an inverting current-mode, and noninverting voltage-mode unity-gain cells, which can be realized by a cascade connection of a current substractor and a voltage follower. Although the CDBA can be realized by using several well-known circuit techniques, one possible practical implementation is given in Figure 2 [4, 11].

Figure 1 

CDBA symbol.

Figure 2 

Implementation of CDBA using two CFAs (AD844s).

3. Proposed CDBA-Based Universal Biquad

Figure 3 shows the proposed voltage-mode universal biquadratic filter, which consists of only two CDBAs, four virtual-grounded resistors, and two capacitors. By straightforward analysis, the single output voltage function realized by this configuration is found to be𝑉_𝑜=𝑠²𝑉₃+(𝑠/𝑅₂𝐶₂)𝑉₂+(1/𝑅₁𝑅₂𝐶₁𝐶₂)𝑉₁𝑠²+(𝑠/𝑅₄𝐶₂)+(1/𝑅₃𝑅₂𝐶₁𝐶₂). (2) From (2), by choosing the component values of 𝑅₁=𝑅₃ and 𝑅₂=𝑅₄, we can see that(1)if 𝑉₁=𝑉_in (an input voltage signal), and 𝑉₂=𝑉₃=0 (grounded), the LP response can be realized with the passband gain 𝐻_LP=1;(2)if 𝑉₂=𝑉_in and 𝑉₁=𝑉₃=0, the BP response can be realized with the passband gain 𝐻_BP=1;(3)if 𝑉₃=𝑉_in and 𝑉₁=𝑉₂=0, the HP response can be realized with the passband gain 𝐻_HP=1;(4)if 𝑉₁=𝑉₃=𝑉_in and 𝑉₂=0, the BS response can be realized with the passband gain 𝐻_BS=1;(5)if 𝑉₁=−𝑉₂=𝑉₃=𝑉_in, the AP response can be realized with the passband gain 𝐻_AP=1.

Figure 3 

Proposed CDBA-based universal biquad filter.

Clearly, the proposed filter can be used as a voltage-mode three-input single-input universal filter that can realize all the standard types of the biquad filter functions.

Also from (2), the natural angular frequency (𝜔_𝑜), the bandwidth (BW), and the quality factor (Q) in all cases are given by𝜔_𝑜=1𝑅_𝐴𝑅_𝐵𝐶₁𝐶₂,1BW=𝑅_𝐵𝐶₂,𝑄=𝑅_𝐵𝐶₂𝑅_𝐴𝐶₁, (3) where 𝑅_𝐴=𝑅₁=𝑅₃ and 𝑅_𝐵=𝑅₂=𝑅₄. Note that the filter parameters 𝜔_𝑜 and BW are adjustable properly by the virtual-grounded resistor 𝑅_𝐴 or the capacitor 𝐶₁, and by the virtual-grounded resistor 𝑅_𝐵 or the capacitor 𝐶₂, in that order.

Taking the nonidealities of the CDBA into account, the relationship of the terminal currents and voltages given in (1) can be rewritten as 𝑖_𝑧𝑣_𝑤𝑣_𝑝𝑣_𝑛=00𝛼_𝑝−𝛼_𝑛.𝑣𝛽00000000000^𝑧𝑖_𝑤𝑖_𝑝𝑖_𝑛, (4) where 𝛼_𝑝=1−𝜀_𝑝 and 𝜀_𝑝(|𝜀_𝑝|≪1) is the current-tracking error from p terminal to z terminal, 𝛼_𝑛=1−𝜀_𝑛 and 𝜀_𝑛(|𝜀_𝑛|≪1) is the current-tracking error from n terminal to z terminal, and 𝛽=1−𝜀_𝑣 and 𝜀_𝑣(|𝜀_𝑣|≪1) is the voltage-tracking error from z terminal to w terminal of the CDBA. Therefore, including these nonidealities into the voltage transfer function, the modified filter parameters are derived as𝐻_LP=𝛼_𝑝1𝛼_𝑛1,𝐻_BP=𝛼_𝑝2𝛽₁𝛼_𝑛2,𝐻_HP=𝛽₂,𝜔_𝑜=𝛼_𝑝2𝛼_𝑛1𝛽₁𝛽₂𝑅_𝐴𝑅_𝐵𝐶₁𝐶₂,𝛼BW=^𝑛2𝛽₂𝑅_𝐵𝐶₂,1𝑄=𝛼_𝑛2𝛼_𝑝2𝛼_𝑛1𝛽₁𝑅_𝐵𝐶₂𝛽₂𝑅_𝐴𝐶₁, (5) where 𝛼_𝑝𝑖, 𝛼_𝑛𝑖, and 𝛽_𝑖 are the parameters 𝛼_𝑝, 𝛼_𝑛, and 𝛽 of the ith CDBA (𝑖=1,2).

The active and passive sensitivities of this universal filter are shown as follows: 𝑆^𝐻LP_𝛼𝑝1=−𝑆^𝐻LP_𝛼𝑛1=1,𝑆^𝐻BP_𝛼𝑝2=−𝑆^𝐻BP_𝛼𝑛2=𝑆^𝐻BP_𝛽1=1,𝑆^𝐻HP_𝛽2𝑆=1,_{𝜔𝑜𝛼𝑝2}=𝑆^𝜔𝑜_𝛼𝑛1=𝑆^𝜔𝑜_𝛽1=𝑆^𝜔𝑜_𝛽2=12,𝑆^BW_𝛼𝑛2=𝑆^BW_𝛽2𝑆=1,_{𝑄𝛼𝑝2}=𝑆^𝑄_𝛼𝑛1=𝑆^𝑄_𝛽1=−𝑆^𝑄_𝛽2=12,𝑆^𝑄_𝛼𝑛2𝑆=−1,_{𝜔𝑜𝑅1}=𝑆^𝜔𝑜_𝑅2=𝑆^𝜔𝑜_𝑅3=𝑆^𝜔𝑜_𝑅4=𝑆^𝜔𝑜_𝐶1=𝑆^𝜔𝑜_𝐶21=−2,𝑆^BW_𝑅2=𝑆^BW_𝑅4=𝑆^BW_𝐶2𝑆=−1,_𝑄𝑅1=−𝑆^𝑄_𝑅2=𝑆^𝑄_𝑅3=−𝑆^𝑄_𝑅4=𝑆^𝑄_𝐶1=−𝑆^𝑄_𝐶2=−1. (6)It is clearly observed from (6) that all of the sensitivities are low.

4. Proposed CDBA-Based Quadrature Oscillator

From the first configuration of Figure 3, by setting 𝑉₁=𝑉₂=𝑉₃=0V (grounded) and connecting the resistor 𝑅₁ between the terminals p and w of the CDBA2, the second proposed CDBA-based sinusoidal quadrature oscillator can be obtained as shown in Figure 4. In this case, the characteristic equation of the second proposed configuration can be given by𝑠²𝐶₁𝐶₂+𝑠𝐶₁(1𝑅₄−1𝑅₁1)+(𝑅₂𝑅₃)=0. (7)The oscillation condition and the oscillation frequency (𝜔_𝑜) of this configuration can be obtained, respectively, as𝑅₁=𝑅₄,𝜔_𝑜=1𝑅₂𝑅₃𝐶₁𝐶₂. (8)Note from (8) that the oscillation condition of the proposed oscillator of Figure 4 can be adjusted by 𝑅₁ or/and 𝑅₄ without affecting the oscillation frequency 𝜔_𝑜.The 𝜔_𝑜 can be adjusted by 𝑅₂ or/and 𝑅₃.Therefore, the oscillation condition and the oscillation frequency of the proposed oscillator circuit can be controlled independently.

Figure 4 

Proposed CDBA-based sinusoidal quadrature oscillator.

From the configuration of Figure 4, the relationship between two quadrature output voltages 𝑉_𝑜1 and 𝑉_𝑜2 can be expressed as𝑉_𝑜2𝑉_𝑜11=−𝑠𝑅₃𝐶₁, (9)where the phase shift is 𝜙=90^∘. This guarantees that the proposed oscillator circuit provides the quadrature outputs 𝑉_𝑜1 and 𝑉_𝑜2.

Taking into account the nonideal CDBAs in Figure 4, the characteristic equation becomes𝑠²𝐶₁𝐶₂+𝑠𝐶₁𝛽₂(𝛼_𝑛2𝑅₄−𝛼_𝑝2𝑅₁𝛼)+(^𝑝2𝛼_𝑛1𝛽₁𝛽₂𝑅₂𝑅₃)=0. (10)The modified oscillation condition and the oscillation frequency (𝜔_𝑜) in this case are obtained as𝛼_𝑛2𝑅₁=𝛼_𝑝2𝑅₄,𝜔_𝑜=𝛼_𝑝2𝛼_𝑛1𝛽₁𝛽₂𝑅₂𝑅₃𝐶₁𝐶₂. (11)(12)It should be noted from (11) and (12) that the oscillation condition and 𝜔_𝑜 of the proposed quadrature oscillator are altered slightly by the effects of the CDBA current- and voltage-tracking errors. However, they can still be controlled independently.

Sensitivities calculated from (12) using relative sensitivity formula are obtained as 𝑆^𝜔𝑜_𝛼𝑝2=𝑆^𝜔𝑜_𝛼𝑛1=𝑆^𝜔𝑜_𝛽1=𝑆^𝜔𝑜_𝛽2=12,𝑆^𝜔𝑜_𝑅2=𝑆^𝜔𝑜_𝑅3=𝑆^𝜔𝑜_𝐶1=𝑆^𝜔𝑜_𝐶21=−2. (13)The circuit has optimum sensitivity performance in the sense that all values are equal to 0.5 in magnitude.

5. Simulation and Experimental Results

In order to confirm the theoretical analysis, all the proposed circuits given in Figures 3 and 4 have been simulated using the PSPICE program. In simulations, the CDBA circuit was constructed with two AD844 ICs of analog devices, as shown in Figure 2. The model parameters of AD844 were taken from the built-in library (AD844/AD), and the supply voltages were taken as ±12 V. Theoretical analysis and simulation results were also verified by experimental testing. For the experiments, the proposed circuits were constructed in laboratory using commercially available current feedback amplifier (CFA) AD844 ICs with the same components used in simulation.

Figure 5 shows the simulated frequency characteristics for the LP, BP, and HP filter functions of the proposed CDBA-based multifunction biquad circuit in Figure 3 with the theory frequency responses. The passive component values were chosen as 𝑅₁=𝑅₂=𝑅₃=𝑅₄=10 kΩ and 𝐶₁=𝐶₂=1 nF, which is designed to obtain a universal filter with a natural angular frequency of 𝑓_𝑜=𝜔_𝑜/2𝜋≅15.91 kHz and the quality factor of 𝑄=1. With the same setting, the simulation and experimental results for the gain and phase responses of the BS and AP filters verifying theory values are, respectively, depicted in Figures 6 and 7. It appears from all figures that the simulation and experimental results are in excellent agreement with the theoretical values.

Figure 5 

Frequency characteristics for LP, BP, and HP responses of the proposed circuit of Figure 3.

(a)

(b)

Figure 6 

Frequency responses of the BS filter, (a) gain response and (b) phase response.

(a)

(b)

Figure 7 

Frequency responses of the AP filter, (a) gain response and (b) phase response.

As an example to demonstrate the workability of the proposed CDBA-based quadrature oscillator circuit in Figure 4, the following passive component values were chosen as 𝑅₁=𝑅₂=𝑅₃=10 kΩ and 𝐶₁=𝐶₂=1 nF. The simulated quadrature output waveforms 𝑉_𝑜1 and 𝑉_𝑜2 of the oscillator are illustrated in Figure 8, where the value of 𝑅₄≅10.5 kΩ was chosen to be slightly larger than 𝑅₁ to ensure that the oscillations will start. The simulated frequency was measured as 15.37 kHz, which is lower than the theoretical value of 15.91 kHz, owing to the parasitic elements 𝑅_𝑥, 𝑧, and 𝐶_𝑧 of the AD844 [12, 13]. Figure 9 shows the simulated frequency spectrums of the quadrature outputs, giving the total harmonic distortion (THD) less than 2.95%. The results of the THD analysis obtained from the simulations are summarized in Table 1.

Table 1 

Total harmonic distortion analysis of the proposed CDBA-based quadrature oscillator of Figure 4.

(a)

(b)

Figure 8 

Simulated output waveforms 𝑉𝑜1 and 𝑉𝑜2 of the proposed oscillator of Figure 4, (a) at transient stage and (b) at steady stage.

Figure 9 

Simulated frequency spectrums of the proposed oscillator of Figure 4.

Figure 10 shows the experimental output waveforms 𝑉_𝑜1 and 𝑉_𝑜2 of Figure 4, and the oscillation frequency 𝑓_𝑜 was measured as 15.52 kHz. The controllability of 𝑓_𝑜 through 𝑅₂ without influencing the oscillation condition is shown in Figure 11.

Figure 10 

Experimental output waveforms 𝑉𝑜1 and 𝑉𝑜2 of the proposed oscillator of Figure 4.

Figure 11 

Variation of the oscillation frequency of Figure 4 obtained by varying the value of 𝑅2.

6. Conclusion

In this paper, a voltage-mode universal biquad filter using two CDBAs, four virtually grounded resistors, and two capacitors has been presented. By appropriately connecting input voltage terminals, the proposed CDBA-based biquadratic filter can realize all the standard types of the biquadratic filter functions. The filter permits orthogonal adjustment of the parameters 𝜔_𝑜 and BW. Furthermore, a CDBA-based quadrature oscillator can be obtained easily by slightly modifying the proposed CDBA-based biquad filter. Its oscillation condition and oscillation frequency can be controlled separately via different virtually grounded resistors. In addition, the sensitivities of all proposed circuits are shown to be low. It is shown that the simulation and experimental results conform very well to the theoretical conclusions.

Acknowledgments

This work was supported by the Commission on Higher Education, Ministry of Education, Thailand, through the Research Group Development Program (CHE-RG-01A), Research Group in Microelectronics for Communications. The authors are also grateful to the editor and reviewers for their valuable comments and helpful suggestions, which substantially improved the quality of the manuscript.