# CDBA-Based Universal Biquad Filter and Quadrature Oscillator

Authors: Worapong Tangsrirat; Tattaya Pukkalanun; Wanlop Surakampontorn
Journal: Active and Passive Electronic Components
 (2008)

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## 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(ωo) 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.


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## Body

## 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 (ωo) 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 Figure1, 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]:(1)[izvwvpvn]=[001−1100000000000].[vziwipin].
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

Figure3 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 be(2)Vo=s2V3+(s/R2C2)V2+(1/R1R2C1C2)V1s2+(s/R4C2)+(1/R3R2C1C2).
From (2), by choosing
the component values of R1=R3 and R2=R4, we can see that(1)
ifV1=Vin (an input voltage signal), and V2=V3=0 (grounded), the LP response can be realized with the passband gain HLP=1;(2)
ifV2=Vin and V1=V3=0, the BP response can be realized with the passband gain HBP=1;(3)
ifV3=Vin and  V1=V2=0, the HP response can be realized with the passband gain HHP=1;(4)
ifV1=V3=Vin and V2=0, the BS response can be realized with the passband
gain HBS=1;(5)
ifV1=−V2=V3=Vin, the AP response can be realized with the passband gain HAP=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 (ωo),
the bandwidth (BW), and the quality factor (Q) in all cases are given by(3)ωo=1RARBC1C2,BW=1RBC2,Q=RBC2RAC1,
where RA=R1=R3 and RB=R2=R4.
Note that the filter parameters ωo and BW are adjustable properly by the virtual-grounded resistor RA or the
capacitor C1, and by the virtual-grounded
resistor RB 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
(4)[izvwvpvn]=[00αp−αnβ00000000000].[vziwipin],
where αp=1−εp and εp(|εp|≪1) is the
current-tracking error from p terminal to z terminal, αn=1−εn and εn(|εn|≪1) is the
current-tracking error from n terminal to z terminal, and β=1−εv and εv(|εv|≪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 as(5)HLP=αp1αn1,HBP=αp2β1αn2,HHP=β2,ωo=αp2αn1β1β2RARBC1C2,BW=αn2β2RBC2,Q=1αn2αp2αn1β1RBC2β2RAC1,
where αpi, αni, and
βi are the parameters αp, αn,
and β of the ith CDBA (i=1,2).The active and passive sensitivities of
this universal filter are shown as follows:(6)Sαp1HLP=−Sαn1HLP=1,Sαp2HBP=−Sαn2HBP=Sβ1HBP=1,Sβ2HH
P=1,Sαp2ωo=Sαn1ωo=Sβ1ωo=Sβ2ωo=12,Sαn2BW=Sβ2BW=1,Sαp2Q=Sαn1Q=Sβ1Q=−Sβ2Q=12,Sαn2Q=−1,SR1ωo=SR2ωo=SR3ωo=SR4ωo=SC1ωo=SC2ωo=−12,SR2BW=SR4BW=SC2BW=−1,SR1Q=−SR2Q=SR3Q=−SR4Q=SC1Q=−SC2Q=−1.It is clearly observed from (6) that all of
the sensitivities are low.


## 4. Proposed CDBA-Based Quadrature  Oscillator

From the first configuration of Figure3, 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 by(7)s2C1C2+sC1(1R4−1R1)+(1R2R3)=0.The oscillation
condition and the oscillation frequency (ωo) of this configuration can be obtained,
respectively, as(8)R1=R4,ωo=1R2R3C1C2.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 ωo.The ωo 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 Figure4,
the relationship between two quadrature output voltages Vo1 and Vo2 can be expressed as(9)Vo2Vo1=−1sR3C1,where the phase shift is ϕ=90°. This guarantees that the proposed oscillator circuit provides the
quadrature outputs Vo1 and Vo2.Taking into account the nonideal CDBAs
in Figure4, the characteristic equation becomes(10)s2C1C2+sC1β2(αn2R4−αp2R1)+(αp2αn1β1β2R2R3)=0.The modified
oscillation condition and the oscillation frequency (ωo) in this case are obtained as(11)αn2R1=αp2R4,(12)ωo=αp2αn1β1β2R2R3C1C2.It should
be noted from (11) and (12) that the oscillation condition and ωo 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
(13)Sαp2ωo=Sαn1ωo=Sβ1ωo=Sβ2ωo=12,SR2ωo=SR3ωo=SC1ωo=SC2ωo=−12.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 Figures3 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.Figure5 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 fo=ωo/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 Figure3.Frequency responses of the BS filter, (a) gain response and (b) phase response.
(a)(b)Frequency responses of the AP filter, (a) gain response and (b) phase response.
(a)(b)As an example to demonstrate the workability of the proposed CDBA-based quadrature
oscillator circuit in Figure4, the following passive component values were
chosen as R1=R2=R3=10 kΩ and C1=C2=1 nF.  The simulated quadrature output waveforms Vo1
and Vo2 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 Rx, z,
and Cz 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 Figure4.
Harmonic no.Frequency (Hz)Fourier componentNormalized componentPhase (Deg)Normalized phase11.592E + 048.264E + 001.000E + 001.232E + 020.000E + 0023.183E + 042.023E-012.447E-021.357E + 021.249E + 0134.775E + 049.652E-021.168E-021.680E + 024.563E + 0146.366E + 046.915E-028.368E-031.418E + 021.854E + 0157.958E + 041.405E-021.701E-03−1.206E + 02−2.438E + 02DC component = −1.699764E-01Total harmonic distortion = 2.948440E + 00 PERCENTSimulated output waveformsVo1 and Vo2 of the proposed oscillator of Figure 4, (a) at transient stage and (b) at steady stage.
(a)(b)Figure 9
Simulated frequency spectrums of the proposed oscillator of Figure4.Figure10 shows the experimental output
waveforms Vo1 and Vo2 of Figure 4, and the oscillation frequency fo was measured as
15.52 kHz.  The controllability of fo through R2 without influencing the oscillation condition is shown in Figure 11.Figure 10
Experimental output waveformsVo1 and Vo2 of the proposed oscillator of Figure 4.Figure 11
Variation of the oscillation frequency of Figure4 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 parametersωo 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.



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*Source: 247171-2008-02-28.xml*