# Third Order Universal Filter Using Single Operational Transresistance Amplifier

Authors: Mourina Ghosh; Sajal K. Paul; Rajiv Kumar Ranjan; Ashish Ranjan
Journal: Journal of Engineering
 (2013)

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## Abstract

This paper proposes a multi-input single-output (MISO) third order voltage mode (VM) universal filter using only one operational transresistance amplifier (OTRA). The proposed circuit realizes low-pass, high-pass, all-pass, band-pass, and notch responses from the same topology. The PSPICE Simulation results using 0.5μm CMOS technology agree well with the theoretical design.


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## Body

## 1. Introduction

Operational transresistance amplifier (OTRA) has emerged as an alternate current mode analog building block [1, 2] since it inherits all the advantages offered by other current mode building blocks such as current conveyor (CC) [3–6], operational transconductance amplifier (OTA) [7], current differencing buffered amplifier (CDBA) [8], differential voltage current conveyor (DVCC) [9]. The input terminals of OTRA are internally grounded, thereby eliminating parasitic capacitances at the input. OTRA has the advantages of a high slew rate and wide bandwidth. It is a high-gain current input voltage output device. Some analog filters using OTRA have been reported in [10–16] and references cited there in. It is well known that higher order analog filters using different kinds of active elements are of considerable interest, as they may be used for some special applications where sharp cut off is desirable and at the same time also useful to implement digital filters. Higher order filters can be obtained by various methods such as cascading lower order filter or state variable method or signal flow graph. Although OTRA based third order filters are reported in the literature [17–19], they all involve more that one OTRA.In this work an attempt is made to propose a third order voltage mode (VM) universal filter employing single OTRA as an active element. All of the five filters, namely lowpass (LPF), highpass (HPF), allpass (APF), bandpass (BPF), notch (NF), can be obtained from the same topology.


## 2. Circuit Description

The circuit symbol of an OTRA is illustrated in Figure1. The characteristic equations of this element can be described as
(1)[VpVnV0]=[000000Rm-Rm0][IpInI0].
A CMOS-based OTRA using 0.5 μm CMOS technology is given in Figure 2 [2].Figure 1
Block diagram of OTRA.Figure 2
Internal circuit of OTRA [2].A generalizednth order filter topology using single OTRA is shown in Figure 3 [20]. The routine analysis gives a voltage transfer function as
(2)Vout(s)Vin(s)=Ya-YbYc-Yd.
The proposed third order universal filter is obtained from Figure 3, if one considers 1/Ya=Za=Ra1∥1/sCa1∥(Ra2+1/sca2), 1/Yb=Zb=1/sCb1∥(Rb+1/scb2), 1/Yc=Zc=Rc∥1/sCc, 1/Yd=Zd=Rd∥1/sCd and splits the Ya terminal into three parts and Yb terminal into two parts as shown in  Figure 4.Figure 3
Generalizednth order filter.Figure 4
Proposed third order universal filter.The routine analysis gives output voltage as(3)Vout=N(s)D(s),
where
(4)N(s)=Rc(1+sCdRd)×{(1+sCb2Rb)(1+sCa2Ra2)V1-sCa1Ra1(1+sCb2Rb)(1+sCa2Ra2)V2-sCa2Ra2(1+sCb2Rb)V3-sCb1Ra1(1+sCb2Rb)(1+sCa2Ra2)V4-sCb2Ra1(1+sCa2Rc)V5},D(s)=Ra1(1+sCb2Rb)(1+sCa2Ra2)×{(1+sCcRc)(1+sCdRd)-sCdRc}.IfCa2Ra2=CdRd, then (4) modifies as
(5)N(s)=Rc×{(1+sCb2Rb)(1+sCa2Ra2)V1×-sCa1Ra1(1+sCb2Rb)(1+sCa2Ra2)V2×-sCa2Ra2(1+sCb2Rb)V3×-sCb1Ra1(1+sCb2Rb)(1+sCa2Ra2)V4×-sCb2Ra1(1+sCa2Rc)V5},D(s)=Ra1(1+sCb2Rb)×{(1+sCcRc)(1+sCdRd)-sCdRc}=Ra1(1+sCb2Rb)×{s2CcRcCdRd+s(CdRd+CcRc-CdRc)+1}=Ra1[s3Cb2RbCcRcCdRd+s2Ra1×{CcRcCdRd+Cb2RbCdRdRa1×+Cb2RbCcRc-Cb2RbCdRc}Ra1+s{Cb2Rb+CcRc+CdRd-CdRc}+1s3Cb2].
The specialization of (4), as shown in Tables 1 and 2, results the transfer function of low-pass filter (LPF), all-pass filter (APF), band-pass filter (BPF), and notch filter (NF) functions. Similarly high-pass (HPF) transfer function is obtained from (4) if Cb2Rb=CdRd.Table 1
The availability of each filer response and corresponding selection ofV1,V2,V3,V4, and V5.
Filter response
InputsVin
V
out
V
1
V
2
V
3
V
4
V
5
Low pass filter (LPF)
1
0
1
0
1
High pass filter (HPF)
0
1
1
0
1
All pass filter (APF)
1
0
1
1
1
Band pass filter (BPF)
0
0
1
0
1
Notch filter (NF)
1
1
1
0
1Table 2
Condition of realization of each filter.
Filter response
Condition
LPF
R
c
R
a2
+
R
a1
R
b
=
R
a2
R
a1,  Ca2Ra2+Cb2Rb+Ca2Ra1=Cb2Ra1
HPF
C
a1
+
C
a2
=
C
b2,  Ca1Ca2Ra2+Ca1Cb2Rb+Ca2Cb2Rb=Cb2Ca2Ra2
APF
C
b1
R
a1
=
C
c
R
c
,
-
(
C
a2
R
a2
+
C
b2
R
b
+
C
a1
R
a1
-
C
b1
R
a1
-
C
b2
R
a1
)
=
C
d
R
d
+
C
c
R
c
+
C
d
R
c
+
C
b2
R
b
,
C
b2
R
b
C
a2
R
a2
+
C
a2
R
a1
C
b2
R
b
-
C
b1
R
a1
C
a2
R
a2
-
C
b1
R
a1
C
a2
R
b
=
C
c
R
c
C
d
R
d
+
C
b2
R
b
C
d
R
d
+
C
b2
R
b
C
c
R
c
-
C
b2
R
b
C
d
R
c
BPF
C
a2
=
C
b2
NF
C
a1
R
a1
=
C
c
R
c
,
C
d
R
d
+
C
b2
R
b
+
C
c
R
c
-
C
d
R
c
=
C
a2
R
a2
+
C
b2
R
b
+
C
a1
R
a1
+
C
a2
R
a1
-
C
b2
R
a1
,
C
b2
R
b
C
a2
R
a2
+
C
a1
R
a1
C
a2
R
a2
+
C
a1
R
a1
C
b2
R
b
+
C
b2
R
a1
C
a2
R
b
=
C
b2
R
a1
C
a2
R
a2The natural frequency and quality factor of the proposed circuit for LPF, APF, BPF, NF can be obtained as(6)ω0=1CcCdCb2RcRbRd3,Q=CdCcCb2RcRbRd3CcRc+CdRd-CdRc+Cb2Rb-CcCdCb2RcRbRd3,
and for HPF as
(7)ω0=1CcCdCa2RcRa2Rd3,Q=CdCcCa2RcRa2Rd3CcRc+CdRd-CdRc+Ca2Ra2-CcCdCa2RcRa2Rd3.
The sensitivity of ω0 with respect to passive elements may be expressed as
(8)SCcω0=SCdω0=SCb2ω0=SCa2ω0=SRcω0=SRbω0=SRdω0=SRa2ω0=-13.
It shows that the proposed circuit offers low sensitivity.


## 3. Effect of Nonidealities of OTRA

In this section the effect of finite transresistance gain(Rm)on the output of the filter circuit is considered and for high frequency applications a passive compensation is employed in the same line as of reference [10, 11]. Ideally the transresistance gain is assumed to approach infinity. However, practically Rm is a frequency dependent finite value. Considering the single pole model for the transresistance gain, Rm can be expressed as
(9)Rm(s)=R01+s/ω0,
where ω0 is the pole frequency and R0 is DC transresistance gain. In the case of high frequency application the transresistance gain Rm(s) reduces to
(10)Rm(s)=1sCP,
where
(11)CP=1R0ω0.
Taking this effect into account the voltage transfer function (2) modifies to
(12)Vout(s)Vin(s)=Ya-YbYc-Yd+sCp,
where sCp is the uncompensated error term. In the case of high frequency applications, passive compensation method may be employed to account for error introduced in (2) which is given in (11). Inspection of circuit of Figure 4  and denominator of (11) indicates that error term sCp may be realized by connecting a capacitor CP in parallel to Yc or Cc for passive compensation. In other words the value of Cc may be decreased by CP to absorb the nonideality effect and thus achieving self-compensation. There is another nonideality due to parasitic resistances and capacitances. Since the input nodes in an OTRA are inherently grounded, the effect of parasites are negligible.


## 4. Simulation Results

The performance of the proposed universal filter is evaluated with CMOS implementation of OTRA as shown in Figure2 [2], with DC power supply voltages VDD=-VSS=1.5V and bias voltage VB=-0.5V. The simulations are performed using PSPICE based on 0.5 μm MOSIS (AGILENT) CMOS technology parameters. Aspect ratios used for different transistors are same as [2].The designed values of resistances and capacitances to obtain multifunction filter output for a cutoff frequency off0=200KHz and quality factor of Q=1 is shown in Table 3. The simulated frequency responses for gain of LPF, HPF, and BPF are shown in Figures 5, 6, and 7, respectively. It is observed that the simulated values closely agree with the theoretical results. Figures 8 and 9 show the frequency responses for both gain and phase for APF and NF, respectively. It is observed that there is deviation of simulated gain at higher frequency for notch filter (NF). To judge the quality of the output, total harmonic distortion is obtained for low-pass, high-pass, and band-pass filter as shown in Figure 10. It is evident that the output distortion is very low and within 0.2% up to 2 volts. Hence it may be claimed that the output is of very good quality and dynamic range is high.Table 3
Designed values of passive components used for VM third order filters.
Filter response
Component values
R
a1 (kΩ)
C
a1 (pF)
R
a2 (kΩ)
C
a2 (pF)
C
b1 (pF)
R
b (kΩ)
C
b2 (pF)
R
c (kΩ)
C
c (pF)
R
d (kΩ)
C
d (pF)
LPF
200
—
200
1.99
—
100
7.96
200
7.96
66.66
5.97
HPF
—
7.96
100
7.96
—
25
15.92
200
7.96
66.66
5.97
APF
200
—
9.52
41.79
7.96
16.66
47.76
200
7.96
66.66
5.97
BPF
—
—
50
7.96
—
100
7.96
200
7.96
66.66
5.97
NF
200
7.96
200
1.99
—
50
15.92
200
7.96
66.66
5.97Figure 5
Frequency response of third order lowpass filter.Figure 6
Frequency response of third order highpass filter.Figure 7
Frequency response of third order unsymmetrical bandpass filter.Figure 8
Frequency response of third order allpass filter.Figure 9
Frequency response of third order unsymmetrical notch filter.Figure 10
Variation of %THD with respect to input voltage amplitude.


## 5. Conclusion

An OTRA based universal third order voltage mode filter is presented. Although a number of passive components matching is required, the proposed circuit offers the following advantageous features: (i) use of single OTRA, (ii) it has low output impedance, hence suitable for cascading, (iii) can implement all the five filter function such as lowpass, highpass, bandpass, allpass, and notch from same topology, (iv) moreover all the capacitors are grounded or virtually grounded, so suitable for monolithic IC implementation, and (v) effect of finite transresistance gain can be reduced by self-compensation method, hence no additional elements needed.



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*Source: 317296-2013-01-10.xml*