# New Topologies of Lossless Grounded Inductor Using OTRA

Authors: Rajeshwari Pandey; Neeta Pandey; Sajal K. Paul; A. Singh; B. Sriram; K. Trivedi
Journal: Journal of Electrical and Computer Engineering
 (2011)

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## Abstract

Two alternate topologies of lossless grounded inductor have been proposed using operational transresistance amplifier (OTRA). Three applications using the proposed inductors are also included.  PSPice simulation and experimental results have been included to demonstrate the performance and verify the theoretical analysis.


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## Body

## 1. Introduction

Recently the operational transresistance amplifier (OTRA) has emerged as alternate important analog building block as it inherits all the advantages offered by  current mode techniques. The OTRA is a high-gain current input voltage output device. The input terminals of OTRA are internally grounded, thereby eliminating response limitations due to parasitic capacitances and resistances at the input. Although the OTRA is commercially available from several sources under the name of current differencing amplifier or Norton amplifier, it has not gained attention until recently. These commercial realizations do not provide internal ground at the input port and allow the input current to flow in one direction only. The former disadvantage limits the functionality of the OTRA where as the later forces to use external DC bias current leading to complex and unattractive designs [1]. Several high-performance CMOS OTRA topologies have been proposed in literature [1–4] leading to growing interest in OTRA-based analog signal processing circuits. In the recent past OTRA has been extensively used as an analog building block for realizing a number of signal processing circuits such as filters [5–8], oscillators [9, 10], multivibrators [11, 12], and immittance simulation circuits [9, 13–15], an application which has been dealt with in this paper. A number of grounded parallel immittance topologies using single OTRA are proposed in [13]. However none of these configurations can realize a lossless grounded inductor. The structure in [14] presents simulation of lossless negative grounded inductance. Lossless grounded inductor simulators using two OTRAs, five resistors, and one capacitor are presented in [9, 15].In this paper two additional topologies of lossless grounded inductor using two OTRAs, five resistors, and one capacitor are reported. In these topologies five passive elements out of six are grounded as compared to four grounded elements in [9, 15]. Some applications of the proposed topologies are also presented.


## 2. Circuit Description

OTRA is a three-terminal device, shown symbolically in Figure1 and its port relations can be characterized by the following matrix:
(1)[VpVnVO]=[000000Rm-Rm0][IpInIO].
For ideal operations the transresistance gain Rm approaches infinity and forces the input currents to be equal. Thus OTRA must be used in a negative feedback configuration.Figure 1
OTRA circuit symbol.The proposed lossless grounded inductor topologies are shown in Figure2. Routine analysis of the circuit of Figure 2(a) results in the following expression for input admittance:
(2)Yin(s)=(G1+G3+G5-G1G5G4)+C1G4sG2G3G5,Yin will be purely inductive if the following condition is met
(3)G1+G3+G5=G1G5G4.
Similarly for inductor topology of Figure 2(b), input admittance is given by
(4)Yin(s)=(G1+G2+G4-G2G4G3)+C1G3sG1G2G5.
This input admittance will be purely inductive provided that
(5)G1+G2+G4=G2G4G3.
The equivalent inductance values along with conditions are given in Table 1.Table 1
Inductors realized by the topologies shown in Figure2.
FigureConditionLeqNon-interactive control ofLeqFigure2(a)G1+G3+G5=G1G5G4C1G4G2G3G5Independent control of value ofLeq through G2Figure2(b)G1+G2+G4=G2G4G3C1G3G1G2G5Independent control of value ofLeq through G5Inductor simulation topologies.
(a)(b)It is clear from Table1 that for both the topologies the inductance value can be controlled independent of condition of realization.The proposed topologies are verified through simulations using the CMOS implementation of the OTRA [4] as given in Figure 3. The SPICE simulation is performed using 0.5 μm CMOS process parameters provided by MOSIS (AGILENT) and supply voltages taken are ±1.5 V. Aspect ratios used for different transistors are given in Table 2.Table 2
Aspect ratio of the transistors.
TransistorW(μm)/L(μm)M1–M3100/2.5M410/2.5M5, M630/2.5M710/2.5M8–M1150/2.5M12, M13100/2.5M1450/0.5Figure 3
CMOS implementation of OTRA [4].An inductor of valueLeq=0.1 mH is designed using the inductor topology of Figure 2(a) with the component values of R1=R2=R3=R5=1 KΩ, R4=3 KΩ, C1=300 pF. The frequency response of the impedance as obtained using this inductor is shown in Figure 4(a). The inset depicts the enlarged view of impedance response in lower frequency range.(a) Impedance versus frequency response of topology of Figure2(a). (b) Impedance versus frequency response of topology of Figure 2(b).
(a)(b)Similarly, Figure4(b) shows the frequency response of the impedance of the inductor of value Leq=10 μH as obtained with component values of R1=R2=R4=R5=1 KΩ, R3=3 KΩ, and C1=30 pF for inductor topology given in Figure 2(b). In the inset, the variation of impedance in lower frequency range is shown.


## 3. Applications

In this section some applications of the proposed topologies have been presented. Both the topologies may be used for constructing filter and oscillator circuits.


### 3.1. High-pass Filter

A high pass filter, as shown in Figure5(a), can be constructed using proposed inductors. The transfer function for high pass response is obtained as
(6)VoVin=s2s2+s/CR+1/LeqC,
where
(7)ω0=1√(LeqC),Q=R√(CLeq).(a) High-pass filter. (b) Frequency response of high pass filter.
(a)(b)The functionality of the high pass filter is verified using the inductor topology of Figure2(a) and designed for lower cutoff frequency of 503.3 KHz. The component values are obtained as R=300 Ω and C=1 nF for Leq=0.1 mH.  The value of Leq=0.1 mH is obtained using inductor topology of Figure 2(a) with component values of R1=R2=R3=R5=1 KΩ, R4=3 KΩ, and C1=300 pF. The frequency response of the filter simulated using PSPICE is depicted in Figure 5(b). Simulated value of lower cutoff frequency is obtained as 505 KHz which is in close agreement to the theoretical value of 503.2 KHz.


### 3.2. Band Pass Filter

The proposed inductor topologies may also be used to obtain band-pass response using the circuit given in Figure6(a). The transfer function for band-pass response is obtained as
(8)VoVin=s/CRs2+s/CR+1/LeqC,
where
(9)ω0=1√(LeqC),Q=R(CLeq).
The theoretical proposition is verified using the topology of Figure 2(b). A band-pass filter is designed having center frequency of 503.3 KHz. The component values are obtained as R=1 KΩ and C=1 nF for Leq=0.1 mH. The value of Leq=0.1 mH is obtained for inductor topology of Figure 2(b) with component values of R1=R2=R4=R5=1 KΩ,R3=3 KΩ, C1=300 pF. The frequency response of the filter simulated using PSPICE is depicted in Figure 6(b). The simulated results are in close agreement with the theoretical prediction.(a) Band-pass filter. (b) Frequency response of band-pass filter.
(a)(b)


### 3.3. LC Oscillator

An LC Oscillator can be realized using the proposed inductor topologies. Figure7(a) shows the schematic of an LC oscillator using topology of Figure 2(a) for which the condition of oscillation (CO) and frequency of oscillation (FO) are obtained as
(10)CO:G1+G3+G5=G1G5G4,(11)FO:fo=12ΠCC1R2R3R5/R4.
Figure 7(b) shows the output of the oscillator for Leq=0.1 mH and C=330 pF. Simulated frequency of oscillation is 860 KHz as against the calculated value of 876.2 KHz with % error of 1.85%.(a) LC oscillator. (b) Output of LC oscillator.
(a)(b)


### 3.4. Experimental Verification

The proposed inductor topologies are also tested experimentally to verify the theory. The OTRA is implemented using two CFOAs (IC AD844AN) as shown in Figure8 with a supply voltage of ±5 V.Figure 8
CFOA based OTRA implementation.The high pass filter of Figure5(a) is prototyped with C=1 nF, R=680 Ω, and Leq=1 mH. The inductor of value Leq=1 mH is implemented using inductor topology of Figure 2(a) with component values of R1=R2=R3=R5=1 KΩ, R4=3 KΩ, and C1=3 nF. Theoretical, simulated and experimental frequency responses are shown in Figure 9. It is observed that the experimental values more or less follow the theoretical and simulated values. The oscillator circuit of Figure 7(a) is also tested experimentally. The output waveform observed on oscilloscope is shown in Figure 10. The observed frequency of oscillation is found to be 872.5 KHz, which is in close agreement to theoretically calculated value of 876.2 KHz.Figure 9
Frequency response of the high-pass filter.Figure 10
Experimental output of LC oscillator.


## 3.1. High-pass Filter

A high pass filter, as shown in Figure5(a), can be constructed using proposed inductors. The transfer function for high pass response is obtained as
(6)VoVin=s2s2+s/CR+1/LeqC,
where
(7)ω0=1√(LeqC),Q=R√(CLeq).(a) High-pass filter. (b) Frequency response of high pass filter.
(a)(b)The functionality of the high pass filter is verified using the inductor topology of Figure2(a) and designed for lower cutoff frequency of 503.3 KHz. The component values are obtained as R=300 Ω and C=1 nF for Leq=0.1 mH.  The value of Leq=0.1 mH is obtained using inductor topology of Figure 2(a) with component values of R1=R2=R3=R5=1 KΩ, R4=3 KΩ, and C1=300 pF. The frequency response of the filter simulated using PSPICE is depicted in Figure 5(b). Simulated value of lower cutoff frequency is obtained as 505 KHz which is in close agreement to the theoretical value of 503.2 KHz.


## 3.2. Band Pass Filter

The proposed inductor topologies may also be used to obtain band-pass response using the circuit given in Figure6(a). The transfer function for band-pass response is obtained as
(8)VoVin=s/CRs2+s/CR+1/LeqC,
where
(9)ω0=1√(LeqC),Q=R(CLeq).
The theoretical proposition is verified using the topology of Figure 2(b). A band-pass filter is designed having center frequency of 503.3 KHz. The component values are obtained as R=1 KΩ and C=1 nF for Leq=0.1 mH. The value of Leq=0.1 mH is obtained for inductor topology of Figure 2(b) with component values of R1=R2=R4=R5=1 KΩ,R3=3 KΩ, C1=300 pF. The frequency response of the filter simulated using PSPICE is depicted in Figure 6(b). The simulated results are in close agreement with the theoretical prediction.(a) Band-pass filter. (b) Frequency response of band-pass filter.
(a)(b)


## 3.3. LC Oscillator

An LC Oscillator can be realized using the proposed inductor topologies. Figure7(a) shows the schematic of an LC oscillator using topology of Figure 2(a) for which the condition of oscillation (CO) and frequency of oscillation (FO) are obtained as
(10)CO:G1+G3+G5=G1G5G4,(11)FO:fo=12ΠCC1R2R3R5/R4.
Figure 7(b) shows the output of the oscillator for Leq=0.1 mH and C=330 pF. Simulated frequency of oscillation is 860 KHz as against the calculated value of 876.2 KHz with % error of 1.85%.(a) LC oscillator. (b) Output of LC oscillator.
(a)(b)


## 3.4. Experimental Verification

The proposed inductor topologies are also tested experimentally to verify the theory. The OTRA is implemented using two CFOAs (IC AD844AN) as shown in Figure8 with a supply voltage of ±5 V.Figure 8
CFOA based OTRA implementation.The high pass filter of Figure5(a) is prototyped with C=1 nF, R=680 Ω, and Leq=1 mH. The inductor of value Leq=1 mH is implemented using inductor topology of Figure 2(a) with component values of R1=R2=R3=R5=1 KΩ, R4=3 KΩ, and C1=3 nF. Theoretical, simulated and experimental frequency responses are shown in Figure 9. It is observed that the experimental values more or less follow the theoretical and simulated values. The oscillator circuit of Figure 7(a) is also tested experimentally. The output waveform observed on oscilloscope is shown in Figure 10. The observed frequency of oscillation is found to be 872.5 KHz, which is in close agreement to theoretically calculated value of 876.2 KHz.Figure 9
Frequency response of the high-pass filter.Figure 10
Experimental output of LC oscillator.


## 4. Conclusion

Two new OTRA-based lossless grounded inductor topologies are presented. A high pass filter, band-pass filter, and an oscillator are realized using the proposed inductor topologies to illustrate their applications. PSpice simulation and experimental results are included to verify the theoretical propositions. It is found that the results obtained are in close agreement with the ideal values. Hence it is expected that the proposed inductors will provide an option to integrated circuit designer where lossless grounded inductor is required.



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*Source: 175130-2011-08-14.xml*