Received 1 February 2016; accepted 19 March 2016; published 23 March 2016
Operational Transresistance Amplifier (OTRA) has received a lot of attention over the past decades. Due to being a differential current controlled voltage source, it has a virtual ground at both the input terminals, which provides the significant advantages of eliminating parasitics at the input ports. Thus, motivated by these advantages, several CMOS OTRA  -  architectures have been proposed in the literature and simultaneously, a large variety of applications of OTRAs have been evolved so far, such as integrators  , immittance simulators  -  , oscillators  -  , square/triangular waveform generators  , monostable/bi-stable multivibrators  , first and second order all pass filters and biquad filters  -  etc.
Although the non-ideal one pole and two pole models  of transresistance gain (Rm) have been employed by several authors in performing a non-ideal analysis of their propositions but to the best of the knowledge of the authors, any explicit use of OTRA-pole in evolving external capacitor-less, active-R circuits using OTRAs has not been reported so far. The main purpose of this paper is, therefore, to report three OTRA-based active-R circuits which realize an inductor, an oscillator and a low-pas/band-pass filter respectively in which the pole of the OTRA has been exploited advantageously to result in circuits which have the interesting feature of employing a low number of total components. The workability of the proposed circuits has been demonstrated by SPICE simulation results based upon CMOS-OTRA implementation using 0.5 µm CMOS technology.
2. The Proposed Circuits
The OTRA can be symbolically shown as in Figure 1, and is characterized by the terminal equation
The two-pole model of the transresistance  of the OTRA is given by
The above expression for Rm(s) can be further modified and written as-
For frequencies lying in the range:, Equation (2) can be approximated as
; where (4)
Ro is the dc resistance of the OTRA.
2.1. Analysis of the Proposed Circuits Using One-Pole Model of an OTRA
Consider now the following:
Figure 1. Symbolic representation of an OTRA.
1) For the circuit of Figure 2 straight forward analysis using given by Equation (4) shows that the input admittance of the circuit is given respectively by
The quality factor of the simulated inductor is found using the expression
2) In Figure 4 we show the new proposed single-resistance-controlled-oscillator (SRCO) circuit.
The straight forward analysis of the proposed circuit using Equation (4) gives the characteristic equation (CE) of the circuit as
Figure 2. Simulated inductor.
Figure 3. Equivalent circuit of simulated inductor.
Figure 4. Fully uncoupled SRCO.
It can be easily verified from the CE given by Equation (7) that the condition of oscillation (C.O.) and the frequency of oscillation (F.O.) are respectively given by
It may be noted that the circuit offers fully uncoupled CO and FO, the former is adjusted by R1 and/or R2 and the latter may be varied by R3 and/or R4. It may be mentioned that any fully uncoupled oscillator using only four resistors along with only two active elements, that too without using any external capacitors, has not been reported in the literature earlier.
3) The last proposition is of an active-R OTRA-based low-pass/band pass filter which has been obtained from the circuit of Figure 4 by changing the polarities of the OTRA terminals and is shown in Figure 5. A straight forward analysis of this circuit using Equation (4) reveals that its transfer functions are given by
where the filter parameters are given by
A novel feature of the proposed band-pass filter is that all the three filter parameters can be found to be electronically tunable  -  and can be controlled independently through various resistors: ωo by R3 while the
Figure 5. Low-pass/band-pass filter.
bandwidth (BW) by R1 and finally, the gain by R2.
2.2. Analysis of the Proposed Circuits Using Two-Pole Model of an OTRA
1) For the circuit of Figure 2 straight forward analysis using given by Equation (3) shows that the input admittance of the circuit is given by
Thus, from Equation (15) the circuit of Figure 2 can be considered as a realization consisting of a resistance (Rp) in parallel with a series combination of a frequency dependent negative capacitance (FDNC), an inductor
and a resistance (Rs) such that, , and. The
equivalent circuit of which is shown in Figure 6.
The quality factor of the simulated inductor considering all the parasitics is found to be using the expression (16)
2) By a straight forward analysis, using Equation (3) we obtain the characteristic equation (CE) of the circuit shown in Figure 4 to be
Neglecting the term s4 and the coefficients involving the term the CE given by Equation (17) can be
Figure 6. Equivalent circuit of simulated inductor.
simplified and rewritten as
Comparing the above equation with the standard form given as
we obtain the following values for the coefficients of the Equation (19) as
For the CE of the form given by (19) the expression for the frequency of oscillation (F.O.) and the condition of oscillation (C.O.) are given respectively as
F.O.: and C.O.: (21)
Thus, for the CE given by (13), the expressions for F.O. and C.O. are found to be
where fo is as given by Equation (9).
The percentage error in the frequency of oscillation can be found using the expression for percentage error =
and the approximate error function is given as
3) A straight forward analysis of the circuit shown in Figure 5 using Equation (3) reveals that its transfer function is given by
Equations (25) and (26) on neglecting the terms containing s4 and s3 can be simplified and the modified transfer functions are thus obtained as follows
From Equations (27) and (28) we obtain the expressions for the filter parameters as
where fo, BW, HoBP and HoLP are given respectively by Equations (12), (13) and (14).
3. SPICE Simulation Results
All the three proposed circuits have been verified through SPICE version 16.0 simulations using CMOS OTRA of  reproduced herein Figure 7 where the aspect ratios of MOSFETs are as given in Table 1 and the model parameters using 0.5 µm CMOS technology provided by MOSIS (AGILENT) are as given in Table 2.
Figure 8 and Figure 9 show the variation of Rp and Lp respectively as compared to their theoretical plots. The proposed inductor was simulated using the component values as R1 = 2.2 KΩ, R2 = 3.3 KΩ and Cp = 1.2 pF resulting in the theoretical value of Rp = 1.32 KΩ and Lp = 8.71 µH which are in close agreement with the simulated values of Rp = 1.34 KΩ and Lp = 8 µH.
The simulated lossy inductor of Figure 2 can be used to realize a band pass filter circuit as shown in Figure 10. A straight forward analysis of the circuit results in the following expression of the transfer function of the circuit.
Figure 7. An exemplary CMOS Implementation of an OTRA  .
Figure 8. Variation of Rp with respect to frequency.
Figure 9. Variation of Leq with respect to frequency.
Table 1. Aspect ratios of the various MOSFETs for the circuit of Figure 7  .
Table 2. Model parameters of NMOS and PMOS transistors.
For the band pass filter shown in Figure 10 we obtain the following filter parameters:
Considering the equivalent circuit of Figure 6 and using it to realize a band pass filter circuit we obtain the transfer function given by
The above equation can be simplified and written as
From Equation (33) we obtain the modified filter parameters as
Figure 10. Simulated inductor used to realize aBand Pass filter.
where, Ho, and fo are respectively given by Equation (31).
Figure 11 shows the simulated result of the band pass filter circuit shown in Figure 10 (an application example of simulated inductor). The circuit was simulated with the choice of components as follows: Ro = Rp = 1.32 KΩ, Lp = 8.712 µH, Co = 4.65 pF resulting in theoretical value of filter parameters as-Ho = 0.5, BW = 326 MHz and fo = 25 MHz as compared to the simulated values Ho = 0.499, BW = 325.7 MHz and fo = 25.16 MHz.
Figure 12 shows the transient response of the proposed oscillator circuit shown in Figure 4 with the component values as R1 = 25.01 KΩ, R2 = 25 KΩ, R3 = R4 = 30 KΩ and Cp = 1.2 pF resulting in theoretical value of oscillation frequency fo = 4.42 MHz which agrees with the simulated result of fo = 4.45 MHz.
For the component values chosen as R1 = 25.01 KΩ, R2 = 25 KΩ, R3 = R4 = 30 KΩ, Cp = 1.2 pF, Ro = 126 MΩ and, using Equations (9) and (22) respectively the oscillation frequency is found to be fo = 4.42 MHz using one-pole model and fo = 4.485 MHz using two-pole model resulting in the percentage error of 1.47%.
Figure 15 shows the SPICE generated frequency response of the band-pass filter with component values chosen as R2 = R3 = R4 = 47 KΩ, R1 = 300 KΩ and Cp = 1.2 pF resulting in theoretical values of fo = 2.82 MHz and BW = 2.78 MHZ which is in accordance with the simulated values of fo = 2.8363 MHz, BW = 2.7554 MHz.
Figure 11. Simulation result of Band pass filter.
Figure 12. Transient response of the proposed SRCO.
Figure 13. Variation of the oscillation frequency with R3.
Figure 14. Variation of the oscillation frequency with R4.
Figure 18 shows the variation of gain of band pass filter with resistance R2 when its value is varied from 280 Ω to 320 KΩ and Figure 19 shows the variation of gain of low pass filter with resistance R3 when its value is varied from 40 KΩ to 80 KΩ. Thus, Figure 18 and Figure 19 respectively show the variability of gain for band-pass and low-pass with respect to R2 and R4.
Figure 15. SPICE generated frequency response of low-pass/band-pass filters.
Figure 16. Effect of temperatureon frequency response of band-pass filter.
Figure 17. Effect of temperature on frequency response of low-pass filter.
Figure 20 and Figure 21 respectively shows the simulation results of the band pass and the low pass response of the circuit shown in Figure 5 considering the one-pole , two-pole model of the OTRA and the SPICE result.
It can be seen that the results obtained from SPICE simulations demonstrate good correspondence with the theoretical values, which confirms the workability of the proposed circuits.
Figure 18. Variation of gain with R2 for the band-pass filter.
Figure 19. Variation of gain with R4 for the low-pass filter.
Figure 20. Simulation result of band-pass filter.
Figure 21. Simulation result of low-pass filter.
4. Concluding Remarks
Three simple circuits using OTRA-pole were proposed: a simulated inductance, a fully uncoupled SRCO and a low-pass/band-pass filter. In these circuits the pole of the OTRA has been exploited advantageously to result in circuits which have the interesting feature of employing a low number of total components. The theory has been validated by SPICE simulation results. The workability of the proposed circuits, thus, demonstrates that the design of active-R circuits using OTRA-pole warrants further investigations.
This work was performed partly at Analog Signal Processing Research Lab., Division of ECE, NSIT, New Delhi and partly at Advanced Analog Signal Processing Lab., Department of ECE, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi.