Full-Phase Operation Transresistance-Mode Precision Full-Wave Rectifier Designs Using Single Operational Transresistance Amplifier

This study proposed the designs of two full-phase operation transresistance-mode (TRM) precision full-wave rectifiers. The first circuit consisted of a single operational transresistance amplifier (OTRA), four diodes, and a resistor. The second scheme was an OTRA combined with a full metal-oxide semiconductor field-effect transistor-based design, which is preferable for integrated circuit implementation because no passive components are used in the circuit topology. Based on our literature review, this is the first study that discussed a full-phase operation transresistance-mode precision full-wave rectifier consisting of a single OTRA and few passive components. In this paper, several previously reported precision full-wave rectifiers consisting of various active devices are first reviewed followed by the proposed OTRA-based transresistance-mode precision full-wave rectifiers and an analysis of nonideal effects. Furthermore, computer simulations and experimental results are presented to verify the validity of the proposed circuits, which were consistent with the theoretical predictions.


Introduction
Rectification circuits have numerous electronic and electrical applications, including communication systems, power conversion circuits, instrument and measurement equipment, and sensor interfaces.Typically, passive-type rectification circuits can be easily constructed using diodes.However, the passive-type rectifiers cannot function accurately when the incoming signal has an amplitude lower than the threshold voltage of the diodes.To process low-level signals for rectification, precision rectifiers (also called active-type rectifiers) must be used.Conventional precision rectifiers were constructed using operational amplifiers (OAs) [1].However, OA-based precision rectifiers operate efficiently only at low frequencies.Thus, the realization of high-performance and versatile precision rectifiers is one of the crucial research topics in the current analog signal processing circuit design research area.According to our review of relevant literature, several previous studies have explored the realization of precision full-wave rectifiers by employing second-generation current conveyors (CCIIs) [2][3][4][5].These contributions have sparked great interest among many researchers to explore various high-performance precision rectifiers until now.In addition to these early reports [2][3][4][5], another possible design for implementing a voltage-mode (VM) CCII-based fullwave rectifier was reported [6].In 2006, the CCII-based resistorless full-wave rectifier was introduced by employing two CCIIs and three n-channel metal-oxide semiconductor field-effect transistors (MOSFETs) [7].To reduce the number of active devices used, an improved version based on a dual-X CCII (DXCCII) and three n-channel MOSFETs was subsequently presented [8].In addition to CCII-based designs, previous studies have focused on using operational transconductance amplifiers (OTAs) to create precision rectifiers.In 1989, a possible design for implementing a voltage-mode (VM) OTA-based precision half-wave rectifier can be found in [9].A VM full-wave rectifier composed of one fully differential input and output operational transconductance amplifier (FDIO-OTA), four diodes, and a resistor was reported in [10].Other feasible circuit topologies for realizing OTAbased full-wave rectifiers were presented in [11].Recently, an OTA-based current-mode (CM) full-wave rectifier was demonstrated [12].The main advantage of these OTA-based solutions [10][11][12] is that the amplitude of the rectified output [17] 4 signal can be tuned electronically by adjusting the bias current of the OTA.In the past several years, current differencing transconductance amplifier (CDTA)-based precision rectifiers have received a considerable amount of attention.
The proposed circuits are characterized by the following features: (1) a tunable rectified output signal amplitude with dual-phase rectification output signal property in a singlecircuit scheme; (2) a resistorless design (the second proposed topology), which has high potential in the integrated circuit manufacturing process; and (3) low-impedance output, which facilitates the use of cascading applications without the need for supplementary buffer circuits.The remainder of this paper is organized as follows.Section 2 presents the functional block and implementation of an OTRA followed by the operation principle of the proposed OTRA-based fullphase operation transresistance-mode precision full-wave rectifiers.Section 3 presents an analysis of the nonideal effects.Section 4 presents the computer simulations and experimental results to demonstrate the effectiveness of the proposed circuits.A conclusion is included in Section 5 to summarize this study.

Proposed Full-Phase Operation Transresistance-Mode Precision Full-Wave Rectifiers
An OTRA is a high-gain transimpedance-type active building block, which was first introduced in 1992 [24]. Figure 1 presents the circuit symbol of an OTRA, which is a threeterminal active device with two internal grounded current input terminals (I + and I − ) and a low-impedance voltage output terminal (V o ).The terminal relations of an OTRA are described in (1), which shows that the output voltage is produced by an inherent transresistance gain (R m ) multiplied by the difference of the input currents (I + and I − ).The transresistance gain (R m ) approaches infinity under an ideal scenario.
[ [ Figure 2(a) presents a CMOS transistor-level implementation of the OTRA [24].In the circuit (Figure 2(a)), (M 1 -M 16 ) simulates a current differencing circuit to generate current as the difference between I + and I − currents.To generate the output voltage (V o ), which is a function of the transresistance gain (R m ) with respect to the difference between the I + and I − currents, a voltage buffer (M 17 -M 22 ) was applied.In addition to the CMOS realization presented in Figure 2(a), an OTRA can be easily implemented using commercially available integrated circuits (AD844s) in a configuration depicted in Figure 2(b) [25].The circuit presented in Figure 2(b) can provide a viable method to fulfill the requirements of the experiments.
Figure 3 presents the circuit diagram and corresponding input and output transfer characteristics of the proposed fullphase operation transresistance-mode precision full-wave rectifier.The circuit is composed of a single OTRA, four diodes, and a resistor.Two diverse operating modes of the circuit (Figure 3(b)) are described as follows.In the circuit mode 1, the switch (S) must be arranged in a manner that a is connected to b.In the positive input current cycle (I in > 0), diode D 3 is activated and diode D 4 is inactivated, thus causing the circuit to produce a positive-polarity rectified output signal at V o .In the negative input current cycle (I in < 0), diode D 3 is inactivated and diode D 4 is activated, resulting in a negative-polarity rectified output signal at V o .Consequently, a positive full-wave rectification output can be obtained when the circuit is operated in mode 1.In a negative full-wave rectification output (mode 2) arrangement, the switch (S) must be arranged in a manner that a is connected to c.In the positive input current cycle (I in > 0), D 1 is activated and D 2 is inactivated, resulting in a negative-polarity rectified output signal at V o .Conversely, in the negative input current cycle (I in < 0), D 1 is inactivated and D 2 is activated, resulting in a positive-polarity rectified output signal at V o .Thus, a negative full-wave rectification output is obtainable through mode 2. A routine circuit analysis yielded the formulas for the input and output in two diverse full-wave rectification modes, which are expressed in ( 2) and (3), respectively.Equations ( 2) and ( 3) indicate that the circuit (Figure 3) performs the fullphase operation full-wave rectification function in a singlecircuit scheme, and the amplitude of the rectified output signal can be tuned using the resistor R.
The circuit presented in Figure 4 is an extended design of the proposed circuit (Figure 3), which is a full MOSFETbased circuit topology.In Figure 4, the MOS transistors (M A and M B ) function as active resistors, which are used to replace the resistor R in the circuit (Figure 3).The formula for the active resistor (M A and M B ) is defined in (4) [26].In (4),  n is the effective channel electronic mobility, C ox is the gateoxide capacitance per unit area, and W and L are the channel width and length, respectively, of M A and M B .Furthermore, in Figure 4, the MOS transistors (M 1 -M 4 ) function as diodes (D 1 -D 4 ) [9].After applying (4) into ( 2) and (3), the formulas for the input and output in two diverse rectification modes are expressed in (5) and (6).The circuit (Figure 4) possesses the following features: (1) it is a resistorless scheme with a full MOSFET-based design and thus has high potential in the integration circuit manufacturing process; (2) it yields a low-output impedance at output V o , which enables cascading application without supplementary buffer circuits; (3) a fullphase operation full-wave rectification function is achievable in a single-circuit scheme by arranging the switch (S) to connect at different terminals; and (4) the amplitude of the rectified output signal can be adjusted by tuning the bias voltages (V a and V b ).Because the proposed rectifiers feature compact circuit topologies and this paper is the first report to present full-phase operation transresistance-mode precision full-wave rectifiers by using OTRAs, the proposed circuits can find many applications in development of industry circuit systems like high-performance power supplies, RF demodulators, AC voltmeters, and watt meters.

Nonideality Effects and Analyses
This section investigates the influences of the nonideal effects on the proposed circuits.Figure 5 presents a sophisticated equivalent circuit model of the applied OTRA (Figure 2(b)) [27].According to the datasheet of AD844 [28], a practical AD844 can be modeled as a positive current conveyor (CC) cascading a voltage buffer (VBF) with parasitic resistances at terminals.In Figure 5, the resistances R x and R z represent the finite series and parallel parasitic resistances at x and z terminals of AD844.The series parasitic resistance R x at x terminal is in the order of several tens of ohms, whereas the parallel parasitic input R z is in the order of a few megaohms.Typical values of these parasitic resistances are R x = 50 Ω and R z = 3 MΩ [28].However, in the ideal case, R x is zero and R z is treated as infinite.By applying the equivalent circuit model Mode 1 (S: Mode 2 (S: (Figure 5) to the circuits (Figures 3 and 4) and repeating the analysis steps presented in Section 2, (2), ( 3), (5), and ( 6) can be modified to ( 7), ( 8), (9), and (10).Equations ( 7), ( 8),  slightly affected the input and output transfer characteristics of the circuits (Figures 3 and 4) to deviate from the analyzed results of the ideal scenario.The following conditions R ≫ R x , R z ≫ R x , and R z ≫ 1 Ω must be satisfied in the design procedure to minimize the influence of these parasitic effects on the circuits.3).V o (mode 1) V o (mode 1)

Computer Simulations and Experimental Results
In this section, several simulations and experimental examples were conducted to verify the validity of the proposed circuits.The circuits (Figures 3 and 4) were simulated using HSPICE program based on the CMOS implementation of the OTRA (Figure 2(a)) by using the TSMC 0.25 m CMOS process parameters.The aspect ratio of all MOS transistors of the OTRA (Figure 2(a)) and the MOS diodes (M 1 -M 4 ; Figure 4) is W/L = 22 m/2.2m.For the circuit (Figure 3), the diodes (D 1 -D 4 ) used in simulation tests were constructed based on 1N914 diode model [29].The supply voltages of the circuit (Figure 2(a)) used were V DD = −V SS = 2.5 V.In the first example, a sinusoidal signal with an amplitude of 100 A and frequency of 100 kHz was applied to the input current (I in ) of the circuits (Figures 3 and 4).For the circuit (Figure 4), the resistor R = 1 kΩ was determined on the basis of (4) by setting the following circuit parameters: (W/L) MA,MB = 10 m/2 m, V a = 1.76 V, and V b = 0.5 V. Figures 6 and  7 display the simulation results of the time waveforms and the corresponding input and output transfer characteristics of the circuit under two diverse rectification output modes.The total power consumption was 2.31 mW.The simulation results were consistent with the theoretical predictions and confirmed that the circuits (Figures 3 and 4) performed a fullphase operation transresistance-mode full-wave rectification function in a single-circuit scheme.
The input and output transfer characteristics of the circuits (Figures 3 and 4) in the wide sweeping range of the input signal are displayed in Figure 8. Simulation results, obtained at a frequency of 1 MHz and by applying an input signal with amplitude of 100 A, revealed that the circuits maintained a high degree of linearity in amplitude for the input current signal ranging from 0 to 250 A.In Figure 8(a), the offset at zero crossing is determined as 586.34 V, whereas in Figure 8(b), the offset at zero crossing is 577.27 V.
To determine the minimal applicable amplitude of the input signal for the circuit (Figure 4), the circuit was arranged at frequencies of 1 Hz, 100 kHz, and 1 MHz and input current signals with diverse amplitudes of 80 and 50 A were applied.The simulation results are presented in Figure 9 and indicated that the circuit reduced the amplitude of the rectified output to a low input signal amplitude.However, in the simulation tests, the circuit did not operate favorably when the amplitude of the input signal was lower than 10 A.
To determine the applicable operating frequency range of the proposed precision rectifiers, the circuit (Figure 4) was used.The simulation results (Figure 10) obtained by applying an input current signal I in = 100 A and frequencies of 1 Hz, 1 MHz, and 5 MHz revealed that when the frequency of the input current signal was increased to 5 MHz, a rectified output with distortion was obtained from the circuit.The simulation results confirmed that the highest operating frequency of the circuit is limited at 5 MHz and thus the circuit can process an input signal at frequencies ranging         ) ＣＨ = 100 A from several hertz to several megahertz.It should be noted that the highest applicable operating frequency limited of the proposed rectifiers depends on the distortion at zero crossing owing to the maximum slew-rate of the output voltage for the active device and turn-on and turn-off transition of diodes and MOSFETs.However, a high slew-rate OTRA and high-speed diodes can be used to extend a higher operating frequency.
With regard to the temperature stabilities of the proposed rectifiers, the circuit (Figure 3) was used.Simulation tests were performed at frequency of 1 MHz and input current signal with amplitude of 100 A at different temperatures over a 25 ∘ C to 75 ∘ C range.Simulation results were recorded in Figure 11 and showed that the output amplitude deviations between the theoretical values and the simulated results are below 1%.In addition to the circuit simulations, a prototype of the circuit (Figure 3) was implemented to conduct experiments using the AD844 integrated circuits (Figure 2(b)) and discrete passive components.The complete hardware implementation of the circuit (Figure 3) for conducting the experiments is presented in Figure 12.In Figure 12, the first AD844 combined with the grounded resistor R in functions as a voltage-to-current converter [30], which was used to produce an input current signal (I in ) as shown in Figure 3. From Figure 12 and according to the terminal behaviors of an AD844 [28], the relationship between I in and V in can be determined as I in = V in /R in .All experiments were performed at supply voltages of ±3.5 V.In the experiment, 1N914 diodes (D 1 -D 4 ) were adopted and the switch (S) was implemented by utilizing commercially available single-pole double-throw switch.Several experiments were conducted as follows.
To determine whether the circuit (Figure 3) performed full-phase operation full-wave rectification in a single-circuit scheme, a sinusoidal signal with an amplitude and a frequency of 100 A and 10 kHz, respectively, was applied to the input I in of the circuit (Figure 3).To obtain an amplitude of the input current signal (I in ) of 100 A, a voltage sinusoidal signal (V in ) with an amplitude of 1 V and R in = 10 kΩ was applied (Figure 12). Figure 13 presents the experimental results of the circuit (Figure 3) that was operated in two diverse full-wave rectified output modes.In this experiment, the value of the resistor R was selected as 1 kΩ.To investigate the minimal applicable amplitude of the input signal for the circuit (Figure 3), the following experiments were conducted.First, the resistors R = 1 kΩ and R in = 10 kΩ were selected.Next, the input current signal I in with a frequency of 10 kHz at distinct amplitudes of 80 A (V in = 0.8 V) and 50 A (V in = 0.5 V) was applied to record the rectified output waveforms.Figure 14 presents the time waveforms at various amplitudes.However, the circuit did not operate favorably when the amplitude of the input current signal (I in ) was lower than 30 A in the experimental tests.These experimental results indicated that the minimal applicable amplitude of the input signal for the circuit (Figure 3) was limited to several tens of micro amperes based on the OTRA (Figure 2(b)) constructed using the commercially integrated circuits (AD844s) on a breadboard.One feature of the proposed circuits (Figures 3  and 4) is that the resistor R can be used to adjust the amplitude of the rectified output waveform.Two test results with the rectified output waveforms of V o = 50 mV (R = 0.5 kΩ) and V o = 200 mV (R = 2 kΩ) at a peak sinusoidal signal and frequency of I in = 100 A and 10 kHz, respectively, are shown in Figures 15 and 16.
For the circuit (Figure 4), the resistor R was determined on the basis of (4) by setting the following parameters: R = 0.5 kΩ (V a = 3.1 V and V b = 0.5 V) and R = 2 kΩ (V a = 1.2 V and V b = 0.5 V).Simulation and experimental results   indicated that the amplitude of the rectified output signal was tunable by varying the resistor R. The maximum slew-rate of the AD844 and the highest applicable operating frequency of the diodes constitute the main obstacle to obtaining the highest applicable frequency in the circuit (Figure 12).The highest applicable operating frequency of the prototype circuit (Figure 12) was limited at approximately 200 kHz in the experimental tests.For comparison, the limited highspeed application of other full-wave rectifiers in Table 1 was also given.The simulation and experimental results supported the theoretical analyses, confirming the feasibility of the proposed circuits.

Conclusions
This study presented two designs of OTRA-based fullphase operation transresistance-mode precision full-wave rectifiers.Unlike the previously OTRA-based voltagemode precision rectifiers, the proposed designs belong to transresistance-mode rectification solutions.Computer simulations and experimental results were presented to validate the efficiency of the circuits, which were consistent with the theoretical analyses.Thus, this study sparks a beginning for research into transresistance-mode precision rectifier design and can provide new possibilities for OTRA device application in current analog signal processing systems.

Figure 3 :Figure 4 :
Figure 3: The proposed full-phase operation transresistance-mode precision full-wave rectifier: (a) circuit diagram and (b) input and output transfer characteristics.
, and(10) indicate that the parasitic resistances R x and R z

Figure 5 :
Figure 5: Equivalent circuit model of the OTRA (Figure 2(b)) with parasitic effects.

Figure 6 :
Figure 6: Simulation results of the input and rectified output waveforms and the corresponding input and output transfer characteristics of the circuit (Figure 3).

Figure 7 :
Figure 7: Simulation results of the input and rectified output waveforms and the corresponding input and output transfer characteristics of the circuit (Figure 4).

Figure 8 :
Figure 8: Simulation results of the input and output transfer characteristics in a wide scope of the (a) circuit in Figure 3 and (b) circuit in Figure 4.

Figure 9 :
Figure 9: Simulation results of the input and rectified output waveforms of the circuit (Figure 4) at frequencies of (a) 1 Hz, (b) 100 kHz, and (c) 1 MHz for various input signal amplitudes.

Figure 12 :
Figure 12: Complete hardware implementation of the circuit (Figure 3) for conducting the experiments.

Figure 13 :
Figure 13: Experimental results of the input and rectified output waveforms and the corresponding input and output transfer characteristics of the circuit (Figure 3).

Figure 14 :
Figure 14: Experimental results of the input and rectified output waveforms of the circuit (Figure 3) for various input signal amplitudes.

Figure 15 :
Figure15: Simulation results of the tunable rectified output amplitude for the circuit (Figure4).

Figure 16 :
Figure16: Experimental results of the tunable rectified output amplitude for the circuit (Figure3).