# Operational Transresistance Amplifier-Based Multiphase Sinusoidal Oscillators

Authors: Rajeshwari Pandey; Neeta Pandey; Mayank Bothra; Sajal K. Paul
Journal: Journal of Electrical and Computer Engineering
 (2011)

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## Abstract

Multiphase sinusoidal oscillator circuits are presented which utilize Operational Transresistance Amplifier (OTRA) as the active element. The first circuit producesn odd-phase oscillations of equal amplitudes and equally spaced in phase. The second circuit is capable of producing n odd- or even- phase oscillations equally spaced in phase. An alternative approach is discussed in the third circuit, which utilizes a single-phase tunable oscillator circuit which is used to inject signals into a phase shifter circuits. An automatic gain control (AGC) circuit has been implemented for the second and third circuit. The circuits are simple to realize and have a low component count. PSPICE simulations have been given to verify the theoretical analysis. The experimental outcome corroborates the theoretical propositions and simulated results.


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## Body

## 1. Introduction

Multiphase sinusoidal oscillators (MSOs) find extensive application in the field of power electronics and communications. In Communications MSO circuits are commonly used in single-sideband generators, phase modulators, and quadrature mixers.  They are also utilized for control of single-phase-to-three-phase PWM converters [1] and for a decoupled dynamic control of a six-phase two-motor drive system [2]. A number of circuits are reported in the literature relating to MSOs [3–17]. The MSOs implemented in [3–7] suffer from complex circuitry. Active R implementations in [8, 9] lack tenability, while the implementation in [10] using an OTA is tunable but has a limited output voltage swing. In [11, 12], MSOs based on Current Conveyor II (CC II) are presented. Their drawback is that they require a JFET and three additional current conveyors for each phase in order to achieve electronic tunability [13]. The structures proposed in [13, 14] utilize translinear CC, and those presented in [15, 16] are based on Current Differencing Transconductance Amplifier (CDTA). Though these circuits can operate at high frequencies and provide electronic tunability, they provide current outputs which need to be converted back to voltage for circuits requiring voltage inputs, which would considerably increase the component count. Moreover, the implementations utilizing parasitic resistors [13, 14] are not very accurate in producing the desired phase shift. The structure proposed in [17] using Current Feedback Operational Amplifier (CFOA) has a good output voltage swing and capable of producing high frequencies but requires an accessible compensation terminal of a CFOA. A major difficulty that is observed with a self-oscillating MSO is the stabilization of the signal amplitude. There are a number of techniques available for controlling amplitude of oscillations [18] for voltage mode op-amps. However, these techniques cannot be applied readily to current mode op-amps.It is well known that the key performance features of current mode technique are inherent wide bandwidth which is virtually independent of closed loop gain, greater linearity, and large dynamic range [19]. Recently Operational Transresistance Amplifier (OTRA) has emerged as an effective alternate analog building block which is a high gain current input, voltage output amplifier [20]. OTRA, being a current processing analog building block, inherits all the advantages of current mode technique and therefore is ideally suited for high-frequency applications [21]. It is also free from parasitic input capacitances and resistances as its input terminals are virtually grounded, and, hence, nonideality problem is less in circuits implemented with OTRA.This paper presents OTRA-based three MSO circuits, which are described in Section2. The first circuit utilizes n OTRAs to produce n odd-phase oscillations of equal amplitudes and equally spaced in phase. The second circuit utilizes (n+1) OTRAs to produce n odd- or even- phase oscillations equally spaced in phase. The third circuit utilizes a single-resistance-controlled (SRCO) sinusoidal oscillator circuit employing a single OTRA [22], whose output is subsequently used to drive a phase shifter network. The phase shifter circuit uses n OTRA-based phase shifter blocks to give a total of (n+1) oscillations. The circuit is tunable and has a low component count. An Automatic Gain Control circuitry (AGC) has also been implemented for the second and third circuits, which helps in the stabilization of the signal amplitude. Section 3 deals with nonideal analysis of the OTRA, while Section 4 gives the simulation and experimental results of the proposed MSO circuits. Conclusion is discussed in Section 5.


## 2. Circuit Description

The circuit symbol of OTRA is shown in Figure1, and port relationship can be given by(1)[V1V2VO]=[000000Rm-Rm0][I1I2IO].
Ideally the transresistance gain Rm approaches infinity [19, 20], and, hence, when used with negative feedback, currents at the two input terminals are forced to be equal. Both of the input terminals are virtually grounded. The output voltage is the difference of two input currents multiplied by transresistance gain (Rm). Inputs of OTRA are current signals while the output is a voltage signal, and, consequently, as both the input and output terminals have low impedances, OTRA is suitable for cascading.Figure 1
OTRA circuit symbol.


### 2.1. Circuit I

The first circuit is shown in Figure2 which produces n odd-phase oscillations. It is based on the scheme discussed in [23] and has been adapted for implementation with OTRA. The OTRAs have been connected in the inverting mode such that the gain G(s) of each block can be expressed as(2)G(s)=(-K1+sCR),
where R2=R4=⋯=R2n=R,R1=R3=⋯=R2n-1=Rx,K=(R/RX),andC1=C2=⋯=Cn=C.Figure 2
Generalized scheme for producingn odd-phase oscillation.From Figure2, the open loop gain L(s) can be expressed as:(3)L(s)=(-K1+sCR)n.
For oscillations to occur, the Barkhausen criterion [18] must be satisfied, hence(4)(-K1+sCR)n=1.
The above equation yields(5)(1+sCR)n+(-1)n+1Kn=0.
Equation (5) will converge only for odd values of n such that n≥3. Thus, the circuit will give rise to equally spaced oscillations having a phase difference of (360/n)°.Consider the case forn=3, then (5) reduces to(6)(1+jω0CR)3+K3=0.
Equating real and imaginary parts of (5) gives the frequency of oscillation (FO) and condition of oscillation (CO) as
(7)FO:f0=√3(2πRC),(8)CO:K=2.
Similarly, for n=5, (5) would reduce to (9)(1+jω0CR)5+K5=0.
Hence, FO and CO can be obtained as(10)FO:f0=0.727(2πRC),CO:K=1.236.
Thus, FO and CO for any value of n can be obtained. It is evident that CO and FO cannot be independently controlled for circuit I. However, the circuit is simple to realize and has a low component count. This circuit produces n odd-phase oscillations of equal amplitudes with a phase difference of (360/n)°.


### 2.2. Circuit II

The second circuit is shown in Figure3. This implementation is capable of producing n odd- or even- phase oscillations. It consists of n cascaded OTRA blocks, each block implementing a noninverting first-order low-pass function. An inverting voltage amplifier with a simple AGC circuit is connected in the feedback loop of the oscillator. The AGC circuit consists of resistors R2n+3,R2n+4 and two diodes D1 and D2.Figure 3
Generalized scheme for producingn odd- or even- phase oscillations with AGC.TakingR2=R4=⋯=R2n=R,R1=R3=R2n-1=Rx,K=(R/RX), and C1=C2=⋯=Cn=C, gain of each block and loop gain can be computed as  (11)G(s)=(K1+sCR),L(s)=(-K1+sCR)nKX,
where
(12)KX=R2n+2R2n+1.KX is effectively maintained at a value 1 with the help of the AGC circuitry. After applying the Barkhausen criterion, the characteristic equation is obtained as (13)(1+sCR)n+Kn=0.
The equation converges for all value of  n≥3, odd or even. As an example for n=4, (12) reduces to(14)(1+jω0CR)4+K4=0.
which gives FO and CO as(15)FO:f0=1(2πRC),CO:K=1.414.
For n=3, the characteristic equation, FO and CO would be the same as (6), (7),  and (8) respectively.InitiallyKX is kept at a value slightly higher than 1 so that the oscillations can begin; once the amplitude crosses a certain threshold, the diodes get switched on and bring down the value of resistance R2n+2, thus bringing down the effective value of KX. This is possible because the input terminals of the OTRA are virtually grounded. Thus, a dynamic equilibrium maintains the value of KX at 1. This circuit is not tunable as CO and FO are not independent. However, the circuit is versatile and can achieve both even- and odd- phase oscillations. The oscillations achieved are equally spaced in phase having a phase difference of (180/n)°.


### 2.3. Circuit III

The third circuit is shown in Figure4. It is based on an SRCO oscillator [22] to which an AGC circuit has been added which works on the same principle as in circuit II.Figure 4
SRCO-based MSO.The open loop gain of this circuit [22] is obtained as (16)L(s)=(sC2R2R3(1-sC1R1)R1+R3+sR1R3(C1+C2)).
Accordingly, the FO and CO for SRCO oscillator are obtained as(17)FO:f0=12πR1+R3C1C2R1R2R3,CO:R2R1=C1C2+1.
It can be observed that by controlling R3 the frequency can be controlled without affecting the CO, which makes the circuit tunable. AGC has been achieved by adjusting R2, which is used to control the loop gain, as seen from (16). At the output of the SRCO with AGC, n subsequent OTRA-based phase shifter blocks can be connected to produce n+1 oscillation. The phase shift produced by each phase shifter block can be given as(18)θ=tan-1⁡(ω0RC).
To add to the flexibility, if the OTRA in the phase shifter block is connected in inverting mode, the phase shift produced will be(19)θ=180°-tan-1⁡(ω0RC).
Hence, the phase shifter can be adjusted to obtain a phase shift of either 0°–90° or 90°–180° depending on its configuration. This circuit provides the flexibility of achieving the desired phase shift without connecting too many active elements.


## 2.1. Circuit I

The first circuit is shown in Figure2 which produces n odd-phase oscillations. It is based on the scheme discussed in [23] and has been adapted for implementation with OTRA. The OTRAs have been connected in the inverting mode such that the gain G(s) of each block can be expressed as(2)G(s)=(-K1+sCR),
where R2=R4=⋯=R2n=R,R1=R3=⋯=R2n-1=Rx,K=(R/RX),andC1=C2=⋯=Cn=C.Figure 2
Generalized scheme for producingn odd-phase oscillation.From Figure2, the open loop gain L(s) can be expressed as:(3)L(s)=(-K1+sCR)n.
For oscillations to occur, the Barkhausen criterion [18] must be satisfied, hence(4)(-K1+sCR)n=1.
The above equation yields(5)(1+sCR)n+(-1)n+1Kn=0.
Equation (5) will converge only for odd values of n such that n≥3. Thus, the circuit will give rise to equally spaced oscillations having a phase difference of (360/n)°.Consider the case forn=3, then (5) reduces to(6)(1+jω0CR)3+K3=0.
Equating real and imaginary parts of (5) gives the frequency of oscillation (FO) and condition of oscillation (CO) as
(7)FO:f0=√3(2πRC),(8)CO:K=2.
Similarly, for n=5, (5) would reduce to (9)(1+jω0CR)5+K5=0.
Hence, FO and CO can be obtained as(10)FO:f0=0.727(2πRC),CO:K=1.236.
Thus, FO and CO for any value of n can be obtained. It is evident that CO and FO cannot be independently controlled for circuit I. However, the circuit is simple to realize and has a low component count. This circuit produces n odd-phase oscillations of equal amplitudes with a phase difference of (360/n)°.


## 2.2. Circuit II

The second circuit is shown in Figure3. This implementation is capable of producing n odd- or even- phase oscillations. It consists of n cascaded OTRA blocks, each block implementing a noninverting first-order low-pass function. An inverting voltage amplifier with a simple AGC circuit is connected in the feedback loop of the oscillator. The AGC circuit consists of resistors R2n+3,R2n+4 and two diodes D1 and D2.Figure 3
Generalized scheme for producingn odd- or even- phase oscillations with AGC.TakingR2=R4=⋯=R2n=R,R1=R3=R2n-1=Rx,K=(R/RX), and C1=C2=⋯=Cn=C, gain of each block and loop gain can be computed as  (11)G(s)=(K1+sCR),L(s)=(-K1+sCR)nKX,
where
(12)KX=R2n+2R2n+1.KX is effectively maintained at a value 1 with the help of the AGC circuitry. After applying the Barkhausen criterion, the characteristic equation is obtained as (13)(1+sCR)n+Kn=0.
The equation converges for all value of  n≥3, odd or even. As an example for n=4, (12) reduces to(14)(1+jω0CR)4+K4=0.
which gives FO and CO as(15)FO:f0=1(2πRC),CO:K=1.414.
For n=3, the characteristic equation, FO and CO would be the same as (6), (7),  and (8) respectively.InitiallyKX is kept at a value slightly higher than 1 so that the oscillations can begin; once the amplitude crosses a certain threshold, the diodes get switched on and bring down the value of resistance R2n+2, thus bringing down the effective value of KX. This is possible because the input terminals of the OTRA are virtually grounded. Thus, a dynamic equilibrium maintains the value of KX at 1. This circuit is not tunable as CO and FO are not independent. However, the circuit is versatile and can achieve both even- and odd- phase oscillations. The oscillations achieved are equally spaced in phase having a phase difference of (180/n)°.


## 2.3. Circuit III

The third circuit is shown in Figure4. It is based on an SRCO oscillator [22] to which an AGC circuit has been added which works on the same principle as in circuit II.Figure 4
SRCO-based MSO.The open loop gain of this circuit [22] is obtained as (16)L(s)=(sC2R2R3(1-sC1R1)R1+R3+sR1R3(C1+C2)).
Accordingly, the FO and CO for SRCO oscillator are obtained as(17)FO:f0=12πR1+R3C1C2R1R2R3,CO:R2R1=C1C2+1.
It can be observed that by controlling R3 the frequency can be controlled without affecting the CO, which makes the circuit tunable. AGC has been achieved by adjusting R2, which is used to control the loop gain, as seen from (16). At the output of the SRCO with AGC, n subsequent OTRA-based phase shifter blocks can be connected to produce n+1 oscillation. The phase shift produced by each phase shifter block can be given as(18)θ=tan-1⁡(ω0RC).
To add to the flexibility, if the OTRA in the phase shifter block is connected in inverting mode, the phase shift produced will be(19)θ=180°-tan-1⁡(ω0RC).
Hence, the phase shifter can be adjusted to obtain a phase shift of either 0°–90° or 90°–180° depending on its configuration. This circuit provides the flexibility of achieving the desired phase shift without connecting too many active elements.


## 3. Nonideality Analysis

The nonidealities associated with OTRA-based circuits may be divided into two groups. The first group concerns with finite transresistance gain, whereas the second one results from practical implementation based on commercially available IC AD 844.


### 3.1. Nonideality due to Finite Transresistance Gain

Here the effect of finite transresistance gain on MSO is considered, and for high-frequency applications self-compensation is employed [20].  Ideally the transresistance gain Rm is assumed to approach infinity. However, practically Rm is a frequency-dependent finite value. Considering a single pole model for the transresistance gain, Rm can be expressed as (20)Rm(s)=(R01+s/ω0).
For high-frequency applications, the transresistance gain Rm(s) reduces to (21)Rm(s)≈(1sCp),whereCp=1R0ω0.
Taking this effect into account (2) modifies to(22)G(s)=(-K1+s(C+Cp)R).
The effect of Cp can be eliminated by preadjusting the value of designed capacitor and thus achieving self-compensation.


### 3.2. Nonideality Attributed to Realization of OTRA Using Commercially Available AD844 [24]

The OTRA can be realized using AD844 CFOA IC as shown in Figure5 [24]. Figure 6 [24] shows the equivalent circuit for nonideal analysis of the circuit presented in Figure 5. The CFOAs have been replaced with current conveyors having finite input resistances (RX) and finite resistance at its Z terminal (RZ). Ideally the input resistance at the X terminal is zero and is infinite at the Z terminal. For the AD844 CFOA the input resistance RX is around 50 Ω and RZ is around 3 MΩ [25]. From Figure 6 various currents can be calculated as follows: (23)IZ1=I+,IX2=ID-I-,ID=IZ1(RZRX+RZ),IZ2=IX2.RZ is the transimpedance of the OTRA. Ideally ID should be equal to IZ1, which can be approximated only if RZ is much greater than RX, which is true for AD844. Also the approximation that the input terminals are virtually grounded will be true only if the external resistance at the input terminal of the OTRA is much larger than RX.Figure 5
OTRA based on AD844.Figure 6
Equivalent circuit of OTRA constructed with AD844.If these two conditions are satisfied, the OTRA constructed with AD844 closely approximates an ideal OTRA. From (23) the output voltage VO, taking into account the above-mentioned approximations, can be calculated as(24)VO=(I+-I-)RZ.


## 3.1. Nonideality due to Finite Transresistance Gain

Here the effect of finite transresistance gain on MSO is considered, and for high-frequency applications self-compensation is employed [20].  Ideally the transresistance gain Rm is assumed to approach infinity. However, practically Rm is a frequency-dependent finite value. Considering a single pole model for the transresistance gain, Rm can be expressed as (20)Rm(s)=(R01+s/ω0).
For high-frequency applications, the transresistance gain Rm(s) reduces to (21)Rm(s)≈(1sCp),whereCp=1R0ω0.
Taking this effect into account (2) modifies to(22)G(s)=(-K1+s(C+Cp)R).
The effect of Cp can be eliminated by preadjusting the value of designed capacitor and thus achieving self-compensation.


## 3.2. Nonideality Attributed to Realization of OTRA Using Commercially Available AD844 [24]

The OTRA can be realized using AD844 CFOA IC as shown in Figure5 [24]. Figure 6 [24] shows the equivalent circuit for nonideal analysis of the circuit presented in Figure 5. The CFOAs have been replaced with current conveyors having finite input resistances (RX) and finite resistance at its Z terminal (RZ). Ideally the input resistance at the X terminal is zero and is infinite at the Z terminal. For the AD844 CFOA the input resistance RX is around 50 Ω and RZ is around 3 MΩ [25]. From Figure 6 various currents can be calculated as follows: (23)IZ1=I+,IX2=ID-I-,ID=IZ1(RZRX+RZ),IZ2=IX2.RZ is the transimpedance of the OTRA. Ideally ID should be equal to IZ1, which can be approximated only if RZ is much greater than RX, which is true for AD844. Also the approximation that the input terminals are virtually grounded will be true only if the external resistance at the input terminal of the OTRA is much larger than RX.Figure 5
OTRA based on AD844.Figure 6
Equivalent circuit of OTRA constructed with AD844.If these two conditions are satisfied, the OTRA constructed with AD844 closely approximates an ideal OTRA. From (23) the output voltage VO, taking into account the above-mentioned approximations, can be calculated as(24)VO=(I+-I-)RZ.


## 4. Simulation and Experimental Results

The proposed circuits have been simulated using PSPICE to validate the theoretical predictions. The OTRA is realized using IC AD 844 as shown in   Figure5. Figure 7(a) shows the simulation results of circuit I having n=3 and component values R1=R3=R5=0.5 kΩ, R2=R4=R6=1 kΩ, and C1=C2=C3=100 pF. The frequency of oscillations achieved was 2.838 MHz against the calculated value of 2.757 MHz having frequency error of 2.93%.  Figure 7(b) shows the simulated and theoretical frequency of oscillation as a function of capacitance (C). It shows that the simulated values deviate slightly from the ideal values at lower frequency range.Simulation result for circuit I.
(a)
Output waveform forn=3, f=2.838 MHz(b)
Frequency error curveFigure8(a) shows the simulation results of circuit II having n=4 and component values R1=R3=R5=R7=0.707 kΩ, R2=R4=R6=R8=R9=R10=1 kΩ, and C1=C2=C3=C4=100 pF. The simulated value achieved was 1.595 MHz against the theoretical value of 1.591 MHz with a frequency error of 0.25%. Figure 8(b) shows the simulated and theoretical frequency of oscillation as a function of capacitance (C).Simulation results for circuit II.
(a)
Output waveforms forn=4, f=1.546 MHz(b)
Frequency error curveFigure9(a) shows the simulation results of circuit III having two phase shifter blocks along with the single phase oscillator (SRCO). The design is obtained for an FO of 2.220 MHz with a phase shift of 45°. The component values chosen are R1=R3=Rp=1 kΩ, R2=2 kΩ, C1=C2=C=50 pF, and R=1.434 kΩ. The frequency observed was 2.220 MHz against the calculated value of 2.361 MHz.  The simulated and theoretical frequency of oscillation as a function of capacitance (C) is shown in Figure 9(b). It may be observed from Figures 7(b), 8(b), and 9(b) that the deviation between the simulated and theoretical FO is minimum in the case of circuit III.Simulation result for circuit III.
(a)
Output waveforms having 2 phase shifter networks of 45° each,f=1.401 MHz(b)
Frequency error curveThe functionality of the proposed MSO circuits is verified through  hardware also. The commercial IC AD844AN is used to implement an OTRA. Supply voltages used are ±5 V. Figure10 shows the experimental results for circuit II, having n=3,  for component values  R1=R3=R5=2.7 kΩ, R2=R4=R6=5.4 kΩ, R8=R9=R10=1 kΩ, and C1=C2=C3=3.3 nF. Observed FO is around 15.7 kHz and is in close agreement with calculated FO of 15.469 kHz. The little variation in experimental values of FO, as seen in Figure 10, from phase to phase may be due to tolerance of the component values.Figure 10
Experimental result for circuit II forn=3.The output of circuit III consisting of SRCO along with two phase shifter blocks is depicted in Figure11 for component values R1=Rp=2.7 kΩ, R2=5.4 kΩ, R3=250 Ω, R=10 kΩ, and C=C1=C2=3.3 nF. These component values result in theoretical FO as 43.4 kHz, and the observed frequency is around 44.72 kHz. The little variation in the experimental values of  FO from phase to phase may be due to tolerance of the component values.Figure 11
Experimental result for circuit III with two phase shifter blocks.


## 5. Conclusion

Three MSO circuits based on OTRAs have been presented. The circuits are versatile and have a wide range of frequency of oscillation. They provide advantages of current mode design techniques and at the same time provide voltage outputs. Hence, they are capable of replacing voltage mode op-amps-based MSO since they can provide higher frequencies of oscillation and are free from the drawbacks of conventional voltage mode op-amps. These circuits are very accurate in providing the desired phase shift without utilizing too many components. They provide high output voltage swing at frequencies in the MHz range. The workability of these circuits has been demonstrated through SPICE simulations and experimental results.



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