Multiphase sinusoidal oscillator circuits are presented which utilize Operational Transresistance Amplifier (OTRA) as the active element. The first circuit produces n 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.

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 Section 2. 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 Figure 1, and port relationship can be given byβŽ‘βŽ’βŽ’βŽ’βŽ£π‘‰1𝑉2π‘‰π‘‚βŽ€βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ’βŽ£π‘…000000π‘šβˆ’π‘…π‘š0⎀βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎣𝐼1𝐼2πΌπ‘‚βŽ€βŽ₯βŽ₯βŽ₯⎦.(1) Ideally the transresistance gain π‘…π‘š 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 (π‘…π‘š). 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.

2.1. Circuit I

The first circuit is shown in Figure 2 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 asξ‚€βˆ’πΎπΊ(s)=1+𝑠𝐢𝑅,(2) where 𝑅2=𝑅4=β‹―=𝑅2𝑛=𝑅,𝑅1=𝑅3=β‹―=𝑅2π‘›βˆ’1=𝑅π‘₯,𝐾=(𝑅/𝑅𝑋),and𝐢1=𝐢2=β‹―=𝐢𝑛=𝐢.

From Figure 2, the open loop gain 𝐿(𝑠) can be expressed as:ξ‚€βˆ’πΎπΏ(𝑠)=1+𝑠𝐢𝑅𝑛.(3) For oscillations to occur, the Barkhausen criterion [18] must be satisfied, henceξ‚€βˆ’πΎξ‚1+𝑠𝐢𝑅𝑛=1.(4) The above equation yields(1+𝑠𝐢𝑅)𝑛+(βˆ’1)𝑛+1𝐾𝑛=0.(5) Equation (5) will converge only for odd values of 𝑛 such that 𝑛β‰₯3. Thus, the circuit will give rise to equally spaced oscillations having a phase difference of (360/𝑛)Β°.

Consider the case for 𝑛=3, then (5) reduces toξ€·1+π‘—πœ”0𝐢𝑅3+𝐾3=0.(6) Equating real and imaginary parts of (5) gives the frequency of oscillation (FO) and condition of oscillation (CO) as FOβˆΆπ‘“0=√3,(2πœ‹π‘…πΆ)(7)CO∢𝐾=2.(8) Similarly, for 𝑛=5, (5) would reduce to ξ€·1+π‘—πœ”0𝐢𝑅5+𝐾5=0.(9) Hence, FO and CO can be obtained asFOβˆΆπ‘“0=0.727,(2πœ‹π‘…πΆ)CO∢𝐾=1.236.(10) 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)Β°.

2.2. Circuit II

The second circuit is shown in Figure 3. 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.

Taking 𝑅2=𝑅4=β‹―=𝑅2𝑛=𝑅,𝑅1=𝑅3=𝑅2π‘›βˆ’1=𝑅π‘₯,𝐾=(𝑅/𝑅𝑋), and 𝐢1=𝐢2=β‹―=𝐢𝑛=𝐢, gain of each block and loop gain can be computed as 𝐾𝐺(𝑠)=,ξ‚€βˆ’πΎ1+𝑠𝐢𝑅𝐿(𝑠)=1+𝑠𝐢𝑅𝑛𝐾𝑋,(11) where 𝐾𝑋=𝑅2𝑛+2𝑅2𝑛+1.(12)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 (1+𝑠𝐢𝑅)𝑛+𝐾𝑛=0.(13) The equation converges for all value of 𝑛β‰₯3, odd or even. As an example for 𝑛=4, (12) reduces toξ€·1+π‘—πœ”0𝐢𝑅4+𝐾4=0.(14) which gives FO and CO asFOβˆΆπ‘“0=1,(2πœ‹π‘…πΆ)CO∢𝐾=1.414.(15) For 𝑛=3, the characteristic equation, FO and CO would be the same as (6), (7),  and (8) respectively.

Initially 𝐾𝑋 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)Β°.

2.3. Circuit III

The third circuit is shown in Figure 4. 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.

The open loop gain of this circuit [22] is obtained as 𝐿(𝑠)=𝑠𝐢2𝑅2𝑅3ξ€·1βˆ’π‘ πΆ1𝑅1𝑅1+𝑅3+𝑠𝑅1𝑅3𝐢1+𝐢2ξ€Έξƒͺ.(16) Accordingly, the FO and CO for SRCO oscillator are obtained asFOβˆΆπ‘“0=1ξƒŽ2πœ‹π‘…1+𝑅3𝐢1𝐢2𝑅1𝑅2𝑅3,𝑅CO∢2𝑅1=𝐢1𝐢2+1.(17) 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 asπœƒ=tanβˆ’1ξ€·πœ”0𝑅𝐢.(18) To add to the flexibility, if the OTRA in the phase shifter block is connected in inverting mode, the phase shift produced will beπœƒ=180βˆ˜βˆ’tanβˆ’1ξ€·πœ”0𝑅𝐢.(19) Hence, the phase shifter can be adjusted to obtain a phase shift of either 0°–90Β° or 90°–180Β° 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 π‘…π‘š is assumed to approach infinity. However, practically π‘…π‘š is a frequency-dependent finite value. Considering a single pole model for the transresistance gain, π‘…π‘š can be expressed as π‘…π‘šξ‚΅π‘…(𝑠)=01+𝑠/πœ”0ξ‚Ά.(20) For high-frequency applications, the transresistance gain π‘…π‘š(𝑠) reduces to π‘…π‘šξ‚΅1(𝑠)β‰ˆπ‘ πΆπ‘ξ‚Ά,where𝐢𝑝=1𝑅0πœ”0.(21) Taking this effect into account (2) modifies to𝐺(𝑠)=βˆ’πΎξ€·1+𝑠𝐢+𝐢𝑝𝑅ξƒͺ.(22) The effect of 𝐢𝑝 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 Figure 5 [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: 𝐼𝑍1=𝐼+,𝐼𝑋2=πΌπ·βˆ’πΌβˆ’,𝐼𝐷=𝐼𝑍1𝑅𝑍𝑅𝑋+𝑅𝑍,𝐼𝑍2=𝐼𝑋2.(23)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.

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 as𝑉𝑂=𝐼+βˆ’πΌβˆ’ξ€Έπ‘…π‘.(24)

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 Figure 5. 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 𝐢1=𝐢2=𝐢3=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 (𝐢). It shows that the simulated values deviate slightly from the ideal values at lower frequency range.

Figure 8(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 𝐢1=𝐢2=𝐢3=𝐢4=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 (𝐢).

Figure 9(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 45Β°. The component values chosen are 𝑅1=𝑅3=𝑅𝑝=1 kΩ, 𝑅2=2 kΩ, 𝐢1=𝐢2=𝐢=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.

The 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 Β±5 V. Figure 10 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 𝐢1=𝐢2=𝐢3=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.

The output of circuit III consisting of SRCO along with two phase shifter blocks is depicted in Figure 11 for component values 𝑅1=𝑅𝑝=2.7 kΩ, 𝑅2=5.4 kΩ, 𝑅3=250 Ω, 𝑅=10 kΩ, and 𝐢=𝐢1=𝐢2=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.

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.