# Current-Mode Third-Order Quadrature Oscillator Using CDTAs

Authors: Jiun-Wei Horng
Journal: Active and Passive Electronic Components
 (2009)

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## Abstract

This paper describes a current-mode third-order quadrature oscillator based on current differencing transconductance amplifiers (CDTAs). Outputs of two current-mode sinusoids with90° phase difference are available in the quadrature oscillator circuit. The oscillation condition and oscillation frequency are orthogonal controllable. The proposed circuit employs only grounded capacitors and is ideal for integration. Simulation results are included to confirm the theoretical analysis.


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## Body

## 1. Introduction

Various new current-mode active building blocks have received considerable attentions owing to their larger dynamic range and wider bandwidth with respect to operational amplifier-based circuits. As a result, current-mode active components have been increasingly used to realize active filters, sinusoidal oscillators, and immittances.Quadrature oscillator is used because the circuit provides two sinusoids with 90° phase difference, as, for example, in telecommunications for quadrature mixers and single-sideband generators or for measurement purposes in vector generators or selective voltmeters. Therefore, quadrature oscillators constitute an important unit in many communication and instrumentation systems [1–17]. Two-integrator loop technique was developed to realize quadrature oscillators by using operational amplifiers or transconductance elements in [1, 2]. Holzel [3] proposed a method for realizing quadrature oscillator consists of two allpass filters and an inverter using operational amplifiers. Keskin et al. [15, 17] proposed two quadrature oscillators that were designed out by the method in [3] using current differencing buffered amplifiers (CDBAs) or current differencing transconductance amplifiers (CDTAs). Ahmed et al. [4] proposed two quadrature oscillator circuits that were realized base on the allpass filters and the noninverting integrators as building blocks using operational transconductance amplifiers (OTAs). This method was also used in [16] to obtain a quadrature oscillator using CDBAs. Soliman [6] describes several quadrature oscillator circuits based on the modification of two-integrator loop technique using current conveyors. Because the high-order network has high accuracy and high-quality factor, it gives good frequency response with low distortion [9–12]. Prommee and Dejhan [9] proposed two third-order quadrature oscillators using OTAs. Horng et al. [10, 11] proposed four voltage-mode third-order quadrature oscillator circuits; each circuit uses three second-generation current conveyors (CCIIs). Maheshwari and Khan [12] proposed a current-mode third-order quadrature oscillator using four CCIIs.In 2003, a new current-mode active element that is called current differencing transconductance amplifier (CDTA) was introduced [18]. Owing to the current conveying property, the CDTA is one of the modifications of the current conveyor (CC). Many applications in the design of active filter [19] and multiphase sinusoidal oscillator [20] using CDTAs as active elements have received considerable attention. A second-order current-mode quadrature oscillator consists of two CDTAs, four resistors, and two capacitors was presented in [17]. However, the capacitors used in this circuit are connected to the input terminals of the CDTAs. Since the input terminals of CDTA have parasitic resistances [20], this quadrature oscillator is not ideal for high-frequency applications. In 2006, Biolek et al. proposed a second-order current-mode quadrature oscillator based on two-integrator loop technique [21]. The main disadvantage of this oscillator is that there is no control on the condition of oscillation.In this paper, a CDTAs-based current-mode third-order quadrature oscillator circuit is presented. The oscillation condition and oscillation frequency of the proposed quadrature oscillator are orthogonal controllable. The proposed quadrature oscillator uses only grounded capacitors. The use of only grounded capacitors is especially interest from the fabrication point of view [22].


## 2. Proposed Circuit

The circuit symbol and the equivalent circuit of the CDTA are shown in Figure1. The terminal characteristic of the CDTA can be described by the following equations [18]:(1)vp=vn=0,iz=ip-in,ix=±gmvz=gmZziz,
where p and n are input terminals, z and x± are output terminals, gm is the transconductance gain, and Zz is external impedance connected at the z terminal. According to the above equation and equivalent circuit of Figure 1(b), the current flowing out of the terminal z(iz) is a difference between the currents through the terminals p and n(ip-in). The voltage drop at the terminal z is transferred to the currents at the terminal xi+ by the transconductance gain (gm), which is electrically controllable by an external bias voltage. These currents that are copied to a general number of output current terminals xj- are equal in magnitude but flow in opposite directions. A possible CMOS-based CDTA circuit realization is given in Figure 2 [17].(a) Circuit symbol of CDTA. (b) Equivalent circuit of CDTA.
(a)(b)Figure 2
CMOS-based CDTA.The CDTAs-based third-order quadrature oscillator is shown in Figure3. The characteristic equation of the circuit in Figure 3 can be expressed as(2)s3C1C2C3+s2C3(C1gm2+C2gm1)+sC3gm1gm2+gm1gm2gm3=0.Figure 3
The proposed current-mode quadrature oscillator.The oscillation condition and oscillation frequency can be obtained as(3)gm3=C3(C1gm2+C2gm1)C1C2,(4)ωo=gm1gm2C1C2.From (3) and (4), the oscillation frequency can be controlled by gm1 or gm2. The oscillation condition can be independently controlled by gm3. From Figure 3, the current transfer function from Io2 to Io1 is(5)Io2(s)Io1(s)=-gm3sC3.
Under sinusoidal steady state, (5) becomes(6)Io2(jω)Io1(jω)=gm3ωC3ej90°.
The phase difference, ϕ, between Io2 and Io1 is(7)ϕ=90°
ensuring the voltages Io2 and Io1 to be in quadrature.The proposed quadrature oscillator employs only grounded capacitors. The use of grounded capacitors is particularly attractive for integrated circuit implementation [22]. From (6), the magnitude of Io2 and Io1 need not the same. For the applications need equal magnitude quadrature outputs, another amplifying circuits are needed.


## 3. Nonideal Effects

Taking the nonidealities of the CDTA into account, Figure4 shows the simplified equivalent circuit that is used to represent the nonideal CDTA [20]. In the figure, αp=1-εp and εp(|εp|≪1) is the current tracking error from the p terminal to the z terminal of the CDTA, αn=1-εn and εn(|εn|≪1) is the current tracking error from the n terminal to the z terminal of the CDTA, and β=1-εi and εi(|εi|≪1) is the output transconductance tracking error from the z terminal to x terminal of the CDTA. Moreover, there are parasitic resistances (Rp and Rn) at terminals p and n and parasitic resistances and capacitances (Rz, Cz and Rx, Cx) from terminals z and x to ground. Reanalysing of the proposed quadrature oscillator in Figure 3 using the nonideal CDTA model and assuming that the operation oscillation frequencies, ω, are very much smaller than 1/CxRn or 1/CxRp and the parasitic resistances at the x terminals are very much greater than the parasitic resistances at p or n terminals of CDTAs, the characteristic equation of Figure 3 becomes
(8)s3C1′C2′C3′+s2[C1′C3′(gm2αn2β21+Gz2)+C2′C3′(gm1αn1β11+Gz1)+C1′C2′Gz3]+s[C3′(gm1gm2αn1αn2β11β21+gm1Gz2αn1β11+gm2Gz1αn2β21+Gz1Gz2)+C1′(gm2Gz3αn2β21+Gz2Gz3)+C2′(gm1Gz3αn1β11+Gz1Gz3)C3′(gm1gm2αn1αn2β11β21+gm1Gz2αn1β11+gm2Gz1αn2β21+Gz1Gz2)]+gm1gm2(gm3αp1αp2αn3β12β22β31+Gz3αn1αn2β11β21)+Gz3(gm1Gz2αn1β11+gm2Gz1αn2β21+Gz1Gz2)=0,
where C1′=C1+Cz1,C2′=C2+Cz2,C3′=C3+Cz3.Figure 4
The nonideal CDTA.The modified oscillation condition and oscillation frequency are(9)gm1gm2A+Gz3BC1′C3′C+C2′C3′D+C1′C2′Gz3=C3′E+C1′F+C2′HC1′C2′C3′,(10)ωo=C3′E+C1′F+C2′HC1′C2′C3′,
where A=(gm3αp1αp2αn3β12β22β31+Gz3αn1αn2β11β21), B=(gm1Gz2αn1β11+gm2Gz1αn2β21+Gz1Gz2), C=(gm2αn2β21+Gz2), D=(gm1αn1β11+Gz1), E=(gm1gm2αn1αn2β11β21+gm1Gz2αn1β11+gm2Gz1αn2β21+Gz1Gz2), F=(gm2Gz3αn2β21+Gz2Gz3), and H=(gm1Gz3αn1β11+Gz1Gz3).Because the values ofα and β are slightly less than unity [23], the parasitic conductances (Gzs) at the z terminals of CDTAs are not zero and the capacitances C1′, C2′, and C3′ are greater than C1, C2, and C3, respectively. From (9) and (10), the oscillation condition and oscillation frequency are deviated from the ideal cases. Therefore, to compensate this effect, we can slightly adjust the gm1 or gm2 values. The oscillation condition still can be independently controlled by gm3. The active and passive sensitivities of the quadrature oscillator are all low and obtained as(11)Sαn1,αn2,β11,β21ωo≅12;Sgm1,gm2ωo≅-SC1′,C2′ωo≅12;SC3′ωo≅0.


## 4. Simulation Results

The quadrature oscillators were simulated using HSPICE. The CMOS CDTA implementation is shown in Figure2 (using 0.18 μm MOSFET from TSMC). The aspect ratios of the MOS transistors were chosen as in Table 1. The multiple current outputs can be easily implemented by adding output branches. Figure 5 represents the current-mode quadrature sinusoidal output waveforms of Figure 3 with C1=C2=C3=10nF, gm1=1.015mS, gm2=232μS, and gm3=1.35mS where gm3 was designed to be larger than the theoretical value to ensure that the oscillations will start. The bias voltages are VB1=0.7V,VB2=-0.7V,V+=1.25V, and V-=-1.25V. The power dissipation is 3.9486 mW. The results of the Io1 and Io2 total harmonic distortion analysis are summarized in Tables 2 and 3, respectively. Figure 6 shows the simulation results of the oscillation frequencies of Figure 3 by varying the value of the transconductance gm1 with C1=C2=C3=10nF, gm2=232μS, and gm3 was varied with gm1 by (3) to ensures that the oscillations will start.Table 1
Aspect ratios of the MOS in Figure2.
MOS transistorsAspect ratio (W/L)M1, M3, M5, M7, M9, M12,27/1.8M21, M22, M23, M24, M26M2, M4, M6, M8, M10, M11,72/1.8M13, M14, M25, M27M15, M16, M17, M18, M19, M2063/1.8Table 2
Total harmonic distortion analysis ofIo1 in Figure 3.
HarmonicFrequencyFourierPhaseno(hz)component(deg)17.3000 k5.3864 u166.4101214.6000 k103.5180 n−107.3535321.9000 k7.0925 n−173.5386429.2000 k50.7076 n−155.1803536.5000 k51.3941 n106.5849643.8000 k32.4016 n−168.3840751.1000 k41.8477 n73.1948858.4000 k8.4629 n120.4927965.7000 k18.1728 n35.0011dc component = −2.329D-09Total harmonic distortion = 2.5713 percentTable 3
Total harmonic distortion analysis ofIo2 in Figure 3.
HarmonicFrequencyFourierPhaseno(hz)component(deg)17.3000 k17.1387 u−103.6257214.6000 k166.8692 n−16.1700321.9000 k11.5300 n−138.9805429.2000 k37.8081 n−53.1517536.5000 k35.5359 n−162.9730643.8000 k20.2288 n−73.4323751.1000 k14.2831 n143.9914858.4000 k4.6312 n177.5460965.7000 k4.5120 n51.6697dc component = 5.725D-06Total harmonic distortion = 1.0327 percentFigure 5
The simulated current-mode quadrature output waveforms of Figure3.Figure 6
Simulation results of the oscillation frequency of Figure3, which is obtained by varying the value of the transconductance gm1.


## 5. Conclusion

In this paper, a new current-mode third-order quadrature oscillator using three CDTA and three grounded capacitors is proposed. Outputs of two sinusoids with 90° phase difference are available in the proposed quadrature oscillator. The oscillation condition and oscillation frequency of the proposed quadrature oscillator are orthogonal controllable. Simulation results verify the theoretical analysis.



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*Source: 789171-2009-12-13.xml*