Current-Mode Third-Order Quadrature Oscillator Using CDTAs

Abstract

This paper describes a current-mode third-order quadrature oscillator based on current differencing transconductance amplifiers (CDTAs). Outputs of two current-mode sinusoids with 90^∘ 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.

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 Figure 1. The terminal characteristic of the CDTA can be described by the following equations [18]:𝑣_𝑝=𝑣_𝑛=0,𝑖_𝑧=𝑖_𝑝−𝑖_𝑛,𝑖_𝑥=±𝑔_𝑚𝑣_𝑧=𝑔_𝑚𝑍_𝑧𝑖_𝑧, (1) where 𝑝 and 𝑛 are input terminals, 𝑧 and 𝑥± are output terminals, 𝑔_𝑚 is the transconductance gain, and 𝑍_𝑧 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(𝑖_𝑧) is a difference between the currents through the terminals 𝑝 and 𝑛(𝑖_𝑝−𝑖_𝑛). The voltage drop at the terminal 𝑧 is transferred to the currents at the terminal 𝑥_𝑖+ by the transconductance gain (𝑔_𝑚), which is electrically controllable by an external bias voltage. These currents that are copied to a general number of output current terminals 𝑥_𝑗− are equal in magnitude but flow in opposite directions. A possible CMOS-based CDTA circuit realization is given in Figure 2 [17].

(a)

(b)

Figure 1 

(a) Circuit symbol of CDTA. (b) Equivalent circuit of CDTA.

Figure 2 

CMOS-based CDTA.

The CDTAs-based third-order quadrature oscillator is shown in Figure 3. The characteristic equation of the circuit in Figure 3 can be expressed as𝑠³𝐶₁𝐶₂𝐶₃+𝑠²𝐶₃(𝐶₁𝑔_𝑚2+𝐶₂𝑔_𝑚1)+𝑠𝐶₃𝑔_𝑚1𝑔_𝑚2+𝑔_𝑚1𝑔_𝑚2𝑔_𝑚3=0. (2)

Figure 3 

The proposed current-mode quadrature oscillator.

The oscillation condition and oscillation frequency can be obtained as𝑔_𝑚3=𝐶₃(𝐶₁𝑔_𝑚2+𝐶₂𝑔_𝑚1)𝐶₁𝐶₂,𝜔_𝑜=𝑔_𝑚1𝑔_𝑚2𝐶₁𝐶₂. (3)(4)

From (3) and (4), the oscillation frequency can be controlled by 𝑔_𝑚1 or 𝑔_𝑚2. The oscillation condition can be independently controlled by 𝑔_𝑚3. From Figure 3, the current transfer function from 𝐼_𝑜2 to 𝐼_𝑜1 is𝐼_𝑜2(𝑠)𝐼_𝑜1𝑔(𝑠)=−^𝑚3𝑠𝐶₃. (5) Under sinusoidal steady state, (5) becomes𝐼_𝑜2(𝑗𝜔)𝐼_𝑜1=𝑔(𝑗𝜔)𝜔𝐶^𝑚3₃𝑒^𝑗90∘. (6) The phase difference, 𝜙, between 𝐼_𝑜2 and 𝐼_𝑜1 is𝜙=90^∘ (7) ensuring the voltages I_o2 and I_o1 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 𝐼_𝑜2 and 𝐼_𝑜1 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, Figure 4 shows the simplified equivalent circuit that is used to represent the nonideal CDTA [20]. In the figure, 𝛼_𝑝=1−𝜀_𝑝 and 𝜀_𝑝(|𝜀_𝑝|≪1) is the current tracking error from the 𝑝 terminal to the 𝑧 terminal of the CDTA, 𝛼_𝑛=1−𝜀_𝑛 and 𝜀_𝑛(|𝜀_𝑛|≪1) is the current tracking error from the 𝑛 terminal to the 𝑧 terminal of the CDTA, and 𝛽=1−𝜀_𝑖 and 𝜀_𝑖(|𝜀_𝑖|≪1) is the output transconductance tracking error from the 𝑧 terminal to 𝑥 terminal of the CDTA. Moreover, there are parasitic resistances (𝑅_𝑝 and 𝑅_𝑛) at terminals 𝑝 and 𝑛 and parasitic resistances and capacitances (𝑅_𝑧, 𝐶_𝑧 and 𝑅_𝑥, 𝐶_𝑥) 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/𝐶_𝑥𝑅_𝑛 or 1/𝐶_𝑥𝑅_𝑝 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 𝑠³𝐶^′₁𝐶^′₂𝐶^′₃+𝑠²[𝐶^′₁𝐶^′₃(𝑔_𝑚2𝛼_𝑛2𝛽₂₁+𝐺_𝑧2)+𝐶^′₂𝐶^′₃(𝑔_𝑚1𝛼_𝑛1𝛽₁₁+𝐺_𝑧1)+𝐶^′₁𝐶^′₂𝐺_𝑧3]+𝑠[𝐶^′₃(𝑔_𝑚1𝑔_𝑚2𝛼_𝑛1𝛼_𝑛2𝛽₁₁𝛽₂₁+𝑔_𝑚1𝐺_𝑧2𝛼_𝑛1𝛽₁₁+𝑔_𝑚2𝐺_𝑧1𝛼_𝑛2𝛽₂₁+𝐺_𝑧1𝐺_𝑧2)+𝐶^′₁(𝑔_𝑚2𝐺_𝑧3𝛼_𝑛2𝛽₂₁+𝐺_𝑧2𝐺_𝑧3)+𝐶^′₂(𝑔_𝑚1𝐺_𝑧3𝛼_𝑛1𝛽₁₁+𝐺_𝑧1𝐺_𝑧3)]+𝑔_𝑚1𝑔_𝑚2(𝑔_𝑚3𝛼_𝑝1𝛼_𝑝2𝛼_𝑛3𝛽₁₂𝛽₂₂𝛽₃₁+𝐺_𝑧3𝛼_𝑛1𝛼_𝑛2𝛽₁₁𝛽₂₁)+𝐺_𝑧3(𝑔_𝑚1𝐺_𝑧2𝛼_𝑛1𝛽₁₁+𝑔_𝑚2𝐺_𝑧1𝛼_𝑛2𝛽₂₁+𝐺_𝑧1𝐺_𝑧2)=0, (8) where 𝐶^′₁=𝐶₁+𝐶_𝑧1,𝐶^′₂=𝐶₂+𝐶_𝑧2,𝐶^′₃=𝐶₃+𝐶_𝑧3.

Figure 4 

The nonideal CDTA.

The modified oscillation condition and oscillation frequency are𝑔_𝑚1𝑔_𝑚2𝓐+𝐺_𝑧3𝓑𝐶^′₁𝐶^′₃𝓒+𝐶^′₂𝐶^′₃𝓓+𝐶^′₁𝐶^′₂𝐺_𝑧3=𝐶^′₃𝓔+𝐶^′₁𝓕+𝐶^′₂𝓗𝐶^′₁𝐶^′₂𝐶^′₃,𝜔_𝑜=𝐶^′₃𝓔+𝐶^′₁𝓕+𝐶^′₂𝓗𝐶^′₁𝐶^′₂𝐶^′₃, (9)(10) where 𝓐=(𝑔_𝑚3𝛼_𝑝1𝛼_𝑝2𝛼_𝑛3𝛽₁₂𝛽₂₂𝛽₃₁+𝐺_𝑧3𝛼_𝑛1𝛼_𝑛2𝛽₁₁𝛽₂₁), 𝓑=(𝑔_𝑚1𝐺_𝑧2𝛼_𝑛1𝛽₁₁+𝑔_𝑚2𝐺_𝑧1𝛼_𝑛2𝛽₂₁+𝐺_𝑧1𝐺_𝑧2), 𝓒=(𝑔_𝑚2𝛼_𝑛2𝛽₂₁+𝐺_𝑧2), 𝓓=(𝑔_𝑚1𝛼_𝑛1𝛽₁₁+𝐺_𝑧1), 𝓔=(𝑔_𝑚1𝑔_𝑚2𝛼_𝑛1𝛼_𝑛2𝛽₁₁𝛽₂₁+𝑔_𝑚1𝐺_𝑧2𝛼_𝑛1𝛽₁₁+𝑔_𝑚2𝐺_𝑧1𝛼_𝑛2𝛽₂₁+𝐺_𝑧1𝐺_𝑧2), 𝓕=(𝑔_𝑚2𝐺_𝑧3𝛼_𝑛2𝛽₂₁+𝐺_𝑧2𝐺_𝑧3), and 𝓗=(𝑔_𝑚1𝐺_𝑧3𝛼_𝑛1𝛽₁₁+𝐺_𝑧1𝐺_𝑧3).

Because the values of 𝛼 and 𝛽 are slightly less than unity [23], the parasitic conductances (𝐺_𝑧𝑠) at the 𝑧 terminals of CDTAs are not zero and the capacitances 𝐶^′₁, 𝐶^′₂, and 𝐶^′₃ 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 𝑔_𝑚1 or 𝑔_𝑚2 values. The oscillation condition still can be independently controlled by 𝑔_𝑚3. The active and passive sensitivities of the quadrature oscillator are all low and obtained as𝑆^𝜔𝑜_{𝛼𝑛1,𝛼𝑛2,𝛽11,𝛽21}≅12;𝑆^𝜔𝑜_{𝑔𝑚1,𝑔𝑚2}≅−𝑆^𝜔𝑜_{𝐶′1,𝐶′2}≅12;𝑆^𝜔𝑜_𝐶′3≅0. (11)

4. Simulation Results

The quadrature oscillators were simulated using HSPICE. The CMOS CDTA implementation is shown in Figure 2 (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, 𝑔_𝑚1=1.015mS, 𝑔_𝑚2=232𝜇S, and 𝑔_𝑚3=1.35mS where 𝑔_𝑚3 was designed to be larger than the theoretical value to ensure that the oscillations will start. The bias voltages are 𝑉_𝐵1=0.7V,𝑉_𝐵2=−0.7V,𝑉+=1.25V, and 𝑉−=−1.25V. The power dissipation is 3.9486 mW. The results of the 𝐼_𝑜1 and 𝐼_𝑜2 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 𝑔_𝑚1 with 𝐶₁=𝐶₂=𝐶₃=10nF, 𝑔_𝑚2=232𝜇S, and 𝑔_𝑚3 was varied with 𝑔_𝑚1 by (3) to ensures that the oscillations will start.

Table 1 

Aspect ratios of the MOS in Figure 2.

Table 2 

Total harmonic distortion analysis of I_o1 in Figure 3.

Table 3 

Total harmonic distortion analysis of I_o2 in Figure 3.

Figure 5 

The simulated current-mode quadrature output waveforms of Figure 3.

Figure 6 

Simulation results of the oscillation frequency of Figure 3, 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.

Acknowledgments

The author would like to thank the reviewers for their suggestions. The National Science Council, China, supported this work under Grant no. NSC 98-2221-E-033-054.