Voltage-Mode Third-Order Quadrature Sinusoidal Oscillator Using VDBAs

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1. Introduction

An oscillator is a very important basic building block, which is frequently used in electrical and electronics engineering applications. Among several types of sinusoidal oscillators, the quadrature oscillators are widely used because they can offer sinusoidal signals with 90˚ phase difference, for example, in telecommunications for quadrature mixers and SSB generators [1] , for measurement purposes in vector generators and selective voltmeters [2] . Because of these applications number of QSOs has been realized employing different active building blocks in the open literature [3] - [8] . Different variety of active building blocks (ABBs) have been introduced in [9] , VDBA is one of them. Since its introduction in [9] , VDBA has been used in many signal processing and signal generation applications. Two biquad filters have been realized in [10] using two VDBAs and two capacitors and a resistor. In [11] VDBA based three lossless and lossy inductance simulators have been proposed employing two or three passive components. Single VDBA based multifunction filter configuration was proposed in [12] using five or six passive components. Quadrature oscillator employing three VDBAs, three capacitors and two resistors was proposed in [13] . The objective of this communication is to present a new voltage-mode TOQSO structure employing two VDBAs, three capacitors and a resistor which is based on a non-inverting VM low pass biquadratic filter and inverting VM integrator in a closed loop. The workability of the proposed configuration is verified by SPICE simulation using 0.18 µm TSMC technology transistor parameters.

2. Proposed Methodology

Figure 1 shows the basic methodology to obtain a TOQSO by cascading a second-order non-inverting low pass filter with an inverting integrator or by cascading an inverting second-order low pass filter with non-inverting integrator. The open loop voltage gain of Figure 1 can be expressed as:

$\frac{{V}_{o}}{{V}_{in}}={T}_{1}\left(s\right){T}_{2}\left(s\right)=\frac{-{a}_{3}}{s\left({a}_{0}{s}^{2}+{a}_{1}s+{a}_{2}\right)}$ (1)

where ${T}_{1}\left(s\right)=\frac{1}{{a}_{0}{s}^{2}+{a}_{1}s+{a}_{2}}$ and ${T}_{2}\left(s\right)=-\frac{{a}_{3}}{s}$

or ${T}_{1}\left(s\right)=-\frac{1}{{a}_{0}{s}^{2}+{a}_{1}s+{a}_{2}}$ and ${T}_{2}\left(s\right)=\frac{{a}_{3}}{s}$

To produced sustained oscillations, V_{o} = V_{in} and hence the characteristic equation can be denoted as

${a}_{0}{s}^{3}+{a}_{1}{s}^{2}+{a}_{2}s+{a}_{3}=0$ (2)

The CO and FO can be deduced from Equation (2) as follows:

$\text{CO}:{a}_{0}{a}_{3}={a}_{1}{a}_{2}$ (3)

$\text{FO}:{\omega}_{0}=\sqrt{\frac{{a}_{3}}{{a}_{1}}}=\sqrt{\frac{{a}_{2}}{{a}_{0}}}$ (4)

3. The Proposed Third-Order Quadrature Oscillator Configuration

The symbolic notation and equivalent model of the VDBA are shown in Figure 2(a) and Figure 2(b) respectively [9] . Using standard notations, the voltage-

Figure 1. Functional block diagram for the realization of third-order quadrature oscillator.

(a) (b)

Figure 2. Symbolic notation (b) equivalent model of VDBA.

Figure 3. New proposed third-order quadrature sinusoidal oscillator.

current relations of VDBA can be described by the following matrix.

$\left(\begin{array}{c}{I}_{p}\\ {I}_{n}\\ {I}_{z}\\ {V}_{w}\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ {g}_{m}& -{g}_{m}& 0& 0\\ 0& 0& \beta & 0\end{array}\right)\left(\begin{array}{c}{V}_{p}\\ {V}_{n}\\ {V}_{z}\\ {I}_{w}\end{array}\right)$ (5)

where β is a non-ideal voltage gain of VDBA. The value of β in an ideal VDBA is unity and g_{m} is the transconductance of the VDBA.

Figure 3 shows the proposed new TOQSO with independent electronic control of both FO and CO.

The expression for characteristic equation (CE) of the circuit of Figure 3 is given by (Detailed explanation of Equation (6) with the help of Figure A1 is given in Appendix):

$\text{CE}:{s}^{3}{C}_{1}{C}_{2}{C}_{3}+{s}^{2}\left(\frac{{C}_{1}{C}_{2}}{{R}_{0}}\right)+s\left(\frac{{C}_{2}{g}_{m1}}{{R}_{0}}\right)+\frac{{g}_{m1}{g}_{m2}}{{R}_{0}}=0$ (6)

The condition of oscillation and the frequency of oscillation can be given as

$\text{CO}:{g}_{m2}{R}_{0}\le \frac{{C}_{2}}{{C}_{3}}$ (7)

$\text{FO}:{\omega}_{0}=\sqrt{\frac{{g}_{m1}}{{R}_{0}{C}_{1}{C}_{3}}}$ (8)

The relationship between V_{o}_{1} and V_{o}_{2} can be obtained as:

$\frac{{V}_{o1}}{{V}_{o2}}=\frac{-j\omega {C}_{2}}{{g}_{{m}_{2}}}=\frac{\omega {C}_{2}}{{g}_{{m}_{2}}}{\text{e}}^{j-{90}^{\circ}}$ (9)

From Equation (9) it is evident that V_{o}_{1} and V_{o}_{2} are in quadrature.

4. Sensitivity Analysis

The sensitivity is an important performance criterion of any circuit structure. The sensitivities of ω_{0} with respect to active and passive elements are given by

${S}_{{C}_{1}}^{{\omega}_{0}}={S}_{{C}_{3}}^{{\omega}_{0}}={S}_{{R}_{0}}^{{\omega}_{0}}=-\frac{1}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}_{{g}_{m1}}^{{\omega}_{0}}=\frac{1}{2}$ (10)

It may be easily observed from Equation (10) that all sensitivities are lower than unity in magnitude, for the proposed third-order quadrature oscillator. It ensures that the sensitivity performance is good.

5. Simulation Results

To confirm theoretical analysis, the proposed TOQSO was simulated using CMOS VDBA (as shown in Figure 4). The CMOS VDBA is implemented using 0.18 µm TSMC real transistor models [14] . The aspect ratios of transistors used in Figure 4 are shown in Table 1. The passive elements were selected as C_{1} = 1.0 nF and C_{2} = 1.0 nF, and R_{0} = 1.66 kΩ. The transconductances of VDBAs were controlled by the bias currents. SPICE generated output waveforms indicating transient and steady state responses of circuit of Figure 3 are shown in Figure 5 and Figure 6 respectively. The results of TOQSO in Figure 5 and Figure 6 show more accuracy than the second order ones. These results, thus, shows the validity of the proposed configuration. Figure 7 shows the output spectrum of circuit

Figure 4. CMOS implementation of VDBA [10] , V_{DD} = V_{SS} = 0.9 V.

Table 1. The aspect ratios of transistors used in Figure 4.

Figure 5. Transient response of proposed third-order quadrature sinusoidal oscillator.

Figure 6. Study state response of proposed third-order quadrature sinusoidal oscillator.

Figure 7. Frequancy response of proposed third-order quadrature sinusoidal oscillator.

shown in Figure 3; whereas the total harmonic distortion (THD) for both the outputs, V_{o1} and V_{o2} are found to be 2.55% and 0.48% respectively. The THD at output V_{o2} is very small. Figure 8 shows the Lissajous pattern for the circuit of Figure 3. The circles are shown in the Figure 8, indicates that two signals are at 90˚ phase difference.

Figure 8. Frequancy response of proposed third-order quadrature sinusoidal oscillator.

6. Conclusion

A new voltage-mode third order quadrature sinusoidal oscillator with independent electronic control of both CO and FO using two VDBAs, three capacitors and a resistor is introduced. The CO can be electronically controlled by transcoductance (g_{m}_{2}) of VDBA_{2} without affecting FO. FO can also be electronically adjusted by transcoductance (g_{m}_{1}) of VDBA_{1} without affecting CO. The proposed TOQSO offers low sensitivities. The circuit exhibits good high frequency performance. One can design TOQSO with single VDBA. Workability of the proposed configuration is verified by SPICE simulation using 0.18 µm TSMC technology.

Appendix

Figure A1. New proposed third-order quadrature sinusoidal oscillator.

KCL at node 1:

${i}_{z1}={g}_{{m}_{1}}({v}_{{p}_{1}}-{v}_{{n}_{1}})={v}_{{z}_{1}}s{C}_{1}$

${g}_{{m}_{1}}({v}_{{p}_{1}}-{v}_{{n}_{1}})={v}_{{z}_{1}}s{C}_{1}$ (a)

KCL at node 2:

${i}_{z2}={g}_{{m}_{2}}(0-{v}_{{n}_{2}})={v}_{{z}_{2}}s{C}_{2}$

$-{g}_{{m}_{2}}{v}_{{z}_{1}}={v}_{{z}_{2}}s{C}_{2},\text{\hspace{0.17em}}$ (b)

KCL at node 3:

${v}_{{n}_{1}}(s{C}_{3}+\frac{1}{{R}_{0}})={v}_{{z}_{2}}s{C}_{3}+{v}_{{z}_{1}}\frac{1}{{R}_{0}}$ (c)

From Equations ((a)-(c)), we will gate CE.

References

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