A miniature tunable quadrature shadow oscillator with orthogonal control

ABSTRACT


INTRODUCTION
The versatility of a quadrature oscillator is well-known in electrical and electronics applications, signal processing, telecommunications, and instrumentation. In modern communication systems, it plays the role of generating reference signals for quadrature amplitude modulation (QAM), single side-band modulation, orthogonal frequency division multiplexing (OFDM), zero-IF and image-reject receivers. In superheterodyne receivers, quadrature oscillators are required in mixers for shifting signals from one frequency range to another [1]. In the field of measurement, quadrature oscillators is the essence for the demodulation section of capacitive rotary encoders [2] used for position and speed measurement. Quadrature oscillators are also employed as signal generators for phase different detectors in measurement systems [3], [4]. It plays a significant role in vector generators and selective voltmeters as well [5]. Therefore, quadrature oscillator circuits are worth going on research and development. It is seen that various literatures of the topic [3]- [21] have been continuously reported over the years.
Among the reported quadrature oscillators current feedback op-amps (CFOA) are employed as the main active components in the circuits published in [3], [5], [7], [9]. However, there is limitation in such 4967 designs for lack of electronic tunability. Besides, the designs require a lot of passive resistors which is unfavorable in the integrated circuit design. In [4], the quadrature oscillator is implemented by using the second generation current conveyor (CCII) but it still lacks electronic tunning capability. A modern analog device, such as current differencing buffer amplifiers (CDBA), is utilized in several quadrature oscillators [6], [8], [13], [14]. However, the frequency of oscillation (FO) cannot be electronically controlled. To overcome this limitation, several passive resistors are replaced with groups of complementary metal-oxidesemiconductor (CMOS) networks [8], but the linear range of tunning parameters is quite narrow. The quadrature oscillators previously reported in [15]- [17], consist of operational transresistance amplifiers (OTRAs) and a lot of discrete passive elements. They suffer from the same restriction that the oscillator parameters can be adjusted by only changing the passive-element values. Although there is an attempt to adjust some parameters of the oscillator via a capacitor, the method is rarely used due to the uncertain properties of the capacitor and the effect of parasitic resistors at higher operating frequency causing undesired frequency-imprecision [22]. Furthermore, the recently proposed quadrature oscillators [10]- [12], realized by the second-generation voltage conveyor (VCII) require a large number of passive components, however, the FO cannot be electronically tuned. Meanwhile, the quadrature oscillator in [20] can be electronically adjusted through the intrinsic resistance of CCCII via dc bias current. Similarly, in the quadrature oscillators [18], [19], tunning the filter parameters is via the transconductances but it has the effect on the filter cell.
Recently, the principle to design a filtering circuit known as a shadow filter has been introduced, [23]- [31]. The characteristics of these shadow filters such as the pole frequency and the quality factor, are obtained using a feedback signal from a filter cell and an external amplifier for tunning. With this new approach, tunning the filter parameters is more convenient and effective than the traditional concept. This is beneficial for minor error compensation after the completion of the integrated-circuit manufacturing process [31]. More recently, this concept had been applied in the design of a shadow sinusoidal oscillator reported in [32], [33], [34]. Among these, the circuit in [32] consists of a filter cell with three inputs, one output, and two external amplifiers based on CFOAs which are available on the shelf in the feedback loop. However, the circuit lacks electronic tunability. Then changing the FO is only by varying a passive resistor in the circuit. Besides, the circuit configuration requires a lot of passive resistors which is unsuitable for monolithic fabrication. Furthermore, only one single sinusoidal signal is obtained as the output.
The design of the shadow oscillator in [33], two cells of integrators with a single input and multiple outputs are functioned as the core circuit. Only the amplitude of the current output can be controlled via an external amplifier. However, tunning the FO relies on changing the parameters within the integrator cells. In the quadrature shadow oscillator [34], a circuit is designed by employing positive feedback of a high Q shadow bandpass filter and two external amplifiers. Although the FO and the condition of oscillation (CO) can be electronically controlled, adjustment on both interacts with each other. Moreover, the circuit needs an excessive number of voltage differential transconductances (VDTAs), up to three sections.
In this literature, a novel scheme of quadrature shadow oscillator is realized by active devices called the VDTAs, along with two resistors and two grounded capacitors. A few advantages are achieved as the total count of VDTAs is decreased comparing with the circuit in [34]. Both the FO and the CO can be adjusted by controlling the dc bias current of the external gain and this is not interfering with the filter cell and suiting the nature of being a shadow oscillator. This tunning feature is attractive for fine-tunning applications and compensation. Besides, under the same filter parameters, tunning in the proposed circuit, the FO can be scaled up more flexibly than that of which depends only on tunning the filter parameters. The results from computer simulation and from the experiment employing a commercial transconductance IC: LM13700 demonstrate and confirm the workability of the proposed work.

METHOD OF VDTA
The proposed quadrature shadow oscillator employs the VDTAs as the main active components to realize a filter cell and the external amplifier. Basic properties of the VDTA are briefly discussed. The VDTA is well known as a universal active building block with a pair of input terminals: and ; and three current output terminals: Z, X+, and X-. The symbol and the internal detail of the VDTA is depicted, respectively, in Figures 1(a) and 1(b). The matrix-representation in (1) shows the relationship among the parameters at all terminals. Meanwhile, the transconductances gm1 and gm2 of the device can be varied by adjusting the dc bias currents. The input-output relationship of the VDTA is described as (1). The transconductance of the VDTA can be approximated as 1 ≃ ( 3 + 4 )/2 and 2 ≃ ( 5 + 8 )/2. The transconductance value of the ℎ transistor can be found as (2): where is the mobility carrier of the NMOS and PMOS transistor; is the gate-oxide capacitance per unit area; is the effective channel width; is the effective channel length; is the bias current of the ℎ transistor.

PROPOSED QUADRATURE SHADOW OSCILLATOR
The proposed quadrature shadow oscillator can be implemented as shown in Figure 2. The circuitry is simple and requires a small number of components, i.e., two VDTAs, two resistors, and two grounded capacitors. VDTA 2 performs the filter cell which gives the bandpass and lowpass response having the following transfer functions.
2 nd filter cell Adjustable gain 1 1 Meanwhile VDTA 1 realizes the external amplifier with the following gain K.
The compound resistor 1 and 2 provides the attenuator gain , were Routine analysis results in the following characteristic equation.
Using the Barkhuizen oscillation criteria, the associated FO and CO are found, respectively as (8) and (9).
From (8), the FO can be adjusted via the external gain K , gm3 and gm4. However, to follow the shadow filter concept, it is intended to tune the FO via only the external gain, i.e., adjusting the dc bias currents IB1 and IB2 of VDTA1. In the filter cell, the values of the bias currents and the capacitors are maintained as constants.
In addition, from (8), the FO of the shadow oscillator is the product of factor√1 + and 0 therefore the FO can be scaled up even if the same filter parameters are of use. Moreover, from (8) and (9), orthogonal controlling the FO and CO is evident. The CO can be set up by choosing the appropriate value of two resistors which should be equal to 4 /( 3 ). Assuming the high input impedance at the VDTA-Vp terminal, the gain can be approximated to .
It is convenient to set up the CO by keeping the dc bias currents of the filter to the same value, i.e., 3 = 4 . As a result, 3  4 , and the term is a reciprocal of the value K. It should be noted that K depending on the bias current of the VDTA is approximated to the ratio of IB1 and IB2. Next, the relationship between the two outputs is found. According to Figure 2, the voltage outputs at the FO can be expressed as (10).
Obviously, the voltage outputs 1 and 2 have 90 phase differences at the operating frequency. Then substitution of (8) into (10), absolute ratio of both outputs at the FO can be found as (11).
From (11), the output signal ratio at the FO depends on the external gain too. Then the sensitivities of the FO related to some important parameters can be found as (12).
It is seen from (12), all the sensitivities of FO are acceptably less than unity.
where and are the transconductance gain error factors with i being the VDTA number and j being the output of the VDTA. Brief analysis discovers that the effect of non-ideality on the FO can be found as (14).
Clearly, the transconductance errors causes the FO to drift. The sensitivity of the FO is then computed, and the results show that all the related sensitivities are still not greater than unity.

The effect of parasitic impedance on the FO
To study the parasitic impedance effects, a non-ideal VDTA model that incorporates the parasitic impedances is considered. The VDTA has parasitic impedances at all terminals. These impedances are represented as virtual grounded resistors and capacitors as shown in Figure 3. In the filter section, the parasitic impedances can be shown in Figure 4.
In the amplifier section, the modified gains including the parasitic impedances can be written as (16).

SIMULATION AND EXPERIMENTATION RESULTS
For evaluation, the proposed quadrature shadow oscillator is simulated using the TSMC 180 nm CMOS model parameters. The implementation of VDTAs employs the CMOS dimensions as those show in Table 1 Figures 5(a) and 5(b). It is clearly seen from the Figure 5 that both outputs can be self-oscillating as expected. Then Figure 6 depicts the two outputs in the steady-state within a short range of enlarged time-scale. Clearly, the proposed circuit generates two sinusoidal signals. In the figures, one has the amplitude of 80 mVp and the other 33 mVp and both having FO at 342 kHz with 90° phase difference.  Additionally, the frequency spectrum of the voltages in Figure 6 are demonstrated in Figure 7. From the spectrum, the fundamental frequency is distinctly situated at 342 kHz. The total harmonic distortions (THD) of Vout1 and Vout2 are found, respectively, as 6.04% and 1.93%. For verification of the 90° phase shift, the two outputs are plotted in the XY-plane and the results are demonstrated in Figure 8. Clearly, the desired 90° phase difference is obtained. Besides, to explore the property of electronic tunability that does not disturb the filter cell, the proposed quadrature shadow oscillator is simulated and only the external gain K is varied from 1-10. This can be set by using 1 = 10 to 100 µA and 2 = 10 µA. Meanwhile, all the other parameters of the filter cell are fixed, i.e., 3 = 4 = 50 µA and 1 = 2 = 0.3 F. The graph in Figure 9 shows the FO appropriately varies from 225 to 395 kHz relating to variation of the external gain K. The effect of various controlling gains on the phase difference between the two outputs are also simulated and the results are given in Figure 10. The proposed circuit is then explored by the Monte Carlo analysis. In this simulation, the capacitor value deviates 5% with Gaussian distribution while the number of simulations is 100 run times and the gain K=4. The circuit parameters are set with 1 = 40 µA, 2 = 10 µA, 3 = 4 = 50 µA, 1 = 2 = 0.3 F. To ensure self-oscillations, the resistors are chosen as 1 =3 kΩ and 2 =3.45 kΩ. Then the histograms of the FOs from the two outputs are shown in Figures 11(a) and 11(b). The histogram of the phase differences between the two outputs is also shown in Figure 11(c).
To confirm the workability, the proposed quadrature oscillator is further explored by actual circuit experiment. In the experiment, commercially available LM13700 operational transconductance amplifiers are employed as the main active elements. Then the schematic diagram is given in Figure 12. The circuit construction on a breadboard is also shown in Figure 13. From the experiment, the parameters in the filter cell are such that the power supplies are set at ±12 V, 1 = 2 = 4.7 F and the two transconductances are 440 μA/V. The external gain is set to 5.7 by using 1 = 266 µA, 2 = 46 µA and 1 = 1.18 kΩ and 2 = 10 kΩ. The oscilloscope screen shown in Figure 14 demonstrates the two sinusoidal output signals, Vout1 the marked trace having the amplitude of 84 mV and the unmarked Vout2 having 60 mV of amplitude. The experiment results in the FO of about 34 kHz with a quadrature phase difference, whereas the theoretical value of the FO is 38 kHz. Then the two signals are plotted in the XY-mode as depicted in Figure 15. From the experiment, the information about the spectrum is also collected. Figure 16 shows the spectrum of the output voltage Vout1. The magnitude of the highest spectrum is greater than 50 dB comparing with the dominant harmonic in the vicinity. It is interesting to note that the simulation results demonstrate that the proposed circuit can produce sinusoidal signals with a quadrature phase-difference and the frequency of the oscillation can be controlled through the circuit parameters. The value of gm in the experimentation associated with the LM13700 can be approximated as about tenfold of the bias currents, i.e.,  10IB A /V. Meanwhile, in the simulation part using the same bias current, gm is obtained as 233 A /V which is smaller than that obtained in the experiment part.
Hence, under the other same condition of gains and capacitors, the FO obtained from the experiment is larger than the FO in simulation part.

PROPERTIES COMPARISON WITH PREVIOUS WORKS
For some little assessment among the various quadrature oscillators including the one having been proposed, a list of important characteristics is given in Table 2. Therefore, considering a number of aspects

CONCLUSION
A new design of quadrature shadow oscillator is proposed. The design presents a few merits as the condition of oscillation and the frequency of oscillation can be independently controlled without disturbing the internal filter cell. To control the CO and the FO is also in orthogonal mode. As the circuit implementation consists of two VDTAs and a few passive components, the FO is tuned via the dc bias current of the external amplifier based on the VDTA realization. Another advantage of the proposed design is that the range of the FO can be easily extended as compared to the previous classical version due to the term 1 K + , where K is the external amplifier gain. This feature is then suitable for applications such as finetunning and the error compensation of mismatch parameters within filter cells after the completion of the integrated-circuit process. Finally, the validity of the proposed work is well-confirmed by both computer simulation and circuit experiment.