© Wuhan University 2022

0 Introduction

The Wien-bridge oscillator is one of the most typical oscillators used in audio and sub-audio frequency ranges, which has been found wide applications in linear and nonlinear circuits^[1]. The Wien-bridge oscillator employing the operation amplifier (Op Amp) is a simple and practical circuit, but it is a voltage-mode circuit rather than a current-mode circuit^[2]. Though the Wien-bridge oscillator using the current conveyor (CCII) is a typical current-mode circuit, its parameter cannot be adjusted electronically^[3]. The Wien-bridge oscillator using the operation transconductance amplifier (OTA) enjoys simple structure and electronic tunability, but it is not a true current-mode circuit because the inputs of the OTA are voltages rather than currents^[4]. The Wien-bridge oscillator proposed by Fabre et al ^[5] can exhibit good performances. However, it suffers from either the use of the floating capacitor or the use of the transconductance mode circuit due to the used OTA.

On the other hand, the second generation current-controlled conveyor (CCCII) employing the bipolar technology was first introduced by Fabre et al in 1996^[5]. Since then, many current-mode circuits based on CCCII, like the analog filters, oscillators, and gyrator circuits, have been investigated, which shows that the CCCII is a very interesting current-mode device. In spite of the fact that oscillators using CCCIIs have been widely reported^[6-14], they are neither Wien-bridge oscillators, nor oscillators with orthogonal adjustment between the resonant frequency and resonant condition. Literature survey shows the CCCII scillators using the traditional Wien-bridge oscillator rather than the grounded-capacitor Wien-bridge oscillator as topology have rarely been reported so far.

In this paper, a current-mode Wien-bridge oscillator employing the CCCII is designed, which utilizes the traditional Wien-bridge oscillator and the adjoint network theory and contains four CCCIIs and two grounded capacitors. Its resonant condition and resonant frequency can be independently and electronically varied by tuning bias currents of the CCCIIs linearly. From the perspective of selectivity, grounding capacitance, and the simplest structure, this oscillator is the simplest, most economical and least distortion circuit. The results from circuit simulation are in agreement with theory.

1 Design

The traditional Wien-bridge oscillator employing single Op Amp reported in Ref. [2] is shown in Fig. 1. This circuit consists of an Op Amp connected in a non-inverting amplifier and a lead-lag network connected as the feedback circuit. It is customary to impose C ₁=C ₂=C, R ₁=R ₂=R, after which the gain of the non-inverting amplifier, the feedback coefficient of the positive feedback network, and the loop gain can be readily calculated as

$A_{\mathrm{u}}=\mathrm{1}+\frac{R_{\mathrm{4}}}{R_{\mathrm{3}}}$(1)

$F_{\mathrm{u}}=\frac{V_{\mathrm{f}}}{V_{\mathrm{o}}}=\frac{\mathrm{1}}{\mathrm{3}+\mathrm{j}(\omega RC-\mathrm{1}/\omega RC)}$(2)

$\mathrm{L}\mathrm{G}=A_{\mathrm{u}}{F}_{\mathrm{u}}=\frac{\mathrm{j}(\mathrm{1}+R_{\mathrm{3}}/R_{\mathrm{4}})\omega RC}{\mathrm{1}-{\omega }^{\mathrm{2}}{R}^{\mathrm{2}}{C}^{\mathrm{2}}+\mathrm{j}\mathrm{3}\omega RC}$(3)

Fig. 1 Traditional Wien-bridge oscillator using Op Amp

The resonant condition (including start-up condition) and resonant frequency for the oscillator have been given by

$R_{\mathrm{4}}\ge \mathrm{2}{R}_{\mathrm{3}},f_{\mathrm{o}}=\frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }RC}$(4)

This is the already-familiar conclusion.

The grounded-capacitor Wien-bridge oscillator using an Op Amp has been reported in Ref. [3]. This circuit can be viewed as a non-inverting amplifier with frequency-dependent gain and a positive frequency-selecting feedback network. When C ₁=C ₂=C, R ₁=R ₂=R, the gain of the non-inverting amplifier, the feedback coefficient of the positive feedback network, and the loop gain can be readily calculated as

$A_{\mathrm{u}}=\frac{V_{\mathrm{o}}}{V_{\mathrm{f}}}=\mathrm{1}+\frac{R_{\mathrm{4}}}{R+\mathrm{1}/\mathrm{j}\omega C}$(5)

$F_{\mathrm{u}}=\frac{V_{\mathrm{f}}}{V_{\mathrm{o}}}=\frac{R//(\mathrm{1}/\mathrm{j}\omega C)}{R_{\mathrm{3}}+R//(\mathrm{1}/\mathrm{j}\omega C)}$(6)

$\begin{array}{l}\mathrm{L}\mathrm{G}=A_{\mathrm{u}}{F}_{\mathrm{u}}\\ =\frac{R[\mathrm{1}+\mathrm{j}\omega (R+R_{\mathrm{4}})C]/(R_{\mathrm{3}}+R)}{\mathrm{1}-{\omega }^{\mathrm{2}}{R}_{\mathrm{3}}{R}^{\mathrm{2}}{C}^{\mathrm{2}}/(R+R_{\mathrm{3}})+\mathrm{j}\omega (R+\mathrm{2}{R}_{\mathrm{3}})RC/(R+R_{\mathrm{3}})}\end{array}$(7)

Substituting (7) into the characteristic equation, 1-A _u F _u=0, we find that the resonant condition and resonant frequency are the same as Eq. (4), showing that although the series and parallel combination of RC is separated, the circuit still belongs to a member of Wien-bridge oscillator family. However, unlike the traditional Wien-bridge oscillator, the loop gain of the grounded-capacitor Wien-bridge oscillator, from (7), is not a band-pass function but a combination of the band-pass function and low-pass function. This could not guarantee that the circuit enjoys less distortion. Consequently, from a selective visual perspective, the traditional Wien-bridge oscillator is superior to the grounded-capacitor Wien-bridge oscillator.

To achieve the grounded-oscillator current-mode oscillator, we need to transform the lead-lag network of Fig. 1 into an adjoint network by the adjoint network theory, as depicted in Fig. 2. According to the adjoint network theory, we can easily find

$F_i=\frac{I_{\mathrm{f}}}{I_{\mathrm{o}}}=F_{\mathrm{u}}=\frac{V_{\mathrm{f}}}{V_{\mathrm{o}}}=\frac{\mathrm{1}}{\mathrm{3}+\mathrm{j}(\omega RC-\mathrm{1}/\omega RC)}$(8)

Fig. 2 Lead-lag network of Fig. 1 (a) RC series and parallel network; (b) Adjoint network of (a)

Here, C ₁=C ₂=C, R ₁=R ₂=R.

Figure 3 shows a current-mode oscillator based on Fig. 2(b), where CCCII₃ and CCCII₄ form the non-inverting current amplifier with gain greater than unity:

$A_i=\mathrm{1}+\frac{R_{x\mathrm{4}}}{R_{x\mathrm{3}}}=\mathrm{1}+\frac{I_{\mathrm{B}\mathrm{3}}}{I_{\mathrm{B}\mathrm{4}}}$(9)

Fig. 3 Designed oscillator with CCCII1 serving a floating resistor R 1 and CCCII2 serving a grounded resistor R 2

where R_{x 3}=V _T/2I _B3, R_{x 4}=V _T/2I _B4, I _B3 and I _B4 are bias currents of CCCII₃ and CCCII₄, respectively, and V _T represents the thermal voltage that is 26 mV at room temperature.

CCCII₁, CCCII₂, C ₁, and C ₂ in Fig. 3 create the positive feedback network to form the feedback coefficient depicted by (8). Substituting (8) and (9) into the characteristic equation of the oscillator, 1-A_iF_i =0, we get

$\mathrm{2}-R_{x\mathrm{4}}/R_{x\mathrm{3}}+\mathrm{j}(\omega R_{x}C-\mathrm{1}/\omega R_{x}C)=\mathrm{0}$(10)

where R $x$=R $x$ ₁=R $x$ ₂=V _T/2I _B, and I _B is the bias current of CCCII₁ or CCCII₂.

The resonant condition (including start-up condition) and resonant frequency are then

$I_{\mathrm{B}\mathrm{3}}\ge \mathrm{2}{I}_{\mathrm{B}\mathrm{4}}$(11)

$f_{\mathrm{o}}=\frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }{R}_{x}C}=\frac{I_{\mathrm{B}}}{\mathrm{\pi }{V}_{\mathrm{T}}C}$(12)

(11) and (12) show that the linear, independent and electronic control of the resonant frequency and resonant condition can be realized as follows; a) tune I _B3 or I _B4 to adjust to the resonant condition, not influencing resonant frequency; b) tune I _B to adjust to the resonant frequency, not influencing resonant condition. (11) and (12) also show that orthogonal electronic control must require four parameters, namely I _B3, I _B4, I _B1, and I _B2 (I _B1=I _B2), so the oscillator needs at least four CCCIIs.

Under sinusoidal steady-state conditions, from Fig.3, we can easily get

$\frac{I_{\mathrm{o}\mathrm{2}}}{I_{\mathrm{o}\mathrm{1}}}=\frac{I_{\mathrm{B}\mathrm{3}}}{I_{\mathrm{B}\mathrm{4}}}=\mathrm{2}$(13)

(13) denotes that the designed oscillator can generate two sinusoidal signals with the same frequency, same phase, and amplitude difference of two times.

2 Verification

To start with, the sub-circuit for a CCCII was established by the transistor model of PR200N and NR200N of the analog device library from NI MULTISIM 11.0 software^[15]. Then the circuit in Fig. 3 was simulated by using ±2.5 V power supplies. When I _B3=12 µA, I _B4=5 µA, then Eq. (11) is satisfied. If setting C ₁=C ₂=100 pF, I _B=5 µA, the design value for f _o from (12) is 612 kHz. The simulation results are shown in Fig. 4, which shows that the actual values for f _o is 568 kHz. The relative error is then -7.2%. I _o2/I _o1 is about 2.

Fig. 4 Initial results of the transient analysis for Fig. 3

When I _B is varied from 5 µA to 25 µA and 50 µA, and I _B3 and I _B4 stay the same, the design value for f _o is adjusted from 612 kHz to 306 kHz and 6.12 MHz, respectively (Fig. 5). Therefore, the controllability of f _o by adjusting I _B is realized. Cursor measurements show that the actual values for f _o is 5.84 MHz when I _B is 50 µA. The relative error is then -4.6%.

Fig. 5 Relationship of resonant frequency and I B

Cursor measurements show that the total harmonic distortions for I _o1 and I _o2 are 0.86% and 1.02%, respectively. Figure 6 shows only the simulated output spectrum for I _o1. In the output waveform, the even-order harmonic components are much lower than the odd-order harmonics due to the fact that the output is positive and negative symmetrical waveforms because of limitations from the output saturation. Consequently, the oscillator can produce two output signals with small distortion.

Fig. 6 Output spectrum of I o1 for the design value of f o=612 kHz

The power consumed by the oscillator is about 3.10 mW. There is no doubt that the simulation results are consistent with the applied theory.

3 Conclusion

The existing current-mode Wien-bridge oscillators using a variety of new types of active devices employ either the grounded-capacitor Wien-bridge oscillator or the traditional Wien-bridge oscillator. The former must use grounded capacitors, which is easy to integrate but has poor selectivity, while the latter must use a floating capacitor, which has good selectivity but is not easy to integrate. So far the current-mode Wien-bridge oscillators using the traditional Wien-bridge oscillator as topology and using grounded capacitors have not been developed, so this is the first paper to give the answer to this question. In addition, the designed oscillator enjoys the following features: orthogonal electronic control of resonant condition and resonant frequency, use of grounded capacitors, good selectivity of frequencies, no externally connected resistors and high output impedances.

References

All Figures

	Fig. 1 Traditional Wien-bridge oscillator using Op Amp
In the text

	Fig. 2 Lead-lag network of Fig. 1 (a) RC series and parallel network; (b) Adjoint network of (a)
In the text

	Fig. 3 Designed oscillator with CCCII1 serving a floating resistor R 1 and CCCII2 serving a grounded resistor R 2
In the text

	Fig. 4 Initial results of the transient analysis for Fig. 3
In the text

	Fig. 5 Relationship of resonant frequency and I B
In the text

	Fig. 6 Output spectrum of I o1 for the design value of f o=612 kHz
In the text