# Derivation of Oscillators from Biquadratic Band Pass Filters Using Circuit Transformations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Proposed Method

_{2}of the obtained circuit would be the complement (COMP) of the original voltage transfer function T

_{1}; i.e., T

_{2}= 1 − T

_{1}. According to inverse transformation by interchanging the output norator and the input voltage source of a circuit (with its transfer function T

_{3}= Y/U), the inverse transfer function (INV) can be obtained as T

_{3}' = 1/T

_{3}= Y'/U'), as shown in Figure 2 [12]. The inverse network can also be obtained by interchanging the output pathological current mirror and the input source of a circuit [19]. To obtain a sinusoidal oscillator from the biquadratic band pass filter, the combination of circuit transformations can be exploited, as the stages shown in Table 1 and Table 2.

**Figure 1.**The complementary transformation for voltage-mode a network. (

**a**) Original network; (

**b**) complementary network of (

**a**).

**Figure 2.**The inverse transformation of a circuit. (

**a**) Original network; (

**b**) inverse network of (

**a**).

**Table 1.**The combination of complementary-inverse transformation (TRANSF) for a voltage-mode network. COMP, complement; INV, inverse transfer function.

Procedure | Stage (0) | Stage (1): | Stage (2): | Stage (3): |
---|---|---|---|---|

COMP of (0) | INV of (1) | COMP of (2) | ||

TRANSF | T | 1-T | $\frac{\text{1}}{\text{1-T}}$ | $\frac{\text{T}}{\text{T-1}}$ |

Procedure | Stage (0) | Stage (1): | Stage (2): | Stage (3): |
---|---|---|---|---|

INV of (0) | COMP of (1) | INV of (2) | ||

TRANSF | T | 1/T | $\frac{\text{T-1}}{\text{T}}$ | $\frac{\text{T}}{\text{T-1}}$ |

_{o}and bandwidth (BW) of the filter can be computed as ω

_{o}= $\sqrt{c}$ and BW = ω

_{o}/Q = b, where Q is the quality factor of the filter. The coefficient b and c are always greater than zero for a stable system.

_{o}= $\sqrt{c}$ and ω

_{o}/Q = (b − a) is derived.

_{f}(s) can be expressed by:

_{o}the loop gain A(s)B(s) is equal to unity, A

_{f}will be infinite. That is, at this frequency, the system defined as an oscillator will have a finite output for zero input signal. Thus, the condition for the feedback loop of Figure 3 to provide sinusoidal oscillations of frequency ω

_{o}is:

_{o}= $\sqrt{c}$.

**Figure 3.**A positive-feedback system with a stable amplifier stage, A, and a frequency-selective network, B.

## 3. Application Examples and Discussion

_{Z}= ±αI

_{X}, V

_{X}= βV

_{Y}, where α = 1 − e

_{i}and e

_{i}denotes the current tracking error, β = 1 − e

_{v}and e

_{v}denotes the voltage tracking error of a current conveyor, respectively, the non-ideal characteristic equation of the oscillator is expressed by Equation (9). Its oscillation condition and oscillation frequency ω

_{o}are given by Equations (10) and (11), respectively. The sensitivities of ω

_{o}to active and passive components are given in Equation (12). It is clear that the obtained oscillator has identical passive sensitivities and has improved active sensitivities compared to the original band pass filter [21]. In addition, the oscillation frequency and the oscillation condition are independently adjustable through with R

_{3}and R

_{1}. In reality, R

_{1}must be slightly greater than R

_{2}/α

_{1}for the startup of oscillation [22].

**Figure 4.**(

**a**) A CCII-based band pass filter; (

**b**) the obtained filter after applying the transformations in Table 1.

_{2}= R

_{3}= R and C

_{1}= C

_{2}= C. It can be found that the oscillation condition and oscillation frequency are orthogonal adjustable. It must be noted that R

_{1}must be slightly less than R/3 for the startup of oscillation.

**Figure 6.**(

**a**) A CCII-based band pass filter; (

**b**) the obtained filter after applying the transformations in Table 2.

_{1}. It is known that the ground node of a nullor-based oscillator can be chosen arbitrarily without affecting the characteristic equation [24]. Therefore, the grounded capacitor oscillator can be obtained by selecting Node A in Figure 7b as the ground. In addition, more oscillators can be obtained by applying adjoint network theorem or RC-CR transformation.

## 4. Simulation Results

_{1}= C

_{2}= C = 0.1 µF and R

_{2}= R

_{3}= 1.6 kΩ. As shown in Figure 8, the oscillation eventually died out with R

_{1}= 1.6 kΩ, and the circuit produced a pure sinusoidal waveform with R

_{1}= 1.68 kΩ. Figure 9 shows the oscillation frequency for various values of C. It confirms the independently tunability of the oscillation frequency. For the oscillator in Figure 7b, we choose C

_{1}= C

_{2}= 27 nF and R

_{2}= R

_{3}= 3.3 kΩ. Figure 10 shows the outputted waveforms with R

_{1}= 1.07 kΩ and R

_{1}= 1.1 kΩ. All if the simulation results are consistent with our prediction in Section 3.

**Figure 8.**The output waveforms for the oscillator in Figure 4b with different oscillation conditions.

**Figure 10.**The output waveforms for the oscillator in Figure 6 with different oscillation conditions.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, H.-Y.; Tran, H.-D.; Nguyen, Q.-M.; Yin, L.-T.; Liu, C.-Y. Derivation of Oscillators from Biquadratic Band Pass Filters Using Circuit Transformations. *Appl. Sci.* **2014**, *4*, 482-492.
https://doi.org/10.3390/app4040482

**AMA Style**

Wang H-Y, Tran H-D, Nguyen Q-M, Yin L-T, Liu C-Y. Derivation of Oscillators from Biquadratic Band Pass Filters Using Circuit Transformations. *Applied Sciences*. 2014; 4(4):482-492.
https://doi.org/10.3390/app4040482

**Chicago/Turabian Style**

Wang, Hung-Yu, Huu-Duy Tran, Quoc-Minh Nguyen, Li-Te Yin, and Chih-Yi Liu. 2014. "Derivation of Oscillators from Biquadratic Band Pass Filters Using Circuit Transformations" *Applied Sciences* 4, no. 4: 482-492.
https://doi.org/10.3390/app4040482