Quadrature Oscillators Using Operational Amplifiers

Abstract

Two new quadrature oscillator circuits using operational amplifiers are presented. Outputs of two sinusoidal signals with 90° phase difference are available in each circuit configuration. Both proposed quadrature oscillators are based on third-order characteristic equations. The oscillation conditions and oscillation frequencies of the proposed quadrature oscillators are orthogonally controllable. The circuits are implemented using the widely available operational amplifiers which results in low output impedance and high current drive capability. Experimental results are included.

1. Introduction

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 [1–7].

Recently, several multiphase oscillators based on operational amplifiers were proposed [6–11]. Two-integrator loop technique was developed to realize quadrature oscillators using operational amplifiers [6]. In 1993 [7], Holzel proposed a new method for realizing quadrature oscillator, which consists of two all-pass filters and one inverter using operational amplifiers. Several multiphase oscillators using operational amplifiers were proposed in [8–11]. However, the quadrature output voltages cannot be obtained from [8–10]. The multiphase sinusoidal oscillator in [11] was constructed by cascading several first-order all-pass networks and unity-gain inverting networks. However, the block diagram of the quadrature oscillators in [11] was the same with [7].

In this paper, two new quadrature oscillator circuits using operational amplifiers are proposed. Outputs of two sinusoidal signals with 90° phase difference are available in each proposed circuit configuration. Both proposed quadrature oscillators are based on third-order characteristic equations. The oscillation conditions and oscillation frequencies of the proposed quadrature oscillators are orthogonally controllable. The circuits are implemented using the widely available operational amplifiers which results in low output impedance, high current drive capability (enabling the systems to drive a variety of loads), simplicity, and low cost.

2. Circuit Description

Figure 1 shows the first proposed quadrature oscillator circuit. The characteristic equation of the circuit can be expressed as𝑠³𝐶₁𝐶₂𝐶₃𝑅₁𝑅₂𝑅₃𝑅₄𝑅₅+𝑠²𝐶₃𝑅₃𝑅₄𝑅₅𝐶₁𝑅₁+𝐶₂𝑅₂+𝑠𝐶₃𝑅₃𝑅₄𝑅₅+𝑅₁𝑅₂=0. (1)

Figure 1 

The first proposed quadrature oscillator circuit.

At 𝑠=𝑗𝜔, by equating the real and imaginary parts with zero, the oscillation condition and oscillation frequency can be obtained as𝑅₃𝑅₄𝑅₅=𝐶₁𝐶₂𝑅₁²𝑅₂²𝐶₃𝐶₁𝑅₁+𝐶₂𝑅₂, (2)𝜔_𝑜=1𝐶₁𝐶₂𝑅₁𝑅₂. (3) From (2) and (3), the oscillation condition and oscillation frequency can be orthogonally controllable.

From Figure 1, the voltage transfer function from 𝑉_𝑜2 to 𝑉_𝑜1is𝑉_𝑜2𝑉_𝑜11=−𝑠𝐶₃𝑅₄. (4) The phase difference, 𝜙, between 𝑉_𝑜2 and 𝑉_𝑜1 is𝜙=90^∘ (5) ensuring the voltage 𝑉_𝑜2 and 𝑉_𝑜1 to be in quadrature. Because the output impedance of the operational amplifier is very small, the two output terminals, 𝑉_𝑜1 and 𝑉_𝑜2, can be directly connected to the next stage, respectively.

The passive sensitivities of the quadrature oscillator in Figure 1 are all low and obtained as𝑆^𝜔𝑜_{𝐶1,𝐶2,𝑅1,𝑅2}1=−2. (6)

Figure 2 shows the second proposed quadrature oscillator circuit. The characteristic equation of the circuit can be expressed as𝑠³𝐶₁𝐶₂𝐶₃𝐶₄𝐶₅𝑅₁𝑅₂𝑅₃+𝑠²𝐶₃𝐶₄𝐶₅𝑅₃𝐶₁𝑅₁+𝐶₂𝑅₂+𝑠𝐶₃𝐶₄𝐶₅𝑅₃+𝐶₁𝐶₂=0. (7)

Figure 2 

The second proposed quadrature oscillator circuit.

At 𝑠=𝑗𝜔, by equating the real and imaginary parts with zero, the oscillation condition and oscillation frequency can be obtained as𝑅₃=𝐶₁²𝐶₂²𝑅₁𝑅₂𝐶₃𝐶₄𝐶₅𝐶₁𝑅₁+𝐶₂𝑅₂, (8)𝜔_𝑜=1𝐶₁𝐶₂𝑅₁𝑅₂. (9) From (8) and (9), the oscillation condition and oscillation frequency can be orthogonally controllable.

From Figure 2, the voltage transfer function from 𝑉_𝑜2 to 𝑉_𝑜1 is𝑉_𝑜2𝑉_𝑜11=−𝑠𝐶₃𝑅₃. (10) The phase difference, 𝜙, between 𝑉_𝑜2 and 𝑉_𝑜1 is𝜙=90^∘ (11) ensuring the voltage 𝑉_𝑜2 and 𝑉_𝑜1 to be in quadrature. Because the output impedance of the operational amplifier is very small, the two output terminals, 𝑉_𝑜1 and 𝑉_𝑜2, can be directly connected to the next stage, respectively.

The passive sensitivities of the quadrature oscillator in Figure 2 are all low and obtained as𝑆^𝜔𝑜_{𝐶1,𝐶2,𝑅1,𝑅2}1=−2. (12)

3. Experimental Results

The quadrature oscillator in Figure 1 was constructed using LF351s. Figure 3 represents the quadrature sinusoidal output waveforms of Figure 1 with 𝐶₁ = 𝐶₂ = 𝐶₃ = 1 nF, 𝑅₁ = 𝑅₂ = 𝑅₄ = 𝑅₅ = 10 kΩ, 𝑅₃ = 4.563 kΩ, and the power supply ±10 V. Figure 4 shows the experimental results of the oscillation frequency of Figure 1 by varying the value of 𝑅 (𝑅 = 𝑅₁ = 𝑅₂ = 𝑅₄ = 𝑅₅) with 𝐶₁ = 𝐶₂ = 𝐶₃ = 1 nF, and 𝑅₃ was varied with 𝑅 by (2) to ensure the oscillations will start.

Figure 3 

The experimental quadrature output waveforms of Figure 1.

Figure 4 

Experimental results of the oscillation frequency of Figure 1, which is obtained by varying the value of R; o o o, experimental results; —, ideal curve.

The quadrature oscillator in Figure 2 was constructed using LF351s. Figure 5 represents the quadrature sinusoidal output waveforms of Figure 2 with 𝐶₁ = 𝐶₂ = 𝐶₃ = 𝐶₄ = 𝐶₅ = 1 nF, 𝑅₁ = 𝑅₂ = 10 kΩ, 𝑅₃ = 4.767 kΩ, and the power supply ±10 V. Figure 6 shows the experimental results of the oscillation frequency of Figure 2 by varying the value of 𝑅 (𝑅 = 𝑅₁ = 𝑅₂) with 𝐶₁ = 𝐶₂ = 𝐶₃ = 𝐶₄ = 𝐶₅ = 1 nF, and 𝑅₃ was varied with 𝑅 by (8) to ensure the oscillations will start.

Figure 5 

The experimental quadrature output waveforms of Figure 2.

Figure 6 

Experimental results of the oscillation frequency of Figure 2, which is obtained by varying the value of R; o o o, experimental results; —, ideal curve.

4. Conclusions

Two new quadrature oscillator circuits based on operational amplifiers are presented. The proposed quadrature oscillators provide the following advantages: (i) two sinusoidal output signals of 90° phase difference are obtained simultaneously in each configuration; (ii) the oscillation conditions and oscillation frequencies are orthogonally controllable; (iii) the output terminals have the advantages of low output impedances and high current drive capability; (iv) simplicity and low cost; (v) the passive sensitivities are low.