Quadrature Oscillators Using Operational Amplifiers

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.


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][2][3][4][5][6][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.

Circuit Description
Figure 1 shows the first proposed quadrature oscillator circuit.The characteristic equation of the circuit can be expressed as (1) At s = jω, by equating the real and imaginary parts with zero, the oscillation condition and oscillation frequency can be obtained as From ( 2) and ( 3), the oscillation condition and oscillation frequency can be orthogonally controllable.
The phase difference, φ, between V o2 and V o1 is 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 as Figure 2 shows the second proposed quadrature oscillator circuit.The characteristic equation of the circuit can be expressed as At s = jω, by equating the real and imaginary parts with zero, the oscillation condition and oscillation frequency can be obtained as From ( 8) and ( 9), the oscillation condition and oscillation frequency can be orthogonally controllable.
From Figure 2, the voltage transfer function from The phase difference, φ, between V o2 and V o1 is 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 as The first proposed quadrature oscillator circuit.
The second proposed quadrature oscillator circuit.

Experimental Results
The quadrature oscillator in Figure 1 was constructed using LF351s.Figure 3 represents the quadrature sinusoidal output waveforms of Figure 1 with 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 nF, and R 3 was varied with R by (2) to ensure the oscillations will start.The quadrature oscillator in Figure 2 was constructed using LF351s.Figure 5 represents the quadrature sinusoidal output waveforms of Figure 2 with 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 nF, and R 3 was varied with R by (8) to ensure the oscillations will start.

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