Designing sine-wave oscillators

Ron Mancini, Texas Instruments -- EDN, 4/27/2000

Figure 1 shows the canonical form of a feedback system, and the following equation describes the performance of any negative feedback system.

Oscillation results from an unstable state because you cannot satisfy the feedback-system transfer function. The equation becomes unstable when (1+A)=0 because A/0 is an undefined state, so the key to designing an oscillator is to insure that A=–1=1180°.

Active and passive components introduce phase shift. Oscillators depend on passive components because passive-component phase shift is accurate and almost drift-free. Active-component phase shift is minimized because it varies with temperature, has a wide initial tolerance, and is device-dependent. Selected amplifiers contribute little or no phase shift at the oscillation frequency. A single-pole RL or RC circuit contributes 90° phase shift per pole; thus, these designs need two poles to achieve oscillation. You cannot consider inductive oscillators for these designs because low-frequency inductors are expensive, heavy, and bulky. You use multiple RC sections instead of inductors in low-frequency-oscillator designs.

The circuit oscillates at the frequency that accumulates –180° phase shift, so phase shift determines the oscillation frequency. The rate of change of phase with frequency, d/df, determines frequency stability. When you cascade buffered RC sections, the phase shift multiplies by the number of sections. Two cascaded RC sections provide 180° phase shift with poor d/df, but four cascaded RC sections provide 180° phase shift (45° per section) with improved frequency stability. Four sections are the maximum number you can use because op amps come in quad packages, and the four-section oscillator yields four sine waves 45° phase-shifted from each other. You can thus use this oscillator to obtain sine/cosine or quadrature sine waves.

Crystal or ceramic resonators make the most stable oscillators because resonators have an extremely high d/df resulting from their nonlinear properties. High-frequency oscillators use resonators, but low-frequency oscillators do not because of size, weight, and cost restrictions.

The oscillator gain must equal one (A+180°) at the oscillation frequency. The circuit can become stable when the gain exceeds one, and oscillations cease even though there is –180° phase shift. At A>1 with a phase shift of –180°, active-device nonlinearity usually reduces the gain to one. Nonlinearity occurs when the amplifier output swings close to either power rail because cutoff or saturation reduces transistor gain. The paradox is that worst-case design practice requires nominal gains exceeding one for manufacturability, and excess gain distorts the output sine wave.

When the nominal gain is too low, oscillations cease under worst-case conditions, and, when the nominal gain is too high, the output waveform is a square wave rather than a sine wave. Phase-shift oscillators have distortion, but they achieve low-distortion output voltages because cascaded RC sections act as distortion filters. Auxiliary methods to control gain range from inserting a nonlinear component into the feedback loop to employing automatic-gain-control loops to limiting external components. Our next column uses this theory to design a phase shift oscillator.

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