# Design of op amp sine wave oscillators

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1 Design of op amp sine wave oscillators By on Mancini Senior Application Specialist, Operational Amplifiers riteria for oscillation The canonical form of a feedback system is shown in Figure, and Equation describes the performance of any feedback system (an amplifier with passive feedback components constitutes a feedback system). Figure. anonical form of a feedback circuit V IN Σ VOUT A VIN Aβ () Oscillation results from an unstable state; i.e., the feedback system can t find a stable state because its transfer function can t be satisfied. Equation becomes unstable when (Aβ) 0 because A/0 is an undefined state. Thus, the key to designing an oscillator is to insure that Aβ (called the Barkhausen criterion), or using complex math the equivalent expression is Aβ 80. The 80 phase shift criterion applies to negative feedback systems, and 0 phase shift applies to positive feedback systems. The output voltage of a feedback system heads for infinite voltage when Aβ. When the output voltage approaches either power rail, the active devices in the amplifiers change gain, causing the value of A to change so the value of Aβ ; thus, the charge to infinite voltage slows down and eventually halts. At this point one of three things can occur. First, nonlinearity in saturation or cutoff can cause the system to become stable and lock up. Second, the initial charge can cause the system to saturate (or cut off) and stay that way for a long time before it becomes linear and heads for the opposite power rail. Third, the system stays linear and reverses direction, heading for the opposite power rail. Alternative two produces highly distorted oscillations (usually quasi square waves), and the resulting oscillators are called relaxation oscillators. Alternative three produces sine wave oscillators. All oscillator circuits were built with TLV27X op amps, 5% resistors, and A β Figure 2. Phase plot of sections Phase Shift, φ (Degrees) % capacitors; hence, component tolerances cause differences between ideal and measured values. Phase shift in oscillators The 80 phase shift in the equation Aβ 80 is introduced by active and passive components. Like any well-designed feedback circuit, oscillators are made dependent on passive component phase shift because it is accurate and almost drift-free. The phase shift contributed by active components is minimized because it varies with temperature, has a wide initial tolerance, and is devicedependent. Amplifiers are selected such that they contribute little or no phase shift at the oscillation frequency. A single pole L or circuit contributes up to 90 phase shift per pole, and because 80 is required for oscillation, at least two poles must be used in oscillator design. An L circuit has two poles; thus, it contributes up to 80 phase shift per pole pair, but L and L oscillators are not considered here because low frequency inductors are expensive, heavy, bulky, and non-ideal. L oscillators are designed in high-frequency applications, beyond the frequency range of voltage feedback op amps, where the inductor size, weight, and cost are less significant. Multiple sections are used in low-frequency oscillator design in lieu of inductors. Phase shift determines the oscillation frequency because the circuit oscillates at the frequency that accumulates 80 phase shift. The rate of change of phase with frequency, dφ/dt, determines frequency stability. When buffered sections (an op amp buffer provides highinput and low-output impedance) are cascaded, the phase shift multiplies by the number of sections, n (see Figure 2). ontinued on next page section 2 sections 3 sections -35 sections Normalized Frequency 33

2 Texas Instruments Incorporated ontinued from previous page Although two cascaded sections provide 80 phase shift, dφ/dt at the oscillator frequency is low, thus oscillators made with two cascaded sections have poor frequency stability. Three equal cascaded filter sections have a higher dφ/dt, and the resulting oscillator has improved frequency stability. Adding a fourth section produces an oscillator with an excellent dφ/dt, thus this is the most stable oscillator configuration. Four sections are the maximum number used because op amps come in quad packages, and the four-section oscillator yields four sine waves that are 5 phase shifted relative to each other, so this oscillator can be used to obtain sine/cosine or quadrature sine waves. rystal or ceramic resonators make the most stable oscillators because resonators have an extremely high dφ/dt resulting from their non-linear properties. esonators are used for high-frequency oscillators, but low-frequency oscillators do not use resonators because of size, weight, and cost restrictions. Op amps are not used with crystal or ceramic resonator oscillators because op amps have low bandwidth. Experience shows that it is more cost-effective to build a high-frequency crystal oscillator and count down the output to obtain a low frequency than it is to use a low-frequency resonator. Gain in oscillators The oscillator gain must equal one (Aβ 80 ) at the oscillation frequency. The circuit becomes stable when the gain exceeds one and oscillations cease. When the gain exceeds one with a phase shift of 80, the active device non-linearity reduces the gain to one. The non-linearity happens when the amplifier swings close to either power rail because cutoff or saturation reduces the active device (transistor) gain. The paradox is that worst-case design practice requires nominal gains exceeding one for manufacturability, but excess gain causes more distortion of the output sine wave. When the gain is too low, oscillations cease under worstcase conditions, and when the gain is too high, the output wave form looks more like a square wave than a sine wave. Distortion is a direct result of excess gain overdriving the amplifier; thus, gain must be carefully controlled in lowdistortion oscillators. Phase-shift oscillators have distortion, but they achieve low-distortion output voltages because cascaded sections act as distortion filters. Also, buffered phase-shift oscillators have low distortion because the gain is controlled and distributed among the buffers. Some circuit configurations (Wien-bridge) or lowdistortion specifications require an auxiliary circuit to adjust the gain. Auxiliary circuits range from inserting a non-linear VETUN VOUT s s where s jω and j. s 3 s s, 3 j ω ω Figure 3. Wien-bridge circuit schematic component in the feedback loop, to automatic gain control (AG) loops, to limiting by external components. Wien-bridge oscillator Figure 3 gives the Wien-bridge circuit configuration. The loop is broken at the positive input, and the return signal is calculated in Equation 2 below. When ω 2πf /, the feedback is in phase (this is positive feedback), and the gain is /3, so oscillation requires an amplifier with (2) F 2 G 20 k 5 V G TLV V Figure. Wien-bridge oscillator with non-linear feedback Lamp L F V -V a gain of 3. When F 2 G, the amplifier gain is 3 and oscillation occurs at f /2π. The circuit oscillated at.65 khz rather than.59 khz with the component values shown in Figure 3, but the distortion 3 Analog and Mixed-Signal Products August 2000 Analog Applications Journal

3 is noticeable. Figure shows a Wien-bridge circuit with non-linear feedback. The lamp resistance, L, is nominally selected as half the feedback resistance, F, at the lamp current established by F and L. The non-linear relationship between the lamp current and resistance keeps output voltage changes small. Some circuits use diode limiting in place of a non-linear feedback component. The diodes reduce the distortion by providing a soft limit for the output voltage. AG must be used when neither of these techniques yields low distortion. A typical Wien-bridge oscillator with an AG circuit is shown in Figure 5. The negative sine wave is sampled by D, and the sample is stored on. and 2 are chosen to center the bias on Q so that ( G Q ) F /2 at the desired output voltage. When the output voltage drifts high, Q increases resistance, thus decreasing the gain. In the oscillator shown in Figure 3, the volt power supply is applied to the positive op amp input to center the output quiescent voltage at V /2. Phase-shift oscillator (one op amp) A phase-shift oscillator can be built with one op amp as shown in Figure 6. The normal assumption is that the phase-shift sections are independent of each other. Then Equation 3 is written: 3 Aβ A (3) s The loop phase shift is 80 when the phase shift of each section is 60, and this occurs when ω 2πf.732/ because the tangent The magnitude of β at this point is (/2) 3, so the gain, A, must be equal to 8 for the system gain to be equal to. The oscillation frequency with the component values shown in Figure 6 is 3.76 khz rather than the calculated Figure 5. Wien-bridge oscillator with AG Q 2 G oscillation frequency of 2.76 khz. Also, the gain required to start oscillation is 26 rather than the calculated gain of 8. These discrepancies are partially due to component variations, but the biggest contributing factor is the incorrect assumption that the sections do not load each other. This circuit configuration was very popular when active components were large and expensive, but now op amps are inexpensive and small and come four in a package, so the single op amp phase-shift oscillator is losing popularity. F V -V D ontinued on next page Figure 6. Phase-shift oscillator (one op amp) F G 55.2 k.5 M 5 V TLV27 35

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