The Wien bridge is one of the simplest and best known oscillators and is used extensively in circuits for audio applications. Figure 19.7 shows the basic Wien bridge circuit configuration. This circuit has only a few components and good frequency stability. The major drawback of the circuit is that the output amplitude is at the rails, saturating the op amp output transistors and causing high output distortion. Taming this distortion is more of a challenge than getting the circuit to oscillate. A couple of ways may be used to minimize this effect, which is covered later. It is now time to analyze this circuit and come up with the transfer function.

The Wien bridge circuit is of the form detailed in Section 19.6. The transfer function for the circuit is created using the technique described in that section. It is readily apparent that Z₁ = R_G, Z₂ = R_F, Z₃ = (R₁ + 1/sC₁), and Z₄ = (R₂‖ 1/sC₂). The loop is broken between the output and Z₁, V_TEST is applied to Z₁, and V_OUT is calculated. The positive feedback voltage, V₊, is calculated first in Equations (19.6) through (19.8). Equation (19.6) shows the simple voltage divider at the noninverting input. Each term is then multiplied by (R₂C₂s + 1) and divided by R₂ to get Equation (19.7).

Substitute s = jω₀, where ω₀ is the oscillation frequency, ω₁ = 1/R₁C₂, and ω₂ = 1/R₂C₁ to get Equation (19.8):

Some interesting relationships now become apparent. The capacitor in the zero, represented by ω₁, and the capacitor in the pole, represented by ω₂, must each contribute 90° of phase shift toward the 180° required for oscillation at ω₀. This requires that C₁ = C₂ and R₁ = R₂. Setting ω₁ and ω₂ equal to ω₀ cancels the frequency terms, ideally removing any change in amplitude with frequency, since the pole and zero negate one another. An overall feedback factor of β = 1/3 is the result, Equation (19.9):

The gain of the negative feedback portion, A, of the circuit must then be set such that |Aβ| = 1, requiring A = 3. R_F must be set to twice the value of R_G to satisfy the condition. The op amp in Figure 19.7 is single supply, so a DC reference voltage, V_REF, must be applied to bias the output for full scale swing and minimal distortion. Applying V_REF to the positive input through R₂ restricts DC current flow to the negative feedback leg of the circuit. V_REF is set at 0.833V to bias the output at the midrail of the single supply, rail to rail input, and output amplifier, or 2.5 V. See Chapter 4 for details on DC biasing single supply op amps. V_REF is shorted to ground for split supply applications.

The final circuit is shown in Figure 19.8, with component values selected to provide an oscillation frequency of ω₀ = 2πf₀, where f₀ = 1/(2πRC) = 19.9 kHz. The circuit oscillated at 1.57 kHz due to slightly varying component values with 2% distortion. This high value is due to the extensive clipping of the output signal at both supply rails, producing several large odd and even harmonics. The feedback resistor was then adjusted ±1%. Figure 19.9 shows the output voltage waveforms. The distortion grows as the saturation increases with increasing R_F, and oscillations cease when R_F is decreased by more than 0.8%.

Applying nonlinear feedback can minimize the distortion inherent in the basic Wien bridge circuit. A nonlinear component, such as an incandescent lamp, can be substituted into the circuit for R_G, as shown in Figure 19.10. The lamp resistance, R_LAMP, is nominally selected as half the feedback resistance, R_F, at the lamp current established by R_F and R_LAMP. When the power is first applied, the lamp is cool and its resistance is small, so the gain is large (>3). The current heats the filament and the resistance increases, lowering the gain. The nonlinear relationship between the lamp current and resistance keeps output voltage changes small. Figure 19.11 shows the output of this amplifier with a distortion of 1% for f_OSC = 1.57 kHz. The distortion for this variation is reduced over the basic circuit by avoiding hard saturation of the op amp transistors.

The impedance of the lamp is due mostly to thermal effects. The output amplitude is then very temperature sensitive and tends to drift. The gain must be set higher than 3 to compensate for any temperature variations, which increases the distortion in the circuit [4]. This type of circuit is useful when the temperature does not fluctuate over a wide range or when used in conjunction with an amplitude limiting circuit.

The lamp has an effective low frequency thermal time constant, t_thermal [5]. As f_OSC approaches t_thermal, distortion is greatly increased. Several lamps can be placed in series to increase t_thermal and reduce distortion. The drawbacks are that the time required for oscillations to stabilize is increased and the output amplitude is reduced.

An automatic gain control circuit must be used when neither of the two previous circuits yield low distortion. A typical Wien bridge oscillator with an AGC circuit is shown in Figure 19.12, with the output waveform of the circuit shown in Figure 19.13. The AGC is used to stabilize the magnitude of the sinusoidal output to an optimum gain level. The JFET serves as the AGC element, providing excellent control because of the wide range of the drain to source resistance (R_DS), which is controlled by the gate voltage. The JFET gate voltage is 0 V when the power is applied, and the JFET turns on with low R_DS. This places R_G2 + R_S + R_DS in parallel with R_G1, raising the gain to 3.05, and oscillations begin and gradually build up. As the output voltage gets large, the negative swing turns the diode on and the sample is stored on C₁, which provides a DC potential to the gate of Q₁. Resistor R₁ limits the current and establishes the time constant for charging C₁, which should be much greater than f_OSC. When the output voltage drifts high, R_DS increases, lowering the gain to a minimum of 2.87 (1 + R_F/R_G1). The output stabilizes when the gain reaches 3. The distortion of the AGC is 0.8%, which is due to slight clipping at the positive rail.

The circuit of Figure 19.12 is biased with V_REF for a single supply amplifier. A zener diode can be placed in series with D₁ to limit the positive swing of the output and reduce distortion. A split supply can be easily implemented by grounding all points connected to V_REF. A wide variety of Wien bridge variations exist to more precisely control the amplitude and allow selectable or even variable oscillation frequencies. Some circuits use diode limiting in place of a nonlinear feedback component. The diodes reduce the distortion by providing a soft limit for the output voltage.

Phase shift oscillators have less distortion than the Wien bridge oscillator, coupled with good frequency stability. A phase shift oscillator can be built with one op amp as shown in Figure 19.14; the resulting output waveform is in Figure 19.15. Three RC sections are cascaded to get the steep dφ/dω slope, as described in Section 19.3, to get a stable oscillation frequency. Any less and the oscillation frequency is high and interferes with the op amp BW limitations.

The normal assumption is that the phase shift sections are independent of each other. Then Equatiotn (19.10) is written:

The loop phase shift is –180° when the phase shift of each section is –60°, and this occurs when ω = 2πf = 1.732/RC because the tangent of 60° = 1.732. The magnitude of β at this point is (1/2)³, so the gain, A, must be equal to 8 for the system gain to be equal to 1.

The oscillation frequency with the component values shown in Figure 19.14 is 3.76 kHz rather than the calculated oscillation frequency of 2.76 kHz. Also, the gain required to start oscillation is 27 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 RC 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, small, and come four in a package, so the single op amp phase shift oscillator is losing popularity. The output distortion is a low 0.46%, considerably less than the Wein bridge circuit without amplitude stabilization.

The buffered phase shift oscillator is much improved over the unbuffered version, the cost being a higher component count. The buffered phase shift oscillator is shown in Figure 19.16 and the resulting output waveform in Figure 19.17. The buffers prevent the RC sections from loading each other, hence the buffered phase shift oscillator performs closer to the calculated frequency and gain. The gain setting resistor, R_G, loads the third RC section. If the fourth buffer in a quad op amp buffers this RC section, the performance becomes ideal. Low distortion sine waves can be obtained from either phase shift oscillator design, but the purest sine wave is taken from the output of the last RC section. This is a high impedance node, so a high impedance input is mandated to prevent loading and frequency shifting with load variations.

The bubba oscillator in Figure 19.18 is another phase shift oscillator, but it takes advantage of the quad op amp package to yield some unique advantages. Four RC sections require 45° phase shift per section, so this oscillator has an excellent dφ/dt, resulting in minimized frequency drift. The RC sections each contribute 45° phase shift, so taking outputs from alternate sections yields low impedance quadrature outputs. When an output is taken from each op amp, the circuit delivers four 45° phase shifted sine waves. The loop equation is given in Equation (19.11). When ω = 1/RCs, Equation (19.11) reduces to (19.12) and (19.13):

The gain, A, must equal 4 for oscillation to occur. The test circuit oscillated at 1.76 kHz rather than the ideal frequency of 1.72 kHz when the gain was 4.17 rather than the ideal gain 4. The output waveform is shown in Figure 19.19. Distortion is 1% for V_OUTSINE and 0.1% for V_OUTCOSINE. With low gain A and low bias current op amps, the gain setting resistor, R_G, does not load the last RC section, thus ensuring oscillator frequency accuracy. Very low distortion sine waves can be obtained from the junction of R and R_G. When low distortion sine waves are required at all outputs, the gain should be distributed among all the op amps. The noninverting input of the gain op amp is biased at 0.5 V to set the quiescent output voltage at 2.5 V for single supply operation and should be ground for split supply op amps. Gain distribution requires biasing of the other op amps, but it has no effect on the oscillator frequency.

The quadrature oscillator shown in Figure 19.20 is another type of phase shift oscillator, but the three RC sections are configured so each section contributes 90° of phase shift. This provides both sine and cosine waveform outputs (the outputs are quadrature, or 90° apart), which is a distinct advantage over other phase shift oscillators. The idea of the quadrature oscillator is to use the fact that the double integral of a sine wave is a negative sine wave of the same frequency and phase. The phase of the second integrator is then inverted and applied as positive feedback to induce oscillation [6].

The loop gain is calculated in Equation (19.14). When R₁C₁ = R₂C₂ = R₃C₃, Equation (19.14) reduces to Equation (19.15). When ω = 1/RC, Equation (19.14) reduces to 1∠–180, so oscillation occurs at ω = 2πf = 1/RC. The test circuit oscillated at 1.65 kHz rather than the calculated 1.59 kHz, as shown in Figure 19.21. This discrepancy is attributed to component variations. Both outputs have relatively high distortion that can be reduced with a gain stabilizing circuit. The sine output had 0.846% distortion and the cosine output had 0.46% distortion. Adjusting the gain can increase the amplitudes. The cost is bandwidth.