Wien bridge oscillator

Classic Wien bridge oscillator

A Wien bridge oscillator is a type of electronic oscillator that generates sine waves. It can generate a large range of frequencies. The oscillator is based on a bridge circuit originally developed by Max Wien in 1891. The bridge comprises four resistors and two capacitors. The oscillator can also be viewed as a positive feedback system combined with a bandpass filter.

The modern circuit is derived from William Hewlett's 1939 Stanford University master's degree thesis. Hewlett figured out how to make the oscillator with a stable output amplitude and low distortion. Hewlett, along with David Packard, co-founded Hewlett-Packard, and Hewlett-Packard's first product was the HP200A, a precision Wien bridge oscillator.

The frequency of oscillation is given by:

$f={\frac {1}{2\pi RC}}$

Background

Wien bridge oscillator is based on a few fundamental circuit concepts. There are three different viewpoints at this exotic circuit solution; all they are right and can be used to explain it.

Positive feedback amplifier with high open-loop gain. Wien bridge oscillator can be considered as a combination of an op-amp and a Wien bridge connected in the positive feedback loop between the op-amp output and the differential input. At the oscillating frequency, the bridge is balanced and has very small transfer ratio. The overall loop gain is a product of the very high op-amp gain and the very low bridge ratio. As the resistive bridge arm is made nonlinear, the loop gain is dynamic - it is bigger than unity when the sine wave changes and equal to unity at the sine peaks where the nonlinear element turns on. The overall feedback can be also considered as composed of two partial feedbacks - a nonlinear negative feedback (the voltage divider connected to the inverting op-amp input) and a frequency-dependent positive feedback (the Wien network connected to the non-inverting input). Thus the feedback voltage applied to the op-amp differential input is a difference between the two partial voltages. In the parts where the sine wave changes, the positive feedback dominates over the negative one; at the sine peaks they become equivalent.

Positive feedback amplifier with small open-loop gain. From another viewpoint, the combination of the voltage divider and the op-amp can be thought as of a non-inverting amplifier with small gain determined by a nonlinear negative feedback (more than three where the sine changes and exactly three at the peaks). This low-gain amplifier is comprised by a positive feedback (the Wien network) having ratio of three at the oscillating frequency. Thus the whole circuit can be viewed as a positive feedback system combined with a bandpass filter.

Negative impedance converter with current inversion. Wien bridge oscillator can be also explained in terms of negative impedance. From this viewpoint, the voltage divider, the series RC network and the op-amp form a negative impedance converter with current inversion (INIC). It converts the complex impedance of the series RC network into negative impedance. At oscillating frequency, it is equal to the impedance of the parallel RC network and compensates it. Actually, because of the nonlinear resistor the net impedance is nonlinear as well. While the sine wave changes, the negative impedance dominates over the positive one; the total impedance is negative and makes the capacitor charge/discharge. At the sine peaks, the two impedances are equivalent and the resulting impedance is infinite; the capacitor stops charging/discharging.

Operation

The circuit operation is revealed in four steps by tracing the "trajectory" of the voltage variation across the grounded capacitor between the two supply rails. There are a few key points (marked in italic below) that help understanding the operation of Wien bridge oscillator.

Increasing. The first assumption is that in the beginning, small initial input voltage exists across the grounded capacitor (as a result of noises); this voltage urges the oscillator to run. It is amplified and returned through the series RC network back to the input; the so increased input voltage is amplified and returned again, and this avalanche-like process continuously repeats. Due to positive feedback, the output voltage is continuously higher than the voltage across the capacitor that continuously charges through the series RC network (figuratively speaking, the output voltage continuously "pulls up" the input voltage). In contrast to a typical regenerative circuit (Schmitt trigger) where this avalanche-like process evolves momentarily, here it advances slowly because the voltage across the capacitor cannot change quickly and the amplifier gain is too small.

At the top. Before the output voltage approaches the positive rail, the nonlinear feedback begins decreasing the gain; the voltage begins slowing its rate of change and the curve begins rounding. At the peak, the loop gain becomes equal to unity and there is no more regeneration. The output voltage is constant positive and equal to the input one; no charging current flows and the voltage across the capacitor stops changing.

Decreasing. The next key point is that the state at the top is not stable. The positive feedback has dynamically brought the voltage to this peak value but cannot hold it there because of the decreased (up to unity) gain. The capacitor begins discharging through the parallel-connected resistor (the upper capacitor prevents the output voltage from charging the bottom capacitor) and the voltage across the capacitor begins decreasing slowly. The nonlinear feedback begins increasing the gain and the regeneration comes gradually into operation. The output voltage is continuously lower than the voltage across the capacitor and it is continuously discharging through the series RC network (the output voltage continuously "pulls down" the input voltage).

At the bottom. Before the output voltage approaches the negative rail, the nonlinear feedback begins decreasing the gain; the voltage begins slowing its rate of change and the curve begins rounding. At the bottom, the loop gain becomes equal to unity and there is no more regeneration. The output voltage is constant negative and equal to the input one; no discharging current is drawn and the voltage across the capacitor stops changing... this state is not stable so the voltage begins increasing slowly again ... and so on and so forth...

Generalization. Sinusoidal oscillations conceive in a Wien bridge oscillator as a result of amplifying and superimposing initial small voltage across a capacitor by means of slight positive feedback (negative impedance). The direction of the voltage change is reversed at the sine peaks by decreasing the loop gain up to unity by means of a nonlinear feedback; thus "the system stays linear and reverses direction, heading for the opposite power rail". Wien bridge oscillations are closely related to relaxation oscillations as both are based on charging and discharging of a single capacitor. In contrast to them, they have a smooth shape of the sine peaks since the amplifier does not change significantly its output voltage at these points because of its extremely low (about unity) gain.

Wien bridge

Bridge circuits were a common way of measuring component values by comparing them to known values. Often an unknown component would be put in one arm of a bridge, and then the bridge would be nulled by adjusting other arms or changing the frequency of the voltage source. See, for example, the Wheatstone bridge.

The Wien bridge is one of many common bridges. Wien's bridge is used for precison measurement of capacitance in terms of resistance and frequency. It was also used to measure frequency.

The Wien bridge does not require equal values of R or C. At some frequency, the reactance of the series R_C–C_C arm will be an exact multiple of the shunt R_D–C_D arm. If the two R_A and R_B arms are adjusted to the same ratio, then the bridge is balanced.

Analysis

Input admittance analysis

The analysis of the circuit will be performed by looking at the circuit from the negative impedance viewpoint. If a voltage source is applied directly to the input of an ideal amplifier with feedback, the input current will be:

$i_{in}={\frac {v_{in}-v_{out}}{Z_{f}}}$

Where $v_{in}$ is the input voltage, $v_{out}$ is the output voltage, and $Z_{f}$ is the feedback impedance. If the voltage gain of the amplifier is defined as:

$A_{v}={\frac {v_{out}}{v_{in}}}$

And the input admittance is defined as:

$Y_{i}={\frac {i_{in}}{v_{in}}}$

Input admittance can be rewritten as:

$Y_{i}={\frac {1-A_{v}}{Z_{f}}}$

For the Wien bridge, Z_f is given by:

$Z_{f}=R+{\frac {1}{j\omega C}}$

$Y_{i}={\frac {\left(1-A_{v}\right)\left(\omega ^{2}C^{2}R+j\omega C\right)}{1+\left(\omega CR\right)^{2}}}$

If $A_{v}$ is greater than 1, the input admittance can be thought as of a negative resistance in parallel with an inductance. The inductance is:

$L_{in}={\frac {\omega ^{2}C^{2}R^{2}+1}{\omega ^{2}C\left(A_{v}-1\right)}}$

As a capacitor with the same value of C is placed in parallel with the input, the circuit has a natural resonance at:

$\omega ={\frac {1}{\sqrt {L_{in}C}}}$

Substituting and solving for inductance yields:

$L_{in}={\frac {R^{2}C}{A_{v}-2}}$

If $A_{v}$ is chosen to be 3:

$L_{in}=R^{2}C$

Substituting this value yields:

$\omega ={\frac {1}{RC}}$

Or:

$f={\frac {1}{2\pi RC}}$

Similarly, the input resistance at the frequency above is:

$R_{in}={\frac {-2R}{A_{v}-1}}$

For $A_{v}$ = 3:

$R_{in}=-R$

The resistor placed in parallel with the amplifier input cancels some of the negative resistance. If the net resistance is negative, amplitude will grow until clipping occurs. Similarly, if the net resistance is positive, oscillation amplitude will decay. If a resistance is added in parallel with exactly the value of R, the net resistance will be infinite and the circuit can sustain stable oscillation at any amplitude allowed by the amplifier.

Notice that increasing the gain makes the net resistance more negative, which increases amplitude. If gain is reduced to exactly 3 when a suitable amplitude is reached, stable, low distortion oscillations will result. Amplitude stabilization circuits typically increase gain until a suitable output amplitude is reached. As long as R, C, and the amplifier are linear, distortion will be minimal.

An alternative approach, with particular reference to frequency stability and selectivity, will be found in Strauss (1970, p. 671) and Hamilton (2003, p. 449).

Amplitude stabilization

The key to Hewlett's low distortion oscillator is effective amplitude stabilization. The amplitude of electronic oscillators tends to increase until clipping or other gain limitation is reached. This leads to high harmonic distortion, which is often undesirable.

Hewlett used an incandescent bulb as a positive temperature coefficient (PTC) thermistor in the oscillator feedback path to limit the gain. The resistance of light bulbs and similar heating elements increases as their temperature increases. If the oscillation frequency is significantly higher than the thermal time constant of the heating element, the radiated power is proportional to the oscillator power. Since heating elements are close to black body radiators, they follow the Stefan-Boltzmann law. The radiated power is proportional to $T^{4}$, so resistance increases at a greater rate than amplitude. If the gain is inversely proportional to the oscillation amplitude, the oscillator gain stage reaches a steady state and operates as a near ideal class A amplifier, achieving very low distortion at the frequency of interest. At lower frequencies the time period of the oscillator approaches the thermal time constant of the thermistor element and the output distortion starts to rise significantly.

Light bulbs have their disadvantages when used as gain control elements in Wien bridge oscillators, most notably a very high sensitivity to vibration due to the bulb's microphonic nature amplitude modulating the oscillator output, and a limitation in high frequency response due to the inductive nature of the coiled filament. Modern Wien bridge oscillators have used other nonlinear elements, such as diodes, thermistors, field effect transistors, or photocells for amplitude stabilization in place of light bulbs. Distortion as low as 0.0003% (3 ppm) can be achieved with modern components unavailable to Hewlett.

Wien bridge oscillators that use thermistors also exhibit "amplitude bounce" when the oscillator frequency is changed. This is due to the low damping factor and long time constant of the crude control loop, and disturbances cause the output amplitude to exhibit a decaying sinusoidal response. This can be used as a rough figure of merit, as the greater the amplitude bounce after a disturbance, the lower the output distortion under steady state conditions.

See also

• Armstrong oscillator
• Clapp oscillator
• Colpitts oscillator
• Hartley oscillator
• Relaxation Oscillator
• Vačkář oscillator

References

• Ferguson, B. W.; Bartlett (July 1928), "The Measurement of Capacitance in Terms of Resistance and Frequency", Bell System Technical Journal, 7: 420
• Hamilton, Scott (2003), An Analog Electronics Companion: basic circuit design for engineers and scientists, Cambridge University Press, ISBN 9780521798389
• Hamilton, Scott (2007), An Analog Electronics Companion: basic circuit design for engineers and scientists and introduction to SPICE simulation, Cambridge University Press, ISBN 9780521687805
• Variable Frequency Oscillation Generator {{citation}}: Unknown parameter |country-code= ignored (help); Unknown parameter |inventor-first= ignored (help); Unknown parameter |inventor-last= ignored (help); Unknown parameter |issue-date= ignored (help); Unknown parameter |patent-number= ignored (help)
• Oliver, Bernard M. (April–June, 1960), "The Effect of μ-Circuit Non-Linearity on the Amplitude Stability of RC Oscillators" (PDF), Hewlett-Packard Journal, 11 (8–10): 1–7 {{citation}}: Check date values in: |date= (help). Shows that amplifier non-linearity is needed for fast amplitude settling of the Wien bridge oscillator.
• Strauss, Leonard (1970), Wave Generation and Shaping (2nd ed.), McGraw-Hill, ISBN 978-0070621619
• Terman, Frederick (1943), Radio Engineers' Handbook, McGraw-Hill
• Williams, Jim (June 1990), Bridge Circuits: Marrying Gain and Balance, Application Note, vol. 43, Linear Technology Inc, p. 29–33, 43
• Williams, Jim, ed. (1991), Analog Circuit Design, Art, Science, and Personalities, Butterworth Heinemann, ISBN 0750696400

External links

• Model 200A Audio Oscillator, 1939, HP Virtual Museum.
• Circuit Analysis, including SPICE simulation.
• Wien Bridge Oscillators using E2POTs, Xicor Application Note AN45