# Wien Bridge Oscillator

A Wien bridge oscillator is a type of electronic oscillator that produces a sine wave . It can generate a large range of frequencies . The oscillator is based on a bridge circuit originally developed by Max Wien in 1891 for the measurement of impedances .  The bridge consists of four resistors and two capacitors . The oscillator can also be viewed as a positive gain amplifier combined with a bandpass filter that provides positive feedback ., Automatic gain control, intentional non-linearity and accidental non-linearity limit the output amplitude in various implementations of the oscillator. In this version of the oscillator, RB is a small incandescent lamp. Typically R1 = R2 = R and C1 = C2 = C. In normal operation, Rb itself heats up to the point where its resistance is Rf/2.

The circuit shown on the right shows a typical implementation of an oscillator with automatic gain control using an incandescent lamp. Under the condition that R 1 =R 2 =R and C 1 =C 2 =C, the frequency of oscillation is given by:

{\displaystyle f_{hz}={\frac {1}{2\pi RC}}}

and the state of steady oscillation is given by

{\displaystyle R_{b}={\frac {R_{f}}{2}}}



## Background

In the 1930s several attempts were made to improve oscillators. Linearity was considered important. The “resistance-stabilized oscillator” had an adjustable feedback resistor; That resistor will be set so that the oscillator just starts (thus setting the loop gain to unity only). The oscillations will continue until the grid of the vacuum tube begins to conduct current, increasing losses and limiting the output amplitude.    Automatic amplitude control was investigated.   Frederick Terman states, “The frequency stability and wave-shape form of any general oscillator can be improved by using an automatic-amplitude-control mechanism to keep the amplitude of the oscillations constant under all conditions.” May go.” 

In 1937, Lord Meacham described using a filament lamp for automatic gain control in bridge oscillators. Also in 1937, Hermann Hosmer Scott described audio oscillators based on various bridges, including the Wien Bridge.

Terman at Stanford University , Harold became interested in Stephen Black’s work on negative feedback,   so he organized a graduate seminar on negative feedback.  Bill Hewlett attended the symposium. Scott’s February 1938 oscillator paper appeared during the symposium. Here is a memoir by Terman: 

And it was changed through push-button. The oscillations were obtained by a simple application of negative feedback.”
In June 1938, Terman, RR Bus, Hewlett and FC Cahill made a presentation about negative feedback at the IRE Convention in New York; In August 1938, a second presentation was made at the IRE Pacific Coast Convention in Portland, OR; The presentation became an IRE paper.  One subject was amplitude control in the Wien bridge oscillator. The oscillator was demonstrated in Portland.  Hewlett, with David Packard , co-founded Hewlett-Packard, and Hewlett-Packard’s first product was the HP200A , a precision Wien bridge oscillator. The first sale took place in January 1939. 

Hewlett’s June 1939 engineer degree thesis used a lamp to control the amplitude of a Wien bridge oscillator.  Hewlett’s oscillator produced a sinusoidal output with a constant amplitude and low distortion .

## Oscillator without automatic gain control

The conventional oscillator circuit is designed so that it begins to oscillate (“start up”) and its amplitude is controlled.

The amplifier uses the oscillator diode on the right to add controlled compression to the output. This can produce total harmonic distortion in the range of 1-5%, depending on how carefully it is trimmed. Schematic of a Wien bridge oscillator using diodes to control amplitude. This circuit typically produces total harmonic distortion in the range of 1-5% depending on how carefully it is trimmed.

For a linear circuit to oscillate, it must satisfy Berkhausen’s conditions : its loop gain must be one and the phase around the loop must be an integer multiple of 360 degrees. Linear oscillator theory does not explain how the oscillator is triggered or how the amplitude is determined. The linear oscillator can support any amplitude.

In practice, the gain of the loop is initially larger than unity. Random noise is present in all circuits, and some of that noise will be near the desired frequency. The gain of more than one loop allows the amplitude of the frequency to increase exponentially each time around the loop. With more than one loop gain, the oscillator will start.

Ideally, the loop gain should be slightly larger than one, but in practice, it is often much larger than one. A large loop gain starts the oscillator quickly. A large loop gain also compensates for the gain variation with temperature and the desired frequency of a tunable oscillator. For the oscillator to start, the loop gain must be greater than one in all possible circumstances.

More than one loop gain has a downside. In theory, the oscillator amplitude will increase without limit. In practice, the amplitude will increase until the output is driven by some limiting factor such as the power supply voltage (the amplifier output runs across the supply rail) or the amplifier output current range. Limiting reduces the effective gain of the amplifier (the effect is called gain compression). In a stationary oscillator, the average loop gain will be one.

Although the limiting action stabilizes the output voltage, it has two important effects: it introduces harmonic distortion and it affects the frequency stability of the oscillator.

The amount of distortion is related to the additional loop gain used for startup. If there is a lot of additional loop gain at small dimensions, then the gain at high instantaneous amplitudes should be low. That means more distortion.

The amount of distortion is also related to the final amplitude of the oscillation. Although the gain of an amplifier is ideally linear, in practice it is nonlinear. The nonlinear transfer function can be expressed as a Taylor series . For smaller dimensions, higher order terms have little effect. For large dimensions, non-linearity is pronounced. Consequently, for low distortion, the output amplitude of the oscillator must be a small fraction of the dynamic range of the amplifier.

## Meacham’s Bridge Stable Oscillator

Lord Meacham revealed the bridge oscillator circuit shown at right in 1938. The circuit was described as having very high frequency stability and very pure sinusoidal output.  Instead of using tube overloading to control the amplitude, Meacham proposed a circuit that sets the loop gain to unity while the amplifier is in its linear region. Meacham’s circuit consisted of a quartz crystal oscillator and a lamp in a Wheatstone bridge .

In Meacham’s circuit, the frequency setting components are in the negative feed back branch of the bridge and the gain control elements are in the positive feed back branch. The crystal, Z4 , operates in series resonance. As such it minimizes the negative feedback on resonance. The particular crystal exhibited an actual resistance of 114 ohms at resonance. At frequencies below resonance, the crystal is capacitive and the gain of the negative feedback branch has a negative phase shift. At frequencies above resonance, the crystal is inductive and the negative feedback branch gainsThere is a positive phase shift. The phase shift passes through zero at the resonant frequency. As the lamp heats up, it reduces the positive feedback. The Q of the crystal in Meacham’s circuit is given as 104,000. At any frequency different from the resonant frequency by a tiny maximal of the crystal’s bandwidth, the negative feedback branch dominates the loop gain and no self-sustaining oscillations can occur except through the narrow bandwidth of the crystal. Simplified schematic of Meacham’s bridge oscillator published in the Bell System Technical Journal, October 1938. The unmarked capacitor has enough capacitance at the signal frequency to be considered a short circuit. Unmarked resistors and inductors are assumed to be suitable values ​​for biasing and loading the vacuum tube. The node labels in this figure are not present in the publication.

In 1944 (following Hewlett’s design), JK Clapp modified Meacham’s circuit to use a vacuum tube phase inverter instead of a transformer to drive the bridge.   A modified Meacham oscillator uses Klap’s phase inverter but substitutes a diode limiter for the tungsten lamp.

## Hewlett’s Oscillator

William R. Hewlett ‘s Wien Bridge Oscillator can be thought of as a combination of a differential amplifier and a Wien bridge, connected in a positive feedback loop between the amplifier output and the differential input. At the oscillation frequency, the bridge is nearly balanced and has a very low transfer ratio. The loop gain is a product of a very high amplifier gain and a very low pull ratio.  In Hewlett’s circuit, the amplifier is implemented by two vacuum tubes. The inverting input tube V of the amplifier is the cathode of 1 and the non-inverting input tube is the control grid of V 2 . To simplify the analysis, for R 1 , R 2 , C 1 and C 2Other than that all other components can be modeled as a non-inverting amplifier with a gain of 1+R f / R b and with high input impedance. R1 , R2 , C1 and C2 form a bandpass filter that is connected to provide positive feedback on the frequency of the oscillations . R B itself heats up and increases negative feedback which reduces the gain of the amplifier until it reaches the point that there is enough gain to sustain sinusoidal oscillation without driving the amplifier. If R1 = R2 and C1 = C2 then at equilibrium Rf /r b = 2 and the amplifier gain is 3. When the circuit is first energized, the lamp is cool and the gain of the circuit is greater than 3. Which ensures startup. The dc bias current of the vacuum tube V1 also flows through the lamp. This does not change the principle of operation of the circuit, but it does reduce the amplitude of the output at equilibrium because the bias current provides part of the heating of the lamp. Hewlett’s thesis drew the following conclusions: Simplified schematic of a Wien bridge oscillator from Hewlett’s US Patent 2,268,872. The unmarked capacitor has enough capacitance to be considered a short circuit at the signal frequency. Unmarked resistors are assumed to be appropriate values ​​for biasing and loading vacuum tubes. The node label and reference in this figure are not identical to those used in the designer patent. The vacuum tubes indicated in Hewlett’s patent were pentodes instead of the triodes shown here.

A resistance-capacity oscillator of the type just described should be suitable for laboratory service. It has an easier to handle beat-frequency oscillator and still has some disadvantages. First, the frequency stability at low frequencies is much better than at the beat-frequency type. No significant placement of parts is required to insure small temperature changes, nor is there a need for a carefully designed detector circuit to prevent interlocking of oscillators. As a result of this, the total load of the oscillator can be kept to a minimum. An oscillator of this type, consisting of a 1-watt amplifier and power supply, weighed only 18 pounds, Which was as opposed to 93 pounds for a typical radio beat-frequency oscillator of comparable performance. The distortion and stability of the output compare favorably with the best beat-frequency oscillators now available. Finally, an oscillator of this type can be designed and manufactured on a similar basis to a commercial broadcast receiver, but with fewer adjustments. Thus it combines the quality of performance with the cheapness of cost to deliver an ideal laboratory oscillator.

## Wien Bridge

Bridge circuits were a common method of measuring component values ​​by comparing known values. Often an unknown component is placed in one arm of the bridge, and then the bridge will be vacated by adjusting the other arms or by changing the frequency of the voltage source (see, for example, Wheatstone bridge).

The Wien bridge is one of several common bridges.  Wiens bridges are used for precise measurement of capacitance in terms of resistance and frequency.  It was also used to measure audio frequencies.

The Wien bridge does not need to be the same value as R or C. Relative to the signal at V out , the phase of the signal at V P is approximately 90 degrees leading at low frequencies and lagging about 90 degrees at high frequencies. At some intermediate frequency, the phase shift will be zero. At that frequency the ratio of Z 1 to Z 2 would be purely real (zero imaginary part). If the ratio of b to f is adjusted in the same ratio, the bridge is balanced and the circuit can sustain oscillation. The circuit will oscillate even though b / fand even though the inverting and non-inverting inputs of the amplifier have different phase shifts. There will always be a frequency at which the total phase shift of each branch of the bridge will be equal. If there is no phase shift in b / f and the phase shift of the inputs of the amplifiers is zero then the bridge is balanced when:

\omega ^{2}={1 \over R_{1}R_{2}C_{1}C_{2}}And{\displaystyle {R_{f} \over R_{b}}={C_{1} \over C_{2}}+{R_{2} \over R_{1}}}

If one chooses 1 = 2 and 1 = 2 then R f = 2 b .

In practice, the values ​​of R and C will never be exactly the same, but the above equations show that for fixed values ​​in the Z 1 and Z 2 impedances, the bridge will be balanced with some and some ratio of b / f .

## Analysis

### Loop gain analyzed from

According to Schilling,  the loop gain of a Wien bridge oscillator, under the condition that R 1 =R 2 =R and C 1 =C 2 =C, is given by

{\displaystyle T=\left({\frac {RCs}{R^{2}C^{2}s^{2}+3RCs+1}}-{\frac {R_{b}}{R_{b}+R_{f}}}\right)A_{0}\,}

where is the frequency-dependent gain of the op-amp (note, the component names in shillings have been replaced with component names in the first figure). Ao

Schilling further says that the condition of oscillation is T=1, which is satisfied by

\omega ={\frac {1}{RC}}\rightarrow f={\frac {1}{2\pi RC}}\,

And

{\frac {R_{f}}{R_{b}}}={\frac {2A_{0}+3}{A_{0}-3}}\,With\lim _{{A_{0}\rightarrow \infty }}{\frac {R_{f}}{R_{b}}}=2\,

Another analysis, especially in terms of frequency stability and selectivity, is in Strauss (1970, p. 671) and Hamilton (2003, p. 449).

### Frequency scheduling network

H(s)={\frac {R_{1}/(1+sC_{1}R_{1})}{R_{1}/(1+sC_{1}R_{1})+R_{2}+1/(sC_{2})}}
H(s)={\frac {sC_{2}R_{1}}{(1+sC_{1}R_{1})(sC_{2}R_{1}/(1+sC_{1}R_{1})+sC_{2}R_{2}+1)}}
H(s)={\frac {sC_{2}R_{1}}{sC_{2}R_{1}+(1+sC_{1}R_{1})(sC_{2}R_{2}+1)}}

H(s)={\frac {sC_{2}R_{1}}{C_{1}C_{2}R_{1}R_{2}s^{2}+(C_{2}R_{1}+C_{2}R_{2}+C_{1}R_{1})s+1}}


Let R=R 1 =R 2 and C=C 1 =C 2

H(s)={\frac {sCR}{C^{2}R^{2}s^{2}+3CRs+1}}

Normalize to CR = 1.

H(s)={\frac {s}{s^{2}+3s+1}}


Thus the frequency determination is zero at 0 and poles in the network or -2.6180 and -0.38197. The resulting root locus traces the unit circle. When the gain is 1, the two real poles meet at -1 and split into a complex pair. At gain 3, the poles cross the imaginary axis. At gain 5, the poles meet on the real axis and split into two real poles.

{\displaystyle -1.5\pm {\frac {\sqrt {5}}{2}}}


## Dimension stabilization

The key to the low distortion oscillation of the Wien bridge oscillator is an amplitude stabilization method that does not use clipping. The idea of ​​using a lamp in a bridge configuration for amplitude stabilization was published by Meacham in 1938.  The amplitude of electronic oscillators continues to increase until clipping or other gain limits are reached. This leads to high harmonic distortion, which is often undesirable.

Hewlett used an incandescent bulb as a power detector, low pass filter and gain control element in the oscillator feedback path to control the output amplitude. The resistance of a light bulb filament (see Resistivity article) increases with increasing temperature. The temperature of the filament depends on the power dissipating in the filament and some other factors. If the period of the oscillator (the inverse of its frequency) is significantly less than the thermal time constant of the filament, the temperature of the filament will remain largely constant over one cycle. The filament resistance will then determine the amplitude of the output signal. If the amplitude increases, the filament heats up and its resistance increases. The circuit is designed so that a large filament resistance reduces the loop gain, Which in turn will reduce the output amplitude. The result is a negative feedback system that stabilizes the output amplitude at a constant value. With this form of amplitude control, the oscillator functions as a near ideal linear system and provides a very low distortion output signal. There is often significant harmonic distortion in the limiting oscillator for amplitude control. At low frequencies, as the time period of the Wien bridge oscillator approaches the thermal time constant of the incandescent bulb, the circuit operation becomes more nonlinear, and the output distortion increases significantly.

Light bulbs have their disadvantages when used as gain control elements in Wien bridge oscillators, particularly the very high sensitivity to vibration due to the miconic nature of the bulb. Due to a limitation in high frequency response filament, and current requirements that exceed the capacity of many op-amps. Modern Wien bridge oscillators have used other non-linear 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 using thermistors exhibit extreme sensitivity to ambient temperature due to the thermistor’s lower operating temperature than an incandescent lamp. 

## Automatic gain control dynamics

Small disturbances in the value of R b cause the major poles to move back and forth on the jω (imaginary) axis. If the poles move to the left half plane, the oscillation rapidly becomes zero. If the poles move in the right half plane, the oscillation accelerates until something limits it. If the disturbance is very small, the magnitude of the equivalent Q is too large so that the amplitude changes slowly. If the disturbance is small and reverses after a short period of time, the envelope follows a ramp. The envelope is almost an integral part of the disturbance. Disturbances in the envelope transfer function roll over at 6 dB/octave and cause a −90° phase shift. Root Locus Plot of the Wien Bridge Oscillator Pole Position for R1 = R2 = 1 and C1 = C2 = 1 vs. K = (R b + R f )/R B. Numerical values ​​of K are shown in purple font. For K=3 the trajectory of the poles is perpendicular to the imaginary (β) axis. For K >> 5, one pole approaches the origin and the other approaches K.

The light bulb has thermal inertia so that its power in resistance transfer function exhibits a single pole low pass filter. The envelope transfer function and bulb transfer function effectively occur in the cascade, so that the control loop effectively has one low pass pole and one pole at zero and a net phase shift of about -180°. This will cause poor transient response in the control loop due to the low phase margin. The output may exhibit squeaking. Bernard M. Oliver  showed that slight compression of the gain by the amplifier reduces the envelope transfer function so that most oscillators show good transient response, except in the rare case where the non-linearities in the vacuum tubes intersect each other producing abnormally Cancels linear amplifier.