Editing Wien bridge oscillator (section)

==Background==
There were several efforts to improve oscillators in the 1930s. Linearity was recognized as important. The "resistance-stabilized oscillator" had an adjustable feedback resistor; that resistor would be set so the oscillator just started (thus setting the loop gain to just over unity). The oscillations would build until the vacuum tube's grid would start conducting current, which would increase losses and limit the output amplitude.<ref>{{harvnb|Terman|1933}}</ref><ref>{{harvnb|Terman|1935|pp=283–289}}</ref><ref>{{harvnb|Terman|1937|pp=371–372}}</ref> Automatic amplitude control was investigated.<ref>{{harvnb|Arguimbau|1933}}</ref><ref>{{harvnb|Groszkowski|1934}}</ref> [[Frederick Terman]] states, "The frequency stability and wave-shape form of any common oscillator can be improved by using an automatic-amplitude-control arrangement to maintain the amplitude of oscillations constant under all conditions."<ref>{{harvnb|Terman|1937|p=370}}</ref>

In 1937, Larned Meacham described using a filament lamp for automatic gain control in bridge oscillators.<ref>{{harvnb|Meacham|1939}}</ref><ref name="Meacham 1938">{{Harvnb|Meacham|1938}}</ref>

Also in 1937, [[Hermon Hosmer Scott]] described audio oscillators based on various bridges including the Wien bridge.<ref>{{harvnb|Scott|1939}}</ref><ref>{{harvnb|Scott|1938}}</ref>

Terman, at [[Stanford University]], was interested in [[Harold Stephen Black]]'s work on negative feedback,<ref>{{harvnb|Black|1934a}}</ref><ref>{{harvnb|Black|1934b}}</ref> so he held a graduate seminar on negative feedback.<ref>{{harvnb|HP|2002}}</ref> [[Bill Hewlett]] attended the seminar. Scott's February 1938 oscillator paper came out during the seminar. Here is a recollection by Terman:<ref>{{harvnb|Sharpe|n.d.}}</ref>
:Fred Terman explains: "To complete the requirements for an Engineer's degree at Stanford, Bill had to prepare a thesis. At that time I had decided to devote an entire quarter of my graduate seminar to the subject of 'negative feedback' I had become interested in this then new technique because it seemed to have great potential for doing many useful things. I would report on some applications I had thought up on negative feedback, and the boys would read recent articles and report to each other on current developments. This seminar was just well started when a paper came out that looked interesting to me. It was by a man from General Radio and dealt with a fixed-frequency audio oscillator in which the frequency was controlled by a resistance-capacitance network, and was changed by means of push-buttons. Oscillations were obtained by an ingenious application of negative feedback."

In June 1938, Terman, R. R. Buss, Hewlett and F. C. Cahill gave a presentation about negative feedback at the IRE Convention in New York; in August 1938, there was a second presentation at the IRE Pacific Coast Convention in Portland, OR; the presentation became an IRE paper.<ref>{{harvnb|Terman|Buss|Hewlett|Cahill|1939}}</ref> One topic was amplitude control in a Wien bridge oscillator. The oscillator was demonstrated in Portland.<ref>{{harvnb|Sharpe|n.d.|p=???}}{{page needed|date=November 2015}}; Packard remembers first demonstration of the 200A in Portland.</ref>  Hewlett, along with [[David Packard]], co-founded [[Hewlett-Packard]], and Hewlett-Packard's first product was the [[HP200A]], a precision Wien bridge oscillator. The first sale was in January 1939.<ref>{{harvnb|Sharpe|n.d.|p=xxx}}{{page needed|date=November 2015}}</ref>

Hewlett's June 1939 engineer's degree thesis used a lamp to control the amplitude of a Wien bridge oscillator.<ref>{{harvtxt|Williams|1991|p=46}} states, "Hewlett may have adapted this technique from Meacham, who published it in 1938 as a way to stabilize a quartz crystal oscillator.  Meacham's paper, "The Bridge Stabilized Oscillator," is in reference number five in Hewlett's thesis."</ref> Hewlett's oscillator produced a sinusoidal output with a stable amplitude and low [[distortion]].<ref>{{Harvnb|Hewlett|1942}}</ref><ref>{{Harvnb|Williams|1991|pp=46–47}}</ref>

===Oscillators without automatic gain control===
[[File:Wien Bridge Oscillator with diode limiting.png|right|thumb|300px|Schematic of a Wien bridge oscillator that uses diodes to control amplitude.  This circuit typically produces total harmonic distortion in the range of 1-5% depending on how carefully it is trimmed.]]

The conventional oscillator circuit is designed so that it will start oscillating ("start up") and that its amplitude will be controlled.

The oscillator at the right uses diodes to add a controlled compression to the amplifier output.  It can produce total harmonic distortion in the range of 1-5%, depending on how carefully it is trimmed.<ref name="Graeme">{{cite book |last1=Graeme |first1=Jerald G. |last2=Tobey |first2=Gene E. |last3=Huelsman |first3=Lawrence P. |year=1971 |title=Operational Amplifiers, Design and Applications |url=https://archive.org/details/operationalampli00grae/page/383 |url-access=registration |edition=1st |publisher=McGraw-Hill |isbn=0-07-064917-0 |pages=[https://archive.org/details/operationalampli00grae/page/383 383–385] }}</ref>

For a linear circuit to oscillate, it must meet the [[Barkhausen stability criterion|Barkhausen conditions]]: its loop gain must be one and the phase around the loop must be an integer multiple of 360 degrees. The linear oscillator theory doesn't address how the oscillator starts up or how the amplitude is determined. The linear oscillator can support any amplitude.

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

Ideally, the loop gain needs to be just a little bigger than one, but in practice, it is often significantly greater than one. A larger loop gain makes the oscillator start quickly. A large loop gain also compensates for gain variations with temperature and the desired frequency of a tunable oscillator. For the oscillator to start, the loop gain must be greater than one under all possible conditions.
<!-- Noise in 1Hz bandwidth -174dBm. If loop gain is 1dB, then 174 loops to get to 0dBm. -->

A loop gain greater than one has a down side. In theory, the oscillator amplitude will increase without limit. In practice, the amplitude will increase until the output runs into some limiting factor such as the power supply voltage (the amplifier output runs into the supply rails) or the amplifier output current limits. The limiting reduces the effective gain of the amplifier (the effect is called gain compression). In a stable oscillator, the average loop gain will be one.

Although the limiting action stabilizes the output voltage, it has two significant effects: it introduces harmonic distortion and it affects the frequency stability of the oscillator.
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The amount of distortion is related to the extra loop gain used for startup. If there's a lot of extra loop gain at small amplitudes, then the gain must decrease more at higher instantaneous amplitudes. That means more distortion.

The amount of distortion is also related to final amplitude of the oscillation. Although an amplifier's gain is ideally linear, in practice it is nonlinear. The nonlinear transfer function can be expressed as a [[Taylor series]]. For small amplitudes, the higher order terms have little effect. For larger amplitudes, the nonlinearity is pronounced. Consequently, for low distortion, the oscillator's output amplitude should be a small fraction of the amplifier's dynamic range.
<!-- Meacham implication. Wireless book. Strauss uses Bessel fcn expansion. -->

===Meacham's bridge stabilized oscillator===
[[File:Meachams bridge oscillator schematic.png|thumb|300px|Simplified schematic of a Meacham's bridge oscillator published in Bell System Technical Journal, Oct 1938. Unmarked capacitors have enough capacitance to be considered short circuits at signal frequency. Unmarked resistors and inductor are considered to be appropriate values for biasing and loading the vacuum tube. Node labels in this figure are not present in the publication.]]

<!-- The information in this section comes from Meacham's paper -->
<!-- Meacham claims that his circuit is new, has very high frequency stability and very pure sinusoidal output -->
<!-- Meacham does not claim that his circuit is better than previous circuits.  Although probably true, it would be Own Research to assert it. -->
<!-- This is own opinion: Meacham's circuit would not make a useful wide frequency range lab instrument because it uses tuned transformers. -->
Larned Meacham disclosed the bridge oscillator circuit shown to the right in 1938. The circuit was described as having very high frequency stability and very pure sinusoidal output.<ref name="Meacham 1938"/>  Instead of using tube overloading to control the amplitude, Meacham proposed a circuit that set the loop gain to unity while the amplifier is in its linear region. Meacham's circuit included a quartz crystal oscillator and a lamp in a [[Wheatstone bridge]].

In Meacham's circuit, the frequency determining components are in the negative feed back branch of the bridge and the gain controlling elements are in the positive feed back branch.  The crystal, Z<sub>4</sub>, operates in series resonance.  As such it minimizes the negative feedback at resonance.  The particular crystal exhibited a real 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 ''gain'' of the negative feedback branch has a positive phase shift. The phase shift goes through zero at the resonant frequency.  As the lamp heats up, it decreases 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 more than a small multiple of the bandwidth of the crystal, the negative feedback branch dominates the loop gain and there can be no self-sustaining oscillation except within the narrow bandwidth of the crystal.

In 1944 (after Hewlett's design), [[James Kilton Clapp|J. K. Clapp]] modified Meacham's circuit to use a vacuum tube phase inverter instead of a transformer to drive the bridge.<ref>{{harvnb|Clapp|1944a}}</ref><ref>{{harvnb|Clapp|1944b}}</ref> A modified Meacham oscillator uses Clapp's phase inverter but substitutes a diode limiter for the tungsten lamp.<ref>{{harvnb|Matthys|1992|pp=53–57}}</ref>

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===Hewlett's oscillator===
[[File:Wien bridge oscillator schematic from Hewletts US patent.png|thumb|300px|Simplified schematic of a Wien bridge oscillator from Hewlett's US patent 2,268,872.  Unmarked capacitors have enough capacitance to be considered short circuits at signal frequency.  Unmarked resistors are considered to be appropriate values for biasing and loading the vacuum tubes.  Node labels and reference designators in this figure are not the same as used in the patent.  The vacuum tubes indicated in Hewlett's patent were pentodes rather than the triodes shown here.]]

[[William Redington Hewlett|William R. Hewlett]]'s Wien bridge oscillator can be considered as a combination of a differential amplifier and a Wien bridge, connected in a positive feedback loop between the amplifier output and differential inputs. At the oscillating frequency, the bridge is almost balanced and has very small transfer ratio. The [[loop gain]] is a product of the very high amplifier gain and the very low bridge ratio.<ref name="Schilling">{{Harvnb|Schilling|Belove|1968|pp=612–614}}</ref>  In Hewlett's circuit, the amplifier is implemented by two vacuum tubes.  The amplifier's inverting input is the cathode of tube V<sub>1</sub> and the non-inverting input is the control grid of tube V<sub>2</sub>.  To simplify analysis, all the components other than R<sub>1</sub>, R<sub>2</sub>, C<sub>1</sub> and C<sub>2</sub> can be modeled as a non-inverting amplifier with a gain of 1+R<sub>f</sub>/R<sub>b</sub> and with a high input impedance.  R<sub>1</sub>, R<sub>2</sub>, C<sub>1</sub> and C<sub>2</sub> form a [[bandpass filter]] which is connected to provide positive feedback at the frequency of oscillation. R<sub>b</sub> self heats and increases the negative feedback which reduces the amplifier gain until the point is reached that there is just enough gain to sustain sinusoidal oscillation without over driving the amplifier.  If R<sub>1</sub> = R<sub>2</sub> and C<sub>1</sub> = C<sub>2</sub> then at equilibrium R<sub>f</sub>/R<sub>b</sub> = 2 and the amplifier gain is 3. When the circuit is first energized, the lamp is cold and the gain of the circuit is greater than 3 which ensures start up. The dc bias current of vacuum tube V1 also flows through the lamp.  This does not change the principles of the circuit's operation, 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 made the following conclusions:<ref>{{harvnb|Hewlett|1939|p=13}}</ref>
: A resistance-capacity oscillator of the type just described should be well suited for laboratory service.  It has the ease of handling of a beat-frequency oscillator and yet few of its disadvantages.  In the first place the frequency stability at low frequencies is much better than is possible with the beat-frequency type.  There need be no critical placements of parts to insure small temperature changes, nor carefully designed detector circuits to prevent interlocking of oscillators.  As a result of this, the overall weight of the oscillator may be kept at a minimum.  An oscillator of this type, including a 1 watt amplifier and power supply, weighed only 18 pounds, in contrast to 93 pounds for the General Radio beat-frequency oscillator of comparable performance.  The distortion and constancy of output compare favorably with the best beat-frequency oscillators now available.  Lastly, an oscillator of this type can be laid out and constructed on the same basis as a commercial broadcast receiver, but with fewer adjustments to make.  It thus combines quality of performance with cheapness of cost to give an ideal laboratory oscillator.

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