Wien bridge oscillator

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.[22]

For a linear circuit to oscillate, it must meet the 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.

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.

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.

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.

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.[9] 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, Z₄, 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), J. K. Clapp modified Meacham's circuit to use a vacuum tube phase inverter instead of a transformer to drive the bridge.[23][24] A modified Meacham oscillator uses Clapp's phase inverter but substitutes a diode limiter for the tungsten lamp.[25]

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 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.[26] In Hewlett's circuit, the amplifier is implemented by two vacuum tubes. The amplifier's inverting input is the cathode of tube V₁ and the non-inverting input is the control grid of tube V₂. To simplify analysis, all the components other than R₁, R₂, C₁ and C₂ can be modeled as a non-inverting amplifier with a gain of 1+R_f/R_b and with a high input impedance. R₁, R₂, C₁ and C₂ form a bandpass filter which is connected to provide positive feedback at the frequency of oscillation. R_b 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 R₁ = R₂ and C₁ = C₂ then at equilibrium R_f/R_b = 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:[27]

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.

According to Schilling,[26] the loop gain of the Wien bridge oscillator, under the condition that R₁=R₂=R and C₁=C₂=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 ${\displaystyle A_{0}\,}$ is the frequency-dependent gain of the op-amp (note, the component names in Schilling have been replaced with the component names in the first figure).

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

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

and

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

Another analysis, with particular reference to frequency stability and selectivity, is in Strauss (1970, p. 671) and Hamilton (2003, p. 449).

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

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

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

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

Let R=R₁=R₂ and C=C₁=C₂

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

Normalize to CR=1.

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

Thus the frequency determining network has a zero at 0 and poles at ${\displaystyle -1.5\pm {\frac {\sqrt {5}}{2}}}$ or −2.6180 and −0.38197.