The Wien Bridge Circuit
An AC Millivoltmeter Circuit
Frequency to Phase Circuit ( fm demodulator application)
Rate of Change Measurement
A Variable Constant Current Generator
A NiCad/NiMH Charger (employing the latter two topics)
A Non-inverting Integrator Circuit

The frequency determining part of the Wien Bridge oscillator may be considered as a potential divider comprising a high pass filter in series with a low pass filter of equal time constant.
It may be shown that at a frequency
f = 1/2πCR the output is in phase with the input Ein and is exactly one third of its amplitude. It follows that if a non-inverting amplifier with a gain of exactly three is connected between the output and the input (shown dotted) an oscillator potentially capable of generating a pure sine wave would be produced.
If the gain is greater than three oscillations will build up in amplitude until limited by the amplifier supplies (square waves) and if the gain is less than three oscillations, once started, will die out.

This problem was addressed by two young MIT graduates Messrs. Hewlett and Packard back in the 1930s. They developed a way of making the gain determining negative feedback power sensitive and thereby to control amplitude and to minimise distortion. They started production and marketing of their sine wave signal generator from a garage and the rest is history.
What they did was to employ the high temperature coefficient of resistance of a lamp filament as a power sensitive resistor incorporating this in a potential divider used to produce the negative feedback which defines a gain of 3 and furthermore to combine this network with the frequency defining network in a Wien Bridge circuit.
The filament was not operated at full power but glowing bright red at about 1100K, a high enough temperature as to be minimally affected by changes of ambient temperature. Hewlett and Packard were of course using valve amplifiers capable of large voltage swings and consequently they were able to use a 6V torch bulb as an element of the feedback network. For a semiconductor circuit something more like a grain of wheat bulb is more appropriate.

The amplitude of the output is the voltage is that which causes the bulb resistance to be half that of Rfb and the power consequently delivered to the bulb is sufficient to make the filament glow.  The value of Rfb may be adjusted over a limited range in order start oscillation and to set a defined output voltage.
Ganged variable capacitors are preferred for tuning for reasons of noise and scale ‘linearity’ but alternatively ganged log potentiometers may be used.

The Wien Bridge may also be employed to measure the total harmonic distortion in a waveform eg. the distortion in the output of an amplifier which is driven by a sinusoidal signal.

Referring to the opening paragraph, the transmission of the frequency selective potential divider relative to its input may be expressed as 0.333 at 0 degrees for the fundamental frequency fo. For harmonics of the fundamental the transmission may be calculated as is tabulated:-

At first appraisal this table does not appear to show anything other than the transmission for harmonics is not dramatically less than that for the fundamental and that a lagging phase angle increases with frequency.
If however the transmissions for the harmonics up to and including the 6th are summed vectorially, the composite transmission is astonishingly equal to unity lagging 45 degrees.
Presence of a seventh harmonic modifies this slightly but it may be said that, if the fundamental is removed, the output of the frequency selective divider approximates closely to the total harmonic content of the waveform applied to the divider.
A graphical representation of which is shown below:-

Measurement of small alternating voltages for display on a moving coil meter is subject to large errors due to rectification losses. Those who have AVO meters will be aware that a separate non linear voltage scales are provided for the lower AC ranges. This non-linearity is necessary to accommodate rectification volt drop which is minimised for the AVO by the use of a copper oxide bridge rectifier and a transformer which steps up low voltages before rectification.
In the above circuit rectification loss is rendered insignificant by negative feedback; permitting very small voltages to be displayed on a ‘linear’ scale with a lower limit being defined by amplifier noise or un-nulled offset.
This circuit is essentially a voltage follower circuit with a meter and bridge rectifier interposed between the amplifier output terminal and the feedback point.
The input waveform is reconstituted after rectification because the high open-loop gain of the operational amplifier commands that the waveforms at its input terminals must be essentially identical. Thus the waveform at its output terminal must accommodate the distortion due to rectification drop.
Waveforms for a sinusoidal input are shown at these points. The rectified average value for an input waveform is displayed by meter and the range resistors define the level which gives a full scale deflection of 1mA.
The approximate rms. value for a sinusoidal input may be displayed if the range resistor values are multiplied by 0.9 to accommodate the form factor.

The following diagram shows how an operational amplifier may be used to produce a constant voltage output signal; the phase angle of which (referred to the input signal) changes with frequency.
The diagram also shows how this response may be used to demodulate a frequency modulated audio signal; a method employed hitherto in wow and flutter meters.

The purpose of this circuit is to produce an output which is proportional to the rate of change of the potential applied to the input terminal. It may be used to detect changes of a nominally fixed voltage irrespective of the magnitude of the voltage. An example of its application might be detection of the small step rise of the terminal voltage of a NiCad or NiMH battery as full charge is reached via constant current charging. This voltage step apparently occurs when the charging current ceases to do the useful work of chemical conversion and the battery becomes slightly resistive.
The circuit is shown in its simplest form and it is arranged to display the rate of positive going voltage voltage excursion on a moving coil meter. It can equally well be used for negative excursions and the output voltage Vr can be used in conjunction with a voltage comparator circuit to trigger a shut down for example. To display, for example, the rate of change expected at the completion of charge of a NiCad battery the time constant CR needs to be quite large and consequently R will be large. The circuit is best suited therefore to fet input operational amplifiers with very low input bias current.

This circuit causes a high source impedance current which is proportional to the input voltage to flow in a load.
It may be employed to supply direct current or pulse current to the load dependent upon the nature of the input voltage signal.

Typical direct current applications might be battery charging or electro-plating and pulse applications might simulate devices which produce charge signals such as nuclear detectors - requiring only a voltage pulse input which mimics the profile of a charge pulse.

Referring to the circuit annotations; the circuit causes a current (Vout1-Vout2)/Rs to flow through the load circuit by virtue of the fact that V+ must be essentially equal to V- due to the high open-loop gain of the amplifier.
Despite the apparent shunt feedback provided by R1 and R2, V- is not at virtual ground potential as might be expected.  Feedback via R3 and R4 forces V- and V+ equally to a negative potential. (Circuit voltages in red are the circuit are those pertaining to Vin = 1V)

In fact the voltage gain (Vout1/Vin) is not given by R2/R1 but (R1 + R2)/R1; which demonstrates that the negative feedback is effectively series feedback.

V- can only equal V+ if there is a voltage (Vout1 – Vout2) across Rs and consequently

i =  - (Vout1 -Vout2)/Rs

It may be shown that  (Vout1 -Vout2) = Vin and consequently i = - Vin/Rs

For direct current applications it is possible to use the circuit in a non inverting mode with Vin applied between common and the end of R4 and with the input end of R1 connected to common. In this case it is necessary to introduce a dominant lag in the amplifier frequency response in order to avoid oscillation. This is achieved by connecting a suitable capacitor in parallel with R2.