Wien bridge oscillator

A Wien bridge oscillator is a type of electronic oscillator that generates sine waves without having any input source. It can output a large range of frequencies. The bridge comprises four resistors and two capacitors. The circuit is based on a network originally developed by Max Wien in 1891. At that time, Wien did not have a means of developing electronic gain so a workable oscillator could not be realized. The modern circuit is derived from William Hewlett's 1939 Stanford University master's degree thesis. Hewlett, along with David Packard co-founded Hewlett-Packard. Their first product was the HP 200A, a precision sine wave oscillator based on the Wien bridge. The 200A is a classic instrument known for its low distortion.

The frequency of oscillation is given by:

$f\; =\; frac\{1\}\{2\; pi\; R\; C\}$

Amplitude stabilization

The key to Hewletts' 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 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 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.

Modern Wien bridge oscillators have used field effect transistors or photocells for amplitude stabilization in place of light bulbs. Distortion as low as 0.0008% (8 parts per million) can be achieved with only modest improvements to Hewlett's original circuit.

Another way of stabilizing the amplitude is by using non-linear elements, such as diodes, to modify the resistance of the negative feedback network. An example of use can be seen at [http://www.linenoise.info/telefonia/generacion-de-funciones-3/] (in Spanish).

Analysis

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:

$Z\_f\; =\; R\; +\; frac\{1\}\{j\; omega\; C\}$

Substituting and rewriting:

$Y\_i\; =\; frac\{left\; (\; 1\; -\; A\_v\; ight\; )\; left\; (omega^2\; C^2\; R\; +\; j\; omega\; C\; ight)\; \}\{1\; +\; left\; (omega\; C\; R\; ight\; )\; ^\; 2\}$

If $A\_v$ is greater than 1, the input admittance is a negative resistance in parallel with an inductance. The inductance is:

$L\_\{in\}\; =\; frac\{omega^2\; C^2\; R^2+1\}\{omega^2\; C\; left\; (A\_v-1\; ight)\}$

If 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:

$omega\; =\; frac\{1\}\{R\; C\}$

Or:

$f\; =\; frac\{1\}\{2\; pi\; R\; C\}$

Similarly, the input resistance at the frequency above is:

$R\_\{in\}\; =\; frac\{-2\; R\}\{A\_v\; -\; 1\}$

For $A\_v$ = 3:

$R\_\{in\}\; =\; -R$

If a resistor is placed in parallel with the amplifier input, it will cancel 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.

References

"Analog Circuit Design, Art, Science, and Personalities", edited by Jim Williams, 1991, Butterword Heinemann ISBN-10: 0750696400

Wien is often mis-spelt 'wein' the german for wine!