# Wienbridge oscillator à la Jim Williams

Jim Williams was probably one of the best and most productive analog designers of the 20th century. He has written many application notes and design briefs in his career as application engineer for Linear Technology. Of course, it featured parts manufactured by his employer. His biggest contribution in the field of analog design, is that he not only showcased his extensive knowledge but also explained in great detail what he did and why he did it and how you can do it yourself. Basically, he was and excellent teacher as well as an engineer. In design note 70 “A Broadband Random Noise Generator”, he uses a current balance to control the amplitude of his generator.
This sine oscillator uses the same approach and this article highlights the math behind it.

Balance is reached when : $I_{D1} = I_{D2}$ $I_{D1} = {V_{ref} – V_{D1} \over R6}$ The average voltage of a sine wave is $${2 \over \pi}$$ . Since only the positive half of the sine wave is used, this should be halved again. The average current through D2 is thus: $I_{D2} = {{ (V_p – V_{D2}) {2 \over \pi } {1 \over 2} } \over R7}$ So if the average currents through both diodes are equal: ${V_{ref} – V_{D1} \over R6} = {{ (V_p – V_{D2}) {2 \over \pi } {1 \over 2} } \over R7}$ Thus ${ { (V_{ref}-V_{D1}) R_7 } \over {R_6 0.637 {1 \over 2} }} + V_{D2}= V_p$ If we fill in the values from the circuit and we assume Vd is 0.65V we get a value of Vp = 6.6V. Which is within 5% of the actual measured value. This is better than expected since the voltage reference has a tolerance of 1% and all resistors have a tolerance of 5%.