We present for the first time analytic solutions for the nonlinear dynamics of a Wien bridge oscillator stabilised by three common methods: an incandescent lamp, signal diodes, and the field effect transistor. The results can be used to optimise oscillator design, and agree well with measurements. The effect of operational amplifier marginal nonlinearity on oscillator performance at high frequencies is clarified. The oscillator circuits and their analysis can be used to demonstrate nonlinear dynamics in the undergraduate laboratory.