Figure 3: Equivalent circuit of the oscillator to be optimized .
A low phase noise is besides the signal properties essential to the design of oscillating electrical circuits. A new and general method to minimize the single-sideband phase noise of free running oscillators reduces the phase noise without requiring additional elements or the manufacture of prototypes. It is based on the description of the signal and noise behavior of an oscillator circuit by the Langevin equations
where are the state variables of the circuit, are the white noise sources and y is a nonlinear noise source denoting the baseband noise. The notation is used. The terms of order are neglected.
The single-sideband phase noise can be simulated by solving (5) with a perturbation theory 
The first term on the right hand side of Eq. (6) describes the phase noise caused by the white noise sources. is the left-sided eigenvector of the fundamental matrix . The matrix denotes the correlation matrix of the white noise sources. The second term of Eq. (6) describes the phase noise caused by the baseband noise. is a coefficient that characterizes the upconversion of the baseband noise to the carrier frequency. The modulation of the noise source due to the oscillation is taken into account as well as the upconversion of the baseband noise caused by the nonlinearities in the circuit. The factor c is derived from baseband noise measurements. The functions and depend on the design parameters of the circuit by the system of nonlinear differential equations
and appropriate boundary conditions. The minimization of the phase noise with respect to the design parameters and subject to Eqs. (7) and (8) is an optimal control problem with as control from a finite dimensional control space and and as state variables of the optimal control problem. The optimization problem is solved numerically by the direct collocation method , .
In an experiment the method is applied to minimize the single-sideband phase noise of a planar integrated free running microwave oscillator at 15 GHz . The equivalent circuit of the oscillator is depicted in Fig. 3. In this special case, five design parameters of the linear network are optimized subject to a system of 20 highly nonlinear differential equations (7), (8). A prototype of the new designed oscillator has been manufactured. A reduction of 10 dB of the phase noise caused by the upconverted baseband noise is measured at a frequence deviation of 10 kHz .