**Figure 3:** Equivalent circuit of the oscillator to be optimized [1].

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 [8]

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** [1],
[14].

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 [1].
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 [1].

Thu May 2 20:47:39 MET DST 1996