Extension of the leeson formula to phase noise calculation in transistor oscillators with complex tanks

For phase noise evaluation in feedback oscillators, the Leeson formula is a useful tool. Nevertheless, a direct application to oscillator circuits with complex feedback tanks can lead to erroneous results because the Leeson formula contains a coefficient, namely, "the loaded quality factor of the circuit," which is not the right coefficient to be used in the expression. It turns out that, in the Leeson formula, this latter coefficient coincides with the loaded Q-factor of the circuit only for specific elemental feedback tanks. We have derived the right coefficient in terms of the energy stored and power dissipated in the circuit. It may be shown that its expression is definitively different from that of the loaded Q-factor of the circuit. We obtain a modified Leeson formula valid for all feedback oscillator circuits. It should be noted that all the electrical models and calculations presented in this paper are valid for FETs as well as for bipolar transistors. A comparison with nonlinear simulations of the phase noise in oscillator circuits with complex feedback tanks demonstrates the validity of the new expression.