Radio Frequency Integrated Circuit Design - Webs

Voltage-Controlled Oscillators
This can be rewritten again with the help of the definition of Q given in
Chapter 4:
| Nout(s)
Nin(s) | 2
= |H1| 2 � 2 o
4Q 2 (��) 2
285
(8.84)
In the special case for which the feedback path is unity, then H1 = H,
and since |H | = 1 near resonance it reduces to
| Nout(s)
Nin(s) | 2
=
� 2 o
4Q 2 (��) 2
(8.85)
Equation (8.85) forms the noise shaping function for the oscillator. In
other words, for a given noise power generated by the transistor amplifier part
of the oscillator, this equation describes the output noise around the tone.
Phase noise is usually quoted as an absolute noise referenced to the carrier
power, so (8.85) should be rewritten to give phase noise as
PN = |N out(s)| 2
2PS
=� |H 1|� o
(2Q��)� 2
� |N in(s)| 2
2PS � (8.86)
where PS is the signal power of the carrier and noting that phase noise is only
half the noise present. The other half is amplitude noise, which is of less interest.
Also, in this approximation, conversion of amplitude noise to phase noise (also
called AM to PM conversion) is ignored. This formula is known as Leeson’s
equation [8].
The one question that remains here is, What exactly is Nin? If the transistor
and bias were assumed to be noiseless, then the only noise present would be
due to the resonator losses. Since the total resonator losses are due to its finite
resistance, which has an available noise power of kT, then
|Nin(s)| 2 = kT (8.87)
The transistors and the bias will add noise to this minimum. Note that
since this is not a simple amplifier with a clearly defined input and output, it
would not be appropriate to define the transistor in terms of a simple noise
figure. Considering the bias noise in the case of the −Gm oscillator, as shown
in Figure 8.22(c), noise will come from the current source when the transistors
Q 1 and Q 2 are switched. If � is the fraction of a cycle for which the transistors