4-16
Stable Frequency (Ideal Oscillator)
Unstable Frequency (Real Oscillator)
Time
F(t)
Time
F(t)
V
1
-1
T1
T2
T3
1
-1
T1
T2
T3
V(t) = V0 sin(2pn0t)
V(t) =[V0 + e(t)] sin[2pn0t + f(t)]
F(t) = 2pn0t
F(t) = 2pn0t + f(t)
                  
V(t) = Oscillator output voltage,     V0 = Nominal peak voltage amplitude
e(t) = Amplitude noise,       n0 = Nominal (or "carrier") frequency
F(t) = Instantaneous phase, and f(t) = Deviation of phase from nominal (i.e., the ideal)
V
Short Term Instability (Noise)
   The output voltage of an ideal oscillator would be a perfect sine wave.  The outputs of all real oscillators deviate from a perfect sine wave due to noise.  The amplitude deviation is represented by e(t), and the phase deviation by f(t).  As frequency is the rate of change of phase, the frequency deviation is (t) - 0 = [1/(2)][d(t)/dt].
   See the next page for another illustration of the amplitude, phase and frequency instabilities.


S. R. Stein, "Frequency and Time - Their Measurement and Characterization," in E. A. Gerber and A. Ballato, Precision Frequency Control, Vol. 2, pp. 191-232, Academic Press, 1985.