INFICON PLO-10i Phase Lock Oscillator - Manual (Page 66)

PLO-10 PHASE LOCK OSCILLATOR
THEORY OF OPERATION 8-18
Or,
Frequency Error = -(1/(2*57.3))(Phase Error, in degrees)(Bandwidth)
For the above ten-ohm crystal, the frequency error caused by a one-degree phase error is
42/114.6 or approximately 0.37 Hz. For a one thousand-ohm crystal, one degree of phase
error results in a 37 Hz error and for a ten thousand-ohm crystal the frequency error is
370 Hz per degree of phase error.
Now, the effective phase error caused by a non-zero quadrature (imaginary) current is
given by the following formula,
Effective Phase error = arctangent (imaginary current/real current)
And since current is proportional to conductance,
Effective Phase error = arctangent (imaginary conductance/real conductance)
The conductance of a one picofarad capacitor at 5 MHz is 31.4 microsiemens. The
conductance of a ten-ohm crystal at resonance is 100 millisiemens.
Effective Phase error = arctangent ((31.4e-6)/(100e-3)) = 0.018 degrees
In other words a one picofarad capacitance unbalance will result in an effective phase
error of only 0.018 degrees when measuring a ten-ohm crystal. However, when
measuring a one thousand-ohm crystal the effective phase error will increase to 1.8
degrees and it will increase to 9 degrees when measuring a five thousand-ohm crystal.
Combining these two errors we can get an idea of the magnitude of the frequency error
caused by imperfect capacitance cancellation.
For a 10 Ω crystal a one picofarad capacitance imbalance results in a 0.018 degree phase
error and a 0.0067 Hz frequency error.
For a 100 Ω crystal, the phase error is 0.18 degrees and the frequency error is 0.67 Hz.
For a 1000 Ω crystal, the phase error is 1.8 degrees and the frequency error is 67 Hz. For
a 5000 Ω crystal, the phase error is 9 degrees and the frequency error is 1,635 Hz.
A two picofarad capacitance imbalance will result in approximately twice the above
error.