INFICON PLO-10i Phase Lock Oscillator - Manual (Page 31)

PLO-10 PHASE LOCK OSCILLATOR
FREQUENCY ERRORS DUE TO IMPERFECT CAPACITANCE
CANCELLATION
5-1
5 FREQUENCY ERRORS DUE TO IMPERFECT CAPACITANCE
CANCELLATION
There are two reasons that proper capacitance cancellation is so important with high
resistance crystals.
The first is that to a first approximation, the frequency error resulting from a given phase
error is proportional to the bandwidth of the crystal. The bandwidth of the crystal is
proportional to the crystal’s resistance. A ten-ohm crystal might typically have a
bandwidth of 42 Hz, while a one thousand-ohm crystal will have a bandwidth of 4,200
Hz. A five thousand-ohm crystal will have a bandwidth of 21,000 Hz. Since the
frequency error for a given phase error is proportional to the bandwidth, a phase error that
would result in a 0.5 Hz frequency error in a ten ohm crystal will cause a 50 Hz error in a
one thousand ohm crystal and 250 Hz error in a five thousand ohm crystal.
The second reason is that the effective phase error caused by a non-zero net quadrature
current is inversely proportional to the real current, which is inversely proportional to the
crystal resistance. In other words, the effective phase error is proportional to the crystal
resistance. For instance, a net unbalance of 1 pfd leads to an effective phase error of 0.02
degrees for a ten ohm crystal, but it leads to a 2 degree error for a one thousand ohm
crystal and a 10 degree error for a five thousand ohm crystal.
Examples:
A ten-ohm, 5 MHz crystal will have a Q (Quality Factor) of about 120,000. The
bandwidth is equal to the crystal frequency divided by Q. Thus, the bandwidth of this
crystal would be about 42 Hz. To a first approximation, near zero phase, the frequency
error per degree of phase error is given by the following formula,
Frequency Error = -½(Phase Error, in radians)(Bandwidth)
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.