INFICON PLO-10i Phase Lock Oscillator - Manual (Page 65)

PLO-10 PHASE LOCK OSCILLATOR
THEORY OF OPERATION 8-17
8.7.2 FREQUENCY ERROR DUE TO PHASE ERROR
Given some finite zero phase error, the resulting frequency error depends on the sensing
crystal’s Q, the higher the Q, the lower the error. For phase errors below 10 degrees the
frequency error is 0.087 PPM per degree for crystals with a Q of 100,000. Thus a one
degree phase error in the PLO results in a 0.44 Hz frequency error for a 5MHz crystal
with a Q of 100,000. For a 5 MHz crystal with a Q of 10,000, the error is 10 time greater
or 4.4 Hz per degree.
Frequency Error/deg = df/f = π/(360*Q)
8.7.3 FREQUENCY ERROR DUE TO IMPERFECT CAPACITANCE CANCELLATION
The effect of imperfect electrode capacitance cancellation can also be viewed as an
equivalent phase error. This error is directly proportional to crystal resistance. The
equivalent phase error due to a non-zero shunt capacitance equal to 1 pfd is one degree
for a crystal with a series resistance of 556 Ω. Since the equivalent phase error is
proportional to the crystal resistance, a 1-pfd residual capacitance error will result in a
10-degree equivalent error for a sensing crystal with a resistance of 5.56 KΩ.
8.8 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)