INFICON RQCM - Quartz Crystal Microbalance Research System - Manual (Page 70)

RQCM – RESEARCH QUARTZ CRYSTAL MICROBALANCE
THEORY OF OPERATION 5-18
5.8 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)
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