INFICON RQCM - Quartz Crystal Microbalance Research System - Manual (Page 68)

RQCM – RESEARCH QUARTZ CRYSTAL MICROBALANCE
THEORY OF OPERATION 5-16
the crystal’s resonant frequency the current leads the voltage and the phase goes to 90 degrees as
the frequency separation continues to increase, see Figure 19. Above the resonant point the
current lags the voltage and the phase go to minus 90 degrees. As the frequency increases
through the resonant frequency, the phase goes from plus 90 through 0 to minus 90. It is
interesting to note that the phase angle is 45 degrees when the VCO frequency is one half of the
crystal’s bandwidth above or below the crystal’s resonant frequency.
The output of the phase detector is fed into an integrator. The integrator accumulates the phase
error such that any positive phase error causes the integrator output to climb; a negative phase
causes the integrator output to fall. With zero phase error the Integrator output holds steady.
The integrator output is connected to the VCO. Thus, if the VCO frequency is initially below the
crystal resonant frequency, the phase will be positive, producing a positive output at the phase
detector. This causes the Integrator output to climb, which causes the VCO frequency to
increase. When the VCO frequency matches the resonant frequency of the crystal the phase will
decrease to zero, the phase detector output will go to zero, the Integrator output will hold steady
and the VCO frequency will be “locked” to the crystal’s resonant frequency.
If the crystal’s resonant frequency moves up or down, a phase difference between the crystal
voltage and current will develop, producing a phase detector output. The non-zero phase detector
output will drive the Integrator output up or down until the phase is zero once again, thus keeping
the VCO frequency locked to the crystal’s resonant frequency.
Once the frequency of the VCO is locked to the series resonant frequency of the crystal, the in-
phase component (at zero phase error, there is no out of phase component) of the crystal current is
demodulated to a DC voltage. This voltage is amplified and converted into resistance value
which the RQCM outputs to the computer.
5.7.1 FREQUENCY ERRORS
The first thing we want to know regarding the performance of the crystal measurement is “What
is the magnitude of the frequency error we can expect from the crystal measurement portion of
the RQCM?”
In any oscillator and sensing crystal system, the error in the frequency measurement, is a function
of both the oscillator and the sensing crystal. The same is true for phase locked loops. Any phase
error will introduce a frequency error and this frequency error will be inversely proportional to
the sensing crystal’s Q. These errors are over and above any change in crystal frequency due to
stress, temperature, adsorption, and humidity changes.
There are four important parameters that determine the frequency error of the PLO and sensing
crystal system or indeed, any oscillator and sensing crystal system. The first two, the zero phase
error and the electrode capacitance cancellation errors, are characteristics of the PLO. The
second two are characteristics of the crystal, the Q of the crystal and the conductance
(1/resistance) of the crystal.
5.7.2 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)