Quartz Crystal Microbalance Theory and Calibration - Stanford ...

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2. A fixed gain amplifier with gain A 1 = 45dB + 20 log (250/200) = 46.94dB (or -
222x.) This inverting amplifier has a bandwidth of 500MHz, and so introduces
very little extraneous phase shift.
3. A source resistance, R s , of 100Ω. This source resistance consists of two series
50Ω resistors, one of which is inside the amplifier A 1 . This source impedance is
reduced by a factor of 4x, to 25Ω, by the 2:1 transformer which follows.
4. An isolation transformer with a 2:1 turns ratio, hence an attenuation of A t =
0.5x. This transformer allows galvanic isolation of the crystal from the oscillator
circuit which is important in electrochemistry applications. In addition to reducing
the source impedance by 4x, the transformer also increases the load impedance
seen at the input of the transformer by 4x, so that when R m =0Ω, the load will be
200Ω.
5. R m , the motional resistance of the crystal at series resonance. R m can vary
from about 10-40Ω for a dry crystal, to about 375Ω for a crystal in water, to about
5kΩ for a crystal in 90% (w/w) glycerol/water solution.
6. A second isolation transformer with a turns ratio of 1:1. This transformer allows
galvanic isolation of the crystal from the oscillator circuit.
7. A load resistance, R L , of 50Ω. The network of R s , R m , and R L provide a
network attenuation, A n , which depends on the crystal’s motional resistance.
A n = R L / ( R s /4 + R m + R L ).
8. An RF amplifier with an adjustable gain, A 2 , of about 4.43x. The gain of this
amplifier, A 2 , is set during calibration to compensate for gain variations of all the
other circuit elements.
9. A low pass filter. This filter is a 5 th order Bessel low pass filter with f c = 3.7MHz,
adjusted so as to provide 180º of phase shift at 5MHz. The phase shift of this
filter, together with the 180º phase shift of the inverting amplifier A 1 , provides the
360º of phase shift necessary for oscillation. The low pass filter is required to
suppress spurious oscillations which would occur due to the high bandwidth of
the loop amplifiers. The low pass filter attenuates a signal at 5MHz by about A f =
-7.8dB (or 0.407x).
The motional resistance of the crystal at series resonance can now be computed.
The product of the gain (or attenuation) of all of the elements around the loop is
exactly one when the circuit is oscillating at constant amplitude. Hence,
A a · A 1 · A t · A n · A 2 · A f = 1
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