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172
THE IMPEDANCE ANALYSIS
The electric energy supplied to the quartz oscillator forces the
quartz to do mechanical work. It is therefore clear that the
electric behavior of the quartz oscillator is closely related to
the material constants and the dimensions of the quartz plate.
For the parameters R0, L0, C0, and C * of the equivalent circuit
(Fig. 4) one fi nds at the fundamental resonance frequency (see
Arnau and Soares 11 ):
[10]
Here, A is the active area of the quartz, i.e. the area of the
metal electrodes, and d is the plate thickness. The following
values apply for an AT-cut quartz 11 :
ε22 = 3.982x10 -11 A 2 s 4 Kg -1 m -3 (permittivity)
ηQ = 9.27x10 -3 Pa s (effective viscosity)
ηQ = 2.947x10 10 N m -2 (piezoelectrically stiffened shear modulus)
e26 = 9.657x10 -2 A s m -2 (piezoelectric constant)
ρQ = 2651 Kg m -3 (density).
A 5 MHz AT quartz has a thickness of d = 3.34x10-4 m (see
equ.1). With electrodes of diameter 6 mm (A = 2.83x10-5 m2 )
one obtains R0 = 14.5 Ω, L0 = 0.0468 H, C0 = 2.17x10-14 F,
and C * = 3.37x10-12 F. In practical work on has to consider that
the capacitances of the connecting wires and of the measuring
instrument contribute to C * . To present a realistic picture, the
admittance of the 5 MHz quartz represented in Fig. 5a/b was
calculated with a value of C * = 40 pF instead of the oscillator’s
intrinsic C * value.
As shown in Fig. 5, one is normally interested in the behavior
of the oscillator over a narrow frequency range near resonance.
Consider the role of the parallel capacitance C * on
the Y{Im}-f curve in this frequency range. According to eq. 4,
C * contributes to the Y{Im}-f curve a nearly constant value of
Y{Im}=ωC * , i.e. C * causes the curve to shift almost parallel to
more positive values. In Fig. 21 the Y{Im}-f curve (a) is calculated
for the quartz in vacuum, as in Fig. 5a, but with C * = 0. The
comparison between the Y{Im}-f curves of Figs. 5a and 21 (a)
illustrates that it is the upward shift caused by C * that leads
to the second frequency at which Y{Im}=0, defi ning the second
resonance frequency, fp.
Fig. 21 Imaginary part of the admittance calculated for a 5 MHz quartz oscillator
with a circular metal electrode of diameter 6 mm: (a) The quartz operates
in vacuum, with the assumption that the parallel capacitance C * = 0; see Fig.
5a for the peak values; (b) calculated curve for the same quartz in contact with
water, with C * = 0; (c) the same 5 MHz quartz in contact with water, with C * =
20 pF; (d) the same 5 MHz quartz in contact with water, with C * = 40 pF.
BOX 2
BUNSEN-MAGAZIN · 9. JAHRGANG · 5/2007
Consider now the Electrochemical QCM. The oscillating quartz
operates with one side in contact with a liquid electrolyte. To
analyze this situation, note that for a Newtonian liquid the mechanical
shear impedance is a complex quantity defi ned by
[11]
The subscript L refers to the contacting liquid phase. Remarkably,
the real and the imaginary component of this shear
wave impedance have both the same value. This implicates
that the corresponding electric impedance corresponds to a
resistor, RL and an inductor, LL, with the following property:
[12]
Unlike true resistors and inductors, the values of RL and LL
are not constants, but depend both on the square root of the
frequency. According to Lucklum, Soares, and Kanazawa 13 ,
the electric impedance of RL is defi ned by:
[13]
Knowing RL, one obtains the corresponding values of LL
with equ. 12. The electric impedance caused by the liquid,
ZL=RL+ jωLL, can be added to the motional arm of the equivalent
circuit (Fig. 4), and the impedance/admittance of the
quartz oscillator in contact with a Newtonian liquid can be
calculated. In Fig 21 imaginary admittance results for the 5
MHz quartz considered before, but now in contact with water,
are represented as curves (b), (c), and (d). For the calculation
of curve (b) the parallel capacitance C * is again set to zero.
The comparison with curve (a) demonstrates the dramatic
damping effect of the liquid: At resonance R0 = 418 Ω while
in vacuum R0 = 14.5 Ω! As C * =0, the Y{Im}-f curve(b) shows
one resonance frequency, which is shifted to a smaller value,
as discussed by Kanazawa and Gordon 14 , see equ. 6.
Curves (c) and (d) demonstrate the dramatic effect of adding
(practically unavoidable) parallel capacitances. Consider fi rst
curve (c) with C * =20 pF. As discussed above, the Y{Im}-f curve
is shifted essentially parallel by ωC * . It crosses the zero
line twice, yielding series and parallel resonances similar to
Fig. 5a. This EQCM system can be operated with a standard
driver.
On the other hand, the admittance curve (d) obtained with
C * =40 pF is shifted to such high values that there is no more
intersection with the zero line. This means, a driver circuit designed
for resonance at zero phase shift (Y{Im}=0) will not
work. In this situation one might make an effort to reduce
the stray capacitances, and to get as close as possible to
the intrinsic C * value of the quartz oscillator. Alternatively, a
modifi ed driver circuit 12 might be employed. Of course, these
considerations demonstrate the advantage of using an impedance
analyzer, by which any response of the quartz oscillator
can be recorded and analyzed conveniently.