Anal. Chem. 1997, 69, 2050-2054
Resonator/Oscillator Response to Liquid Loading Stephen J. Martin,* James J. Spates,† Kurt O. Wessendorf, and Thomas W. Schneider‡
Microsensor Research and Development Department, Sandia National Laboratories, Albuquerque, New Mexico 87185-1425 Robert J. Huber
Department of Electrical Engineering, University of Utah, Salt Lake City, Utah 84112
The resonant frequency of a thickness-shear mode resonator operated in contact with a fluid was measured with a network analyzer and with an oscillator circuit. The network analyzer measures changes in the device’s intrinsic resonant frequency, which varies linearly with (Gη)1/2, where G and η are liquid density and viscosity, respectively. The resonator/oscillator combination, however, responds differently to liquid loading than the resonator alone. By applying the operating constraints of the oscillator to an equivalent-circuit model for the liquid-loaded resonator, the response of the resonator/ oscillator pair can be determined. By properly tuning the resonator/oscillator pair, the dynamic range of the response can be extended and made more linear, closely tracking the response of the resonator alone. This allows the system to measure higher viscosity and higher density liquids with greater accuracy. A quartz resonator typically consists of a thin disk of quartz with metal film electrodes patterned on both sides (Figure 1). The electric field generated by applying a voltage across the electrodes couples to mechanical displacement in the piezoelectric quartz. The quartz crystal resonates at a frequency determined by the crystal thickness and the type of mode that is excited. Both longitudinal and shear mode resonators function in the gas phase to measure surface mass accumulation. For liquid sensing applications, however, longitudinal-mode devices are prohibitively damped: the surface-normal displacement generates compressional waves in the liquid, causing excessive energy loss from the resonator. Shear-mode devices, however, couple less strongly to the contacting fluid and can be successfully operated as liquidphase sensors.1,2 Shear-mode resonators have been used in liquid sensing applications to measure either surface mass accumulation or properties of the contacting fluid. A rigid surface layer causes a decrease in resonant frequency in proportion to the areal mass density of the layer.3,4 The mass sensitivity has been shown to be nearly the same in liquid as in air or vacuum, enabling the device to function as a general-purpose gravimetric detector. This capability has enabled a number of liquid-phase sensing applications.5,6 In addition, the sensitivity of the device to fluid properties
Figure 1. Top (a) and side (b) views of a quartz TSM resonator.
has allowed these devices to be used as fluid monitors.7 Applications include in situ monitoring of lubricant and petroleum properties.8 To interpret the response of resonator fluid monitors, it is necessary to relate the device response to fluid properties. Kanazawa and Gordon have shown that the series resonant frequency decreases proportionally with (Fη)1/2, where F and η are liquid density and viscosity, respectively.9 Muramatsu et al.10 and Tiean et al.11 have demonstrated that resonance damping, described in terms of a motional resistance, also increases proportionally with (Fη)1/2. Reed et al. derived the electrical response of a resonator with a generalized viscoelastic layer, of which fluid loading is a subcase.12 Martin et al. derived an equivalent-circuit model that describes the electrical characteristics of the mass- and/or liquid-loaded device.13 These models all show adequate agreement with measurements made on liquidloaded resonators, provided (1) the device surface is sufficiently smooth14 and (2) the device is measured in isolation, i.e., using a test instrument rather than an oscillator.
On contract from Ktech Corp., Albuquerque, NM 87110. Present address: Science Applications International Corp., Rockville, MD 20850. (1) Numura, T.; Minemura, A. Nippon Kagaku Kaishi 1980, 1621-1625. (2) Konash, P. L.; Bastiaans, G. J. Anal. Chem. 1980, 52, 1929-1931. (3) Sauerbrey, G. Z. Phys. 1959, 155, 206-222. (4) Stockbridge, C. D. In Vacuum Microbalance Techniques; Behrndt, K. H., Ed.; Plenum: New York 1966; Vol. 5, pp 193-205.
(5) Bruckenstein, S.; Shay, M. J. J. Electroanal. Chem. 1985, 188, 131-136. (6) Ward, M. D.; Buttry, D. A. Science 1990, 249, 1000-1007. (7) Martin, S. J.; Frye, G. C.; Wessendorf, K. O. Sens. Actuators A 1994, 44, 209-218. (8) Cernosek, R. W.; Martin, S. J.; Wessendorf, K. O.; Terry, M. D.; Rumpf, A. N. Proceedings Of The Sensor Expo; Cleveland, OH, 1994; pp 527-539. (9) Kanazawa, K. K.; Gordon, J. G., II. Anal. Chem. 1985, 57, 1770-1771. (10) Muramatsu, H.; Tamiya, E.; Karube, I. Anal. Chem. 1988, 60, 2142-2146. (11) Tiean, Z.; Liehua, N.; Shouzhuo, Y. Electroanal. Chem. Interfacial Electrochem. 1990, 1-18. (12) Reed, C. E.; Kanazawa, K. K.; Kaufman, J. H. J. Appl. Phys. 1990, 68, 19932001.
2050 Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
S0003-2700(96)01194-8 CCC: $14.00
† ‡
© 1997 American Chemical Society
The response of the resonator to fluid loading can be determined using test instruments such as the impedance analyzer or network analyzer. Measurements of electrical impedance or admittance made over a range of frequencies near resonance have been shown to “fully” characterize the device response.10,15 In implementing practical sensors, however, it is desirable to use a simple oscillator circuit, as opposed to a test instrument, to measure the device response. An oscillator circuit consists of an amplifier with positive feedback that uses the resonator as the frequency-control element. A number of oscillator circuits have been designed to operate shear-mode resonators in the liquid phase.16,17 Ideally, the oscillator should (1) accurately track the resonant frequency and provide an output of this parameter, (2) vary loop gain to compensate for liquid loading and provide an output indicating crystal damping, and (3) have sufficient gain and stability to sustain oscillation in highly viscous media. Many oscillator circuits operate in a highly nonlinear fashion, and the frequency of oscillation may be a strong function of oscillator nonlinearities. Therefore they are not well controlled by the resonator parameters of interest. Consequently, when the resonator is operated in an oscillator circuit, the response of the combination may not track the resonator response as determined by a network analyzer. That is, the resonator/oscillator combination does not behave the same as the resonator alone, particularly with liquid loading, in which crystal damping varies. Thus, measurements of liquid-induced frequency changes made using an oscillator-driven resonator commonly do not agree with models derived for the thickness-shear mode (TSM) resonator alone.18 If the oscillator is capable of sustaining oscillation at high liquid damping (high Fη), the discrepancy is especially pronounced. In this paper, we examine the effect of liquid loading on (1) the resonant frequency of a resonator alone, as determined by network analyzer measurements, and (2) the oscillation frequency and feedback voltage of a resonator/oscillator combination. We begin by considering an equivalent-circuit model for the liquidloaded resonator, noting how changes in resonant frequency are calculated and measured for the resonator alone. We then examine the constraints under which the oscillator functions. By applying these constraints to the resonator model, one can determine the behavior of the resonator/oscillator under liquid loading. THEORY The fundamental thickness-shear mode, commonly excited in AT-cut quartz, is illustrated in Figure 2. Displacement is maximum at the surfaces and varies sinusoidally across the thickness. When operated in contact with a fluid, the surface motion generates plane-parallel laminar flow in the fluid (Figure 2). Solution of the Navier-Stokes equation for a Newtonian fluid gives the particle velocity in the x direction, vx, as a function of distance y from the surface:21 (13) Martin, S. J.; Granstaff, V. E.; Frye, G. C. Anal. Chem. 1991, 63, 22722281. (14) Martin, S. J.; Frye, G. C.; Ricco A. J.; Senturia, S. D. Anal. Chem. 1993, 65, 2910-2922. (15) Arlin, K. L.; Thompson, M. Anal. Chem. 1990, 62, 1514-1519. (16) Wessendorf, K. O. Proc. 1993 Frequency Control Symp.; IEEE: New York, 1993; pp 711-717. (17) Barnes, C. Sens. Actuators A 1991, 29, 59-69. (18) Spates J. J.; Martin S. J.; Wessendorf K. O.; Huber R. J. Electrochem. Soc. 1994, 668, 1052-1053 (abstract). (19) White, F. M. Viscous Fluid Flow; McGraw-Hill: New York, 1991; section 3-5.1.
Figure 2. Cross-sectional view of a smooth TSM resonator with the upper surface in contact with a liquid. Shear motion of the smooth surface causes a thin layer of the contacting liquid to be viscously entrained.
vx(y,t) ) vxoe-y/δ cos(ωt - y/δ)
(1)
where vxo is the surface particle velocity and ω ) 2πf, where f is the oscillation frequency. Equation 1 represents a critically damped shear wave that is radiated into the contacting fluid by the oscillating resonator surface; the decay length δ of this wave is given by9
δ ) (2η/ωF)1/2
(2)
The mechanical interaction between the TSM resonator and a contacting fluid influences the electrical response of the device. This serves as a basis for using the resonator to measure fluid properties. Equivalent-Circuit Model for the Liquid-Loaded Resonator. The electrical response of the device in liquid contact has been described by an equivalent-circuit model (Figure 3) consisting of a “static” capacitance (Co*) in parallel with a “motional” branch (L1, C1, R1, L2, R2).13 The static capacitance arises from the electrodes located on opposite sides of the dielectric quartz resonator (Co) and from parasitic capacitances external to the resonator (Cp): Co* ) Co + Cp. The motional branch arises from electrically excited mechanical motion in the piezoelectric crystal. The static capacitance dominates the electrical behavior away from resonance, while the motional branch dominates near resonance. The unperturbed (dry) device response is determined by the elements Co*, L1, C1, and R1. By measuring the electrical response of the unperturbed resonator over a range of frequencies near resonance and fitting the equivalent-circuit model to these data, the values for Co*, L1, C1, and R1 can be determined. When the TSM resonator is operated in contact with a liquid, liquid coupling to the surface causes an increase in the motional impedance, represented by the motional inductance (L2) and resistance (R2) in the equivalent-circuit model:13
R2 ) ωsL2 )
( )
nωsL1 2ωsFη Nπ µqFq
1/2
(3)
where n is the number of sides in contact with liquid, N is the harmonic number, ωs is the angular frequency at series resonance, (20) For AT-cut quartz, Fq ) 2.651 g/cm3, µq ) 2.947 × 1011 dyn/cm2, K2 ) 7.74 × 10-3, and ηq ) 3.5 × 10-3 g/(cm‚s). (21) Weast R. C., Ed. Handbook Of Chemistry And Physics; CRC: Boca Raton, FL, 1989-1990; D234.
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Figure 4. Oscillator circuit designed to operate the TSM resonator in liquid media and provide two outputs: V01 (rf output) indicates series resonant frequency fs; V02 (dc output) is proportional to motional resistance Rm. ALC represents automatic level control circuitry. Figure 3. Equivalent-circuit model to describe the electrical characteristics (for ω near ωs) of a TSM resonator with liquid loading.
and Fq and µq are the density and shear stiffness of quartz, respectively.20 The motional elements L2 and R2 represent kinetic energy storage and power dissipation, respectively, in the contacting fluid; each is proportional to (Fη)1/2. If we define the series resonant frequency, fs, as the frequency at which the motional reactance vanishes, i.e., the motional impedance becomes real, we obtain
fs )
1
(4)
2πx(L1 + L2)C1
Then from eqs 3 and 4, we calculate the change in fs (∆fs) caused by liquid loading:13,14
∆fs = -
( )
L2fs 2nfs2 Fη )2L1 N µ F 4πfs
x
q q
1/2
(5)
As defined, ∆fs varies with (Fη)1/2 in agreement with the prediction of Kanazawa and Gordon.9 This change in series resonant frequency arises only from changes in the motional inductance L2 and not from changes in the motional resistance R2. We will see below that the frequency of a resonator/oscillator pair generally does not respond as described by eq 5 under liquidloading conditions but is also dependent upon changes in R2. Oscillator Circuit. An oscillator circuit has an amplifier with positive feedback provided to cause oscillation. Use of the resonator as the frequency control element causes oscillation to be sustained at a frequency largely, but not completely, determined by the resonator. This oscillator output is used to track the resonant frequency of the resonator. Wessendorf has described an oscillator (Figure 4) that provides a second feedback loop that varies the amplifier gain to maintain a constant oscillation level.16 This automatic level control (ALC) circuitry compensates for fluid damping of the resonator and allows oscillation to be sustained in highly viscous media. The ALC feedback voltage (V02), proportional to the total motional resistance Rm ) (R1 + R2), is taken as a second output to indicate crystal damping. Since R2 is proportional to (Fη)1/2, it serves as a convenient indicator of fluid properties. Table I lists the component values used in the 2052
Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
Table 1. Component Values for the Oscillator Circuit of Figure 4 Cc Cf Cs Cr Lc Lt
449 pF 0.001 µF 0.01 µF 0.01 µF 2.2 µH variable
Rc Re Rf Rs Rt Vs
154 Ω 549 Ω 1.05 kΩ 75 Ω variable 5 V dc
oscillator described by Figure 4. The primary feedback loop of the oscillator causes the oscillation frequency to vary, controlling the loop phase shift to 0° and unity gain. For the loop phase to be 0°, the resonator’s impedance phase angle φ0 will be close to 0°, independent of the resonator’s impedance magnitude. Thus, the oscillator operating constraint can be stated as follows: oscillation frequency will change under liquid loading to maintain constant resonator impedance-phase angle. We note that this constraint is different from the one used to derive eq 5, which could be stated as, the resonant frequency fs is the frequency at which the capacitive impedance and inductive impedance cancel (eq 4). Response of Resonator/Oscillator Combination. By combining the circuit model for the oscillator (Figure 4) with the equivalent-circuit model for the liquid-loaded shear-mode resonator (Figure 3), and simulating the overall responses as L2 and R2 vary under liquid loading (eq 3), one could determine the response of the resonator/oscillator combination to liquid loading. This can be done more simply by applying the oscillator operating constraint to the equivalent-circuit model. The electrical impedance of the liquid-loaded resonator is
Z ) Z2/(Y1Z2 + 1)
(6)
where Y1 is the admittance of the “static” elements from the resonator equivalent circuit plus the tuning inductor Lt and resistor Rt:
Y1 )
1 1 + + jωCo* Rt jωLt
(7)
and Z2 is the impedance of the “motional” elements (Figure 3):
Z2 ) (R1 + R2) + jω(L1 + L2) + (1/jωC1)
(8)
Applying the oscillator operating constraint that constant resonator impedance phase φo must be maintained gives
∠Z ) φo
(9)
where ∠Z denotes the phase angle of Z and φo is a constant. Equations 3 and 6-9 determine the frequency of oscillation for a given value of liquid loading: L2 and R2 vary with (Fη)1/2 according to eq 3; eqs 6-9 then determine the frequency at which the resonator/oscillator pair will operate. These equations also determine the range of liquid properties over which the resonator/ oscillator pair functions; oscillation ceases for Fη values for which eq 9 has no solution. EXPERIMENTAL SECTION The first experiment consists of measuring the response of the resonator alone using a network analyzer (HP 8751A) under varying liquid-loading conditions. Several glycerol solutions were mixed with concentrations in water ranging from 0 to 92% by weight. The viscosity of these solutions was measured at temperatures slightly above and below room temperature using a rotating cup viscometer (Physica Viscolab). Solution densities were determined from literature values.21 The solutions have a (Fη)1/2 range of 0.1-1.6 g cm-2 s-1/2. The network analyzer measured the electrical admittance (Y ) vs excitation frequency (f ) over a frequency range near the fundamental resonance (5 MHz). First, the response of the unperturbed (dry) device was measured. The equivalent-circuit model (Figure 3) was fit to the data (with L2 ) R2 ) 0) to determine the “unperturbed elements” Co*, L1, C1, and R1. Next, Y vs f measurements were made of the resonator with the network analyzer as the glycerol solutions were placed on one side of the device. A plastic housing was used to contain fluid in contact with one side. The equivalent-circuit model (with unperturbed elements fixed at values determined above) was then fit to each set of liquid-loaded measurements to determine L2 and R2. The series resonant frequency fs was then determined from eq 4. The second experiment consisted of measuring the change in oscillation frequency and damping of the resonator/oscillator pair as liquid loading was varied. The oscillation frequency was measured at oscillator output V01 (Figure 4) with a frequency counter (HP Model 5384A), while the ALC output (V02) was measured with a multimeter (HP Model 3478A). The temperature of the glycerol solutions was monitored during the measurement with a thermocouple and scanning thermometer (Keithley Model 740). A measurement of reference frequency and ALC voltage were recorded for the dry device. Frequency and ALC voltage were then measured as the glycerol solutions were placed on one side of the device. Measurements with the fluids were repeated as two tuning parameters were varied: (1) the resonator impedance phase φo and (2) the tuning inductance Lt. Since this inductor effectively “tunes out” a portion of the static capacitance Co*, the effect of varying Lt can be described in terms of the “net static capacitance”, Cn:
Cn ) Co + Cp - (1/ωs2Lt)
(10)
Co, as stated previously, is the resonator capacitance; any additional capacitance external to the resonator is lumped together to form
Figure 5. Variation in frequency vs the liquid-loading parameter (Fη)1/2 for the series resonant frequency of an isolated resonator measured (9, dashed line best fit) with a network analyzer and the resonator/oscillator combination measured (b) and calculated (solid lines, eqs 3 and 6-9) for several values of the net static capacitance Cn.
Cp. The parasitic capacitance Cp arises primarily from the oscillator printed circuit board and transistor (Q2) capacitance from base to emitter (Cbe). The impedance phase angle at which the resonator/oscillator operates is controlled by the feedback elements Rf and Cf and by the capacitance from the base to the collector (Cbc) of transistor Q2. From ref 16, the phase of the resonator is shown to be approximately equal to the phase of the feedback elements. The phase φo is approximately -5° when Rf ≈ 1.05 kΩ, Cf ≈ 0.001 µF, Cbc ≈ 3 pF, and the oscillator is operating at 5 MHz. Additional capacitance is added in parallel with Rf and Cf (see Figure 4) to change the phase angle. RESULTS AND DISCUSSION Figure 5 shows the variation in the series resonant frequency (squares, dashed line) of the TSM resonator alone vs the liquid loading parameter (Fη)1/2. The series resonant frequency was determined from network analyzer measurements, as described in the Experimental Section. The shift in series resonant frequency varies linearly with (Fη)1/2, in agreement with eq 5. Figure 5 also shows the variation in the frequency of the resonator/oscillator combination vs the liquid loading parameter (Fη)1/2. The circles are data measured for three values of the net capacitance Cn. The solid lines are calculated using eqs 3 and 6-9; fitting the model to the data determines best-fit values for Cn. The oscillator was set to track an impedance phase angle φo ) -5°. For small values of (Fη)1/2, the oscillation frequency tracks the series resonance of the device regardless of the value of Cn. At larger values of (Fη)1/2, the oscillation frequency deviates from series resonance substantially. The range of (Fη)1/2 values over which the oscillator tracks series resonance can be varied by tuning the net capacitance Cn. For large Cn, oscillation frequency begins to deviate from fs (falling below the dashed line in Figure 5) at low values of (Fη)1/2. As the net capacitance is reduced, oscillation frequency tracks fs to higher values of liquid loading. Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
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Figure 6. Variation in frequency vs the liquid-loading parameter (Fη)1/2 for the series resonant frequency of an isolated resonator measured (9, dashed line best fit) with a network analyzer and the resonator/oscillator combination measured (b) and calculated (solid lines, eqs 3 and 6-9) for two values of the resonator impedance phase setting φo.
If the resonator impedance phase angle were zero, the oscillator would track fs exactly when Cn ) 0. With φo ) -5°, oscillation frequency lies above the dashed line (below fs) for small values of Cn. It is clear that when the oscillator is well tuned (i.e., Cn is small), it tracks the response of the resonator well. Figure 6 shows the shift in the frequency of the resonator/ oscillator combination (circles and solid lines) vs the liquid loading parameter (Fη)1/2 as the impedance phase angle φo tracked by the oscillator was varied. The net static capacitance was held nearly constant at ∼8.6 pF. For larger negative phase angles, the oscillation frequency falls above the dashed line for a wide range of liquid-loading values (Fη)1/2. For the net capacitance shown in Figure 6, the oscillator tracks fs better at high values of liquid loading with φo ≈ -20° than with φo ) -5°. It is clear that the precise manner in which the oscillation frequency tracks fs depends on the combination of net capacitance and phase angle. Figure 7 illustrates the shift in the ALC feedback voltage vs the liquid-loading parameter (Fη)1/2. The phase angle was held constant at φo ) -5°, and the net capacitance (Cn) fixed at two values, 3.1 and 8.7 pF. When Cn is 3.1 pF, the response is nearly linear with (Fη)1/2 and the oscillator operates over twice the (Fη)1/2 range (or 4 times the Fη range) as when Cn ) 8.7 pF. These data demonstrate the improvement in dynamic range and linearity that is obtained if the resonator/oscillator combination is tuned to reduce Cn. Figure 7 also shows that the ALC feedback voltage shift, which is proportional to R2, can be used as an alternative to frequency shift in determining fluid properties. This is possible because of the relationship of L2 to R2 given by eq 3. Using the ALC voltage shift is beneficial because measurement circuitry is significantly simplified. CONCLUSION The response of the liquid-loaded resonator alone can be well characterized by network analyzer measurements; these measurements indicate a frequency shift that agrees with the simple model of eq 5. When operated in an oscillator, however, the nonlinear 2054 Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
Figure 7. ALC feedback voltage (V02) shift vs the liquid-loading parameter (Fη)1/2 measured for two different values of net capacitance with resonator impedance phase setting φo ) -5°.
response causes the resonator/oscillator pair to behave differently from the resonator alone. For the oscillator shown (Figure 4), the oscillation frequency varies to maintain constant resonator impedance phase. Application of this constraint to the equivalentcircuit model for the liquid-loaded resonator gives a set of equations that determines oscillation frequency as a function of liquid properties (density and viscosity). The manner in which frequency varies with liquid loading depends on the combination of oscillator impedance phase setting φo and net capacitance Cn across the resonator. In the ideal case, both φo and Cn would be zero and oscillation frequency would exactly track the series resonant frequency of the device. In practice, however, the oscillator can at best be tuned to small negative φo values. Using a tuning inductor to minimize the net capacitance Cn, the resonator/oscillator combination tracks fs quite closely. Thus, by modeling the resonator/oscillator combination and optimally tuning the appropriate parameters, the response is made more linear and the range of fluid properties over which it operates is extended. This tuning enables the oscillator circuit to be used in place of more expensive test instruments for measuring device response. ACKNOWLEDGMENT The authors thank Dr. R. Cernosek of Sandia National Laboratories for helpful discussions and L. Casaus of Sandia National Laboratories and K. Rice of Team Specialty Products for technical assistance. This work was supported by the United States Department of Energy under Contract DE-AC04-94AL850000. Sandia is a multiprogram laboratory operated by Sandia Corp., a Lockheed Martin Co., for the United States Department of Energy.
Received for review November 25, 1996. Accepted March 7, 1997.X AC961194X X
Abstract published in Advance ACS Abstracts, May 1, 1997.
