Series resonant frequency

INFICON has developed a new technology for overcoming these constraints on the active oscillator. The new system constantly analyzes Ihe response of the crystal fo an applied frequency not only fo determine the (series) resonance frequency, but also fo ensure that the quartz oscillates in the desired mode. The new system is insensitive te mode hopping and the resultant inaccuracy. It is fast and precise. The crystal frequency is determined 10 times a second w/ith an accuracy to less than 0.0005 Hz. 

Fig. 33.5. The plot of the series resonance frequency of AT-cut crystal covered by neutravidin and with immobilized biotinylated 32-mer DNA aptamer selective to heparin binding site of thrombin as a function of thrombin and HSA concentration, respectively. The fundamental frequency of the crystal was 9MHz. The frequency was determined by HP4395A Network-Spectrum analyzer (Hewlet Packard, Colorado Springs, CO, USA). The crystal was placed in a flow cell developed by Thompson et al. [37] (experiment was performed by I. Grman in M. Thompson laboratory [75]).

The reactance of the mechanical arm is zero at the series resonance frequency fs when jcoLi = 1/jcoCi the parallel resonance/p occurs when the currents flowing in the two arms are in antiphase, that is when j(cuLi — 1/coCi)-1 = —1/jcuCo, or when co = (Co + CflUC 112. 

The planar coupling coefficient kp is related to the parallel and series resonant frequencies by... 

Figure 12.4 depicts a typical admittance parametric plot for the quartz crystal resonator. Note that the effect of the static capacitance C0 in the parallel branch is to shift the admittance circle upward by resonance frequency top which now depends on C0, in addition to the series resonance frequency to, = 2irfa. Changes in the resonance frequency are related to changes in the equivalent inductance L and broadening of the admittance curve near resonance (decrease in the circle diameter l/R in Fig. 12.4) are related to equivalent resistance R. 

The series resonant frequency fs is defined as the frequency at which the motional reactance is zero, i.e.,... 

Since the series resonant frequency is defined as the point where motional inductance and capacitance resonate, the motional inductance Li causes a shift in series resonant frequency (relative to the unperturbed case) given by ... 

The motional inductance La, representing the kinetic energy of die entrained liquid layer (with effective thickness 812), leads to a decrease in the series resonant frequency [14,17] from Equation 3.23 in agreement with the prediction of Kanazawa and Gordon [18] ... 

Example 3.4 Calculate the liquid decay length S, motional resistance Rz, and change in series resonant frequency Af, caused by placing water in contact with one face of a 5 MHz TSM resonator having C =5 pF. For quartz [23] 7.74 X 10 , p, = 2.65... 

Abstract Oscillators are the standard interface circuits for quartz crystal resonator sensors. When applying these sensors in gases a large set of circuits is available, which can be adapted to particular applications. In liquid applications viscous damping accompanied by a significant loss in the Q factor of the resonator requires specific solutions. We summarize major design rules and discuss approved solutions. We especially address the series resonance frequency and motional resistance determination and parallel capacitance compensation. We furthermore introduce recent developments in network analysis and impulse excitation technique for more sophisticated applications. Impedance analysis especially allows a more complete characterization of the sensor and can nowadays be... 

In consequence of - Co, the series resonance frequency at conductivity maximum, G ... 

Ls, Cs, Rs, and Co determine resonance frequencies of the crystal. Considering Fig. 8, the series resonance frequency, ff, and the parallel resonance... 

Fig. 12 Characteristic resonance frequencies of quartz crystal resonators, shown in the locus of impedance, Z = R+jX (a), and admittance, Y = G+jB (b). is the parallel resonant frequency/p at Umax, O is the parallel resonant frequency/a at X = 0, O is the parallel resonant frequency/ at 2 max, i Ih series resonance frequency/s at Gmax, is the series resonant frequency/r at = 0, and is the series resonant frequency/m...

Equation Eq. 17 clearly explains the dilemma of the series resonance frequency at zero phase depends on the equivalent resistance. 

The oscillator circuit in Fig. 13c, applying a non-inverting amplifier, works as a series resonance oscillator where the quartz fulfils the phase condition at series resonance frequency. 

Because of the parallel capacitance the zero phase frequency deviates from the series resonance frequency, Eq. 14, as discussed in Sect. 3.1.1, therefore series resonance oscillators oscillate at a frequency/osc fs- The series resonance frequency/s is not accessible with standard oscillator concepts without compensation of the parallel capacitance, Cq. The phase of a quartz crystal with compensation of Co becomes (pm = where again... 

The quartz crystal should be operated in the neighborhood of its series resonance frequency, since alterations in Co or Cext have much lower effects on resonance frequency than on parallel resonance (Table 2). Another essential reason for operating at series resonance is that the quartz impedance is in the range of RF-technique impedance (50 f2), which minimizes the effect of interference signal coupling. 

The Lever oscillator [39], Fig. 16, allows the application of series resonance configurations with one-side quartz electrode grounding. Since the effect of parasitic capacitance is minimized and simple shielding is possible, this circuit configuration is especially suited for under-liquid QCM. Besides the series resonance frequency, the series resonance resistance Rs can be measured. For this purpose the Lever oscillator allows a largely transistor current gain-independent measurement of the resistance. An automatic level control provides a signal proportional to Rs. 

The concept behind the design of the oscillator shown in Fig. 22 ensures continuous measurement and automatic compensation of the parallel capacitance Cq, while the quartz crystal is simultaneously and independently driven at its zero-phase frequency [47,48]. Provided that the capacitance compensation is effective, the zero-phase frequency is always equal to the sensor series resonance frequency/s, irrespective of the load. 

Ml, II and the VCO form a phase-locked loop feedback system. The multiplier makes a synchronous detection of (V2 - Vi) at the frequency/h. The output of II drives the VCO so that the output frequency/h constantly adjusts to the frequency where the admittance of the motional arm Ys of the sensor is real, i.e., to the series resonance frequency/s. Therefore, the oscillator output frequency/out =/h is continuously tracking/. 

This approach is dedicated to the measurement of hquid viscosity by determining the real part of the sensor admittance at series resonance frequency. According to this concept, one terminal of the sensor is fed with the (constant-level) output of a VCO. The resonator current I is measured by connecting a transimpedance amphfier at the second terminal. Due to the low input impedance of the transimpedance amphfier, the entire VCO output voltage is apphed to the sensor. Parasitic capacitances from the sensor terminals to ground (e.g., due the shielding of the connection cables) are on one side... 

Oscillator circuits are a cost-efficient alternative to impedance analysis and ring-down [12,13]. Naturally, most sensors rim on oscillator circuits. Some advanced circuits provide a measure of the dissipation (such as the peak resistance, Ri, see Sect. 6) in addition to the frequency. Most oscillators operate on one harmonic only. Oscillators can be more stable than ring-down and impedance analysis because the latter two techniques periodically turn the crystal on and off in one way or another, whereas oscillators just run quietly on one fixed frequency. If the signal-to-noise ratio is the primary concern, no technique can beat oscillators. There is one pitfall with the use of oscillators worth mentioning the theory below pertains to the series resonance frequency (simply called resonance frequency). The output frequency of an oscillator circuit, on the other hand, usually is not the series resonance frequency (Fig. 2). For instance, phase-locked-loop oscillators keep the phase constant. Many oscillators run at the zero-phase frequency (B = 0, Fig. 2). Importantly, the difference between the zero-phase frequency and the series resonance frequency changes if the bandwidth or the parallel capacitance change (Sect. 6). The... 

Fig. 2 Impedance analysis is based on the conductance curve of the crystal. The central parameters of measiuement are the resonance frequency, /r, and the half-band-half-width, r. The insert shows the admittance diagram in the complex plane of the admittance Y(a>) = G(a>) + iB(a>). The series resonance frequency, /r, corresponds to the peak of the conductance. The frequency corresponding to B = 0 is the zero-phase frequency...

We use complex resonance frequencies, where the real part, /r, is the series resonance frequency and the imaginary part, F, is half the bandwidth at half maximum of the resonance (half-band-half-width, HBH width, also termed bandwidth for short). In the following, we comment on why—and under what conditions—the imaginary part of the resonance frequency is equal to the half-band-half-width [37]. 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

Even though a/ii is an entirely acoustic quantity, the series resonance frequency is affected by the value of the electrical capacitance, Co, because of the element Z = - (j) /(icoCo), which introduces piezoelectric stiffening into the acoustic branch. Piezoelectricity adds a negative capacitor into the mechanical branch of the circuit. 

Typically Co is much less than Ci, the maximum magnitude of acceptance exists at near This series resonance frequency is the one generally used in the measurement. 

When a quartz resonator is in contact with a viscous liquid, viscous coupling is operative (Figure 2-(b)). The added effect of viscous liquid modifies the equivalent circuit to include both the mass loading density component (Ll), and a resistive viscosity component (RJ of the liquid. The increase of the total value of the resistor (Ri + RJ, results in reducing the radii in admittance circles, which affects the stability of oscillation. The series resonance frequency shifts its value when a QCM is immersed in the liquid. The new resonance frequency due to the increased inductance from the viscous liquid is,... 

When a mass is added on the surface and the mass is negligibly small, the series resonance frequency change (Af) due to the mass added on the surface is given by,... 

In the simplest format, usually only the series resonant frequency is measured. This method, known as quartz crystal microbalance (QCM), was developed by Sauerbrey (1959), who derived a relation between changes in resonant frequency, Afs, and changes in the surface mass density, p ... 

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