Series resonance frequency change

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... 

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,... 

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. 

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... 

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 ... 

Figure 5.14 Changes in the series resonant frequency A/j (black) and motional resistance h.Rm (gray) following various modifications of the quartz crystal surface and interaction of the aptamer with thrombin. BF, buffer TH, thrombin. Experiments have been performed at 23 1°C. [Adapted from Hianik et al. (2008), with permission of Ben-tham Science Publishers.]...

From changes in the series resonant frequency of 80 Hz due to binding of the thrombin to the aptamer at 30 nM, it is possible to calculate the surface density of the protein. Using equation (5.1) and the molecular mass of a-thrombin (33.6 kD), we can calculate the number of thrombin molecules attached to the aptamer layer Athrombin = 5 x 10 (here again a correction factor of 2 was used). On the other hand, on the base of the crystal strucmre of the thrombin (Malkowski et al., 1997) the cross-sectional area of this molecule is approximately 20 mn. Therefore, approximately 1.0 x 10 thrombin molecules are required to cover the electrode surface. Thus, the value obtained suggests that at 30 nM the thrombin covers a substantial part of the sensor surface. However, further decrease in the frequency after addition of a higher thrombin concentration may demonstrate the aggregation of thrombin molecules and the formation of a multilayer strucmre. 

The frequency in which the series resonant circuit changes from capacitive to inductive behavior is called the self-resonance frequency (Equation 9.27). At this frequency, fhe imaginary part of the impedance is zero. Because the... 

Film thicknesses of the Glassclad-RC adhesion layers were measured with a Rudolph Research Auto-El ellipsometer. Conductance spectra (10,11) were obtained with a Hewlett-Packard 4192A impedance analyzer. Real-time measurements of the oscillation frequency of the QCM were made with a broadband oscillator circuit [10] built at UW and powered by a HP dual output power supply, a Philips PM6654 frequency counter, and a Kipp 2 nen XYY recorder. Note that the broadband oscillator circuit is designed to track the series resonant frequency of the QCM resonator in real time as its mass changes due to metal binding, while the impedance analyzer is used to characterize the entire resonance spectrum of the resonator. 

All unite developed up to now are based on use of an active oscillator, as shown schematically in Fig, 6.5. This circuit keeps the crystal actively in resonance so that any type of oscillation duration or frequency measurement can be carried out. In this type of circuit the oscillation is maintained as long as sufficient energy is provided by the amplifier to compensate for losses in the crystal oscillation circuit and the crystal can effect the necessary phase shift. The basic stability of the crystal oscillator is created through the sudden phase change that takes place near the series resonance point even with a small change in crystal frequency, see Fig. 6.6. 

Normally an oscillator circuit Is designed such that the crystal requires a phase shift of 0 degrees to permit work at the series resonance point. Long-and short-term frequency stability are properties of crystal oscillators because very small frequency differences are needed to maintain the phase shift necessary for the oscillation. The frequency stability Is ensured through the quartz crystal, even If there are long-term shifts In the electrical values that are caused by phase jitter due to temperature, ageing or short-term noise. If mass Is added to the crystal. Its electrical properties change. 

In the above example, by changing the capacitor bank to a 500-kVAR unit, the resonance frequency is increased to 490 Hz, or the 8.2 harmonic. This frequency is potentially less troublesome. (The reader is encouraged to work out the calculations.) In addition, the transformer and the capacitor bank may also form a series resonance circuit as viewed from the power source. This condition can cause a large voltage rise on the 480-V bus with unwanted results. Prior to installing a capacitor bank, it is important to perform a harmonic analysis to ensure that resonance frequencies do not coincide with any of the characteristic harmonic frequencies of the power system. 

Upon addition of an enzyme solution, a rapid decrease of resonator frequency is observed followed by a long period of the slower decrease (Fig. la). The adsorption of HRP on a PSS layer is essentially completed after 5-10 min. At the same time, the changes in series resonance resistance, which are proportional to the square root of density and viscosity of the media near the resonance surface [5], do not exceed 2 Ohm. It allows to consider the formed enzyme layers as a rigid and use the Sauerbrey equation to calculate the mass of the adsorbed HRP. 

In a series of seminal papers [63, 64], Heller and co-workers showed by a time-dependent approach that in pre-resonance conditions, when only the very short time dynamics on the excited state must be considered, the first equality in Eq. 8.75 stiU holds, even when the independent-mode model is not valid and the second equality in Eq. 8.75 does not hold. In tme resonance cases and staying within the harmonic approximation, Duschinsky mixings and frequency changes must be considered and this can be done by direct application of Eq. 8.46. 

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