Electric circuits resonance

Cady in World War II realized that such a mechanical resonance of a vibrating crystal could be used in frequency control. This discovery had an important influence on radio communications.Alternating electric fields, such as those generated by the radio tubes of the time, were applied to plates of piezoelectric crystals and the expansions and contractions of the plates were caused to react on electrical circuits. If the natural frequency of the mechanical vibration of the quartz plate coincided with the frequency of oscillation of the electric circuit, resonance between the two took place and energy was acquired by the mechanical oscillators. Later. Rochelle salt and barium titanate, which are each both ferroelectric and piezoelectric, were used. ° In ferroelectric crystals, the polarization or dipole moment is reversed or reoriented upon application of an electric field. Ferroelasticity is another property displayed by some crystals in which stress can cause the interconversion between two stable orientational states. These physical properties of crystals are of great use in modern technology. 

In electrical circuits die above analysis can be applied by adding an alternating voltage of angular frequency co in serts with the circuit -shown jp Bg. 3. However, the results in (bis case are normally less dramatic. In fact the condition of resonance, at which... 

In a quadrupole mass spectrometer, the ions pass into a path between four rods that are attached to an electric circuit that applies a range of frequencies to the rods. Ions resonate in the quadrupole until a certain frequency, which depends on their mass and charge, is reached and then the ions exit the quadrupole and are measured. A diagram of a quadrupole mass spectrometer is given in Section 13.2.3, Figure 13.5. 

The great advantage of this method of describing the state of affairs is that the mutual action of two atoms, which in the earlier forms of the quantum theory was very indeterminate, can be calculated by combining their wave equations, in a manner more or less analogous to that which would be used to calculate the resonance of two oscillating electric circuits. 

The output of the model is then compared with the output of the real device and the individual elements are iteratively adjusted. When a good fit is obtained, the model is tested. It is a very important step, because the robustness of this procedure must be characterized by establishing the range of validity of the model, for the frequency and amplitude of the excitation signal, as well as for the range of values of the individual circuit elements. The wider the validity range, the more accurate is the representation of the real device by its model. The flowchart for building the equivalent electrical circuit model is shown in Fig. 4.11, and the equivalent electrical circuit of a QCM harmonic oscillator is shown in Fig. 4.12. Close to its resonance,... 

Measurements of sound velocity at ultrasonic frequencies are usually made by an acoustic interferometer. An example of this apparatus11 is shown in Fig. 2. An optically flat piezo-quartz crystal is set into oscillation by an appropriate electrical circuit, which is coupled to an accurate means of measuring electrical power consumption. A reflector, consisting of a bronze piston with an optically flat head parallel to the oscillating face of the quartz, is moved slowly towards or away from the quartz by a micrometer screw. The electrical power consumption shows successive fluctuations as the distance between quartz and reflector varies between positions of resonance and non-resonance of the gas column. Measurement of the distance between resonance positions gives a value for A/2, and if /... 

When a sound source is turned on in an enclosure, it excites one or more of the normal modes of the room. When the source is turned off, the modes continue to resonate their stored energy, each decaying at a separate rate determined by the mode s damping constant, which depends on the absorption of the room. This is entirely analogous to an electrical circuit containing many parallel resonances [Beranek, 1986], Each mode has a resonance curve associated with it, whose inxquality factor (Q) depends on the damping constant. 

Piezoelectric crystals, notably quartz, are used to control or limit the operating frequency of electrical circuits. A well-known example is their use in quartz clocks . The fact that a dielectric body vibrating at a resonant frequency can absorb considerably more energy than at other frequencies provides the basis for piezoelectric wave filters. The equivalent circuit for a piezoelectric body vibrating at frequencies close to a natural frequency is given in Fig. 6.3. At resonance the impedance due to L, and C falls to zero and, provided that Rx is small, the overall impedance is small. 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

The advantage of network analysers is the possibility of impedance measurement near resonance with evaluation of the parameters R, L, C and C0 and test of the equivalent electrical circuit. However frequency response and network analysers are relatively slow with 1-10 s per measurement in typical experiments. A new generation of faster instruments has come to the market like the HP E5100 Network Analyzer with 40 (is per point in the impedance spectrum which allows us to obtain the impedance of the system in less than 10 ms. 

The selection of impedance or admittance for presentation of experimental results and data analysis is dependent on the type of equivalent electric circuit. For instance, for the analysis of -> charge-transfer processes and -> double-layer charging, the impedance may be preferred, while for the resonance circuits (e.g., in piezometric systems) the admittance may offer advantages. 

Fig. 5.11 The basic electrical circuit of the Hartshorn and Ward (1936) resonance method of dielectric measurement.

It is our thesis that the loop-gap lumped circuit resonator introduced recently by us will eventually supplant microwave cavity resonators in ESR spectroscopy except for a few specialized applications [53,291-293], Figure 24 (from Ref. 291) shows this resonator. In a sense, this is a hybrid structure midway between low-frequency lumped circuits where a capacitor and an inductor are connected by a transmission line, and high-frequency distributed circuit cavity resonators where the electric and magnetic... 

The AT-cut quartz resonator can be modeled mechanically as a body containing mass, compliance, and resistance. Figurel-a) shows the mechanical vibration motion depicting the vibration of the quartz resonator. An electrical network called an equivalent electrical circuit consisting of inductive, capacitive and resistive components can represent this mechanical model. Figure... 

The equivalent electrical circuit has the advantage of expressing the mechanical properties of a quartz resonator during oscillation easily by the use of simple electrical network analysis. Impedance analysis can elucidate the properties of the quartz resonator as well as the interaction of the crystal with the contacting medium. Impedance analysis involves the measurement of current over a specified range of frequencies at a known voltage. The admittance (Y), the reciprocal of impedance, in the equivalent circuit shown in Fig.l can be expressed as shown in eqn.(4) [11],... 

The portion of the electrical circuit representation of a system that delivers the oscillatory resonance behavior at one or more rescmance frequencies ( resonance and antiresonance frequencies). 

The equivalent electrical circuit, rearranged under the influence of an apphed physical field, is considered as a parallel resonant circuit coupled to another circuit such as an antenna output circuit Thus, in Figure 15.4c, Wj, Cd, La, and Ra correspond to the circuit elements each Wd represents active emitter-coupled oscillator and Cd, Ld, and Rd, represent passive capacitive, inductive, and resistive elements respectively. The subscript d is related to the particular droplet diameter, that is, the droplet under consideration. Now, again the initial electromagnetic oscillation is represented by... 

In practice and by analogy of resonating electrical circuits, the resonance factor, or simply the Q-factor, is given by... 

The second type of electric circuit makes use of a well-known analogy between alternating-current networks containing inductances and capacitances and coupled mechanical systems. The single junction network shown in Fig. 9-5, for example, has a resonant frequency v given by... 

The twentieth century saw two major advances in time measurement the development of quartz clocks, which used electric circuits to generate constant electrical vibrations in quartz crystals, and the invention of atomic clocks, which take advanti e of the natural resonance frequency of atoms to create... 

Time-domain methods are often used to characterize linear circuits, and can also be used to describe resonance. When an electrical circuit exhibits an undamped oscillatory or slightly damped behavior it is said to be in resonance, and the waveforms of the voltages and currents in the circuit can oscillate indefinitely. 

This can be verified by substituting the expression for v t) into the differential equation model and performing the indicated operations. The fact that v t) can be shown to have this form indicates that it is possible for this circuit to sustain oscillatory voltage and current waveforms indefinitely. When the parametric expression for v(t) is substituted into the differential equation model the value of co that is compatible with the solution of the equation is revealed to be ct) = l/.-/(LC). This is an example of the important fact that the frequency at which an electrical circuit exhibits resonance is determined by the physical value of its components. The remaining parameters ofv(t), K, and (p are determined by the initial energy stored in the circuit (i.e., the boundary conditions for the solution to the differential equation model of the behavior of the circuit s voltage). 

LC circuit An electrical circuit with both inductance (L) and capacitance (C) that is resonant at a particular frequency. 

The conference persuaded Cady to turn his interest to piezoelectricity. In 1919 Cady initiated the study of resonators and the first report on piezoelectric resonator was presented to the American Physical Society in 1921. He proposed the piezoelectric quartz resonator as a frequency standard or a filter. Cady showed how to connect a resonating quartz crystal to an electrical oscillator and in this way to achieve frequency stability. Studies of properties of crystal resonator represented by its equivalent electrical circuit were undertaken by Butterworth, Dye, Van Dyke and Mason. They led to a better imderstanding of crystal resonators used in filters and... 

Figure 46. Equivalent electrical circuit (A) and resonance frequency of a mounted piezoelectric crystal oscillator (B) 23I], [239]...

---