Actual characteristic impedance

Rewriting these equations, the actual characteristic impedance and admittance (in phase domain) are given by... 

Table 1.3 shows the actual characteristic impedance of the horizontal line in Figure 1.27 for the vertical twin-circuit line illustrated in Figure 1.25. It is clear that the characteristic impedance is significandy frequency dependent. The variation reaches about 10% for selfimpedance and about 50% for mutual impedance in the frequency range of 100 Hz to 1 MHz. The characteristic impedance decreases as frequency increases and tends to approach the value as in the case of the perfecdy conducting earth and conductor. 

It is clear that the actual characteristic impedance is constant independent of frequency. 

The actual characteristic impedance [Zq] is obtained from these equations in the following form ... 

Fig. A.4 Case IV Given two arbitrary points 82 and 83, the transfonvation circle for Zi can be drawn immediately with center on the real axis assuming that the characteristic impedance Zi is real. Next draw a tangent from the origin O touching the transformation drcle at point T. The length ofOT = Zi. This approach is only valid in the rectangular coordinate system unless actual numbers are read in the Smith chart.

In the previous section we considered the conditions under which mechanical resonances would occur in a TSM resonator. In considering only the mechanical properties of the crystal, however, we neglected consideration of how these resonances would actually be excited or detected. The device uses a piezoelectric substrate material in which the electric field generated between electrodes couples to mechanical displacement. This allows electrical excitation and detection of mechanical resonances. In constructing a practical sensor, changes in resonant frequency of the device are measured electrically. The electrical characteristics of the resonator can be described in terms of an equivalent-circuit model that describes the impedance (ratio of applied voltage to current) or admittance (reciprocal of impedance) over a range of frequencies near resonance. 

Besides the fact that the medium resists the passage of a pressure wave (impedance), an elastic medium possesses another complicating characteristic. A purely sinusoidal pressure wave travels with a characteristic velocity in a medium, i.e., the phase velocity c (= f A). When any simple wave at any frequency travels through a medium at the same phase velocity, this medium is said to be non-dispersive. Actually, a medium is more or less dispersive so that when a multifrequency wave pulse travels in such a medium, the pulse spreads out. 

The nonideal characteristics of a transformer include core and winding losses, presence of leakage fluxes, and finite permeability of the core. Hence, the actual model should include the physical representations of these nonideal characteristics. This is shown in Fig. 10.105(b), where the shunt magnetization and core loss components have been ignored for the sake of convenience. Such approximations are common in transformer analysis and only cause trivial inaccuracies in computation. Figure 10.106 shows the equivalent circuit of Fig. 10.105(b) with the secondary-side impedance referred to the primary side. 

The shape of the ac impedance plots may deviate from that expected for the simple RC and Warburg elements. There are different reasons for deviations. Typical reasons are rough siufaces, constriction resistance, and distribution of elements with different characteristic parameters, mainly in the bulk. The constriction resistance is due to a smaller contact area of the electrode than the nominal electrode area. At low frequencies the capacitance reflects the actual contact area, while at high frequencies the capacitance reflects the area of the electrode material which may be larger. Thus the contact caimot be described by a single capacitance. It has also been shown that for a MIEC electrode the impedance of transfer of oxygen from the gas phase into the MIEC and the impedance of diffusion inside the MIEC, though coupled in series, do not yield separated parts in the Cole-Cole plot. 

Debye relaxation time — A stationary ion is surrounded by an equally stationary ionic cloud only thermal movement causes any change in the actual position of a participating ion. Upon application of an external electric field the ions will move. At sufficiently high frequencies / of an AC field (1// < r) the symmetry cannot be maintained anymore. The characteristic relaxation time r is called Debye relaxation time, the effect is also called Debye-Falkenhagen effect. In the absence of a complete and perfectly shaped ionic cloud movement of the ions is less impeded by the ionic cloud, thus electrolytic conductivity should increase. Above frequencies / --10 to 10 this increase has been observed, accordingly the Debye relaxation time is r 10 s. 

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