Electrical length transmission lines

It is therefore apparent that in passing from high to low freqnency in a system of this kind, there is an additional impedance dne to concentration polarization. Macdonald and HnU [1984] considered this effect on the electrical response of this type of system. Under many drcnmstances, the presence of concentration polarization might be confnsed with an interface impedance. At different ratios of mobiU-ties of anions and cations, either diffnsion-Iike response (finite-length transmission line behavior) or parallel capacitative-resistive behavior may appear. Ac impedance methods have been nsed to determine ionic transference nnmbers in polymeric electrolytes using this principle (Sorensen and Jacobsen [1982]). 

But an electrolytic cell or dielectric test sample is always finite in extent, and its electrical response often exhibits two generic types of distributed response, requiring the appearance of distributed elements in the equivalent circuit used to fit IS data. The first type, that discussed above, appears just because of the finite extent of the system, even when all system properties are homogeneous and space-invariant. Diffusion can lead to a distributed circuit element (the analog of a finite-length transmission line) of this type. When a circuit element is distributed, it is found that its impedance cannot be exactly expressed as the combination of a finite number of ideal circuit elements, except possibly in certain limiting cases. 

Commercial wind turbines are composed of several parts on the top of a tower that can be in excess of 100 m tall, three giant blades of 30 meters or more in length rotate a horizontal shaft, turning a generator that converts mechanical energy into roughly 3 MW electricity. Inside the tower there is a cable that carries electricity to transmission lines and a computer system that controls the rotation and direction of the blades. 

In the first case, if we had considered a safe line length of 250 km, this would become 500 km for a symmetrical line. Figure 24.18 illustrates such a condition. Depending upon the length and type of line, a line length compensation may be required. Most transmission lines are seen to be within permissible lengths and only a few may require such a compensation. Nevertheless, it may be worth reducing the phase displacement between E, and to less than 15° electrical, to further iinprove the quality and stability level of power transmission. 

For digital circuits, the electrical length of the line can be defined relative to the signal rise time. A common rule of thumb for simple lines with no branches is that interconnections must be treated as transmission lines when the round-trip transit time of the signal, 2t pdZ, exceeds the signal rise time, tr (37). This rule defines a critical line length (Zc) given by... 

We can generalize the analogy by considering the viscoelastic materials as a continuum where the theory of transmission lines can be applied. In this way, a continuous distribution of passive elements such as springs and dash-pots can be used to model the viscoelastic behavior of materials. Thus the relevant equations for a mechanical transmission line can be written following the same patterns as those in electrical transmission lines. By representing the impedance and admittance per unit of length by g and j respectively, one has... 

The quantities or elements cpi and unit length (flcm ) corresponding to the whole electrode area A, and is an impedance length (flcm ). The overall impedance is isomorphous to that of a transmission line. Regarding the electrical potential distribution, the simple assumption is made that an ac potential can be defined in each phase and 02 which is, at each frequency, a unique function of position x no radial distribution of potential into the pores or solid particles is considered. It follows that the ac potential difference between the two phases at a point X, i.e., the overvoltage in interfacial reactions, is 02 - 0i ... 

Microwave in Microfiuidics, Fig. 3 Distributed element model for transmission lines. On a transmission line segment with a length of Ax, the electrical signal experiences a series inductance due to the magnetic field generated by the current, a series resistance due to the conductive losses over the conductive wire, a parallel capacitance due to the electric field developed between the two conductive lines, and a parallel conductance due to the current leaked through the insulating material... 

Eqnation (121) has analogies in both heat condutdion and electrical circuit theory. Consider the semiinfinite transmission line composed only of resistors and capacitors (Figure 2.1.12). If r is the resistance per unit length and c is the capacitance per unit length, then... 

Sah [1970] introduced the use of networks of electrical elements of infinitesimal size to describe charge carrier motion and generation/recombination in semiconductors. Barker [1975] noted that the Nemst-Planck-Poisson equation system for an unsupported binary electrolyte could be represented by a three-rail transmission line (Figure 2.2.8fl), in which a central conductor with a fixed capacitive reactance per unit length is connected by shunt capacitances to two resistive rails representing the individual ion conductivities. Electrical potentials measured between points on the central rail correspond to electrostatic potential differences between the corresponding points in the cell while potentials computed for the resistive rails correspond to differences in electrochemical potential. This idea was further developed by Brumleve and Buck [1978], and by Franceschetti [1994] who noted that nothing in principle prevents extension of the model to two or three dimensional systems. 

Here functions R(v) and C(v) can be obtained by piecewise-linear interpolation of the dependence of R and C parameters obtained by fitting the experimental spectra at different voltages (such as in Figure 4.5.4) to the impedance function in Eq. (10). Any other suitable smooth interpolation can be used. The impedance function has to be expressed in terms of electric parameters, as described in Section 4.5.1.3. For use in a discretized equivalent circuit, the values obtained from the fit have to be divided or multiplied by the number of chains, depending on the series or parallel position of the electric element. So, for series resistors it has to be divided, and for parallel, multiplied. It should be considered that the low-frequency limit of Re Z), used as a fitting parameter in the equation, is not always a simple sum of the discrete elements that constitute a transmission line. In particular, in Eq. (10) the Ra is 1/3 of the specific resistance multiplied by the transmission line length, as can be seen from Eq. (8). Therefore resistance of single chain shown in Eig 4.55 will be Ra 3/N. 

Fig. 4.10 Typical application of zinc anode grounding cells. Zinc cell capacity to be installed at each location is based on information concerning maximum fault current and fault duration obtained from operators of electric transmission line. Necessity for grounding cells between pipeline and electric transmission line to be based on a study of the length of exposure, electric transmission line voltage, spacing between pipeline and electric line, soil resistivity, pipeline coating condition, and so on (Downing, 1964).

The waveguide discontinuities shown in Fig. 4.23(a) to Fig. 4.23(f) illustrate most clearly the use of E and H field disturbances to realize capacitive and inductive components. An E-plane discontinuity (Fig. 4.23(a)) can be modeled approximately by a frequency-dependent capacitor. H-plane discontinuities (Fig. 4.23(b) and Fig. 4.23(c)) resemble inductors as does the circular his of Fig. 4.23(d). The resonant waveguide iris of Fig. 4.23(e) disturbs both the E and H fields and can be modeled by a parallel LC resonant circuit near the frequency of resonance. Posts in waveguide are used both as reactive elements (Fig. 4.23(f)) and to mount active devices (Fig. 4.23(g)). The equivalent chcuits of microstrip discontinuities (Fig. 4.23(k) to Fig. 4.23(o)) are again modeled by capacitive elements if the E field is interrupted and by inductive elements if the H field (or current) is interrupted. The stub shown in Fig. 4.23(j) presents a short chcuit to the through transmission line when the length of the stub is A. /4. When the stubs are electrically short (shunt capacitances in the through transmission Hne. 

Notice, however, the point on the graph marked at sHghtly less than 1/4-wavelength. This length yields an impedance of 600-800 2 and is ideal for the PA-plate circuit. It is necessary, therefore, to physically foreshorten the shorted coaxial transmission-line cavity to provide the correct plate impedance. Shortening the line also is a requirement for resonating the tube s output capacity, because the capacity shunts the transmission line and electrically lengthens it. 

From the transmission line approximation for the current distribution, the far electric fields can be computed. From this the Poynting vector, (watts per square meter), can be integrated over a large spherical surface to find the total radiated power. The radiation resistance at the feed point is then found from this radiated power and the known current at the feed point. The near fields dictate the imaginary part of the impedance. Some typical values of radiation resistance for various elements, which have small diameter, are given in Table 13.1. The section on dipole characteristics gives the input impedance of dipoles of various lengths computed from the latest moment method formulation described in Sec. 13.1.4. 

Uniform transmission line A transmission line with electrical characteristics that are identical, per unit length, over its entire length. 

The electric breakdown process in liquids occurs on a nanosecond time scale. Fast voltage and current measurement systems are required. A coaxial discharge circuit fulfills most of these requirements. The propagation time per unit length of a transmission line is given by... 

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