Impedance formulas

What if the battery B is replaced by a capacitor C (Figure 8.11(b)) With the sine wave superimposed, the current will be phase-shifted. From a single frequency measurement, we can find the RC values by the impedance formula Z = R + l/jwC. 

The deductive knowledge is represented as a taxonomy of production rules that implements the knowledge relevant to the applicable electrical laws and formulae, such as Ohm law, Kirchhoff laws, equivalent impedance formulae, etc. The system uses the symbolic method (electrical quantities represented as complex numbers) in its problem solving activity. 

Furthermore, the impedance of a depressed Nyquist plot due to surface roughness, dielectric inhomogeneities and diffusion is defined as the Cole-Cole impedance formula [33-36]. Hence,... 

Problems with Existing Impedance Formulas Used in Circuit Theory-Based Approaches... 

Knowing the low-frequency and high-frequency formulas, the internal impedance formula at any frequency can be given in the following form by applying Rolle s averaging theorem [2] ... 

Similar to Example 1.1, the impedance formula is rewritten in the following form so that only a real number calculation is necessary ... 

There are a number of papers on nonuniform lines [30,31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48-49]. EMC-related transients or surges in a gas-insulated substation and on a tower involve nonuniform lines, such as short-line, nonparallel, and vertical conductors. Pollaczek s [7], Caron s [8], and Sunde s [50] impedance formulas for an overhead line are well known and have been widely used in the analysis of the transients mentioned earlier. However, it is not well known that these formulas were derived assuming an infinitely long and thin conductor, that is, a uniform and homogeneous line. Thus, impedance formulas are restricted to the uniform line where the concept of per-unit-length impedance is applicable. 

This section explains impedance and admittance formulas of nonuniform lines, such as finite-length horizontal and vertical conductors based on a plane wave assumption. The formulas are applied to analyze a transient on a nonuniform line by an existing circuit theory-based simulation tool such as the EMTP [9,11]. The impedance formula is derived based on Neumann s inductance formula by applying the idea of complex penetration depth explained earlier. The admittance is obtained from the impedance assuming the wave propagation velocity is the same as the light velocity in free space in the same manner as an existing admittance formula, which is almost always used in steady-state and transient analyses on an overhead line. 

From this observation, it should be clear that the impedance formulas in Equations 1.245 and 1.246 are the most generalized forms for a horizontal conductor, although they are approximate formulas based on the concept of penetration depth. 

Let us consider the vertical multiconductor system illustrated in Figure 1.61. In the same manner as the finite-length horizontal conductor, the following impedance formula is obtained ... 

Under conditions in which Equations 8.1 and 8.2 are not satisfied, only Kikuchi s and Wedepohl s impedance formulas are applicable [8, 9-10]. These require more advanced numerical integration than that applied to Pollaczek s formula. 

Impedance formulas for inclined and nonparallel conductors have been proposed in References 6, 7, and 14. Since the formulas have been derived from the idea of a complex... 

Figure 1.22 illustrates a 500 kV horizontal transmission line, and Table 1.1 shows the frequency dependence of its impedance. It is observed that the resistance increases nearly proportional to where f is frequency. On the contrary, the inductance decreases as f increases. The previously mentioned phenomena can be explained analytically based on the approximate impedance formula in Equations 1.7 and 1.15 ... 

This statement applies to thin-film electrodes. If the electrode is thick it is necessary to include the extra mass in the acoustic impedance formula. 

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