Carsons Impedance

Nowadays, the advancement in computing capabilities makes it possible to calculate an infinite integral, and various methods of evaluating Pollaczek s 

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Carson et al. °° showed that phase-sensitive detection measurements with a single reference signal biases the error structure of the impedance data due to errors introduced when the square-wave reference signal is in pheise with the measured signal. Modem phase-sensitive detection instruments employ more than one reference signal and may thereby avoid this imdesired correlation. 

The magnitude of the stocheistic errors in impedance measurements depends on the selection of experimental parameters as detailed in Chapter 8. The simulation results described by Carson et a 00,25i,255 particular provide insight into differences between commonly used impedance instrumentation, including methods based on Fourier analysis and on phase-sensitive detection. ... 

S. L. Carson, M. E. Orazem, O. D. Crisalle, and L. H. Garcia-Rubio, "On the Error Structure of Impedance Measurements Series Expansions," Journal of The Electrochemical Society, 150 (2003) E501-E511. 

An important step prior to starting voltage analysis is the determination of line impedance. Quite often, impedance tables are used for standard conductors. However, for greater accuracy, Carson s equations are recommended (Kersting, 1992). 

A modification of Carson s equations is now used widely in calculating self-and mutual impedances in ohm per mile. The equations are... 

Apparatus has been developed in the Bureau of Standards (F. Carson (i) also (7)) for measuring the permeability of papers, fibreboards, and leathers (section C of Fig. 132). Instruments have also been described (8) for measuring permeabilities of fabrics and textiles, of even smaller impedance to air flow. Fig. 134 illustrates the permeameter of Schiefer and Best (9). The fabric pressure gauge is inclined at a slope of 1 in 10, which allows the difference in pressure between the chamber A and... 

There were no computers in the 1920s thus, it was impossible to use Pollaczek s impedance [8]. Carson derived the same formula as Pollaczek, neglecting the earth permittivity (i.e., =... 

Again, assuming x is infinite, Pollaczek s, Carson s, or Sunde s impedance is obtained ... 

This equation is identical to the impedance derived for an infinite horizontal conductor in Reference 6, which is an approximation of Carson s earth-return impedance, as already explained in Section I.2.2.2. 

FIGURE 1.62 Self-inductance and capacitance of a finite horizontal conductor, (a) Inductance, (b) Capacitance, exp. experimental result, cal.-fin. calculated result of finite line impedance, cal.-inf. calculated result of Carson s infinite line impedance. 

Table 1.11 shows measured and calculated results of the surge impedance of a horizontal conductor. It is clear that the proposed formula shows more accuracy than Carson s. The accuracy of the proposed formula increases as x/h decreases, corresponding to the characteristic of the inductance. A similar observation has been made in different measurements in Reference 44. 

Even in the case of Pg = 100 Q m, a transient of a 10 ns time region cannot be simulated by Pollaczek s and Carson s impedances [8, 9, 10-11]. It should be noted that most frequency-dependent line models are not applicable in these models because they are based on Pollaczek s and Carson s impedances. 

Pollaczek s and Carson s impedances are for a horizontal conductor. In reality, there are a number of nonhorizontal conductors, such as vertical and inclined ones. Although many papers have been published on the impedance of vertical conductors such as transmission towers, it is still not clear if the proposed formulas are correct. The empirical formula in Reference 12 is almost identical to an anal5 cal formula [13], which agrees quite well with the measured results. However, the anal5 cal formula requires further investigation to confirm if the derivation is correct. 

Earth is stratified, as is well-known, and its resistivity varies significantly at the top layer depending on the weather and climate. The earth-return impedance of an overhead conductor above the stratified earth was derived in Reference 16, and the stratified-earth effect was investigated in Reference 17. The stratified-earth effect may be far more significant than the accurate evaluation of the homogenous earth-return impedance of Pollaczek and Carson, and this requires further investigation. 

Earth resistivity, as mentioned earlier, is weather/climate dependent. The resistivity after the rains is lower than that measured during dry days. Also, it may be frequency dependent. The frequency dependence of earth permittivity may be far more significant than that of earth resistivity. Furthermore, water (H2O), which is a dominant factor for earth permittivity, is extremely temperature dependent [18]. As a result, the error due to the uncertainty of earth resistivity and permittivity might be far greater than that due to the incompleteness of the earth-return impedance derived by Carson and Pollaczek. This should be remembered as a physical reality that is important in engineering practice. 

In fhe 1920s, fhere was no compufer, and fhus if was impossible to use Pollaczek s impedance. Carson derived fhe same formula as fhe Pollaczek s one neglecfing fhe earfh permiffivify, fhaf is, e = eo in Equation 1.23, and furfher he derived a series expansion of fhe infinite integral in Equafion 1.21. The defail of Carson s expansion formula is explained in many publications, for example. Ref. [10]. 

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