Homogeneous Impedance and Admittance

3 Homogeneous Model of a Cross-Bonded Cable 3.2.3.1 Homogeneous Impedance and Admittance 

Section 3.2.1 discussed the impedance and admittance of a single-phase cable. This section discusses the impedance and admittance of a cross-bonded three-phase cable and how 6x6 impedance and admittance matrices can be reduced to 4 x 4 impedance and admittance matrices. 

The second subscripts c and s denote the core and sheath, respectively. And the third subscripts a, b, and c express the phases. The other voltage and current vectors (Vt )/ (4 ) have the same form as (Vj ). 

The sheath sectionalizing joint is mathematically expressed by a rotation matrix [1 ]  

Defining voltage difference between nodes k-1 and k, the following 

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In addition to the impedance, other derived quantities, such as the dielectric modulus (M), the complex dielectric constant (e), or susceptibility (x), can be calculated from the IS measurements their interrelations have been tabulated elsewhere (Macdonald 1987). Complementary information on the dielectric response of a given system can be obtained from the different impedance plots and the related magnitudes. It is important to point out the different nature of these magnitudes extensive or sample geometry dependent in the case of impedance or admittance, and intensive or characteristic of homogeneous materials in the case of conductivity and the dielectric constant. 

Table 3.3a shows the calculated results of the impedance, the admittance, the modal attenuation constant, and the propagation velocity on the solidly bonded case Table 3.3b shows the results on the cross-bonded case with the homogeneous model at frequency f = 100 kHz. 

Based upon Faraday s work, James Clerk Maxwell published his famous equations in 1873. He more specifically calculated the resistance of a homogeneous suspension of uniform spheres (also coated, two-phase spheres) as a function of the volume concentration of the spheres. This is the basic mathematical model for cell suspensions and tissues still used today. However, it was not Maxwell himself who in 1873 formulated the four equations we know today as Maxwell s equations. Maxwell used the concept of quaternions, and the equations did not have the modern form of compactness he used 20 equations and 20 variables. It was Oliver Heaviside (1850—1925) who first expressed them in the form we know today. It was also Heaviside who coined the terms impedance (1886), conductance (1885), permeability (1885), admittance (1887), and permittance, which later became susceptance. 

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