Solidly bonded cable

Setup for measuring sequence currents for a solidly bonded cable (a) zero sequence current and (b) positive sequence current. 

Eliminating sheath current (Is) in (3.60), core current (I) can be derived as 

Using (3.66), core current ( ) in (3.61) can be eliminated, which yields 

Solving (3.68) for Vs and eliminating Vs from (3.66), the zero sequence current is found as 

Core current (I) is obtained from (3.63) and (3.70). Once the core current is found, the positive sequence current can be calculated as 

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The core admittance submatrix [F ] is a diagonal matrix determined by the capacitances between the cores and the sheath, because a sheath encloses a core. The admittance submatrix of the cores for the cross-bonded cable is identical to the solidly bonded cable ... 

In this section, we derive theoretical formulas of the sequence currents for the majority of underground cable systems that is, a cross-bonded cable that has more than two major sections. We also derive theoretical formulas for a solidly bonded cable considering the increased use of submarine cables. 

Table 3.1 shows that the positive-sequence impedance is smaller for a solidly bonded cable than for a cross-bonded cable, and the positive-sequence current is larger for a solidly bonded cable. Because of this size differential, the return current flows only through the... 

The phase angle of the zero-sequence current mentioned in Table 3.1 demonstrates that grounding resistance at substations in both cross-bonded and solidly bonded cables significandy affects the zero-sequence current. As a result, there is litde difference in the zero-sequence impedance of the cross-bonded cable and the solidly bonded cable. The results indicate the importance of obtaining an accurate grounding resistance at the substations to derive accurate zero-sequence impedances of cable systems. 

The capacitance matrix looks similar to the impedance matrix in Table 3.3a-2. The capacitance between the core and sheath of the homogeneous model is identical to that of the solidly bonded cable. The equivalent capacitance Q4 of the cross-bonded cable in Table 3.3b-2 is given as the sum of the elements as shown in Equations 3.44 and 3.45. 

In the solidly bonded cable, the first three modes (columns) shown in Table 3.3 a-3 express coaxial-propagation modes (that is, the core-to-sheath mode [2]). The other modes (columns) correspond to one of the transformation matrices of an untransposed three-phase overhead line [2]. In this case, mode 4 expresses an earth-return mode and modes 5 and 6 correspond to aerial modes. 

Modes 1-3 in the solidly bonded cable shown in Table 3.3a-4 are coaxial modes and are the same as mode 3 in the homogeneous cross-bonded cable model. Although the attenuations of the inter-core modes (modes 1 and 2) shown in Table 3.3b-4 are almost identical to that of the coaxial mode of the solidly bonded cable, the velocities are lower. The velocity of the coaxial mode is determined by the permittivity of the main insulator = 2.3 shown in Table 3.2 ... 

Because the cable is installed in a tunnel (that is, the cable is in air), the attenuation constant and the propagation velocity of mode 4 (earth return) and modes 5 and 6 (the first and second inter-sheath) in the solidly bonded cable show similar characteristics to those of an overhead line [1]. The attenuation constants of modes 5 and 6 are much smaller and the propagation velocity is much greater in the solidly bonded cable than those in the other modes. 

Figure 3.14d shows the transient response of the core voltage in a solidly bonded cable. It shows a stair-like waveform with a length of 70 ps. This length is determined by the round-trip time shown in Equation 3.89. Sheath voltages of the solidly bonded cable are much smaller than those of the cross-bonded cable. The results indicate that not all cross-bonded cables can be simplified by a solidly bonded cable from the viewpoint of not only the sheath voltages but also the core voltages. 

Grounding of a cable, (a) Single-bonded cable and (b) solidly bonded cable. 

Figure 3.9 shows a sequence current measurement circuit for a solidly bonded cable. The following equations are given from the 6x6 impedance matrix in (3.47) and Figure 3.9 ... 

In Table 3.3a-l, the upper 3x3 matrix expresses three-phase core impedance [ZJ, the right upper and the left lower matrices are three-phase core to sheath [ZJ and sheath to core impedance [Z J = [ZJ and the lower 3x3 matrix is three-phase sheath impedances [ZJ. The upper line is the resistance [Q/km] and the lower one is the inductance [mH/km]. The [ZJ matrix of the solidly bonded cable in Table 3.3a-l is the same as that of fhe cross-bonded cable in Table 3.3b-l. The impedance in the last column,... 

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