Sheath voltages

Meijer and Goedheer [174] have developed a ID model with which, among other things, the dc self-bias and the sheath voltages can be calculated. It... 

The sheaths between the conductive plasma and the walls behave like capacitors, so a relation exists between the sheath voltages and the electrode areas A ... 

Fig. 8 is for a symmetric reactor, i.e., one in which the two electrodes have the same area. Then the time-average potential distribution is symmetric as well, and there is no DC bias developed. In asymmetric reactors the time-average potential distribution looks much like the one in Fig. 6. In fact, assuming capacitive voltage division between the two sheaths, the time-average sheath voltage scales with the inverse electrode area ratio as [15]...

The exponent has the theoretical value of = 4 [27], but a value of n 1-2 has been determined experimentally [28]. In any case, Eq. (3) shows that the smaller electrode develops a larger sheath voltage, hence it receives stronger ion bombardment. 

For fixed sheath voltage and ion current, the sheath thickness scales as 5 oc, i.e., a weak dependence on pressure. 

When an+ 1 ions traverse the sheath in a short time compared to the field oscillations. Under this condition, an ion traversing the sheath experiences the sheath voltage prevailing at the time the ion entered the sheath. In the absence of collisions, the lED function will reflect precisely the variation of the sheath voltage with time. This quasi steady-state condition of cut+ 1 is satisfied for low RF frequencies or short ion transit times, i.e., thin sheaths (low sheath voltage or small Debye length). 

In capacitively coupled systems (Fig. 7) the ion bombardment energy increases with decreasing frequency (Fig. 36). This is because the sheath voltage increases at lower frequency [184]. Hence a larger fraction of the applied power is dissipated for accelerating ions in the sheath rather than for producing radicals in the bulk plasma. 

Fig. 36. Dependence of the maximum ion bombardment energy on the excitation frequency in a capaci-tively coupled chlorine discharge at a pressure of 0.3 torr. At low frequencies ions traverse the sheath in a short time compared to the period of the field. The maximum ion energy is then equal to the peak of the sheath voltage. At high frequencies, the ion transit time is many RF cycles ions respond only to the time-average sheath voltage. The transition happens at the ion plasma frequency which for the conditions of this Figure is about I MHz. After [184],...

Matrix and Child Law Sheaths. Calculate the sizes of the matrix and Child law sheaths for non-thermal plasma with electron temperature 3 eV, electron density lO cm, and sheath voltage 300 V Compare the results obtained for the models of matrix and Child law sheaths. 

The a-to-y transition in APPJ happens because of the Townsend breakdown (4-4) of the sheath, which occurs when ion density and sheath voltage exceed the critical values ( p(cnt) = 3 xl0"cm , Fs= 300 V see Chirokov, 2005 Raizer et al., 1995). Chirokov... 

Figure 3.6 illustrates a major section of a cross-bonded cable. The bold solid line and the broken line express the core and sheath, respectively. The sheaths are grounded through grounding impedance Zg at both sides of the major section. The core and sheath voltages and Vk and currents Ij and jjJ at the kth cross-bonded node are related as in the following equations ... 

The lengths of the minor sections can have imbalances due to constraints on the locations of joints. The imbalances are designed to be as small as possible, since they increase sheath currents and raise sheath voltages. When a cable system has multiple major sections, the overall balance is considered for minimizing sheath currents. As a result, when a cable system has more than two major sections, sheath currents are generally balanced among three conductors, which allows for the reduction from three metallic sheaths to one conductor [4,5]. 

Since three-phase sheath conductors are short-circuited and grounded in every major section (as illustrated in Figure 3.6), the sheath voltages of the three phases are equal at each... 

As mentioned earlier, it is a common practice for underground cable systems that are longer than approximately 2 km to cross-bond the metallic sheaths of three-phase cables to simultaneously reduce sheath currents and suppress sheath voltages [3]. Submarine cables. 

Assuming the grounding resistance at substations Rg, the sheath voltage Vj can be obtained by ... 

Calculated transient sheath voltages on a tunnel-installed cable (a) cross-bonded and (b) mixed. 

The maximum sheath voltage at the first cross-bonded joint shown in Figure 3.15a becomes 0.05 pu, which is about 40% of the core voltage at the time. This voltage is generated by a reflection at the cross-bonded joint. This is an inherent characteristic of the cross-bonded... 

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. 

Continuous voltage limitation is the limitation of the sheath voltage induced by the normal load flow in phase conductors without any faults. It is enforced by government or district regulations in many countries and differs in each area based on said regulations. This limitation was enforced for the safety of the maintenance crews who may come into contact with the sheath circuit. Even if this limitation is not enforced, utilities follow their own standards for continuous voltage limitation. 

When only power-frequency components are considered, SLG faults and three-phase faults are studied using theoretical formulas. Some utilities study SLG faults and three-phase faults using EMTP in order to consider transient components of the sheath voltage. Switching surges rarely become an issue for the sheath overvoltage. 

A three-phase cable system consisting of three single-core (SC) cables becomes a six-conductor circuit. If its sheath voltages can be neglected, the cable can be expressed by a three-phase circuit. The voltage drop due to the cable-series impedance is expressed by the following equation ... 

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