**2.1. Partial discharge pulse properties**

**Figure 1.** A typical high-voltage extruded solid dielectric cable. A, encapsulating jacket; B, neutral wires; C, ground shield; D, dielectric; E, conductor shield; F, conductor; the conductor shield and ground shield are semiconductive. Usually, they are made by adding carbon black into a polymer and the particle size of the carbon black ranges from 15 to 50 nm. The basic function of this configuration is to confine the electric field within the cable and obtain a symmetri‐

Most failures occurring in the shielded dielectric cables are related to partial discharge. Partial discharges are localized breakdowns in a small portion of the insulations, which can be solid or fluid electrical insulation. When high voltage is applied to high-voltage equipment, defects introduced during the manufacturing process such as contained insulation cracks, contained insulation surfaces, or voids can all lead to partial discharges. In some cases, even without any defects, aging can cause the degradation of the insulator leading to partial discharges. PD makes damage to the equipment, and equipment with PD occurring within will eventually fail after a certain time depending on the strength of the PD if proper treatments are not applied. It's important to monitor partial discharges in high-voltage systems and if PD is detected, appropriate actions should be taken to prevent sudden failures, which can cause big blackouts.

Figure 2 gives us some basic ideas on how partial discharges occur. The cavity in the insulation normal contains gas which could be ionized when the electrical field exceeds the cutoff strength. When this happens, electromagnetic waves in the radiofrequencies are generated along with light, heat, noise, and possibly gas. With appropriate technologies and by detecting the HF radio signals, the magnitude as well as the location of partial discharges can be

**Figure 2.** A "typical" partial discharge mechanism. The cavity in the insulation normally contains gas, which could be

identified and used for assessing the health status of the shielded dielectric cable.

cal radial distribution of the electric within the dielectric [1].

94 Advanced Electromagnetic Waves

ionized when the electrical field exceeds the cutoff strength.

**1.2. Partial discharge in shielded dielectric power cables**

As discussed above, PD can create high-frequency electromagnetic waves in the radiofre‐ quencies, heat, light, gas, etc. A lot of research has been conducted to study for detecting PD using sensors for heat, light, gas, etc. In this chapter, we focus on radiofrequency highfrequency PD signal detection as it provides a way to detect PD over long distances, which makes it useful in many cases. To effectively detect and analyze PD, first we have to understand the PD pulse. The authors will analyze the PD pulse spectrums, frequency properties, etc. Note that PD occurring in different high-voltage apparatus varies significantly due to the insulation properties and electric stress needed for triggering PD. Since this chapter focuses on PD in shielded dielectric cables, we will focus on the PD pulse properties in shielded dielectric cables.

Because the formation of electron avalanches is within the nanosecond range, the PD event is associated with a very fast current pulse. As of today, there is no direct way to measure the PD current pulse. A lot of theories have been proposed to simulate and study the current pulse and different measuring techniques were developed to measure PD current pulses. The PD is caused by the flow of the electrons and ions. The moving speed of the electrons is much faster than the ions. Some early theoretical simulations [2–5] predicted that the pulse current in the voids of the shielded dielectric cables has a rising time and pulse widths in the nanosecond range followed by a long-duration, low-magnitude pulse. Figure 3 shows one theoretically predicted PD current pulse. With the high-speed oscilloscope available, some PD experiments were conducted by Fujimoto and Boggs (1981) and Boggs and Stone (1982) [4] and the experimental results agree with the theoretical simulation results.

**Figure 3.** One theoretically predicted PD current pulse [2–5].

There is no standard spectrum chart for PD pulses in the power cables because under different situations, the PD pulse can be in different shapes leading to different energy spectrums. Some experimental results [6] show that the partial discharge spectrum has a peak point at about 100 MHz when measuring PD at a distance very close to the PD source. When the partial discharge pulse propagates in the high-voltage cable, its high-frequency components are attenuated by the losses majorly caused by the semiconducting materials and, after a 100-meter trip, the peak frequency of the spectrum can be reduced to 20–30 MHz.

### **2.2. Shielded dielectric cable HF attenuation properties**

When the electromagnetic PD wave propagates in the cable, its HF components are signifi‐ cantly attenuated by the shielded cables. How far the pulse can travel and be detected by the PD detection devices heavily relies on the attenuation properties of the cable. A lot of research [7–17] has been conducted to study different aspects of the HF attenuation features of the shielded dielectric cables. From Figure 1, the HF attenuation is caused by various components. One measurement result [9] shown in Figure 4 gives HF attenuation losses for different components for different frequencies. From Figure 4, it can be seen that for this measurement, conductor and neutral wire skin effect losses dominate for low frequencies up to 5 MHz. After 5 MHz, grounding shielding losses as well as the dielectric losses start to play a more important role. This can be explained by the fact that at low frequencies, the capacitive current is low thus the current passing through the dielectric, conductor and ground shielding is low leading to relatively low losses. When frequencies are higher, the larger capacitive current flowing through the resistive component of the conductor and ground shielding makes big losses.

**Figure 4.** Measurement results for one shielded cable [9].

### *2.2.1. Shield HF property measurements*

From the above analysis, it can be seen that the dielectric properties of the shielding are critical to HF losses of the shielding dielectric cables. The dielectric properties of the cable semicon‐ ducting shielding can be measured by a HF impedance analyzer. Figure 5 [10 and 17] shows a typical measurement configuration for HF dielectric properties (dielectric constant and

conductivity) of the cable shielding. Conducting electrodes with small surface resistivity, often metal paint, are applied to the two surfaces of the shield material. Normally, contact with the electrodes is made along one edge of the sample. There is a voltage drop across the "conduct‐ ing" electrodes, which can cause errors in the dielectric properties measurement when the current flowing through the sample and the nonperfect electrode resulting in possible measurement errors. shows a typical measurement configuration for HF dielectric properties (dielectric constant and conductivity) of the cable shielding. Conducting electrodes with small surface resistivity, often metal paint, are applied to the two surfaces of the shield material. Normally, contact with the electrodes is made along one edge of the sample. There is a voltage drop across the "conducting" electrodes, which can cause errors in the dielectric properties measurement when the current flowing through the sample and the nonperfect electrode resulting in possible measurement errors.

semiconducting shielding can be measured by a HF impedance analyzer. Figure 5 [10 and 17]

Figure 4. Measurement results for one shielded cable [9].

**2.2.1 Shield HF property measurements** 

**2.2. Shielded dielectric cable HF attenuation properties**

96 Advanced Electromagnetic Waves

**Figure 4.** Measurement results for one shielded cable [9].

*2.2.1. Shield HF property measurements*

When the electromagnetic PD wave propagates in the cable, its HF components are signifi‐ cantly attenuated by the shielded cables. How far the pulse can travel and be detected by the PD detection devices heavily relies on the attenuation properties of the cable. A lot of research [7–17] has been conducted to study different aspects of the HF attenuation features of the shielded dielectric cables. From Figure 1, the HF attenuation is caused by various components. One measurement result [9] shown in Figure 4 gives HF attenuation losses for different components for different frequencies. From Figure 4, it can be seen that for this measurement, conductor and neutral wire skin effect losses dominate for low frequencies up to 5 MHz. After 5 MHz, grounding shielding losses as well as the dielectric losses start to play a more important role. This can be explained by the fact that at low frequencies, the capacitive current is low thus the current passing through the dielectric, conductor and ground shielding is low leading to relatively low losses. When frequencies are higher, the larger capacitive current flowing through the resistive component of the conductor and ground shielding makes big losses.

From the above analysis, it can be seen that the dielectric properties of the shielding are critical to HF losses of the shielding dielectric cables. The dielectric properties of the cable semicon‐ ducting shielding can be measured by a HF impedance analyzer. Figure 5 [10 and 17] shows a typical measurement configuration for HF dielectric properties (dielectric constant and

Figure 5. Typical measurement method for HF dielectric properties of the cable shielding and a simplified sample geometry and lumped element representation [10 and 17]. **Figure 5.** Typical measurement method for HF dielectric properties of the cable shielding and a simplified sample ge‐ ometry and lumped element representation [10 and 17].

The measured sample impedance is normally interpreted as resulting from the dielectric constant and conductivity of the sample if we assume that the losses and voltage drop in the electrodes are negligible [7 and 10]. The applied electrode has a finite conductivity (normal The measured sample impedance is normally interpreted as resulting from the dielectric constant and conductivity of the sample if we assume that the losses and voltage drop in the electrodes are negligible [7 and 10]. The applied electrode has a finite conductivity (normal silver paint) resulting in changes in the measured loss leading to errors in predicting the shield properties. These errors increase with frequency because the current through the electrodes increase significantly with frequency as a result of the large dielectric constant of the shield material. If we assume that voltage is applied along one edge of the upper and lower surface of a rectangular sample, as shown in Figure 5, and define *I*1(*x*) is the current as a function of distance from that edge on one surface (electrode), then the magnitude of the current in the other surface is also *I*1(*x*) with an opposite direction. Likewise, if the voltage on energized surface as a function of distance from that edge is *U*1(*x*), then the voltage on the grounded surface is *U*o-*U*1(*x*). *U*o is the applied voltage. From the geometry shown in Figure 5, the following equations can be derived

$$\begin{aligned} I\_{1}(\mathbf{x}) &= -\frac{\mathcal{U}\_{0}\sqrt{\frac{Ab}{2\rho}\bigg\{\exp\left(x\sqrt{\frac{2\rho A}{b}}\right) - \exp\left[(2a-x)\sqrt{\frac{2\rho A}{b}}\right]\right\}}{\left[1 + \exp\left(2a\sqrt{\frac{2\rho A}{b}}\right)\right]}\\ \mathcal{U}\_{1}(\mathbf{x}) &= \frac{\mathcal{U}\_{0}}{2}\left\langle \frac{\exp\left(x\sqrt{\frac{2\rho A}{b}}\right) + \exp\left[(2a-x)\sqrt{\frac{2\rho A}{b}}\right] + \exp\left(2a\sqrt{\frac{2\rho A}{b}}\right) + 1}{1 + \exp\left(2a\sqrt{\frac{2\rho A}{b}}\right)} \right\rangle\\ A &= \frac{b(\sigma + jo\alpha x\_{0})}{d} \end{aligned} \tag{1}$$

where *d* is the sample thickness, *ρ* is the electrode surface resistivity in Ω/sq, *a* is the sample length, *b* is the sample width, *ω* is the angular frequency, and *σ* and *ε* are the conductivity and dielectric constant of the sample, respectively. Furthermore, we can derive the measured conductivity and relative dielectric constant as [17]

$$
\sigma\_p = \frac{(\mathcal{D}P\_{\text{electrode}} + P\_{\text{shield}})d^2}{V\mathcal{U}\_0^2} \tag{2}
$$

and

$$
\varepsilon\_p = \frac{dI\_c}{\mathcal{U}\_0 a \varepsilon\_0 ab} \tag{3}
$$

From Eq. 2, it can be seen that the power losses *P*electrode caused by the electrodes cause errors as we are measuring losses by the shield *P*shield, not losses by the electrodes. On the other hand, there is a voltage drop along the electrode due to the larger current and finite conductivity of the electrode resulting in some areas where there is no current flowing through. Thus, the measured losses can be larger or smaller than the actual shield losses. The shield conductivity property measurements affect cable high-frequency attenuation calculations a lot, thus it's very important that we minimize the errors caused by the electrode resistance. A first-order estimate of the maximum frequency to which accurate measurements are likely by using the DC conductivity and low frequency shield dielectric constant to roughly estimate the maximum frequency at which the impedance of the shield is comparable to that of the sample electrodes resulting in Equation 4 for a rectangle sample and Equation 5 for a circular sample [10 and 17]:

$$f\_{\max} = \frac{\sqrt{-4\sigma^2 \rho^2 a^2 + \left(\frac{d}{a}\right)^2}}{4\pi a \rho \varepsilon \varepsilon\_0} \tag{4}$$

$$f\_{\max1} = \frac{\sqrt{-\sigma^2 \rho^2 \ln\left(\frac{r\_2}{r\_1}\right)^2 + \left(\frac{d}{r\_2^2 - r\_1^2}\right)^2}}{2\pi \ln\left(\frac{r\_2}{r\_1}\right)\rho \varepsilon \varepsilon\_0} \tag{5}$$

where *r*2 is the "conducting" electrode radius, *r*1 is the contact radius at the center of the electrode, and *d* is the sample thickness. Figure 6 shows the measured ground shield relative dielectric constant and conductivity for a shield power cable as well as the calculated and measured HF attenuation for the shielded cables.

**Figure 6.** The measured ground shield relative dielectric constant and conductivity for a shielded power cable.

### *2.2.2. Shielded cable HF attenuation calculations*

of a rectangular sample, as shown in Figure 5, and define *I*1(*x*) is the current as a function of distance from that edge on one surface (electrode), then the magnitude of the current in the other surface is also *I*1(*x*) with an opposite direction. Likewise, if the voltage on energized surface as a function of distance from that edge is *U*1(*x*), then the voltage on the grounded surface is *U*o-*U*1(*x*). *U*o is the applied voltage. From the geometry shown in Figure 5, the

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98 Advanced Electromagnetic Waves

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From Figure 4, there are losses from different components of the shielded power cable. The losses come from insulation, skin effect, conduct shield, and ground shield. For high frequen‐ cies (5–20 MHz), the loss in the ground shield could dominate the shield cable loss [11,12]. This chapter discusses the loss calculations in the ground shield as it dominates losses at high frequency, which is critical for HF PD pulse propagation, detection and analysis. The shielded power cable can have different geometries, configurations, and number of neutral wires. The chapter won't cover all cable configurations. Two typical shielded cables will be studied.
