Question 7.7: DIELECTRIC LOSS AND FREQUENCY Calculate the heat generated p...
DIELECTRIC LOSS AND FREQUENCY Calculate the heat generated per second due to dielectric loss per cm³ of cross-linked polyethylene, XLPE (typical power cable insulator), and alumina, Al_{2}O_{3} (typical substrate in thin- and thick-film electronics), at 60 Hz and 1 MHz at a field of 100 kV cm^{−1}. Their properties are given in Table 7.4. What is your conclusion?
The power dissipated per unit volume is
W_{ vol }=(2 \pi f) E^{2} \varepsilon_{o} \varepsilon_{r}^{\prime} \tan \delta
We can calculate W_{vol} by substituting the properties of individual dielectrics at the given frequency f. For example, for XLPE at 60 Hz,
\begin{aligned} W_{ vol } &=(2 \pi 60 Hz )\left(100 \times 10^{3} \times 10^{2} V m^{-1 } \right)^{2}\left(8.85 \times 10^{-12} F m ^{-1}\right)(2.3)\left(3 \times 10^{-4}\right) \\ &=230 W m \end{aligned}
We can convert this into per cm³ by
W_{ vol }^{\prime}=\frac{W_{ vol }}{10^{6}}=0.230 mW cm ^{-3}
which is shown in Table 7.4.
From similar calculations we can obtain the heat generated per second per cm³ as shown in Table 7.4. The heats at 60 Hz are small. The thermal conductivity of the insulation and its connecting electrodes can remove the heat without substantially increasing the temperature of the insulation. At 1 MHz, the heats generated are not trivial. One has to remove 5.12 W of heat from 1 cm³ of XLPE and 47.3 W from 1 cm³ of alumina. The thermal conductivity \kappa of XLPE is about 0.005 W cm^{−1} K^{−1}, whereas that of alumina is almost 100 times larger, 0.33 W cm^{−1} K^{−1}. The poor thermal conductivity of polyethylene means that 5.12 W of heat cannot be conducted away easily and it will raise the temperature of the insulation until dielectric breakdown ensues. In the case of alumina, 47.3 W of heat will substantially increase the temperature. Dielectric loss is the mechanism by which microwave ovens heat food.
Dielectric heating at high frequencies is used in industrial applications such as heating plastics and drying wood.
| Table 7.4 Dielectric loss per unit volume for two insulators (\kappa is the thermal conductivity) | |||||||
| f = 60 Hz | f = 1 MHz | ||||||
| Material | \varepsilon_{r}^{\prime} | tan δ | Loss (mW cm^{-3}) |
\varepsilon_{r}^{\prime} | tan δ | Loss (mW cm^{-3}) |
\kappa (W cm^{-1} K^{-1}) |
| XLPE | 2.3 | 3 × 10^{−4} | 0.230 | 2.3 | 4 × 10^{−4} | 5.12 | 0.005 |
| Alumina | 8.5 | 1 × 10^{−3} | 2.84 | 8.5 | 1 × 10^{−3} | 47.3 | 0.33 |