Loss tangent and dielectric power loss

Loss tangent

In an ideal dielectric the current leads the voltage by exactly 90°, so no power is lost. In a real dielectric, polarization cannot follow an AC field instantly, so part of the energy is turned into heat. This is described by a complex relative permittivity

$$\varepsilon_r = \varepsilon_r' - j\varepsilon_r''$$

where $\varepsilon_r'$ is the ordinary (storage) part and $\varepsilon_r''$ is the loss part. The loss tangent is

$$\tan\delta = \frac{\varepsilon_r''}{\varepsilon_r'}$$

$\delta$ is the angle by which the current falls short of leading the voltage by 90°. Good insulators have $\tan\delta \sim 10^{-4}$ (polyethylene); lossy ones are 0.01–0.1.

      I_total
        ^   /
  I_cap |  /
  (wC'V)| /  delta
        |/________> V
          I_loss (wC''V)

Power loss per unit volume

Consider a parallel-plate capacitor with area $A$, spacing $d$, filled with the dielectric, and an applied voltage $V = V_0 e^{j\omega t}$.

Capacitance with complex permittivity:

$$C = \frac{\varepsilon_0(\varepsilon_r' - j\varepsilon_r'')A}{d}$$

Current:

$$I = j\omega C V = \frac{\omega\varepsilon_0 A}{d}\left(j\varepsilon_r' + \varepsilon_r''\right)V$$

The term $\frac{\omega\varepsilon_0\varepsilon_r'' A}{d}V$ is in phase with $V$ and causes power loss; the $j\varepsilon_r'$ term is the lossless capacitive current. So the dielectric acts like a capacitor with a parallel conductance

$$G_p = \frac{\omega\varepsilon_0\varepsilon_r'' A}{d}$$

Power dissipated (using rms voltage $V_{rms}$):

$$W = G_p V_{rms}^2 = \frac{\omega\varepsilon_0\varepsilon_r'' A}{d}V_{rms}^2$$

Divide by the volume $Ad$ and use $E_{rms} = V_{rms}/d$:

$$W_{vol} = \frac{W}{Ad} = \omega\varepsilon_0\varepsilon_r'' E_{rms}^2$$

Since $\varepsilon_r'' = \varepsilon_r'\tan\delta$:

$$\boxed{W_{vol} = \omega\, \varepsilon_0\, \varepsilon_r'\, E_{rms}^2 \tan\delta}$$

Conclusion

The dielectric loss per unit volume is directly proportional to:

• the frequency $\omega = 2\pi f$ of the applied field,
• the loss tangent $\tan\delta$ (or loss factor $\varepsilon_r'\tan\delta$),
• the square of the field strength.

So insulation for high-frequency or high-voltage use (cables, RF capacitors) must have a very low $\tan\delta$. The same heating is used usefully in microwave ovens and dielectric heating of plastics and wood.