Measurement of dielectric constant and loss factor

The dielectric constant ($\varepsilon_r$) of an insulating material is the ratio of the capacitance of a specimen to its capacitance with vacuum (air) as dielectric. The loss factor is $\varepsilon_r\tan\delta$, where $\tan\delta$ (dissipation factor) is the ratio of the loss (resistive) current to the charging (capacitive) current. Both are measured by bridge methods at power and higher frequencies; the most common is the high-voltage Schering bridge.

  Ix     Ic (leading by 90 deg)
   \     |
    \    |   delta = angle between
     \   |   Ix and Ic
      \d |
       \ |
  ------+--------> Ir (in phase with V)

$$\tan\delta = \frac{I_R}{I_C}, \qquad \text{loss } P = V^2\omega C\tan\delta$$

1. High-voltage Schering bridge

              HV supply (V)
              /           \
      Arm 1  /             \ Arm 2
    Cx + Rx                 Cs (standard,
   (specimen)               loss-free)
          |                   |
          A------[ D ]--------B
          |                   |
      Arm 3: R3        Arm 4: R4 || C4
          \                  /
            ----- earth ----

• Arm 1: test specimen, represented as $C_x$ in series with $R_x$.
• Arm 2: standard compressed-gas capacitor $C_s$ (loss-free).
• Arm 3: variable non-inductive resistance $R_3$.
• Arm 4: fixed resistance $R_4$ in parallel with a variable capacitor $C_4$.
• D: vibration galvanometer or electronic null detector.
• Arms 3 and 4 are at low voltage (near earth), so adjustments are safe; protective gaps across them guard against a breakdown of $C_x$ or $C_s$. Earthed shields (Wagner earth) remove errors due to stray capacitance.

Balance condition $Z_1 Z_4 = Z_2 Z_3$:

$$\left(R_x + \frac{1}{j\omega C_x}\right)\frac{R_4}{1 + j\omega C_4R_4} = \frac{R_3}{j\omega C_s}$$

Equating real and imaginary parts:

$$\begin{aligned} C_x &= C_s\,\frac{R_4}{R_3} \\ R_x &= R_3\,\frac{C_4}{C_s} \\ \tan\delta &= \omega C_x R_x = \omega C_4 R_4 \end{aligned}$$

Usually $R_4$ is a fixed value such as $\frac{1000}{\pi}\ \Omega$. At 50 Hz, $\omega R_4 = 100\pi\times\frac{1000}{\pi} = 10^5\ \Omega$, so $\tan\delta = 0.1\,C_4$ (with $C_4$ in μF), and the $C_4$ dial can be calibrated directly in $\tan\delta$. (With $R_4 = \frac{10^4}{\pi}\ \Omega$, $\tan\delta = C_4$ in μF.)

Dielectric constant: with specimen thickness $d$ and electrode area $A$, $C_0 = \varepsilon_0 A/d$, and

$$\varepsilon_r = \frac{C_x}{C_0} = \frac{C_x\,d}{\varepsilon_0 A}, \qquad \text{loss factor} = \varepsilon_r\tan\delta$$

A three-terminal (guard-ring) electrode system is used so that surface leakage and edge fringing do not affect the result.

2. Transformer ratio-arm bridge

• Uses a precision ratio transformer (ratio arms) instead of resistive arms; balance is set by turns ratios $N_s/N_x$.
• $C_x = C_s N_s/N_x$ and $\tan\delta$ from the conductance balance.
• Advantages: very high accuracy, unaffected by stray capacitances to earth, can test earthed specimens, works over a wide frequency range.

3. Other methods

• Resonance (Q-meter) method for high frequencies (kHz–MHz): change in resonance capacitance gives $C_x$, change in Q gives $\tan\delta$.
• Measurements versus voltage and temperature: $\tan\delta$ rising with voltage ("tip-up") indicates voids and partial discharge; rising with temperature indicates moisture and ageing.