High voltage measurement by electrostatic voltmeter

An electrostatic voltmeter measures voltage from the attractive force between two charged electrodes. Since the force depends on $V^2$, it reads the true rms value of AC (of any waveform) and the value of DC, and it draws almost no current.

Principle

For two parallel plates of area $A$ separated by $x$, the stored energy is $W = \frac12 CV^2$ with $C = \varepsilon_0 A/x$. The force between the plates is

$$\begin{aligned} F &= \frac{dW}{dx} = \frac12 V^2\frac{dC}{dx} \\ &= -\frac12\,\frac{\varepsilon_0 A V^2}{x^2}\quad(\text{attraction}) \end{aligned}$$

For AC, $V = V_m\sin\omega t$, and the moving system responds to the average force:

$$F_{av} = \frac{\varepsilon_0 A}{2x^2}\cdot\frac1T\int_0^T v^2\,dt = \frac{\varepsilon_0 A}{2x^2}V_{rms}^2$$

So the deflection measures $V_{rms}$.

Construction (attracted-disc type)

          HV electrode  (V)
   ===============================
                 |  gap x
   ======  [ movable disc ]  =====
   guard     |  (earthed)    guard
   ring      |               ring
             +-- spring / balance
             +-- mirror -> lamp & scale
                 |
               earth

• High-voltage electrode: a large plane disc (or sphere) connected to the voltage to be measured.
• Low-voltage electrode: an earthed plane with a guard ring; a small movable disc sits in a hole at its centre, flush with the plane.
• The guard ring keeps the field over the movable disc uniform, so the simple parallel-plate formula holds and edge effects are removed.
• The movable disc is held by a spring or a balance beam. Its small movement is magnified by a mirror and light beam on a scale, or the force is balanced by weights or a restoring spring (absolute instruments such as the Kelvin attracted-disc electrometer).
• The whole system may be enclosed in compressed gas (N₂, SF₆) or vacuum to raise the range.

Working

1. Voltage is applied between the HV electrode and the earthed plane.
2. The movable disc is attracted towards the HV electrode with force proportional to $V^2$.
3. The disc moves until the spring force balances the electrostatic force; the deflection is read on the scale, calibrated directly in kV.
4. For range change, the gap $x$ is adjusted (larger gap for higher voltage).

Features

Ranges of about 600 kV in air and up to about 1000 kV with compressed gas insulation are achieved.