Moving-Coil Galvanometer

What is modelled

A rectangular coil of N turns and area A hangs in a radial magnetic field B, so the torque from a current i is G i with G = N A B whatever the angle. A torsion fibre gives a restoring torque kθ and eddy-current and air damping give c dθ/dt. A short pulse of charge Q sets the coil moving; its first maximum swing is proportional to Q, which makes the instrument ballistic.

Equations

• J θ″ + c θ′ + k θ = G i(t), natural frequency ω0 = √(k/J), damping ratio ζ = c / (2√(kJ)).
• Ballistic first throw θmax = G Q e−ζω0tp sin(ωdtp) / (J ωd).
• Steady DC deflection θ = G i / k; AC amplitude G i / |k − Jω² + i c ω|.
• Logarithmic decrement ln(θn/θn+1) = 2πζ / √(1 − ζ²).

Checks run in the page

• RK4 first throw for a 1 ms pulse against the closed form.
• Settled DC deflection against G i / k.
• Steady AC amplitude against the frequency response.
• Free-ring log decrement against the damping ratio.

Honest simplifications

A linear torsion spring, ideal radial field, viscous damping and a rectangular current pulse; no thermal drift, no coil resistance or circuit damping, and the scale is idealised. Values are illustrative of the principle, not of one instrument.

• Choose Lightly damped ringing and watch many swings decay; compare the log decrement.
• Choose DC deflection and see the coil settle at G i / k.
• Choose AC near resonance and sweep the frequency to find the peak.