Geophones: coil-magnet physics

Part 2 — Receivers

Learning objectives

Derive the geophone transfer function from the driven damped-oscillator equation
Read a geophone spec sheet: interpret natural frequency f₀ and damping ratio ζ
Explain why geophones are useless below f₀ but extraordinarily sensitive above it
Recognise the phase shift at f₀ (−90°) and at high frequency (→ 0° for velocity input)

A geophone is a mechanical velocity transducer. The physics is a mass on a spring inside a case, with a coil wound around the mass (or fixed to the case) and a permanent magnet fixed to the case (or to the mass). When ground motion shakes the case, the spring-supported mass lags behind because of inertia; the resulting relative velocity between coil and magnet generates a voltage by electromagnetic induction.

The equation of motion

Let x(t) be ground displacement and y(t) be the displacement of the mass relative to the case. Newton’s second law on the mass, with a spring constant k and damper c:

m \ddot y + c \dot y + k y = -m \ddot x

The output voltage is proportional to the relative velocity \dot y. Define the natural frequency f₀ = (1/2π)√(k/m) and the damping ratio ζ = c / (2√(km)). Taking the Fourier transform and solving for the transfer from ground velocity to output gives

H(f) = \frac{(f/f_0)^2}{(1-(f/f_0)^2) + 2\,i\,\zeta\,(f/f_0)}

which is the canonical second-order velocity-sensor response.

What the transfer function tells you

Below f₀: |H| ~ (f/f₀)². At f = f₀/2, response is one quarter of flat; at f = f₀/4, one sixteenth. The geophone is approximately blind to sub-f₀ frequencies.
At f = f₀: phase is −90° regardless of ζ. Magnitude depends on damping: for ζ = 0.7, |H(f₀)| ≈ 0.71; for ζ = 0.3, |H(f₀)| ≈ 1.67 (peaked).
Above f₀: |H| → 1, phase → 0°. This is the flat region the geophone is designed to live in.

Real-world geophone specs

A typical land exploration geophone is the 10 Hz model: f₀ = 10 Hz, ζ = 0.7 (set by the "shunt" resistor across the coil — changing the resistor changes the electromagnetic damping and hence ζ). Recent models have dropped to 4.5 Hz or even 2 Hz to compete with MEMS for low-frequency content; those lower-f₀ designs require a larger mass or a softer spring, making them physically bigger and more sensitive to tilt.

The shunt resistor also sets sensitivity, measured in V/(m/s). Typical: 28–80 V/(m/s) for exploration-grade geophones. Higher sensitivity gives better SNR at low signal levels but saturates earlier. Every geophone spec sheet carries three numbers: f₀, ζ, and sensitivity. Know them cold.

Why damping = 0.7

Critical damping is ζ = 1: no overshoot but slow response. ζ = 0.7 is the flattest magnitude response compatible with not peaking at f₀. Lower ζ gives more gain at f₀ but rings; higher ζ is stable but rolls off later. 0.7 is the industry sweet spot — you will see it on every geophone spec you ever read.

References

Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.
Mougenot, D. (2013). MEMS-based 3C accelerometers for land seismic acquisition. The Leading Edge, 32(4), 388–396.