# Effect of coil inductance on geophone equation

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 7 221 - 252 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem

If we wish to take into account the small inductance term ${\displaystyle L\ {\rm {d}}^{3}i/{\rm {d}}t^{3}}$ in the geophone equation (7.9a), show that, for a harmonic wave, it can be included approximately in the term involving the damping factor ${\displaystyle h}$ in equation (7.9b).

### Background

The moving-coil electromagnetic geophone is widely used for land seismic work. The basic elements are a case with attached magnet and a coil suspended in the magnetic field by means of springs in such a way that, when an arriving seismic wave moves the case vertically, the inertia of the coil causes it to remain relatively stationary, that is, it moves in a different manner from that of the case; the relative motion between the coil and the magnetic field induces a voltage in the coil that is a replica of the seismic signal that caused the ground motion.

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Let ${\displaystyle m}$, ${\displaystyle r}$, ${\displaystyle n}$, and ${\displaystyle i}$ be the mass, radius, number of turns, and current in the coil, ${\displaystyle R}$ and ${\displaystyle L}$ the resistance and inductance of the coil; ${\displaystyle H}$ is the magnetic field strength, ${\displaystyle \tau }$ the mechanical damping factor (the damping force being ${\displaystyle \tau \times }$ vertical velocity of the coil relative to ${\displaystyle H}$), ${\displaystyle K=2\pi nrH=}$ electromagnetic force on the coil per unit current, ${\displaystyle z}$ the vertical displacement of the geophone case. Using these symbols, the geophone equation (see Sheriff and Geldart, 1995, 218–20) is

 {\displaystyle {\begin{aligned}L{\frac {{\rm {d}}^{3}i}{{\rm {d}}t^{3}}}+\left(R+{\frac {L\tau }{m}}\right){\frac {{\rm {d}}^{2}i}{{\rm {d}}t^{2}}}+\left({\frac {SL+\tau R+K^{2}}{m}}\right){\frac {{\rm {d}}i}{{\rm {d}}t}}+\left({\frac {SR}{m}}\right)i=K{\frac {{\rm {d}}^{3_{Z}}}{{\rm {d}}t^{3}}}\end{aligned}}} (7.9a)

It is usually assumed that ${\displaystyle L}$ is small enough that it can be set equal to zero. In this case the geophone equation reduces to

 {\displaystyle {\begin{aligned}{\frac {{\rm {d}}^{2}i}{{\rm {d}}t^{2}}}+2h\omega _{0}{\frac {{\rm {d}}i}{{\rm {d}}t}}+\omega _{0}^{2}i=\left({\frac {K}{R}}\right){\frac {{\rm {d}}^{3}i}{{\rm {d}}t^{3}}},\end{aligned}}} (7.9b)

where ${\displaystyle \omega _{0}={\sqrt {S/m}}=2\pi f_{0}={}}$ natural frequency of oscillation of the coil and

{\displaystyle {\begin{aligned}2h\omega _{0}=\left({\frac {\tau }{m}}+{\frac {K^{2}}{mR}}\right).\end{aligned}}}

The quantity ${\displaystyle {\textit {h}}}$ is called the damping factor for the following reason. If ${\displaystyle h=0}$ in equation (7.9b), the equation becomes that for undamped simple harmonic motion; if the geophone is at rest and the coil is set in motion, it oscillates forever (theoretically) with undiminished amplitude. However, if ${\displaystyle h\neq 0}$, the amplitude decreases steadily by a fixed proportion each cycle; this decrease in amplitude is called damping [see also equation (2.18b)].

### Solution

If in equation (7.9a) we set ${\displaystyle L=0}$ in all terms except the first, we get equation (7.9b) plus the additional term ${\displaystyle L\left({\rm {d}}^{3}i/{\rm {d}}t^{3}\right)}$. Assuming input of the form ${\displaystyle i=i_{0}\cos \omega t}$, we have ${\displaystyle {\rm {d}}i/{\rm {d}}t=-\omega \sin \omega t}$, ${\displaystyle {\rm {d}}^{3}i/{\rm {d}}t^{3}=\omega ^{3}\sin \omega t}$ hence ${\displaystyle {\rm {d}}^{3}i/{\rm {d}}t^{3}=\omega ^{2}{\rm {d}}i/{\rm {d}}t}$. Therefore we can take the term ${\displaystyle L{\rm {d}}^{3}i/{\rm {d}}t^{3}}$ in equation (7.9b) into account by changing the coefficient of ${\displaystyle {\rm {d}}i/{\rm {d}}t}$ to ${\displaystyle (2h\omega _{0}+\omega ^{2}L)}$.