Effect of coil inductance on geophone equation

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

Problem

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

$^{\rm {TM}}$ Flexichoc and Vaporchoc are trademarks of Compagnie Géneralé de Géophysique; Maxipulse is a trademark of Western Geophysical Company.

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

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

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

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

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

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