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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Table of Contents (book)

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• Effect of station angle on location errors
• Transit satellite navigation
• Effective penetration of profiler sources
• Directivity of linear sources
• Sosie method
• Energy from an air-gun array
• Dominant frequencies of marine sources
• Effect of coil inductance on geophone equation
• Streamer feathering due to cross-currents
• Filtering effect of geophones and amplifiers
• Filter effects on waveshape
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