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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm d}i/{\rm d}t=-\omega \sin \omega t , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm d}^{3} i/{\rm d}t^{3} =\omega ^{3} \sin \omega t hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm d}^{3} i/{\rm d}t^{3} =\omega ^{2} {\rm d}i/{\rm d}t . Therefore we can take the term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): L{\rm d}^{3} i/{\rm d}t^{3} in equation (7.9b) into account by changing the coefficient of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm d}i/{\rm d}t to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (2h\omega _{0} + \omega ^{2} L) .

Continue reading

Previous section	Next section
Dominant frequencies of marine sources	Streamer feathering due to cross-currents
Previous chapter	Next chapter
Characteristics of seismic events	Reflection field methods
Table of Contents (book)

Also in this chapter

External links

find literature about
Effect of coil inductance on geophone equation