Applied Geophysics : Seismics

Seismometer-Galvanometer Documentation

( Seismometer Details : see Seismometer Documentation, in separate window )

- Principles of Operation -

In a seismometer - galvanometer seismograph
  the velocity transducer ( coil-magnet-assemly ) of a seismometer
is connected
  to the driving coil of a galvanometer
through
  a resistive T element.

The response of the system is charakterized
  by the natural period and the damping of both seismometer ( T_s and α_s ) and galvanometer ( T_g and α_g )
and
  by a dimensionless "coupling coefficient" σ^2 ( denoted α^2 in some publications, range 0...+1 ), describing the reciprocal influence of the two electro-mechanical devices.

The frequency response to ground velocity corresponds to the product of
  a 2-pole high pass ( output voltage of seismometer with respect to ground velocity )
  and
  a 2-pole low pass ( angular deflection of galvanometer coil / mirror with respect to driving current )
with
  an additional quadratic term proportional to σ^2 in the resulting denominator polynomial of 4th degree
  and
  a constant factor proportional to σ ( = sqrt(σ^2) ), determining the amplification of the system.

This results in a 4th order band pass with
  corner periods of approx. T_s and T_g ( for moderate damping values α_s, α_g ≤ 1 )
  and
  slopes of 12 dB / octave outside the pass band,
and
  a "distorsion" within the pass band, caused by the σ^2 term, and noticeable for values of σ^2 above approx. 0.5.

Values of the parameters T_s, α_s, T_g, α_g and σ^2 for commonly used systems are listed in Manual of Seismological Observatory Practice ( 1979 Edition, Chap. Instruments, 1 Type of Instruments ),
the amplitude response curves with respect to ground displacement are displayed in fig. 1.1 of the manual.

- Table of Contents -

> Equations of Motion
>   Seismometer
>   Galvanometer

> Resistive Coupling

> Laplace Transforms
>   Seismometer
>   Galvanometer

> Transfer Function
>   Coupling Coefficient
>   Amplification

> Parameter Adjustment
>   Example Rd_s = Rd_g
>   Example Rd_s ≠ Rd_g

> Applets

- Equations of Motion -

Seismometer :

The equilibrum of all external and internal forces acting on the moving mass of a seismometer with velocity transducer ( coil - magnet assembly ) leads to an equation of motion :
with the parameters of the mechanical system and the velocity transducer :
relating the time functions :

( see Seismometer Documentation, in separate window )

Galvanometer :

In a galvanometer a coil ( axis horizontal ) is suspended in the magnetic field of a permanent magnet at two vertical torsion wires ( or ribbons, above and below the coil )

A current through the coil produces a momentum resulting in a angular deflection of the coil from its zero-postion.
A light beam reflected by a mirror fixed to the coil is focused to a scale or to photographic paper allows to observe / record the coil deflection.

The angular deflection is decribed by the equation of motion :
with the parameters of the mechanical system and the coil - magnet assembly :
relating the time functions :

Equations of Motion Table of Contents Top of Page

- Resistive Coupling -

The seismometer part of the circuit is characterized by the parameters of the velocity transducer
the galvanometer part by the parameters of the galvanometer coil, i.e.
and the external resistance T by
where all resistances are assumed to be purely resistive.

The voltages induced in the seismometer and galvanometer coil
lead to currents in seismometer and galvanometer coils
where the total resistances seen from the seismometer and from the galvanometer coil determine the resp. damping
and the coupling is described by the resistance
with the abbreviations

Resistive Coupling Table of Contents Top of Page

- Laplace Transforms -

The application of the Laplace transformation to the equations of motion replaces the functions of time and their drevatives by the corresponding functions of the complex frequency variable
neglecting the initial values x(+0), x'(+0) etc. :

Seismometer :

Substituting the current
and dividing the equation of motion by m leads to
with the commonly used abbreviations
and a coefficient
determining the additional acceleration proportional to the angular velocity of the galvanometer coil and acting on the seismometer mass due to the coupling circuit.

Galvanometer :

Substituting the current
and dividing the equation of motion by Θ leads to
with the commonly used abbreviations
and a coefficient
determining the angular acceleration proportional to the velocity of the seismometer mass and acting on the galvanometer coil due to the coupling circuit.

Laplace Transforms Table of Contents Top of Page

- Transfer Function -

Eliminating X(s) from the Laplace transformed equations of motion,
and
multiplying the angular deflection Φ(s) of the galvanometer by the length 2r of the light beam to pass to the the photographic recorded amplitude Y(s), leads to :

A [s] is an amplification factor
and
H(s) is a dimensionless transfer function,
  corresponding to the product of the transfer functions of the galvanometer ( 2nd order low pass ) and the seismometer ( 2nd order high pass )
with
  an additional coupling term - K_s ∗ K_g ∗ s^2 in the denominator.

Coupling Coefficient :

The variables in the product K_s ∗ K_g can be reordered to relate the coupling term to the seismometer and galvanometer damping :

The coupling coefficient σ^2 :

is limited to the electromagnetic part of damping in proportion to the total damping of both instruments :

and can be reduced by a factor, depending on the resistive network :

Amplification :

The amplification factor A [s] :
can be split up into
  the sensitivity G_s [Vs/m] of the seismometer
and
  the static amplification of the galvanometer :
derived from the static trace excursion y_0, recorded for a dc current i_0 :

The resistance term of the amplification factor A and the static amplification can be combined into an effective amplification of the galvanometer
leading to
where only σ is affected by adjustments of coupling, provided the damping values of both instruments are kept unchanged.

Laplace Transforms Table of Contents Top of Page

- Parameter Adjustment -

First of all the damping resistances of both instruments for the damping values prescribed / desired
have to be calculated, if all parameters involved are known, or determined in separate experiments from the free motion of the instruments
( see Seismometer Documentation or Seismometer Calibration, with slightly modified procedures to meet the requirements of a galvanometer ).

There are several restrictions to be considered when adusting the coupling circuit :

the ( trivial ) conditions
limit the range of possible damping values to

and from
  ( see : Resistive Coupling, above )

follows
limitting the range of possible values of
  the coupling coefficient and the amplification of the system.

Finally the often stated demand for a "negligible" small value of the coupling coefficient
  leads to a corresponding decrease of the amplification of the system.

If the coil and damping resistances of both instruments are known / determined, the external resistances of the coupling T for a given factor q_σ can be calculated from :
  ( see Coupling Coefficient and Resistive Coupling )

leading to
where the choice of q_σ has to comply with min ( Re_s, Re_g ) ≥ 0.

Example Rd_s = Rd_g = Rd :

With Rd_s = Rd_g = Rd the above equations are reduced to

Rd ≥ Rc_s + Rc_g enables q_σ = 1 ( Re_0 -> &infin ), leading to a single series resistance
whereas q_σ < 1 leads to a symmetric resistance T :
affording either
or

Example Rd_s ≠ Rd_g :

The evaluation of the above equations can be simplified by normalizing the resistance values to the damping resistances Rd_s and Rd_g :

The ranges of usefull values q_σ
  for given quotients Rc_s / Rd_s, Rc_g / Rd_g and Rd_g / Rd_s
can be observed in the applet Seismometer-Galvanometer Coupling, where
  the functions R_s / Rd_s and R_g / Rd_g are displayed for q_σ = 0 ... 1
with
  Rc_s / Rd_s and Rc_g / Rd_g as horizontal lines.

Screen Shot :

( green : usefull range of q_σ, cyan : actually selected value of q_σ )

Parameter Adjustment Table of Contents Top of Page

- Applets -

The applet Seismometer-Galvanometer Coupling, shows
  a schematic diagram of the coupling network,
  an interactive graphic display of normalized coupling parameters
and allows
  to set relevant seismometer and galvanometer parameters in a dialogue area
  and
  to adjust the coupling network in the graphic display.

Response Functions of Seismographs, shows
  amplitude and phase of the transferfunction with respect to ground displacement, velocity and acceleraton,
and
  the response in the time domain to some impulsive ground motions
for 7 seismometer - galvanometer systems ( + 6 direct recording and digital seismographs ).

Comparison of Seismographs, compares
  amplitude and phase of the transferfunction with respect to ground displacement
and
  the response in the time domain to some impulsive ground displacements and to the ground displacements of some earthquakes, recorded at the station CLZ.
for up to 3 of 7 seismometer - galvanometer systems ( + 6 direct recording and digital seismographs ).

The parameter values of T_s, α_s, T_g, α_g and σ^2 used for the applets are taken from Manual of Seismological Observatory Practice ( 1979 Edition, Chap. Instruments, 1 Type of Instruments ),
where unfortunately no instrument specific parameters ( i.e. G_s, m, G_g, Θ or Vstat_g ) are listed.

Therefore the parameters G_s and Veff_g, determining the amplification factor A [s] are chosen to fit the response curves with respect to ground displacement, displayed in fig. 1.1 of the manual.

Rev. 12-nov-2012

Comments to Fritz Keller
( ned gschempfd isch globd gnueg )

Table of Contents Top of Page