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
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- 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
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- 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
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- 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
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- 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
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- 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 )
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