### Passive Velocity Transducer (Geophone) Responses:

Relationships Between Resistors, Datalogger Input Impedance and Response Parameters

For a passive velocity sensor, the sensitivity and damping (and therefore the poles and zeros)
can depend on:

- coil resistance,
- resistive load on the sensor (shunt resisitor and datalogger/amplifier input impedance),
- mass,
- natural frequency, and
- mechanical damping.

**Why It's Important**
The best approach to generating a passive sensor response is to calibrate the
sensor to find current values for the properties listed above and calculate the
response parameters from those values. This page explains how to do this.

The next best approach is to use sensor-specific values from your manufacturer
for the properties listed above to calculate response parameters.

Nominal responses given in this library are accurate only for sensors with the
exact resistor values listed and assume that the amplifier (datalogger) input impedance
is at least 2 orders of magnitude larger than the total resistance of the disconnected
sensor. Only when this is true will the sensor damping and sensitivity remain the same
whether the datalogger is attached or not and the nominal response values suffice.
If the impedance contrast is smaller than this, the datalogger preamp will influence
the sensor electronics, changing damping and therefore its poles.

**How to Calculate the Effective Damping, Poles and Sensitivity**
Let:

- bt -
*total damping* = bo + bc, where
- bo -
*open circuit damping due to the mechanical properties of the sensor*, and
- bc -
*current damping due to the electrical properties of the sensor*

- Rt -
*total resistance* = Rc + Rload, where
- Rc -
*feedback coil resistance*
- Rload -
*parallel sum of Rs and Zamp* = (Rs * Zamp) / (Rs + Zamp) where
- Rs -
*shunt resistor*, and
- Zamp =
*input impedance of the preamp or datalogger*

In cases where there is no shunt resistor (Rs), Rload = Zamp. Similarly, if there is a shunt
resistor, but no datalogger is attached, Rload = Rs.
In addition to bo, Rc, Rs (if nonzero) and bc, a calibration sheet will also have these values:

- fo -
*natural frequency (Hz)*
- m -
*mass (Kg)*
- Go =
*instrinsic sensitivity* = sqrt( 4 * pi * fo * m * Rt * bc )

To calculate the poles, you have to find the total damping, which depends on the total
resistance where

- bc = (Go**2) / (4 * pi * fo * m * Rt)

Then the poles will be
- p1 = (-2 * pi * fo * bt) + j * (2 * pi * fo * sqrt( 1 - bt**2 ) )
- p2 = (-2 * pi * fo * bt) - j * (2 * pi * fo * sqrt( 1 - bt**2 )

There will be two zeros at zero.
The effective sensitivity (Ge) due to resistances in the sensor and the input impedance of the amplifier
or datalogger is
*Mary Templeton 3/7/2013*