### 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:

1. coil resistance,
2. resistive load on the sensor (shunt resisitor and datalogger/amplifier input impedance),
3. mass,
4. natural frequency, and
5. 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
• Ge = Go * (Rload / Rt)

Mary Templeton 3/7/2013