Calibration

ADU-08e

HF Board 08e

The theoretical transfer function for the HF-channel is given below:

\(F_{HF - Channel} = G_{1} \cdot F_{1} \cdot F_{2} \cdot F_{3} \cdot F_{4}\)

with

\(G_{1} = 1 \ or\ 4, 8, 16\) depending on gain setting of first stage

\(F_{1} = \frac{1}{1 + P_{1}}\); \(P_{1} = i \cdot \frac{f}{338 kHz}\)

\(F_{2} = \frac{1}{1 + P_{2}}\); \(P_{2} = i \cdot \frac{f \cdot G_{1} }{100 MHz}\)

\(F_{3} = \frac{1}{1 + P_{3}}\); \(P_{3} = i \cdot \frac{f}{1.59 MHz}\)

\(F_{4} = \frac{P_{4}}{1 + P_{4}}\); \(P_{4} = i\frac{f}{482 Hz}\) if high-pass is switched on.

\(F_{4} = 1\) if high-pass is switched off.

(gains and input divider are calibrated into the LSB, you don’t see them)

LF Board 08e

\(F_{LF - Channel} = G_{1} \cdot G_{2} (\cdot G_{3}) \cdot F_{1} \cdot F_{2} \cdot F_{3} \cdot F_{4}\)

\(G_{1} = 1 \ or\ 4, 8, 16\) depending on gain stage 1 settings

\(G_{2} = 1 \ or\ 4, 8, 16, 32, 64\) depending on gain stage 2 settings, inside the ADC; (32, 64 not set by software, only manually)

Gains and input divider do not appear in the ats file, they are calibrated into the LSB

\(G_{3} = 2\) is a fixed gain and can’t be changed; this gain is invisible in the time series

\(F_{1} = \frac{1}{1 + P_{1}}\); \(P_{1} = i\frac{f}{318 kHz}\)

\(F_{2} = \frac{1}{1 + P_{2}}\); \(P_{2} = i\frac{f \cdot G_1 }{2 MHz}\)

\(F_{3} = \frac{1}{1 + 1.414 \cdot P_{3} + P_{3}^{2}}\); \(P_{3} = i \cdot \frac{f}{4Hz}\) if 4 Hz Low-pass is switched on, else 1

and

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot \frac{f}{10.5 kHz}\) if RF-2 on & DIV-8 = on (default for coil, not used for E)

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot \frac{f}{30 kHz}\) if RF-1 on & DIV-8 = on (not used for E)

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 7.27E^{-9}Hz\) if RF-2 on & DIV-1 = on (default for electrodes < 1500 Ohm contact resistance, default for buffer electrodes, not used for H) )

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 470E^{-12}Hz\) if RF-1 on & DIV-1 = on ( electrodes > 1500 Ohm contact resistance, not used for H)

in old manuals the equations were written like
\(P_{4} = i \cdot \frac{f}{\frac{0.159}{(R_{sensor} + 200) \cdot 7.27E^{-9}Hz}}\)
\(P_{4} = i \cdot \frac{f}{\frac{0.159}{(R_{sensor} + 200) \cdot 470E^{-12}Hz}}\)

ADU-10e

LF Board 10e

\(F_{LF - Channel} = [G_{1}] \cdot F_{1} \cdot F_{4}\)

\(G_{1} = 1 \ or\ 4, 8, 16, 32, 64\)
The \(G_{1}\) (inside ADC) is a chopper stabilized gain. The SW shall set 1 as default, for E this means \(\pm\) 2.5 V input range.
Since the gain \(G_{1}\) and the input divider are inside the LSB (ATS format) or included in the mV data stream (ATSS) we neglect the gain.

\(F_{1} = \frac{1}{1 + P_{1}}\); \(P_{1} = i\frac{f}{318 kHz}\)

and

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot \frac{f}{7.8 kHz}\) if DIV-8 = on (default for coil, not used for E)

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 6.8E^{-9}Hz\) if DIV-1 = on (default for electrodes, not used for H) )

(in old manuals the equation was written like \(P_{4} = i \cdot \frac{f}{\frac{0.159}{(R_{sensor} + 200) \cdot 6.8E^{-9}Hz}}\) )

red dotted line 2000 Hz
For H up to 2 kHz, the low-pass filter of the board can be considered negligible. ADU-10e Low-Pass Filter Response for H

For E the low-pass filter of the board has a more significant effect for higher frequencies and high resistance surfaces.

ADU-10e Low-Pass Filter Response for E, gain 1

So at 5 k\(\Omega\) and 2000 Hz in the spectra, the decay is 0.9.

ADU-11e

To correct the low pass character of the ADU-11e1, the following transfer functions are used:

Board

\(F_{LF - Channel} = [G_{2}] \cdot F_{1} \cdot F_{2} = [G_{2}] \cdot \frac{1}{1 + P_{1}} \cdot \frac{1}{1 + P_{2}}\)

\(G_{2} = 1 \ or\ 4, 8, 16, 32, 64\) (G1 was on old boards, not existing here, G2 is inside ADC)
Since the gain \(G_{1}\) and the input divider are inside the LSB (ATS format) or included in the mV data stream (ATSS) we neglect the gain.

\(F_{1} = \frac{1}{1 + P_{1}}\); \(F_{2} = \frac{1}{1 + P_{2}}\)

For H and DIV-8 on, always RF-1 (default for coil, not used for E)
\(P_{1} = i \cdot 2 \pi f \cdot 2200 \cdot 1 \cdot E^{-9}\)

\(P_{2} = i \cdot 2 \pi f \cdot 1100 \cdot 1 \cdot E^{-9}\)

For E and DIV-1 on (default for electrodes, not used for H)

  • RF1 \(C=1.0 \cdot E^{-10}\) < 1500 Ohm contact resistance, default for buffer electrodes
  • RF2 \(C=1.1 \cdot E^{-9}\)   > 1500 Ohm contact resistance

\(P_{1} = i \cdot 2 \pi f \cdot (R_{sensor} + 2200) \cdot 1 \cdot E^{-9}\)

\(P_{2} = i \cdot 2 \pi f \cdot (\frac{R_{sensor}}{2} + 1100) \cdot C\)

ADC

Gaussian low-pass filter is applied to the signal: \[e^{-\left(\frac{f}{f_c}\right)^{2}}\]

sampling rate cut-off frequency
1024 Hz 393 Hz
2048 Hz 786 Hz
4096 Hz 1570 Hz
8192 Hz 3140 Hz
16384 Hz 6261 Hz
32768 Hz 12370 Hz
65536 Hz 23600 Hz
131072 Hz 41000 Hz

In a plot natural logarithm of the normalized signal lnnormalized versus f² the Gaussian low-pass filter will appear as a straight line with a slope of \(\frac{-1}{f_c²}.\)

With fc = 0.38 * fsampling rate works perfectly up to 32 kHz and even 64 kHz. At 131 kHz the approximation starts to deviate significantly.

Gaussian Linearization
22.4k label is placed at 22.4k² “real world”

ADU-07e

HF Board 07e

The theoretical transfer function for the HF-channel is given below:

\(F_{HF - Channel} = G_{1} \cdot G_{2} \cdot F_{1} \cdot F_{2} \cdot F_{3}\)

with

\(G_{1} = 1 \ or\ 8\) depending on gain setting of first stage

\(G_{2} = 1 \ or\ 8 \ or\ 64\) depending on gain setting of second stage

\(F_{1} = \frac{1}{1 + P_{1}}\); \(P_{1} = i \cdot \frac{f}{7.23 MHz}\) if \(G_1 \ne 1\)

\(F_{2} = \frac{1}{1 + P_{2}}\); \(P_{2} = i \cdot \frac{f}{7.23 MHz}\) if \(G_2 \ne 1\)

\(F_{3} = \frac{P_{3}}{1 + P_{3}}\); \(P_{4} = i\frac{f}{1 Hz}\) if high-pass is switched on.

(gains and input divider are calibrated into the LSB, you don’t see them)

LF Board 07e

\(F_{LF - Channel} = G_{1} \cdot G_{2} \cdot F_{1} \cdot F_{2} \cdot F_{3} \cdot F_{4}\)

\(G_{1} = 1 \ or\ 2, 4, 8, 16, 32, 64\) depending on gain stage 1 settings

\(G_{2} = 1 \ or\ 2, 4, 8, 16, 32, 64\) depending on gain stage 2 settings, inside the ADC; (32, 64 not set by software, only manually)

\(F_{1} = \frac{1}{1 + P_{1}}\); \(P_{1} = i \cdot \frac{f}{4 kHz}\) // to be checked if \(G_1 \ne 1\)

\(F_{2} = \frac{1}{1 + P_{2}}\); \(P_{2} = i \cdot \frac{f}{21.2 kHz}\) // to be checked

\(F_{3} = \frac{1}{1 + 1.414 \cdot P_{3} + P_{3}^{2}}\); \(P_{3} = i \cdot \frac{f}{4Hz}\) if 4 Hz Low-pass is switched on

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 2.2E^{-11}Hz\) if RF-1 on & DIV-1 = on (default for electrodes < 600 Ohm contact resistance not used for H, prefer RF-2) )

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 1.22E^{-10}Hz\) if RF-2 on & DIV-1 = on (default for electrodes < 1500 Ohm contact resistance, default for buffer electrodes, not used for H) )

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 6.822E^{-9}Hz\) if RF-3 on & DIV-1 = on (default for electrodes 1500-2500 Ohm contact resistance, prefer RF-4 not used for H) )

\(F_{4} = \frac{1}{1 + P_{4}}\); \(P_{4} = i \cdot 2 \pi f \cdot (R_{sensor} + 200) \cdot 6.922E^{-9}Hz\) if RF-4 on & DIV-1 = on (default for electrodes > 1500 Ohm contact resistance) )

// missing DIV-8

MF Board 07e

tbd.

MFS Coils

OLD Calibration Files are normalized by f!
e.g. the MFSXXX.txt files.

The JSON files are using mV (as the time series data) and are not normalized by f. 
If you normalize the calibration you end up with a constant \(\frac{200 mV}{nT \cdot Hz}\) and 90° phase below 0.1 Hz for the MFS-06e and \(\frac{20 mV}{nT \cdot Hz}\) and 90° for the MFS-07e.

The term \(P_{1}\) is the cut-off frequency generated by the feedback coil.
The other terms \(P_{2}\), \(P_{3}\), \(P_{4}\) are result of the pre-amplifier components23.

MFS-06e

\(P_{1} = i \cdot \frac{1}{4 Hz}, \enspace P_{2} = i \cdot \frac{1}{9645 Hz} \enspace\) \(P_{3} = i \cdot \frac{1}{0.72 Hz},\enspace P_{4} = i \cdot \frac{1}{23897 Hz}\)

Chopper on

\(F_{on}(f)\,\left[\frac{mV}{nT}\right] = 800 \cdot \frac{P_1}{1+P_1} \cdot \frac{1}{1+P_2} \cdot \frac{1}{1+P_4}\)

Chopper off

\(F_{off}(f)\,\left[\frac{mV}{nT}\right] = 800 \cdot \frac{P_1}{1+P_1} \cdot \frac{1}{1+P_2} \cdot \frac{P_3}{1+P_3} \cdot \frac{1}{1+P_4}\)

MFS-07e

\(P_{1} = i \cdot \frac{1}{32 Hz}, \enspace P_{2} = i \cdot \frac{1}{45150 Hz} \enspace\)

\(P_{3} = i \cdot \frac{1}{0.72 Hz},\enspace P_{4} = i \cdot \frac{1}{49735 Hz}\)

Chopper on

\(F_{on}(f)\,\left[\frac{mV}{nT}\right] = 640 \cdot \frac{P_1}{1+P_1} \cdot \frac{1}{1+P_2} \cdot \frac{1}{1+P_4}\)

Chopper off

\(F_{off}(f)\,\left[\frac{mV}{nT}\right] = 640 \cdot \frac{P_1}{1+P_1} \cdot \frac{1}{1+P_2} \cdot \frac{P_3}{1+P_3} \cdot \frac{1}{1+P_4}\)

MFS-12e

\(P_{1} = i \cdot \frac{1}{16 Hz}, \enspace P_{2} = i \cdot \frac{1}{9645 Hz} \enspace\) \(\enspace P_{4} = i \cdot \frac{1}{50048 Hz}\)

Switch free

In the old days we used copper on == LF mode and chopper off == HF mode.
Now we have a switch free design, so only one transfer function is used.
However, the amplifier still has a chopper stage, which is always on, but invisible in the transfer function.
The transfer function is a mix of both modes:
\(F(f)\,\left[\frac{mV}{nT}\right] = 800 \cdot \frac{P_1}{1+P_1} \cdot \frac{1}{1+P_2} \cdot \frac{1}{1+P_4}\)

MFS-14e

tdb.

SHFT coils

SHFT-02e

Only a basic approximation of the transfer function is available:

\(P_{1} = i \cdot \frac{f}{300,000 Hz}\)

\(F(f)\,\left[\frac{mV}{nT}\right] = 50 \cdot \frac{1}{1 + P_1}\)

YOU MUST USE THE CALIBRATION FILES FOR THIS SENSOR

SHFT-03e

YOU MUST USE THE CALIBRATION FILES FOR THIS SENSOR

Fluxgates

Fluxgate magnetometers from Bartington Instruments just need to be scaled.

FGS-02e

Fluxgate with 75,000 nT range, Geomag-01.

\(F(f)\,\left[\frac{mV}{nT}\right] = 7.5 \cdot 10^{-4}\)

FGS-03e

Fluxgate with 100,000 nT range, Bartington Mag-03.

\(F(f)\,\left[\frac{mV}{nT}\right] = 1 \cdot 10^{-4}\)

FGS-04e

Fluxgate with 100,000 nT range, Bartington Mag-13 MCL (MC = cylindrical core, L = low noise).
That is a newer version of the FGS-03e, with a better noise performance.

\(F(f)\,\left[\frac{mV}{nT}\right] = 1 \cdot 10^{-4}\)

FGS-05e

Fluxgate with 75,000 nT range, Bartington (never sold)
You may get an out of drift here.

\(F(f)\,\left[\frac{mV}{nT}\right] = 1.43 \cdot 10^{-4}\)

Footnotes

  1. AMD AD7771 mit SINC5↩︎

  2. \(P_{4}\) is tuned during calibration; so 23897 (for MFS-06e) is not fixed; however \(P_{4}\)
    has almost no influence in the calculation of the transfer function.↩︎

  3. MFS-6e :Rev. 1p4, MFS-7e :Rev. 1.2↩︎