AWESOME and HUMSNIFFER Calibration

Ryan Said

November 5, 2008

1 Amplitude

The total magnetic ux through a loop is

Φ =∫S

~B · d~s (1)

(Recall that the induced voltage is given by Faraday's law:)

Vind = − d

dtΦ (2)

The denition of self inductance is

L =ΦI

(3)

For a loop antenna with N windings and area A, Φ = BNA. So (3) becomes

L =BNA

I(4)

If we know the current I through the antenna, and given antenna values L, N , and A, we can calculatethe incident magnetic eld strength B in Teslas. Our preamp cards contain a calibration circuit that injectsa known current Ical in series with the antenna. From (4), this gives

Bcal =LIcalNA

(5)

If we record data when Ical is injected, we can then relate the measured amplitude acal to the magneticeld Bcal. The calibration number, the factor that converts measured amplitudes to the incident magneticeld strength, is simply Bcal

acal. The next two subsections describe the calibration circuits in the HUMSNIFFER

and AWESOME preamps.

1.1 HUMSNIFFER calibration circuit

The HUMSNIFFER box contains a calibration circuit which generates a 1V peak-to-peak square wave fromthe Test Point to ground. This signal passes through a voltage divider to become Vcal. Using Rcal = 10kΩ,we can readily nd Ical = Vcal/Rcal. The voltage divider scales the (one-sided) voltage at the Cal test pointby 22.1/(22.1+49900).

The square wave has a period of ≈ 1/5000 seconds, so the fundamental harmonic is at ≈ 5 kHz withVcal = 22.1

22.1+499001V2

4π = 280µV , or 200µV rms. There are odd harmonics at 15, 25, 35 kHz ... with

amplitudes scaled by 1/3, 1/5, 1/7, ... with respect to the fundamental.

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1.2 AWESOME calibration circuit

The AWESOME preamp card contains a microprocessor which generates a 10-bit psuedo-random numbersequence (PRNS). The bit width is T1 = 1/256e3 seconds, and the sequence repeats after 1023 bits, or afterT2 = 1023 ∗ T1 seconds. Measuring the voltage at +Cal Test Point with respect to ground, a 1 bit isrepresented by a positive voltage of A = 1V, and a 0 bit is represented by 0 volts (so 1V peak-to-peakw.r.t. ground). If the dierential voltage between the + and - Cal Test points is measured, the PRNS

should alternate between ±1V , with a voltage swing of 2A = 2V.As with the HUMSNIFFER calibration circuit, this signal passes through a voltage divider to become

Vcal. Again we have Rcal = 10kΩ, and we nd Ical = Vcal/Rcal. The voltage divider scales the (one-sided)voltage at +Cal Test Point by 100/(100+2200).

The PRNS generates tones every 1/T2=250.244 Hz. Given a one-sided Vcal amplitude ofA= A*100/(100+2200)= 0.0434V of the PRNS, we wish to determine the amplitude of each tone.

The PRNS Vcal can be thought of as a square pulse of width T1 and height A convolved with a sequenceof delta functions each spaced by T1 with weights alternating between 0 and 1 according to the PRNS. Thedelta train sequence repeats every 1023∗T1 seconds. The Fourier Transform will therefore be a sinc functionmultiplied by a delta train with impulses every 1

T2. Noting that the Fourier Transform of a square pulse

centered at 0 with width T1 is proportional to sinc(f ∗ T1) and that the frequency domain is sampled everydf = 1

T2= 1

1023∗T1, we nd that the Fourier Series of the PRNS x(t) is given by

F.S.x(t)[k] ≡ fk =

A2 k = 0(X2

)sinc

(k

1023

)k 6= 0

(6)

where the DC component was determined by direct integration, and X ∗ sinc(k/1023) is the one-sidedamplitude of the harmonic at k/(1023 ∗ T1) = k ∗ 250.244 Hz. Using Parseval's theorem:

1T

∫T

|f(t)|2dt =∞∑

k=−∞

|fk|2 (7)

and noting that the PRNS takes a +A value half of the time (and 0 elsewhere), we have

A2

2=A2

4+(X

2

)2∑k 6=0

sinc2(

k

1023

)(8)

Using ∑∞k=−∞ sinc2(kb) = 1

b , b ≤ 1 (9)

(which is readily derived by applying Parseval's theorem to the inverse Fourier Series of sinc(kb)), andtherefore ∑

k 6=0 sinc2(kb) = 1

b − 1, b ≤ 1 (10)

we have (A

2

)2

=(X

2

)2

(1023− 1) (11)

Hence the one-sided amplitude of each tone isX = 1V√1022∗ 100

2300 = 1.36mV , orX = 1.36/√

2 = .96mV ≈ 1mVrms. Note that there will be a small roll-o in amplitude between DC and the Nyquist frequency due to thesinc envelope of the Fourier series. At 50 kHz, the amplitude of each tone is lower by−20log10(sinc(50e3/256e3)) =0.55dB with respect to tones near DC.

• NOTE: in Calibrate_VLF, A was mistakenly set to .5 V (the dierential voltage between ± Cal TestPoints was assumed to be 1V), yielding calibration values that are 6dB too low.

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1.3 Measuring Peak Amplitude

When extracting calibration amplitudes (which make up acal), it will be necessary to measure the peakamplitude at specic frequencies. To measure the single-sided peak amplitude at a particular frequency:

• Window with some window w

• Take a DFT of the result (for speed, use an FFT with a power of 2)

• Divide by the sum of the window taps

• Scale by 2 (for single-sided amplitude), except at DC and Nyquist

• Interpolate to nd the peak amplitude and frequency

Specically, windowing x[n] with a window w[n] oset by l samples, we normalize the DFT as such:

A[k] = F DFTx[n]w[n−l][k]∑n w[n] , F =

1, k = 0, N22, otherwise

(12)

to recover the single-sided amplitude A[k] at frequency kfs

N , where N is the DFT length and fs is the samplingfrequency.

Using a Gaussian window increases the accuracy of quadratic interpolation in the (dB) amplitude.

1.4 Power Spectral Density

If the data signal x[n] is calibrated and carries units pT, then the periodogram P at frequency kfs

N is givenby

P(kfs

N

)= F

fs

|DFTx[n]w[n−l][k]|2∑n|w[n]|2

[pT 2

Hz

], F =

1, k = 0, N22, otherwise

(13)

and so√P carries units of pT√

Hz. To recover the amplitude of a deterministic tone in pT,

√P must be

integrated over the relevant bandwidth.An estimate of the Power Spectral Density (PSD) is obtained by taking the average of many successive

periodograms, where each Pl is calculated with a dierent window oset l:

PSD =∑l PlL

(14)

where L is the total number of sample periodograms used. This technique for estimating the PSD is knownas Welch's method.

1.5 Receiver Sensitivity

Figure 1 shows the measured, uncalibrated Power Spectral Density from several hardware setups using theAWESOME receiver. With the minimum value set to ∆ = 1 (the AWESOME receiver records values with16-bit sampling, covering the range −215 to 215 − 1 in increments of 1), the one-sided quantization noiseoor (QNF) is

QNF = 10log10

(∆2

121

fs/2

)= −57.8

[dB

pT√Hz

](15)

shown by the dotted black line. The blue curve shows the measured noise oor, obtained using Welch'smethod, of the receiver with no lter card. This noise oor is generated from a noise source after the ltercard. The green curve shows the noise oor with the lter card inserted into the preamp, but with no preampconnection. This noise oor is generated by a noise source introduced in the anti-aliasing lter card whicheectively reduces the dynamic range of the system by ∼ -40 - -57 = 17 dB, or by ∼ 3 bits, limiting theoverall dynamic range to ∼ 80 dB.

3

0 10 20 30 40 50−60

−50

−40

−30

−20

−10

0

dB

-raw

/sq

rt(H

z)

Frequency [kHz]

Quantization Noise Floor

No Filter CardFilter card only (no Preamp)

Preamp: 0 dB Gain

Preamp: 10 dB Gain

Preamp: 20 dB Gain

Preamp: 30 dB Gain

AWESOME Receiver Noise Floor

Figure 1: AWESOME receiver noise oor

The next four plots (red, cyan, magenta, and yellow) show the uncalibrated (raw) noise oor with thepreamp connected at the four possible gain settings: 0, 10, 20, and 30 dB, respectively. At the 0 dB gainsetting, the noise oor is dominated by the lter card above 20 kHz and by the preamp card below 20 kHz.Increasing the gain in the preamp by 10 dB, we see that the noise oor increases by ∼ 7 dB below 20 kHzand ∼ 3 dB above 20 kHz. Since the input signal is amplied by 10 dB, we will have an improvement of 3dB (7 dB) below (above) 20 kHz in the signal to noise ratio (SNR). The dynamic range will also be reducedby 7 dB (3 dB) below (above) 20 kHz. Increasing the gain switch to 20 dB, the noise oor is uniformlyincreased by 10 dB, and so the SNR will remain the same while the dynamic range is reduced by 10 dB.This happens again going to the 30 dB gain setting.

The optimal gain setting will therefore be application dependent. If dynamic range is of primary im-portance, then the 0dB gain setting should be used. If sensitivity is more important, than the 10 dB gainsetting should be used. Little is gained by going to the 20 dB and 30 dB gain settings.

The PSD curves drawn in Figure 1 correspond to raw uncalibrated data. The input-referred sensitivity iscalculated by scaling each frequency component by the calibration number Bcal

acal, where Bcal depends on the

antenna A area and number of windings N , as in equation (5). Figure 2 shows the input-referred sensitivityusing the 0dB gain setting for two antenna congurations: T1 (A = 1.69 m2, N = 12) and T2 (A = 17.64m2, N = 6). The sensitivity of T2 is 20log10

(17.64∗61.69∗12

)= 14.4dB better than T1. For reference, the red curve

shows the natural noise oor, and the dotted blue and green curves show the theoretical sensitivities for thetwo antenna sizes. The frequency axis is plotted on a log scale to highlight the features.

Increasing N ∗ A of the antenna improves sensitivity and SNR. If the local measured noise oor is closeto the receiver sensitivity, then that site will benet from an antenna with a larger N ∗ A. Note, too,that increasing N ∗ A increases the gain, thereby reducing the maximum peak amplitude before receiversaturation.

Figures 3 and 4 plot the measured PSD in blue, together with the natural noise oor for reference andthe measured receiver sensitivity in magenta, at Tucson, AZ and Juneau, AK, respectively. At Tucson, asmaller T1 antenna is used, but the measured background noise is still higher than the receiver sensitivity.

4

1 50.5 10 50

Frequency [kHz]

Receiver Sensitivity for 0 dB Gain

T1

T1

T2

T2

Figure 2: AWESOME receiver sensitivity

5

Power Spectral Density (PSD)

Figure 3: Measured PSD and sensitivity at Tucson

At Juneau, the larger T2 antenna is used, allowing for measuring a natural noise oor which is ∼15 dB lowerthan the sensitivity of the receiver at Tucson.

The Marantz PMD671 recorder also has an internal noise source after the input amplication, so thegain on the recorder front-end must be set suciently high so that this internal noise source does not limitthe HUMSNIFFER measured spectrum.

1.6 A Note on Windowing

The measured spectral dynamic range will be limited by the relative side lobe height of the windowingfunction w. For example, if a rectangular window is used, the rst side lobe is only 13 dB down from themain-lobe peak. While the side lobe level rolls o at approximately 6 dB per octave, the measured spectrumwill still have a much lower maximum dynamic range than the receiver. In general, the windowing functionshould be chosen to ensure that the side lobe level does not limit the measured dynamic range. A Kaiserwindow with a high β provides a convenient trade-o between main lobe width and side lobe height.

2 System Response

Figure 5 plots the measured system impulse response, in units of rawpT , for one channel. The Broadband

Analysis Tutorial goes through the mechanics of the amplitude and phase calibrations in greater detail.The attenuation below 1 kHz and above 48 kHz are reected in the receiver sensitivity in Figure 2.

Figure 6 shows the corresponding inverse system response, which may be convolved with the raw datarecord to produce calibrated data. The lter taps in Figure 6 carry units of pT

raw .

6

Power Spectral Density (PSD)

Figure 4: Measured PSD and sensitivity at Juneau

Time [usec]

Mag

nit

ud

e [d

B-r

aw/p

T]

Filt

er T

aps

[raw

/pT

]

Un

wra

pp

ed P

has

e [d

eg]

Frequency [Hz]

0

Measured System Response

Figure 5: AWESOME receiver system response

7

Time [usec]

Mag

nit

ud

e [d

B-p

T/ra

w] 0

Un

wra

pp

ed P

has

e [d

eg]

Pre-filter

Filt

er t

aps

[pT/

raw

]

Frequency [Hz]

Figure 6: AWESOME receiver inverse response

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