VLF Receiver Calibration using Thermal Noise

Using the thermal noise of a resistor to calibrate a remote VLF receiver front end.

Introduction

With some installations it is inconvenient to calibrate a receiver by means of a signal generator in the normal way. It may be time consuming to visit the front end, or a signal generator capable of operating in the field may not be available. We can exploit the fact that a typical VLF E-field receiver is sensitive enough to easily pick up the thermal noise from a suitable resistor. Indeed, in VLF receivers which use a JFET for the front end, the overall system noise is dominated by one such resistor - the gate bias resistor.

The thermal noise EMF of a resistor can be calculated very accurately. The mean square noise voltage present across the open-circuit resistor is given by

   V^2 = 4 * k * T * R * BW

where k is Boltzmann's constant, 1.38 * 10^-23, T is the absolute temperature, R is the resistance, and BW is the bandwidth over which the mean square voltage is taken.

When connected into a circuit, the noise voltage is reduced by the voltage drop due to noise current flow out of the resistor. In the gate circuit of the front-end JFET, the bias resistor is shunted by the parallel combination of the gate capacitance, plus any circuit capacitance, plus the antenna capacitance if connected.

If C is the total shunt capacitance and R is the bias resistor value, then the total circuit impedance to the flow of noise current is

 Z = sqrt( R^2 + XC^2)

where

 XC = 1/(2 * pi * F * C)

The resulting RMS noise current is therefore

  sqrt( 4 * k * T * R * BW)/sqrt( R^2 + XC^2)

and the noise voltage at the terminals of the resistor is reduced to

 XC * sqrt( 4 * k * T * R * BW)/sqrt( R^2 + XC^2)

in a bandwidth BW centered on the frequency F.

The graph on the right shows the predicted noise voltage for two different values of gate bias resistor, 470k and 10M, for the case where the total shunt capacitance is 37pF. Below a corner frequency given by

   1/(2 * pi * R * C)

the noise amplitude is constant. Above this corner frequency, the noise reduces by 6dB per octave.

Some measured noise spectra are shown on the left. The red trace is obtained by shorting the gate to ground, thus removing all the thermal noise from before the FET gate. The residual noise is the sum of all the amplifier and soundcard noise, and the quantisation noise of the soundcard A/D conversion. This we will refer to as the system noise.

The thermal noise from the resistors is several times the amplitude of the system noise, and can therefore be used as a good calibration source. By taking two spectra, one with the gate grounded, and the other with the antenna disconnected, we can deduce the sensitivity of the receiver as a function of frequency.

Procedure

The procedure is roughly as follows:-

Set the bias resistor to a low value R1 (in my case 450k ohms, by adding a 470k resistor across the existing 10M bias resistor) and record the amplitude spectrum N(f) as a function of frequency.
Take a second spectrum, using the same sample rate, gain, and FFT settings, but this time with the gate shorted to ground, to obtain the system noise function NS(f).
Let the actual thermal noise voltage due to R be NT(f). Then bearing in mind that noise voltages add quadratically, we have the basic equations:-
```
  N(f)^2 = NS(f)^2 + K(f)^2 * NT(f)^2
```
where K(f) is a calibration function. In other words, for each frequency, the square of the measured spectrum is equal to the sum of the squares of the system noise and resistor thermal noise spectra. The thermal noise function NT(f) is calculated as above, and the purpose of the calibration is to determine the function K(f).
Rearrange the previous equation to give
```
 K(f) = sqrt( N(f)^2 - NS(f)^2 )/NT(f)
```

The function K(f) gives the number of units of soundcard A/D ouput per volt of front-end gate voltage.

Results

The digital output of the soundcard conversion can be processed in software to produce a calibrated waveform. The raw soundcard output is passed through a Fourier transform. Each complex bin amplitude is then divided by the K(f) for that frequency, giving a calibrated spectrum which is then reverse transformed.

The result is a time domain waveform accurately calibrated in units of volts at the FET gate. A further straightforward correction can be applied to the spectra to give the antenna terminal voltage, or the E-field strength, as desired.

The calibration will only be accurate over a range of frequencies for which the resistor noise clearly exceeds the system noise, and for which there is little or no interference. In my case, this limited the calibration to the range 500Hz to 15kHz.

I obtained the following results after calibration with the above procedure. The voltages given are gate voltages. The open-circuit antenna terminal voltage would be about 3 or 4 times the gate voltage. The antenna is a 2m pipe, 40mm diameter, with the base at 30cm above ground, placed in an open location.

Signal Bandwidth Measured Theoretical Comments

Thermal noise, 10Meg resistor 500Hz-13kHz 6.2uV RMS 6.8uV RMS Bias resistor shunted by 37pF

Thermal noise, 450k resistance 500Hz-9kHz 7.1uV RMS 7.0 uV RMS Bias resistor 10M shunted by 470k and 37pF

System noise 500Hz-9kHz 1.4uV RMS N/A Measured with gate grounded

Mains hum 500Hz-15kHz 150uV RMS, 500uV peak N/A Quite a spiky waveform

Wind noise 500Hz-2kHz 10uV RMS N/A The hiss when a stiff breeze blows across the antenna tube

Daytime VLF background hiss 500Hz-15kHz About 5-15uV RMS N/A Sampled in between the sferics

Daytime sferics 500Hz-15kHz 100-500uV peak N/A That's the background patter. Occasional sferics peaking at 2-3mV.

Whistler, typical S2 500Hz-15kHz 10-20uV RMS N/A Amplitude fluctuates as the frequency descends

Signal	Bandwidth	Measured	Theoretical	Comments
Thermal noise, 10Meg resistor	500Hz-13kHz	6.2uV RMS	6.8uV RMS	Bias resistor shunted by 37pF
Thermal noise, 450k resistance	500Hz-9kHz	7.1uV RMS	7.0 uV RMS	Bias resistor 10M shunted by 470k and 37pF
System noise	500Hz-9kHz	1.4uV RMS	N/A	Measured with gate grounded
Mains hum	500Hz-15kHz	150uV RMS, 500uV peak	N/A	Quite a spiky waveform
Wind noise	500Hz-2kHz	10uV RMS	N/A	The hiss when a stiff breeze blows across the antenna tube
Daytime VLF background hiss	500Hz-15kHz	About 5-15uV RMS	N/A	Sampled in between the sferics
Daytime sferics	500Hz-15kHz	100-500uV peak	N/A	That's the background patter. Occasional sferics peaking at 2-3mV.
Whistler, typical S2	500Hz-15kHz	10-20uV RMS	N/A	Amplitude fluctuates as the frequency descends

The 10% error obtained when the calibration is checked against the theoretical noise of the 10M bias resistor is entirely due to frequency components in the range 10kHz to 13kHz. In this range the calibration is not so good because the resistor noise amplitude during calibration was only a little above the system noise. When the measured and theoretical spectra are compared, the calibration error for frequencies in the range 500Hz to 8kHz is seen to be 2% or better.

The nightime background hiss is proving difficult to measure due to the density of sferics, but is about 10-15uV.

Paul Nicholson, vlf0308@abelian.org