Chapter 5

Induction subsystem

Coils, drivers, receive chain, digitiser and excitation for full-tensor measurement.

The induction subsystem measures the complex response of buried targets at twelve frequencies, from up to three transmit directions and at three receivers. It is the part of the platform that identifies what a target is made of and what shape it has.

5.1Requirements

Frequency coverage that contains the quadrature peaks of coin-sized non-ferrous targets (about 1 to 6 kHz for 2 to 5 mm copper and aluminium).
Phase-coherent sampling of all transmit current references and all receivers.
Input-referred noise low enough that the ground, not the electronics, limits detection range.
A coil arrangement that allows the full polarizability tensor to be recovered.

5.2Why no radio-frequency front end is needed

Radio-frequency design rules (transmission lines, impedance matching, mixers) apply when conductors are comparable to the wavelength, usually taken as longer than \(\lambda/10\). A 2 m coil cable reaches that length at 15 MHz. In the induction band, below 19 kHz, \(\lambda/10\) is more than 1.5 km. All connections are therefore ordinary lumped circuits. The analog-to-digital converter samples the band directly, so no local oscillator or intermediate frequency is needed. The first amplifier is specified by its input noise density, as in audio design.

Figure 5.1.Wire length at which transmission-line effects begin (\(\lambda/10\)) as a function of frequency. The induction band lies four decades below the limit for a 2 m cable; the radar band lies above it.

5.3Transmit chain

Each transmit coil is driven by a class-D audio power amplifier through a series resistor, which keeps the load impedance above the amplifier's minimum rating at every tone. The coil is not tuned with a capacitor: a resonant coil would emphasise one frequency and suppress the others. A small sense resistor in series with each coil is read by a codec input. Dividing every receiver signal by this current reference removes gain and phase drift in the digital-to-analog converter and the amplifier. With 20 turns, 12 cm radius and 1 A, the coil's magnetic moment is 0.9 A·m².

5.4Receive coils

Table 5.1.Reference receive coil.

Parameter	Value	Basis
Turns, diameter	100, 12 cm	Trade-off between signal and self-resonance
Inductance	4.58 mH	\(L = \mu_0 N^2 r\,[\ln(8r/a) - 2]\), wire diameter 0.3 mm
Winding resistance	12.8 Ω	37.7 m of wire; thermal noise 0.46 nV/√Hz
Self-resonance	429 kHz	Assumed winding capacitance 30 pF; 23 times the highest tone
Tuning	none	Flat response across all tones
Electrostatic shield	slotted foil	Blocks capacitive pickup; the slot prevents a shorted turn

The coil can be read in voltage mode, where the output rises in proportion to frequency, or by a transimpedance amplifier, which gives a flat response above 446 Hz. Voltage mode needs no extra components and its slope is removed by calibration. The transimpedance stage is kept as an option in case the lowest tones turn out to be limited by dynamic range.

Figure 5.2.Response of the reference coil in voltage mode and transimpedance mode, normalised at 10 kHz. Across the tone band the voltage-mode response changes by a factor of about 100.

5.5Digitiser: CS42448

The CS42448 provides six 24-bit analog-to-digital converters and eight digital-to-analog converters behind one TDM interface. All channels run from one master clock, so inter-channel phase is fixed by construction. The board available to the project ("6IN8OUT") has line-level, AC-coupled inputs without microphone preamplifiers.

Table 5.2.CS42448 properties relevant to the design. Noise and full-scale values are derived from the datasheet and will be measured in T1.3.

Property	Value	Consequence
Converters	6 in, 8 out, 24-bit	Sufficient for 3 TX, 3 current references and 3 RX
Clocking	one master clock, TDM	All channels sample-synchronous
ADC dynamic range	about 105 dB (A-weighted)	Input noise about 80 nV/√Hz at 2 V_rms full scale
Input type	line level, AC-coupled	An external low-noise stage is required per receiver
Sample rate	up to 192 kHz; 48 kHz planned	96 kHz to be tested (T1.3)

Figure 5.3.Channel assignment in the two build phases. Phase A uses one transmitter and leaves an input for the GNSS pulse. Phase B uses all six inputs for the full-tensor arrangement; the pulse is then summed onto the first current reference. A spare output provides a permanent calibration tone.

5.6Noise budget

The input noise of the codec, about 80 nV/√Hz, is roughly 36 times higher than that of a good microphone preamplifier. The echo of a 1 cm copper sphere at 20 cm depth is about 20.5 µV at the coil and falls as \(1/d^6\).

Figure 5.4.Echo of a 1 cm copper sphere against the noise floors. With 4.3 s averaging the codec alone is adequate. For single 85 ms blocks during movement its floor coincides with the estimated ground response. A ×33 preamplifier restores the margin.

Each receiver therefore gets a non-inverting ×33 stage with a low-noise operational amplifier. With an OPA1612 the input-referred noise of the chain is 2.6 nV/√Hz; with an NE5532 it is 5.5 nV/√Hz. In both cases the ground response, estimated at about 300 times the electronic floor, sets the practical detection range of about 40 cm for coin-sized targets.

5.7Excitation

All tones are transmitted at once as a multi-tone waveform. At 48 kHz and a block length of 4096 samples the frequency resolution is 11.72 Hz and one block lasts 85.3 ms. Twelve tones are placed on exact FFT bins with logarithmic spacing:

Table 5.3.Tone plan for 48 kHz sampling.

Quantity	Value
FFT bins (N = 4096)	16, 24, 37, 56, 85, 130, 197, 300, 456, 693, 1053, 1600
Frequencies	188, 281, 434, 656, 996 Hz; 1.52, 2.31, 3.52, 5.34, 8.12, 12.3, 18.75 kHz
Crest factor	4.90 with zero phases; 3.06 with optimised phases (4.1 dB more power per tone)
Phases (rad)	2.589, 5.287, 1.692, 2.610, 3.376, 2.940, 1.805, 1.120, 0.966, 3.592, 5.042, 0.208

Figure 5.5.(a) Tone comb. (b, c) One block of the transmit waveform with zero phases and with optimised phases, scaled to equal power. The peak value, which limits the amplifier, falls from 4.90 to 3.06 times the RMS value.

Simultaneous transmission is not free. At equal amplifier peak, twelve simultaneous tones give about 6.7 dB less signal-to-noise ratio per tone than stepping a single tone through the same frequencies in the same total time. We therefore use stepped tones for static measurements and simultaneous tones when the sensor moves. For moving acquisition the block length is reduced to 1024 samples (21.3 ms, about 1 cm of travel at walking speed).

Figure 5.6.Quadrature response of the reference targets compared with the band available at 48 kHz. The peaks of coin-sized copper and aluminium targets lie inside the band. Stainless steel and the ferrous peak lie above it; ferrous discrimination does not depend on the peak.

5.8Coil configurations

The tensor \(\mathbf M\) is symmetric and has six independent elements per frequency. One transmitter excites the target along one direction \(\mathbf t\); three receivers then measure the vector \(\mathbf M\mathbf t\), which is one column of the tensor. Three transmitters along independent directions give all columns and over-determine the six unknowns.

Figure 5.7.Information obtained with one and with three transmitters. Colours in the right-hand matrix mark pairs of equal elements.

Table 5.4.Coil configurations.

Configuration	Out / in	Location	Tensor from one position	Use
1 TX + 1 RX	1 / 2	Only by scanning	1 element	Conventional detector
1 TX + 3 RX	1 / 4	Yes	One column	Phase A; full tensor by moving the sensor (Chapter 7)
3 TX + 3 RX	3 / 6	Yes	Full tensor	Phase B

The three transmit coils are oriented along x, y and z. Coils lying side by side in one plane produce nearly parallel fields at a deep target and add little information. Orthogonal coils also have almost no mutual coupling. All three transmit at the same time on interleaved tone sets (for example bins 16, 24, … for coil x; 17, 25, … for y; 18, 26, … for z). Tones on distinct FFT bins are orthogonal over one block, so a single FFT separates the three transmitters.

Two of the receive coils can also be operated as a gradiometer. Their outputs are subtracted, so that distant fields such as mains hum and the bulk ground response cancel, while the field of a nearby target is retained.

Figure 5.8.Gradiometer. Distant fields reach both receive coils equally and cancel in the difference. The dipole field of a nearby target falls as \(1/r^3\) and is much stronger in the lower coil.

5.9Coupling and nulling

The field of the transmit coil at a receive coil is 10⁵ to 10⁶ times larger than a target echo. This direct coupling has to be reduced geometrically, by overlapping the coils so that the net flux through the receiver is close to zero, before any electronics can help. The design target is a residual of 10⁻³ or less at all twelve tones simultaneously (task T2.2). The remaining coupling is measured without a target and subtracted as a background spectrum. This nulling is the most demanding analog task of the project.