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Susceptibility measurement setup

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Enthalpy:
Signal, noise, electronics.

If the sample rotates at 5Hz on R=0.1m, the enter and exit voltage bu_mps can be 20ms wide and spaced by 40ms, while 30ms or more are needed to integrate them completely. Helmholtz' region of uniform induction provides a time of zero voltage between the bu_mps where integration can stop and start.

Software can integrate each bu_mp over 30ms and compute the difference. Integration times and their distance are usefully multiples of 20ms to reject interferences at 50Hz, or 16.67ms at 60Hz, and the repetition period an odd multiple. As the sample's rotation or oscillation frequency can vary, a separate sensor can tell the instantaneous position and software determine the best start and duration of the integration windows.

With the example times, the integration picks noise from 10Hz to 40Hz, with complicated limits. The window can be smoothened a bit, but we make metrology here.

A few TL1028 make a good differential amplifier (instrumentation amplifiers are about as good: AD8229, AD8428, AD8429 and competitors)
http://www.analog.com/media/en/technical-documentation/data-sheets/1028fd.pdf
Noise per input is 1nV/sqrt(Hz) and 3pA/sqrt(Hz) at these frequencies. Each half-coil is essentially resistive: 340Ω and at 40Hz j200Ω. After integration and enter-exit difference, the differential noise is 24nV.

Two 2000 turn coils pick 44nV×s from a 1cm3 χ=10-5 sample, or mean 1.5µV over 30ms. The enter-exit difference is 2.9µV or 100× the noise voltage. A resolution of Δχ=0.01×10-5 results from averaging few measures. The sample can be smaller.

The analog circuit shall provide limited and fixed low-pass filtering. This leaves the mean value of a bu_mp untouched, provided that the time of zero voltage between the bu_mps is kept. At identical transition and selectivity, inverse Chebychev and elliptic filters have a faster and quieter time response than Chebychev and Butterworth, don't believe books.

The considered amplifiers have an offset smaller than a strong signal. The analog circuit can pass the DC to avoid measurement errors and to settle quickly despite the slow signals. Enter-exit bu_mps difference by software removes the DC and low frequency components.

A PC distributes irregular >10A in unshielded cables and its processor draws >100A. It's a bad source of magnetic and conducted interferences. If box and distance don't suffice, consider a microcontroller instead.

Marc Schaefer, aka Enthalpy

Enthalpy:

--- Quote from: Enthalpy on July 28, 2018, 01:15:15 PM ---I trust today's computation.
--- End quote ---

I shouldn't have, because if now I compute the vector product properly using the sine and not the cosine, I get a signal twice as big that equals the other computation on July 22, 2018, 03:15:06 PM.


--- Quote from: Enthalpy on July 29, 2018, 08:14:51 AM ---the repetition period an odd multiple (of the mains' period)
--- End quote ---

But this doesn't reject interferences. If P×Q measures are averaged, they should start at P different angles from the start of a mains' period, the angles being equally spread over a turn. Examples:

* 2Q measures can start at 10° and 190° from the start of a mains' period.
* 3Q at 20°, 140°, 260°.
* 6Q at 1°, 31°, 61°, 91°, 121°, 151°.This squashes the interferences at the mains' frequency and its harmonics not multiple of P (it's a sum of the roots of 1 in the complex plane) so a big P has some usefulness. Exact angles improve the rejection, so the measures should start when the sample's speed is stable, and even at big P×Q, precise timing improves over sampling asynchronous with the mains.

Marc Schaefer, aka Enthalpy

Enthalpy:
Most sources show that the Helmholtz coil provides an induction uniform along the z axis because this is simple algebra. But how uniform is the induction radially?

A less simple algebraic solution must exist off-axis, possibly with Bessel functions. Here I prefer to show more generally that, due to its properties in vacuum, if the induction is uniform along the symmetry axis, it's uniform radially too, and the deviation can be estimated. With approximation signs on the sketch, but a mathematician would do it cleanly with Taylor series and Cauchy remainder.

Here the on-axis induction Bz(0,z) is maximally flat, varying as βz4 approximately. div(B)=0 links the axial variation of Bz with the radial component Br near the axis, which is small, and if β=0 it's zero. curl(B) aka rot(B)=0 links the axial variation of Br with the axial component Bz near the axis, which varies slowly with r, and if β=0 it's uniform.

So how wide can a sample be? Let's take a sphere centred on r=z=0 with radius Z. The radial variation is like r2z2 but on the sphere r2+z2=Z2, so the radial variation is maximum for r2=z2=Z2/2, or +0.75βZ4, while the axial variation is -0.25βZ4 there and -1.00βZ4 at z=Z. That is, the sphere is a reasonable boundary. With the previous 2Z=1cm it provides 0.5cm3 to the sample.

I've deduced function values on a surface or volume from the values on a line. That's common with a differential equation as we have here, and may even be accurate far from the line. We could have written Bz(0,z) with more powers of z, or as a Fourier series...

I suppose a link with holomorph functions. In a vacuum domain not surrounding a current, div(B)=rot(B)=0 lets define a scalar potential grad(ψ)=B with Δψ=0. Within a φ=const plane, Δψ=0 makes ψ the real part of a holomorph function, so knowing its values on a line fully defines it on the domain.

Marc Schaefer, aka Enthalpy

Enthalpy:
On 07/29/18 02:14 PM, I meant "the sample rotates at 2.5Hz", not 5Hz.

5Hz is possible too. The duration and distance of the integration windows can still be multiples of 20ms or 16.67ms, and the signal-to-noise improves by 3dB for each pulse pair and by 6dB if averaging over an identical duration.

Enthalpy:
Some materials, mainly man-made ones like polymers and technical ceramics, have homogeneous susceptibility and can be machined to accurate shape. Then, as the deviations of the induction have opposite signs along the axis and the radius, special proportions of the sample can let the deviations compensate an other. The measure is more accurate, and the sample can be bigger.

Using the radial gradient of the induction obtained on 08/04/18, the image computes the compensation condition as Z/R=D/L=sqrt(2/5)~0.633 for a cylinder, within a domain where the on-axis deviation is written as -βz4, and the residue for a sphere of radius Z.

The sphere isn't bad, provided it's accurately shaped and homogeneous: it compensates by 0.086, or to 0.086% if βZ4=1%.

For every shape, an accurate position is necessary.

A cylindrical or spherical container filled with a liquid or even a powder could keep the good properties if the filling is complete and uniform.

Marc Schaefer, aka Enthalpy

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