Proof: We prove this equation under more general assumptions. Define

$F_{s}(a_{u},a_{v})=\|\widehat{\zeta}_{u}-\widehat{\zeta}_{v}\|^{-s}+\|\widehat{\zeta}_{u}-(0,0,1)\|^{-s}+\|\widehat{\zeta}_{v}-(0,0,1)\|^{-s},$
(116)

Where

$\zeta_{u}=(-\sqrt{3}/3-a_{u},0),\hskip 10.0pt\zeta_{v}=(\sqrt{3}/3+a_{v}).$
(117)

Our assumption $d>\sqrt{3}/3$ lets us take $a_{u},a_{v}>0$. Next for
$0<b_{u}<a_{u}$ and $0<b_{v}\leq a_{v}$ define

$E(s)=F_{s}(a_{u},a_{v})-F_{s}(b_{u},b_{v})$ (118)

We will show that $E(s)\geq 0$ when $s\geq 2$. It suffices to prove this
result in the intermediate case when $a_{u}=b_{u}$ or $a_{v}=b_{v}$ because
then we can apply the intermediate result twice to get the general case.
Without loss of generality we consider the case when $a_{v}=b_{v}$ and
$b_{u}<a_{u}$. We think of $\zeta_{u}$ as moving from its old location defined
by $a_{u}$ inward to its new location defined by $b_{u}$.

With the file LemmaB22.m we compute that

$\frac{\partial F_{2}}{\partial a_{u}}\bigg{|}_{(a_{u},a_{v})},\hskip
30.0pt-\frac{\partial F_{-2}}{\partial a_{u}}\bigg{|}_{(a_{u},a_{v})}$

are both rational functions of $a_{u},a_{v}$ with all positive coefficients.
Hence $E(2)>0$ and $E(-2)<0$.

When $a_{u}=a_{v}=0$ the points $\widehat{\zeta}_{u},\widehat{\zeta}_{v}$ and
$(0,0,1)$ make an equilateral triangle on a great circle. Hence, when
$a_{u},a_{v},b_{u},b_{v}>0$ the point $\widehat{\zeta}_{u}$ is closer to
$(0,0,1)$ than it is to $\widehat{\zeta}_{v}$ both in its old location and in
its new location. The inward motion of the point $\zeta_{u}$ increases the
shorter (corresponding spherical) distance and decreases the longer
(corresponding spherical) distance. More to the point, our move decreases the
longer inverse-distance and increases the shorter inverse-distance. Thus the
sign sequence (§10.1) for $E(s)$ is $+,-.-,+$. By Descartes’ Lemma, $E(s)$
changes sign at most twice and also $E(s)>0$ when $|s|$ is sufficiently large.

Since $E(-2)<0$ as see that $E$ changes sign on $(-\infty,-2)$. If $E$ changes
sign on $(2,\infty)$ then in fact $E$ changes sign twice because it starts and
ends positive on this interval. But then $E$ changes sign $3$ times, a
contradiction. Hence $E(s)\geq 0$ for $s\geq 2$. $\spadesuit$

###  11.4 Proof of Lemma B3

By symmetry, it suffices to prove Lemma B3 for the list $\\{(0,1),(1,2)\\}$.
Let $D$ denote the set of triples of points
$(q_{0},q_{1},q_{2})\in(\mbox{\boldmath{$R$}}^{2})^{3}$ such that there is
some $q_{3}$ such that $q_{0},q_{1},q_{2},q_{3}\in\Upsilon^{\prime}$. The
symmetrization operation is given by
$(q_{0},q_{1},q_{2})\to(q_{1}^{\prime},q_{2}^{\prime},q_{3}^{\prime})$, where

$q^{\prime}_{0}=\bigg{(}\frac{q_{01}-q_{21}}{2},q_{02}\bigg{)},\hskip
30.0ptq^{\prime}_{1}=(0,q_{21}),\hskip
30.0ptq^{\prime}_{2}=\bigg{(}\frac{q_{21}-q_{01}}{2},q_{22}\bigg{)},\hskip
30.0pt$ (119)

Note that
$\|q_{0}^{\prime}-q_{1}^{\prime}\|=\|q_{2}^{\prime}-q_{1}^{\prime}\|$.
Therefore, by the Convexity Lemma, it suffices to prove that
$\\{(0,1),(1,2)\\}$ is good for the parameter $s=2$.

We define

$[u,v]t=u(1-t)+vt.$ (120)

The map $t\to[u,v]t$ maps $[0,1]$ to $[u,v]$.

For all $4$ choices of signs we define
$\phi_{\pm,\pm}:[0,1]^{5}\to(\mbox{\boldmath{$R$}}^{2})^{3}$ as follows:

$\phi_{\pm,\pm}(a,b,c,d,e)=q_{0}(a,d,\pm b),q_{1}(\pm e,c),q_{2}(a,d,\pm b),$
(121)

where

$512q_{0}(a,d,\pm b)=([416,498]a+49e,\pm 16b).$ $512q_{1}(\pm d,c)=(\pm
32d,[348+465]c)$ $512q_{2}(a,d,\pm b)=([-416,-498]a+49e,\pm 16b).$

In these coordinates, horizontal symmetrization is the map
$(a,b,c,d,e)\to(a,b,c,0,0)$.

######  Lemma 11.7 (B31)

We have

$D\subset\phi_{+,+}([0,1]^{5})\cup\phi_{+,-}([0,1]^{5})\cup\phi_{-,+}([0,1]^{5})\cup\phi_{-.-}([0,1]^{5}).$

Proof: Let $D_{ij}$ denote the set of possible coordinates $q_{ij}$ that can
arise for points in $D$. This, for instance $D_{01}=[-16,16]/512$. Let
$D_{ij}^{*}$ denote the set of possible coordinates $q_{ij}$ that can arise
from the union of our parametrizations. By construction $D_{i2}\subset
D_{i2}^{*}$ for $i=0,1,2$ and $D_{11}\subset D_{11}^{*}$.

Remembering that we have $q_{01}\geq|q_{21}|$, we see that the set of points
pairs $(q_{01},q_{21})$ satisfying all the conditions for inclusion in $D$
lies in the triangle $X$ with vertices $(498,-498)$ and $(498,-400)$ and
$(432,-400)$. At the same time, the set of pairs
$(512)(p^{*}_{01},p^{*}_{21})$ that we can reach with our parametrization is
the rectangle $X^{*}$ with vertices

$(498,-498),\hskip 12.0pt(416,-416),\hskip 12.0pt(498,-498)+(49,49),\hskip
12.0pt(416,-416)+(49,49).$

We have $X\subset X^{*}$ because

$(432,-400)=(416,-416)+(16,16),\hskip 18.0pt(498,-400)=(449,-449)+(49,49).$

This completes the proof. $\spadesuit$

Using our coordinates above, we define

$F_{\pm,\pm}(a,b,c,d,e)=\|\widehat{p}_{0}-\widehat{p}_{1}\|^{-2}+\|\widehat{p}_{2}-\widehat{p}_{1}\|^{-2},$
$\Phi_{\pm,\pm}(a,b,c,d,e)={\rm
num\/}_{+}(F_{\pm,\pm}(a,b,c,d,e)-F_{\pm,\pm}(a,b,c,0,0)).$ (122)

######  Lemma 11.8 (B32)

For any sign choice $\Phi_{\pm,\pm}>0$ on $(0,1)^{5}$.

Proof: Let $\Phi$ be any of the $4$ polynomials. The file LemmaB32.m computes
that

  * •

$F$ and $\Phi_{d}$ and $\Phi_{e}$ are zero when $d=e=0$.

  * •

$\Phi_{d}+2\Phi_{e}$ and $\Phi_{dd}$ and $\Phi_{ee}$ are weak positive
dominant and hence non-negative on $[0,1]^{5}$.

Let $Q_{d}\subset[0,1]^{5}$ be the sub-cube where $d=0$. Let $\phi(d)$ be the
restriction of $\Phi$ to a line segment which starts at some point
$(a,b,c,0,0)$ and moves parallel to $(0,0,0,1,0)$. By Lemma B321 we have
$\phi(0)=\phi^{\prime}(0)$ and also $\phi^{\prime\prime}(d)\geq 0$. Hence
$\phi(d)\geq 0$ for $d\geq 0$. Hence $\Phi\geq 0$ on $Q_{d}$. A similar
argument shows that likewise $\Phi\geq 0$ on $Q_{e}$. Any point in $(0,1)^{5}$
can be joined to a point in $Q_{d}\cup Q_{e}$ by a line segment $L$ which is
parallel to the vector $(0,0,0,1,2)$. By Lemma B3121, $\Phi$ increases along
such a line segment as we move out of $Q_{d}\cup Q_{e}$. Hence $\Phi\geq 0$ on
$[0,1]^{5}$. $\spadesuit$

Lemma B312 implies $F_{\pm}(a,b,c,d,e)\geq F_{\pm}(a,b,c,0,0)$. See §10.2.
Lemma B3 thus follows from Lemma B31 and Lemma B32.

###  11.5 Proof of Lemma B4

By symmetry, it suffices to prove that $\\{(0,1),(3,1)\\}$ is good for all
$s\geq 12$. For ease of notation set $q_{k}=p^{\prime\prime}_{k}$. Let $D$ be
the set of configurations $(q_{0},q_{1},q_{3})$ such that
$(q_{0},q_{1},q_{2},q_{3})\in\Upsilon^{\prime\prime}$ when $q_{2}$ is the
reflection of $q_{0}$ in the $y$-axis. By symmetry, it suffices to treat the
case when $q_{02}\geq 0$. We let $D_{\pm}\subset D$ denote those
configurations with $\pm(q_{12}+q_{32})\geq 0$. Obviously $D=D_{+}\cup D_{-}$.

As in Equation 120, let $[u,v]t=u(1-t)+vt$. Similar to the horizontal case, we
define $\phi_{\pm}(a,b,c,d)=(q_{0}(b,d),q_{1}(a,c),q_{3}(a,c))$, where

$512q_{0}(b,d)=([416,498]b,16d).$ $512q_{1}(a,c)=(0,-[348,465]a\pm 59c).$
$512q_{3}(a,c)=(0,+[348,465]a\pm 59c).$

In these coordinates, the symmetrization operation is $(a,b,c,d)\to(a,b,0,0)$.

######  Lemma 11.9 (B41)

$D_{\pm}\subset\phi_{\pm}([0,1]^{4})$.

Proof: This is just like the proof of Lemma B21. The only non-obvious point is
why every pair $(p_{12},p_{32})$ is reached by the map $\phi_{\pm}$. The
essential point is that for configurations in $D_{\pm}$ we have
$512|p_{12}+p_{32}|\leq 2\times 59$. $\spadesuit$

Following the same idea as in the proof of Lemma B3, we define

$F_{s,\pm}(a,b,c,d)=\|\Sigma^{-1}(q_{0})-\Sigma^{-1}(q_{1})\|^{-s}+\|\Sigma^{-1}(q_{0})-\Sigma^{-1}(q_{3})\|^{-s},$
(123) $\Phi_{s,\pm}(a,b,c,d)={\rm
num\/}_{+}(F_{s,\pm}(a,b,c,d)-F_{s,\pm}(a,b,0,0)).$ (124)

The points on the right side of Equation 123 are coordinatized by the map
$\phi_{\pm}$. We can finish the proof by showing that $\phi_{2,+}\geq 0$ and
$\phi_{12,-}\geq 0$ on $[0,1]^{4}$. The Convexity Lemma then takes care of all
exponents greater than $2$ on $D_{+}$ and all exponents greater than $12$ on
$D_{-}$.

######  Lemma 11.10 (B42)

$\Phi_{2,+}\geq 0$ on $[0,1]^{4}$.

Proof: Let $\Phi=\Phi_{2,+}$. Let $\Phi|_{c=0}$ denote the polynomial we get
by setting $c=0$. Etc. Let $\Phi_{c}=\partial\Phi/\partial c$, etc. The
Mathematica file LemmaB42.m computes that $\Phi|_{c=0}$ and $\Phi|_{d=0}$ and
$\Phi_{c}+\Phi_{d}$ are weak positive dominant. Hence $\Phi\geq 0$ when $c=0$
or $d=0$ and the directional derivative of $\Phi$ in the direction $(0,0,1,1)$
is non-negative. This suffices to show that $\Phi\geq 0$ on $[0,1]^{4}$.
$\spadesuit$

######  Lemma 11.11 (B43)

$\Phi_{12,-}\geq 0$ on $[0,1]^{4}$.

Proof: The file LemmaB43.m has the calculations for our argument. Let
$\Phi=\Phi_{12,-}$. This monster has $102218$ terms and we simplify it
carefully.

Let $M$ denote the maximum coefficient of $\Phi$. We let $\Phi^{*}$ be the
polynomial we get by taking each coefficient of $c$ of $\Phi$ and replacing it
with the greatest integer less than $10^{10}c/M$. This has the effect of
killing off about half the terms of $\Phi$, namely the positive terms that are
less than $10^{-10}M$. The “small” negative coefficients are changed to $-1$.
The polynomial $10^{10}\Phi-M\Phi^{*}$ has all non-negative coefficients.
Hence, if $\Phi^{*}\geq 0$ on $[0,1]^{4}$ so is $M\Phi\geq 0$ and so is
$10^{10}\Phi$ and finally so is $\Phi$. Now $\Phi^{*}$ has $37760$ monomials
in which the coefficient is $-1$. We check that each such monomial is
divisible by one of $c^{2}$ or $d^{2}$ or $cd$. We therefore define

$\Psi=\Phi^{**}-37760(c^{2}+d^{2}+cd),$

where $\Phi^{**}$ is obtained from $\Phi^{*}$ by setting all the $(-1)$
monomials to $0$. We have $\Psi\leq\Phi^{*}$ on $[0,1]^{4}$. Hence, if $\Psi$
is non-negative on $[0,1]^{4}$ then so is $\Phi$. We have reduced the problem
to showing that $\Psi\geq 0$ on $[0,1]^{4}$. The polynomial $\Psi$ has $5743$
terms, which is more manageable.

Again we let $F_{a}=\partial F/\partial a$, etc. We check that $\Psi_{aaa}$ is
weak positive dominant and hence non-negative on $[0,1]^{4}$. This massive
calculation reduces us to showing that the restrictions $\Psi|_{a=0}$ and
$\Psi_{a}|_{a=0}$ and $\Psi_{aa}|_{a=0}$ are all non-negative on $[0,1]^{3}$.
Letting $F$ be any of these $3$ functions, we consider

$F|_{c=0},\hskip 30.0ptF|_{d=0}\hskip 30.0pt4F_{c}+F_{d},$ (125)

We show that all three functions are weak positive dominant for
$\Psi_{a}|_{a=0}$ and $\Psi_{aa}|_{a=0}$. This shows that $\Psi_{a}|_{a=0}$
and $\Psi_{aa}|_{a=0}$ are non-negative on $[0,1]^{3}$. Concerning the choice
$F=\Psi|_{a=0}$, all that remains is showing (in some other way) that
$G=4F_{c}+F_{d}\geq 0$.

We check that $G_{d}$ is weak positive dominant and hence non-negative on
$[0,1]^{3}$. This reduces us to showing that $H=G|_{d=0}$ is non-negative on
$[0,1]^{2}$. Here $H$ is a $2$-variable polynomial in $b,c$. We check that the
two subdivisions $S_{b,0}(H)$ and $S_{b,1}(H)$ are weak positive dominant.
This proves that $H$ is non-negative on $[0,1]^{2}$. $\spadesuit$

Remark: The proof of Lemma B43 is pretty crazy. Along with my computer code, I
include a PDF document having an alternate (but longer) proof that involves
much smaller polynomials.

##  12 Endgame: Proof of Lemma C1

The proof here uses the material in §10.

We also recall some notation that we use repeatedly. $\Sigma^{-1}$ is inverse
stereographic projection, as in Equation 8. We often write
$\widehat{p}=\Sigma^{-1}(p)$ for $p\in\mbox{\boldmath{$R$}}^{2}$. The sets
$\Psi_{4}$ and $\widehat{\Psi}_{4}$ are defined by

$64\Psi_{4}=[43,64]^{2},\hskip 30.0pt64\widehat{\Psi}_{4}=[55,56]^{2}.$ (126)

The sets $\Psi_{8}$ and $\widehat{\Psi}_{8}$ respectively are the main
diagonals of $\Psi_{4}$ and $\widehat{\Psi}_{4}$. A point $(x,y)$ in these
domains defines the avatar with $-p_{2}=p_{0}=(x,0)$ and $-p_{1}=p_{3}=(0,y)$;
we define ${\cal E\/}_{s}(x,y)$ to be the $R_{s}$ potential of the
corresponding configuration in $S^{2}$.

###  12.1 Reduction to Two Halves

We have the symmetrization operation $\sigma:\widehat{\Psi}_{4}\to\Psi_{8}$
given by

$\sigma(x,y)=(z,z),\hskip 30.0ptz=\frac{x+y+(x-y)^{2}}{2}.$ (127)

Lemma C1 says ${\cal E\/}_{s}\circ\sigma\leq{\cal E\/}_{s}$ on
$\widehat{\Psi}_{4}$ for all $s\in[14,16]$, with equality iff $x=y$. Using
notation special to this chapter, we write ${\cal
E\/}_{s}(x,y)=G_{s}(x,y)+H_{s}(x,y)$, where

$G_{s}(x,y)=\|\widehat{p}_{0},\widehat{p}_{2}\|^{-s}+\|\widehat{p}_{1},\widehat{p}_{3}\|^{-s},$
$H_{s}(x,y)=2\|\widehat{p}_{0},(0,0,1)\|^{-s}+2\|\widehat{p}_{1},(0,0,1)\|^{-s}+4\|\widehat{p}_{0},\widehat{p}_{1}\|^{-s}.$
(128)

Lemma C1 follows immediately from Lemmas C11 and C12.

######  Lemma 12.1 (C11)

$G_{s}(x,y)\geq G_{s}(z,z)$ for $s\geq 2$ and $(x,y)\in\widehat{\Psi}_{4}$.
When $s>2$ we get equality if and only if $x=y$.

Proof: By the Convexity Lemma from §10.1 it suffices for us to prove that
$G_{2}(x,y)\geq G_{2}(z,z)$ for all $x,y\in\Psi_{4}$. Let
$\phi:[0,1]^{2}\to\widehat{\Psi}_{4}$ be the affine isomorphism whose linear
part is a positive diagonal matrix. Define

$\Phi={\rm num\/}_{+}(G_{2}\circ\phi-G_{2}\circ\sigma\circ\phi).$ (129)

The file LemmaC11.m computes that $\Phi(a,b)=(a-b)^{2}\Phi^{*}$, where
$\Phi^{*}$ is weak positive dominant. Hence $\Phi^{*}>0$ on $(0,1)^{2}$. This
does it. $\spadesuit$

######  Lemma 12.2 (C12)

$H_{s}(x,y)\geq H_{s}(z,z)$ for $s\in[14,16]$ and
$(x,y)\in\widehat{\Psi}_{4}$.

###  12.2 Proof of Lemma C12

######  Lemma 12.3 (C121)

The following is true:

  1. 1.

$H_{2}(x,y)\leq H_{2}(z,z)$ for all $(x,y)\in\widehat{\Psi}_{4}$.

  2. 2.

$H_{14}(x,y)\geq H_{14}(z,z)$ for all $(x,y)\in\widehat{\Psi}_{4}$.

  3. 3.

$H_{16}(x,y)\geq H_{16}(z,z)$ for all $(x,y)\in\widehat{\Psi}_{4}$.

We get strict inequalities for points in the interior of
$\widehat{\Psi}_{4}-\Psi_{8}$.

Proof: For integers $k=2,14,16$ define

$\Phi_{k}={\rm num\/}_{+}(H_{k}\circ\phi-H_{k}\circ\sigma\circ\phi).$ (130)

An algebraic miracle happens. The file LemmaC121.m computes that

  1. 1.

$-\Phi_{2}(a,b)=(a-b)^{2}\Phi^{*}_{2}(a,b)$ and $\Phi^{*}_{2}$ is weak
positive dominant.

  2. 2.

$\Phi_{14}(a,b)=(a-b)^{2}\Phi^{*}_{14}(a,b)$ and $\Phi^{*}_{14}$ is weak
positive dominant.

  3. 3.

$\Phi_{16}(a,b)=(a-b)^{2}\Phi^{*}_{16}(a,b)$ and $\Phi^{*}_{16}$ is weak
positive dominant.

This does it. $\spadesuit$

Now suppose there is some $(x,y)\in\widehat{\Psi}_{4}-\Psi_{8}$ and some
$s_{0}\in(14,16)$ such that $H_{s_{0}}(x,y)<H_{s_{0}}(z,z)$. Perturbing, we
can assume that $(x,y)$ lies in the interior of $\widehat{\Psi}_{4}-\Psi_{8}$.
Let $p_{0},p_{1},p_{2},p_{3}$ and
$p^{\prime}_{0},p^{\prime}_{1},p^{\prime}_{2},p_{3}^{\prime}$ respectively be
the configurations corresponding to $(x,y)$ and $(z,z)$. Define

  1. 1.

$r_{01}=\|\Sigma^{-1}(p_{0})-\Sigma^{-1}(p_{1})\|^{-1}$.

  2. 2.

$r_{0}=\|\Sigma^{-1}(p_{0})-(0,0,1)\|^{-1}$ and
$r_{1}=\|\Sigma^{-1}(p_{1})-(0,0,1)\|^{-1}$.

  3. 3.

$r^{\prime}_{01}=\|\Sigma^{-1}(p^{\prime}_{0})-\Sigma^{-1}(p_{1}^{\prime})\|^{-1}$.

  4. 4.

$r^{\prime}_{0}=\|\Sigma^{-1}(p^{\prime}_{0})-(0,0,1)\|^{-1}=\|\Sigma^{-1}(p^{\prime}_{1})-(0,0,1)\|^{-1}$.

Replacing $(x,y)$ by $(y,x)$ if necessary, we arrange that $r_{0}<r_{1}$. The
purpose of the next result is to get us into shape to apply Descartes’ Lemma
from §10.1.

######  Lemma 12.4 (C122)

$r_{0},r_{1},r^{\prime}_{0}<1/\sqrt{2}<r_{01},r_{01}^{\prime}$ and
$r_{01}<r_{01}^{\prime}$.

Proof: We have $x,y,z\in(0,1)$. We compute

$(1/2)-r_{0}^{2}=\frac{1-x^{2}}{4}>0,\hskip
15.0pt(1/2)-r_{1}^{2}=\frac{1-y^{2}}{4}>0,\hskip
15.0pt(1/2)-(r_{0}^{\prime})^{2}=\frac{1-z^{2}}{4}>0,$
$(r_{01})^{2}-(1/2)=\frac{(1-x^{2})(1-y^{2})}{4(x^{2}+y^{2})}>0,\hskip
20.0pt(r_{01}^{\prime})^{2}-(1/2)=\frac{(1-z^{2})^{2}}{8z^{2}}>0.$

This proves the first statement.

For the second statement, we define
$J=\|\widehat{p}_{0}-\widehat{p}_{1}\|^{-2}=r_{01}^{2}$ and then define $\Phi$
in terms of $J$ just as in Equation 130. The file LemmaC122.m computes that
$\Phi(a,b)=-(a-b)^{2}\Phi^{*}(a,b)$ where $\Phi^{*}$ is weak positive
dominant. Hence $\Phi^{*}>0$ on $(0,1)^{2}$. Hence $\Phi<0$ on $(0,1)^{2}$.
Hence $J(z,z)>J(x,y)$. But this implies that $r_{01}<r^{\prime}_{01}$.
$\spadesuit$

We now deduce Lemma C12 from Lemma C121 and Lemma C122. We fix $(x,y)$ and
$(z,z)=\sigma(x,y)$ and define

$h(s):=H_{s}-H_{s}\circ\sigma=+2r_{0}^{s}-4(r^{\prime}_{0})^{s}+2r_{1}^{s}+4r_{01}^{s}-4(r^{\prime}_{01})^{s}$
(131)

Now we observe the following consequences of Lemma C121:

  * •

$h(2)<0$ and $h(14)>0$. Hence $h$ changes sign in $(2,14)$.

  * •

$h(14)>0$ and $h(s_{0})<0$. Hence $h$ changes sign in $(14,s_{0})$.

  * •

$h(s_{0})<0$ and $h(16)>0$. Hence $h$ changes sign in $(s_{0},16)$.

  * •

$h(s)<0$ for $s$ sufficiently large because the term $-4(r^{\prime}_{01})^{s}$
eventually dominates. Hence $h$ changes sign in $(16,\infty)$.

Hence $h$ vanishes at least $4$ times. By Descartes’ Lemma, the sign sequence
must change signs at least $4$ times. Lemma C122 implies that the sign
sequence must be one of

$-,+,+,+,-,\hskip 30.0pt+,-,+,+,-,\hskip 30.0pt+,+,-,+,-.$

In no case does it change sign at least $4$ times. This is a contradiction.
The proof of Lemma C12 is done, but we remark that more analysis would show
that Equation 131 has the terms in the correct order and the middle sign
seqence is correct.

##  13 Endgame: Proof of Lemma C2

###  13.1 The Goal

Recall that the point $(1,\sqrt{3}/3)$ represents the TBP avatar. We define

$\Theta(s,x,y)={\cal E\/}_{s}(x,y)-{\cal E\/}(1,\sqrt{3}/3).$ (132)

Lemma C2 is equivalent to the following statements

  1. 1.

If $s\in[15,15_{+}]$ then $\Theta$ has a unique minimum in
$\widehat{\Psi}_{8}$.

  2. 2.

$\Theta>0$ on $[13,15]\times\Psi_{4}$ and on
$[15,15_{+}]\times(\Psi_{4}-\widehat{\Psi}_{4}).$

###  13.2 Proof of Statement 1

Let $\Theta_{x}$ be the partial derivative of $\Theta$ with respect to $x$,
etc. Statement 1 is equivalent to the statement that the single variable
function $f(x)=\Theta(s,t,t)$ has only one minimum for $s\in I$. Here
$64I=[55,56]$.

######  Lemma 13.1 (C21)

For all $s\in[13,15_{+}]$ and $(x,y)\in\Psi_{4}$ the quantities
$\Theta_{xx},\Theta_{yy},\Theta_{xy}$ are all positive.

We prove this result below. By the Chain Rule,

$f_{tt}=\Theta_{xx}+\Theta_{yy}+2\Theta_{xy}>0$ (133)

Hence $f$ is a convex function on $I$. Hence $f$ has a unique minimum in $I$.

###  13.3 Proof of Statement 2

The proof of Statement 2 is a divide-and-conquer calculation. We first explain
some details of the calculation.

The Expanding Out Method: This is a repeat of the definition in §9.3. Suppose
we want to establish an inequality like
$(\frac{a}{b})^{\frac{p}{q}}<\frac{c}{d}$, where every number involved is a
positive integer. This inequality is true iff $b^{p}c^{q}-a^{p}d^{q}>0.$ We
check this using exact integer arithmetic. The same idea works with $(>)$ in
place of $(<)$. We call this the expanding out method.

Rational Approximation Method: More generally, we will want to verify
inequalities like

$\sum_{i=1}^{10}b_{i}^{-s}-\sum_{i=1}^{10}a_{i}^{-s/2}>C.$ (134)

where all $a_{i}$ belong to the set $\\{2,3,4\\}$, and $b_{i},c,s$ are all
rational. more specifically $s\in[13,15_{+}]$ will be a dyadic rational and
$c$ will be positive. The expression on the left will be ${\cal
E\/}_{s}(p)-{\cal E\/}_{s}(p_{0})$ for various choices of $p$, and the
constant $C$ is related to the error term we define below.

Here is how we handle expressions like this. For each index
$i\in\\{1,...,10\\}$ we produce rational numbers $A_{i}$ and $B_{i}$ such that

$A_{i}^{s/2}>a_{i}\hskip 30.0ptB_{i}^{s}<b_{i}.$ (135)

We use the expanding out method to check these inequalities. We then check
that

$\sum_{i=1}^{10}B_{i}-\sum_{i=1}^{10}A_{i}>C.$ (136)

This last calculation is again done with integer arithmetic. Equations 135 and
136 together imply Equation 134. Logically speaking, the way that we produce
the rational $A_{i}$ and $B_{i}$ does not matter, but let us explain how we
find them in practice. For $A_{i}$ we compute $2^{32}a_{i}^{-s/2}$ and round
the result up to the nearest integer $N_{i}$. We then set
$A_{i}=N_{i}/2^{32}$. We produce $B_{i}$ in a similar way. When we have
verified Equation 134 in this manner we say that we have used the rational
approximation method to verify Equation 134. We will only need to make
verifications like this on the order of $20000$ times.

Error Estimate: We say that a block is a rectangular solid, having the
following form:

$X=I\times Q\subset[0,16]\times[0,1]^{2},$ (137)

where $I$ is a dyadic interval and $Q$ is a dyadic square. We define $|X|_{1}$
to be the length of $I$ and $|X|_{2}$ to be the side length of $Q$.

######  Lemma 13.2 (C22)

For any block $X\subset[13,16]\times\Psi_{4}$,

$\min_{X}\Theta\geq\min_{v(X)}\Theta-(|X|_{1}^{2}/512+|X|_{2}^{2}).$

Here $v(X)$ denotes the vertex set of $X$. Thus, to show that $\Theta|_{X}>0$
we just need to show that

$\Theta_{v(X)}>\frac{|X|_{1}^{2}}{512}+|X|_{2}^{2}.$

Grading a Block: We perform the following pass/fail evaluation of $X$.

  1. 1.

If $I\subset[0,13]$ or $I\subset[15_{+},16]$ or $Q\cap\Psi_{4}=\emptyset$, we
pass $X$ because $X$ is irrelevant to the calculation.

  2. 2.

If $s_{0}\geq 15$ and $Q\subset\widehat{\Psi}_{4}$ we pass $X$.

  3. 3.

$s_{0}<13$ and $s_{1}>13$ we fail $X$ because we don’t want to make any
computations which involve exponents less than $13$.

  4. 4.

If $X$ has not been passed or failed, we try to use the rational approximation
method to verify that $\Theta(v)>|X|_{1}^{2}/512-|X|_{2}^{2}$ for each vertex
$v$ of $X$. If we succeed at this, then we pass $X$. Otherwise we fail $X$.

To prove Statement 2 above it suffices to find a partition of
$[0,16]\times[0,1]^{2}$ into blocks which all pass the evaluation.

Subdivision: Let $X=I\times Q$. Here is the rule we use to subdivide $X$: If
$16|X|_{2}>|X|_{1}$ we subdivide $X$ along $Q$ dyadically, into $4$ pieces.
Otherwise we subdivide $X$ along $I$, into two pieces. This method takes
advantage of the lopsided form of Lemma C22 and produces a small partition.

The Main Algorithm: We perform the following algorithm.

  1. 1.

We start with a list $L$ of blocks. Initially $L$ has the single member
$\\{0,16\\}\times\\{0,1\\}^{2}$.

  2. 2.

We let $B$ be the last block on $L$. We grade $B$. If $B$ passes, we delete
$B$ from $L$. If $L=\emptyset$ then HALT. If $B$ fails, we delete $B$ from $L$
and append to $L$ the subdivision of $B$. Then we go back to Step 1.

######  Lemma 13.3 (C23)

When the algorithm runs it halts.

Proof: As for the other calculations, I used a 2017 iMac Pro with a 3.2 GHz
Intel Zeon W processor, running the Mojave operating system. When I run the
algorithm, it halts with success after $21655$ steps and in about $1$ minute.
The partition it produces has $14502$ blocks. $\spadesuit$

This proves Statement 2.

###  13.4 Proof of Lemma C21

We will show that $\Theta_{xx}>0$ and $\Theta_{xy}>0$ on
$[13,16]\times\Psi_{4}$. The case of $\Theta_{yy}$ follows from the case of
$\Theta_{xx}$ and symmetry. Setting $u=s/2$ we compute

${\cal E\/}_{s}(x,y)=A(s,x)+A(s,y)+2B(s,x)+2B(s,y)+4C(s,x,y),$ (138)
$A(x)=a(x)^{u},\hskip 30.0ptB(x)=b(x)^{u},\hskip 30.0ptC(x)=c(x)^{u},$
$a(x)=\frac{(1+x^{2})^{2}}{16x^{2}}\hskip 15.0ptb(x)=\frac{1+x^{2}}{4}\hskip
15.0ptc(x,y)=\frac{(1+x^{2})(1+y^{2})}{4(x^{2}+y^{2})}$

Hence

$\Theta_{xx}=A_{xx}+2B_{xx}+4C_{xx},\hskip 30.0pt\Theta_{xy}=C_{xy}.$ (139)

For each choice of $F=A,B,C$ we have

$F_{xx}=u(u-1)f^{u-2}f_{x}^{2}+uf^{u-1}f_{xx},\hskip
14.0ptC_{xy}=u(u-1)c^{u-2}c_{x}c_{y}+uc^{u-1}c_{xy}.$ (140)

(The second equation is just relevant for $C$.) We compute

$a_{xx}=\frac{3+x^{4}}{8x^{4}}>0,\hskip 20.0ptb_{xx}=\frac{1}{2},\hskip
20.0ptc_{xx}=\frac{(1-y^{4})(3x^{2}-y^{2})}{2(x^{2}+y^{2})^{3}}\geq 0.$
$c_{x}=\frac{x(y^{4}-1)}{2(x^{2}+y^{2})^{2}}<0,\hskip
10.0ptc_{y}=\frac{y(x^{4}-1)}{2(x^{2}+y^{2})^{2}}<0,\hskip
10.0ptc_{xy}=\frac{2xy(1+x^{2}y^{2})}{(x^{2}+y^{2})^{3}}>0.$

Equation 140 combines with all this to prove that $\Theta_{xx}>0$ and
$\Theta_{xy}>0$ on $[13,16]\times\Psi_{4}$.

###  13.5 Proof of Lemma C22

######  Lemma 13.4 (C221)

$|\Theta_{xx}|,|\Theta_{yy}|\leq 4$ on $[13,16]\times\Psi_{4}$.

Proof: By symmetry it suffices to prove this for $\Theta_{xx}$. We already
know $\Theta_{xx}>0$ on our domain. We use the notation from §13.4. An easy
exercise in calculus shows that $f\in(0,3/5)$ on $\Psi_{4}$ for each
$f=a,b,c$. From this bound, we see that the expression in Equation 140 is
decreasing as a function of $u$ for $u\geq 6$. (Recall that $u=s/2$.) Hence it
suffices to prove that $4-\Theta_{xx}\geq 0$ on $\\{12\\}\times[43/64,1]^{2}$.

We define $\phi(t)=(43/64)(1-t)+t.$ The file LemmaC221.m computes that for
$s=12$ the polynomial $\Phi={\rm num\/}_{+}(4-\Theta_{xx}\circ\phi)$ is weak
positive dominant and hence non-negative on $[0,1]^{2}$. Hence
$4-\Theta_{xx}\geq 0$ when $s=12$ and $(x,y)\in\Psi_{4}$. $\spadesuit$

######  Lemma 13.5 (C222)

$|\Theta_{ss}|\leq 1/64$ on $[13,16]\times\Psi_{4}$.

Proof: Let $\psi(s)=b^{-s}$. Let $\beta=(1.3,\sqrt{2},\sqrt{3})$ and
$\gamma=(440,753,4184)$. We first establish the following bound:

$0<\min_{b\geq\beta_{j}}\psi_{ss}(s,b)\leq 1/\gamma_{j},\hskip
30.0ptj=1,2,3,\hskip 30.0pt\forall s\geq 13.$ (141)

As a function of $s$, and for $b>1$ fixed, $\psi_{ss}(s,b)=b^{-s}\log(b)^{2}$
is decreasing. Hence, it suffices to prove Equation 141 when $s=13$. Choose
$b\geq 1.3$. The equation $\psi_{ssb}(13,b)=0$ has its unique solution in
$[1,\infty)$ at the value $b=\exp(2/13)<1.3$. Moreover, the function
$\psi_{ss}(13,b)$ tends to $0$ as $b\to\infty$. Hence the restriction of the
function $b\to\psi_{ss}(13,b)$ to $[b,\infty)$ takes its maximum value at $b$.
Evaluating at $b=1.3,\sqrt{2},\sqrt{3}$ we get Equation 141.

For $x,y\in[43/64,1]$ we easily check the inequalities

$A(-1,x)\geq 3,\hskip 15.0ptB(-1,x)\geq 2,\hskip
15.0ptC(-1,x,y)\geq(1.3)^{2}.$

The quantities on the left are the square distances of the various pairs of
points in the corresponding configuration on $S^{2}$. From this analysis we
conclude that the $10$ distances associated to a $5$-point configuration
parametrized by a point in $\Psi_{4}$ exceed $1.3$, and at least $6$ of them
exceed $\sqrt{2}$, and at least $2$ of them exceed $\sqrt{3}$. The same
obviously holds for the TBP.

Now, $10$ of the $20$ terms comprising $\Theta_{ss}(s,x,y)$ are positive and
$10$ are negative. Also, for the terms of the same sign, all $10$ of them are
less than $1/440$, and at least $6$ of them are less than $1/753$, and at
least $2$ of them are less than $1/4184$. Hence, by Equation 141, we have the
final bound $|\Theta_{ss}|\leq(4/440)+(4/753)+(2/4184)<1/64.$ $\spadesuit$

Write $I=[s_{0},s_{1}]$ and $Q=[x_{0},x_{1}]\times[y_{0},y_{1}]$. Choose
$(s,x,y)\in X=I\times Q$. Taylor’s Theorem with remainder tells that for any
function $f:[a,b]\to\mbox{\boldmath{$R$}}$ and any $x\in[a,b]$ we have

$f(x)\geq\min(f(a),f(b))-\frac{1}{8}\max_{[a,b]}|f^{\prime\prime}|.$

Applying this result $3$ times, using Lemmas C221 and C222, we have

$\Theta(s,x,y)\geq\min_{i}\Theta(s_{i},x,y)-|I|/512\geq\min_{i,j}\Theta(s_{i},x_{j},y)-|I|/512-|x_{0}-x_{1}|/2\geq$
$\min_{i,j,k}\Theta(s_{i},x_{j},y_{j})-|I|/512-|x_{0}-x_{1}|/2-|y_{0}-y_{1}|/2=\min_{v(X)}\Theta-|X|_{1}/512-|X|_{2}.$

This completes the proof of Lemma C22.

##  14 Endgame: Proof of Lemma C3

###  14.1 Reduction to a Simpler Statement

We carry over the notation from the previous two chapters. In particular, we
define $\Theta$ as in Equation 132. We use the same equation for ${\cal
E\/}_{s}(t,t)$, in terms of the functions $A,B,C$, as in Equation 138. Recall
that $I=[55,56]/64$. Let $t_{0}=55/64$ be the left endpoint of $I$. We claim
that

$\Theta_{stt}(15,t,t)<0,\hskip 30.0pt\forall t\in I.$ (142)

We compute that

$\Theta_{st}(15,t_{0},t_{0})<0,\hskip
30.0pt\Theta_{s}(15,t_{0},t_{0})<-2^{-7},$ (143)

and these conditions combine with Equation 142 to show that

$\Theta_{s}(15,t,t)<-2^{-7}.\hskip 30.0pt\forall t\in I.$ (144)

######  Lemma 14.1 (C31)

$|\Theta_{ss}|\leq 2^{-6}$ on $[13,16]\times\Psi_{4}$.

This is just Lemma C222. By Lemma C31 we have

$|\Theta_{ss}|\times|15_{+}-15|\leq 2^{-6}\times\frac{25}{512}<2^{-7}.$ (145)

Hence $\Theta_{s}(s,t,t)$ varies by less than $2^{-7}$ as $s$ ranges in
$[15,15_{+}]$. Hence $\Theta_{s}(s,t,t)<0$ for all $s\in[15,15_{+}]$ and all
$t\in I$. This is Lemma C3.

Now we prove Equation 142. The file LemmaC3.m does the calculations for this
proof. Because the $s$-energy of the TBP does not depend on the $t$-variable,
we have

$\Theta_{stt}(15,t,t)=2A_{stt}|_{s=15}+4B_{stt}|_{s=15}+4C_{stt}|_{s=15}.$
(146)

Call the three functions on the right $\alpha(t)$, $\beta(t)$, and
$\gamma(t)$. To finish the proof, we just need to see that each of these is
negative in $I$. We write $f\sim f^{*}$ if

$\frac{f}{f^{*}}=2^{u}t^{v}(1+t^{2})^{w}(2+t^{2}+t^{-2})^{x}$

for exponents $u,v,w,x\in\mbox{\boldmath{$R$}}$. In this case, $f$ and $f^{*}$
have the same sign.

######  Lemma 14.2 (C32)

$\beta<0$ on $I$.

Proof: Taking $(u,v,w,x)=(-14,0,11/2,0)$ we have $\beta\sim-\beta^{*}$,

$\beta^{*}(t)=(-2+30\log(2))+t^{2}(-58+420\log(2))-15(1+14t^{2})\log(1+t^{2}).$

Noting that $\log(2)=0.69...$ we eyeball $\beta^{*}$ and see that it is
positive for $t\in I$. The term $+420\log(2)t^{2}$ dominates. Hence $\beta<0$
on $I$. $\spadesuit$

######  Lemma 14.3 (C33)

$\gamma<0$ on $I$.

Proof: Taking $(u,v,w,x)=(-41/2,-16,12,1/2)$ we have $\gamma\sim-\gamma^{*}$,

$\gamma^{*}(t)=(-31+360\log(2))+\underline{t^{2}(56-585\log(2))}+t^{4}(-29+315\log(2))+$
$15(-8+13t^{2}-7t^{4})\log(2+t^{2}+t^{-2}).$

We have $\gamma^{*}(55/64)>2^{4}$ and we estimate easily that
$\gamma^{*}_{t}>-2^{10}$ on $I$. Only the underlined term has negative
derivative in $I$. Noting that $I$ has length $2^{-6}$, we see that
$\gamma^{*}$ cannot decrease more than $2^{4}$ as we move from $x_{0}$ to any
other point of $I$. Hence $\gamma^{*}>0$ on $I$. Hence $\gamma<0$ on $I$.
$\spadesuit$

######  Lemma 14.4 (C34)

$\alpha<0$ on $I$.

Proof: Taking $(u,v,w,x)=(-29,-14,10,3/2)$ we have $\alpha\sim-\alpha*$,

$\alpha^{*}(t)=\gamma^{*}(t)+\delta^{*}(t),\hskip 20.0pt\delta^{*}(t)=15\log
2\times(8-13t^{2}+7t^{4}).$

We see easily that $\delta^{*}>0$ on $I$. So, from Lemma C33, we have
$\alpha^{*}>0$ on $I$. Hence $\alpha<0$ on $I$. $\spadesuit$

##  15 References

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# Evaluation of noise limits of a precision ADC for direct digital signal
integration of magnetic measurements

R. Cavazzana  M. Gottado  A. Rigoni  Consorzio RFX, Corso Stati Uniti 4,
Padova 35127, Italy

###### Abstract

The characterization of laboratory plasma instabilities, magnetic reconnection
and turbulence associated phenomena, require the simultaneous signal sampling
from arrays of magnetic sensors (hundreds or even thousands) to obtain spatial
resolution, with several hundred kHz for time resolution. Magnetic
measurements based on pick-up (Mirnov) coil are quite common in experimental
pulsed devices for plasma research, thanks to their simplicity and
reliability. Being the signal from this type of sensor proportional to the
time variation of the magnetic field ($dB/dt$), it has to be time-integrated
to recover the instant value of magnetic field. Depending on the required
integration time usually either analog integrators or chopped integrators are
used. However these solutions tend to limit the frequency bandwidth in the kHz
range, they are not easy to design and build, and require additional fast
channels to directly acquire the $dB/dt$ signal to recover the plasma dynamic
features at higher frequencies. In this paper we evaluate the feasibility of
using a direct single channel precision ADC to allow the simultaneous
acquisition of $dB/dt$ measurements and to provide the integrated B
measurement by means of digital integration on the new RFX-mod2 device. On
this purpose we interfaced an existing ADC-module to a Xilinx Zynq FPGA, in
order to evaluate the intrinsic noise and to investigate the feasible
integration window of this configuration. The result opens the door to a
compact, cost-effective and reliable acquisition system, usable for
simultaneous real-time control and transient signal recording, scalable from
tenths to thousands channels, applicable to a broad class of pulsed plasma
experimental devices.

##  I Introduction

RFX-mod Sonato  _et al._ (2003) Riva  _et al._ (2018) is a medium size
toroidal plasma multi-configuration machine (major radius $R=2.0m$, minor
radius $a=0.46m$, operated up to 2 MA current reversed field pinch (RFP)
configuration or 0.5 T tokamak). The experiment is now being further upgraded
to RFX-mod2 Peruzzo  _et al._ (2018): one of the major step foreseen is a
substantial improvement of the magnetic measurement system. The total number
of the new magnetic pick-up coil sensors will be increased with the aim of
improved spatial resolution Marchiori  _et al._ (2017a). Moreover the sensors
will be moved inside the vacuum vessel widening their usable signal bandwidth
up to 200 kHz. These desirable characteristics imply demanding requirements
for the acquisition system. The extension of the present analog integration
system Pomaro and Basso (2005), followed by two separate sets of ADC channels,
one for precision off-line transient data sampling and one for real-time
control, would be costly, requiring several rack cubicles and severely
limiting the useful bandwidth. Thus a more compact and cost effective solution
is being investigated. A possible one could be based on the ATCA-MIMO-ISOL
Carvalho  _et al._ (2010) architecture originally developed for the plasma
column vertical stabilization of the JET tokamak experiment and already tested
on RFX-mod Manduchi  _et al._ (2012). In principle this architecture could be
suitable for performing the numeric integration need for real-time plasma
control, while simultaneously recording the dB/dt signals suitable for the
study of the fast MHD dynamic processes taking place into the plasma Zuin  _et
al._ (2009) Innocente  _et al._ (2014). The acquisition channels of this
architecture are isolated input modules using precision 18-bit, 2 MSamples/s
SAR ADC coupled to an FPGA which then routes the data flow to the
processing/storage servers through a PCI-e interface. These characteristics of
high sensitivity and high serial throughput, combined with the galvanic
insulation of the input section of each module make this solution very
attractive. However for our purposes, the ATCA/PCI-e architecture is still
expensive, complex to manage, and requires substantial software and firmware
development. With this in mind we decided to switch to simpler and more
flexible hosting interface, while trying to keep the existing isolated ADC
module. The use of the Xilinx Zynq system on chip, integrating both FPGA and
CPU cores, allows to set-up a self-contained acquisition board with a drastic
simplification of the system, providing both the autonomous local storage for
transient recording function, as well as the ability to send real-time
decimated and integrated data to the control servers through standard gigabit
Ethernet interface. To assess the feasibility of this proposal, we developed
an interface adapter between the MIMO ISOL ADC module and a commercial
Redpitaya Zynq board, and then we investigated the limits of the direct
numeric integration technique with presently available electronic components.
The goal is to check to what extent a directly acquired signal from the RFX-
mod2 probes can be digitally integrated. Even if applied to this specific
case, the results obtained here could be used as guidelines for other similar
applications as well. The paper is organized as follows: the dynamic range is
defined at first from the assumption of the lowest and the highest signal
expected. This has been set led by the experience of RFX-mod operation, using
the collected data on the existing analog system as a baseline. The structure
of the ADC and the FPGA interface developed to gather data are then briefly
described. A characterization of the acquired signal is than provided to
verify the feasibility of this approach.

##  II Expected signal range

The dynamic range requirements for the signal acquisition system ask for a
careful evaluation of both the minimum detectable and the maximum expected
signal, keeping in mind that the signal has to be numerically integrated. The
maximum level of the signals we are going to acquire can be straightforwardly
derived from experimental measurements already carried out on RFX-mod, which
can be used to estimate the level of the signal once the probe characteristics
and its measurement chain are defined. The probes used up to now on RFX-mod
Bolzonella  _et al._ (2003) Pomaro and Basso (2005) present a typical sensing
area of 0,025 m2 and proved to offer a good compromise between compactness,
signal sensitivity and usable bandwidth. The new probes of RFX-mod 2 will
maintain these basic geometric and electric features, while the materials and
the manufacturing process will be modified to comply with the vacuum
installation Marchiori  _et al._ (2017b). An example of the expected $dB/dt$
signal source and its spectrum, obtained by wide-band in-vessel probes
installed on RFX-mod, are shown in Fig.1. This signal will be later used, to
set the maximum signal level.

Figure 1: dB/dt signal of the toroidal field component from an in-vessel wide
bandwidth probe from a reversed field pinch pulse at 1.5 MA in RFXmod-2 (right
signal, left spectrum).

On the other hand the “floor” signal requirement involves the sensitivity to
slowly changing magnetic field at low intensity. Magnetized plasmas are prone
to amplify the spatial resonant component of an external field. Even an
unwanted small field variation (the so called “field errors”) can have large
effects on plasma behavior [compass scan]. The ability to detect these
features is thus needed for example to allow the characterization of
misalignments of the sensors and the structure of these small field
components. As a comparison case, the performance of noise level and
integrator drift of the actual conditioning system of RFX-mod is shown in Fig.
2. The final integration error is of the order of 0.6 mT, with the typical
signal range to be measured which spans from 10 mT to 0.5 T. Most of the
integration drift error can be corrected in post-processing, but it is
necessary to keep it into an acceptable range to allow the use of these
signals for real-time control purposes.

Figure 2: Example of integrated signals (left) and distribution of final
integration error of the analog integrators of RFX-mod after 10 sec, from
toroidal field sensors on 8 “dry” pulses (1510 signals),(right).

As the minimum detectable reference signal, we choose a trapezoidal 1 mT field
with a slow trapezoidal evolution: 1 sec. ramp, 1 sec. state, 1 sec. ramp-down
is shown on Fig. 3.

Figure 3: desired minimum detectable signal scenario

The low signal level limit and the integration capability are severely
affected by the $1/f$ noise of the circuit, which is poorly documented in the
manufactures’ ADC data sheets and sometimes not well understood in the
literature. These issues will be discussed later

##  III ADC module structure

Figure 4: block schematic of the ADC module

The analog to digital conversion inside each of the ATCA-MIMO-ISOL board ADC
modules is delegated to an Analog Devices AD7641, a 18-bit SAR converter that
acquires signals from a fully differential input in the range 2.048 V at the
maximum rate of 2 MSamples/s. A block scheme of the module is shown in Fig. 4:
the analog input is initially filtered by a 1 pole, 100 kHz passive component
connected to a differential amplifier THS4520 used as input range adapter; the
AD7641 is configured to operate using serial communication protocol and the
digital signals are delivered to the FPGA by means of a digital isolator. The
power is supplied through an isolated DC/DC converter which is synchronized
with the ADC conversion start command in order to minimize the noise spikes
generated during power switching phase.

##  IV ADC digital interface

The module has been subsequently interfaced with a standard Xilinx Zynq FPGA
board, developing a custom HDL code with the purpose of streaming data output
to a proper embedded GNU Linux driver. Fig. 5 shows the schematic design of
the bus connections of the FPGA logic.

Figure 5: block scheme of the implemented Zynq Xilinx AXI driver

Two main IP symbolic components are visible: the first one from the left of
the sketch represents the processor entity, the second is related to the
module digital acquisition. The processor unit presents two bus connections,
externally linked to the DRR and fixed I/O hardware components, and one
internally connected by the AXI_GP0 general purpose bus to the other custom
logics. Here all the IRQ and clock connections have been hidden to improve the
schema readability but all the components are clocked by 125MHz Zynq internal
multiplier and the FIFO drives an interrupt of the PS to inform the driver of
the status changes. The custom logic named scc52460_fifo encloses all the
implemented design exploiting a 32-bit Xilinx AXI-Stream FIFO core with the
input connected to a chain of custom cores performing in sequential order: the
acquisition from module LVDS serial port, a configurable decimation of
acquired data, and a configurable length packetization of the output stream.
The communication protocol selected in the schematic hardware design of ADC
module is the AD7641 serial master WARP mode connection with a further
external flip-flop connected to the clock output that makes the signal
transaction dual edge synchronized (rising and falling). This aims at
achieving, likewise the DDR protocol specification, the same throughput with
half of the frequency that in this way matches the one of the signal data. The
read implementation approach is then a double buffers with two process that
have the sensibility to the respectively rising and falling edge of the
SCLK_in clock input port. On the other side a proper kernel module has been
developed implementing the Xilinx FIFO core communication protocol axi (2016),
in this way the kernel spools data from the FIFO buffer to the user allocated
memory. Other solutions could be certainly applied to read data from the
device, for example the Zynq is equipped with two high speed embedded DMA
channels that are able to write data directly into DDR relieving the CPU of
the mem-copy load. The proper transfer architecture should be always chosen in
relation with the overall project complexity and goals. The system has been
then tested for long lasting acquisition pulses at full speed 2 MSPS output
and it resulted to be able to work properly in continuous mode.

##  V ADC module analog characteristics

The module frequency response has been characterized by means of an HP33120A
generator, with two different techniques: a) providing a series of sine waves
at fixed frequency at 10 Vpp and b) with a white noise output. The result is
shown in Fig. 6, where the effect of the 100 kHz low pass input filter and the
residual aliasing are clearly visible. In the following the Welch method has
been used to compute an estimate of the power spectral density (WPSD i.e.
Welch Power Spectral Density hereafter) by successively dividing the data into
overlapping segments, computing a modified periodogram for each segment and
finally averaging the obtained data.

Figure 6: Frequency response of ADC module (spectral levels are amplitude
normalized).

In view of integrating the input signal, we carried out a detailed
characterization of the noise seen by the ADC, for every one of the different
configurations explored. Each run consisted of 8 noise snapshots, each one
lasting 40 s (8∙107 samples). In order to reduce extraneous components, known
noise sources have been either switched-off or kept far from the ADC board.
The modules used are configured to have an input signal range of 10.3 V. The
noise has been characterized in time (Fig. 7) and frequency domains (Fig. 9)
marked with red line. The noise generated by the DC/DC converter is visible in
the time domain as a series of synchronous spikes (7a), which are reflected in
a histogram of the noise amplitude with three peaks (7b).

Figure 7: Noise sampling (left) and noise histogram (100,000 samples, right)
of the module in standard configuration with 50 Ohm terminated input. Note the
repetitive synchronous spikes and the resulting multimodal distribution due to
the DC/DC converter residual noise.

The frequency domain analysis shows two peaks due to isolation DC/DC
converter, one at 500 kHz and its 1 MHz harmonic at Nyquist frequency, while
at low frequency the spectrum is dominated by the 1/f noise. In order to
better identify the noise sources inside the circuit, the DC/DC converter has
been replaced by a linear power supply. Although the histogram of the sampled
noise becomes comparable to the ADC specifications the spectrum improvement
remains restricted to specific high frequencies that are easily filtered out.

Figure 8: Noise sampling (left) and noise histogram (100,000 samples, right)
with 50 Ohm terminated input and the module fed by linear power supply.

In the final circuit configuration the input amplifier has been removed as
well, this time obtaining a reduction of the 1/f noise by a factor 20. A
comparison of the overall noise spectrum with the final configuration is
proposed in (Fig. 9), where the red spectrum represents the complete noise
recorded without any intervention in the module, and the blue spectrum is the
result of the signal acquisition with the DC/DC converter and the linear
supply removed.

Figure 9: Noise spectra (spectral levels are power normalized) after
disconnecting the input amplifier; left panel circuit fed by its DC/DC
converter, right panel by linear power supply – note the suppression of the
frequency peak at 500 KHz.

##  VI Integration of noise and expected drift error

As already mentioned in the chapter 1, one of the main goals of the present
work is the complete replacement of existing analog integrators (Fig. 2) with
direct numerical integration of the digitally acquired signal. For short time
integration, the noise spectrum involved is in the high frequency range, i.e.
dominated by the white noise. In this case the numerical integration of a
sampled sequence $s_{i}$ is a well defined case, being equivalent to a
discrete random walk in which the distribution of the steps $s_{i}$ has zero
mean with a finite variance $\sigma^{2}$. Thus given the final integral sum
after $n$ steps $S_{n}=\sum_{i=0}^{n}s_{i}$, its variance is simply
proportional to number of steps: $E|S_{n}^{2}|=\sigma^{2}n$.

In the case at issue the integration time lasts several seconds and the
desired frequency range extends far below the typical 1/f corner of standard
electronic components. The integration is then dominated by the $1/f$ noise
level. Contrary to the white noise, for this case there is no simple relation
linking the noise level to the standard deviation of the final integration
step. For this reason the problem has been tackled by means of a statistical
analysis based on experimental data. We performed the numerical integration on
a number of samples [quanti?] on the different circuit configuration
considered. As shown if Fig. 10, the estimated variance of the final value of
the integrated signal after 10s with the ADC module in standard configuration,
is about 4 ADC $\text{count}\cdot\text{sec}$.

Figure 10: Noise integrated signals from the module in standard configuration

This translates in an equivalent final error for the magnetic probes of 5 mT.
As appears from Fig. 11, besides its expected final value, the integration
error applied to the reference magnetic measurement case makes practically
impossible the reconstruction of the magnetic signal.

Figure 11: Expected signals with a probe connected directly connected to the
input module, with the minimum reference input signal. The signal appears
indistinguishable from the noise.

The case with the ADC without the input amplifier is reported in Fig. 12,
where the final error standard deviation is 0.33 ADC counts ∙ sec, which ten
times lower than the previous case.

Figure 12: Noise integrated signals from the module with the input amplifier
disconnected, fed by the DC/DC converter.

By adding the equivalent magnetic input reference signal to the noise, the
desired signal can be successfully identified with an overall acceptable error
and further corrected by suitable post processing.

Figure 13: Expected signals with the noise generated solely by the ADC input,
with the minimum reference input signal. The integrated signal is now clearly
visible.

This last test reported, i.e. the ADC without any input amplifier and with a
linear power supply, results in a standard deviation of 0.316 shown Fig. 14.
This implies that for practical purposes the use of synchronous DC/DC
converter does not affect the ADC conversion, at least with the purpose of
numerical integration.

Figure 14: Histogram of the noise integrated signals from the module with the
input amplifier disconnected, fed by the linear power supply. Note that there
is no practical gain in comparison with Fig. 12.

As briefly mentioned computational methods can be further applied to finally
recover the actual input signal from a very low frequency drift. A commonly
naive solution is to compute a standard linear interpolation of the noise
evaluated outside the interval of interest, where the magnetic field is known
to be steady, and subtract the interpolated offset during integration. Such
operation can be easily applied on the signal in Fig. 13, while is likely to
fail on the case of Fig. 11. This situation can be ascribed to the presence of
1/f noise components at relatively high frequencies, close to the signal band.

##  VII Tests with magnetic probe

As a final proof of concept the complete signal acquisition chain has been
deployed in a test bench environment to check if circuit noise is compatible
with the actual RFX probes. The test circuit used consists of a solenoid
inducing a controlled field to the magnetic probe. This was made by wrapping
40 turns of copper conductor of 2mm diameter in a coil that was estimated to
present a 0.1mH inductance. In order to obtain the correct impedance to the
circuit, a further resistive load of 2.5 Ohm was connected in series capable
of dissipating a power of 100W, supplying in test by 4A it will dissipate 40W.
The maximum current applicable to the total resistive load is 6.32A.

Figure 15: manual

The test result is shown in Fig 15, where the signal provided to the magnetic
probe was manually regulated using an external potentiometer changing the
current flowing into the generator coil. The direct acquired signal is
reported on the curve at the high part of the image while the integrated
result lies on the plot below. Here a successful magnetic field reconstruction
can be directly appreciated measuring a 1.25 mT fluctuation within a time
interval of about 3 s. The possible noise interpolation could exploit the
leverage of the signal drift of about 1 mT measured along the entire 10 s
signal to a reconstruction improvement of few $microT$.

##  VIII Conclusions

A complete acquisition chain has been built testing the feasibility of the
numerical integration approach to recover the RFX magnetic field from
solenoids probes. A complete isolated ADC module has been interfaced to an
embedded SoC DAQ board exploiting a Xilinx FPGA real-time numerical
elaboration. In conclusion it has been shown that a direct digital integration
appears feasible, even if it seems to require the state of the art components
to recover high dynamic range and wide frequency spectrum signals at the same
time, with a single channel. Both a simulated setup and an experimental
testbed has been set up to evaluate the ADC noise impact on the integration
results. Even using the best available commercial components, an accurate
evaluation of the ADC characteristics must lead the choice of technology
matching the desired input dynamic range and the bandwidth attenuation. The
need of high quality components has been shown to be related with the 1/f low
drift noise mainly coming from the ADC and the input amplifier electronics.
Although the 1/f noise seems to be the main issue for the integrated signals
in long duration pulses, the high frequency oscillation due to the insulation
DC/DC converter does not appear to create severe artifacts to the acquisition
that can be reconstructed by digital filters synchronized with the acquisition
system.

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