= Parallel-path model of a superconducting ceramic
{title2=$R_{\mathrm{eff}}=R_n r/(R_n+r)$}
Consider an original <electrical circuit> model between two equipotential cross-sections. A normal path has <electrical resistance> $R_n>0$. Let $b\in\{0,1\}$ indicate whether an additional path connects those same cross-sections; when connected it has <electrical resistance> $r\geq0$. Ideal voltage contacts measure those cross-sections. This collapses a spatial network to two parallel branches; it is not a fit to the <Bednorz-Müller experiment>.
For $r>0$, both branches have the same <electric potential difference> $V$. <Ohm's law> and the <Kirchhoff node law> give $I=V/R_n+bV/r$, hence
$$
R_{\mathrm{eff}}=\begin{cases}
R_n,&b=0,\\
\dfrac{R_n r}{R_n+r},&b=1,\ r>0,\\
0,&b=1,\ r=0.
\end{cases}
$$
The last case follows directly from the ideal short's constraint $V=0$, provided it can carry the imposed <electric current> below its <superconducting critical current>. It does not follow by substituting $r=0$ into $V/r$. A normal bridge in series with otherwise superconducting pieces belongs in $r$; superconducting grains alone need not short the measured interval.
\Image[/superconducting-ceramic-connectivity-and-volume-fraction.png]
{title=Equal superconducting volume fractions can have different transport connectivity}
{description=Original chosen geometry and circuit example, not measured data or a historical fit. Top: blue superconducting regions between orange equipotential ends, taken uniform through the thickness, have superconducting <phase fraction> $f=0.08$. Bottom: two stipulated connectivity thresholds, $f_c=0.08$ and $0.60$, produce different apparent <electrical resistance> drops. Ideal zero values are displayed at a chosen instrument floor; these curves predict neither <magnetic susceptibility> nor a universal <percolation> threshold.}
Choose $R_n=1\,\Omega$. A connected bridge with $r=0.01\,\Omega$ gives $R_{\mathrm{eff}}=1/101\,\Omega\simeq9.901\,\mathrm{m\Omega}$, almost a hundredfold drop despite finite <electrical resistance>. An ideal connected superconducting filament instead gives zero in this model even when its volume is a small part of the specimen. Isolated superconducting islands can have the same superconducting <phase fraction> $f$ while leaving $b=0$. For the illustrative disconnected geometry we stipulate that the normal conducting network retains $R_n$.
To plot a chosen connectivity sequence, set $b(f)=0$ for $f<f_c$ and $b(f)=1$ for $f\geq f_c$, with $r=0$ on the connected branch. The two displayed values of $f_c$ describe different chosen architectures. A real <percolation> calculation would need an explicit network and spatial distribution; the scalar <phase fraction> $f$ does not determine $b$ by itself.
An instrument with unresolved <electric potential difference> bounded by $\delta V$ reports only $R_{\mathrm{eff}}\lesssim\delta V/|I|$. For chosen $I=1\,\mathrm{mA}$ and $\delta V=100\,\mathrm{nV}$ this floor is $10^{-4}\,\Omega$, assuming offsets and noise are included in the bound. The figure clips smaller model values to that floor. Increasing $|I|$ improves this formal bound only while <Joule heating> and the path's <superconducting critical current> remain controlled. None of these transport statements determines the superconducting <phase fraction> or establishes complete <Meissner effect> shielding.
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