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Named field expressions — formal definitions

All quantities below are registered as HFSS Fields Calculator named expressions by linc.calculator.register_all. Each one can be evaluated at any $(\text{solution},\ f,\ \phi)$ via linc.evaluate.eval_one.

0. Conventions

HFSS Driven Modal stores the complex E-field phasor $\vec{\tilde E}(\vec r)$ at every saved-field frequency. The real-time field at phase $\phi$ is

$$ \vec E(\vec r, \phi) ;=; \mathrm{Re}!\left{ \vec{\tilde E}(\vec r) , e^{,i\phi} \right}. $$

For an oriented curve $C$ with unit tangent $\hat t(\ell)$,

$$ V_C(\phi) ;=; \int_C \vec E(\vec r, \phi) \cdot d\vec\ell ;=; \int_C \vec E(\vec r, \phi) \cdot \hat t , d\ell, \qquad \tilde V_C ;=; \int_C \vec{\tilde E}(\vec r) \cdot d\vec\ell. $$

Note $V_C(\phi) = \mathrm{Re}{\tilde V_C,e^{i\phi}}$ and $V_C(0) = \mathrm{Re},\tilde V_C$.

1. voltage_<line> — phase-0 junction voltage

$$ V_i^{(0)} ;=; \int_{C_i} \vec E(\vec r, 0) \cdot d\vec\ell ;=; \mathrm{Re},\tilde V_{C_i}. $$

Real-valued. Sign is meaningful (the two junctions can be in-phase or anti-phase). Calculator stack:

E -> 0 -> AtPhase -> line -> Tangent -> Dot -> line -> Integrate

2. <line>_sign — voltage sign

$$ \sigma_i ;=; \frac{V_i^{(0)}}{,\bigl|V_i^{(0)}\bigr|,} ;\in; {-1,,+1}. $$

Calculator stack:

voltage_<line> -> voltage_<line> -> Abs -> /

3. signed_voltage_<line> — signed phasor magnitude

The true phasor amplitude is

$$ \bigl|\tilde V_{C_i}\bigr| ;=; \sqrt{, \Bigl(\textstyle\int_{C_i} \mathrm{Re},\vec{\tilde E}!\cdot d\vec\ell\Bigr)^{!2} ;+; \Bigl(\textstyle\int_{C_i} \mathrm{Im},\vec{\tilde E}!\cdot d\vec\ell\Bigr)^{!2} ,}. $$

The signed magnitude re-attaches the phase-0 sign:

$$ \hat V_i ;=; \sigma_i , \bigl|\tilde V_{C_i}\bigr|. $$

Without re-signing, $|V_1| + |V_2|$ could never cancel even for a perfectly antisymmetric drive. The sign carries the phase-relationship between $V_1$ and $V_2$ that the absolute value throws away.

4. JJ_flux — SQUID-loop flux in units of Φ₀

For a SQUID-loop sheet $S$ with unit normal $\hat n$,

$$ \frac{\Phi}{\Phi_0} ;=; \frac{1}{\Phi_0} \int_{S} \bigl|,\vec{\tilde B}(\vec r) \cdot \hat n,\bigr|, dA ;=; \frac{1}{\Phi_0} \int_{S} \bigl|,\mu(\vec r),\vec{\tilde H}_{\text{smooth}}(\vec r)\cdot\hat n,\bigr|, dA, $$

with $\Phi_0 = h/2e \approx 2.0678 \times 10^{-15}\ \text{Wb}$. The calculator first smooths $\vec{\tilde H}$ to reduce mesh-edge artefacts, then multiplies by the local permeability $\mu$ to get $\vec{\tilde B}$, takes the normal component, the complex magnitude, integrates, and divides by $\Phi_0$. Calculator stack:

H -> Smooth -> ClcMaterial(mu, mult) -> surf -> Normal -> Dot -> CmplxMag
  -> surf -> Integrate -> 2.0678e-15 -> /

5. asym_voltage — drive asymmetry

$$ \boxed{;; \eta_{\text{asym}} ;=; \frac{\bigl|,\hat V_1 + \hat V_2,\bigr|}{\bigl|,\hat V_1 - \hat V_2,\bigr|} ;=; \frac{|\text{common-mode amplitude}|}{|\text{differential amplitude}|}. ;;} $$

$\eta_{\text{asym}}$ drive character
$\to 0$ pure differential ($V_1 \approx -V_2$) — wanted inductive flux mode
$\sim 1$ balanced — common and differential equal
$\gg 1$ common-mode dominates — capacitive parasitic, wrong mode excited

Calculator stack:

signed_V1 -> signed_V2 -> +
  signed_V1 -> signed_V2 -> - 
  -> / -> Abs

Why this diagnoses the linearized SQUID

A linearized-SQUID coupler requires a purely antisymmetric drive: the flux line's $\vec{\tilde E}$ should make $\hat V_1 \approx -\hat V_2$, so $\hat V_1 + \hat V_2 \to 0$ and $\eta_{\text{asym}} \to 0$. That's the inductive AC-flux mode that threads the SQUID loop.

A capacitive parasitic instead drives both pads to the same potential ($\hat V_1 \approx \hat V_2 \approx V_{\text{cm}}$), making $\hat V_1 - \hat V_2 \to 0$ and $\eta_{\text{asym}} \to \infty$. So $\eta_{\text{asym}}(f)$ tells you exactly where the geometry stops behaving inductively.