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Vapor-Cooled Current Lead Calculator engineering
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Vapor-Cooled Current Lead Calculator

Superconducting magnet engineering: Calculate McFee optimum current lead geometry ($(I \cdot L / A)_{opt}$), cold-end conduction heat leak, and liquid helium boil-off consumption.

Operating Current & Lead Parameters

Optimum Lead Sizing & Cryogen Boil-Off

Optimum Cross-Section $A$
--
$\text{mm}^2$ (copper conductor)
Optimum Ratio $(IL/A)$
--
$\text{A}/\text{m}$ (McFee criterion)
Heat Leak at 4.2 K (Per Lead)
--
W (cold end conduction)
Specific Heat Leak $q/I$
--
mW / Ampere
LHe Boil-Off Consumption
--
liters / hour (total leads)
Conduction-Only Leak (Uncooled)
--
W (without vapor cooling)

Temperature Profile Along Current Lead

Position Along Lead $x / L$ (Cold 0 to Hot 1) Temperature $T$ (K) Optimum Vapor-Cooled Lead ($dT/dx|_{x=0} \approx 0$) Thermal Runaway / Burnout ($I > I_{opt}$)
In an optimally sized vapor-cooled lead, escaping cold vapor absorbs all internal Joule heat, flattening the temperature gradient at the cold end ($dT/dx|_{x=0} \to 0$).

The McFee Optimum Lead Formulation

Current leads introduce heat to a cryostat through combined conduction and Joule heating. Dr. R. McFee (1959) showed that an optimal balance occurs at a specific aspect ratio $(I L / A)$:

$$\frac{d}{dx}\left( k(T) A \frac{dT}{dx} \right) + \frac{I^2 \rho(T)}{A} - \dot{m} C_p \frac{dT}{dx} = 0$$ $$\left(\frac{I \cdot L}{A}\right)_{opt} \approx 2.8 \times 10^5 \text{ A/m} \quad (\text{for OFHC Copper})$$

Vapor cooling reduces the 4.2 K heat leak from $42 \text{ mW/A}$ (conduction only) down to $1.04 \text{ mW/A}$—a $40\times$ reduction! Modern High-Temperature Superconducting (HTS) leads use YBCO from 4.2 K to 77 K, dropping the 4.2 K heat leak by another order of magnitude ($< 0.1 \text{ mW/A}$).

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Frequently Asked Questions