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##### From Explosively Driven Pulsed Power: Helical Magnetic Flux Compression Generators

Bruce L. Freeman and Andreas A. Neuber

## 3.1 Magnetic Flux Diffusion

Magnetic flux diffusion into the metal conductors of a magnetic flux compression generator is the most basic loss of magnetic flux in these systems. As long as the magnetic diffusion is such that the resistivity of the conductors may be treated as a constant for times of interest, the diffusion is considered linear. If the diffusion becomes large enough to affect the conductivity of the metals during time scales of interest (ohmic heating temperature increase increased resistivity increased diffusion), the diffusion is considered non-linear in nature.

### 3.1.1 Linear Magnetic Flux Diffusion

For deriving the magnetic flux diffusion into a conductor as a function of time, we begin with the general Maxwell equations:

 (3.1)
 (3.2)
 (3.3)
 (3.4)

where

 (3.5)

with

 (3.6)

where ? 0 = 4 ? 10 -7 Henry/meter, and

 (3.7)

with

 (3.8)

where ? 0 = 8.85 10 -12 Farads/meter.

To complete this description, Ohm's Law is

 (3.9)

where j is the current density, ? is the conductivity, and ? is the resistivity, the reciprocal of the conductivity. The resistivity is introduced and will be carried in the derivation to emphasize the more prevalent usage of the resistivity over the conductivity in the laboratory environment.

Now, we want to obtain an equation that describes the generalized propagation of the magnetic field into a conductor. By substituting (3.9) into (3.3) and taking the

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