# Static fields in a dynamic code

The Aether code (scheduled for release on September 30) culminates 12 years of my research on fast numerical methods for electrodynamics. The frequency-domain techniques were adapted from a program called RedWall that I developed for NIST. The time-domain strategies were tested in EMP3.

The application of Aether to pulsed-power devices is particularly interesting to me because I spend a couple decades designing high-voltage generators and doing research with them. One of the main simulation issues posed by my colleagues was how to model a static-charged device (like a Blumlein line) with a code where pulses are created by time-varying currents. The initial state has H equal zero everywhere and nonzero values of E.

In the past I found some partially-successful approaches:

• Treat the state of a charged transmission line as overlapping traveling waves with amplitude V0/2 and generate the positive-going wave with a current source.
• Set initial values of electric field from known formulas.

These methods worked only for special structures with simple electric field variations. The second method usually resulted in high-frequency transients because of the discontinuous initial fields.

Two new features in Aether completely resolve the static-charge issue.

1. The code can import electrostatic solutions from HiPhi to set the initial electric field distribution. There is an exact correspondence to the structure, making it possible to treat structures like capacitors where fields are not described by a simple formula. The continuous field variations eliminate the problem of spurious transients.
2. The user can specify time-dependent conductivities to represent switches. For a physically-valid solution, I found that it was essential that the electrostatic field correspond exactly to the initial state of the electromagnetic solution. The implication is that all switch elements must be open-circuits at t = 0.0 (i.e., σ(0) = 0.0). An additional benefit of conductivity variation is that it is possible to model the response time of real switch elements.

The first figure below shows a benchmark test with vacuum parallel-plate transmission lines. The solution includes a charge line (25 cm in length) and terminated output line separated by a switch region. The left-hand figure shows equipotential lines of the HiPhi solution with a continuous field distribution across the switch region. The initial electric field in the line is Ey = 50 V/m. In the Aether solution, the conductivity of the switch region changes smoothly from 0.0 S/m to 25.0 S/m in 1.0 ns. In the final state, the switch impedance is 25 times smaller than the line impedance. The figure on the right shows |E| in the Aether solution at t = 1.0 ns. As expected, a voltage wave with Ey = 25 V/m moves into the output line and a wave with Ey = -25 V/m moves into the charge line.

The next figure shows a three-dimensional example that is a real test of the code. The goal is to determine the effect of strong voltage reversal on the lifetime of a dielectric sheet. The plate and inductor are charged to a high static voltage. A switch shorts the electrodes, generating a oscillating voltage across the sheet. The assembly is immersed in transformer oil inside a grounded tank. Three-dimensional numerical methods are essential for this calculation for three reasons:

1. The initial electric field distribution is quite complex.
2. The effect of the tank walls and electrodes makes it difficult to estimate lumped element parameters.
3. Transit-time effects play a large role in determining the electric field in the test insulator.

In the electromagnetic solution, the switch has an initial conductivity of zero and rises to 25.0 S/m in 5 ns. The time-variation of electric field inside the dielectric is plotted in the third figure. The waveform is approximately a sinusoidal function, as expected for an LC circuit. In contrast to the lumped-element solution, the electric field is larger on the negative cycle. This result reflects transit-time effects with pulse interference. Plots of |H| at different times show that the magnetic field energy moves back and forth along the helical inductor. We could determine capacitance and inductance values of the assembly with the RF mode of Aether. For a rough estimate, note that the dielectric gap has spacing 0.01 m and approximate area 7.85E-4 m2. The dimensions correspond to a capacitance of about 40.0 pF. The oscillation period is 20 ns, implying an inductance of 0.25 μH.