# Foundations: fields, time signals, and grids

Start here for the Yee update, propagation, power flow, ring-up/ringdown, and numerical-grid behavior.

[← Back to the complete example atlas](../index.md)

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:::{grid-item-card} Vacuum dipole — finite energy + PML decay
:link: ../cases/analytical_vacuum_energy
:link-type: doc

<span class="metric-pass">PASS</span> · benchmark page · analytical · fields_finite
:::
:::{grid-item-card} Pulsed vs CW multilayer transmission consistency
:link: ../cases/analytical_pulsed_vs_cw
:link-type: doc

<span class="metric-pass">PASS</span> · benchmark page · analytical · max_rel_err
:::
:::{grid-item-card} Plane-wave phase velocity + impedance (vacuum & dielectric)
:link: ../cases/analytical_plane_wave
:link-type: doc

<span class="metric-pass">PASS</span> · benchmark page · analytical · pytest_exit
:::
:::{grid-item-card} Phasor vs time-domain Poynting flux consistency
:link: ../cases/analytical_phasor_poynting
:link-type: doc

<span class="metric-pass">PASS</span> · benchmark page · analytical · pytest_exit
:::
:::{grid-item-card} Mildly stretched grid vs uniform
:link: ../cases/analytical_nonuniform_grid
:link-type: doc

<span class="metric-pass">PASS</span> · benchmark page · analytical · pytest_exit
:::
::::


```{toctree}
:hidden:
:maxdepth: 1

../cases/analytical_vacuum_energy
../cases/analytical_pulsed_vs_cw
../cases/analytical_plane_wave
../cases/analytical_phasor_poynting
../cases/analytical_nonuniform_grid
```