Fields

The four equations, in words

\[\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}\]

\[\oint \vec{B} \cdot d\vec{A} = 0\]

\[\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}\]

\[\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}\]

Currents always run in loops

Don't ask "where does the signal go?"

Ask "where does it come back?"

The capacitor falls out

The inductor falls out

Every wire is both

The antenna does not radiate

What this buys you on-chip

• On-chip inductors: small loops, lossy substrate - the fine print charges twice
• Decoupling is loop design, not capacitor selection
• Ground bounce is Figure 1(b) at chip scale
• A radio is this chapter, run on purpose

Summary

• Circuit theory is Maxwell with the fine print deleted; the fine print still bills you
• Charge conservation means every current closes a loop - the return path is not optional, only its location is
• A capacitor is a deliberately good gap: displacement current carries the loop across
• An inductor is a deliberately good loop: every loop encloses flux, and a changing current fights its own flux
• At high frequency the return current hugs the signal wire, because the smallest loop has the smallest inductance
• The antenna metal only sets boundary conditions: field loops that detach beyond lambda/2pi carry the power, and radiation resistance is their receipt

Would you like to know more?

• Feynman Lectures on Physics, Volume II, chapter 18
• Howard Johnson & Martin Graham, High-Speed Digital Design: A Handbook of Black Magic