Power Integrity Analysis and Management for Integrated Circuits

Appendix B. Derivation of the Helmholtz Equation for Planar Circuits

B.1 Derivation of the Helmholtz Equation for Planar Circuits

Equation (5-24), the Helmholtz equation for planar circuits, represents a combination of Faraday’s law [Equation (5-15a)] and Ampere’s circuit law [Equation (5-15b)] with other assumptions related to planar circuits.

Applying the curl operator to both sides of Faraday’s law, the following equation is obtained:

∇ × ∇ × E = −jωμ∇×H

The vector identity for the curl of curl is used to obtain:

∇(∇•E) − ∇²E = −jωμ∇×H

Ampere’s circuital law is used to substitute for ∇ × H in the above equation:

∇(∇•E) − ∇²E = −jωμ(j + jωεE)

In the absence of any free charge, the ∇ •E term vanishes according to Gauss’s law [Equation (5-15c) ...