Via coupling within parallel rectangular planes

Convenient formulas which characterize via or probe coupling are presented for both frequency-domain mutual impedance and time-domain voltage waveforms. These expressions are valid for arbitrary via locations and take into account all edge reflections. Skin effect is included within the frequency domain formulas. These results have application to power/ground plane "bounce" in digital circuits. They also provide insight into more general transient behavior. A direct closed-form Fourier inversion back into the time domain is detailed for the negligible loss case. To determine the behavior for finite rise-time excitations, the step response is integrated term-by-term to obtain a ramp response. Responses of other excitations, such as a finite rise-time step current, and various pulse currents, are easily obtained through superposition of appropriately delayed ramp responses. This process is extended to include piecewise-quadratic current excitation waveforms, thereby avoiding all slope discontinuities. These somewhat more involved expressions give "smoother" results which may have greater esthetic appeal.