Complex frequency versus complex propagation constant modeling and Q-balancing in periodic structures

The paper compares the complex frequency modeling (eigenmode analysis) and the complex propagation constant modeling (drivenmode analysis) of periodic structures based on a generalized unit cell model, and derives the mathematical relation between these two approaches. Moreover, it shows that only under the condition of Q-balancing, the phase-frequency responses (dispersion diagrams) calculated by the two approaches are identical. The Q-balancing condition corresponds to the Heaviside condition for distortionless uniform lossy transmission lines, although it applies here to the different context of periodic structures. This study may provide deeper insight and understanding into the eigenmode analysis of periodic structures, which is commonly used by commercial simulators.