A periodic lossless line can consist of equal-length sections with impedances . Its semi-infinite input is a fixed point of the two-section impedance transformation. The physical solution selects a forward-energy or decaying Bloch wave. At special frequencies where the cell matrix is the identity, the fixed-point equation alone loses uniqueness; limiting absorption or a termination specifies the response.
For a real-frequency lossless reciprocal periodic line, a cell transfer matrix has determinant one. If half its trace lies outside , the Bloch multipliers are real reciprocal numbers rather than unit-modulus propagation factors. The physical interior field decays and carries no net power to infinity. A real reference line is then totally reflected by a semi-infinite structure; a finite structure can still tunnel.
The voltage-to-current ratio of a selected Bloch eigenmode at a specified position within a periodic cell. It depends on the chosen cell boundary, unlike the cell's propagation eigenvalues. In a pass band a physical semi-infinite mode carries positive real power; in an ideal stop band its input impedance is reactive and the mode decays into the line.
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