{'fraction_non_alphanumeric': 0.013029315960912053, 'fraction_numerical': 0.0, 'mean_word_length': 5.402777777777778, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Efficient Numerical Modal Solutions for RF Propagation in Lossy Circular Waveguides

The propagation of electromagnetic waves is reconsidered in the context of cylindrical tunnels through material of uniform electrical properties. In the absence of internal conductors, the wave solutions reduce to well-known analytic expressions, requiring the solution of a single transcendental equation in complex phase space. Any given tunnel configuration has a wide range of solutions in phase and frequency space. Identifying these solutions can be a significant numerical challenge. Both the Newton-Raphson method and the winding number technique have been described in previous publications, and each can be prone to missing solutions. This paper reviews these two solution techniques and develops a combination of the two to provide a fast and efficient procedure for locating the roots in well-defined regions of phase space.