What is the difference between a binomial and a Chebyshev multi-section transformer?

Binomial (maximally flat) and Chebyshev (equal-ripple) are two design approaches for multi-section impedance transformers, each optimizing a different aspect of the frequency response: (1) Binomial transformer: the section impedances are chosen so that all derivatives of the reflection coefficient are zero at the center frequency. This produces the flattest possible response at midband. The response rolls off monotonically toward the band edges (no ripple). Bandwidth is narrower than Chebyshev for the same number of sections. Best for applications requiring the flattest possible midband response (e.g., precision measurement systems). (2) Chebyshev transformer: the section impedances are chosen to produce equal-amplitude ripple in the passband. The maximum reflection coefficient (VSWR) is the same at every ripple peak. This provides wider bandwidth than binomial for the same number of sections and maximum ripple level. The trade-off: the midband response has ripple (it is not perfectly flat). Best for applications where bandwidth is the priority and small ripple is acceptable. (3) Comparison (3 sections, 50 to 100 ohms, max Gamma = 0.05): binomial: bandwidth ≈ 70% (VSWR < 1.105). The response is flat at midband, rises monotonically to Gamma = 0.05 at the band edges. Chebyshev: bandwidth ≈ 100% (VSWR < 1.105). The response ripples between Gamma = 0 and Gamma = 0.05 three times across the passband. The Chebyshev provides 43% more bandwidth by allowing the small ripple. (4) Design formulas: binomial section impedances: ln(Z_n/Z_(n-1)) = 2^(-N) × C(N,n) × ln(Z_L/Z_S). Chebyshev section impedances: computed using Chebyshev polynomial synthesis (more complex; typically computed using software or design tables).