Final answer:
The number of quarter-wave transformers needed for impedance matching between a 150Ω load and a 50Ω transmission line is not directly provided; additional context or formulas are required to calculate the exact number based on specified reflection coefficients and passband.
Step-by-step explanation:
The task here is to determine the number of quarter-wave transformers that should be inserted between a 150Ω load and a 50Ω transmission line to achieve a specified bandwidth and reflection coefficient. The conditions given are such that over a passband π/5≤βl≤π/5, the magnitude of the reflection coefficient |Γ| should be less than or equal to 0.125.
Though the exact solution involves using Smith charts or complex impedance matching calculations, we can infer from the information provided that the content loaded N quarter-wave transformers will act to match the impedance between the 50Ω transmission line and the 150Ω load. Each transformer serves to step the impedance progressively, minimizing reflections.
Regrettably, the data provided in the fragments does not contain enough information to directly answer the original question. We can't specify the number of transformers without additional context or formulas that relate the desired reflection coefficient and passband to the number of transformers.