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i think that the quarter wave transformer or quarter wave T.L is only used in matcing impedances...that if we want to match a tl of z0 to another on of z2 we have to put a quarter wave t.l inbetween which have an impedance of
z1=sqrt(z0*z2)..
u have to think of it from multiple reflections point of view, when the wave is incident on the interface of zo and z1 line, some of it reflects back while the transmitted wave will move for a quarter of wavelengh till reaches the z1 and z2 interface, then some of if reflects back again moving another quarter wavelengh till reaches the first interface again with a net 180 phase, so it can cancel with the other reflected waves in the z1 t.l giving zero reflections back to the source...that's why we use quarter wave.
u can read about it in Pozar refrence,chapter 2,multiple reflections.
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