1. Input impedance Z_i for quarter wave transmission line

A transmission line is said to be quarter wave transmission line when its length equals quarter wavelength.

That is                                             l=λ/4

Or                                                   l=(2n-1)λ/4 (odd multiple of λ/4)

Or                                                   βl=2π/λ * λ/4                                              [ β=2π/λ]

=π/2

β   l=(2n-1)λ/2

therefore                                  tan βl= tan [(2n-1) π/2] +- ∞

as                                                           Z_i=Z₀[ Z_r+j Z₀ tan βl/ Z₀+j tan βl]

divide numerator and denominator by j tan βl

Z_i=Z₀[ Z_r/j tan βL+ Z₀/Z₀/j tan βl+Z_r]

As                             βl= π/2

Thus                                  Z_i=Z₀[0+Z₀]/(0+Z_r)

Thus                                  Z_i=Z_u² / Z_R

Thus quarter waves loss-less line transform the load impedance (Z_t) to input terminals as its inverse multiplied by the square of Z₀ . It is also called as quarter wave transformer. An open circuit quarter wave line appears as short circuit at the input terminals and short circuit appears as open circuit.

2. Input Impedance Z_i for half-wave transmission line:-

A ransmission line is said to be half- wave transmission line when its length equal half wavelength.

That is                              l=λ/2

Βl= 2π/λ*λ/2

=π

When   l=nλ/2 (integral multiple of λ/2)

Then                                  βl=nπ

Thus                           tan  βl= tan(nπ)

= 0

As                                   Z_i=Z₀[Z_r+jZ₀tanβl/Z₀+jZ_rtanβl]

Substituting value of tan βl=tan nπ=0 in above expression,we get

Z_i=Z_R

Thus,half wave lossless line transforms the Z_Rto Z_i without any change.