# Input impedance for quarter wave and half wave transmission line

1. Input impedance Zi 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                                                           Zi=Z0[ Zr+j Z0 tan βl/ Z0+j tan βl]

divide numerator and denominator by j tan βl

Zi=Z0[ Zr/j tan βL+ Z0/Z0/j tan βl+Zr]

As                             βl= π/2

Thus                                  Zi=Z0[0+Z0]/(0+Zr)

Thus                                  Zi=Zu2 / ZR

Thus quarter waves loss-less line transform the load impedance (Zt) 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 Zi 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

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

Then                                  βl=nπ

Thus                           tan  βl= tan(nπ)

= 0

As                                   Zi=Z0[Zr+jZ0tanβl/Z0+jZrtanβl]

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

Zi=ZR

Thus,half wave lossless line transforms the ZRto Zi without any change.