# Input impedance for quarter wave and half wave transmission line

**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}=Z_{0}[ Z_{r}+j Z_{0} tan βl/ Z_{0}+j tan βl]

divide numerator and denominator by j tan βl

Z_{i}=Z_{0}[ Z_{r}/j tan βL+ Z_{0}/Z_{0}/j tan βl+Z_{r}]

As βl= π/2

Thus Z_{i}=Z_{0}[0+Z_{0}]/(0+Z_{r})

Thus Z_{i}=Z_{u}^{2} / 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 Z_{0} . 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}=Z_{0}[Z_{r}+jZ_{0}tanβl/Z_{0}+jZ_{r}tanβl]

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_{R}to Z_{i} without any change.