Lossless transmission line question

[James123]

Homework Statement

A 50 Ω lossless transmission line of length 0.4λ is terminated in a load of (40 + j30) Ω. Determine, using the equation given below, the input impedance to the line.

Homework Equations

The Attempt at a Solution

Zo= 50
Zl= 50∠36.87 or 40+j30
βι= 144° (this is known due to a previous question)

Zin= 50 x ((50∠36.87)*cos144°) + (j50*sin144°)/(50*cos144°) + ((j50∠36.87)*sin144°

=50 x (-40.5∠36.87 + j29.4)/(-40.5 + j29.4∠36.87)

Here I convert to rectangular and get:

=50 x ((-32.4 - j24.3) + j29.4))/(40.5 + (j23.52 + 17.64)

=50 x (-32.4 + j5.1)/(-22.86 + 23.52)

=50 x (0.8+j0.6) = 40 +j30 Ω

This question has been done on other threads before but I couldn't make sense of them.

Can anyone tell me why I've arrived back at the Load impedance? and where am I supposed to go from here?

Any help is appreciated,

Thanks!

 

[scottdave]

The 144° is from 0.4λ, correct?

 

[scottdave]

Also, notice how the magnitude of the load and line compare. What would happen if they were different?

 

[James123]

Hi, yes that's where I got the 144° from.

My heads honestly scrambled on this, if they were different there would have been something between the two on the line? i.e resistance?

 

[rude man]

where am I supposed to go from here?

? You already went where you were supposed to go, assuming your math is correct.