Question of a problem in Antenna Theory by Balanis

Attached is a copy of the question and the solution given. I don't know where in the book say

\[Z_{in}(\frac{\lambda}{2})=\frac{1}{2}Z_{in}(\lambda)\]

In page 191 right below equation (4-105), it said at kh=0 Rr is half the value of an l'=2l. But this whole section is for vertical infintesimal dipole only. Can anyone tell me what's going on?

Thanks

Alan0354 said:

Attached is a copy of the question and the solution given. I don't know where in the book say

\[Z_{in}(\frac{\lambda}{2})=\frac{1}{2}Z_{in}(\lambda)\]

In page 191 right below equation (4-105), it said at kh=0 Rr is half the value of an l'=2l. But this whole section is for vertical infintesimal dipole only. Can anyone tell me what's going on?

Thanks

Click to expand...

Anyone please?

Alan0354 said:

Attached is a copy of the question and the solution given. I don't know where in the book say

\[Z_{in}(\frac{\lambda}{2})=\frac{1}{2}Z_{in}(\lambda)\]

In page 191 right below equation (4-105), it said at kh=0 Rr is half the value of an l'=2l. But this whole section is for vertical infintesimal dipole only. Can anyone tell me what's going on?

Thanks

Click to expand...

From reading about monopole over ground feeding at the bottom, seems like this can even applies to λ/2 Dipole with h=0( the bottom of the lower wire at ground plane.). The reason is because the current distribution of a λ/2 dipole is same as the top half of the λ dipole. So the ground plane provide the image of the lower half of the λ dipole.

From this reasoning, this should work for dipole of any length.