Half, Quarter and Eight-wave lines

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Eight-wave line (or) $\frac{\lambda }{8}$ -line:-

Consider a eight-wave line with length of the wave length as $\frac{\lambda }{8}$ .

from the input impedance of a Transmission line

$Z_{s}$ (or) $Z_{input} = \frac{Z_{O}(Z_{R}+Z_{O}\tan h\gamma l)}{(Z_{O}+Z_{R}\tan h\gamma l)}$ .

for a loss-less line $\alpha =0$ , $\gamma = j\beta$ .

then the input impedance changes to $Z_{input} = \frac{Z_{O}(Z_{R}+Z_{O}\tan (hj\beta l))}{(Z_{O}+Z_{R}\tan (hj\beta l))}$ .

$Z_{input} = \frac{Z_{O}(Z_{R}+jZ_{O}\tan \beta l)}{(Z_{O}+jZ_{R}\tan \beta l)}$ . since $\tan (j\beta l) =j \tan \beta l$ .

now by substituting $\beta l=\frac{2\pi }{\lambda }.\frac{\lambda }{8}\Rightarrow \beta l=\frac{\pi }{4}$ .

$Z_{input} = \frac{Z_{O}(Z_{R}+jZ_{O}\tan \frac{\pi }{4} )}{(Z_{O}+jZ_{R}\tan \frac{\pi }{4} )}$ .

$\therefore Z_{input} = Z_{O}$ .

i.e, The eight wave line , the input impedance is the Characteristic impedance ( $Z_{O}$ ) .

Half-wave line (or) $\frac{\lambda }{2}$ -line:-

Consider a Half-wave line with length of the wave length as $\frac{\lambda }{2}$ .

from the input impedance of a Transmission line

$Z_{s}$ (or) $Z_{input} = \frac{Z_{O}(Z_{R}+Z_{O}\tan h\gamma l)}{(Z_{O}+Z_{R}\tan h\gamma l)}$ .

for a loss-less line $\alpha =0$ , $\gamma = j\beta$ .

then the input impedance changes to $Z_{input} = \frac{Z_{O}(Z_{R}+Z_{O}\tan (hj\beta l))}{(Z_{O}+Z_{R}\tan (hj\beta l))}$ .

$Z_{input} = \frac{Z_{O}(Z_{R}+jZ_{O}\tan \beta l)}{(Z_{O}+jZ_{R}\tan \beta l)}$ . since $\tan (j\beta l) =j \tan \beta l$ .

now by substituting

i.e, The Half wave line , the input impedance is the load impedance () , this line repeats it’s terminating impedance, hence it is called as one-to-one transformer.

the main application of line is to connect a load to source where both of them can’t made adjacent in such case , we may connect a parallel Half-wave line at the load end.

Quarter-wave line (or) -line:-

Consider a Quarter-wave line with length of the wave length as .

from the input impedance of a Transmission line

$Z_{s}$ (or) $Z_{input} = \frac{Z_{O}(Z_{R}+Z_{O}\tan h\gamma l)}{(Z_{O}+Z_{R}\tan h\gamma l)}$ .

for a loss-less line $\alpha =0$ , $\gamma = j\beta$ .

then the input impedance changes to $Z_{input} = \frac{Z_{O}(Z_{R}+Z_{O}\tan (hj\beta l))}{(Z_{O}+Z_{R}\tan (hj\beta l))}$ .

$Z_{input} = \frac{Z_{O}(Z_{R}+jZ_{O}\tan \beta l)}{(Z_{O}+jZ_{R}\tan \beta l)}$ . since $\tan (j\beta l) =j \tan \beta l$ .

now by substituting $\beta l=\frac{2\pi }{\lambda }.\frac{\lambda }{4}\Rightarrow \Rightarrow \beta l=\frac{\pi }{2}$ .

$Z_{input} = \frac{Z_{O}(\frac{Z_{R}}{\tan \beta l}+jZ_{O})}{(\frac{Z_{O}}{\tan \beta l}+jZ_{R})}$ .

$Z_{input} = \frac{Z_{O}(\frac{Z_{R}}{\tan \frac{\pi }{2}}+jZ_{O})}{(\frac{Z_{O}}{\tan \frac{\pi }{2}}+jZ_{R})}$ .

$Z_{input} = \frac{Z_{O}(jZ_{O})}{(jZ_{R})}$ .

$Z_{input} = \frac{Z_{O}^{2}}{Z_{R}}$ .

$\therefore Z_{O} = \sqrt{Z_{input}Z_{R}}$ .

if $Z_{O}$ is constant then $Z_{s}\propto \frac{1}{Z_{R}}$ .

i.e, The Quarter wave transformer transforms a load impedance $Z_{R}< Z_{O}$ into a value $Z_{S}$ which is larger than $Z_{O}$ and vice-versa.

sometimes it is called as impedance inverter if $Z_{R}$ is pure resistive $Z_{S}$ also becomes resistive.

This Quarter wave transformer is useful when it is desired to transform a resistance into a different resistance value either larger (or) smaller.

The desired value of $Z_{O}$ can be obtained by choosing proper value of the ratio of spacing of the line conductors to their diameters because the physical dimensions can not practically have an infinite range of values where as the transformer ratio of the $\frac{\lambda }{4}$ transformer is subject to the practical limitation.

it is very useful device because of its simplicity and the ease with which it’s behavior is calculated .

if the length is an odd multiple of $\frac{\lambda }{4}$ will have same transformation properties but when the length is made larger the sensitivity to a small change of frequency becomes larger.

since input impedance is inversely proportional to $Z_{R}$ . If $Z_{R}$ is high , $Z_{S}$ will be low and vice-versa.

if $Z_{R}$ is capacitive , $Z_{S}$ will be inductive and vice-versa. if $Z_{R}$ is resistive , $Z_{S}$ will be resistive and vice-versa.

Thus depending up on $Z_{R}$ , a $\frac{\lambda }{4}$ transformer acts as step-up (or) step-down impedance transformer and that is why it is being called as impedance inverter.

a $\frac{\lambda }{4}$ transformer is disadvantageous , as it is sensitive to change in frequency because a new frequency section will no longer be $\frac{\lambda }{4}$ in length.

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