When the sending end voltage and current are numerically equal to the receiving end voltage and current respectively, then the line is called

For a long transmission line,

\(\left[ {\begin{array}{*{20}{c}} {{V_S}}\\ {{I_S}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cosh \gamma \ell }&{{Z_c}\sinh \gamma \ell }\\ {\frac{1}{{{Z_c}}}\sinh \gamma \ell }&{\cosh \gamma \ell } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_R}}\\ {{I_R}} \end{array}} \right]\)

Now, for an overhead line, G and R are neglected.

γ = Propagation constant

\(= j\omega \sqrt {LC}\)

\(cosh\;\gamma \ell = cosh\;j\omega \ell \;\sqrt {LC} = cos\;\omega \ell \;\sqrt {LC} \)

\(\sinh \gamma \ell = \sinh j\omega \ell \sqrt {LC} = jsin\;\omega \ell \sqrt {LC} \)

Sending end voltage (V_s) = Receiving end voltage (V_R)

And sending end current (I_S) = Receiving end current (I_R)

If |V_S| = |V_R| and |I_S| = |I_R|

Then \(\omega \ell \sqrt {LC} = n\pi ,\;n = 1,2,3 \ldots\)

Hence the line is a tuned line.