Question

The correction factors ${\mathbf{F}}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{sinh}\left(\mathrm{\gamma l}\right)}{\mathbf{\gamma l}}$ and ${\mathbf{F}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{tanh}\left(\frac{\mathrm{\gamma l}}{2}\right)}{\frac{\mathbf{\gamma l}}{\mathbf{2}}}$ which are complex numbers, have the units of _______

Short answer

Therefore, the correction factors are “dimensions less constants (per unit).”

Step-by-step solution

Determine the expression for the correction factors.

The expression for $\mathbf{sinh}\left(\mathrm{\gamma l}\right)$ is,

$\mathbf{sin}\left(\mathrm{\gamma l}\right)\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{\gamma l}}\mathbf{-}{\mathbf{e}}^{\mathbf{-}\mathbf{\gamma l}}}{\mathbf{2}}$ …… (1)

Here, $\mathbf{\gamma }$ is the propagation constant and l is the length of the transmission line.

The expression for role="math" localid="1656747258187" $\mathbf{tanh}\mathbf{\left(}\frac{\mathbf{\gamma l}}{\mathbf{2}}\mathbf{\right)}$ is,

$\mathbf{tanh}\left(\mathrm{\gamma l}\right)\mathbf{=}\frac{{\mathbf{e}}^{\frac{\mathbf{\gamma l}}{\mathbf{2}}}\mathbf{-}{\mathbf{e}}^{\mathbf{-}\frac{\mathbf{\gamma l}}{\mathbf{2}}}}{{\mathbf{e}}^{\frac{\mathbf{\gamma l}}{\mathbf{2}}}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\frac{\mathbf{\gamma l}}{\mathbf{2}}}}$ …… (2)

Determine the unit of the correction factors.

The correction factor$F_1$ is given by,

$F_1=\frac{\sin \left(\gamma l\right)}{\gamma l}$

Substitute$\frac{e^{\gamma l}-e^{-\gamma l}}{2}$ for$\sin \left(\gamma l\right)$ into above expression,

$F_1=\frac{e^{\gamma l}-e^{-\gamma l}}{2\left(\gamma l\right)}$

The correction factor$F_1$ is given by,

$F_1=\frac{\tan \left(\frac{\gamma l}{2}\right)}{\frac{\gamma l}{2}}$

Substitute $\frac{e^{\frac{\gamma l}{2}}-e^{-\frac{\gamma l}{2}}}{e^{\frac{\gamma l}{2}}+e^{-\frac{\gamma l}{2}}}$for $\tan \left(\frac{\gamma l}{2}\right)$into above expression,

$\begin{array}{l}{F}_2=\frac{\frac{e^{\frac{\gamma l}{2}}-e^{-\frac{\gamma l}{2}}}{e^{\frac{\gamma l}{2}}+e^{-\frac{\gamma l}{2}}}}{\frac{\gamma l}{2}}\\ F_2=\frac{e^{\frac{\gamma l}{2}}-e^{-\frac{\gamma l}{2}}}{\left(e^{\frac{\gamma l}{2}}+e^{-\frac{\gamma l}{2}}\right)\left(\frac{\gamma l}{2}\right)}\end{array}$

From the above discussion, the correction factors are dimensions less constants (per unit).