Talk:Travelling Wave Line Model

Footnotes

Derivation of Voltage Equation

The general solution to the transmission line wave equations are:

$I(x,t)=i^{+}(x-vt)+i^{-}(x+vt)\,$

$V(x,t)=v^{+}(x-vt)+v^{-}(x+vt)\,$

Where $v={\frac {1}{\sqrt {LC}}}\,$ is the velocity of propagation (m/s).

$i^{+}(t)\,$ and $i^{-}(t)\,$ are arbitrary current functions of time.

$v^{+}(t)\,$ and $v^{-}(t)\,$ are arbitrary voltage functions of time.

The Laplace Transform of the above equations are:

$I(x,s)=I^{+}(s)e^{-{\frac {sx}{v}}}+I^{-}(s)e^{\frac {sx}{v}}\,$ ... Equ. (1)

$V(x,s)=V^{+}(s)e^{-{\frac {sx}{v}}}+V^{-}(s)e^{\frac {sx}{v}}\,$ ... Equ. (2)

Recall the Telegrapher's equation for current:

${\frac {\partial I(x,t)}{\partial x}}=-C{\frac {\partial V(x,t)}{\partial t}}\,$

Taking the Laplace Transform of this equation, we get:

${\frac {\partial I(x,s)}{\partial x}}=-sCV(x,s)\,$ ... Equ. (3)

Substituting Equations (1) and (2) into Equ. (3):

${\frac {\partial }{\partial x}}\left[I^{+}(s)e^{-{\frac {sx}{v}}}+I^{-}(s)e^{\frac {sx}{v}}\right]=-sC\left[V^{+}(s)e^{-{\frac {sx}{v}}}+V^{-}(s)e^{\frac {sx}{v}}\right]\,$

We can differentiate the left-hand side:

${\frac {s}{v}}\left[-I^{+}(s)e^{-{\frac {sx}{v}}}+I^{-}(s)e^{\frac {sx}{v}}\right]=-sC\left[V^{+}(s)e^{-{\frac {sx}{v}}}+V^{-}(s)e^{\frac {sx}{v}}\right]\,$

Equating the $e^{-{\frac {sx}{v}}}$ and $e^{\frac {sx}{v}}$ terms on both sides, we get the following pair of equations:

$-{\frac {1}{v}}I^{+}(s)=-CV^{+}(s)\,$

${\frac {1}{v}}I^{-}(s)=-CV^{-}(s)\,$

Solving for voltage yields the following:

$V^{+}(s)=I^{+}(s)Z_{c}\,$ ... Equ. (4)

$V^{-}(s)=-I^{-}(s)Z_{c}\,$ ... Equ. (5)

Where $Z_{c}={\frac {1}{vC}}={\frac {\sqrt {LC}}{C}}={\sqrt {\frac {L}{C}}}$ is the characteristic impedance (Ohms)

We can use the above equations to re-write the voltage equation of Equ. (2) in terms of current as follows:

$V(x,s)=Z_{c}\left[I^{+}(s)e^{-{\frac {sx}{v}}}-I^{-}(s)e^{\frac {sx}{v}}\right]\,$

Finally, taking the inverse Laplace Transform of the equation above, we get:

$V(x,t)=Z_{c}\left[i^{+}(x-vt)-i^{-}(x+vt)\right]\,$