Long Transmission Lines

\[\begin{aligned} dV_{x} & =I_{x}zdx\Rightarrow\dfrac{dV_{x}}{dx}=zI_{x}\\ dI_{x} & =V_{x}ydx\Rightarrow\dfrac{dI_{x}}{dx}=yV_{x} \end{aligned}\]

\[\dfrac{d^{2}V_{x}}{dx^{2}}=\dfrac{dI_{x}}{dx}z=yV_{x}z\] \[V_{x}=C_{1}e^{\gamma x}+C_{2}e^{-\gamma x}\]

where \(\boxed{\gamma=\sqrt{yz}}\) \[\dfrac{dV_{x}}{dx}=C_{1}\gamma e^{\gamma x}-C_{2}\gamma e^{-\gamma x}=zI_{x}\] \[\therefore I_{x}=\dfrac{C_{1}}{Z_{c}}e^{\gamma x}-\dfrac{C_{2}}{Z_{c}}e^{-\gamma x}\] where \[\boxed{Z_{c}=\sqrt{\left(\dfrac{z}{y}\right)}}\]

\[\begin{aligned} V_{x}&=C_{1}e^{\gamma x}+C_{2}e^{-\gamma x}\\ I_{x}&=\dfrac{C_{1}}{Z_{c}}e^{\gamma x}-\dfrac{C_{2}}{Z_{c}}e^{-\gamma x} \end{aligned}\] Applying boundary condition: \[\begin{array}{c} V_{R}=C_{1}+C_{2}\\ I_{R}=\dfrac{1}{Z_{c}}\left(C_{1}-C_{2}\right) \end{array}\Rightarrow\begin{cases} C_{1}=\dfrac{1}{2}\left(V_{R}+I_{R}Z_{c}\right)\\ C_{2}=\dfrac{1}{2}\left(V_{R}-I_{R}Z_{c}\right) \end{cases}\] \[\begin{aligned} V_{x} & =\left(\dfrac{V_{R}+I_{R}Z_{c}}{2}\right)e^{\gamma x}+\left(\dfrac{V_{R}-I_{R}Z_{c}}{2}\right)e^{-\gamma x}\\ I_{x} & =\left(\dfrac{V_{R}/Z_{c}+I_{R}}{2}\right)e^{\gamma x}+\left(\dfrac{V_{R}/Z_{c}-I_{R}}{2}\right)e^{-\gamma x} \end{aligned}\]

\[\begin{aligned} V_{x} & =V_{R}\left(\dfrac{e^{\gamma x}+e^{-\gamma x}}{2}\right)+I_{R}Z_{c}\left(\dfrac{e^{\gamma x}-e^{-\gamma x}}{2}\right)\\ &=V_{R}cosh\left(\gamma x\right)+I_{R}Z_{c}sinh\left(\gamma x\right) \end{aligned}\]

\[\begin{aligned} I_{x} & =V_{R}\dfrac{1}{Z_{c}}\left(\dfrac{e^{\gamma x}-e^{-\gamma x}}{2}\right)e^{\gamma x}+I_{R}\left(\dfrac{e^{\gamma x}+e^{-\gamma x}}{2}\right)\\ &=I_{R}cosh\left(\gamma x\right)+V_{R}\dfrac{1}{Z_{c}}sinh\left(\gamma x\right) \end{aligned}\] when \(x = l, \quad V_{x} = V_{s}, \quad I_{x} = I_{s}\)

$$\boxed{ \left[\begin{array}{c} V_S \\ I_S \end{array}\right]=\left[\begin{array}{cc} \cosh (\gamma l) & Z_c \sinh (\gamma l) \\ \frac{1}{Z_c} \sinh (\gamma l) & \cosh (\gamma l) \end{array}\right]\left[\begin{array}{c} V_R \\ I_R \end{array}\right] }$$

\[\begin{aligned} cosh\left(\alpha l+j\beta l\right) & =cosh\left(\alpha l\right)cos\left(\beta l\right)+jsinh\left(\alpha l\right)sin\left(\beta l\right)\\ sinh\left(\alpha l+j\beta l\right) & =sinh\left(\alpha l\right)cos\left(\beta l\right)+jcosh\left(\alpha l\right)sin\left(\beta l\right) \end{aligned}\]

The trigonometric values can be looked from standard tables.

\[\begin{aligned} cosh\left(\gamma l\right) & =1+\dfrac{\gamma^{2}l^{2}}{2!}+\dfrac{\gamma^{4}l^{4}}{4!}+\cdots\approx\left(1+\dfrac{YZ}{2}\right)\\ sinh\left(\gamma l\right) & =\gamma l+\dfrac{\gamma^{3}l^{3}}{3!}+\dfrac{\gamma^{5}l^{5}}{5!}+\cdots\approx\sqrt{YZ}\left(1+\dfrac{YZ}{6}\right) \end{aligned}\] This series converges rapidly for values of \(\gamma l\) and can be conveniently approximated \[\begin{aligned} A & =D\approx\left(1+\dfrac{YZ}{2}\right)\\ B & \approx Z\left(1+\dfrac{YZ}{6}\right)\\ C & \approx Y\left(1+\dfrac{YZ}{6}\right) \end{aligned}\]

\[\begin{aligned} cosh\left(\alpha l+j\beta l\right) & =\dfrac{1}{2}\left(e^{\alpha l}\angle\beta l+e^{-\alpha l}\angle-\beta l\right)\\ sinh\left(\alpha l+j\beta l\right) & =\dfrac{1}{2}\left(e^{\alpha l}\angle\beta l-e^{-\alpha l}\angle-\beta l\right) \end{aligned}\]

Instantaneous voltage, \(v_{x}\left(t\right) =v_{x1}+v_{x2}\)

\[\begin{cases} v_{x1} & =\sqrt{2}\left|\dfrac{V_{R}+I_{R}Z_{c}}{2}\right|e^{\alpha x}cos\left(\omega t+\beta x+\phi_{1}\right)\\ v_{x2} & =\sqrt{2}\left|\dfrac{V_{R}-I_{R}Z_{c}}{2}\right|e^{-\alpha x}cos\left(\omega t-\beta x+\phi_{2}\right) \end{cases}\]

where: \[\begin{aligned} \phi_{1} & = \angle\left(V_{R}+I_{R}Z_{c}\right)\\ \phi_{2} & = \angle\left(V_{R}-I_{R}Z_{c}\right) \end{aligned}\]

• \(v_{x}\left(t\right)\) is a function of two variables - time & distance

• Represents two travelling waves: Incident and reflected waves