I'm trying to understand power transfer in transmission lines.
Consider a voltage source with complex voltage \$V_t\$ and internal impedance \$Z_t\$. Suppose we connect it to a transmission line with complex impedance \$Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}}\$, where \$R\$ and \$L\$ are the series resistance and inductance per unit length and \$G\$ and \$C\$ are the shunt condutance and capacitance per unit length, respectively. Note that we do not assume \$Z_0\$ to be purely real, since we include potential losses on the line. At the other end, at distance \$d\$, we connect a load with impedance \$Z_L\$. Our goal is to maximize the active power generated in the load.
1)On the one hand, I know that taking \$Z_L=Z_0\$ makes the reflection coefficient \$\rho = \frac{Z_L-Z_0}{Z_L+Z_0}\$ equal to zero and the transmission coefficient \$\tau = \frac{2Z_L}{Z_L+Z_0}\$ maximal, \$\tau = 2\$.
2)On the other hand, by impedance transformation the impedance seen by the voltage source at the terminals of the transmission is given by \$Z_{in} = Z_0\frac{Z_L+Z_0\tanh(\gamma d)}{Z_0+Z_L\tanh(\gamma d)}\$. Here \$\gamma = \sqrt{(R+j\omega L)(G+j\omega C)}\$. One might then think that we should take \$Z_{in} = Z_t^*\$ (conjugate), as we normally do for maximum power transfer.
Which of these is right? In a set of lecture notes I'm using, the appear to claim both (?). On the one hand, they claim in the text that 1) is right, since it minimizes reflections. On the other hand, in an exercise, they claim that 2) is right.
By the way, is there any way to nicely typeset math formulae here?
EDIT Regarding Janka's comment:
The voltage is \$V(d)=A_0^+e^{-\gamma d}+A_0^-e^{\gamma d}\$.
The current is \$I(d)=\frac{1}{Z_0}(A_0^+e^{-\gamma d}-A_0^-e^{\gamma d})\$
Here \$A_0^+\$ and \$A_0^-\$ denote the amplitude of the forward and backward wave, respectively.
We must have \$Z_L=V(d)/I(d)\$. Then we find \$A_0^- = \frac{Z_L-Z_0}{Z_L+Z_0}A_0^+e^{-2\gamma d}\$. If \$Z_L = Z_0\$, we get \$A_0^-=0\$.