A simple AC-driven RLC series circuit has equivalent impedance $$Z = R + i\left( \omega L - \frac{1}{\omega C} \right),$$ where $R$ is the resitance, $L$ is the inductance, $C$ is the capacitance, and $\omega$ is the driving frequency.
Now consider connecting an infinite ladder of such circuits to form a transmission line, which has characteristic impedance $$Z = \sqrt{\frac{\omega L + iR}{\omega C + i G}},$$ where $\omega$ is again the driving frequency, $R$ and $L$ are now the resistance and inductance per unit length along each conductor, $C$ is the capacitance per unit length between the two conductors, and $G$ is the conductance per unit length of the dielectric separating the two conductors.
In the simple circuit, energy loss comes from resistive heating from current passing through the resistor $R$. An ideal lossless circuit has $R = 0$, so the impedance is pure imaginary.
In the transmission line, losses come from resistive heating across the resistor as well as dielectric loss from current leaking across the dielectric that separates the two conductors. So an ideal lossless circuit has $R = G = 0$, and the characteristic impedance is pure real (and also independent of the driving frequency).
This is a little counterintuitive to me. In a simple RLC circuit, the lossy elements contribute to the real part of the impedance, and the lossless version has pure imaginary impedance. But in a transmission line, it's the opposite: the lossy components contribute to the imaginary part of the impedance, and the lossless version has pure real impedance. Why the opposite behavior between the two cases?
I know that ultimately, the answer is just "A single RLC circuit and an infinite ladder of RLC circuits are two very different circuit topologies, so the math just works out differently." But is there a more physically intuitive way to understand this difference?
In particular, what is the characteristic impedance $Z_n$ of a long but finite ladder of n lossless LC circuits? Is it pure imaginary, as in the $n = 1$ case, or pure real, as in the $n = \infty$ case, or does it have both nonzero real and complex parts? (I know that the answer will depend on the boundary condition at the far end of the line, but I assume that the asymptotic behavior for large $n$ will be independent of the boundary condition.) My guess is that as $n$ increases, the modulus $|Z_n|$ interpolates between $\omega L - 1/(\omega C)$ and $\sqrt{L/C}$, and the argument $\arg(Z_n)$ decreases monotonically from $\pm\pi/2$ to 0.
(Final question: in the equation for the transmission line impedance, which branch of the complex square root function do we choose?)