Question #89630

An infinite discrete transmission line is constructed along the negative x-axis, with nodes located at $x_n = -n a$ for $n = 0, 1, 2, \dots \infty$, where the lattice spacing is $a = 2 \text{ mm}$. Between adjacent nodes $n$ and $n+1$, the series element consists of two inductors $L_1 = 150 \text{ } \mu\text{H}$ and $L_2 = 300 \text{ } \mu\text{H}$ connected in parallel. At each node $n \ge 1$, there is a shunt branch to ground consisting of two capacitors $C_1 = 15 \text{ nF}$ and $C_2 = 30 \text{ nF}$ connected in series. A harmonic voltage signal $V(0, t) = V_0 \cos(\omega t)$ is applied at the input node $x_0 = 0$, where the angular frequency is $\omega = 10^6 \text{ rad/s}$. Assuming the signal propagates in the steady state, determine the phase velocity vector $\vec{v}_p$ and the group velocity vector $\vec{v}_g$ of the wave.