Understanding Ohm's law(?) for transmission lines
Your formula is just the voltage divider formula. \$E_b\$ is the input voltage, which is divided between the source resistance \$R_b\$ and the effective input impedance \$Z_{in1}\$.

\$V_b\$ will always be a complex number (since real numbers are a subset of the complex number). The imaginary part will be non-zero whenever \$Z_{in1}\$'s imaginary part is nonzero. This will almost always be true. There are a few special cases where it won't, like if \$Z_{c1}=Z_{c2}\$ and \$R_l=Z_{c2}\$, or if you use a special combination of transmission line lengths and load impedances that happens to add up to give a reflection with 0 or 180 degree phase.