The imperfect second-order PLL is defined as a PLL with the loop filter F(s) = s + / s + in which

The imperfect second-order PLL is defined as a PLL with the loop filter

F(s) = s + α / s + λα

in which λ is the offset of the pole from the origin relative to the zero location. In practical implementations λ is small but often cannot be neglected. Use the linear model of the PlLL and derive the transfer function for Θ(s)/Ф(s).  Derive expressions for ω_n and ξ in terms of K_t, α, and λ.