Consider the complex line bundle over $S^1$: $$ E = \left\{(\alpha,z\begin{pmatrix}1\\e^{i\theta_\alpha}\end{pmatrix}): \alpha\in \mathbb{R}/{2\pi}\mathbb{Z}, \rho_\alpha e^{i\theta_\alpha}=a+be^{i\alpha}, z\in\mathbb{C} \right\}\subset \mathbb{R}/{2\pi}\mathbb{Z} \times \mathbb{C}^2 $$ where $a,b\in\mathbb{R}$. There are two rather different cases for the behavior of $\theta_\alpha$: $a<b$ and $a>b$.
This bundle is used to compute some invariant in physics. It appears to be trivial for $a>b$ and not trivial for $a<b$. I do not understand why this is the case as $$ (\alpha,z\begin{pmatrix}1\\e^{i\theta_\alpha}\end{pmatrix}) \mapsto (\alpha,z) $$ seems a decent global trivialization.