# Research Update 3/16/16

Going back to the math of it was a good call. I should really have been viewing this as a system of bounded equations to analyze all along. If we made a feedback loop with some forward gain $G$, and then in feedback: a gain before the VCO $B$, the VCO as $k_{vco}/s$, then a gain after $A$, we can write the transfer function as:

$$ \dfrac{V_{out}}{V_{in}} = \dfrac{G}{1+B\frac{k_{vco}}{s}A} = \dfrac{sG}{s+GBAk_{vco}} $$

We can model the VCO noise as white noise as a function of frequency (just a single pole system with some offset). We need to optimize the loop so that:

- maximize $H(s)$
- minimize $NTF_{VCO}$

as well as

- Ensuring $\Delta \Theta$ is always within bounds for the phase detector
- Ensuring our forward gain $G$ out is not outside of the range of the oscillator