x= \frac{1}{1-x} \int_{T_{b}}^{T_{c}} \!\! dT \,G_{t\,int}(T)
\label{eq:PhaseSep_equation}$$ Using this simplification, we obtain curves of $R_{t}$ versus $T_{b}$ for a CRESST-like device for several values of $V_{b}$, as shown in Fig. \[fig:RTb\_PhaseSep\].

![[]{data-label="fig:RTb_PhaseSep"}](figs/Gen/IV_CRESST_alphainfty.pdf){width="3.3in"}

If not for phase separation, all of these curves would be infinitely sharp step functions centered at $T_{b}= T_{c}$. With separation though, we observe significant degradation in device sensitivity that worsens with increasing $V_{b}$. Basically, a phase separated TES acts as if it has a very large $\beta$ from a DC perspective. This exact behavior is seen in CRESST devices [@CRESST_2001_PhononDet; @Emilija_2008_thesis].

Naively, one would think that this suppression in signal amplitude would severely affect phonon power resolution at all frequency scales. However, this is not the case. At low frequencies, thermal fluctuations across $G_{tb}$ completely dominate the noise in a non-phase-separated TES (Fig. \[fig:Spt-Sit\] top), and thus both signal and noise are suppressed equally. Whereas at high frequencies, the total noise is dominated by Johnson noise across the TES and thus the qualitative net effect is a suppression in $\omega_{S/N \, \delta \! P_{t}}$. Further, energy that is absorbed by the TES electron system in portions of the TES that are either fully normal or superconducting must 