curves taken over a range of bath temperatures. If the electrothermal feedback is strong, the TES temperature is essentially constant within the transition at $T_C$, allowing $K$ and $n$ to be found.

Figure \[fig:PT\_Inset\_PV\](a) shows thermal power against the voltage across the TES, $V_{TES}$, for device 3R$_b$, for a set of bath temperatures, $T_{B}$. The topmost curve corresponds to the lowest bath temperature used, $T_{B0}$. The power is essentially constant across the voltage range for which the bilayer is in its transition, indicating the presence of strong electrothermal feedback. This plot is representative of all devices tested, and demonstrates that the presence of the phononic filters in the legs (4 for each device), and associated Nb wiring layer, does not introduce artifacts into the operation of the device.

The power averaged over the transition region is shown in Fig. \[fig:PT\_Inset\_PV\](b) as a function of $T_B$ for devices 3IC (green), 3I$_b$ (red), and 3R$_b$ (magenta). Figure \[fig:PT\_Inset\_PV\](b) displays a clear reduction in the power transmitted through the triple phononic filters relative to the straight leg control, with further improved attenuation for the ring resonator 3R$_b$ over the interferometer 3I$_b$. This behaviour is reproduced almost identically in devices 3I$_a$ and 3R$_a$, For 