---+ Loop Gain Equations
Motivation
For voltage biased TES bolometers, knowing the steepness of the transition is very important. The most natural parametrization of that steepness would be a thermal one, dR/dT. However, when dealing with dark bolometers in a constant temperature bath, it is more useful to define the steepness of the transition using the parameter L: the loop gain of the electrothermal feedback (ETF). It is defined as
%BEGINLATEX%
\begin{equation}
\mathcal{L} = \frac{\alpha P_e}{G T_{bolo}} = \frac{dR}{dT} \frac{V^2}{G R^2}
\end{equation}
%ENDLATEX%
The Power Balance Equation
For a dark bolometer
%BEGINLATEX%
\begin{equation}
\frac{V^2}{R} = G \Delta T
\end{equation}
%ENDLATEX%
It is useful to consider a additional small electrical signal and expand this equation. Equating the first order terms:
%BEGINLATEX%
\begin{equation}
\frac{2 V \delta V}{R} - \frac{V^2}{R^2} \delta R = G \frac{\delta R}{\frac{dR}{dT}} = \frac{V^2}{\mathcal{L} R^2} \delta R
\end{equation}
\begin{equation}
\frac{2 V \delta V}{R} = \frac{V^2}{R^2} \delta R \frac{1+\mathcal{L}}{\mathcal{L}}
\end{equation}
%ENDLATEX%
The power balance only needs to hold on time scales that are relatively long compared to the TES time constant. Letting angled brackets denote applying a "single pole" low pass filter:
%BEGINLATEX%
\begin{equation}
\delta R = 2 R \frac{< V \delta V >}{< V^2 >} \frac{\mathcal{L}}{1 + \mathcal{L}}
\end{equation}
%ENDLATEX%
Four ways of Measuring Loop Gain
1. Optical Responsivity
The optical responsivity is given by:
%BEGINLATEX%
\begin{equation}
\frac{dI}{dP_{opt}} = - \frac{1}{V} \frac{\mathcal{L}}{1 + \mathcal{L}}
\end{equation}
%ENDLATEX%
2. Time Constant
The bolometer time constant is sped up by the loop gain as follows:
%BEGINLATEX%
\begin{equation}
\tau_{eff} = \frac{\tau_0}{1+\mathcal{L}}
\end{equation}
%ENDLATEX%
3. Electrothermal feedback - Small signal measurement
We can apply a small helper voltage separated from the carrier (sinusoidal bias voltage) by a small frequency offset. This is referred to as a sideband. The electrothermal feedback causes a suppression of the sideband, but creates a sideband on the opposite side of the carrier in frequency space.
If we let the carrier frequency be omega and the helper frequency be (omega + d omega); the carrier voltage be of amplitude V and the helper voltage be of amplitude (epsilon V) then the first order perturbation to the current is predicted to be:
%BEGINLATEX%
\begin{equation}
\delta I = \frac{\epsilon | V |}{R} \left[ \frac{1}{1 + \mathcal{L}} \mathrm{cos}( (\omega + \delta \omega) t) - \frac{\mathcal{L}}{1+\mathcal{L}} \mathrm{cos}((\omega-\delta \omega)t) \right]
\end{equation}
%ENDLATEX%
4. IV Curve
Apparently the loop gain is given by
%BEGINLATEX%
\begin{equation}
R = \frac{V}{I}
\end{equation}
\begin{equation}
Z = \frac{\partial V}{\partial I}
\end{equation}
\begin{equation}
\mathcal{L} = \frac{Z-R}{Z+R}
\end{equation}
%ENDLATEX%
so it can be read straight off the IV curve. I need to think about this more.
--
TijmenDeHaan - 02 Mar 2010
This topic: ColdFeedback
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Topic revision: r1 - 2011-06-13 - TijmenDeHaan