---+ Strong Electrothermal Feedback

Introduction

I am trying to simulate a bolometer using Simulink (an addon to MATLAB).

I start from equation (3.6) in Trevor's thesis, which can schematically be written as. (I'm not sure about the derivative and I make some assumptions about G=const)

%BEGINLATEX%

\begin{equation} \left( P_{sky} + \frac{V_{bias}^2}{R} \right) \frac{1}{1 + i \omega \tau}= G \Delta T \end{equation}

%ENDLATEX%

where G and C are constants and all other variables can change with time.

So this equation gives a way to relate inputs P_sky and V_bias to temperature T and TES resistance R. An additional equation relating R to T is required: the shape of the transition.

The arctan function will do for now. The { R , T } closure relation is then:

%BEGINLATEX% \begin{equation} \frac{R}{1 \Omega} = \frac{1}{2} + \frac{1}{\pi} \mathrm{arctan}\left( \frac{T-T_c}{T_0} \right) \end{equation} %ENDLATEX%

where T₀ is some parameter determining the steepness of the slope.

Or choosing to work in terms of resistance, we can solve for T in order to substitute it into the { P , R , T } relation.

%BEGINLATEX% \[ T = T_c + T_0 \mathrm{tan}\left( \pi \left[ \frac{R}{1 \Omega} - \frac{1}{2} \right] \right) \] %ENDLATEX%

Substituting:

%BEGINLATEX% \begin{equation} \mathrm{Low Pass} \left( P_{sky} + \frac{V_{bias}^2}{R} \right) = G \left[T_c + T_0 \mathrm{tan}\left( \pi \left[ \frac{R}{1 \Omega} - \frac{1}{2} \right] \right) \right] \end{equation} %ENDLATEX%

This equation has no closed form for R. Two options are:

• Use a root finder to solve the equation for R at each timestep.
• Approximate the transition by a linear function (Taylor expansion about an assumed stable point) and solve for R analytically.

Parameters: First TES Simulation

I used order 1 parameters for frequencies, voltages and power values to avoid numerical issues. The transition used is the arctan function with Tc=0.05, Tc=1 such that the maximal alpha=40 occurs at R=0.5 and T=Tc=1.

Results

The Simulink simulation looks as follows:

where the "Scope" looks at the bolometer resistance. Varying the strength of the sky signal results in the following plots.

So the bias voltage is keeping the TES at R >= 0.5 and the sky signal can linearly alter the resistance up to R >> 0.5 where the TES saturates at R = 1.

Bolo I-V curve

The current as a function of carrier voltage amplitude is shown in the following plot. The axes should be:

• X axis: Carrier Voltage (V)
• Y axis: Current (A)

Note the inverse relation between I-V at low voltages, the turnaround and the linear behaviour when the bolometer resistance saturates at 1 Ohm. At the lowest voltages, the input impedance of the current meter takes over and the relation becomes linear at a resistance of 0.1 Ohm + R_bolo ( T = 0 )

The feature at the high V end of the plot is an artifact (from when I started the simulation).

Amplifier Input Impedence

The input impedence of the amplifier is a non-ideality. The above I-V curve has R_input = 0.1 Ohm. The following I-V curves have R_input = 0.2 Ohm (top) and R_input = 0.4 Ohm (bottom).

The I_*_V = const part of the curve is the interesting part, since that is where the power applied to the bolometer is constant. For large input impedances, the curve deviates from I_*_V = const behaviour. That deviation implies deviation from strong electrothermal feedback.

-Tijmen

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This topic: ColdFeedback > WebHome > StrongElectrothermalFeedback Topic revision: r9 - 2011-11-17 - TijmenDeHaan

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