Difference between revisions of "HFI time response model"

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== Detection chain ==
+
== LFER4 model (PR1 /2013 release) ==
<span style="color:red">(Francesco Piacentini)</span>
 
  
This entire section has been moved to [[HFI_detection_chain | this page]].
+
Here we describe the "Low frequency excess response" 4 model.
 
+
If we write the input signal (power) on a bolometer as
===Principles of the readout electronics ===
+
<math>\label{bol_in}
 
+
s_0(t)=e^{i\omega t}</math>,
See figure [Principles of readout electronics]. The bolometer is biased by a square wave AC current obtained by the differentiation of a triangular voltage through a load capacitance, in a completely differential architecture. The presence of the stray capacitance due to losses of charge in the wiring requires a correction of the shape of the square bias current by a transient voltage. Thus the bias voltage generation is controlled by the two parameters I-bias and T-bias that express the amplitude of the triangular and transient voltage. The compensation voltage added to the bolometric signal to optimize the dynamic of the chain is controlled by the V-bal parameter.
+
the bolometer physical impedance can be written as
 
+
<math>\label{bol_out} s(t)=e^{i\omega t}F(\omega)</math>,
Parameters of the Readout Unit can be set to optimize the detectors performance.
+
where <math>\omega</math> is the angular frequency of the signal and <math>F(\omega)</math> is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of four single pole low-pass filters: <math>\label{bol_tf}
 
+
F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i}</math>.
The modulation frequency of the AC bias system, fmod of the square bias current
+
The modulation of the signal is performed with a square wave, written here as a composition of sine waves of decreasing amplitude: <math>\label{sigmod}
can be tuned from 70 Hz to 112 Hz by the telecommand parameters:
 
Nsample, which defines the number of samples per half period of modulated signal,
 
fdiv which determines the sampling frequency of the ADC.
 
 
 
The optimal frequencies are around 90 Hz.
 
 
 
 
 
Each channel has its own settings for the following parameters:
 
I-bias, amplitude of the triangular bias voltage;
 
 
 
T-bias, amplitude of the transient bias voltage;
 
 
 
V-bal, amplitude of the square compensation voltage;
 
 
 
G-amp, value of the programmable gain of the REU [1/3, 1, 3, 7.6];
 
 
 
N-blank, number of blanked samples at the beginning of halfperiod not taken into account during integration of the signal;
 
 
 
S-phase, phase shift when computing the integrated signal.
 
 
 
All these parameters influence the effective response of the
 
detection chains, and were optimized during the calibration
 
campaigns and confirmed during the calibration and performance
 
verification (CPV) phase following the launch of
 
Planck. The scientific signal is provided by the integral of the
 
signal on each half-period, between limits fixed by the S-phase
 
and N-blank parameters.
 
 
 
 
 
The interaction of modulated readout electronics with semiconductor
 
bolometers is rather different from that of a classical
 
DC bias readout (Jones 1953). The differences were seen during
 
the calibration of the HFI, although the readout electronics
 
was designed to mimic the operation
 
of a DC biased bolometric system as far as possible. With the
 
AC readout the maximum of responsivity is lower and is obtained
 
for higher bias current in the bolometer with respect to the DC model.
 
This is caused by the stray capacitance in the
 
wiring which has negligible effects for a DC bias and a major
 
effect for an AC bias. In our case, a stray capacitance of 150 pF
 
induces increases of NEP ranging from 4% (857 GHz bolometers)
 
to 10% (100 GHz bolometers) and also affects the HFI time
 
response. Details of the effect of the HFI AC bias system into
 
the time response of the detectors are discussed in the
 
[http://www.sciops.esa.int/wikiSI/planckpla/index.php?title=HFI_design,_qualification,_and_performance&instance=Planck_PLA_ES#Time_response| Time Response] Section.
 
 
 
=== JFETs===
 
 
 
Given the high impedance of the bolometers and the length of the connecting cables, it it is  essential that the impedance of the signal is lowered as close as possible to the detectors. In our system this is accomplished by means of JFET source followers, located in boxes connected to the 50 K stage.
 
The JFET box has been designed, developed and tested in the Observational Cosmology group in the Physics Department of the University of Rome "La Sapienza" (Brienza D. et al 2006).
 
There are two JFETs per channel, since the readout is fully differential, and they provide a current amplification of the signal while keeping the voltage amplification very close to unity.
 
 
 
Inside the box, the JFETs are mounted on a thermally insulated plate with an active temperature control to keep them at the optimal temperature of 110 K. With a dissipated power lower than 240 mW, mainly produced by the JFETs and the source resistors, we obtained a noise power spectral density of less than 3 nV Hz1/2 for the frequency range of interest. This increases the total noise ofall bolometer channels by less than 5%.
 
 
 
 
 
===Data compression===
 
 
 
The output of the readout electronics unit (REU) consists of one
 
value for each of the 72 science channels for each half-period of
 
modulation. This number, <math>S_{REU}</math> , is the sum
 
of the 40 16-bit ADC signal values obtained within the given
 
half-period. The data processor unit (DPU) performs a lossy
 
quantization of <math>S_{REU}</math>.
 
 
 
We define a compression slice of 254 <math>S_{REU}</math> values, corresponding
 
to about 1.4 s of observation for each detector and to a
 
strip of sky about 8 degrees long. The mean <math>\langle S_{REU} \rangle</math> of the data within
 
each compression slice is computed, and data are demodulated
 
using this mean:
 
 
 
<math>S_{demod,i} = (S_{REU,i} − \langle S_{REU} \rangle ) ∗ (−1)^{i}</math>
 
 
 
where <math>1 < i < 254</math> is the running index within the compression slice.
 
 
 
The mean <math>\langle S_{demod} \rangle</math> of the demodulated data <math>S_{demod,i}</math>
 
is computed and subtracted, and the resulting data slice is quantized
 
according to a step size Q that is fixed per detector:
 
 
 
<math> S_{DPU,i} = \mbox{round} \left[( S_{demod,i} − \langle S_{demod} \rangle) /Q \right ] </math>
 
 
 
This is the lossy part of the algorithm: the required compression
 
factor, obtained through the tuning of the quantization step Q,
 
adds a measure of noise to the data. Assuming Gaussian white
 
noise with standard deviation <math>\sigma</math>, a quantization setting of
 
<math>\sigma</math>/Q = 2 adds 1% to the noise (Pajot et al. 2010; Pratt 1978).
 
The value of <math>\sigma</math> was determined at the end of the CPV phase after
 
subtraction of the signal from the timeline.
 
 
 
The two means
 
<math>\langle S_{REU} \rangle</math>
 
and
 
<math>\langle S_{demod} \rangle</math>
 
are computed as
 
32-bit words and sent through the telemetry, together with the
 
<math>S_{DPU,i}</math> values.
 
Variable-length encoding of the <math>S_{DPU,i}</math> values is
 
performed on board, and the inverse decoding is applied on
 
ground.
 
 
 
 
 
Optimal use of the bandpass available for the downlink (75 kb/s 
 
average for HFI science) was obtained initially by using a value
 
of Q = <math>\sigma</math>/2.5 for all bolometer signals.
 
After 12 December 2009, only for the 857 GHz detectors, the
 
value was reset to Q = <math>\sigma</math>/2.0 to avoid data loss
 
due to exceeding the limit of downlink rate.
 
 
 
===Time response===
 
 
 
 
 
 
 
The HFI bolometers and readout electronics have a finite response time to changes in incident optical power.  The bolometers are thermal detectors of radiation whose response time is determined by the thermal circuit defined by the heat capacity of the detector and thermal conductivity.
 
 
 
Due to Planck's nearly constant scan rate, the time response is degenerate with the optical beam.  However, because of the long time scale effects present in the time response, the  time response is deconvolved from the data in the processing of the HFI data (see [[TOI processing|TOI processing]]).
 
 
 
The time response of the HFI bolometers and readout electronics is modeled as a Fourier domain transfer function (called the LFER4 model) consisting of the product of an bolometer thermal response <math>F(\omega)</math> and an electronics response <math>H'(\omega)</math>.  
 
 
 
<math>\label{LFER4def}TF^{LFER4}(\omega) = F(\omega) H'(\omega)</math>
 
 
 
=== LFER4 model ===
 
 
 
If we write the input signal (power) on a bolometer as <math>\label{bol_in}
 
s_0(t)=e^{i\omega t}
 
</math> the bolometer physical impedance can be written as: <math>\label{bol_out}
 
s(t)=e^{i\omega t}F(\omega)
 
</math> where <math>\omega</math> is the angular frequency of the signal and <math>F(\omega)</math> is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of 4 single pole low pass filters: <math>\label{bol_tf}
 
F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i}
 
</math> The modulation of the signal is done with a square wave, written here as a composition of sine waves of decreasing amplitude: <math>\label{sigmod}
 
 
s'(t)=e^{i\omega t}F(\omega)\sum_{k=0}^{\infty} \frac{e^{i\omega_r(2k+1)t}-e^{-i\omega_r(2k+1)t}}{2i(2k+1)}
 
s'(t)=e^{i\omega t}F(\omega)\sum_{k=0}^{\infty} \frac{e^{i\omega_r(2k+1)t}-e^{-i\omega_r(2k+1)t}}{2i(2k+1)}
</math> where we have used the Euler relation <math>\sin x=(e^{ix}-e^{-ix})/2i</math> and <math>\omega_r</math> is the angular frequency of the square wave. The modulation frequency is <math>f_{mod} = \omega_r/2\pi</math> and was set to <math>f_{mod} = 90.18759 </math>Hz in flight. This signal is then filtered by the complex electronic transfer function <math>H(\omega)</math>. Setting: <math>\omega_k^+=\omega+(2k+1)\omega_r</math> <math>\omega_k^-=\omega-(2k+1)\omega_r</math> we have: <math>\label{sigele}
+
</math>, where we have used the Euler relation <math>\sin x=(e^{ix}-e^{-ix})/2i</math> and <math>\omega_r</math> is the angular frequency of the square wave. The modulation frequency is <math>f_{\rm mod} = \omega_r/2\pi</math> and was set to <math>f_{\rm mod} = 90.18759 </math>Hz in flight. This signal is then filtered by the complex electronic transfer function <math>H(\omega)</math>. Setting <math>\omega_k^+=\omega+(2k+1)\omega_r</math> <math>\omega_k^-=\omega-(2k+1)\omega_r</math> we have <math>\label{sigele}
 
\Sigma(t)=\sum_{k=0}^\infty\frac{F(\omega)}{2i(2k+1)}\left[H(\omega_k^+)e^{i\omega_k^+t}-H(\omega_k^-)e^{i\omega_k^-t}\right]
 
\Sigma(t)=\sum_{k=0}^\infty\frac{F(\omega)}{2i(2k+1)}\left[H(\omega_k^+)e^{i\omega_k^+t}-H(\omega_k^-)e^{i\omega_k^-t}\right]
</math> This signal is then sampled at high frequency (<math>2 f_{mod} NS</math>). <math>NS</math> is one of the parameters of the HFI electronics and corresponds to the number of high frequency samples in each modulation semi-period. In order to obtain an output signal sampled every <math>\pi/\omega_r</math> seconds, we must integrate on a semiperiod, as done in the HFI readout. To also include a time shift <math>\Delta t</math>, the integral is calculated between <math>n\pi/\omega_r+\Delta t</math> and <math>(n+1)\pi/\omega_r+\Delta t</math> (with <math>T=2 \pi/\omega_r</math> period of the modulation). The time shift <math>\Delta t</math> is encoded in the HFI electronics by the parameter <math>S_{phase}</math>, with the relation <math>\Delta t = S_{phase}/NS/f_{mod} </math>.
+
</math>. This signal is then sampled at high frequency, (<math>2 f_{\rm mod} N_{\rm S}</math>). Here <math>N_{\rm S}</math> is one of the parameters of the HFI electronics and corresponds to the number of high frequency samples in each modulation semi-period. In order to obtain an output signal sampled every <math>\pi/\omega_r</math> seconds, we must integrate on a semiperiod, as done in the HFI readout. To also include a time shift <math>\Delta t</math>, the integral is calculated between <math>n\pi/\omega_r+\Delta t</math> and <math>(n+1)\pi/\omega_r+\Delta t</math> (with <math>T=2 \pi/\omega_r</math> period of the modulation). The time shift <math>\Delta t</math> is encoded in the HFI electronics by the parameter <math>S_{\rm phase}</math>, with the relation <math>\Delta t = S_{\rm phase}/N_{\rm S}/f_{\rm mod} </math>.
  
After integration, the <math>n</math>-sample of a bolometer can be written as <math>\label{eqn:output}
+
After integration, the <i>n</i>-sample of a bolometer can be written as <math>\label{eqn:output}
 
Y(t_n) = (-1)^n F(\omega) H'(\omega) e^{i t_n \omega}
 
Y(t_n) = (-1)^n F(\omega) H'(\omega) e^{i t_n \omega}
</math> where <math>\label{tfele}
+
</math>, where <math>\label{tfele}
 
H'(\omega) = \frac 12 \sum_{k=0}^\infty
 
H'(\omega) = \frac 12 \sum_{k=0}^\infty
 
e^{-i(\frac{\pi\omega}{2\omega_r}+\omega\Delta t)} \Bigg[
 
e^{-i(\frac{\pi\omega}{2\omega_r}+\omega\Delta t)} \Bigg[
Line 152: Line 23:
 
  \left(1-e^{\frac{i\omega_k^+\pi}{\omega_r}}\right)  
 
  \left(1-e^{\frac{i\omega_k^+\pi}{\omega_r}}\right)  
 
\\ - \frac{H(\omega_k^-)e^{i\omega_k^- \Delta t}}{(2k+1)\omega_k^-}  \left(1-e^{\frac{i\omega_k^-\pi}{\omega_r}}\right)
 
\\ - \frac{H(\omega_k^-)e^{i\omega_k^- \Delta t}}{(2k+1)\omega_k^-}  \left(1-e^{\frac{i\omega_k^-\pi}{\omega_r}}\right)
\Bigg]
+
\Bigg].
 
</math>
 
</math>
  
The output signal in equation eqn:output can be demodulated (thus removing the <math>(-1)^n</math>) and compared to the input signal in equation bol_in. The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, <math>H'(\omega)</math>: <math>TF(\omega) = F(\omega) H'(\omega)
+
The output signal in equation eqn:output can be demodulated (thus removing the (-1)<sup><i>n</i></sup>) and compared to the input signal in equation bol_in. The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, <math>H'(\omega)</math>: <math>TF(\omega) = F(\omega) H'(\omega)</math>.
</math>
 
  
The shape of <math>H(\omega)</math> is obtained combining low and high-pass filters with Sallen Key topologies (with their respective time constants) and accounting also for the stray capacitance low pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic band-pass function <math>H(\omega) = h_0*h_1*h_2*h_3*h_4*h_{5}</math> are listed in table table:readout_electronics_filters.
+
The shape of <math>H(\omega)</math> is obtained combining low- and high-pass filters with Sallen-Key topologies (with their respective time constants) and accounting also for the stray capacitance low-pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic band-pass function <math>H(\omega) = h_0*h_1*h_2*h_3*h_4*h_{5}</math> are listed in the following table.
  
=== Parameters of LFER4 model ===
+
<center>
 
+
{| class="wikitable"  align="center" style="text-align:left" border="1" cellpadding="5" cellspacing="0"
The LFER4 model has are a total of 10 parameters(<math>A_1</math>,<math>A_2</math>,<math>A_3</math>,<math>A_4</math>,<math>\tau_1</math>,<math>\tau_2</math>,<math>\tau_3</math>,<math>\tau_4</math>,<math>S_{phase}</math>,<math>\tau_{stray}</math>) 9 of which are independent, for each bolometer. The free parameters of the LFER4 model are determined using in-flight data in the following ways:
+
|+ '''HFI electronics filter sequence.'''  Here we define <math>s = i \omega</math>.
 
+
|- bgcolor="ffdead"
* <math>S_{phase}</math> is fixed at the value of the REU setting.
+
! Filter || Parameters || Function
* <math>\tau_{stray}</math> is measured during the QEC test during CPV.
 
* <math>A_1</math>, <math>\tau_1</math>, <math>A_2</math>, <math>\tau_2</math> are fit forcing the compactness of the scanning beam.
 
* <math>A_3</math>, <math>\tau_3</math>, <math>A_{4}</math> <math>\tau_4</math> are fit by forcing agreement of survey 2 and survey 1 maps.
 
* The overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole. 
 
 
 
The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters listed here in table table:LFER4pars.
 
 
 
 
 
===HFI electronics filter sequence ===
 
 
 
{| class="wikitable" style="text-align: left; border-collapse: collapse; border-width: 1px; border-style: solid; border-color: #000"
 
|+HFI electronics filter sequence. We define $s = i \omega$
 
 
|-
 
|-
! style="border-style: solid; border-width: 1px"|Filter
+
|0. Stray capacitance low-pass filter || <math>\tau_{\rm stray}= R_{\rm bolo} C_{\rm stray}</math> || <math>h_0 = \frac{1}{1.0+\tau_{\rm stray}*s}</math>
! style="border-style: solid; border-width: 1px"|Parameters
 
! style="border-style: solid; border-width: 1px"|Function
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|0. Stray capacitance low pass filter
+
|1. Low-pass filter || <math>R_1=1</math>k<math>\Omega</math> <br /> <math>C_1=100</math>nF || <math>h_1 = \frac{2.0+R_1*C_1*s}{2.0*(1.0+R_1*C_1*s)}</math>
|style="border-style: solid; border-width: 1px"|<math>\tau_{stray}= R_{bolo} C_{stray}</math>
 
|style="border-style: solid; border-width: 1px"|<math>h_0 = \frac{1}{1.0+\tau_{stray}*s}</math>
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|1. Low pass filter
+
|2. Sallen-Key high-pass filter || <math>R_2=51</math>k<math>\Omega</math><br /> <math>C_2=1\mu</math>F || <math>h_2= \frac{(R_2*C_2*s)^2}{(1.0+R_2*C_2*s)^2}</math>3
|style="border-style: solid; border-width: 1px"|<math>R_1=1</math>k<math>\Omega</math> <br /> <math>C_1=100</math>nF
 
|style="border-style: solid; border-width: 1px"|<math>h_1 = \frac{2.0+R_1*C_1*s}{2.0*(1.0+R_1*C_1*s)}</math>
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|2. Sallen Key high pass filter
+
|3. Sign reverse with gain || || <math>h_3=-5.1</math>
|style="border-style: solid; border-width: 1px"|<math>R_2=51</math>k<math>\Omega</math><br /> <math>C_2=1\mu</math>
 
|style="border-style: solid; border-width: 1px"|<math>h_2= \frac{(R_2*C_2*s)^2}{(1.0+R_2*C_2*s)^2}</math>3
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|3. Sign reverse with gain
+
|4. Single pole low-pass filter with gain || <math>R_4=10</math>k<math>\Omega</math><br /> <math>C_4=10</math>nF || <math>h_4= \frac{1.5}{1.0+R_4*C_4*s}</math>
|style="border-style: solid; border-width: 1px"|
 
|style="border-style: solid; border-width: 1px"|<math>h_3=-5.1</math>
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|4. Single pole low pass filter with gain
+
|5. Single pole high-pass filter coupled to a Sallen-Key low-pass filter || <math>R_9=18.7</math>k<math>\Omega</math><br /><math>R_{12}=37.4</math>k<math>\Omega</math><br /> <math>C=10.0</math>nF<br /><math>R_{78}=510</math>k<math>\Omega</math><br /> <math>C_{18}=1.0\mu</math>F<br /><math>K_3 = R_9^2*R_{78}*R_{12}^2*C^2*C_{18}</math><br /> <math>K_2 = R_9*R_{12}^2*R_{78}*C^2+R_{9}^2*R_{12}^2*C^2+R_9*R_{12}^2*R_{78}*C_{18}*C</math><br /> <math>K_1 =R_9*R_{12}^2*C+R_{12}*R_{78}*R_9*C_{18}</math> || <math>h_{5} = \frac{2.0*R_{12}*R_9*R_{78}*C_{18}*s}{s^3*K_3 +  
|style="border-style: solid; border-width: 1px"|<math>R_4=10</math>k<math>\Omega</math><br /> <math>C_4=10</math>nF
 
|style="border-style: solid; border-width: 1px"|<math>h_4= \frac{1.5}{1.0+R_4*C_4*s}</math>
 
|-
 
|style="border-style: solid; border-width: 1px"|5. Single pole high pass filter coupled to a Sallen Key low pass filter
 
|style="border-style: solid; border-width: 1px"|<math>R_9=18.7</math>k<math>\Omega</math><br /><math>R_{12}=37.4</math>k<math>\Omega</math><br /> <math>C=10.0</math>nF<br /><math>R_{78}=510</math>k<math>\Omega</math><br /> <math>C_{18}=1.0\mu</math>F<br /><math>K_3 = R_9^2*R_{78}*R_{12}^2*C^2*C_{18}</math><br /> <math>K_2 = R_9*R_{12}^2*R_{78}*C^2+R_{9}^2*R_{12}^2*C^2+R_9*R_{12}^2*R_{78}*C_{18}*C</math><br /> <math>K_1 =R_9*R_{12}^2*C+R_{12}*R_{78}*R_9*C_{18}</math>
 
|style="border-style: solid; border-width: 1px"|<math>h_{5} = \frac{2.0*R_{12}*R_9*R_{78}*C_{18}*s}{s^3*K_3 +  
 
 
       s^2*K_2+  
 
       s^2*K_2+  
 
       s*K_1 + R_{12}*R_9 } </math>
 
       s*K_1 + R_{12}*R_9 } </math>
  
 
|}
 
|}
 +
</center>
 +
 +
=== Parameters of LFER4 model ===
 +
 +
The LFER4 model has are a total of 10 parameters for each bolometer (<math>A_1</math>,<math>A_2</math>,<math>A_3</math>,<math>A_4</math>,<math>\tau_1</math>,<math>\tau_2</math>,<math>\tau_3</math>,<math>\tau_4</math>,<math>S_{\rm phase}</math>,<math>\tau_{\rm stray}</math>) nine of which are independent. The free parameters of the LFER4 model are determined using in-flight data in the following ways:
  
 +
* <math>S_{\rm phase}</math> is fixed at the value of the REU setting;
 +
* <math>\tau_{\rm stray}</math> is measured during the QEC test of the CPV phase;
 +
* <math>A_1</math>, <math>\tau_1</math>, <math>A_2</math>, <math>\tau_2</math> are fit by forcing the compactness of the scanning beam;
 +
* <math>A_3</math>, <math>\tau_3</math>, <math>A_{4}</math> <math>\tau_4</math> are fit by forcing agreement of Survey 2 and Survey 1 maps;
 +
* the overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole. 
  
 +
The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters are listed below.
  
{| class="wikitable" style="text-align: left; align: center; border-collapse: collapse; border-width: 1px; border-style: solid; border-color: #000"
+
<center>
|+ Parameters for LFER4 model.
+
{| class="wikitable"  align="center" style="text-align:center" border="1" cellpadding="5" cellspacing="0"  width=500px
|-
+
|+ '''LFER4 model parameters'''
|style="border-style: solid; border-width: 1px"|Bolometer
+
|- bgcolor="ffdead"
|style="border-style: solid; border-width: 1px"|<math>A_1</math>
+
! Bolometer || <math>A_1</math> || <math>\tau_1</math> (s) || <math>A_2</math> || <math>\tau_2</math> (s) || <math>A_3</math> || <math>\tau_3</math> (s) || <math>A_4</math> || <math>\tau_4</math> (s) || <math>\tau_{\rm stray}</math> (s) || <math>S_{\rm phase}</math> (s)
|style="border-style: solid; border-width: 1px"|<math>\tau_1</math> (s)
 
|style="border-style: solid; border-width: 1px"|<math>A_2</math>
 
|style="border-style: solid; border-width: 1px"|<math>\tau_2</math> (s)
 
|style="border-style: solid; border-width: 1px"|<math>A_3</math>
 
|style="border-style: solid; border-width: 1px"|<math>\tau_3</math> (s)
 
|style="border-style: solid; border-width: 1px"|<math>A_4</math>
 
|style="border-style: solid; border-width: 1px"|<math>\tau_4</math> (s)
 
|style="border-style: solid; border-width: 1px"|<math>\tau_{stray}</math> (s)
 
|style="border-style: solid; border-width: 1px"|<math>S_{phase}</math> (s)
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|100-1a
+
| 100-1a|| 0.392|| 0.01|| 0.534|| 0.0209|| 0.0656|| 0.0513|| 0.00833|| 0.572|| 0.00159|| 0.00139
|style="border-style: solid; border-width: 1px"|0.392
 
|style="border-style: solid; border-width: 1px"|0.01
 
|style="border-style: solid; border-width: 1px"|0.534
 
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|style="border-style: solid; border-width: 1px"|0.512
 
|style="border-style: solid; border-width: 1px"|0.00703
 
|style="border-style: solid; border-width: 1px"|0.41
 
|style="border-style: solid; border-width: 1px"|0.00703
 
|style="border-style: solid; border-width: 1px"|0.0639
 
|style="border-style: solid; border-width: 1px"|0.0272
 
|style="border-style: solid; border-width: 1px"|0.0139
 
|style="border-style: solid; border-width: 1px"|0.232
 
|style="border-style: solid; border-width: 1px"|0.00173
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|217-1
+
| 217-1|| 0.0136|| 0.00346|| 0.956|| 0.00346|| 0.0271|| 0.0233|| 0.00359|| 1.98|| 0.00159|| 0.00111
|style="border-style: solid; border-width: 1px"|0.0136
 
|style="border-style: solid; border-width: 1px"|0.00346
 
|style="border-style: solid; border-width: 1px"|0.956
 
|style="border-style: solid; border-width: 1px"|0.00346
 
|style="border-style: solid; border-width: 1px"|0.0271
 
|style="border-style: solid; border-width: 1px"|0.0233
 
|style="border-style: solid; border-width: 1px"|0.00359
 
|style="border-style: solid; border-width: 1px"|1.98
 
|style="border-style: solid; border-width: 1px"|0.00159
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|217-2
+
| 217-2|| 0.978|| 0.00352|| 0.014|| 0.0261|| 0.00614|| 0.042|| 0.00194|| 0.686|| 0.0016|| 0.00125
|style="border-style: solid; border-width: 1px"|0.978
 
|style="border-style: solid; border-width: 1px"|0.00352
 
|style="border-style: solid; border-width: 1px"|0.014
 
|style="border-style: solid; border-width: 1px"|0.0261
 
|style="border-style: solid; border-width: 1px"|0.00614
 
|style="border-style: solid; border-width: 1px"|0.042
 
|style="border-style: solid; border-width: 1px"|0.00194
 
|style="border-style: solid; border-width: 1px"|0.686
 
|style="border-style: solid; border-width: 1px"|0.0016
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|217-3
+
| 217-3|| 0.932|| 0.00355|| 0.0336|| 0.00355|| 0.0292|| 0.0324|| 0.00491|| 0.279|| 0.00174|| 0.00125
|style="border-style: solid; border-width: 1px"|0.932
 
|style="border-style: solid; border-width: 1px"|0.00355
 
|style="border-style: solid; border-width: 1px"|0.0336
 
|style="border-style: solid; border-width: 1px"|0.00355
 
|style="border-style: solid; border-width: 1px"|0.0292
 
|style="border-style: solid; border-width: 1px"|0.0324
 
|style="border-style: solid; border-width: 1px"|0.00491
 
|style="border-style: solid; border-width: 1px"|0.279
 
|style="border-style: solid; border-width: 1px"|0.00174
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|217-4
+
| 217-4|| 0.658|| 0.00135|| 0.32|| 0.00555|| 0.0174|| 0.0268|| 0.00424|| 0.473|| 0.00171|| 0.00111
|style="border-style: solid; border-width: 1px"|0.658
 
|style="border-style: solid; border-width: 1px"|0.00135
 
|style="border-style: solid; border-width: 1px"|0.32
 
|style="border-style: solid; border-width: 1px"|0.00555
 
|style="border-style: solid; border-width: 1px"|0.0174
 
|style="border-style: solid; border-width: 1px"|0.0268
 
|style="border-style: solid; border-width: 1px"|0.00424
 
|style="border-style: solid; border-width: 1px"|0.473
 
|style="border-style: solid; border-width: 1px"|0.00171
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-3a
+
| 353-3a|| 0.554|| 0.00704|| 0.36|| 0.00704|| 0.0699|| 0.0305|| 0.0163|| 0.344|| 0.0017|| 0.00125
|style="border-style: solid; border-width: 1px"|0.554
 
|style="border-style: solid; border-width: 1px"|0.00704
 
|style="border-style: solid; border-width: 1px"|0.36
 
|style="border-style: solid; border-width: 1px"|0.00704
 
|style="border-style: solid; border-width: 1px"|0.0699
 
|style="border-style: solid; border-width: 1px"|0.0305
 
|style="border-style: solid; border-width: 1px"|0.0163
 
|style="border-style: solid; border-width: 1px"|0.344
 
|style="border-style: solid; border-width: 1px"|0.0017
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-3b
+
| 353-3b|| 0.219|| 0.00268|| 0.671|| 0.00695|| 0.0977|| 0.0238|| 0.0119|| 0.289|| 0.00157|| 0.00111
|style="border-style: solid; border-width: 1px"|0.219
 
|style="border-style: solid; border-width: 1px"|0.00268
 
|style="border-style: solid; border-width: 1px"|0.671
 
|style="border-style: solid; border-width: 1px"|0.00695
 
|style="border-style: solid; border-width: 1px"|0.0977
 
|style="border-style: solid; border-width: 1px"|0.0238
 
|style="border-style: solid; border-width: 1px"|0.0119
 
|style="border-style: solid; border-width: 1px"|0.289
 
|style="border-style: solid; border-width: 1px"|0.00157
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-4a
+
| 353-4a|| 0.768|| 0.00473|| 0.198|| 0.00993|| 0.0283|| 0.0505|| 0.00628|| 0.536|| 0.00181|| 0.00125
|style="border-style: solid; border-width: 1px"|0.768
 
|style="border-style: solid; border-width: 1px"|0.00473
 
|style="border-style: solid; border-width: 1px"|0.198
 
|style="border-style: solid; border-width: 1px"|0.00993
 
|style="border-style: solid; border-width: 1px"|0.0283
 
|style="border-style: solid; border-width: 1px"|0.0505
 
|style="border-style: solid; border-width: 1px"|0.00628
 
|style="border-style: solid; border-width: 1px"|0.536
 
|style="border-style: solid; border-width: 1px"|0.00181
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-4b
+
| 353-4b|| 0.684|| 0.00454|| 0.224|| 0.0108|| 0.0774|| 0.08|| 0.0149|| 0.267|| 0.00166|| 0.00111
|style="border-style: solid; border-width: 1px"|0.684
 
|style="border-style: solid; border-width: 1px"|0.00454
 
|style="border-style: solid; border-width: 1px"|0.224
 
|style="border-style: solid; border-width: 1px"|0.0108
 
|style="border-style: solid; border-width: 1px"|0.0774
 
|style="border-style: solid; border-width: 1px"|0.08
 
|style="border-style: solid; border-width: 1px"|0.0149
 
|style="border-style: solid; border-width: 1px"|0.267
 
|style="border-style: solid; border-width: 1px"|0.00166
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-5a
+
| 353-5a|| 0.767|| 0.00596|| 0.159|| 0.0124|| 0.0628|| 0.0303|| 0.0109|| 0.357|| 0.00156|| 0.00111
|style="border-style: solid; border-width: 1px"|0.767
 
|style="border-style: solid; border-width: 1px"|0.00596
 
|style="border-style: solid; border-width: 1px"|0.159
 
|style="border-style: solid; border-width: 1px"|0.0124
 
|style="border-style: solid; border-width: 1px"|0.0628
 
|style="border-style: solid; border-width: 1px"|0.0303
 
|style="border-style: solid; border-width: 1px"|0.0109
 
|style="border-style: solid; border-width: 1px"|0.357
 
|style="border-style: solid; border-width: 1px"|0.00156
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-5b
+
| 353-5b|| 0.832|| 0.00619|| 0.126|| 0.0111|| 0.0324|| 0.035|| 0.0096|| 0.397|| 0.00166|| 0.00111
|style="border-style: solid; border-width: 1px"|0.832
 
|style="border-style: solid; border-width: 1px"|0.00619
 
|style="border-style: solid; border-width: 1px"|0.126
 
|style="border-style: solid; border-width: 1px"|0.0111
 
|style="border-style: solid; border-width: 1px"|0.0324
 
|style="border-style: solid; border-width: 1px"|0.035
 
|style="border-style: solid; border-width: 1px"|0.0096
 
|style="border-style: solid; border-width: 1px"|0.397
 
|style="border-style: solid; border-width: 1px"|0.00166
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-6a
+
| 353-6a|| 0.0487|| 0.00176|| 0.855|| 0.006|| 0.0856|| 0.0216|| 0.0105|| 0.222|| 0.00199|| 0.00125
|style="border-style: solid; border-width: 1px"|0.0487
 
|style="border-style: solid; border-width: 1px"|0.00176
 
|style="border-style: solid; border-width: 1px"|0.855
 
|style="border-style: solid; border-width: 1px"|0.006
 
|style="border-style: solid; border-width: 1px"|0.0856
 
|style="border-style: solid; border-width: 1px"|0.0216
 
|style="border-style: solid; border-width: 1px"|0.0105
 
|style="border-style: solid; border-width: 1px"|0.222
 
|style="border-style: solid; border-width: 1px"|0.00199
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-6b
+
| 353-6b|| 0.829|| 0.00561|| 0.127|| 0.00561|| 0.0373|| 0.0252|| 0.00696|| 0.36|| 0.00228|| 0.00111
|style="border-style: solid; border-width: 1px"|0.829
 
|style="border-style: solid; border-width: 1px"|0.00561
 
|style="border-style: solid; border-width: 1px"|0.127
 
|style="border-style: solid; border-width: 1px"|0.00561
 
|style="border-style: solid; border-width: 1px"|0.0373
 
|style="border-style: solid; border-width: 1px"|0.0252
 
|style="border-style: solid; border-width: 1px"|0.00696
 
|style="border-style: solid; border-width: 1px"|0.36
 
|style="border-style: solid; border-width: 1px"|0.00228
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-1
+
| 353-1|| 0.41|| 0.000743|| 0.502|| 0.00422|| 0.0811|| 0.0177|| 0.0063|| 0.329|| 0.00132|| 0.00097
|style="border-style: solid; border-width: 1px"|0.41
 
|style="border-style: solid; border-width: 1px"|0.000743
 
|style="border-style: solid; border-width: 1px"|0.502
 
|style="border-style: solid; border-width: 1px"|0.00422
 
|style="border-style: solid; border-width: 1px"|0.0811
 
|style="border-style: solid; border-width: 1px"|0.0177
 
|style="border-style: solid; border-width: 1px"|0.0063
 
|style="border-style: solid; border-width: 1px"|0.329
 
|style="border-style: solid; border-width: 1px"|0.00132
 
|style="border-style: solid; border-width: 1px"|0.00097
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-2
+
| 353-2|| 0.747|| 0.00309|| 0.225|| 0.00726|| 0.0252|| 0.0447|| 0.00267|| 0.513|| 0.00154|| 0.00097
|style="border-style: solid; border-width: 1px"|0.747
 
|style="border-style: solid; border-width: 1px"|0.00309
 
|style="border-style: solid; border-width: 1px"|0.225
 
|style="border-style: solid; border-width: 1px"|0.00726
 
|style="border-style: solid; border-width: 1px"|0.0252
 
|style="border-style: solid; border-width: 1px"|0.0447
 
|style="border-style: solid; border-width: 1px"|0.00267
 
|style="border-style: solid; border-width: 1px"|0.513
 
|style="border-style: solid; border-width: 1px"|0.00154
 
|style="border-style: solid; border-width: 1px"|0.00097
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-7
+
| 353-7|| 0.448|| 0.0009|| 0.537|| 0.0041|| 0.0122|| 0.0273|| 0.00346|| 0.433|| 0.00178|| 0.00125
|style="border-style: solid; border-width: 1px"|0.448
 
|style="border-style: solid; border-width: 1px"|0.0009
 
|style="border-style: solid; border-width: 1px"|0.537
 
|style="border-style: solid; border-width: 1px"|0.0041
 
|style="border-style: solid; border-width: 1px"|0.0122
 
|style="border-style: solid; border-width: 1px"|0.0273
 
|style="border-style: solid; border-width: 1px"|0.00346
 
|style="border-style: solid; border-width: 1px"|0.433
 
|style="border-style: solid; border-width: 1px"|0.00178
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|353-8
+
| 353-8|| 0.718|| 0.00223|| 0.261|| 0.00608|| 0.0165|| 0.038|| 0.00408|| 0.268|| 0.00177|| 0.00111
|style="border-style: solid; border-width: 1px"|0.718
 
|style="border-style: solid; border-width: 1px"|0.00223
 
|style="border-style: solid; border-width: 1px"|0.261
 
|style="border-style: solid; border-width: 1px"|0.00608
 
|style="border-style: solid; border-width: 1px"|0.0165
 
|style="border-style: solid; border-width: 1px"|0.038
 
|style="border-style: solid; border-width: 1px"|0.00408
 
|style="border-style: solid; border-width: 1px"|0.268
 
|style="border-style: solid; border-width: 1px"|0.00177
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|545-1
+
| 545-1|| 0.991|| 0.00293|| 0.00743|| 0.026|| 0.00139|| 2.6||   0||   0|| 0.00216|| 0.00111
|style="border-style: solid; border-width: 1px"|0.991
 
|style="border-style: solid; border-width: 1px"|0.00293
 
|style="border-style: solid; border-width: 1px"|0.00743
 
|style="border-style: solid; border-width: 1px"|0.026
 
|style="border-style: solid; border-width: 1px"|0.00139
 
|style="border-style: solid; border-width: 1px"| 2.6
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|0.00216
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|545-2
+
| 545-2|| 0.985|| 0.00277|| 0.0128|| 0.024|| 0.00246|| 2.8||   0||   0|| 0.00187|| 0.00097
|style="border-style: solid; border-width: 1px"|0.985
 
|style="border-style: solid; border-width: 1px"|0.00277
 
|style="border-style: solid; border-width: 1px"|0.0128
 
|style="border-style: solid; border-width: 1px"|0.024
 
|style="border-style: solid; border-width: 1px"|0.00246
 
|style="border-style: solid; border-width: 1px"| 2.8
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|0.00187
 
|style="border-style: solid; border-width: 1px"|0.00097
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|545-4
+
| 545-4|| 0.972|| 0.003|| 0.0277|| 0.025|| 0.000777|| 2.5||   0||   0|| 0.00222|| 0.00111
|style="border-style: solid; border-width: 1px"|0.972
 
|style="border-style: solid; border-width: 1px"|0.003
 
|style="border-style: solid; border-width: 1px"|0.0277
 
|style="border-style: solid; border-width: 1px"|0.025
 
|style="border-style: solid; border-width: 1px"|0.000777
 
|style="border-style: solid; border-width: 1px"| 2.5
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|0.00222
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|857-1
+
| 857-1|| 0.974|| 0.00338|| 0.0229|| 0.025|| 0.00349|| 2.2||   0||   0|| 0.00176|| 0.00111
|style="border-style: solid; border-width: 1px"|0.974
 
|style="border-style: solid; border-width: 1px"|0.00338
 
|style="border-style: solid; border-width: 1px"|0.0229
 
|style="border-style: solid; border-width: 1px"|0.025
 
|style="border-style: solid; border-width: 1px"|0.00349
 
|style="border-style: solid; border-width: 1px"| 2.2
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|0.00176
 
|style="border-style: solid; border-width: 1px"|0.00111
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|857-2
+
| 857-2|| 0.84|| 0.00148|| 0.158|| 0.00656|| 0.00249|| 3.2||   0||   0|| 0.0022|| 0.00125
|style="border-style: solid; border-width: 1px"|0.84
 
|style="border-style: solid; border-width: 1px"|0.00148
 
|style="border-style: solid; border-width: 1px"|0.158
 
|style="border-style: solid; border-width: 1px"|0.00656
 
|style="border-style: solid; border-width: 1px"|0.00249
 
|style="border-style: solid; border-width: 1px"| 3.2
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|   0
 
|style="border-style: solid; border-width: 1px"|0.0022
 
|style="border-style: solid; border-width: 1px"|0.00125
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|857-3
+
| 857-3|| 0.36|| 4.22e-05|| 0.627|| 0.0024|| 0.0111|| 0.017|| 0.002|| 1.9|| 0.00152|| 0.00126
|style="border-style: solid; border-width: 1px"|0.36
 
|style="border-style: solid; border-width: 1px"|4.22e-05
 
|style="border-style: solid; border-width: 1px"|0.627
 
|style="border-style: solid; border-width: 1px"|0.0024
 
|style="border-style: solid; border-width: 1px"|0.0111
 
|style="border-style: solid; border-width: 1px"|0.017
 
|style="border-style: solid; border-width: 1px"|0.002
 
|style="border-style: solid; border-width: 1px"| 1.9
 
|style="border-style: solid; border-width: 1px"|0.00152
 
|style="border-style: solid; border-width: 1px"|0.00126
 
 
|-
 
|-
|style="border-style: solid; border-width: 1px"|857-4
+
| 857-4|| 0.278|| 0.0004|| 0.719|| 0.00392|| 0.00162|| 0.09|| 0.00152|| 0.8|| 0.00149|| 0.000558
|style="border-style: solid; border-width: 1px"|0.278
 
|style="border-style: solid; border-width: 1px"|0.0004
 
|style="border-style: solid; border-width: 1px"|0.719
 
|style="border-style: solid; border-width: 1px"|0.00392
 
|style="border-style: solid; border-width: 1px"|0.00162
 
|style="border-style: solid; border-width: 1px"|0.09
 
|style="border-style: solid; border-width: 1px"|0.00152
 
|style="border-style: solid; border-width: 1px"| 0.8
 
|style="border-style: solid; border-width: 1px"|0.00149
 
|style="border-style: solid; border-width: 1px"|0.000558
 
 
|}
 
|}
 +
</center>
 +
 +
 +
== PR2 (2015) and PR3 (2018) Model ==
 +
 +
For the data associated to the PR2 (2015) and PR3 (2018) releases, the bolometer time transfer function is described by the sum of five single-pole low-pass functions, each with a time constant τi and an associated amplitude ai. The parameters for the model are in the table below (from {{PlanckPapers|planck2014-a08}}).
 +
 +
[[Image:2015_time_response_params.png|frameless|500px|center|Parameters of the time response model used for PR2 and PR3]]
 +
 +
== References ==
 +
<References />
 +
 +
[[Category:HFI design, qualification and performance|012]]

Latest revision as of 09:15, 22 June 2018

LFER4 model (PR1 /2013 release)[edit]

Here we describe the "Low frequency excess response" 4 model. If we write the input signal (power) on a bolometer as [math]\label{bol_in} s_0(t)=e^{i\omega t}[/math], the bolometer physical impedance can be written as [math]\label{bol_out} s(t)=e^{i\omega t}F(\omega)[/math], where [math]\omega[/math] is the angular frequency of the signal and [math]F(\omega)[/math] is the complex intrinsic bolometer transfer function. For HFI the bolometer transfer function is modelled as the sum of four single pole low-pass filters: [math]\label{bol_tf} F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i}[/math]. The modulation of the signal is performed with a square wave, written here as a composition of sine waves of decreasing amplitude: [math]\label{sigmod} s'(t)=e^{i\omega t}F(\omega)\sum_{k=0}^{\infty} \frac{e^{i\omega_r(2k+1)t}-e^{-i\omega_r(2k+1)t}}{2i(2k+1)} [/math], where we have used the Euler relation [math]\sin x=(e^{ix}-e^{-ix})/2i[/math] and [math]\omega_r[/math] is the angular frequency of the square wave. The modulation frequency is [math]f_{\rm mod} = \omega_r/2\pi[/math] and was set to [math]f_{\rm mod} = 90.18759 [/math]Hz in flight. This signal is then filtered by the complex electronic transfer function [math]H(\omega)[/math]. Setting [math]\omega_k^+=\omega+(2k+1)\omega_r[/math] [math]\omega_k^-=\omega-(2k+1)\omega_r[/math] we have [math]\label{sigele} \Sigma(t)=\sum_{k=0}^\infty\frac{F(\omega)}{2i(2k+1)}\left[H(\omega_k^+)e^{i\omega_k^+t}-H(\omega_k^-)e^{i\omega_k^-t}\right] [/math]. This signal is then sampled at high frequency, ([math]2 f_{\rm mod} N_{\rm S}[/math]). Here [math]N_{\rm S}[/math] is one of the parameters of the HFI electronics and corresponds to the number of high frequency samples in each modulation semi-period. In order to obtain an output signal sampled every [math]\pi/\omega_r[/math] seconds, we must integrate on a semiperiod, as done in the HFI readout. To also include a time shift [math]\Delta t[/math], the integral is calculated between [math]n\pi/\omega_r+\Delta t[/math] and [math](n+1)\pi/\omega_r+\Delta t[/math] (with [math]T=2 \pi/\omega_r[/math] period of the modulation). The time shift [math]\Delta t[/math] is encoded in the HFI electronics by the parameter [math]S_{\rm phase}[/math], with the relation [math]\Delta t = S_{\rm phase}/N_{\rm S}/f_{\rm mod} [/math].

After integration, the n-sample of a bolometer can be written as [math]\label{eqn:output} Y(t_n) = (-1)^n F(\omega) H'(\omega) e^{i t_n \omega} [/math], where [math]\label{tfele} H'(\omega) = \frac 12 \sum_{k=0}^\infty e^{-i(\frac{\pi\omega}{2\omega_r}+\omega\Delta t)} \Bigg[ \frac{H(\omega_k^+)e^{i\omega_k^+ \Delta t}}{(2k+1)\omega_k^+} \left(1-e^{\frac{i\omega_k^+\pi}{\omega_r}}\right) \\ - \frac{H(\omega_k^-)e^{i\omega_k^- \Delta t}}{(2k+1)\omega_k^-} \left(1-e^{\frac{i\omega_k^-\pi}{\omega_r}}\right) \Bigg]. [/math]

The output signal in equation eqn:output can be demodulated (thus removing the (-1)n) and compared to the input signal in equation bol_in. The overall transfer function is composed of the bolometer transfer function and the effective electronics transfer function, [math]H'(\omega)[/math]: [math]TF(\omega) = F(\omega) H'(\omega)[/math].

The shape of [math]H(\omega)[/math] is obtained combining low- and high-pass filters with Sallen-Key topologies (with their respective time constants) and accounting also for the stray capacitance low-pass filter given by the bolometer impedance combined with the stray capacitance of the cables. The sequence of filters that define the electronic band-pass function [math]H(\omega) = h_0*h_1*h_2*h_3*h_4*h_{5}[/math] are listed in the following table.

HFI electronics filter sequence. Here we define [math]s = i \omega[/math].
Filter Parameters Function
0. Stray capacitance low-pass filter [math]\tau_{\rm stray}= R_{\rm bolo} C_{\rm stray}[/math] [math]h_0 = \frac{1}{1.0+\tau_{\rm stray}*s}[/math]
1. Low-pass filter [math]R_1=1[/math]k[math]\Omega[/math]
[math]C_1=100[/math]nF
[math]h_1 = \frac{2.0+R_1*C_1*s}{2.0*(1.0+R_1*C_1*s)}[/math]
2. Sallen-Key high-pass filter [math]R_2=51[/math]k[math]\Omega[/math]
[math]C_2=1\mu[/math]F
[math]h_2= \frac{(R_2*C_2*s)^2}{(1.0+R_2*C_2*s)^2}[/math]3
3. Sign reverse with gain [math]h_3=-5.1[/math]
4. Single pole low-pass filter with gain [math]R_4=10[/math]k[math]\Omega[/math]
[math]C_4=10[/math]nF
[math]h_4= \frac{1.5}{1.0+R_4*C_4*s}[/math]
5. Single pole high-pass filter coupled to a Sallen-Key low-pass filter [math]R_9=18.7[/math]k[math]\Omega[/math]
[math]R_{12}=37.4[/math]k[math]\Omega[/math]
[math]C=10.0[/math]nF
[math]R_{78}=510[/math]k[math]\Omega[/math]
[math]C_{18}=1.0\mu[/math]F
[math]K_3 = R_9^2*R_{78}*R_{12}^2*C^2*C_{18}[/math]
[math]K_2 = R_9*R_{12}^2*R_{78}*C^2+R_{9}^2*R_{12}^2*C^2+R_9*R_{12}^2*R_{78}*C_{18}*C[/math]
[math]K_1 =R_9*R_{12}^2*C+R_{12}*R_{78}*R_9*C_{18}[/math]
[math]h_{5} = \frac{2.0*R_{12}*R_9*R_{78}*C_{18}*s}{s^3*K_3 + s^2*K_2+ s*K_1 + R_{12}*R_9 } [/math]

Parameters of LFER4 model[edit]

The LFER4 model has are a total of 10 parameters for each bolometer ([math]A_1[/math],[math]A_2[/math],[math]A_3[/math],[math]A_4[/math],[math]\tau_1[/math],[math]\tau_2[/math],[math]\tau_3[/math],[math]\tau_4[/math],[math]S_{\rm phase}[/math],[math]\tau_{\rm stray}[/math]) nine of which are independent. The free parameters of the LFER4 model are determined using in-flight data in the following ways:

  • [math]S_{\rm phase}[/math] is fixed at the value of the REU setting;
  • [math]\tau_{\rm stray}[/math] is measured during the QEC test of the CPV phase;
  • [math]A_1[/math], [math]\tau_1[/math], [math]A_2[/math], [math]\tau_2[/math] are fit by forcing the compactness of the scanning beam;
  • [math]A_3[/math], [math]\tau_3[/math], [math]A_{4}[/math] [math]\tau_4[/math] are fit by forcing agreement of Survey 2 and Survey 1 maps;
  • the overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole.

The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters are listed below.

LFER4 model parameters
Bolometer [math]A_1[/math] [math]\tau_1[/math] (s) [math]A_2[/math] [math]\tau_2[/math] (s) [math]A_3[/math] [math]\tau_3[/math] (s) [math]A_4[/math] [math]\tau_4[/math] (s) [math]\tau_{\rm stray}[/math] (s) [math]S_{\rm phase}[/math] (s)
100-1a 0.392 0.01 0.534 0.0209 0.0656 0.0513 0.00833 0.572 0.00159 0.00139
100-1b 0.484 0.0103 0.463 0.0192 0.0451 0.0714 0.00808 0.594 0.00149 0.00139
100-2a 0.474 0.00684 0.421 0.0136 0.0942 0.0376 0.0106 0.346 0.00132 0.00125
100-2b 0.126 0.00584 0.717 0.0151 0.142 0.0351 0.0145 0.293 0.00138 0.00125
100-3a 0.744 0.00539 0.223 0.0147 0.0262 0.0586 0.00636 0.907 0.00142 0.00125
100-3b 0.608 0.00548 0.352 0.0155 0.0321 0.0636 0.00821 0.504 0.00166 0.00125
100-4a 0.411 0.0082 0.514 0.0178 0.0581 0.0579 0.0168 0.37 0.00125 0.00125
100-4b 0.687 0.0113 0.282 0.0243 0.0218 0.062 0.00875 0.431 0.00138 0.00139
143-1a 0.817 0.00447 0.144 0.0121 0.0293 0.0387 0.0101 0.472 0.00142 0.00125
143-1b 0.49 0.00472 0.333 0.0156 0.134 0.0481 0.0435 0.27 0.00149 0.00125
143-2a 0.909 0.0047 0.0763 0.017 0.00634 0.1 0.00871 0.363 0.00148 0.00125
143-2b 0.912 0.00524 0.0509 0.0167 0.0244 0.0265 0.0123 0.295 0.00146 0.00125
143-3a 0.681 0.00419 0.273 0.00956 0.0345 0.0348 0.0115 0.317 0.00145 0.00125
143-3b 0.82 0.00448 0.131 0.0132 0.0354 0.0351 0.0133 0.283 0.00161 0.000832
143-4a 0.914 0.00569 0.072 0.0189 0.00602 0.0482 0.00756 0.225 0.00159 0.00125
143-4b 0.428 0.00606 0.508 0.00606 0.0554 0.0227 0.00882 0.084 0.00182 0.00125
143-5 0.491 0.00664 0.397 0.00664 0.0962 0.0264 0.0156 0.336 0.00202 0.00139
143-6 0.518 0.00551 0.409 0.00551 0.0614 0.0266 0.0116 0.314 0.00153 0.00111
143-7 0.414 0.00543 0.562 0.00543 0.0185 0.0449 0.00545 0.314 0.00186 0.00139
217-5a 0.905 0.00669 0.0797 0.0216 0.00585 0.0658 0.00986 0.342 0.00157 0.00111
217-5b 0.925 0.00576 0.061 0.018 0.00513 0.0656 0.0094 0.287 0.00187 0.00125
217-6a 0.844 0.00645 0.0675 0.0197 0.0737 0.0316 0.0147 0.297 0.00154 0.00125
217-6b 0.284 0.00623 0.666 0.00623 0.0384 0.024 0.0117 0.15 0.00146 0.00111
217-7a 0.343 0.00548 0.574 0.00548 0.0717 0.023 0.0107 0.32 0.00152 0.00139
217-7b 0.846 0.00507 0.127 0.0144 0.0131 0.0479 0.0133 0.311 0.00151 0.00139
217-8a 0.496 0.00722 0.439 0.00722 0.0521 0.0325 0.0128 0.382 0.00179 0.00111
217-8b 0.512 0.00703 0.41 0.00703 0.0639 0.0272 0.0139 0.232 0.00173 0.00125
217-1 0.0136 0.00346 0.956 0.00346 0.0271 0.0233 0.00359 1.98 0.00159 0.00111
217-2 0.978 0.00352 0.014 0.0261 0.00614 0.042 0.00194 0.686 0.0016 0.00125
217-3 0.932 0.00355 0.0336 0.00355 0.0292 0.0324 0.00491 0.279 0.00174 0.00125
217-4 0.658 0.00135 0.32 0.00555 0.0174 0.0268 0.00424 0.473 0.00171 0.00111
353-3a 0.554 0.00704 0.36 0.00704 0.0699 0.0305 0.0163 0.344 0.0017 0.00125
353-3b 0.219 0.00268 0.671 0.00695 0.0977 0.0238 0.0119 0.289 0.00157 0.00111
353-4a 0.768 0.00473 0.198 0.00993 0.0283 0.0505 0.00628 0.536 0.00181 0.00125
353-4b 0.684 0.00454 0.224 0.0108 0.0774 0.08 0.0149 0.267 0.00166 0.00111
353-5a 0.767 0.00596 0.159 0.0124 0.0628 0.0303 0.0109 0.357 0.00156 0.00111
353-5b 0.832 0.00619 0.126 0.0111 0.0324 0.035 0.0096 0.397 0.00166 0.00111
353-6a 0.0487 0.00176 0.855 0.006 0.0856 0.0216 0.0105 0.222 0.00199 0.00125
353-6b 0.829 0.00561 0.127 0.00561 0.0373 0.0252 0.00696 0.36 0.00228 0.00111
353-1 0.41 0.000743 0.502 0.00422 0.0811 0.0177 0.0063 0.329 0.00132 0.00097
353-2 0.747 0.00309 0.225 0.00726 0.0252 0.0447 0.00267 0.513 0.00154 0.00097
353-7 0.448 0.0009 0.537 0.0041 0.0122 0.0273 0.00346 0.433 0.00178 0.00125
353-8 0.718 0.00223 0.261 0.00608 0.0165 0.038 0.00408 0.268 0.00177 0.00111
545-1 0.991 0.00293 0.00743 0.026 0.00139 2.6 0 0 0.00216 0.00111
545-2 0.985 0.00277 0.0128 0.024 0.00246 2.8 0 0 0.00187 0.00097
545-4 0.972 0.003 0.0277 0.025 0.000777 2.5 0 0 0.00222 0.00111
857-1 0.974 0.00338 0.0229 0.025 0.00349 2.2 0 0 0.00176 0.00111
857-2 0.84 0.00148 0.158 0.00656 0.00249 3.2 0 0 0.0022 0.00125
857-3 0.36 4.22e-05 0.627 0.0024 0.0111 0.017 0.002 1.9 0.00152 0.00126
857-4 0.278 0.0004 0.719 0.00392 0.00162 0.09 0.00152 0.8 0.00149 0.000558


PR2 (2015) and PR3 (2018) Model[edit]

For the data associated to the PR2 (2015) and PR3 (2018) releases, the bolometer time transfer function is described by the sum of five single-pole low-pass functions, each with a time constant τi and an associated amplitude ai. The parameters for the model are in the table below (from Planck-2015-A07[1]).

Parameters of the time response model used for PR2 and PR3

References[edit]

(Planck) High Frequency Instrument

Readout Electronic Unit

Calibration and Performance Verification