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| − | == Time response ==
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| − | 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.
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| − | 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]]).
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| − | 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>.
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| − | <math>\label{LFER4def}TF^{LFER4}(\omega) = F(\omega) H'(\omega)</math>
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| | == LFER4 model == | | == LFER4 model == |
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| − | If we write the input signal (power) on a bolometer as <math>\label{bol_in} | + | Here we describe the "Low frequency excess response" 4 model. |
| − | s_0(t)=e^{i\omega t} | + | If we write the input signal (power) on a bolometer as |
| − | </math> the bolometer physical impedance can be written as: <math>\label{bol_out} | + | <math>\label{bol_in} |
| − | s(t)=e^{i\omega t}F(\omega) | + | s_0(t)=e^{i\omega t}</math>, |
| − | </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} | + | the bolometer physical impedance can be written as |
| − | F(\omega) = \sum_{i=0,4} \frac{a_i}{1 + i\omega\tau_i} | + | <math>\label{bol_out} s(t)=e^{i\omega t}F(\omega)</math>, |
| − | </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} | + | 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)} | | 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>. |
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| − | 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[ |
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| | \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> |
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| − | 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>
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| − | | |
| − | 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.
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| − | | |
| − | == Parameters of LFER4 model ==
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| − | | |
| − | 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:
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| − | * <math>S_{phase}</math> is fixed at the value of the REU setting.
| + | 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. |
| − | * <math>\tau_{stray}</math> is measured during the QEC test during CPV.
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| − | * <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.
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| − | * <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.
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| − | * The overall normalization of the LFER4 model is forced to be 1.0 at the signal frequency of the dipole.
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| − | The details of determining the model parameters are given in (reference P03c paper) and the best-fit parameters listed here in table table:LFER4pars.
| + | <center> |
| − | | + | {| class="wikitable" align="center" style="text-align:left" border="1" cellpadding="5" cellspacing="0" |
| − | | + | |+ '''HFI electronics filter sequence.''' Here we define <math>s = i \omega</math>. |
| − | ==HFI electronics filter sequence ==
| + | |- bgcolor="ffdead" |
| − | | + | ! Filter || Parameters || Function |
| − | {| class="wikitable" style="text-align: left; border-collapse: collapse; border-width: 1px; border-style: solid; border-color: #000" | |
| − | |+HFI electronics filter sequence. We define $s = i \omega$ | |
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| − | ! 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
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| − | ! style="border-style: solid; border-width: 1px"|Function
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| − | |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>
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| | |- | | |- |
| − | |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
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| − | |style="border-style: solid; border-width: 1px"|<math>h_4= \frac{1.5}{1.0+R_4*C_4*s}</math>
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| − | |-
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| − | |style="border-style: solid; border-width: 1px"|5. Single pole high pass filter coupled to a Sallen Key low pass filter
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| − | |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 | |
| − | |style="border-style: solid; border-width: 1px"|0.0209 | |
| − | |style="border-style: solid; border-width: 1px"|0.0656 | |
| − | |style="border-style: solid; border-width: 1px"|0.0513 | |
| − | |style="border-style: solid; border-width: 1px"|0.00833 | |
| − | |style="border-style: solid; border-width: 1px"|0.572 | |
| − | |style="border-style: solid; border-width: 1px"|0.00159 | |
| − | |style="border-style: solid; border-width: 1px"|0.00139 | |
| | |- | | |- |
| − | |style="border-style: solid; border-width: 1px"|100-1b
| + | | 100-1b|| 0.484|| 0.0103|| 0.463|| 0.0192|| 0.0451|| 0.0714|| 0.00808|| 0.594|| 0.00149|| 0.00139 |
| − | |style="border-style: solid; border-width: 1px"|0.484 | |
| − | |style="border-style: solid; border-width: 1px"|0.0103 | |
| − | |style="border-style: solid; border-width: 1px"|0.463 | |
| − | |style="border-style: solid; border-width: 1px"|0.0192 | |
| − | |style="border-style: solid; border-width: 1px"|0.0451 | |
| − | |style="border-style: solid; border-width: 1px"|0.0714 | |
| − | |style="border-style: solid; border-width: 1px"|0.00808 | |
| − | |style="border-style: solid; border-width: 1px"|0.594 | |
| − | |style="border-style: solid; border-width: 1px"|0.00149 | |
| − | |style="border-style: solid; border-width: 1px"|0.00139 | |
| | |- | | |- |
| − | |style="border-style: solid; border-width: 1px"|100-2a
| + | | 100-2a|| 0.474|| 0.00684|| 0.421|| 0.0136|| 0.0942|| 0.0376|| 0.0106|| 0.346|| 0.00132|| 0.00125 |
| − | |style="border-style: solid; border-width: 1px"|0.474 | |
| − | |style="border-style: solid; border-width: 1px"|0.00684 | |
| − | |style="border-style: solid; border-width: 1px"|0.421 | |
| − | |style="border-style: solid; border-width: 1px"|0.0136 | |
| − | |style="border-style: solid; border-width: 1px"|0.0942 | |
| − | |style="border-style: solid; border-width: 1px"|0.0376 | |
| − | |style="border-style: solid; border-width: 1px"|0.0106 | |
| − | |style="border-style: solid; border-width: 1px"|0.346 | |
| − | |style="border-style: solid; border-width: 1px"|0.00132 | |
| − | |style="border-style: solid; border-width: 1px"|0.00125 | |
| | |- | | |- |
| − | |style="border-style: solid; border-width: 1px"|100-2b
| + | | 100-2b|| 0.126|| 0.00584|| 0.717|| 0.0151|| 0.142|| 0.0351|| 0.0145|| 0.293|| 0.00138|| 0.00125 |
| − | |style="border-style: solid; border-width: 1px"|0.126 | |
| − | |style="border-style: solid; border-width: 1px"|0.00584 | |
| − | |style="border-style: solid; border-width: 1px"|0.717 | |
| − | |style="border-style: solid; border-width: 1px"|0.0151 | |
| − | |style="border-style: solid; border-width: 1px"|0.142 | |
| − | |style="border-style: solid; border-width: 1px"|0.0351 | |
| − | |style="border-style: solid; border-width: 1px"|0.0145 | |
| − | |style="border-style: solid; border-width: 1px"|0.293 | |
| − | |style="border-style: solid; border-width: 1px"|0.00138 | |
| − | |style="border-style: solid; border-width: 1px"|0.00125 | |
| | |- | | |- |
| − | |style="border-style: solid; border-width: 1px"|100-3a
| + | | 100-3a|| 0.744|| 0.00539|| 0.223|| 0.0147|| 0.0262|| 0.0586|| 0.00636|| 0.907|| 0.00142|| 0.00125 |
| − | |style="border-style: solid; border-width: 1px"|0.744 | |
| − | |style="border-style: solid; border-width: 1px"|0.00539 | |
| − | |style="border-style: solid; border-width: 1px"|0.223 | |
| − | |style="border-style: solid; border-width: 1px"|0.0147 | |
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| − | |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> |
| | + | [[Category:HFI design, qualification and performance|012]] |
