text:
40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
 40m Log, Page 337 of 348 Not logged in
 New entries since: Wed Dec 31 16:00:00 1969
ID Date Author Type Category Subject
9059   Fri Aug 23 21:01:38 2013 Alex ColeHowToElectronicsAutomated Photodetector Frequency Response System

This post describes how to use the Automated Photodetector Frequency Response System.

On the mechanical side, turn on:

-the diode laser (in rack 1Y1)

-the RF Switch (in rack 1Y1)

-the reference PD (under the POY table)

-the AG4395A Network Analyzer

The NA’s RF output should go to the laser’s modulation input, the reference PD’s output should go to the NA’s R input, and the RF Switch Chassis’s output (which is the combination of the two switches’ COM channels using a splitter) should go to the NA’s A input.

Once this is done, navigate into /users/alex.cole and run PDFR.sh. This script collects data for each photodetector under consideration by switching using a python script and communicating with the NA via GPIB. It then sends all the data to RF.m, which fits the functions, plots the latest data against canonical data, and saves the plots to file.

The fitting function, fit.m, also outputs peak frequency to the command line. This function uses PD name data (e.g. ‘REFL33’) to choose an interval with minimal noise to fit.

The main script prompts the user to press enter after each NA sweep to make sure that measurements don’t get interrupted/put out of order by RF switching.

Once you're done, you should turn off the laser, NA, RF Switch, and reference PD.

Troubleshooting

Sometimes, the NA throws up and doesn’t feel like running a particular sweep. If this happens, it’s a good idea to keep the matlab script from trying to analyze this PD’s data. Do this by opening up RF.m and commenting out the calls to ‘fit’ and ‘canonical’ for that PD.

If fit.m complains about a particular set of data, it is often the case that the N/P ratio (where N is order of approximation and P is number of points in the interval) is too high. You can fix this by reducing N or making the PD’s frequency range (chosen in the fnew_idx line) larger.

Choosing a single PD

If you only want to grab the transfer function for one PD, first look up which switch input it belongs to. This information is contained in /users/alex.cole/switchList. To turn the switch to a particular input, type something like:

python rf.py “ch7”

This command uses TCP/IP to tell the switch to look at channel 7. Switch input numbers range from 1 to 16, though not all of them are in use.

Once the switch is looking at the correct input, you can run a sweep and download the data by typing /opt/rtcds/caltech/c1/scripts/general/netgpibdata/NWAG4395A -s 1000000 -e 500000000 -c 499000000 -f [filestem for output] -d [path of directory for output] -i 192.168.113.108 -g 10 -x 15.

8795   Wed Jul 3 11:07:17 2013 AlexSummary Photodetector Characterization

[Alex, Koji]

We characterized Koji's BBPD MOD for REFL165 (see attachment).

First, we calibrated the Agilent 4395 Network Analyzer (NA) to account for differences in cable features between the Ref PD and Test PD connections. This was done using the 'Cal' softkey on the NA.

Then we performed transimpedance measurements for the test PD and reference PD relative to the RF output of the NA and relative to each other (see 2nd attachment. Note that the NA's RF output is split and sent to both the IR Laser and the NA's Ref input).

Next, we made DC measurements of the outputs of the photodetectors to estimate the photocurrent distribution of the transimpedance setup (like the 2nd attachment, but with the outputs of the PDs going to a multimeter). By photocurrent distribution, we mean how the beamsplitter and respective quantum efficiencies/generalized impedance/etc. of the PDs influence how much current flows through each PD at with a DC input.

Finally, we measured the output noise as a function of photocurrent (like the 2nd attachment, but with a lightbulb instead of the IR Laser). Input voltages for the lightbulb ranged from 0mV to 6V. Data was downloaded from the NA using netgpibdata from the scripts directory. Analysis is currently in progress; graphs to come soon.

Attachment 1: BBPD_PCB.pdf
Attachment 2: transimpedance_measurement.pdf
8806   Mon Jul 8 16:27:49 2013 AlexUpdate Planned rack additions

Alex and Eric

For the photodetector frequency response automation project, we plan to add modules to rack 1y1 as shown in the attached picture (Note: boxes are approximately to scale).

The RF switch will choose which photodetector's output is sent to the Agilent 4395A Network Analyzer.

The Diode Laser Module is powered by Laser Power Supply, will be modulated by the Network Analyzer and will be output to a 1x16 optical splitter which is already mounted in another rack (not pictured).

The Transformer Module has not been built yet.

We would like to install the power supply and the laser module tomorrow and will not begin routing fibers and cables until we post a drawing in the elog.

Also, our reference photoreceiver arrived today.

Attachment 1: Annotated_Rack_1y1.pdf
8829   Thu Jul 11 12:00:50 2013 AlexUpdate Planned rack additions

[Eric, Alex]

We mounted our Laser Module and Laser Power Source in rack 1y1. We plan to add our RF Switch and Transformer Module to the rack, as pictured. (Note: drawn-in boxes in picture are approximately to scale.) Note that the panel of knobs which the gray boxes overlap is obsolete and will soon be removed.

Attachment 1: Annotated_Rack_1y1_-_update.pdf
8849   Mon Jul 15 16:44:46 2013 AlexUpdateOMCOMC North Safety

[Eric Alex]

We are planning on testing our laser module soon, so we have added aluminum foil and a safety announcement to the door of OMC North. The safety announcement is as pictured in the attachment.

Attachment 1: photo_2_(1).JPG
8850   Mon Jul 15 16:51:37 2013 AlexConfiguration Planned AS Table addition

[Eric, Alex]

We are planning to add our reference PD to the southern third of the AS Table as pictured in the attachment. The power supply will go under the table.

17422   Wed Jan 25 16:58:19 2023 AlexUpdateCamerasRecording CCD cameras

Thus far, the software needed for the Magewell video encoder has been successfully installed on Donatella. OBS studio has also been installed and works correctly. OBS will be the video recording software that can be interfaced via command line once the SDI video encoder starts working. (https://github.com/muesli/obs-cli)

So far, the camera can not be connected to the Magewell encoder. The encoder continues to have a pulsing error light that indicates "no signal" or "signal not locked". I have begun testing on a secondary camera, directly connected to the Magewell encoder with similar errors. This may be able to be resolved once more information about the camera and its specifications/resolution is uncovered. At this time I have not found any details on the LCL-902K by Watec that was given to me by Koji. I will begin looking into the model used in the 40 meter next.

17441   Wed Feb 1 16:53:55 2023 AlexSummaryGeneralShadowing Anchal on developing a change for the c1ioo CDS computer

During my time shadowing Anchal, we discussed the need for digital control systems on the suspension systems for the 40 meter optics. The controls and diagnostics system (CDS) allows us to develop our own feedback controls and filters for the suspension systems by taking in analog signals from the shadow sensors. The feedback control system developed in the CDS then utilizes the OSEM actuators to dampen harmonic motion and noise on the suspension lines. While improving these feedback loops is an ongoing challenge, it is a problem that is likely non-linear, meaning the system must be understood on a much higher level to make further improvements. This brings us to the new addition of a wavefront sensor in the 40m lab, which will allow for constant monitoring of the active wavefront in the interferometer. The wavefront will soon be used for gathering training data for a neural net that will help further analyze the non-linear effects within the suspension and damping system. What Anchal was working on today was an update within a CDS model for clioo to allow for the integration of the wavefront sensor such that he may use a switch to change between connections in the mode cleaner and the arm cavity. The CDS models may be edited and updated using Matlab/Simulink to arrange blocks and code in a robust and visual manner. The final system designed in Simulink can then be saved and compiled using the real-time code generator (RCG), which cross-compiles the Simulink file into C code that can be read by the CDS system to assign inputs, outputs, and various logic or algorithms for filtering.

17459   Thu Feb 9 11:07:38 2023 AlexUpdateCDSAdding callibration filters to c1sus

Today I updated the ETMY suspension model to include 4 new filters at the output of the position, pitch, yaw and side summers and before the "To Coil Matrix". The library that was changed and updated is "sus_single_control_new". These callibration filters are labeled in the orange box as POSCAL, PITCAL, YAWCAL, and SIDECAL. The four filters are important as the will allow us to callibrate the position and side from counts to micro meters, and pitch and yaw from counts to micro radians.

The next steps for utilizing this update will be

- create a few experiments to find the callibration constants for the 4 degrees of freedom

- edit and update the ETMY Suspension screen to include selectable filter boxes to implement the callibration constants

For future reference, preform the update and restore the models to their previous states you may use the following:

to install the models we ssh into the computers running the ETMY suspension models (for example)

ssh c1sus

then for each model using the suspension library (we used c1sus c1mcs c1scx c1scy c1su2 c1su3) do

rtcds build-install c1sus

the watchdogs will need to be shut down for c1sus and the model will need to be restarted next

rtcds restart c1sus

Now we restore the filter values to the last saved point (about an hour before the update) in cds folder

python burtRestoreRTSepics.py -m c1sus c1mcs c1scx c1scy c1su2 c1su3 -o 10

Last we reset the watchdogs again using the following script in SUS>medm

python resetFromWatchdogTrip.py MC1 MC2 MC3 BS PRM SRM ITMX ITMY ETMX ETMY

17471   Thu Feb 16 23:54:11 2023 AlexUpdateDaily ProgressYaw and Pitch Calibration constants for ETMY op-lev

This work was done by Ancal and I.

To recallibrate the op-lev for ETMY, a python script was first written to calculate the change in distance in x or y that the photodiode array will see when the mirror incurs a change in yaw or pitch. The python script approximates d by integrating, using a reimann sum, the area under a gaussian curve, given by I(r)= Iexp(−2r2/ 2w(z)2), where r is the radial position, and w(z) is the waist (radius) size of the gaussian beam where power reaches 1/eof its maximum. The distance d, is the difference from the center of the gaussian to the point at which the beam profile has a normalized area under the curve equal to that of the percent of the beam profile showing on one half of the circular photodiode array.

Above, the gaussian is related to the translation of the beam profile on the photodiode where the area calculated under the curve of the gaussian, is equivalent to the ratio of the beam profile in 2 adjacent quaters of the photodiode array.

The gaussian, is directly related to the waist size of the laser beam profile, and thus a beam profiler was used to calculate the waist size over an average of 100 takes. Due to the thickness of the beam profiler, we were unable to get a direct measurement of the size of the beam at the exact location of the photodiode. Instead, we took two seperate measurements while moving the profiler 1 inch further away from the photodiode and back calculated the average size of the beam at the photodiode assuming that at this distance away from the source, the beams width would expand linearly. This provided a 2*waist size of 1625 ± 40 um.

image above displays the laser beam profiler used to approximate the waist size of the op-lev laser.

The physically calculated translation of the beam profile, d, can then be used to determine the overall angle, theta, that the mirror has moved to create this offset. The relation between distance and theta is Theta = d/2R, where R is the length from the mirror surface to the photodiode. R was then measured by hand over the optics table, and estimated to the best of our ability using the accurate autocad drawings of ETMY. This provided us with an R length of 1.76 ± 0.02 meters.

Image above shows the current system in place for converting the photodiode counts into microradians. The calibration constant is implemented at the last green filter boxes for pitch and yaw.

Lastly, to calculate the callibration constants, a series of tests were run on the ETMY suspeneded mirror. First, a time averaged value of the photodiode counts was taken with the mirror locked in place. Next, pitch and yaw were adjusted by 10 counts seperately, and the photodiode outputs recorded. This was done again but by moving the mirror 50 counts in pitch and yaw (seperately). The final result of the difference of the calculated theta values over the difference of pitch or yaw counts provided the following callibration constants:

Pitch moved +10 counts: 131 ± 5 cts/urad

Pitch moved +50 counts: 155 ± 5 cts/urad

Yaw moved +10 counts: 237 ± 5 cts/urad

Yaw moved +50 counts: 241 ± 5 cts/urad

Given our results, we believe that the values found for our 50 count translation to be the best approximation of the calibration constant due to its movement being more significant than that of the change seen from adjusting yaw or pitch by only 10 counts.

Next steps will be to update the values in the controls system and improve the python script to be more autonomous rather than a a step by step calculation.

17477   Wed Feb 22 23:40:48 2023 AlexUpdateCalibrationAdding calibration constants for sus matrix and filter control buttons to the sus control screen

The callibration constants were updated for the oplev pitch and yaw. The values were changed as denoted in 17471 were:

To make these changes for the oplev callibration constants I went to ETMY - SELECTED OPLEV SERVO BOX

I then opened OLMATRIX and turned off PITCH and YAW servos in the ETMY SUSPENSION SCREEN such that the system does not attempt to actively make corrections while values are being changed.

Then I adjusted the matrix to include our updated calibration constants and reinitiated the oplev ptich and yaw servo's

This updated the calibration constants for everything

The next change that was made was the addition of the calibration filters for position, pitch, yaw and side into the sitemap view for the suspension systems.

Adding calibration filters will allow us to callibrate the pos, pitch, yaw, and side to true values of urad and umeters (see 17459)

The final screen may be seen bellow (the updated area is outlined in red):

When each of the filter buttons is clicked, the following screen will now appear (circled in yellow is the calibration constant gain we will be calculating and entering into the system):

To create the edits to the controls screen we must complete the following process

We can edit the original screen - right click > evaluate > edit this screen

Then I adjusted the width of the overall screen, and moved the right half of the modules over to the right so I could fit in some filter buttons. I then Navigated to the c1ioo wfs master screen using the open feature to copy a pre existing filter module

I then adjusted the filter module and its contents to correspond to the features and autogenerated model files from RTCDS

There was some rearranging and adjusting needed to get these files in place first. The autogenerated files from the RTCDS can be found in dir = "/opt/rtcds/caltech/c1/medm/c1sus/"

We copied these files to dir = "/opt/rtcds/userapps/trunk/sus/c1/medm/templates/NEW_SUS_SCREENS/"

The directory we placed them in is where the models for c1 sus can be found that are referenced by the sitemap suspension monitor screen

Each file was then opened in Vscode and a few changes were made such that the specific naming values referenced by the different screens of the sitemap and different optics, are replaced by the overarching values seen in each instance of the screens.

There are approximately 50 referenced file names of "C1:SUS-BS_PITCAL" etc. In each instance we made the following changes:

"-BS" was changed to "-$(OPTIC)" "C1:" was changed to "$(IFO):"

The new strings should read "$(IFO):SUS-$(OPTIC)_PITCAL"

Once this change was made we can now right click on the filter module box, click on "Label/Name/Args" button

In the display file, we must add the path name for the calibration directory "/opt/rtcds/userapps/trunk/sus/c1/medm/templates/NEW_SUS_SCREENS/SUS_POSCAL.adl"

And for the arguments box we will enter OPTIC=$(OPTIC), IFO=$(IFO)

You can also copy and paste the directory names in the file boxes using right click copy from the file manager and paste into the box using a single click of the mouse scroller wheel

Lastly, the PV limits were changed for each number output right click value box > PV limits > Precision > Source changed to "Default" with a value of 1.

The shown value of the position, pitch, yaw, and side was then changed to show the output from the newly added filter. This is done also by right clicking the value box and adjusting the "Readback Channel".

Value changed from "$(IFO):SUS-$(OPTIC)_TO_COIL_1_#_INMON" to the outputs from the filters which are

"$(IFO):SUS-$(OPTIC)_POSCAL_OUTMON" (for others changing POSCAL to the appropriate variable)

This is how to edit and add the Medm screens for single suspension optics into the sitemap IFO SUS screen

Lastly, Tomohiro and I worked on acquiring 6 data sets from DC stepping through adjustments in pitch and yaw for MC1, MC2 and MC3. These datasets will be fit quadratically and combined with more tests dine by AC driving the stepper motors tomorrow to find the calibration constants for the mirrors.

Attachment 2: InkedScreenshot_2023-02-22_18-28-41.jpg
Attachment 3: InkedScreenshot_2023-02-22_18-29-00.jpg
17481   Fri Feb 24 13:29:16 2023 AlexSummaryIMCUpdated angular actuation calibration for IMC mirrors

Tomohiro, Anchal, and I did the following to make updates to the calibration constants for pitch and yaw on MC1, MC2 and MC3.

To acquire the data used for fitting a curve respective to the change in counts per change in mirror pitch and yaw, we utilized some code that Anchal has already developed.

The scripts used to take time averaged data points of the IMC mirrors can be found by entering the command s into a terminal window to enter the scripts folder. Then enter the path "SUS/angActCal" The following scripts will be found there to be used for this experiment: angActCal.py & parabolaFit.py To take data we used the angActCal.py function with set values for the time averaging = 5 s, settle time = 5 s, and adjusted the offset such that we would acquire approximately 20 data points given our ASC Bias limits. We defined the limits for each plot based on where the transmission fall off from the maximum value reached an average range of 10,000 counts. The "readChannel" for each was the "C1:IOO-MC_TRANS_SUMFILT_OUTPUT" and can be found from the site map at IOO>Lock MC> see MC2_TRANS The adjustment channels for Pitch and yaw on each IMC mirror were entered as the offset value found in the IMC screen at ALIGNMENT OFFSETS > BIASPIT/BIASYAW > OFFSET For the code to work, the offset switch must be turned on. parabolaFit.py The data from MC1, MC2, and MC3 for pitch and yaw was saved to individual text files which were then entered into the parabolaFit.py function to get the results seen in attachment 1 and 2. The above images show the printout from the plot fitting function and one of the graphs produced.  Optic ACT Fit curve factor for DC (1/cts^2) MC1 PIT 2.41 +/- 0.01 e-3 MC1 YAW 4.12 +/- 0.02 e-3 MC2 PIT 5.75 +/- 0.03 e-3 MC2 YAW 8.48 +/- 0.13 e-3 MC3 PIT 1.83 +/- 0.03 e-3 MC3 YAW 4.52 +/- 0.05 e-3 From the fitted curve values we then derived the equations that will soon be described further by Tomohiro (see entry _____) to arrive at the final callibration constants.  Optic ACT Callibration constant at DC (urad/cts) MC1 PIT 12.66 +/- 0.03 MC1 YAW 6.64 +/- 0.02 MC2 PIT lock6.83 +/- 0.02 MC2 YAW 4.69 +/- 0.04 MC3 PIT 11.03 +/- 0.09 MC3 YAW 6.96 +/- 0.04 Final Calibration Constants for MC1, MC2, & MC3 We then utilized our calculated calibration constants (as seen bellow) to adjust the following filter parameters in the IMC control panel. To make the updates such that the IMC screens show the correct urad values at the output of the filter banks, we must do the following steps to MC1, MC2, and MC3: First, to make changes to our calibration filters, we must first shut off the pitch and yaw feedback loop controls. TO do so for the Lock Filters, we will set the pitch and yaw SUS ASC inputs to 0 but entering the sitemap > IOO > C1IOO_WFS_MASTER Nex head to action at the top right, and we can select "MC WFS relief 60s", this will relieve the values from the pitch and yaw inputs to the 40m Mode Cleaner Alignment settings to save the overall alignment and allow us to turn off the WFS servos to make the necessary adjustments on the lock filters. Once we have waited a sufficient amount of time for the values on the ASC inputs to hover around 0, select Turn WFS ON/OFF button and choose "Turn OFF MCWFS Servo" Next, we will press on the "on/off" button (see attachment 3 - circled in orange) for pitch and yaw found in just the LOCK FILTERS windows. Once these are off we will stay in the same screen and adjust the gain values (boxed in yellow) for pitch and yaw. Next, we will take the current value and divide it by the newly found corresponding calibration constant. This is to adjust for the changes we will be making on the output end of the filter banks such that all values in the feedback controls are normalized to the same scale. The changes made here can be seen bellow:  Damp Filter Orig Damp Filter NEW Lock Filter Orig Lock Filter New MC1 PIT 40.0 3.160 1.0 0.079 MC1 YAW 40.0 6.024 1.0 0.151 MC2 PIT 5.0 0.732 1.0 0.146 MC2 YAW 5.0 1.066 1.0 0.213 MC3 PIT 3.0 0.272 1.0 0.091 MC3 YAW 5.0 0.718 1.0 0.144 Now that these changes have been made in the damp and lock filter banks, with the pitch and yaw feedback loops STILL OFF, we may adjust the newly made calibration filters for pitch and yaw (as seen in attachment 4). The "P" and "Y" filters may be opened (boxed in red) and we may adjust the gain (circled in yellow). Because each of these filters have just been created, the value is set to 1. This value can be completely replaced with the calibration constant found in our table above. Thus we will now change MC1 Pitch to have a "gain" of 12.66 and so forth. Once each of the calibration filters have been updated, you may go back into the damp filters and reinitiate the feedback loops. Once all values have been entered, This concludes the updating of the IMC filter calibration constants at DC. Attachment 1: angActCal_C1-SUS-MC1_BIASPIT_OFFSET_to_C1-IOO-MC_TRANS_SUMFILT_OUT_1361152703.png Attachment 2: Screenshot_2023-02-23_16-47-58.png Attachment 3: InkedScreenshot_2023-02-23_17-02-13.jpg Attachment 4: InkedScreenshot_2023-02-23_17-02-49.jpg 17484 Sun Feb 26 00:13:55 2023 AlexConfigurationASCIOO MC PIT/YAW gain change The following changes were made to the WFS MASTER IMC Pitch and Yaw gains: Gain values for the pitch and yaw on MC1, MC2, and MC3 filters on the SUS ASC inputs have been carried over to the WFS MASTER output filters. This was done such that Tomohiro and I could take AC measurements at an oscillation freq of 77 Hz on the pitch and yaw mirrors, while being sure that the amplitude of the AC signal being applied to each mirror is the same. The filters on the WFS output will have gains changed from 1.0 to the previously mentioned calibration values described in ELOG 17481 The values calculated for each filter were inverses of the callibration constants. The filters at the SUS ASC inputs were modified to read gain values of 1.0 again. See the table bellow for the values passed to each filter. In summary: originally IOO-MC1,2,3_PIT/YAW_GAIN = 1.0. Now: MC1,2,3_ASCPIT/ASCYAW_GAIN >> IOO-MC1,2,3_PIT/YAW_GAIN IOO-MC1,2,3_PIT/YAW_GAIN >> 1.0 17486 Wed Mar 1 17:13:38 2023 AlexUpdateIMCTransfer Function for IMC mirrors using sine sweep The following work has been done by Tomohiro, Anchal and I: To acquire the transfer functions for each of the IMC mirrors, we utilized diaggui, the CDS Diagnostic Test tool. We would like to measure the open loop transfer function, which is the ratio of In1 and In2, corresponding to before and after the injection point of the excitation signal. A sinusoidal excitation signal was swept from 0,2 Hz to 5 Hz and includes 11 data points from an average of 10 cylcles per point. NOTE: the WFS gain must be adjusted from 1.0 to 4.0 for these measurements (this is the slider underneath the "Turn WFS ON/OFF" button in C1:IOO_WFS_MASTER. For the three sets of data taken for Pitch in WFS1, WFS2, and MC2 Trans, the amplitude of the excitation wave was 30,000. In each measurement, the injection point is "C1:IOO-X_EXCMON", where X is the WFS or MC2 + Pitch or Yaw. We will be conculding our measurements tomorrow and will report the findings for YAW in WFS1, WFS2, and MC Trans2. Attachment 1: WFS1_PIT_OLTF.pdf Attachment 2: WFS2_PIT_OLTF.pdf Attachment 3: MC2_TRANS_PIT_OLTF.pdf 17500 Thu Mar 9 10:29:15 2023 AlexUpdateIMCStep response test on MC1, MC2, and MC3 YAW Tomohiro, Anchal and I completed the following processs for acquiring a new Output Yaw matrix for the "C1IOO_WFS_OUTMATRIX". To did this by following the same process in 17493 but instead of adding our offsets in the WFS1, WFS2 and MC Trans filter banks, offsets were added at the end of the feedback loop at the optics, MC1, MC2 and MC3 YAW. Optimal offset values were found such that the offset change did not disrupt the output WFS transmission signal by more than about a one thousand counts. Each limit was set to come close to this limit. Our final offset values were:  Optic Offset Value MC1 55 MC2 15 MC3 35 The step response was than observed in Diaggui, but the entire 800 s run was unable to be viewed at once. We then utilized our python script from the previous step response data that we took to develop the following: The measured response from stepping the optics was: $\begin{pmatrix} 1.31\pm0.24 & 54.2\pm1.3 & -0.28\pm0.03\\ -2.13\pm0.23 & -20.7\pm1.6 & 1.11\pm0.03\\ 1.82\pm0.27 & -25.8\pm1.5 & 0.16\pm0.03\\ \end{pmatrix} \begin{pmatrix} MC_{1Y}\\ MC_{2Y}\\ MC_{3Y}\\ \end{pmatrix} = \begin{pmatrix} WFS_{1Y}\\ WFS_{2Y}\\ MC_{2Y-TRANS}\\ \end{pmatrix}$ *The values in this matrix represent the number of counts/offset count. Thus all ovalues found from the step response were divided by the number of counts on each offset. To find the new yaw matrix, we then take the inverse of the step response output matrix to get: $\begin{pmatrix} MC_{1Y}\\ MC_{2Y}\\ MC_{3Y}\\ \end{pmatrix} = \begin{pmatrix} 0.188 & -0.009 & 0.403 \\ 0.017 & 0.005 & -0.006 \\ 0.689 & 0.987 & 0.656 \end{pmatrix} \begin{pmatrix} WFS_{1Y}\\ WFS_{2Y}\\ MC_{2Y-TRANS}\\ \end{pmatrix}$ The results from the step response may also be seen graphically in attachment 1. The first plot shows all 3 response signals. Then each following plot shows the individual signals and the step responses overlayed for each one. The plots also include horizontal lines that represent the average for the stepped signals and the average of the signal at rest along with shading for their associated uncertainties. This was then tested in C1IOO_WFS_BASIS Yaw matrix, and at first did not work well. The WFS1 Yaw output would rail toward the limits. To fix this, the sign of the gain was flipped (from 0.5 to -0.5) which seemed to solve this issue. This was then transmitted to the matrix to give: $\begin{pmatrix} MC_{1Y}\\ MC_{2Y}\\ MC_{3Y}\\ \end{pmatrix} = \begin{pmatrix} -0.188 & -0.009 & 0.403 \\ - 0.017 & 0.005 & -0.006 \\ -0.689 & 0.987 & 0.656 \end{pmatrix} \begin{pmatrix} WFS_{1Y}\\ WFS_{2Y}\\ MC_{2Y-TRANS}\\ \end{pmatrix}$ This did not solve all issues, the overall ouput signals from the WFS filters still seemed to have large fluctuations. I then began adjusting the gains of the WFS1, WFS2 and MC Trans yaw output filters and achieved much steadier signals. The following table describes the current best gain valuse for our Yaw matrix:  Sensor Gain Value WFS1 YAW 5.94 WFS2 YAW 6.44 MC TRANS YAW 1.9 The results from our found matrix and gain changes can be seen on the left of attachement 2 that displays the ouputs on the Error Signal Monitor. The original output yaw matrix signals can be seen on the right hand side. There is work to still be done on adressing these issues, but overall this may be improved by some additional changes in the gains on each channel. Attachment 1: step_response_080323.pdf Attachment 2: Screenshot_2023-03-08_18-17-35.png 17501 Thu Mar 9 14:22:24 2023 AlexUpdateComputer Scripts / ProgramsUpdate to toggleWFSoffsets.py for step response testing I have pushed changes made to the toggleWFSoffsets.py script to the git. This file may be found in: "/opt/rtcds/caltech/c1/Git/40m/scripts/MC/WFS/" The changes made are: Updated the script to allow for toggling step responses on either optics or sensors (default = optics), chosen by user The script orignally asked user to make any last changes to the offsets before hitting enter to run without displaying the new changes. Now the script checks for changes made by the user to the offsets and displays them if detected. If no changes are made, the code starts running the steps. 17505 Mon Mar 13 15:37:13 2023 AlexUpdateIMCStep Response of newly diagonalizing YAW output matrix From the work that Anchal has completed for diagnolizing the YAW ouput matrix, a step response was taken of this new matrix using our previous methodolgies and the following results: The step response can be seen plotted in attachment 1. The off diagonal terms of this new matrix sum to 1.24, which is a large decrease from the current matrix and the matrices that were tested from our previous step responses. Tomohiro and I are now currently working futher to configure the UGF's for YAW given this new output matrix. UPDATE: Tomohiro and I have completed testing the YAW Sensor outputs with broadband noise injection and have confirmed that gains currently set on each filter module (which is 1.0 for WFS1, WFS2, and MC Trans) provides us with adequate UGF's. As seen bellow in attachment 2-3, WFS1 and WFS2 have UGF's between 2 and 3 Hz. MC Trans can be seen in attachment 4 and has been confirmed to have a UGF around 0.1 Hz. Finally, attachment 5 displays the off diagnolized sums and uncertainties for each of our previous step response results and the newest result (labeled "new") for Anchal's OUTPUT YAW matrix. The first graph in blue displays the overall sum and uncertainty related to each step response taken. Then in the following 3 plots, the sum's and uncertaintes for each sensor are displayed individually for each step response test. For reference: New: corresponds to Anchal's YAW OUPUT MATRIX D0: refers to the previously implemented matrix, prior to any testint or updates D1: refers to the matrix that was computed based off of the first test Tomohiro and I performed D2: refers to the matrix computed as a secondary result from D1. This matrix was thought to provide a lower off diagonal sum, but did not. This thoroughly displays our results such that the newly computed matrix from Anchal is much more diagnolized then that of the step response matrices Tomohiro and I have computed. Attachment 1: step_response_YAW_130323.pdf Attachment 2: WFS1_YAW_OLTF_NI.pdf Attachment 3: WFS2_YAW_OLTF_NI.pdf Attachment 4: MC2_YAW_OLTF_NI.pdf Attachment 5: Mar13_Dfactor.pdf 17507 Tue Mar 14 11:34:05 2023 AlexHowToComputer Scripts / ProgramsSummary Pages Restart If the summary pages go down, it could be from a break in the script or some small error. The first remedy for fixing this can be to remove the cron jobs in the que and restart the "gw_daily_summary.sub" and "gw_daily_summary_rerun.sub" scripts.  For more information on how to do this, follow instructions found in the wiki. 17515 Tue Mar 21 18:41:12 2023 AlexUpdateIMCDither Lines set on MC1, MC2, MC3 for the night With Anchal's help, I have setup dither lines for Rana on MC1,2,3 that will be running overnight. The oscilations were set on MC1,2,3, oscillator screens. The following table describes the current setup:  Mirror Frequency Amplitude MC1 21.12 Hz 2000 MC2 25.52 Hz 1000 MC3 27.27 Hz 2500 These frequencies and amplitudes were set on LOCKIN1 for each MC1,2,3. The output filters matrix for MC1,2,3 was also updated to reflect the degree of freedom being tested: PITCH. The frequencies were picked to avoid the dewhitening frequency: 28Hz, and the Bounce/Roll frequencies: 16 Hz & 24 Hz. Furthermore, decimal value frequencies were utilized to avoid the multiples of 1 Hz. The oscilators were originally started at 1363480200 and will be turned off at 1363535157. See attachment 1 for the plot of the power spectrum. This test is done to find the beam offset for pitch. Attachment 1: 21032023_Dither_lines_plot 17516 Wed Mar 22 15:51:44 2023 AlexUpdateIMCBeam offset calculation for MC1,2,3 from dither results I have organized the resulting data from running dither lines on MC1,2,3. The data has been collected from diaggui as shown in attachment 1.  Mirror $f_l$ Avg Re (+/- 1000) Avg Im (+/- 1000) Peak Power ($\delta f$) Cts/urad MC1 21.12 7000 4000 8062 12.66 MC2 25.52 13000 10000 16401 6.83 MC3 27.27 4000 -600 4044 11.03 Next using the following equations we can find $\Delta Y$: $\Delta L = \Delta Y \cdot \theta_{AC}$ Where $\Delta L$ is the change in length in result of the dithering and $\Delta Y$ is the overall change in beam spot position Delta L can be calculated by: $\Delta L = \frac{\delta f}{v_{laser}} \cdot L_{IMC}$ where $\delta f$ is the peak power of the line frequency and is found by taking the square root of the magnitude of the Real and imaginary terms, $v_{laser}$ is frequency the laser light is traveling at (281 THz) and $L_{IMC}$ is the lenght of the IMC (13.5 meters). $\theta_{AC}$ can then be calculated by: $\theta_{AC} = \theta_{DC}/f_l^2$ where $\theta_{AC}$ is the angle at which the mirror was shaken at a given frequency. We can find $\theta_{DC}$ by converting the amplitude of the frequency that the mirror was shaken at and converting it into radians using the conversion constants found here: 17481. $\theta_{AC}$ is then shown to be found by this angle diveded by the line frequency. The final values are calculated and displayed bellow:  Mirror $\theta_{DC}$ $\theta_{AC}$ $\Delta L$ $\Delta Y$ MC1 157.9 urad 0.35 urad 0.38 nm 1.08 mm MC2 146.4 urad 0.23 urad 0.78 nm 3.39 mm MC3 226.7 urad 0.31 urad 0.19 nm 0.61 mm Attachment 1: 22032023_Dither_lines_demod_MC1_21-12.pdf 1947 Tue Aug 25 23:16:09 2009 Alberto, ranaConfigurationComputerselog moved in to the cvs path In nodus, I moved the elog from /export to /cvs/cds/caltech. So now it is in the cvs path instead of a local directory on nodus. For a while, I'll leave a copy of the old directory containing the logbook subdirectory where it was. If everything works fine, I'll delete that. I also updated the reboot instructions in the wiki. some of it also is now in the SVN. 2887 Thu May 6 17:47:01 2010 Alberto, kiwamu, Jc The 3rd (aka The Drigg)OmnistructureTMIMinutes from the Lab Organization Commitee meeting Today we met and we finally come up with a lot of cool, clever, brilliant, outstanding ideas to organize the lab. You can find them on the Wiki page created for the occasion. http://lhocds.ligo-wa.caltech.edu:8000/40m/40m_Internals/Lab_Organization Enjoy! 3049 Fri Jun 4 11:32:51 2010 Alberto, kiwamuUpdateIOOMC MMT1 Mirror Tests [Alberto, Kiwamu] Last Wednesday, we measured the beam profile after the MC mode matching telescope n.1 (MMT1). We found that the reflected beam had an irregular profile when observed with the beam scan. Fringes also appeared on an IR card. We thought that such effect could be due to interference of the main reflected beam with the beam reflected by the back surface of the mirror. To test the hypothesis we checked the transmitted and the reflected beams of a spare optic identical to MMT1. (This was the same optic that got dropped during the cleaning/baking process.) We tested it on the PSL table, using a 200mW beam coming from the new 2W Innolight laser. To maximize the separation between the two beams, we tested MMT1 at 45 degrees. The setup we used is shown here: We looked at the beam reflected by MMT1 about 5 meters from the mirror. At that distance the beam spot had a size of about 1-2cm. it didn't look perfectly round, but it showed no fringes, as it had happened with original MMT1 inside the MC chamber. At the transmission, the second ghost beam due to the back surface reflection (see picture above) was very week. In order to be able to see it on an IR card, we had to increase the laser pumping current from 1A to about 1.5A. We are now thinking of a way to measure the relative power between the two. The problem is that they run very close to each other and it's not easy to resolve them with a power meter or a photodiode. 1193 Thu Dec 18 19:15:54 2008 Alberto, YoichiConfigurationSUSMode Cleaner Cavity Alignment  Quote: This morning I found the MC locked to the 10 mode. When I locked it on the 00 mode, it was unstable and eventually it always got locked to the wrong mode. I looked at the Drift Mon MEDM screen, which shows a reference record for position, pitch and yaw of each mirror, and I found that the MC optics were in a different status. Moving the sliders of the mirrors' actuators, I brought them back to the reference position. Then the lock got engaged and it was stable, although the MC reflection from the photodiode, with the wave front sensors (WFS) off, was about 2V. That's higher than the 0.5V the it could get when we aligned the cavity and the input periscope last time. With the WFS on, the reflection dropped to 0.3V and, so far, the the cavity has been stably locked. This evening the mode cleaner was again locking on a higher mode so we tweaked the mirrors' actuators by their sliders on the MEDM screen until we improved the reflection to 0.3V. Then we went inside and, on the AS table, we centered the beam on the wave front sensors. Now the mode cleaner is locked, the reflection is less than 0.3V and the transmission about 3V, tha is it is in ideal conditions. We'll see if it holds. 1195 Fri Dec 19 11:29:16 2008 Alberto, YoichiConfigurationMZMZ Trans PD Lately, it seems that the matching of the input beam to the Mode Cleaner has changed. Also, it is drifting such that it has become necessary to continuously adjust the MC cavity alignment for it to lock properly. Looking for causes we stopped on the Mach Zehnder. We found that the monitor channel: C1:PSL-MZ_MZTRANSPD which supposedly reads the voltage from some photodiode measuring the transmitted power from the Mach Zehnder, is totally unreliable and actually not related to any beam at all. Blocking either the MZ input or output beam does not change the channel's readout. The reflection channel readout responds well, so it seems ok. 2477 Tue Jan 5 10:26:32 2010 Alberto, SteveOmnistructureEnvironmentAdded new wall cable-racks we hung two new WALL cable racks. One is on the pillar next to the Sp table, the other is next to the PSL computer rack. To do that we had to drill holes in the wall since the simple screws weren't strong enough to keep them up. One of the racks, the yellow, is dedicated to 4-pin lemos and other thick cables. 2106 Fri Oct 16 16:44:39 2009 Alberto, SanjitUpdateComputerselog restarted This afternoon the elog crashed. We just restarted it. 1162 Tue Nov 25 18:38:03 2008 Alberto, RobUpdatePSLMC Periscope Alignment This morning when I came in I found the MC cleaner unlocked and the autolocker script could not lock it. The reflected beam was quite off and showed in the bottom left corner of the IMCR camera. After turning off the WFS locking, I started slightly changing the alignment of the steering mirrors on the MC periscope, waiting for the LSC servo to lock the cavity. It didn't work. At some point I lost the beam from the IMCR camera and that is how someone might have found it when I left it for about one hour. When I came back and tried again adjusting the steering mirrors, I noticed that the autolocker was working and was trying to lock the cavity. After just a bit of adjustment, the MC got easily locked. After that, I spent a couple of hours trying to improve the alignment of the periscope to minimize the reflection and maximize the transmission. I started with a transmission of 0.4 V but, despite all the tweaking (I used the technique of turning both yaw knobs at the same time), I couldn't get more than 1.2 V (and 2.4 V at the reflection) if only the LSC servo was on. Looking at the camera, I moved the beam around to look for a more favorable spot but the MC wouldn't lock with the beam in other places. Maybe I could do better or maybe not because the cavity is not aligned. I'm going to try again tomorrow. 1166 Tue Dec 2 17:56:56 2008 Alberto, RanaConfigurationPSLMC Alignment In the attempt to maximize the Mode Cleaner transmission and minimize the reflection from the steering mirrors of the MC periscope, we could not get more ~2 V at the MC Trans PD and ~ 0.5 V at MC REFL_DC. As it turned out from the SUS Drift Monitor, the reason was that the MC optics had been somehow displaced from the optimal position. After restoring the reference position values for the mirrors and tweaking again the periscope, we got ~3V at the MC TransPD and 0.5V at the reflection. The beam was then probably clipped at the REFL PD so that we had to adjust the alignment of one of the BS in the transmitted beam path on the AS table. We also zeroed the WFS PDs, but not before reducing the power from the MZ, for their QPDs not to saturate. After relocking, the transmission was 3V and the reflection ~0.3V. The beam isnow centered on the Trans PD and REFL PD and the Mode Cleaner locked. More details on the procedure will follow. 1883 Mon Aug 10 20:49:13 2009 Alberto, RanaUpdatePSLPMC Mode Matching Lenses Tuning Rana, Alberto This afternoon we tried to improve the mode matching of the beam to the PMC. To do that we tuned the positions of the two lenses on the PSL table that come before the PMC. We moved the first lens back an forth the without noticing any improvement on the PMC transmitted and reflected power. Then we moved the first backwards by about one cm (the order is set according to how the beam propagates). That made the things worse so we moved also the second lens in the same direction so that the distance in between the two didn't change significantly. After that, and some more adjustments on the steering mirrors all we could gain was about 0.2V on the PMC transmission. We suspect that after the problems with the laser chiller of two months ago, the beam size changed and so the mode matching optics is not adequate anymore. We have to replace the mode matching lenses with other ones. 2468 Wed Dec 30 18:01:03 2009 Alberto, RanaUpdateGeneralAll watchdogs tripped this morning  WQuote: This morning I found all the watchdogs had tripped during the night. I restored them all. I can't damp ITMX. I noticed that its driving matrix is all 1s and -1s as the the right values had been lost in some previous burtrestoring. Rana fixed the problem. He found that the side damping was saturating. He lowered the gain a little for a while, waited for the the damping to slow down the optic and then he brought the gain back where it was. He also upadted the MEDM screen snapshot. 3029 Wed Jun 2 01:47:28 2010 Alberto, KiwamuUpdateIOOmode measurement of new input optics The mode profile of the new input optics was measured. Although the distance between each optic was not exactly the same as the design because of narrow space, we measured the profile after the curved mirror (MMT1) that Jenne and Kevin put in the last week. (interference from MMT1) Below is a sketch of the current optical path inside of the chamber. In the beginning of this measurement, the angle between the incident and the reflection on MMT1 (denoted as theta on the sketch) was relatively big (~40deg) although MMT1 was actually made for 0deg incident. At that time we found a spatially large interference imposed on the Gaussian beam at the beam scan. This is not good for mode measurement This bad interference can be caused by an extra reflection from the back surface of MMT1 because the interference completely vanished by removing MMT1 . In order to reduce the interference we decreased the angle theta as small as possible. Actually we made it less than 10deg which was our best due to narrow space. Now the interference got less and the spot looks better. The picture below shows an example of the beam shape taken by using the beam scan. Top panel represents the horizontal mode and bottom panel represents the vertical mode. You can see some bumps caused by the interference on the horizontal mode, these bumps may lead to overestimation of the horizontal spot size . (result) The above plot shows the result of the mode measurement. Here are the parameter obtained by fitting. The data is also attached as attachment:4  waist size for vertical w0v [mm] 0.509 +/-0.0237 waist size for horizontal w0h [mm] 0.537 +/- 0.0150 waist position from MMT1 for vertical xv[m] -2.91 +/- 0.214 waist position from MMT1 for horizontal xh[m] -2.90 +/- 0.127 Attachment 3: MMT1_.dat.zip 3046 Thu Jun 3 14:40:28 2010 Alberto, KiwamuUpdateIOOmode measurement of new input optics  Quote: For the record, we wanted to check whether the fringes on the beam spot were caused by SM2 (see diagram above). We tried two different mirrors for SM2, The first was one of the flat, 45 degree ones that were already on the BS table. The last, which is the one currently in place, was inside the plastic box with the clean optics that Jenne left us . The fringes were present in both cases. 1261 Fri Jan 30 17:30:31 2009 Alberto, JosephbConfigurationComputersNew computer Ottavia set up Alberto, Joseph, Today we installed the computer that some time ago Joe bought for his GigE cameras. It was baptized "OTTAVIA". Ottavia is black, weighs about 20 lbs and it's all her sister, Allegra (who also pays for bad taste in picking names). She runs an Intel Core 2 Quad and has 4GB of RAM. We expect much from her. Some typical post-natal operations were necessary. 1) Editing of the user ID • By means of the command "./usermod -u 1001 controls" we set the user ID of the user controls to 1001, as it is supposed to be. 2) Connection to the Martian network • Ottavia was given IP address 131.215.113.097 by editing the file /etc/sysconfig/networ-scripts/ifcfg-eth0 (we also edited the netmask and the gateway address as in the Wiki) • In linux1, which serves as name server, in the directory /var/named/chroot/var/named, we modified both the IP-to-name and name-to-IP register files 131.215.113.in-addr.arpa.zone and 131.215.11in-addr.martian.zone. • We set the file /etc/resolv.conf so that the OS knows who is the name server. 3) Mounting of the /cvs/cds path • We created locally the empty directories /cvs/cds • We edited the files /etc/fstab adding the line "linux1:/home/cds /cvs/cds nfs rw,bg,soft 0 0" • We implemented the common variables of the controls environment by sourcing the cshrc.40m: in the file /home/controls/.cshrc we added the two lines "source /cvs/cds/caltech/cshrc.40m" and "setenv PATH{PATH}:/cvs/cds/caltech/apps/linux64/matlab/bin/"
2490   Fri Jan 8 20:13:49 2010 Alberto, JoBConfigurationComputersThe 40m Kaiser Permanent Reboot Marathon
This morning after Alex and Jo's tinkering with Megatron the RFM network crashed and it brought down also some computers. The effect was that it was not possible to lock the mode cleaner anymore.
A few computers crashed and things didn't come back to their origianl state.
After an endless day of rebooting and fixing problems with the single front ends (in particular with c1susvme1), eventually the mode cleaner got locked again.
Among my weapons I also used the Nuclear Option (TM).
Maybe I'll include more details in a future elog entry.
Anyway, in the end I burtrestored everything to Jan 8 2009 at 9:00.

2971   Fri May 21 16:41:38 2010 Alberto, JoUpdateComputersIt's a boy!

Today the new Dell computer for the GSCS (General SURF Computing Side) arrived.

We put it together and hooked it up to a monitor. And guess what? It works!

I'm totally impressed by how the Windows get blurred on Windows 7 when you move them around. Good job Microsoft! Totally worth 5 years of R&D.

2520   Mon Jan 18 09:44:36 2010 Alberto, BobOmnistructureEnvironmentNo rain water infiltrations so far

It has rained continuously for the last 24 hours. Bob walked through the lab looking for possible water infiltrations. The floor looked dry: no puddles or leaks anywhere so far.

131   Wed Nov 28 16:18:15 2007 AlbertoMetaphysicsEnvironmentso clean you can eat on it
I tidied up the desks in the lab, brought the Spectrum Analyzers back to the Salumeria (you don't want to know about that), sorted a lot of stuff and boxed up what I didn't know (you can find it in a couple of carton boxes on the table).
The blackmail with the pie might not work next time.

Alberto
Attachment 1: DSC_0180.JPG
Attachment 2: DSC_0181.JPG
142   Thu Nov 29 18:10:13 2007 AlbertoHowToComputer Scripts / ProgramsGPIB Scripts
I've spent a lot of time trying to configurate the GPIB-USB interface for the HP4195. After installing 1) the Agilent libraries, 2) the drivers, 3) the matlab Instrument Toolbox, 4) Jamie script, 5) Alice's script the computer can see the HP but still they can't 'talk' to each other.
I give up. I asked Alice Wang how she managed to get data. I'm not sure she used the GPIB interace. Rob said she might have used the old fashion floppy disks that we can't read anymore here.
I would really appreciate any suggestion by anyone who happened to have the same problems.
246   Thu Jan 17 18:22:14 2008 AlbertoUpdateElectronicsRF Monitor Band-pass Filter
After we finalized the schematic for the RF monitor board based on buffered LC resonators, on Richard Abbott's suggestion to avoid the complication brought in by the fast op-amps, we gave another chance to the a passive configuration of the band-pass filter based on a Chebyshev topology. Rich and Ben gave me an old but very powerful software tool to design that kind of filters and showed me the way to circumvent many hassles in making RF test boards.

I made a test circuit for the 166MHz line (see attached schematic), using tunable inductors. The TF are also attached.
We get more than 20 dB of isolation after 33MHz (with a loss of only few dB at the resonance - it could be less), which is enough for all the other frequencies (33,133,199 MHz) but we would like more for the 166. We are going to add one or two extra orders to the filter.

We also have to understand the spike at about 320Mhz and eventually somehow get rid of it.

Alberto
Attachment 1: RF166Mhz.png
Attachment 2: Chebyshevb.png
Attachment 3: Chebyshev2b.png
248   Fri Jan 18 11:53:50 2008 AlbertoUpdateElectronicsRF Monitor Band-pass Filter
The response is asymmetric and on the left side of the peak, we have at least 33dB within 33Mhz, which is enough for all the frequencies. We probably don't need an higher order filter but just low pass filters in series.

The spike at 320MHz doesn't depend on the circuit board. It's either the cables, their connection, or the splitters.

Note that the frequency of this test circuit has still to be tuned exactly at 166MHz (now it's 149).

Alberto

 Quote: After we finalized the schematic for the RF monitor board based on buffered LC resonators, on Richard Abbott's suggestion to avoid the complication brought in by the fast op-amps, we gave another chance to the a passive configuration of the band-pass filter based on a Chebyshev topology. Rich and Ben gave me an old but very powerful software tool to design that kind of filters and showed me the way to circumvent many hassles in making RF test boards. I made a test circuit for the 166MHz line (see attached schematic), using tunable inductors. The TF are also attached. We get more than 20 dB of isolation after 33MHz (with a loss of only few dB at the resonance - it could be less), which is enough for all the other frequencies (33,133,199 MHz) but we would like more for the 166. We are going to add one or two extra orders to the filter. We also have to understand the spike at about 320Mhz and eventually somehow get rid of it. Alberto
Attachment 1: Chebyshevb.png
268   Fri Jan 25 15:53:59 2008 AlbertoUpdateElectronics40 dB from the 3rd order Chebyschev
I managed to tune the 7 knobs in the 3rd order Chebyshev bandpass filter obtaining the tranfer function attached to this entry. We have now 40 dB of attenuation between 166 Mhz and 133 and 199. With this tuning the insertion loss is rather high. We need a better one.

Alberto
Attachment 1: 166MhzElog.png
285   Wed Jan 30 11:49:30 2008 AlbertoSummaryElectronicsRF monitor's filters final schematics and transfer functions
These are the final schematics for the 6th order Chebyshev filters of the RF monitor board. I'm also attaching the TF as I measured. The tuning is probably not optimal, less insertion noise could be achieved.
Attachment 1: 33Melog30Jan08.png
Attachment 2: 133Melog30Jan08.png
Attachment 3: 166Melog30Jan08.png
Attachment 4: 199Melog30Jan08.png
Attachment 5: 33elog30Jan08.png
Attachment 6: 133elog30Jan08.png
Attachment 7: 166elog30Jan08.png
Attachment 8: 199elog30Jan08.png
314   Wed Feb 13 11:41:00 2008 AlbertoUpdateElectronicsSome characterization of the RF Monitor Box (StocMon)
I'm attaching a table with some measurements and the power spectrum from the pd to help evaluate the numbers.

The box output ranges from 0.5V to 2.1V. The coefficient between power and voltage is negative so higher voltage means lower power.

The red numbers are the outputs from each channel at their resonant frequencies. As one can see these are not very well centered on the dynamic range of the power detectors.

The cross coupling seems to be not a problem.

Even if the 166 filter, which handles the smallest of the frequencies and is also the most lossy (for construction reason), mounts a preamplifier, the output is still rather small. this explain also the high bias due to the noise amplification at the maximum power (13dB). A better insertion loss either remaking the filter or re-tuning that one would simplify many problems, i.e. there is not much room in the metal pomona box to fit the amplifier. I might want to consider, after everything else is ready and if I have time before leaving next week, to work on a new 166 filter.
Attachment 1: CircuitCharacterization.png
Attachment 2: alberto.spectrum3.png
321   Mon Feb 18 12:04:39 2008 AlbertoUpdateElectronicsRF Monitor (StocMon)
I put the amplifiers next to the monitor on the PSL table, layed the power and the RF SMA cables out to the rack. I'm powering the box and the amplifiers with the power supply, waiting for someone to show me tomorrow how to connect it to the Sorensen (Steve, Ben?).

I'm ready to hook up the channels into EPICS.
Attachment 1: DSC_0443.JPG
331   Fri Feb 22 08:29:07 2008 AlbertoUpdateElectronicsRF Monitor (StocMon)

 Quote: I put the amplifiers next to the monitor on the PSL table, layed the power and the RF SMA cables out to the rack. I'm powering the box and the amplifiers with the power supply, waiting for someone to show me tomorrow how to connect it to the Sorensen (Steve, Ben?). I'm ready to hook up the channels into EPICS.

Me and Ben Abbot were plugging the cables that power that RF Monitor box into the PSL rack when inadvertently we made some arcs spark between the pins on the back of one of the ADC. Somehow that made the laser shut down although the MOPA stayed on. We also notice some smell of burn.

Later on, after several failed attempts, Rob, Ben and Steve could restart the laser. It took some times because the written procedure to start the chiller is not very precise.
332   Fri Feb 22 08:33:18 2008 AlbertoUpdateElectronicsRF Monitor (StocMon)

 Quote: I put the amplifiers next to the monitor on the PSL table, layed the power and the RF SMA cables out to the rack. I'm powering the box and the amplifiers with the power supply, waiting for someone to show me tomorrow how to connect it to the Sorensen (Steve, Ben?). I'm ready to hook up the channels into EPICS.

With Ben, we hooked up the RF Monitor box into the PSL rack and created 4 EPICS channels for the outputs:

C1:IOO_RF_STOC_MON_33
C1:IOO_RF_STOC_MON_133
C1:IOO_RF_STOC_MON_166
C1:IOO_RF_STOC_MON_199

The power cable bringing +15V to the preamplifier on the PSL table should be replaced eventually.
545   Thu Jun 19 15:52:06 2008 AlbertoConfigurationComputersMeasure of the current absorbed by the new Megatron Computer
Together with Rich Abbot, sam Abbot and I measured the current absorbed by the new Megatron computer that we installed yesterday in the 1Y3 rack. The computer alone absorbs 8.1A at the startup and then goes down to 5.9A at regime. The rest of the rack took 5.2A without the computer so the all rack needs 13.3 at the startup and the 11.1A.

We also measured the current for the 1Y6 rack where an other similar Sun machine has been installed as temporary frame builder and we get 6.5A.

Alberto, Rich and Sam Abbot
555   Mon Jun 23 21:51:19 2008 AlbertoUpdateGeneralArm Cavity Length Measurement
We measured the arm cavity lengths sweeping the ETM mirror position and looking at the reflected demodulated output. We excited the mirror by a sine wave of 0.2 Hz and amplitude of 30000 counts. From the time series of the occurrences of the resonances of the sidebands and of the carrier we evaluated the free spectral range of the cavities and thus the lengths. The details of the procedure are explained in the attached document. As discussed in it, for each cavity we obtain two possible values of the length depending on which of the sideband resonances is that corresponding to the upper sideband and which corresponds to the lower one instead. The numbers are:
Lx=(38.30 +/- 0.08)m / (38.45 +/- 0.08)m
Ly=(38.16 +/- 0.08)m / (38.70 +/- 0.08)m

Since the difference between the two possibilities is quite large, we should be able to decide which one is correct by somehow measuring directly the cavity length. We want to try it tomorrow by a tape meter.

Alberto and Koji
Attachment 1: 40mLengthMeasure.pdf
621   Wed Jul 2 06:46:05 2008 AlbertoConfigurationGeneralNPRO on to warm up
This morning I turned on the NPRO on the AP table so that it can warm up for a few hours before I start using it today.
The flipping mirror is down so no beam is injected in to the IFO.

Alberto
ELOG V3.1.3-


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 4.3: Applications of Derivatives

Difficulty Level: At Grade Created by: CK-12

## Related Rates

CONTENT

This lesson is a kind of tour through implicit differentiation. It may be best to understand and teach these ideas based upon the idea of differentiation as a kind of operator. That is, if we are given some equality among any number of different variables like:

\begin{align*}a^2+b^2=c^2\end{align*}

then we can operate on both sides of this equation with the derivative operator \begin{align*}\frac{d}{da}\end{align*} to see how everything changes if \begin{align*}a\end{align*} is varied. This would produce:

\begin{align*}2a + 2b \frac{db}{da} = 2c \frac{dc}{da} \Rightarrow \frac{dc}{da} = \frac{a}{c} + \frac{b}{c} \frac{db}{da} \end{align*}

This equality should be interpreted as explaining how the rates of change for the side lengths \begin{align*}c\end{align*} and \begin{align*}b\end{align*} with respect to changes in a are related to one-another.

Alternatively, we could have operated on both sides with the derivative operator \begin{align*}\frac{d}{db}\end{align*} or \begin{align*}\frac{d}{dc}\end{align*} although these would have produced similar results. On the other hand, we could operate on the entire equality by the derivative operator \begin{align*}\frac{d}{dt}\end{align*} which measures the changes in the variables as some outside parameter \begin{align*}t\end{align*} is changed. This would produce the result in the book. The essential point is that any equation also gives a variety of equations relating the rates of change for all the variables involved.

PROCESS

Related rates are readily applicable to real-world problems. For example, how does the radius of a balloon or a car tire change as air is pumped into it? How fast does the amount of carbon-dioxide in the atmosphere increase for a given rate human population increase? How much does a star become brighter as we increase its magnification? All of these questions are questions of related rates, and can be used to make this material more interesting.

A nice way to teach this is to have students actually experiment with the formulas. For example, the students could be split into groups of three to study the formula for a right triangle. One member would be the referee and would need some kind of a stopwatch. The other two students would stand in one corner and walk away from each other along the wall measuring the length to the next corner in steps. Ideally they should coordinate so that their step size is about the same.

The distances along the wall and the diagonal distance between the students at opposite corners should be recorded. Then the referee should count or clap off seconds and the other two students should walk along the wall back towards the corner. One should walk a rate of one step per second and the other at a rate of one step every two seconds. After \begin{align*}4 \mathrm {seconds}\end{align*}, the students should stop and again measure the straight-line distance between them as well as the remaining distance to the corner. Then they should proceed again in the same way for another These numbers taken together should satisfy the equation:

\begin{align*}a \frac{da}{dt} + b \frac{db}{dt} = c \frac{dc}{dt}\end{align*}

where \begin{align*}a\end{align*} is the distance from one student to the wall, \begin{align*}\frac{da}{dt}\end{align*} is his/her walking rate (either \begin{align*}1\end{align*} or \begin{align*}2\end{align*} steps/sec), \begin{align*}b\end{align*} is the distance from one student to the wall, \begin{align*}\frac{db}{dt}\end{align*} is his/her walking rate, \begin{align*}c\end{align*} is the diagonal straight-line distance between the two and \begin{align*}\frac{dc}{dt}\end{align*} is this distance’s rate of change.

PRODUCTS

Word problems are the classic approach to testing this material, but it should be recognized that some students are much better at these than others. For the students who struggle putting words into equations, it may be best to test this material purely using equations. Questions like the following are good for these students:

1. Suppose the variables \begin{align*}x, y\end{align*}, and \begin{align*}z\end{align*} are related by: \begin{align*}x^2-2 \mathrm {Sin}(z)+e^y = \frac{12}{z}\end{align*}

a. What is \begin{align*}\frac{dz}{dx}\end{align*} in terms of \begin{align*}\frac{dy}{dx}\end{align*} and the variables \begin{align*}x, y\end{align*} and \begin{align*}z\end{align*}?

b. When \begin{align*}x=1,y=0\end{align*}, and \begin{align*}z=\pi\end{align*}, the variables \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are being increased in time at a rate of \begin{align*}1\end{align*}. What is the rate of change \begin{align*}\frac{dz}{dt}\end{align*} at this time?

2. The volume of a sphere \begin{align*}V\end{align*} is related to its radius \begin{align*}r\end{align*} by \begin{align*}V = \frac{4}{3} \ \pi r^3\end{align*}. The surface area is actually rate of change in volume as \begin{align*}r\end{align*} changes: \begin{align*}A = \frac{dV}{dr}\end{align*}.

a. Find a formula for \begin{align*}A\end{align*}.

b. If the radius of a unit sphere \begin{align*}(r=1)\end{align*} is increasing at a rate of \begin{align*}10 \;\mathrm{m/s}\end{align*}, how fast are the volume and area increasing? That is, find \begin{align*}\frac{dV}{dt}\end{align*} and \begin{align*}\frac{dA}{dt}\end{align*} when \begin{align*}r=1\end{align*} and \begin{align*}\frac{dr}{dt}=10\end{align*}.

## Extrema and the Mean Value Theorem

CONTENT

In order to teach effectively about extrema, Rolle’s Theorem, and the mean value theorem it is necessary to use geometric intuition and good pictures throughout. The language in the description of a maximum and a minimum is quite complicated but the concept could not be clearer. Forgetting endpoints for a moment, a function has a maximum at\begin{align*}x\end{align*}if its graph is peaked and a minimum if its graph has a trough. To get technical, maxima and minima can also occur at the boundaries where the graph could end at a high point or a low point. However this can only occur if the domain includes some boundary point, and this is undesirable anyway with everyday calculus functions since the function cannot be differentiable there. For example, the function \begin{align*}f(x)=\sqrt{x}\end{align*} may be defined with the domain \begin{align*}x \ge 0\end{align*} so that the boundary point \begin{align*}x=0\end{align*} is included. Then the point \begin{align*}x=0\end{align*} is easily seen to be a minimum of the function although the function fails to be differentiable there. Much more useful and interesting are interior maxima and minima.

The key to understanding everything in this lesson is that the function goes from increasing to decreasing at an interior maximum, and from decreasing to increasing at an interior minimum. This means that the derivative goes from positive to negative in the former, and from negative to positive in the latter. However, for a continuous derivative this is only possible if it is zero at the point in question. That is, a differentiable maximum/minimum must have horizontal tangent line with \begin{align*}\mathrm {slope} = 0\end{align*}. The proof of this “theorem” is given in the text but is intuitive in the following sense: If the derivative were positive at a max or min then the slope at the point would be positive and so the function would be increasing through the point in question. That is, the values would be slightly smaller to the left (so it couldn’t be a minimum) and slightly larger to the right (so it couldn’t be a maximum). The same argument with some elements reversed shows that if the derivative is negative the point again can neither be a maximum nor a minimum. So we are left with the fact that the derivative must be identically zero. This is a very simple geometric concept and students will have the best chance at understanding it in this context. The proof itself will make much more sense when it is clear that this picture is just being put into symbols.

Rolle’s Theorem is not proven in the book but the idea is extremely geometric and the opportunity to show a simple picture in class should not be missed. The point is that if a nice (differentiable) function over \begin{align*}[a,b]\end{align*} starts and ends at the same height \begin{align*}f(a)=f(b)\end{align*}, then it is either flat between or else changed directions somewhere in between. In both cases the derivative is zero somewhere in \begin{align*}[a,b]\end{align*}, either throughout if it is flat or at the point where it changes direction (a maximum or a minimum). The picture is:

The mean value theorem is proven by creating a function from \begin{align*}f\end{align*} that satisfies Rolle’s Theorem. Essentially we tilt the function \begin{align*}f(x)\end{align*} into a new function \begin{align*}g(x)\end{align*} that satisfies \begin{align*}g(a)=g(b)\end{align*}. The text describes the function \begin{align*}g(x)\end{align*} as the difference between \begin{align*}f(x)\end{align*} and the line joining \begin{align*}f(a)\end{align*} and \begin{align*}f(b)\end{align*}. An alternative way of explaining this function to students is to illustrate that our goal is to tilt \begin{align*}f\end{align*} so that we get a function whose values at \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are equal. This amounts to subtracting off the change between \begin{align*}a\end{align*} and \begin{align*}b\end{align*} which can be done by subtracting any function which undergoes the same change between \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. The simplest choice is to subtract any line with slope \begin{align*}m= \frac{f(b)-f(a)}{(b-a)}\end{align*} such as the line \begin{align*}y=m \ x\end{align*}. In the proof given the line connecting \begin{align*}f(a)\end{align*} and \begin{align*}f(b)\end{align*} is chosen instead, which clearly also has slope \begin{align*}m\end{align*}.

PROCESS

Teaching this material can be a little tricky since it is largely conceptual. The best technique may be to have students pair off so that one person will first explain how to prove that if a point is an interior extrema then its derivative is zero. The other person will then explain how to prove the mean value theorem. The partner who is explaining should have only paper and pencil whereas the other person can help him/her along with the help of the book. This will help students become more comfortable with the logical progression of these proofs.

PRODUCTS

Students can be tested on this material by finding maxima and minima and by sketching a great many plots. This will be particularly useful with the next few chapters in mind. By having the goal in mind that they are to use derivatives in order to plot functions, students will be far more likely to understand what follows.

## The First Derivative Test

CONTENT

The content of this chapter really is purely geometric and has the best chance at getting through to students in this way. It is quite simple, when the derivative is positive the function is increasing and when it is negative, it is decreasing. This is actually precisely what the derivative measures, whether a function is increasing or decreasing. Furthermore, it should be quite intuitive that a function increases to a maximum then decreases away from it, and that the opposite is true of a minimum.

PROCESS

This material can be conveyed to students by having them think before studying the formal ideas. They should consider some complicated looking function like

\begin{align*}f(x) = \frac{x}{\text{log}(x)}\end{align*}

We find that its derivative:

\begin{align*}f'(x) = \frac{\text{log}(x) - 1}{[\text{log}(x)]^2}\end{align*}

is zero whenever \begin{align*}\mathrm{log}(x) - 1 = 0\end{align*}. This is true only when \begin{align*}x=e\end{align*} so that means that\begin{align*}x=e\end{align*} is a critical point of the function \begin{align*}f(x)\end{align*}.

Students should then be asked how they can tell if\begin{align*}x=e\end{align*}is a maximum, a minimum, or a saddle point. After thinking about this they should either graph the function or discuss what is happening for \begin{align*}x\end{align*} close to \begin{align*}x=e\end{align*} in terms of the derivative. In the former case the graph of \begin{align*}f(x)=\frac{x}{\mathrm{log}(x)}\end{align*} should look like the following:

This plot makes it clear that\begin{align*}x=e\end{align*}is a minimum. In the latter case the students may notice that for \begin{align*}x < e, f'(x)\end{align*} has a negative numerator and positive denominator and is therefore negative. This means that \begin{align*}f\end{align*} is decreasing to the left of \begin{align*}x=e\end{align*}. On the other hand for \begin{align*}x>e, f'(x)\end{align*} has a positive numerator and a negative denominator and so is positive. This means \begin{align*}f\end{align*} is increasing to the right, so the point \begin{align*}x=e\end{align*} is a minimum. This is a great project for pairs. One student can plot the function to see that \begin{align*}x=e\end{align*} is a minimum and the other student can talk about the derivative.

PRODUCTS

A very nice little riddle is solved by the function above. Consider asking students to solve the problem:

Using the following steps instead of your calculator, determine which is bigger: \begin{align*}e^{\pi}\end{align*} or \begin{align*}\pi^e\end{align*}?

a. Start with the equation \begin{align*}e^{\pi}\neq \pi^e\end{align*}, take the log of both sides, and simplify:

i. Answer: \begin{align*}\pi \neq e \mathrm{log}(\pi)\end{align*}

b. Now manipulate this inequality so that one side involves only \begin{align*}\pi\end{align*} and the other side involves only \begin{align*}e\end{align*}:

i. Answer: \begin{align*}\frac{\pi}{\text{log}(\pi)}\neq e\end{align*}

c. Now notice that by replacing \begin{align*}\pi\end{align*} by the variable \begin{align*}x\end{align*} we can consider this side of the inequality to be a function of \begin{align*}x\end{align*}. What are the critical point/s of this function and are they maxima or minima?

i. Answer: We solved this above, the function \begin{align*}f(x) = \frac{x}{\text{log}(x)}\end{align*} has the critical point\begin{align*}x=e\end{align*} only. This, we determined, was a minimum!

d. Now for the punch line, what is the value of this function at its critical point/s?

i. Answer: At \begin{align*}x=e, f(x) = \frac{e}{\text{log}\ (e)} = e\end{align*}

e. How do these value/s compare with the function’s value at \begin{align*}x=\pi\end{align*}?

i. Answer: \begin{align*}x=e\end{align*}is a minimum, so \begin{align*}f(\pi)>f(e)\end{align*} or \begin{align*}\frac{\pi}{log(\pi)} >e\end{align*} or \begin{align*}\pi > e \mathrm{log}(\pi)\end{align*} or taking the exponential of both sides: \begin{align*}e^{\pi} > \pi^e\end{align*}

f. What are the numerical values of \begin{align*}e^{\pi}\end{align*} and \begin{align*}\pi^e\end{align*}?

i. Answer: \begin{align*}e^{\pi} \cong 23.1407, \pi^e \cong 22.4592\end{align*}

## The Second Derivative Test

CONTENT

Notice that in order to determine whether critical values were maxima, minima, or saddle points we appealed to how the derivative was changing. If the derivative was positive beforehand and then negative after, i.e. decreasing, then the point was a maximum. If on the other hand the derivative was negative beforehand and then positive after, i.e. increasing, then the point was a minimum. If on the other hand the sign did not change, i.e. the derivative was constant; the point was a saddle point. The quick way to check how something is changing is to take its derivative. So if we want to see how the derivative is changing, we should look at its derivative: The second derivative!

When the second derivative is positive the first derivative is increasing and this gives the function a concave up shape. If this is true at a critical point then the critical point must be a minimum. When the second derivative is negative the first derivative is decreasing and this gives the function a concave down shape. If this is true at a critical point then the critical point must be a maximum. If the second derivative is zero at a critical point then the function may have a maximum, minimum, or saddle point there.

PROCESS

Recall the complicated function we examined earlier:

\begin{align*}f(x) = \frac{x}{\text{log}(x)}\end{align*}

we used its first derivative:

\begin{align*}f'(x) = \frac{\text{log}(x) - 1}{[\text{log}(x)]^2}\end{align*}

to see that it had a critical point at \begin{align*}x=e\end{align*}. We can conclusively classify this critical point by looking at how the derivative is changing at \begin{align*}x=e\end{align*}. This is determined by finding \begin{align*}f''(e)\end{align*} and to get this we will need to calculate that:

\begin{align*}f''(x) = \frac{2 - \text{log}(x)}{x [ \text{log}(x)]^3}\end{align*}

From this we see easily that:

\begin{align*}f''(e) = \frac{2 - \text{log}(e)}{e[\text{log}(e)]^3} = \frac{2 - 1}{e 1^3} = \frac{1}{e} \cong 0.37\end{align*}

So the second derivative is positive at the critical point\begin{align*}x=e\end{align*}which means that \begin{align*}f'(x)\end{align*} is increasing there (going from negative to positive) and so\begin{align*}x=e\end{align*}is a minimum.

PRODUCTS

Students can be tested on this material with a large variety of word problems asking them to formulate a function and then classify its critical points. For students who do not work well with word problems it may be better to simply give functions directly and ask students to classify their critical points using the first and second derivatives. As a look ahead, again, it is useful to have students draw plots of the functions as well, using the information that they have determined.

## Limits at Infinity

CONTENT

In order to understand the behavior of functions, it is often important to look more closely near points they are not defined, and examine how they behave after they run off your graph paper. A function like \begin{align*}f(x)= \frac{x^2-1}{x-1}\end{align*} is undefined at \begin{align*}x=1\end{align*} however we see clearly that because \begin{align*}\lim_{x \to 1} f(x) = 2\end{align*} the function doesn’t do anything very exciting there. On the other hand, the simple function \begin{align*}f(x) = \frac{1}{x}\end{align*} is defined at \begin{align*}x=0\end{align*} and the limits:

\begin{align*}\lim_{x \to 0^+} f(x) & = \infty \\ \lim_{x \to 0^-} f(x) & = - \infty\end{align*}

tell us that to the left of the origin the function tends down towards negative infinity and to the right coming in towards zero it tends up to positive infinity.

We can also look at how functions behave for very very large values of\begin{align*}x\end{align*}by examining the limits: \begin{align*}\lim_{x \to \pm \infty} f(x)\end{align*}. The simplest case is if this limit is zero or some other constant number. Then we see that the function simply approaches this horizontal line. This is the case with the function \begin{align*}f(x) = \frac{1}{x}\end{align*} or with the function \begin{align*}f(x) = C+ \frac{1}{x^2}\end{align*} for example. However if the limit is \begin{align*}\pm \infty\end{align*} then we have to be a little more careful. For example, consider the function from above again:

\begin{align*}f(x)= \frac{x^2-1}{x-1}\end{align*}

Although we see clearly that \begin{align*}\lim_{x \to \infty} = \infty\end{align*}, there is more that can be said. For large values of \begin{align*}x\end{align*}, the constant values of \begin{align*}1\end{align*} in the numerator and in the denominator are irrelevant so the function begins to look more and more like:

\begin{align*}\lim_{x \to \infty} \frac{x^2 -1}{x-1} = \lim_{x \to \infty} \frac{x^2}{x} = \lim_{x \to \infty} x\end{align*}

That is to say that in this limit, for large \begin{align*}x\end{align*}, the function begins to look a lot like the straight-line function \begin{align*}g(x)=x\end{align*}. This is sometimes described by saying that \begin{align*}f(x)\end{align*} “behaves like\begin{align*}x\end{align*}or diverges like\begin{align*}x\end{align*}for large \begin{align*}x\end{align*}.” It is also sometimes said that \begin{align*}f(x)\end{align*} has a diagonal asymptote in the line \begin{align*}g(x)=x\end{align*}.

PROCESS

With an eye towards what follows, students may find it useful to practice this material by dividing into small groups and analyzing a function. Each group could be given a function like \begin{align*}f(x)= x + \mathrm {Sin}(x) + \frac{1}{x^2}\end{align*} and the group should be asked to thoroughly analyze their function. Find all of its critical points, classify these, examine all of its limits and points of discontinuity, and describe if there is any asymptotic behavior. For example, the function given has an infinite number of local maxima and minima corresponding to each half-integer multiple of \begin{align*}\pi\end{align*}, will diverge to plus \begin{align*}\infty\end{align*} from both sides at zero, and will asymptotically diverge like \begin{align*}g(x)=x\end{align*} for values of \begin{align*}x\end{align*} that are from zero.

PRODUCTS

This material is readily tested in a way that leads nicely towards using derivatives and limits to graph functions. Students should be given some function with a variety of critical points and discontinuities and asked to consider how the function behaves near these points. This should be done by taking limits if the derivative is not defined or one and two derivatives where it is defined and checking for their signs. Then the student should be asked to analyze how the function behaves for values of\begin{align*}x\end{align*}that are far from the origin by examining limits.

## Analyzing the Graph of a Function

CONTENT

This section is essentially a culmination of the ideas that students have been considering thus far. By putting together all of the tools they have been given, it should be much easier to get a feeling for how a function’s plot will look ahead of time.

PROCESS

This section should be taught by having students repeat the kinds of exercises they have done earlier. The class could be split into pairs and each team could thoroughly analyze some complicated function by finding all the critical points, classifying each, and examining the limits at discontinuities and with each tail. It would be nice if each pair could present their function to the class on the board so that everyone could see the variety of functions.

PRODUCTS

This material is best tested by having students thoroughly analyze at least one function with some interesting behavior. The best functions are usually rational functions.

## Optimization

CONTENT

Optimization problems are probably the best way to make calculus seem important. This content should be described as one of the most important things that you hope to convey to students in the course. The tools they have learned thus far will be consolidated in this section to provide an enormous application.

PROCESS

The best way to teach this material is to provide students with a simple question. Suppose you are CEO of Starbucks and are trying to decide how expensive a small coffee should cost to obtain the most profit. Clearly if the coffee is free the profit will be zero. On the other hand, if the price is too high nobody will buy it and the profit will also be zero. So there must be some intermediate price that is not too high and not too low so that the profit is the greatest. This is a problem of optimization, and it can be solved using calculus.

If students are interested, tell them that by analyzing years and years of data you have modeled the profit \begin{align*}y\end{align*} for different prices \begin{align*}x\end{align*} with the function:

\begin{align*}y = \frac{x^3}{1 + e^{x^2}}\end{align*}

It is clear that this function satisfies our expectations that profit \begin{align*}y\end{align*} should be zero for \begin{align*}x= \0\end{align*} and that it should become zero for very large values of\begin{align*}x\end{align*}as well. This can be shown by using L’Hopital’s rule to find the limit at large \begin{align*}x\end{align*}.

In order to determine the optimal price, we must use calculus. We take a derivative of the function to see that:

\begin{align*}y'= \frac{3x^2+e^{x^2}(3x^2-2x^4)}{(1+e^{x^2})^2}\end{align*}

In order to find the maximum, we would like to look for critical points. That is, we want to look for prices \begin{align*}x\end{align*} that make the derivative vanish. Since the denominator is never zero, the fraction is zero whenever its numerator \begin{align*}3x^2+e^{x^2}(3x^2-2x^4)=0\end{align*}.

Now, this equation cannot be solved by algebra or by any other exact means. So we must resort to something like Newton’s Method. So we let:

\begin{align*}f(x)=3x^2+e^{x^2}(3x^2-2x^4)\end{align*}

and we calculate that:

\begin{align*}f'(x)=6x+e^{x^2}(6x-2x^3-4x^5)\end{align*}

If we guess that a good price for coffee might be close to \begin{align*}\2\end{align*} then we might set \begin{align*}x_0=2\end{align*} and use Newton’s Method. We find that:

\begin{align*}x_0 & = \2 \\ x_1 & = x_0 - \frac{f(x_0 )}{f'(x_0 )} = \1.8499 \\ x_2 & = x_1- \frac{f(x_1 )}{f'(x_1 )} = \1.70397 \\ & \qquad \qquad \ \ \ldots \\ \lim_{n \to \infty} x_n & = \1.32607\end{align*}

So the derivative \begin{align*}y'\end{align*} is zero for \begin{align*}x \cong \1.33\end{align*}. Furthermore, we see that the second derivative:

\begin{align*}y'' = \frac{2x (3 + e^{x^2}(6 - 7x^2 - 2x^4 + e^{x^2}(3 - 7x^2 + 2x^4)))}{(1 + e^{x^2})^3}\end{align*}

at \begin{align*}x=1.33\end{align*} is about \begin{align*}-1.47\end{align*} so the critical point is a maximum. The price for a small coffee that will maximize profit is \begin{align*}\1.33\end{align*}.

PRODUCTS

There are some great problems in optimization for students who are good at word problems and for students who are not. Word problems can be avoided by asking for specific properties of an explicit function or by simply presenting a word problem with the equations explicitly written out.

## Approximation Errors

CONTENT

This lesson is best explained by first saying that essentially any function that we’ll use in calculus is exactly equal to its Taylor Polynomial as \begin{align*} n \rightarrow \infty\end{align*}. That is, we have that for any “nice” (infinitely differentiable) function we can pick a point \begin{align*}x_0\end{align*} and we have for all points that are in the same part of the domain as \begin{align*}x_0\end{align*} that:

\begin{align*}f(x)=f(x_0 )+f'(x_0 )(x-x_0 )+ \frac{f''(x_0 )}{2!} (x-x_0 )^2 + \frac{f'''(x_0 )}{3!} (x-x_0 )^3 + \ldots\end{align*}

where the dots are meant to imply that this summation goes on and on without ever ending. The fact is that the terms are guaranteed to eventually get smaller and smaller because of the factorial in the denominator, no matter how big \begin{align*}x-x_0\end{align*} may be.

An approximation just means that instead of taking all the infinite number of terms, we cut off the series somewhere. The remaining infinite terms added up are called the “tail” of the series and the approximation is good when this tail is small. That is because the tail represents the difference between the actual function and the approximate function. Thus, we make the approximation better and better when we take more terms or when \begin{align*}x-x_0\end{align*} is small since the tail of the series will be smaller in each case.

PROCESS

Students could be split into groups of three that are each given a function and a value for \begin{align*}x_0\end{align*}. For example, one group could be given \begin{align*}f(x)=\mathrm {Sin}(x)\end{align*} and \begin{align*}x_0 = \pi\end{align*}, and another could be given \begin{align*}f(x)= \sqrt{x}\end{align*} and \begin{align*}x_0=9\end{align*}. Then the groups should designate tasks to each member: One member will be the artist, one will be the derivator, and one will be the calculator.

The calculator will begin by making a table with a series of points near \begin{align*}x_0\end{align*} and the function’s values there, like the example below:

\begin{align*}& x && f(x)=\sqrt{x} && L(x)=f(x_0 ) + f'(x_0 )(x-x_0) && Q(x)=f(x_0 )+f'(x_0 )(x-x_0)+ \frac{(f''(x_0 )}{2} (x-x_0 )^2 \\ &&&&&=3 + \frac{1}{6}(x-3) && =3+ \frac{1}{6} (x-3)- \frac{1}{216}(x-3)^2\\ & 8.7 && 2.95 && 2.94958 && 2.94958 \\ & 8.8 && 2.96667 && 2.96648 && 2.96648 \\ & 8.9 && 2.98333 && 2.98329 && 2.98329\\ & 9 && 3 && 3 && 3 \\ & 9.1 && 3.01667 && 3.01662 && 3.01662 \\ & 9.2 && 3.03333 && 3.03315 && 3.03315 \\ & 9.3 && 3.05 && 3.04959 && 3.04959\end{align*}

Then the artist will work with this table to draw a careful graph of the function on graph paper near the point \begin{align*}x_0\end{align*} while the mathematician determines the equation for the linearization of the group’s function near the point \begin{align*}x_0\end{align*} and the quadratic approximation as in the table above.

The calculator will calculate and fill in the column of the table for the same \begin{align*}x-\end{align*}values using the linearization and quadratic approximation, and the artist will draw in a plot of the approximations. Enough points should be used with enough spacing to be able to see the approximations deviate from the function. This may require a very fine scale, a large spacing, or a great many points. This will be the task of the group as a whole to figure out what window to use and how many points to include.

PRODUCTS

Students can be tested on this material by having them create approximations for functions with various degrees of accuracy.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Show Hide Details
Description
Authors:
Tags:
Subjects:

format: plain


text:
# Surrogate relaxation

Jump to: navigation, search

Solving the surrogate dual problem, see the supplement on duals. For given multipliers $LaTeX: \textstyle (u,v) (u \ge 0),$ the surrogate dual objective is the relaxed mathematical program:

$LaTeX: \max \left \{f(x) : x \in X, ug(x) \le 0, vh(x) = 0 \right \}.$

(More generally, the surrogate could be a nonlinear combination of the constraints.)


text:
# Solving a System of Equations

Systems of linear equations take place when there is more than one related math expression. For example, in $$y = 3x + 7$$, there is only one line with all the points on that line representing the solution set for the above equation.

When you are given 2 equations in the same question, and asked to solve for a unique answer, you can visualize the problem as be two lines on the same xy-plane. The following two equations are graphed on the same xy-plane:

$$y = 3x + 5$$ $$y = - x$$

The solution set to any equation is the place where BOTH equations meet on the xy-plane. This meeting place is called the Point of Intersection. If you have a linear equation and a quadratic equation on the same xy-plane, there may be TWO POINTS where the graph of each equation will meet or intersect. Here's a geometric view:

Here is a sample of two equations with two unknown variables:

### Example

$$x + y = 10$$ $$3x + 2y = 20$$

There are three methods to solve our sample question.

• 1) We can solve it graphically
• 2) We can solve it algebraically
• 3) We can also solve it through algebraic elimination

I will solve the question using all 3 methods. Method 1: Solve Graphically:

To solve graphically, it is best to write BOTH equations in the slope-intercept form or in the form: $$y = mx + b$$ where m = the slope and b = the y-intercept as your first step. So, $$x + y = 10$$ becomes $$y = - x + 10$$ (slope-intercept form). Next, $$3x + 2y = 20$$ becomes $$y = -\frac{3x}{2} + 10$$ when written in slope-intercept form.

Then, graph the two lines, leading to the point of intersection. After graphing these lines, you'll find that BOTH equations meet at point (0,10). Point (0,10) means that if you plug x = 0 and y = 10 into BOTH original equations, you will find that it solves both equations. Here's what these two equations look like on the xy-plane:

### Method 2: Solve algebraically

Steps:

1) Solve for eaither x or y in the first equation ($$x + y = 10$$). I will solve for y. So, $$x + y = 10$$ becomes $$y = -x + 10$$.

2) Plug the value of y (that is, -x + 10) in the second equation to find x. Our second equation was $$3x + 2y = 20$$ and, after substituting, becomes $$3x + 2(-x + 10 ) = 20$$

Next: Solve for x.

$$3x -2x + 20 = 20$$ $$x + 20 = 20$$ $$x = 0$$

3) Plug x = 0 into EITHER original equations to find the value of y. I will use our second equation.

$$3x + 2y = 20$$ $$3(0) + 2y = 20$$ $$0 + 2y = 20$$ $$y = 10$$

So, our point of intersection is once again (0,10).

### Method 3: Algebraic Elimination

This method deals with matching the variables to ELIMINATE or do away with one. Keep in mind that it is your choice which variable you want to eliminate first.

GOAL: Eliminate x and solve for y or vice-versa. Let's go back to our original equations.

In our second 3x + 2y = 20, you can eliminate 3x by multiplying -3 by EVERY term in our first equation (x + y = 10).

x + y = 10
3x + 2y = 20

-3(x) + -3(y) = -3(10)
3x + 2y = 20

-3x + -3y = -30
3x + 2y = 20

NOTICE that -3x and 3x are eliminated. See it? See why? Here's why: A negative PLUS a positive = ZERO.

We now have this:

-3y = -30
2y = 20

-3y + 2y = -30 + 20

-y = -10

y = 10.

Next: To find x, we plug y = 10 into EITHER of the original equations. By now you should see that our answer for x will be ZERO.

Here it is:

I will use x + y = 10

x + 10 = 10

x = 0.

Do you see what I see? Yes, I found the SAME point of intersection again, which is (0,10).

By Mr. Feliz
(c) 2005


text:
# Equation of a curve

1. Mar 17, 2009

### locachola17

1. The problem statement, all variables and given/known data

A curve passes through the point (0, 9) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?

2. Relevant equations

3. The attempt at a solution

2. Mar 17, 2009

### gabbagabbahey

Hi locachola, welcome to PF!

We're not here to do your homework for you, we're here to help you learn. You must show some attempt at a solution, in order to receive help....

3. Mar 17, 2009

### lanedance

Hi locachola17 wlecome to PF

The idea of the forum is to help with your working... so any ideas on how to get started?

as a hint, the line
"has the property that the slope of the curve at every point P is twice the y-coordinate of P"
can you write this line as an equation?

4. Mar 17, 2009

### locachola17

okay well at the point (0,9) the y-coordinate is 9
if the slope is twice the y-coordinate, the slope is 18?!

5. Mar 17, 2009

### gabbagabbahey

Sure, but what about at any other point (x,y) on the curve? What does that tell you about dy/dx for this curve?

6. Mar 17, 2009

### lanedance

thats for that single point....

but the question says its true for every point on the line... how would you write this to show it for every point on the line...?

remeber the slope is given by the derivative y'(x) = dy/dx

7. Mar 17, 2009

### locachola17

dy/dx is 2y

8. Mar 17, 2009

### gabbagabbahey

Right, so y(x)=____?

9. Mar 17, 2009

### gabbagabbahey

No it doesn't...It says for every point on the "curve"....the curve is not a line

10. Mar 17, 2009

### locachola17

y(x)=y^2

11. Mar 17, 2009

### gabbagabbahey

That makes no sense, you have a separable differential equation for y(x): $$\frac{dy(x)}{dx}=2y(x)$$....how do you usually solve a separable DE?

12. Mar 17, 2009

### locachola17

you integrate both sides?

13. Mar 17, 2009

### gabbagabbahey

How do you determine $$\int y(x)dx$$ when you don't know what y(x) is?

No, you separate varaiables first and then integrate:

$$\frac{dy}{dx}=2y\implies \frac{dy}{y}=2dx$$

Now you can integrate....


text:
# Thread: I need help /w equations of some lines

1. ## I need help /w equations of some lines

Hey everyone!

I can't seem to solve these two questions. Any help is appreciated!

1) A line has direction angles of 60 degrees, 45 degrees and 60 degrees. It passes through the point (1, -2, 5). Determine the vector equation of this line.

2) Find the vector equation of a line that intersects the following 2 lines at 90 degrees.
Line 1: (x,y,z) = (4, 8, -1) + a(2, 3, -4)
Line 2: (x, y,z) = (7, 2, -1) + a(6, 1, 2)

-Mike

2. Originally Posted by Mike49214
...

2) Find the vector equation of a line that intersects the following 2 lines at 90 degrees.
Line 1: (x,y,z) = (4, 8, -1) + a(2, 3, -4)
Line 2: (x, y,z) = (7, 2, -1) + a(6, 1, 2)

-Mike
1. The direction vectors of the lines are different therefore the lines are not parallel.
2. An attempt to calculate the point of intersection doesn't yield a result therefore the lines are skewed.
3. The cross-product of the direction vectors yield the direction perpendicular to both lines:

$(2,3,-4) \times (6,1,2) = (10, -28, -16)$

4. Now use this direction vector and any arbitrary point on $L_1$ to get the equation of a family of parallel lines:

$p: (x,y,z) = q \cdot (10,-28,-16)+\left((4, 8, -1) + r(2, 3, -4)\right)$ ....... r is considered to be constant with a certain line of the family.

5. One (and only one) member of this family will intersect the line $L_2$:

$q \cdot (10,-28,-16)+\left((4, 8, -1) + r(2, 3, -4)\right) = (7, 2, -1) + s(6, 1, 2)$

6. Solve this system of linear equations for r, s and q.
I've got $q=-\frac{33}{190}~\wedge~s=-\frac{48}{95}~\wedge ~=-\frac{36}{95}$

7. The 2 points on different lines which have the smallest distance are:

$P\left(\frac{449}{95}~,~\frac{154}{95}~,~ -\frac{167}{95}\right)$ and $Q\left(\frac{284}{95}~,~\frac{616}{95}~,~ \frac{97}{95}\right)$

8. Take one of the two points to calculate the equation of the line:

$p: (x,y,z) = \left(\frac{449}{95}~,~\frac{154}{95}~,~ -\frac{167}{95}\right) + t \cdot (10,-28,-16)$

My results do look really ugly, so better check my arithmetic. (If they are miraculously correct you are allowed to think two or three ugly words about the author of this question)

3. Originally Posted by earboth
...
I've gone over it like 3 times and I can't see any errors in your calculations. Thanks a lot for the help, I appreciate it.

I am still having trouble with the first question though, if anyone would be nice enough to lend a hand.

4. Originally Posted by Mike49214
I've gone over it like 3 times and I can't see any errors in your calculations. Thanks a lot for the help, I appreciate it.

I am still having trouble with the first question though, if anyone would be nice enough to lend a hand.
Nobody of the forum has answered your question because it isn't understandable:

You mention three angles. Which geometric objects form the angles? The line and axes? (if yes which axis belongs to which angle?), planes? (if yes which plane belongs to which angle?)

... and if you could provide a rough sketch that would really help us to help you.