# Users guide for the Field II program

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Users guide for the Field II program Release 3.24, May 12, 214 Jørgen Arendt Jensen May 21, 214 Jørgen Arendt Jensen May 21, 214 Department of Electrical Engineering, Build. 349, Technical University of Denmark DK-28 Lyngby, Denmark Web:

2 CONTENTS 1 Introduction 3 2 Program organization 5 3 Method of simulation The spatial impulse response Simulation Focusing and apodization Attenuation Installation 9 5 Description of Matlab procedures List of current procedures Procedures for Field initialization Procedures for transducer definition Procedures for element manipulation Procedures for field calculation Examples Phased array imaging Linear array imaging Flow data generation Bibliography 66 i

3 ii

4 LIST OF FIGURES 5.1 Concave, round transducer with a radius of 8 mm divided into 1 by 1 mm mathematical elements Rectangles for a convex array with Rconvex equal to 2 mm Rectangles for an elevation focused, convex array with Rfocus equal to 1 mm and Rconvex equal to 3 mm Rectangles for an elevation focused, multi-row, convex array with Rfocus equal to 7 mm and Rconvex equal to 3 mm Rectangles for an elevation focused, linear array with Rfocus equal to 15 mm Rectangles for an elevation focused, multi-row linear array with Rfocus equal to 1 mm and 5 rows Display of the geometry and apodization of a linear array transducer Rectangles for a 16 elements linear array transducer Geomtery of multi-row linear array transducer. Currently x and y has been switched Rectangles for a 16 by 5 elements multi-row transducer Piston transducer with a radius of 8 mm divided into 1 by 1 mm mathematical elements Fully populated two-dimensional array with 11 by 13 elements Partially populated two-dimensional array with 23 elements Linear array transducer with a fixed apodization of the mathematical elements Intensity profile for linear array transducer with an elevation focus lens Example of calculated response when using different physical element excitations Received voltage traces from the individual elements of a 16 elements linear array transducer, when transmitting with three different elements Received voltage traces from the individual elements of a linear array transducer (top) and the sum of all the individual responses (bottom)

5 2

6 CHAPTER ONE Introduction This is the user guide for the version 3.24 of May 12, 214 of the Field II program. This version of the program runs under Matlab and can simulate all kinds ultrasound transducers and the associated images. The focusing and apodization of the transducers can be controlled dynamically, and it is, thus, possible to simulate all kinds of ultrasound imaging systems. The versions can also be used for synthetic aperture imaging. The program is free for use, if you make a proper reference to the papers describing the program, when you publish results from its use. The reference are [1] and [2]. Also the name of the program (Field II) should be mentioned in the publication. Some unfortunately forget this, and the program will only stay in the public domain, if people continue to properly acknowledge its use. This guide is intended as a presentation of the currently available routines. It includes a few examples and gives a small amount of background information. It is, however, not intended as an introduction to ultrasound scanning, and the reader should consult the extensive literature on this. The program executables can be downloaded from the Web-site for the program: It currently exist for a number of platforms like Windows, Linux, Mac OS-X. Versions for other operating systems are generally discontinued due to lack of demand. A parallel Pro version also exits. Information about this can be found on the web-site above. The web site also contains more extensive examples than are given in this guide, and up-to-date references and papers can also be found on the web-site. The manual is made as a clickable pdf document with hyperlinks. All links are indicated in blue, and when clicked on will lead to the indicated references, which can be a web-site, figure, equation, etc. The manual is organized as follows: Chapter 2 gives an overview of the organization of the program and how it is connected to Matlab. Chapter 4 details the installation from the programs on the web-site. A listing of all procedures callable in the program is given in Chapter 5 and finally a few examples are given in 6. More can be found on the web. Jørgen Arendt Jensen May 21, 214 Department of Electrical Engineering, Build. 349, Technical University of Denmark DK-28 Lyngby, Denmark 1 Older versions can be found on the web-site for Matlab 4-7 3

7 4

8 CHAPTER TWO Program organization The program consists of a C program and a number of Matlab m-functions that calls this program. All calculations are performed by the C program, and all data is kept by the C program. Three types of m-functions are found. The are used for initializing the program, defining and manipulating transducers, and for performing calculations. The initializing routines are preceeded by field, the transducer commands by xdc, and the calculation routines by calc. Help on use of the routines can be obtained by typing help <routine name>. Each of the routines are described in the following section and then three examples of use are given. The first shows how a phased array image is generated, the second simulates a flow system, and the last example is for a linear array system. The last example uses a computer generated phantom. The m-file for this phantom is also given in the example section. 5

9 6

10 CHAPTER THREE Method of simulation 3.1 The spatial impulse response The Field program system uses the concept of spatial impulse responses as developed by Tupholme and Stepanishen in a series of papers [9, 1, 11]. The approach relies on linear systems theory to find the ultrasound field for both the pulsed and continuous wave case. This is done through the spatial impulse response. This response gives the emitted ultrasound field at a specific point in space as function of time, when the transducer is excitated by a Dirac delta function. The field for any kind of excitation can then be found by just convolving the spatial impulse response with the excitation function. The impulse response will vary as a function of position relative to the transducer, hence the name spatial impulse response. The received response from a small oscillating sphere can be found by acoustic reciprocity. The spatial impulse response equals the received response for a spherical wave emitted by a point. The total received response in pulseecho can, thus, be found by convolving the transducer excitation function with the spatial impulse response of the emitting aperture, with the spatial impulse response of the receiving aperture, and then taking into account the electromechanical transfer function of the transducer to yield the received voltage trace. An explanation and rigorous proof of this can be found in [14] and [15]. Any excitation can be used, since linear systems theory is used. The result for the continuous wave case is found by Fourier transforming the spatial impulse response for the given frequency. The approach taken here can, thus, yield all the diffent commenly found ultrasound fields for linear propagation. 3.2 Simulation A number of different authors have calculated the spatial impulse response for different transducer geometries. But in general it is difficult to calculate a solution, and especially if apodization of the transducer is taken into account. Here the transducer surface does not vibrate as a piston, e.g. the edges might vibarte less then the center. The simulation program circumvents this problem by dividing the transducer surface into squares and the sum the response of these squares to yield the response. Thereby any tranducer geometry and any apodization can be simulated. The approach is described in [1]. The time for one simulation is also of major concern. As the squares making up the tranducer apertue are small, it is appropriate to use a far-field approximation, making simulation simple. Another issue in keeping the simulation time down is to use a low sampling frequency. Often spatial impulse responses are calculated using sampling frequencies in the GHz range due to the sharp discontinuities of the responses. These discontinuities are handled in the Field programs by accurately keeping track of the time position of the responses and uses the integrated spatial impulse response as an intermediate step in the calculations. Thereby no energy is lost in the response, which is far more important than having an exact shape of the spatial impulse response. Hereby the Field program ususally does better using 1 MHz sampling and approximate calculations, than using the exact analytic expression and GHz sampling. 7

11 3.3 Focusing and apodization The focusing and apodization is handled in the program through time lines as: Focusing: From time Focus at x 1, y 1, z 1 t 1 x 1, y 1, z 1 t 2 x 2, y 2, z 2.. Apodization: From time Apodize with a 1,1, a 1,2, a 1,Ne t 1 a 1,1, a 1,2, a 1,Ne t 2 a 2,1, a 2,2, a 2,Ne t 3 a 3,1, a 3,2, a 3,Ne.. For each focal zone there is an associated focal point and the time from which this focus is used. The arrival time from the field point to the physical transducer element is used for deciding which focus is used. The focusing can also be set to be dynamic, so that the focus is changed as a function of time and thereby depth. The focusing is then set as a direction defined by two angles and a starting point on the aperture. All the time values for focusing are calculated relative to a point on the aperture. Initially this is set to (,, ). It can be set to other values through the procedure xdc center focus. This is used in linear array imaging, where the origin of the emitted and received beam is moved over the aperture. The focusing values are calculated by: t i = 1 ( ) (x c x f ) c 2 + (y c y f ) 2 + (z c z f ) 2 (x i x f ) 2 + (y i y f ) 2 + (z i z f ) 2 (3.1) where (x f, y f, z f ) is the position of the focal point, (x c, y c, z c ) is the reference center point on the aperture for the focus as set by xdc center focus, (x i, y i, z i ) is the center for the physical element number i, c is the speed of sound, and t i is the calculated delay time. The value is then quantized, if that is set for the aperture. The time line method is employed for the apodization, where the time decides which apodization vector is used. The vector holds one apodization value for each physical element. 3.4 Attenuation Frequency dependent attenuation can be included in the simulation by using the procedure set field. The attenuation is included through a frequency dependent term and a frequency independent term. The frequency dependent term is linearized through a center frequency att f, so that the attenuation is zero db at att f. This is done to make the inclusion of the attenuation computationally efficient. The variation in distance over the aperture of the frequency dependent attenuation is usually not significant, and therefore only the frequency independent attenuation is varied over the aperture. The frequency dependent attenuation is then included on the response by using the mean distance to the aperture. The attenuation is assumed to be minimum phase. 8 Chapter 3. Method of simulation

12 CHAPTER FOUR Installation The excutable code for the program can be obtained free of charge from the web-site: Here the mex-file to run under Matlab and the m-files for calling the mex-files can be found. Versions are currently found for Linux (Intel processors, 32 and 64 bits), MAC OS and Windows 32 and 64 bits. The latest version is only found for 64 bits processors. Older versions are found for HP-UX (PA-RISC processors), SUN (OS4.1 and Solaris), DEC ALPHA, Silicon Graphics, IBM AIX, but they are no longer supported. Matlab 8. or higher is required to run the program, but older versions from Matlab 5 and on are also found on the web-site. The individual files can be found at the web-site along with compressed Unix-style tar-files. A zip file also exits for the windows version. The tar-file should be downloaded to the directory, that must hold the files. The file is then extracted by writing: gzip -d <name_of_tar_file>.tar.z tar -xvf <name_of_tar_file>.tar to uncompress and extract the file. The tar-file can then be deleted. The program can now be run from this directory or from an other directory by writing: addpath( /home/user/field_ii/m_files ); field_init where /home/user/field II/m files contains the Field II m-files. This ensure that the directory is included in the Matlab search path, and the user-written m-files can then be placed in a separate directory. 9

13 1

14 CHAPTER FIVE Description of Matlab procedures 5.1 List of current procedures General commands Function name Purpose Page field debug Initialize debugging 13 field end Terminate the Field II program system and release the storage 13 field guide Display the Field II users guide in Acrobat reader 13 field info Display information about the state of the Field II program system 13 field init Initialize the Field II program system 14 set sampling Set the sampling frequency the system uses 15 set field Set various parameters for the program 16 Transducer commands Function name Purpose Page xdc apodization Create an apodization time line for an aperture. 17 xdc baffle Set the baffle condition for the aperture. 17 xdc center focus Set the origin for the dynamic focusing line. 18 xdc concave Define a concave aperture. 18 xdc convert Convert rectangular description to triangular description. 19 xdc convex array Create a convex array transducer. 2 xdc convex focused array Create an elevation focused convex array transducer. 2 xdc convex focused multirow Create an elevation focused convex, multi-row transducer. 22 xdc dynamic focus Use dynamic focusing for an aperture 23 xdc excitation Set the excitation pulse of an aperture. 24 xdc focus Create a focus time line for an aperture. 25 xdc focused array Create an elevation focused linear array transducer. 25 xdc focused multirow Create an elevation focused linear, multi-row transducer. 26 xdc focus times Creating a focus time line for an aperture with all delay 28 values supplied by the user. xdc free Free storage occupied by an aperture. 28 xdc get Get information about an aperture. 28 xdc impulse Set the impulse response of an aperture. 3 xdc linear array Create a linear array transducer. 31 xdc linear multirow Create a linear multi-row array transducer

15 Function name Purpose Page xdc lines Create an aperture bounded by a set of lines. 34 xdc piston Define a round, flat aperture. 36 xdc quantization Set quantization of the phase delays. 37 xdc rectangles Procedure for creating an aperture consisting of rectangles. 38 xdc show Show information about an aperture. 39 xdc times focus Creating a focus time line for an aperture with all delay values supplied 4 by the user. xdc triangles Make a multi-element aperture consisting of triangles. 41 xdc 2d array Create a two-dimensional array transducer. 41 Element manipulation commands Function name Purpose Page ele apodization Set the apodization for individual mathematical elements. 45 ele delay Set the delay for individual mathematical elements. 46 ele waveform Set the waveform for individual physical elements. 47 Field calculation commands Function name Purpose Page calc h Calculate the spatial impulse response. 5 calc hhp Calculate the pulse echo field. 51 calc hp Calculate the emitted field. 52 calc scat Calculate the received signal from a collection of scatterers. 53 calc scat all Calculate the received signals from a collection of scatterers for all 53 transmit and receive elements in the aperture. calc scat multi Calculate the received signals from a collection of scatterers for all the elements in the aperture Chapter 5. Description of Matlab procedures

16 5.2 Procedures for Field initialization Field II user s guide field debug Purpose: Procedure for initialize debugging. This will print out various information about the programs inner working. Initially no debugging is done. Calling: field debug(state) Input: State - 1: debugging, : no debugging. Output: none. Field II user s guide field end Purpose: Procedure for terminating the Field II program system and releasing the storage. Calling: field end ; Input: none. Output: none. Field II user s guide field guide Purpose: Procedure for displaying the Field II users guide (this guide) using the Adobe acrobat reader. Calling: field guide Input: Output: none. The Field II guide is displayed in a separate window using acrobat reader. Note that the Adobe pdf reader must be installed on the system, and it must be accessible under Matlab under the name acroread. The users guide should be in the search path of Matlab, preferrably in the same directory as the m-files for Field II with the name users guide.pdf. Field II user s guide field info Purpose: Procedure for showing information about the Field II program. The information is printed in the Matlab window. Calling: field info 5.2. Procedures for Field initialization 13

17 Input: Output: None. Information is printed in the Matlab window. For boolean variables a value of 1 indicates true and for false. Example: Print the information: field_info Current Field II configuration: Version 2.88, December 14, 21 (Matlab version) Number of apertures in operation: 3 Apertures to be defined uses rectangles: Apertures to be defined uses triangles: Apertures to be defined uses bounding lines: 1 Program uses accuracte time calculation for rectangles: Program uses fast integration for lines and triangles: 1 Speed of sound: 154. m/s Sampling frequency: 1. MHz Whether to use attenuation: Frequency independent attenuation is:. db/m Frequency dependent attenuation around. MHz is. db/[m Hz] Constant tau_m used in attenuation calculation: 2. Number of bytes reserved: 1224 Maximum number of bytes that has been reserved: Number for next signal to be used: 6656 Internal state of the program: 1 Debug mode enabled: Last calculation type done: Whether calculation time should be shown: 1 Seconds between showing times 5 s A boolean value of 1 indicates true, indicates false Field II user s guide field init Purpose: Procedure for initializing the Field II program system. Must be the first routine that is called before using the system. Calling: field init (suppress) ; Input: suppress An optional argument suppress with a value of zero can be given to suppress the display of the initial field screen. No ACII ouput will be given, if the argument is -1. Debug messages will be written if enable by field debug, and all error messages will also be printed. Output: none. 14 Chapter 5. Description of Matlab procedures

18 Initial values: The following initial values are used by the program after field init has been called: Variable Content Value c Speed of sound 154 m/s fs Sampling frequency Hz show times Whether to print information about the time taken for the calculation (yes = any positive numer). A number large than 2 is taken as the time in seconds between the printing of estimates. debug Whether to show debuging information (no) use att Whether to use attenuation (no) att Frequency independent attenuation. db/m. freq att Frequency dependent attenuation in around the center frequency. db/[m Hz] att f att f Attenuation center frequency in Hz. Hz use rectangles Whether to use rectangles for describing apertures 1 (yes) use triangles Whether to use triangles for describing apertures (no) use lines Whether to use lines for describing apertures (no) no ascii output Whether ASCII output is not printed (no, output is printed) fast integration Whether to use fast integration for bound lines and triangles (no) Initially the program is set to use rectangles for the modeling of transducers. All of the options can be changed by the procedure set field. Example: Include the Field II m-files in Matlab s search path and start the Field II simulation system: path(path, /home/user/field_ii/m_files ); field_init Field II user s guide set sampling Purpose: Set the sampling frequency the system uses. Remember that the pulses used in all apertures must be reset for the new sampling frequency to take effect. This procedure has been superseed by set field, and it is for portability reasons better to use set field. Calling: set sampling (fs); Input: fs - The new sampling frequency. Output: none Procedures for Field initialization 15

19 Field II user s guide set field Purpose: Set various parameters that determins the function of the program. Calling: set field (option name, value); Input: use att Whether to use attenuation (<> for attenuation) att Frequency independent attenuation in db/m. freq att Frequency dependent attenuation in db/[m Hz] around the center frequency att f. att f Attenuation center frequency in Hz. debug Whether to print debug information (1 = yes) c Set the speed of sound in m/s. fs Set the sampling frequency. show time Show calculation times during calculation. (yes = any positive numer). A number large than 2 is taken as the time in seconds between the printing of estimates. use rectangles Use rectangles for the apertures. (1 = yes) use triangles Use triangles for describing apertures. (1 = yes) use lines Use lines for describing apertures. (1 = yes) fast integration Whether to use fast integration (1) of the responses for bound lines and triangles. Fast integration uses a simple trapezoidal time integration of the responses, else a Romberg integration, as described in Numerical Receipes, are used. Output: none. Example: Set the attenuation to 1.5 db/cm and.5 db/[mhz cm] around 3 MHz and use this: set_field ( att,1.5*1); set_field ( Freq_att,.5*1/1e6); set_field ( att_f,3e6); set_field ( use_att,1); Note that the frequency independent and the frequency dependent terms should correspond, so that the frequency independent attenuation is the same as the frequency dependent term at the center frequency set. This is ensured if att = Freq att*att f, else the attenuation can be too big or too low at large depths in tissue. 16 Chapter 5. Description of Matlab procedures

20 5.3 Procedures for transducer definition Field II user s guide xdc apodization Purpose: Procedure for creating an apodization time line for an aperture Calling: xdc apodization (Th, times, values); Input: Th Pointer to the transducer aperture. times Time after which the associated apodization is valid. values Apodization values. Matrix with one row for each time value and a number of columns equal to the number of physical elements in the aperture. At least one apodization value in each row must be different from zero. Output: none. Field II user s guide xdc baffle Purpose: Procedure for setting the baffle condition for the aperture. Calling: xdc baffle (Th, soft baffle); Input: Th Pointer to the transducer aperture. soft baffle Whether to use the soft-baffle condition: 1 - using soft baffle - using rigid baffle (default for apertures) Output: none. Implementation: For a soft baffle, in which the pressure on the baffle surface is zero, the Rayleigh-Sommerfeld integral is used instead of the standard Rayleigh integral. This is: h s ( r 1, t) = S δ(t r1 r2 c ) cos ϕ ds (5.1) 2π r 1 r 2 Here cos ϕ is the angle between the line through the field point orthogonal to the aperture plane and the radius of the spherical wave. The angles ϕ is fixed for a given radius of the projected spherical wave and thus for a given time. It is given by cos ϕ = z p R = z p (5.2) ct I can be shown that h s ( r 1, t) = z p ct h( r 1, t). (5.3) where h( r 1, t) is the standard spatial impulse response. The spatial impulse response for the soft baffle case is, thus, be found from the normal spatial impulse response by multiplying with z p /(ct), which is the method employed by the Field II program. Example: 5.3. Procedures for transducer definition 17

21 Create a 16 elements linear array, and divide the physical elements into 2 by 3 mathematical elements to increase the accuracy of the simulation. Then set the soft-baffle boundary condition. % Set initial parameters height=5/1; % Height of element [m] width=1/1; % Width of element [m] kerf=width/4; % Distance between transducer elements [m] N_elements=16; % Number of elements focus=[ 4]/1; % Initial electronic focus % Define the transducer Th = xdc_linear_array (N_elements, width, height, kerf, 2, 3, focus); % Set the soft-baffle option xdc_soft_baffle (Th, 1); Field II user s guide xdc center focus Purpose: Procedure for setting the center point for the focusing. This point is used as a reference for calculating the focusing delay times and as a starting point for dynamic focusing. Calling: xdc center focus (Th, point); Input: Th Pointer to the transducer aperture. point Focus center point. Output: none. Field II user s guide xdc concave Purpose: Procedure for creating a concave transducer Calling: Th = xdc concave (radius, focal radius, ele size); Input: radius Radius of physical elements. focal radius Focal radius. ele size Size of mathematical elements. Output: Th A pointer to this transducer aperture. Example of transducer definition: Create a concave, round transducer with an 8 mm radius and a focal radius of 2 mm and divided it into 1 mm mathematical elements. % Set initial parameters 18 Chapter 5. Description of Matlab procedures

22 8 6 4 z [mm] y [mm] y [mm] x [mm] z [mm] 1 5 y [mm] 5 5 x [mm] 5 Figure 5.1: Concave, round transducer with a radius of 8 mm divided into 1 by 1 mm mathematical elements. R=8/1; % Radius of transducer Rfocal=2/1; % Focal radius of transducer ele_size=1/1; % Size of mathematical elements % Define the transducer Th = xdc_concave (R, Rfocal, ele_size); The resulting transducer is shown in Fig Field II user s guide xdc convert Purpose: Procedure for converting an aperture from a rectangular description to a triangular description. Calling: xdc convert (Th); Input: Th Aperture to be converted. Output: None. Note: The number of mathematical elements gets to be twice as large since one rectangle is modeled by two triangles Procedures for transducer definition 19

23 Field II user s guide xdc convex array Purpose: Procedure for creating a convex array aperture. Calling: Th = xdc convex array (no elements, width, height, kerf, Rconvex, no sub x, no sub y, focus); Input: no elements Number of physical elements. width Width in x-direction of elements. height Width in y-direction of elements. kerf Distance in x-direction between elements. Rconvex Convex radius. no sub x Number of sub-divisions in x-direction of elements. no sub y Number of sub-divisions in y-direction of elements. focus[] Fixed focus for array (x,y,z). Vector with three elements. Output: Th A pointer to this transducer aperture. Example of transducer definition: Create a 16 element convex array with a convex radius of 2 mm: % Set initial parameters height=5/1; % Height of element [m] width=1/1; % Width of element [m] kerf=width/4; % Distance between transducer elements [m] N_elements=16; % Number of elements Rconvex=2/1; % Convex radius [m] focus=[ 4]/1; % Initial electronic focus % Define the transducer Th = xdc_convex_array (N_elements, width, height, kerf, Rconvex, 1, 5, focus); Note that the radii are quite small in order to show the aperture curvature. The resulting aperture is shown below. Field II user s guide xdc convex focused array Purpose: Procedure for creating a mechanical elevation focused convex array aperture. Calling: Th = xdc convex focused array (no elements, width, height, kerf, Rconvex, Rfocus, no sub x, no sub y, focus); 2 Chapter 5. Description of Matlab procedures

24 z [mm] y [mm] y [mm] x [mm] z [mm] x [mm] z [mm] y [mm] 5 x [mm] 5 Figure 5.2: Rectangles for a convex array with Rconvex equal to 2 mm. Input: no elements Number of physical elements. width Width in x-direction of elements. height Width in y-direction of elements. kerf Distance in x-direction between elements. Rconvex Convex radius. Rfocus Radius of elevation focus. no sub x Number of sub-divisions in x-direction of elements. no sub y Number of sub-divisions in y-direction of elements. focus[] Fixed focus for array (x,y,z). Vector with three elements. Output: Th A pointer to this transducer aperture. Limitations: The kerf and width of the elements must lie within the range: π*rconvex<=(kerf*(no elements- 1)+width*no elements). Also all parameters for physical dimensions (width, height, kerf, Rconvex, Rfocus) must be positive. Example of transducer definition: : Create a 32 element elevation focused, convex array with an elevation focus at 1 mm and a convex radius of 3 mm: % Set initial parameters height=1/1; % Height of element [m] width=1.9/1; % Width of element [m] kerf=width/2; % Distance between transducer elements [m] N_elements=32; % Number of elements Rfocus=5/1; % Elevation focus [m] Rconvex=3/1; % Convex radius [m] focus=[ 7]/1; % Initial electronic focus % Define the transducer 5.3. Procedures for transducer definition 21

### Introduction to acoustic imaging

Introduction to acoustic imaging Contents 1 Propagation of acoustic waves 3 1.1 Wave types.......................................... 3 1.2 Mathematical formulation.................................. 4 1.3

### Physics 202 Problems - Week 8 Worked Problems Chapter 25: 7, 23, 36, 62, 72

Physics 202 Problems - Week 8 Worked Problems Chapter 25: 7, 23, 36, 62, 72 Problem 25.7) A light beam traveling in the negative z direction has a magnetic field B = (2.32 10 9 T )ˆx + ( 4.02 10 9 T )ŷ

### Manual for simulation of EB processing. Software ModeRTL

1 Manual for simulation of EB processing Software ModeRTL How to get results. Software ModeRTL. Software ModeRTL consists of five thematic modules and service blocks. (See Fig.1). Analytic module is intended

### Reflection and Refraction

Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,

### CONCEPT-II. Overview of demo examples

CONCEPT-II CONCEPT-II is a frequency domain method of moment (MoM) code, under development at the Institute of Electromagnetic Theory at the Technische Universität Hamburg-Harburg (www.tet.tuhh.de). Overview

### Third Harmonic Imaging using Pulse Inversion

Paper presented at the IEEE International Ultrasonics Symposium, Orlando Florida, 211: Third Harmonic Imaging using Pulse Inversion Joachim Hee Rasmussen, Yigang Du and Jørgen Arendt Jensen Center for

### DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS RAYLEIGH-SOMMERFELD DIFFRACTION INTEGRAL OF THE FIRST KIND

DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS RAYLEIGH-SOMMERFELD DIFFRACTION INTEGRAL OF THE FIRST KIND THE THREE-DIMENSIONAL DISTRIBUTION OF THE RADIANT FLUX DENSITY AT THE FOCUS OF A CONVERGENCE BEAM

### DESIGN AND EVALUATION OF PROBE WITH THREE DEGREE- OF-FREEDOM FOR NON-DESTRUCTIVE TEST USING THREE- DIMENSIONAL FINITE ELEMENT METHOD

DESIGN AND EVALUATION OF PROBE WITH THREE DEGREE- OF-FREEDOM FOR NON-DESTRUCTIVE TEST USING THREE- DIMENSIONAL FINITE ELEMENT METHOD Masafumi Aoyanagi Graduate School of Systems and Information Engineering,

### Using the Siemens S65 Display

Using the Siemens S65 Display by Christian Kranz, October 2005 ( http://www.superkranz.de/christian/s65_display/displayindex.html ) ( PDF by Benjamin Metz, 01 st November 2005 ) About the Display: Siemens

### DMU Space Analysis. Preface What's New Getting Started Basic Tasks Workbench Description Customizing Glossary Index

DMU Space Analysis Preface What's New Getting Started Basic Tasks Workbench Description Customizing Glossary Index Dassault Systèmes 1994-99. All rights reserved. Preface DMU Space Analysis is a CAD-independent

### Physics 25 Exam 3 November 3, 2009

1. A long, straight wire carries a current I. If the magnetic field at a distance d from the wire has magnitude B, what would be the the magnitude of the magnetic field at a distance d/3 from the wire,

### Grazing incidence wavefront sensing and verification of X-ray optics performance

Grazing incidence wavefront sensing and verification of X-ray optics performance Timo T. Saha, Scott Rohrbach, and William W. Zhang, NASA Goddard Space Flight Center, Greenbelt, Md 20771 Evaluation of

### Lab Exercise: Antenna Basics

Experiment Objectives: Lab Exercise: Antenna Basics In this lab you will analyze received radar signals from the P400 to gain an understanding of the basics of electromagnetic antennas. This lab will demonstrate

### TVM 4155 Numerical modelling and hydraulics 10. March 2014. OpenFOAM homework

TVM 4155 Numerical modelling and hydraulics 10. March 2014 OpenFOAM homework OpenFOAM is the most popular open-source CFD program in the world today. In this homework we will use the program to determine

### CHAPTER 3: DIGITAL IMAGING IN DIAGNOSTIC RADIOLOGY. 3.1 Basic Concepts of Digital Imaging

Physics of Medical X-Ray Imaging (1) Chapter 3 CHAPTER 3: DIGITAL IMAGING IN DIAGNOSTIC RADIOLOGY 3.1 Basic Concepts of Digital Imaging Unlike conventional radiography that generates images on film through

### MANUAL FOR RX700 LR and NR

MANUAL FOR RX700 LR and NR 2013, November 11 Revision/ updates Date, updates, and person Revision 1.2 03-12-2013, By Patrick M Affected pages, ETC ALL Content Revision/ updates... 1 Preface... 2 Technical

### Direct and Reflected: Understanding the Truth with Y-S 3

Direct and Reflected: Understanding the Truth with Y-S 3 -Speaker System Design Guide- December 2008 2008 Yamaha Corporation 1 Introduction Y-S 3 is a speaker system design software application. It is

### Sound Waves. PHYS102 Previous Exam Problems CHAPTER. Sound waves Interference of sound waves Intensity & level Resonance in tubes Doppler effect

PHYS102 Previous Exam Problems CHAPTER 17 Sound Waves Sound waves Interference of sound waves Intensity & level Resonance in tubes Doppler effect If the speed of sound in air is not given in the problem,

### INTERFERENCE OF SOUND WAVES

1/2016 Sound 1/8 INTERFERENCE OF SOUND WAVES PURPOSE: To measure the wavelength, frequency, and propagation speed of ultrasonic sound waves and to observe interference phenomena with ultrasonic sound waves.

### Algorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure

Fifth Grade CCSS Math Vocabulary Word List *Terms with an asterisk are meant for teacher knowledge only students need to learn the concept but not necessarily the term. Addend Any number being added Algorithm

### Advancements in High Frequency, High Resolution Acoustic Micro Imaging for Thin Silicon Applications

Advancements in High Frequency, High Resolution Acoustic Micro Imaging for Thin Silicon Applications Janet E. Semmens Sonoscan, Inc. 2149 E. Pratt Boulevard Elk Grove Village, IL 60007 USA Phone: (847)

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### Transmission Line and Back Loaded Horn Physics

Introduction By Martin J. King, 3/29/3 Copyright 23 by Martin J. King. All Rights Reserved. In order to differentiate between a transmission line and a back loaded horn, it is really important to understand

### Lab 3: Introduction to Data Acquisition Cards

Lab 3: Introduction to Data Acquisition Cards INTRODUCTION: In this lab, you will be building a VI to display the input measured on a channel. However, within your own VI you will use LabVIEW supplied

### 9 Measurements on Transmission Lines

Measurements on Transmission Lines Power and Attenuation Measurements Although a variety of instruments measure power, the most accurate instrument is a power meter and a power sensor. The sensor is an

### SainSmart DDS Series User Manual (Software)

SainSmart DDS Series User Manual (Software) For Oscilloscope, Signal Generator and Logic Analyzer Updated software and user manual can be downloaded from http://www.sainsmart.com/tools-equipments/oscilloscope-dso/sainsmart-dds.html

### Random Map Generator v1.0 User s Guide

Random Map Generator v1.0 User s Guide Jonathan Teutenberg 2003 1 Map Generation Overview...4 1.1 Command Line...4 1.2 Operation Flow...4 2 Map Initialisation...5 2.1 Initialisation Parameters...5 -w xxxxxxx...5

### Drawing a histogram using Excel

Drawing a histogram using Excel STEP 1: Examine the data to decide how many class intervals you need and what the class boundaries should be. (In an assignment you may be told what class boundaries to

### Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.

Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal

### GeoGebra. 10 lessons. Gerrit Stols

GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter

### Simulation Study of Real Time 3-D Synthetic Aperture Sequential Beamforming for Ultrasound Imaging

Downloaded from orbit.dtu.dk on: Jan 28, 2016 Simulation Study of Real Time 3-D Synthetic Aperture Sequential Beamforming for Ultrasound Imaging Hemmsen, Martin Christian; Rasmussen, Morten Fischer; Stuart,

### Rectangular Prisms Dimensions

Rectangular Prisms Dimensions 5 Rectangular prisms are (3-D) three-dimensional figures, which means they have three dimensions: a length, a width, and a height. The length of this rectangular prism is

### ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's

### http://school-maths.com Gerrit Stols

For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

### CNC-STEP. "LaserProbe4500" 3D laser scanning system Instruction manual

LaserProbe4500 CNC-STEP "LaserProbe4500" 3D laser scanning system Instruction manual 2 Hylewicz CNC-Technik Siemensstrasse 13-15 D-47608 Geldern Fon.: +49 (0) 2831 133236 E-Mail: info@cnc-step.com Website:

Edinburgh COLLEGE of ART ARCHITECTURE 3D Modelling in AutoCAD - tutorial exercise The screen The graphics area This is the part of the screen in which the drawing will be created. The command prompt area

### 5: Magnitude 6: Convert to Polar 7: Convert to Rectangular

TI-NSPIRE CALCULATOR MENUS 1: Tools > 1: Define 2: Recall Definition --------------- 3: Delete Variable 4: Clear a-z 5: Clear History --------------- 6: Insert Comment 2: Number > 1: Convert to Decimal

### ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT

www.arpapress.com/volumes/vol12issue2/ijrras_12_2_22.pdf ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT M. C. Anumaka Department of Electrical Electronics Engineering,

### 0 Introduction to Data Analysis Using an Excel Spreadsheet

Experiment 0 Introduction to Data Analysis Using an Excel Spreadsheet I. Purpose The purpose of this introductory lab is to teach you a few basic things about how to use an EXCEL 2010 spreadsheet to do

### A Guide to Acousto-Optic Modulators

A Guide to Acousto-Optic Modulators D. J. McCarron December 7, 2007 1 Introduction Acousto-optic modulators (AOMs) are useful devices which allow the frequency, intensity and direction of a laser beam

Section 5. : Horn Physics Section 5. : Horn Physics By Martin J. King, 6/29/8 Copyright 28 by Martin J. King. All Rights Reserved. Before discussing the design of a horn loaded loudspeaker system, it is

### Beamforming and hardware design for a multichannel front-end integrated circuit for real-time 3D catheter-based ultrasonic imaging.

Beamforming and hardware design for a multichannel front-end integrated circuit for real-time 3D catheter-based ultrasonic imaging. Ira O. Wygant a, Mustafa Karaman b, Ömer Oralkana, and Butrus T. Khuri-Yakub

### LabVIEW Day 1 Basics. Vern Lindberg. 1 The Look of LabVIEW

LabVIEW Day 1 Basics Vern Lindberg LabVIEW first shipped in 1986, with very basic objects in place. As it has grown (currently to Version 10.0) higher level objects such as Express VIs have entered, additional

### Getting Started With DraftSight A Guide For AEC Users

Getting Started With DraftSight A Guide For AEC Users DraftSight.com Facebook.com/DraftSight Welcome to DraftSight a valuable tool for any AEC professional! DraftSight is more than a free, professional-grade

### Structural Integrity Analysis

Structural Integrity Analysis 1. STRESS CONCENTRATION Igor Kokcharov 1.1 STRESSES AND CONCENTRATORS 1.1.1 Stress An applied external force F causes inner forces in the carrying structure. Inner forces

### Background: A CCD consists of an array of individual detector sites, as suggested by the sketch below: Array of independent detector sites

Q: Can I use PixelScope to measure the MTF of a CCD camera? Can I use PixelScope to map the actual sensitive area of a pixel? A: Yes to both. This note explains how. Approach This can be a complicated

### Visualization of 2D Domains

Visualization of 2D Domains This part of the visualization package is intended to supply a simple graphical interface for 2- dimensional finite element data structures. Furthermore, it is used as the low

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Optiffuser. High-performance, high bandwidth lightweight 1D diffuser.

Optiffuser High-performance, high bandwidth lightweight 1D diffuser. General product information The Optiffuser comes in packs of four panels. Two positives and two negatives (see page 5) per package.

### Instruction Manual Service Program ULTRA-PROG-IR

Instruction Manual Service Program ULTRA-PROG-IR Parameterizing Software for Ultrasonic Sensors with Infrared Interface Contents 1 Installation of the Software ULTRA-PROG-IR... 4 1.1 System Requirements...

### Passive Optical Resonators

Passive Optical Resonators Optical Cavities and Feedback Back mirror I 0 I 1 Laser medium with gain, G Output mirror I 3 R = 100% R < 100% I 2 Cavities are essential components of the lasers. They provide

### CS1112 Spring 2014 Project 4. Objectives. 3 Pixelation for Identity Protection. due Thursday, 3/27, at 11pm

CS1112 Spring 2014 Project 4 due Thursday, 3/27, at 11pm You must work either on your own or with one partner. If you work with a partner you must first register as a group in CMS and then submit your

### An Introduction to Point Pattern Analysis using CrimeStat

Introduction An Introduction to Point Pattern Analysis using CrimeStat Luc Anselin Spatial Analysis Laboratory Department of Agricultural and Consumer Economics University of Illinois, Urbana-Champaign

### Lecture PowerPoints. Chapter 23 Physics: Principles with Applications, 7th edition Giancoli

Lecture PowerPoints Chapter 23 Physics: Principles with Applications, 7th edition Giancoli This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching

### Plotting: Customizing the Graph

Plotting: Customizing the Graph Data Plots: General Tips Making a Data Plot Active Within a graph layer, only one data plot can be active. A data plot must be set active before you can use the Data Selector

### Universal Simple Control, USC-1

Universal Simple Control, USC-1 Data and Event Logging with the USB Flash Drive DATA-PAK The USC-1 universal simple voltage regulator control uses a flash drive to store data. Then a propriety Data and

### CSE168 Computer Graphics II, Rendering. Spring 2006 Matthias Zwicker

CSE168 Computer Graphics II, Rendering Spring 2006 Matthias Zwicker Last time Sampling and aliasing Aliasing Moire patterns Aliasing Sufficiently sampled Insufficiently sampled [R. Cook ] Fourier analysis

### (a) (i) Label the diagram of the human eye to show the lens, retina and optic nerve.

Practice Test: 28 marks (42 minutes) Additional Problem: 37 marks (56 minutes) 1. This question is about the human eye. (a) (i) Label the diagram of the human eye to show the lens, retina and optic nerve.

### Vector storage and access; algorithms in GIS. This is lecture 6

Vector storage and access; algorithms in GIS This is lecture 6 Vector data storage and access Vectors are built from points, line and areas. (x,y) Surface: (x,y,z) Vector data access Access to vector

### Impedance 50 (75 connectors via adapters)

VECTOR NETWORK ANALYZER PLANAR TR1300/1 DATA SHEET Frequency range: 300 khz to 1.3 GHz Measured parameters: S11, S21 Dynamic range of transmission measurement magnitude: 130 db Measurement time per point:

### Quickstart for Desktop Version

Quickstart for Desktop Version What is GeoGebra? Dynamic Mathematics Software in one easy-to-use package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra,

### Lab Exercise 1: Acoustic Waves

Lab Exercise 1: Acoustic Waves Contents 1-1 PRE-LAB ASSIGNMENT................. 2 1-3.1 Spreading Factor: Spherical Waves........ 2 1-3.2 Interference In 3-D................. 3 1-4 EQUIPMENT........................

### 1051-232 Imaging Systems Laboratory II. Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002

05-232 Imaging Systems Laboratory II Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002 Abstract: For designing the optics of an imaging system, one of the main types of tools used today is optical

### Teacher Notes. Exploration: The Determinant of a Matrix. Learning outcomes addressed. Lesson Context. Lesson Launch.

Exploration Teacher Notes Exploration: The Determinant of a Matrix Learning outcomes addressed 3.7 Calculate the determinant of a matrix. 3.8 Evaluate a determinant to calculate the area of a triangle

### Data Communications Prof. Ajit Pal Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture # 03 Data and Signal

(Refer Slide Time: 00:01:23) Data Communications Prof. Ajit Pal Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture # 03 Data and Signal Hello viewers welcome

### Transmit Sub-aperture Optimization in MSTA Ultrasound Imaging Method

Transmit Sub-aperture Optimization in MSTA Ultrasound Imaging Method YuriyTasinkevych, Ihor Trots, AndrzejNowicki, Marcin Lewandowski Abstract The paper presents the optimization problem for the multi-element

### A Simple Introduction to Interference

Course PHYSICS260 Assignment 4 Due at 11:00pm on Wednesday, February 27, 2008 A Simple Introduction to Interference Description: Interference is discussed for pulses on strings and then for sinusoidal

### Appendix C: Graphs. Vern Lindberg

Vern Lindberg 1 Making Graphs A picture is worth a thousand words. Graphical presentation of data is a vital tool in the sciences and engineering. Good graphs convey a great deal of information and can

### Digital Signal Processing For Radar Applications

Digital Signal Processing For Radar Applications Altera Corporation Radar: RAdio Detection And Ranging Need a directional radio beam Measure time between transmit pulse and receive pulse Find Distance:

### Accurate Measurement of the Mains Electricity Frequency

Accurate Measurement of the Mains Electricity Frequency Dogan Ibrahim Near East University, Faculty of Engineering, Lefkosa, TRNC dogan@neu.edu.tr Abstract The frequency of the mains electricity supply

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### INTERFERENCE OF SOUND WAVES

2011 Interference - 1 INTERFERENCE OF SOUND WAVES The objectives of this experiment are: To measure the wavelength, frequency, and propagation speed of ultrasonic sound waves. To observe interference phenomena

### Tutorial 2: Using Excel in Data Analysis

Tutorial 2: Using Excel in Data Analysis This tutorial guide addresses several issues particularly relevant in the context of the level 1 Physics lab sessions at Durham: organising your work sheet neatly,

### Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

### Fourier Analysis. Evan Sheridan, Chris Kervick, Tom Power Novemeber

Fourier Analysis Evan Sheridan, Chris Kervick, Tom Power 11367741 Novemeber 19 01 Abstract Various properties of the Fourier Transform are investigated using the Cassy Lab software, a microphone, electrical

### The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

### A pulse is a collection of cycles that travel together. the cycles ( on or transmit time), and. the dead time ( off or receive time)

chapter 2 Pulsed Ultrasound In diagnostic ultrasound imaging, short bursts, or pulses, of acoustic energy are used to create anatomic images. Continuous wave sound cannot create anatomic images. Analogy

### Detailed information about Gerber, NC Drill and NC Route.

Title: Product: Summary: Manufacturing Output OrCAD / Allegro PCB Editor Detailed information about Gerber, NC Drill and NC Route. Author/Date: Beate Wilke / 27.05.2009 Table of Contents 1 Gerber... 2

### Coefficient of Potential and Capacitance

Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that

### B3. Short Time Fourier Transform (STFT)

B3. Short Time Fourier Transform (STFT) Objectives: Understand the concept of a time varying frequency spectrum and the spectrogram Understand the effect of different windows on the spectrogram; Understand

### MATLAB DFS. Interface Library. User Guide

MATLAB DFS Interface Library User Guide DHI Water Environment Health Agern Allé 5 DK-2970 Hørsholm Denmark Tel: +45 4516 9200 Fax: +45 4516 9292 E-mail: dhi@dhigroup.com Web: www.dhigroup.com 2007-03-07/MATLABDFS_INTERFACELIBRARY_USERGUIDE.DOC/JGR/HKH/2007Manuals/lsm

### Wave field visualization using scanned hydrophone measurements

header for SPIE use Wave field visualization using scanned hydrophone measurements Alan R. Selfridge Ultrasonic Devices Inc., 50 Ocean View Road, Los Gatos, CA 95033 Correspondence: Email: selfridge@ultrasonic.com;

### Applying a circular load. Immediate and consolidation settlement. Deformed contours. Query points and query lines. Graph query.

Quick Start Tutorial 1-1 Quick Start Tutorial This quick start tutorial will cover some of the basic features of Settle3D. A circular load is applied to a single soil layer and settlements are examined.

### Matlab Tutorial: Basics

Matlab Tutorial: Basics Topics: 1. Downloading Matlab 2. Opening Matlab 3. Entering Data 4. The Command Window 5. m-files 6. Running m-files 7. Add directory to path 8. Getting help with Matlab functions

### Cabcharge Australia Limited, 2005. Cabcharge TMS Help Manual

Cabcharge TMS Help Manual Cabcharge Australia Limited, 2005 p1 p2 Table of Contents Welcome to TMS 5 A Brief Overview 6 Getting Started 8 System Requirements 8 Downloading Statement Data 9 Set up your

### Using Microsoft Excel to Plot and Analyze Kinetic Data

Entering and Formatting Data Using Microsoft Excel to Plot and Analyze Kinetic Data Open Excel. Set up the spreadsheet page (Sheet 1) so that anyone who reads it will understand the page (Figure 1). Type

### Financial Econometrics MFE MATLAB Introduction. Kevin Sheppard University of Oxford

Financial Econometrics MFE MATLAB Introduction Kevin Sheppard University of Oxford October 21, 2013 2007-2013 Kevin Sheppard 2 Contents Introduction i 1 Getting Started 1 2 Basic Input and Operators 5

### The Basics of FEA Procedure

CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring

### Chapter 22: Electric Flux and Gauss s Law

22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

DSP First Laboratory Exercise #9 Sampling and Zooming of Images In this lab we study the application of FIR ltering to the image zooming problem, where lowpass lters are used to do the interpolation needed

### Calibration and Use of a Strain-Gage-Instrumented Beam: Density Determination and Weight-Flow-Rate Measurement

Chapter 2 Calibration and Use of a Strain-Gage-Instrumented Beam: Density Determination and Weight-Flow-Rate Measurement 2.1 Introduction and Objectives This laboratory exercise involves the static calibration

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### GEOGEBRA3D ANDREAS LINDNER

South Bohemia Mathematical Letters Volume 21, (2013), No. 1, 47 58. GEOGEBRA3D ANDREAS LINDNER Abstract. Originally GeoGebra was designed as a program for the dynamic combination of geometry and algebra.

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### Mathematics Common Core Cluster. Mathematics Common Core Standard. Domain

Mathematics Common Core Domain Mathematics Common Core Cluster Mathematics Common Core Standard Number System Know that there are numbers that are not rational, and approximate them by rational numbers.