ۥ-@-A?d"~~~~(4~Y:4rx BODE v 2.31 sg  by Thomas Reitmayr Copyright by AktionsGemeinschaft Table of Contents VERZEICHNIS \o1. Introduction 1 1.1. Notice 1 1.2. Parts of this Version 1 1.3. System Requirements 1 1.4. Installation 1 2. BODE.LIB 2 2.1. General Description 2 2.2. Format of Input 2 2.3. Restrictions to Input 4 2.4. The Menu 4 2.5. Analysis of a Exemplary Function 4 2.5.1. Input 5 2.5.2. Program Execution 5 2.5.3. Result: Presentation of the Function 6 2.5.4. Result: Marginal Values 6 2.5.5. Result: Documentation of Zeros and Poles 7 2.5.6. Result: Gain Characteristic 8 2.5.7. Result: Phase Characteristic 9 2.6. Known Weaknesses of the Program 10 3. Usage of LISTER 10 4. Deinstallation of the Programs 10 5. Appendix 11 5.1. History of Construction of BODE v2.31sg 11 5.2. Where to Get Further Information? 12  1. Introduction 1.1. Notice The programs BODE.LIB, BODEE.LIB, BODEGX.LIB, BODEGXE.LIB, and LISTER.LIB as well as its documentations are provided "as are", and are subjected to change without notice. AktionsGemeinschaft (AG) and the author Thomas Reitmayr make no warranty of any kind with regard to the software or documentation, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose. AG and the author shall not be liable for any error or for incidental or consequential damages in connection with the furnishing, performance, or use of this software and documentation. The programs except LISTER.LIB are copyrighted by the AktionsGemeinschaft (AG) of the technical university of Vienna. Customers may freely reproduce and distribute this material. Sale of this material is prohibited without prior written permission of the AG. LISTER.LIB is provided as freeware. 1.2. Parts of this Version This package contains a program for analysing transfer functions of linear time-invariant systems and drawing the corresponding bode-diagram. There is also an auxiliary program (LISTER) included as freeware to display the results of BODE. This program can be used in connection with the bode-analyser or independently in own applications. 1.3. System Requirements HP48 S(X) or G(X) with at least 25 KB free memory. Therefore it is not recommented to run the programs on a calculater equipped only with the basic memory size of 32 KByte. 1.4. Installation a) Delete all installed versions of BODE and LISTER on your calculater. b) Download the files BODE.LIB and LISTER.LIB into your HP48 (e.g. using KERMIT on your PC or from another HP48) c) Recall the contents of the menu entry "BODE.LIB" to the stack and purge this entry. Then store the library into a (not merged!) port (e.g. use the command "0 STO" for port 0) d) Carry out point c for LISTER.LIB too. e) Turn your calculater off and on again. In case of correct installation of BODE, it reports version number, author, and copyright in a short message each time performing a warm-start (e.g. throug ON-C). 2. BODE.LIB 2.1. General Description SONDZEICHEN 183 \f "Symbol" \s 12 \h Detailed analysis of all zeros and poles of the given transfer function. SONDZEICHEN 183 \f "Symbol" \s 12 \h Construction of the multiplied and the factored form of the transfer function. SONDZEICHEN 183 \f "Symbol" \s 12 \h Graphical output of a linear approximation and the smooth characteristic of gain and phase. SONDZEICHEN 183 \f "Symbol" \s 12 \h Axes of graphs labeled in powers of ten (frequency SONDZEICHEN 118 \f "SymbolProp BT"), decibel (gain) und radiants (phase). SONDZEICHEN 183 \f "Symbol" \s 12 \h Arbitrary dimension of the analysis protocol through organization in text pages and scrollable graphical pages. SONDZEICHEN 183 \f "Symbol" \s 12 \h Comfortable and flexible kinds of input. SONDZEICHEN 183 \f "Symbol" \s 12 \h High execution speed, depending on the function to be analysed between 25 and 30 seconds (SX) or 15 and 20 seconds (GX) SONDZEICHEN 183 \f "Symbol" \s 12 \h Preservation of the last calculated data, which can be recalled at any time. SONDZEICHEN 183 \f "Symbol" \s 12 \h Special menu entry for multiplieing and factoring a polynomial. 2.2. Format of Input Before starting an analysis of a transfer function, this one has to be available to the program in a suitable format: SONDZEICHEN 183 \f "Symbol" \s 12 \h A single constant is entered as simple number. SONDZEICHEN 183 \f "Symbol" \s 12 \h The basis of the following points is the expression of a fully multiplied polynomial EINBETTEN Equation  which is done through specifieing the coefficients ai. These are provided to the program in the following manner: EINBETTEN Equation  i.e. the coefficients are expressed as list with the highest order coefficient first. Important: Coefficients ai = 0 must not be left out as well! Example: EINBETTEN Equation  SONDZEICHEN 183 \f "Symbol" \s 12 \h If a polynomial is provided in partially factored form, the seperate polynomials can be entered without being multiplied first. EINBETTEN Equation  is to be entered as EINBETTEN Equation  Example: EINBETTEN Equation  SONDZEICHEN 183 \f "Symbol" \s 12 \h Due to further simplification constant factors can be inserted at any place within the first brace-level. EINBETTEN Equation  can be entered as EINBETTEN Equation  Example: EINBETTEN Equation  After the different forms for specifieing a polynomial are known, the required input of arbitrary transfer functions is show. In case of linear time-invariant systems, the function to be analysed exists as fraction of two polynomials: EINBETTEN Equation  According to the rules mentioned above the numerator is typed in first and then the denominator. Thus the topmost stack-levels look like this: #2: { numerator } #1: { denominator } That's it! Now just select the menu of the BODE-library and execute BODE (see chapter 2.4). 2.3. Restrictions to Input At the beginning of program execution the entered data are checked, whereby the following situations are recognized as inadmissible and terminate the program with an error message: SONDZEICHEN 183 \f "Symbol" \s 12 \h There is no or only one polynomial on the stack. SONDZEICHEN 183 \f "Symbol" \s 12 \h At least one polynomial is repugnant to the rules described above. SONDZEICHEN 183 \f "Symbol" \s 12 \h A constant factor is zero. SONDZEICHEN 183 \f "Symbol" \s 12 \h A polynomial only consists of coefficients equal to zero. SONDZEICHEN 183 \f "Symbol" \s 12 \h A number is not real but complex. SONDZEICHEN 183 \f "Symbol" \s 12 \h The transfer function represents a pure interator or differentiator. These do not have any singularity at a point different to zero, which is necessary to dimension the graphs. 2.4. The Menu To start on of the programs of BODE, the menu of the library must be selected. The bottom line of the display shows like that: EINBETTEN CPaint4 \s \* FormatVerbinden The different items of the menu can be described as follows: SONDZEICHEN 183 \f "Symbol" \s 12 \h BODE: Actually starts the analysis of a transfer function; for format of necessary parameters see above. SONDZEICHEN 183 \f "Symbol" \s 12 \h BSHOW: Shows the result of the last performed analysis. SONDZEICHEN 183 \f "Symbol" \s 12 \h BCLEAR: Clears the last analysis protocol. Thus between 1 and 4 KByte are released depending on the complexity of the previous function. SONDZEICHEN 183 \f "Symbol" \s 12 \h ABOUTBOD: Shows the name of the program, version, author and copyright; only for information. SONDZEICHEN 183 \f "Symbol" \s 12 \h FACTOR: Multiplies and factors polynomials. The parameter is one polynomial formatted in the usual way, which is multiplied by the program if necessary and then factored in two steps. These three results are prepared graphically and displayed. SONDZEICHEN 183 \f "Symbol" \s 12 \h CROOT: Calculates all roots of a polynomial, i.e. complex roots are possible too. The calculated values are collected to a list object. This menu entry will be interesting especially for S(X)-owners, because in G(X) versions there already exists a similar command named PROOT. 2.5. Analysis of a Exemplary Function In the following sections the result of a performed analysis is commented. The used transfer function is EINBETTEN Equation  2.5.1. Input After input the stack presents as follows:  This picture already shows the BODE-menu. The program is invoked through pressing key [A]. 2.5.2. Program Execution After about one second, in which the correctness of the parameters is checked, the display is turned off, to reduce execution time by 11 %. The progress of calculation can be watched though the blinking symbols (annuntiators) on the upper edge of the display. During this state the program can be terminated by pressing the ON-key, the data calculated up to there can be recalled with BSHOW, as usual. In that case an additional last page in the protocol indicates the incomplete analysis. 2.5.3. Result: Presentation of the Function The first page in the analysis protocol shows the multiplied form of the given transfer function.  On the following page the factored form is presented. 2nd degree terms with complex roots will not factored anymore, because that is not necessary for analysis.  2.5.4. Result: Marginal Values The third page shows, which frequency area is interesting for analysis and plotting of the transfer function. This calculation is done accourding to the rule, that at least one decade should be plotted beyond the frequency of the outermost poles or zeros. Furthermore there is given an asymptotical approximation for the function at s SONDZEICHEN 174 \f "SymbolProp BT" 0.  The following page presents the marginal values of the left edge of the specified area (10-2 s-1 in our example).  2.5.5. Result: Documentation of Zeros and Poles The pages are sorted in a way that zeros are shown first and then poles. The description of zeros is done in the same manner as of poles, therefore all following statements apply to both kinds of roots. Below the kind of root (zero of pole) the radian frequency SONDZEICHEN 110 \f "SymbolProp BT" (or SONDZEICHEN 119 \f "Symbol") of that point is given ("SONDZEICHEN 110 \f "SymbolProp BT" = xx.xx s-1"). Then in case of 2nd degree terms a notice is shown and additionally the mathematical expression of that term. In case of real roots only the multiplicity of that point is returned (1-fold, ). "SONDZEICHEN 68 \f "SymbolProp BT"gradient" gives the gradient's additive change in the gain characteristic in decibel/decade. For the reason of the special kind of transfer function only multiples of 20 dB/dec. are possible. Analogous "SONDZEICHEN 68 \f "SymbolProp BT"phase" shows the additive change in the approximation of the phase characteristic in radiants. There are multiples of SONDZEICHEN 112 \f "SymbolProp BT"/2 allowed only. The last line in the page provides the deviation of the approximations to the real characteristics at the current point. This value must be consumed very carefully, because it is valid only if the function has this single root and not two or three. But in practice it is a good approximation if there are no roots in immediate neighbourhood of the examined point. Here there are the three pictures of our analysis:    2.5.6. Result: Gain Characteristic In the graph of the gain characteristic the abscissa is labelled in powers of ten and the ordinate in decibel. Approximation and smooth curve are plotted over each other, but they are distinguishable clearly. In order to show pictures which exceed the boundaries of the calculater's display, too, the graph can be scrolled with the arrow keys (see below). The size of the graphs is limited only by the amount of free memory.  2.5.7. Result: Phase Characteristic There mainly the things mentioned above are valid. The ordinate here is labelled in radiants. Especially it has to be considered that the deviation between the approximated and the real characteristic is quite large, because an approximation of order zero is used.  2.6. Known Weaknesses of the Program This program is not perfect too, therefore there is a list of weaknesses which should not cause severe problems in general. SONDZEICHEN 183 \f "Symbol" \s 12 \h Roots with a multiplicity of more than 2 leed to inaccuracy in calculating the roots. The analysis of the different points can show wrong results (e.g. square term instead of 2-fold), but generally the characteristics are not affected. SONDZEICHEN 183 \f "Symbol" \s 12 \h In general it has to be considered that the protocol shows rounded values, i.e. { 1 .00001 } appears as s+0 on the first page! SONDZEICHEN 183 \f "Symbol" \s 12 \h Equal terms in numerator and denominator are not reduced. The phase characteristic shows at that point a useless vertical line. 3. Usage of LISTER Because the BODE-program needs an auxiliary program named LISTER to display the analysis protocol, the following table shows its simple operation. function key action  SONDZEICHEN 184 \f "SymbolProp BT" (divide) jump to first page  SONDZEICHEN 43 \f "SymbolProp BT" (plus) jump to last page  SONDZEICHEN 180 \f "SymbolProp BT" (multipliy) one page up  SONDZEICHEN 45 \f "SymbolProp BT" (minus) one page down  arrow keys scroll graph  shift-right + arrow keys jump to top, bottom, left or right edge of the graph  F (menu key) toggle display of page numbers  OFF turn off the calculater (no program termination!)   Every other key or key-combination leeds to the termination of LISTER. 4. Deinstallation of the Programs If you want to remove these programs from your calculater to install new version (can not image other reasons :-), do the following: a) Type in for BODE to be removed: :x:1228 for LISTER to be removed: :x:1230 (x has to be replace by the specific port number!) b) Press ENTER again to double the previous line c) Execute DETACH d) Execute PURGE It is recommended to run BCLEAR before removing the BODE-library to delete the protocol of the last analysis. Otherwise these data reside in memory and can not be deleted until BODE is installed again! 5. Appendix 5.1. History of Construction of BODE v2.31sg v2.31sg - april 1995: + moved all language dependent strings into string-table + created parallel English version BODEE.LIB and BODEGXE.LIB v2.30sg - february 1995: + improved error handling + added my own graphic packers PKGROB/UPKGROB + documentation written (this here!) v2.20sg - december 1994: + created G(x)-version v2.20g + routine for displaying polynomials expanded to complex numbers + factorizer made available through "FACTOR" + rooter made available through "CROOT" v2.17s - november 1994: + now pre-factored polynomials can be entered too v2.13s - november 1994: - PGROB/UPGROB removed because of bugs (graphic stored uncompressed now) v2.10s - october 1994: + complex routines for calculating the characteristic replaced by real number algorithm (SONDZEICHEN 174 \f "Symbol" faster) + included graphic packer/unpacker PGROB/UPGROB by Erik Bryntse. + average execution time 25 - 30 seconds (SX)! - removed some bugs in displaying the polynomials on page 1/2 v2.00s - september 1994: + translated into system-RPL and stored as library + LCD turns off for higher speed (about 11 %) + average execution time 45 - 50 seconds (SX)! - auxiliary program POLY removed v1.20sg - may 1994: + included factored form of function into protocol + parallel version v1.20g for G(X)-series v1.00s - april 1994: + program written in user-RPL and stored as directory + complete analysis of zeros and poles + characteristic of gain and phase plotted + POLY v3.2 by Wayne Scott used for finding roots + average execution time 4 - 5 minutes (SX) 5.2. Where to Get Further Information? If you have any suggestions or questions feel free to contact me at: Email: e9225490@stud1.tuwien.ac.at WWW: http://stud1.tuwien.ac.at/~e9225490/ SEITE3 Thomas Reitmayr BODE v 2.31 sg AKTUALDAT \@ "MMMM jjjj"June 1995 SEITE3 v/=nst"v}/=nst} /=nstJ:] %  4%%   %C (Iz:4F |@  .1  ` & MathType@8Times New Roman+-!P!s!a!s!a !s !a]!s!a!!s !a Times New Roman-!n!nn !n !n4Times New Roman+-!(l!)pSymbol-!=!+B !+!+!+!+  Symbol-!- !-g  Times New Roman+-!1y !1!2!2!1!0MT Extra-!L"System-x60:KW T  .1    & MathTypePTimes New Roman-!a!a!a !as !a Times New Roman-!n!n Symbol-!- Times New Roman-!1!2w !18 !0MT Extra-!L{ Fences0-!l{ !q"System-::"+   .1   & MathTypeP@ Times New Roman+-!s@i!s@0!s@ !s Times New RomanM-!4 !3!2@Times New Roman+-!5@!104@ !100@7!1@!5@!104@!100@!0@Symbol-!+@!+@ !+@!@!@>!;XFences.-!l;!q"System-z :(4N6 @  .1  $ & MathType@8Times New Romanf-!P!sR!a!s9 !ao !s!a!s!a,!b!s!b !!s"!b Times New RomanM-!nE!n !n !n!m0!m4Times New Romanf-!(l!)!(!)!(4$!)pSymbol-!=#!+ !+!+!+8!+!+!!+}  Symbol-!- !-  Times New RomanM-!1 !1R!1z!0 !1#!0TMT Extra-!L!L"System-':6P| < .1  ` & MathType`Times New Romanf-!a~!a!aC !a !b!b!b!bm Times New Roman-!nC!n !m!m Symbol-!-:!-< Times New Roman-!1G !1 !0!1!16!0hMT Extra-!L!LFences)-!l !qg !l!q !m!r"System-Z:(N  ~ .1  $ & MathType`@ Times New RomanM-!s@X!s@!s@ !s@Times New Roman+-!(@!)@h !(@ !) 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