From comp.sys.hp48 Wed Mar 4 19:08:16 1992 Path: seq!ecsgate!mcnc!gatech!swrinde!zaphod.mps.ohio-state.edu!sol.ctr.columbia.edu!ira.uka.de!sun.rhrk.uni-kl.de!hammes From: hammes@rhrk.uni-kl.de (Stefan Hammes [Informatik]) Newsgroups: comp.sys.hp48 Subject: What is a Bode plotter - The answer Message-ID: <1992Feb21.222025.29025@rhrk.uni-kl.de> Date: 21 Feb 92 22:20:25 GMT Organization: University of Kaiserslautern, Germany Lines: 142 *********************************************************************** * * * This is posted for (and written by) a friend of mine, who cannot * * post. Please send all replies to him. * * * *********************************************************************** tsarver@uunet.UU.NET (Tom Sarver) asked: >>> What's a bode plotter? I have a B.A. in math, but I don't pretend to know every mathematical term. Could you give me a few sentences about plotting bodes (or bode plotting) and what your program does in relation to this concept? Thanks, --Tom BTW, wouldn't hurt to post your description to comp.sys.hp48. <<< Well, my english isn't the best but I try to explain it ! The bode plotter program is especially developed for use in control systems. Look at the following picture. It shows a block diagram of a typical control system. (A home heating system) --------- ----------- Desired W xd | | Y | Heating | X House ---->( )----->| Valve |----->| |------> temperature ^ | | | system | | temperature - | --------- ----------- | | | | | | ----------- | | | Thermal | | --------| |<----------------- | sensor | ----------- The components of a control system are diverse in nature and may include electrical, mechanical, thermal, and fluidic devices. The differential equations for these devices are obtained using the basic laws of physics. (...) Laplace transformations of these differential equations yields an algebraic equation, in terms of the complex frequency variables s. (...) For example, consider a simple electrical circuit shown below, in which we apply an input voltage v1(t) to an RC network. R ( )-----/\/\/\--------------------( ) + | + | C --- v1(t) --- v2(t) | - | - ( )-------------------------------( ) The output voltage v2(t) is related to the input through the differential equation dv2 v1(t) = RC --- + v2(t) dt which has been obtained by applying Kirchoff's voltage law. Taking the Laplace transform of both sides of the equation above, assuming zero initial conditions, we have V1(s) = RCsV2(s) + V2(s) Solving this equation, we get the transfer function V2(s) 1 F(s) = ----- = ------- | s = i*w (omega), CR = constant V1(s) 1 + sCR omega is the radian frequency (w = 2*pi*f) With such a transfer function we can feed the bode plotter program. The resulting plots are called Bode plots, honoring H.W. Bode, who used them in the study of feedback amplifieres. The plots require semilog graph paper, where the logarithmic scale is used for the omega axis. Two graphs are required, one for the gain in decibels, defined as 20*log(abs(F)) plotted against frequency on the log scale, and the other for the phase shift in degrees plotted against frequency on the log scale. The simplification in Bode plots result partly due to the basic advantage of logarithmic representation that multiplication and division are replaced by addition and subtraction, respectively. There are several other advantages but this would be to much for this explanation. The logarithmic scale, used for frequency in the Bode plots, has some interesting properties. First we observe that the logarithmic scale is nonlinear; that is, the distance between 1 and 2 is greater than the distance between 2 and 3, and so on. As a result, use of this scale enables us to cover a greater range of frequencies. Semilog graph paper comes in one, two, three, or four cycles, indicating the range of coverage. For example, a two-cycle graph paper has the range from 1 to 10 and 10 to 100. It is interesting to note that on the log scale, the distance between 1 and 10 is equal to the distance between to 100. This distance is called a decade. In fact, the distance between k and 10k, where k is any positive (nonzero) number is equal to the decade. This follows because 'log 10k - log k = log 10 = 1' Similarly, the distance between k and 2k is equal to the constant log 2, and is called an octave. It may be noted that we cannot locate the point omega = 0 on the log scale since log 0 = - infinite. Consider the transfer function 1 F(s) = ----- 1 + s To simulate the semilog graph paper for a magnitude plot, the HP-48SX has 'only' to plot the following EQ. EQ: '20*log(abs(inv(1+i*alog(w))))' This all is done by the bode plotter programm. That's all for now. ************************************************************************** * * * * |) /\ |/ * Organization: Fachhochschule Osnabrueck * * |\./--\.|\ * Fachbereich Elektrotechnik * * * * * Rainer A. Kemme **************************************************** * Sedanstrasse 16 * * * W4500 Osnabrueck * * * Germany * Internet : kemme@jupiter.rz.uni-osnabrueck.de * * Ph.: +49 541 61252 * * * FAX: +49 2651 48845 * * * * * **************************************************************************