w—r——.____ ' This is to eenlfg that the thesis entitled ”Replacement of a Distributed RC Line by Bumped Parameters Ior a ThermalAEleetrical Analogue." presented by Laimons Freimanls has been accepted towards fulfillment of the requirements for '__l_4§_degree mml Engineering . Major progesor‘ ‘ Date October 8, 1952 0-169 . .1 l'- . n‘ ' I . 1‘ t‘ lit-J " - l l ‘ ,. {4‘3“ . 4 f' ’1." " d "l," . .¢ 4) b" \‘él'r ‘ ;' . c- ' :21 ‘5. [:1 -‘.A'"\ .1 .5 ,'i€,._;hc'.'. 1‘! ‘KLCW‘F: . q 515%. a» “ J M .:- .7", ‘ p1,. “' , ".W‘v .'.‘J A ‘ 1 ’ ‘5 " '? #2" k '- 1, , . ‘. I" h ' I) i, -' r Mfg/H L“ "‘ hot”. i if "'h ' ' .‘u 1‘ '0'" ‘f ‘ > ' 1 O 11"» I . . . f l .w ‘ 1‘ 3‘].ng “ ,. . ‘ ‘. ' . ‘ 7”: E :1 f ’ An“! N . 1‘ , DJ ', v ’ r m , S; .7 3:0" .‘ '6 ‘, ‘ ._ . ,. . .M " it. .2. 2-) _. .3. u . . ‘ "I ~ c"- qt .pc .- ‘ "'k’.‘ l 1.... l.- ’ ‘1 ' ' a '.(I,‘ , . .1 1' 'I ; .,:.- rw " f“ . 1 :4 -", 1 “9-. , r d' Q. . ._ I, . '1 ‘I' -. 4'1 .. rt ‘ ~ . ‘. ',,a “’ A" ‘(V . .3 ' v «<‘ ’1')“ ‘ V ~f ‘ ‘3 1. . A ‘ '3' r 9' _y ‘xx . ' f l ‘ ‘ pr 5*“- I -.;.._ ‘1’... . a \__—__. J‘nn _.— ;- .‘i; "'”“M v». .- .- be“ ‘- . '7. REPLACEMENT OF.A DISTRIBUTED R.C. LINE BY LUHPED PARthPiRS FOR A THERMAL-ELEC?RICAL aNALOGUE by Laimons Vreimanis W A THESIS Submitted to the Scnool of Graduate Studies of Michigan State College of agriculture and Applied Science in partial fulfillment of the requirements for the degree 0! MASTER OF SCIENCE Department of Electrical Engineering 1952 THESIS A CKNO WLED GIa’IE NTS The author wishes to express his sincere thanks to Dr. J. A. Strelzoff for introducing him to the prob- lem of analogues and encouraging him to carry on gradu- ate studies at Michigan State College He is also greatly indebted to Dr. J. S. Frame for his valuable help in the mathematical derivations. “3”} 0 TABLE OF CONTENTS Introduction P. Temperature Control P. Methcis for Obtaining Transfer Function P. The Electric Analogy Method P. Replacing Distributed Parameters P. Statement of Problem P. Solution for Characteristic Termination P. Introduction of Different Terminal Conditions P. General P. Solution for the Distributed Line P. Solution for the Lumped Line P. Solution for One Lumped Section P. Solution by Laplace Transform P. Conditions for Replacement by One Section P. Examples in Heat Transfer P. Conditions for Replacement by "m" Sections P. General P. Equality of Residues P. Equality of Roots P. Simplified Case Where NS'O P. Simplified Case Where N-e-oo P. Conclusions P. Bibliography . P. 20 20 22 27 32 52 35 57 38 58 39 42 45 46 49 52 INTRODUCTION Temperature Control In order to design a controller. it is necessary to know the dynamical characteristics of the system to be con- trolled. In a thermal.system. this means to know the ten- perature-time response of the controlled medium. as the heat supply is varied. The dynamical characteristics of a thermal system can be expressed mathematically in terms of a transfer function in Laplace transform. Such a transfer function - G is used in the theory of servomechanisms and regulating systems. (1. 3. 7). For a thermal system. it would contain a certain number of negative real roots. If the transfer function of the system is known. a pro- per ccntroller can be applied and the resulting equation in- vestigated as to the expected error and stability under dif- ferent load and reference input variations. An alternate method. used in temperature control. is to obtain a graph which represents the temperature—time varia- tion of the medium as heat is suddenly applied. This graph is called "process reaction curve" (9). ' In general. all natural processes are capable of repre- sentation by an exponential curve. If_the transfer fuuation has more than one root. a so called "transfer lag" would ex- ist as a result of superimposed echnential functions. Fig. 1 Process Reaction ' Curve without Transfer Lac (1 root) Fig. 2 Process Reaction Curve with Transfer Lag (more than one root) A large transfer lag makes the control problem difficult. whereas it is quite simple if no transfer lag exists. Therefore. to decide what type what type of controller should be used. the transfer function or process reaction curve should be determined. transfer function is known. The latter may be plotted if the METHODS FOR OBTAINING TRANSFER FUNCTIONS To obtain a transfer function or process reaction curve for a thermal process. an experiment may be performed on the system. thus determining the reaction curve by actual measure- ments of the temperature at certain time intervals. Sometimes. however. this can not be done and other mes thods have to be employed. If the thermal characteristics of the materials in the system as well as the dimensions and operating conditions are known. an analytical solution may be attempted. Thermal conductivities as well as specific heats would have to be cone sidered since a transient solution is required. The calcula- tion is extremely involved. however. and has been developed for only very simple cases. Besides the mathematical-analytical method. three other methods of analysis are known. 1). A graphical method. first devised by Schmidt (24). It is suitable only for very simple cases and only after mak- ing a number of simplifying assumptions. Even then. it's use is rather cumbersome. 2). A numerical method. known as Southwell's relaxation method and applied to heat flow problems by Emmons (11). The method is easy to learnzand - within its range of applicability - is very useful. It becomes quite involved in case of changing thermal properties and if applied to cyclic heating. 3). An electric analogy method. which is very versatile -4- and subject to fewer limitations than the other methods. It's main drawback is its' rather expensive equipment (available at the Department of Mechanical Engineering. Columbia Univer- sity). THE ELECTRIC ANALOGY METHOD The method has been first devised by Beuken (2). and in- troduced into the United States by Paschkis (20. 21. 22). The method rests on the fundamental similarity between the flow of heat within a rigid body and that of a charge in a noninductive electric circuit. Conservation of the scalar quantity. charge. corresponds to conservation of heat: The scalar point function. electric potential. corresponds to the scalar point function. temperature. The concept "electric capacity of a conductor". corresponds to the concept "ther- mal capacity of a portion of mass". There is a direct identity in form between the defining equations for thermal and electrical resistance and thermal and electrical capacity. The temperature distribution in a body. at any time. is given by Fourier's general law of heat conduction. which is a partial differential equation derived by the usual methods considering an infinitely small cube. Schlack (23. p. 29). It is: so K a___e__ 3:9 24s) 31‘ pc ‘bx‘ By‘ ‘32“ where temperature at any point given by the coordinates x. ’03 time coefficient of thermal conductivity specific heat of unit mass ‘ankx CD mass per unit volume -5- The distribution of conductivity and specific heat in space is assumed to be uniform and continuous. The solutions of this equation are very complex. They .may be found in books on advanced calculus and heat transfer. Carslaw (5) in his work. which probably is the most compre- hensive development of the analytical approach. gives three- dimensional solutions for rectangular parallelepiped. cylin- der. sphere and cone. Even after the solutions of this equa- tion have been obtained. it is necessary to be extremely careful in selecting the proper solution to fit the condi- tions of the problem. It is possible. however. to represent a three-dimension- al problem by a combination of one-dimensional systems. Such a method is indicated by Paschkis (20) where the thermal-elec- trical analogy is developed on one-dimensional basis which then could be used to represent two or three-dimensional sys- tems if necessary. For one-dimensional heat flow the Fourier's equation re- duces to: ‘36 K ‘3‘9 32‘ /°C 'BX‘ One dimensional heat flow would take place. for instance. along an insulated rod or across an infinite plate. The analogous electrical system would be a conductor with resistance and capacitance uniformly distributed along it's length and having negligible leakage and inductance. TTTTTTT TTT Fig. 3 Distributed R-C Line The differential equation describing the potential at various points of the conductor at different times is: av_ / 2w ~.* 31‘ RC 'ax‘ where: V potential at distance x and time t R = distributed resistance ohms/unit length C : distributed capacitance farads/unit length Comparing the two equations. it is recognized that if the ten- perature is considered to be analogous to potential. the qual- __,tityP/: is analogous to the quantity --’-- in the 'electrical 'RC system. The analogous quantity for heat flow.$g. can be estab- lished considering the defining equation for heat conductivi- ty. The heat flow across a unit surface per unit time at any point x is: -8- where: thermal conductivity K .256; " t t d1 t t i t 2" - empera ure gra en a pon x The current density at a point x in a conductor is: -- 3V 1. -- - 6 —— “ax where: 6 : specific conductivity of the material 3V ’ otential radiant at oint I 2x " P 3 P For one dimensional systems where Z- is uniform and the dis- tribution of 61s uniform and continuous [sun—LEV ’R ex I If K is analogous to '27- then heat flow per unit time. (3'. is the analogous quantity to current i. K D The antit -- was analo cue to—— . Therefore the analogue for capacitance per unit distance C would be the specific heat per unit volume PC. The analogous quantities thus established are given be- low: V - potential volts 9- temperatu're [0F ] I - current amps. q- heat flow [-54%] _ _ volume 1w C Capacitance farads/meterfc specific heat [fl G - conductance mhos/meter K- heat conductivity R - resistance ohms/meter -'-- inverse conductivity K GL" load conductance mhos h- surface coefficient 1' 3f“ J 13h“ f )7“ °F /)r of heat transfer ff“. 0;», ohms _l__ inverse surface R - t L load resis ance coefficient -9- Now an electrical analogue can be drawn for the one-dimen- sional heat flow problem. For the case of the insulated rod (Fig. 4) , fl 1 . \ s ‘- Q , ,’ ~ --“ ’8», -‘_-‘\ -Q‘A“\ ./“‘ .z ; 5 7‘ A 6941‘ («1‘90 ¢§EWOH1- ’Kinob ‘: c. <2un;y fear Fig. 4 Insulated Rod with Outflow Conductance the electrical analogue would be an R C cable terminated in a load resistance (Fig. 5). T T T T 791...; Fig. 5 R C Line Terminated in RL Two essentially different cases will be considered: l).The temperature of the heated end of the rod is a known function of time. 2).The heat input at the heated end of the rod is a known function of time. The problem in both cases would be to find the tempera- ture of the far end as a function of time. In the first case in the electrical analogue. a poten- tial V(t) should be applied and the potential across the -10- load resistance measured. For the second problem. a current I(t) would have to be applied and the potential on the load resistance measured. The second case is of much greater importance in con- trol problems wherever electrical heating is used because the heat input would be known. It could be directly derived from the power input in the heating elements. To obtain a process reaction curve. which gives the tan- perature response.at a certain point to a sudden application of heat. from the electrical analogue. an unit step current should be applied on the input terminals and the potential across the load resistance measured and plotted versus time. If it is preferable to use voltage as the input function. an alternative method. as suggested by Brown (3). may be used. This method utilizes the dual of a R G cable. . I 1 a o No I o o ' I o 0'; I o o 1 b o'c’u. o I'I'IVI 0 mm o o'o’o'o o" a" . I .1- 0 e ‘. "J 4/“! Fig. 6 Dual for R C Cable The unit step heat input in this case would be analogous to unit step voltage on the input terminals. and the tempera- ture at the outer surface to the current through load induc- tance. -11- This way Brown (3) avoids the necessity to apply constant current. It is of advantage if the problem is solved analy- tically. as Brown (3) does. because almost all of the solu- tions in electrical circuit theory are on this basis. If a laboratory model is to be used. however. this method seems to have thedisadvantage. that inductances usually would not be available with such accuracies as capacitors. and a certain resistance in them would not be avoidable. It is proposed. therefore. to use the R 0 analogue and apply a unit step current. Paschkis (20) describes such a constant current device used in the permanent electric model at Columbia University. This device provides. by means of electron tubes. any cur- rent value between O.lmA and SOmA which can be set and main- tained constant throughout the experiment. regardless of ap- parent changes in resistance resulting from the loading of the condensers. The voltages are measured by recording milli- voltmeters fed by two stage amplifiers. This way the current drawn from the circuit by the amplifier is less than 10"9 amperes. which may be neglected as compared with the leakage currents through the insulation and other parts of the model. REPLACING DISTRIBUTED PARAMETERS This far the distributed one-dimensional heat flow has been represented by a distributed electrical system - a R C cable. For an actual problem in laboratory. it would be diffi- cult to construct a cable for each problem. The parameters could not be easily varied and interconnections to represent a two or three dimensional heat flow could not be readily made. The natural suggestion would be to replace the distribu- ted R C line by a cascade of lumped R C circuits. This is what actually is done in all the models described in litera- ture (10. 12. 18. 20. 21. 22. 26). TTTTITITTTT T T T Fig. 7 Distributed and Lumped R C Circuits The question may arise as to how many lumped sections should be used to represent a certain distributed system. Paschkis (20) states that the smaller the "lumps" the -15- the more perfect will be the representation of the actual ca- ble by lumped cable and that a feasible compromise is usually possible in practice. In the experiment described (20) the analogue representing pipe insulation. is divided in five sec- tions. so chosen as to conform to the positions of the there mocouples in an experiment performed on the same insulation by Perry and Berggren at the University of California McCann (18) indicates that the size of elements into which the distributed system should be divided would be governed by the configuration of the body. boundary conditions. and the required accuracy of the solution. Tribus (26) investigating the ice protection problem di- vides the guide vane of an airplane in eight sections and the propeller in five sections. and uses a lumped R C network for each section. interconnecting the separate networks in such a way as to represent a two-dimensional heat flow. This work is interesting because he uses non-linear networks to represent parameters that are not constant but varies with temperature. Such as the convection from an exposed surface. Eckman (10) is investigating automatic control problems by electrical analogy and using three R C networks in cascade to represent a multiple capacity process. Each capacitor h: shunted by a resistor to represent the demand. Fig. 8 Lumped R C Network with Resistors Representing Demand -14- An equivalent network is used to represent the thermocouple and well. Hornfeck (17) states that a thin thermometer socket could be represented by a single lumped R C section. whereas for a heavy socket it would not be permissible. He suggests that two sections be used. The differential equation of such a sys- tem is easily handled analytically. and according to the author (17). gives reasonably good correlation between the experiment- al and calculated response. (In the discussion of the article (17) Paschkis questions the validity of using only two sections and indicates that it probably would be necessary to use several sections for the protecting socket as well as for the internal element. Since the resulting equations would not be manageable mathematically. the electric-analOgy method (20) is suggested. In the authors closure (17) Hornfeck states that if a di- mensionless parameter m (ratio of the socket film resistance to the internal resistance) is either much smaller or much larger than unity. the response reduces to the simple exponen- tial function. It should be noted here that the circuit used to repre- sent the thermometer is essentially different from that used for heat conduction through a wall. . 4? 9.. e I 9 «919 T4 I62 I T T Fig. 9 R 0 Circuit to Re- Fig.10 R C Circuit to Represent present Heat Flow through Heat Flow to a Wall with Out- a Thick Thermometer Wall flow Resistance -15- Ahrdent (1) indicates that the analysis of a thermal sys- tem.may be simplified assuming that certain elements are es- sentially either resistive or capacitative, and that a distri- buted R 0 system, the determinantal equation of which would have infinite number of poles on the negative real axis. could be represented by a lumped system neglecting the poles at real high values of S. because the value of residues at these high values of S diminishes. Eyres (12) have solved a heat conduction problem replac- ing the distributed system by six lumped sections. and solv— ing the resulting differential equation on a differential an- alyzer. The same problem was solved analytically for the db- tributed system and the maximum deviation of the two solutions found to be 0.4%. Since six lumped sections result in a differential equation of the sixth order which requires quite an elaborate set up on the differential analyzer and rather long time to carry the solution through, another solution was performed using only three lumped sections. The increase in the maximum error was slight (from 0.4% to 1.5%). It was con- cluded that probably using only two lumped sections. the ernor would be within acceptable limits. It was indicated that the error is approximately proportional to the inverse of the square of number of sections. STATEMENT OF PROBLEM The question may arise. whether it would be possible to obtain a criteria for the different factors that govern the error if certain numbers of lumped sections are used to represent a distributed system. Since. using the already established analogies. a con- version from an electrical to a thermal system can be readi- ly made. it is pr0posed to attempt a solution for the elec- trical circuit. because there is not a true lumped parame- ter in a thermal system. Taking the simple example of one dimensional heat flow in an insulated rod for the case more important to control problems. where the heat input is varied. the problem in the electrical system.may be stated as follows: «(the A) e I“ T TITTT T a“. Fig. 11 The Distributed R C Line 72, 7?. 1’: Q—I— 5T Fig. 12 The Lumped Line : E {odd If a line of finite length. having distributed resis- tance and capacitance and terminated in a load resistance. is to be replaced by a number of equal lumped sections. such that the total resistance and capacitance of the lumped sec- tions is the same as the total resistance and capacitance of the line. what is the smallest number of sections that diould be used to have the voltage response on the output terminals -17- of the lumped line to a unit step current input within a prescribed error of that of the distributed line. Stated in mathematical terms: Rload (d) " Blend (1) Bline (C1) Rline ((')= (n ' 1)Rs ‘Cline (d) : Cline (I) g (n ' 2)08 R81 3 R82 2 R3(n - 1) Cal = Caz g Cs(n - 2) For 1(t) = u(t) the response for the distributed line is edm and for the lumped line 6" (t). Find nmin such that 64(1)- 9M0 é J Where the subscript 0/ designates the distributed line; I’ - the lumped line; 3 - parameter value of one section; n - the number of nodes; (n-l) the number of equal resistors; (n - 2) the number of equal ca- pacitors in the lumped line; cf - the maximum allowable er- ror. The unit step input is chosen because it is very common in discontinuous control problems and response to any other function can be derived from the unit step response. apply- ing the principle of superposition. (L4) SOLUTION FOR CHARACTERISTIC TERMINATION A similar problem is discussed by Guillemin (16). He states the problem as follows: if an artificial line repre- senting the long line is to be constructed. one should be able to determine the number of sections and the parameter values of each for a given finite length of line. frequency range. and maximum allowable error. Guillemin shows that such an artificial line is physically realizable and that if the num- ber of structures becomes infinite. the artificial line heads to the uniform line itself. An expression for the decimal error in transfer impedance and propagation function is de- J, ”33v 2';- (»'-)‘ (12 +JLwXG*J'C“') These errors increase with frequency being smallest at zero veloped. frequency. while for a given frequency they increase as the square of the line length and decrease as the reciprocal of the square of the number of cascaded sections. When the maxi- .mum errors in ZT and are specified of the same magnitude. than the error of ZT will govern the design. The expression is derived on a steady state basis for a given frequency range. It could be extended to a transiait case if a unit step function is applied by means of Fourier's integral. Unfortunately. however. it can not be used for the pro- blem at hand. Guillemin does not consider the terminal con- ditions which. it is believed. would have considerable effect upon the error if the line is terminated in an impedance much different from the characteristic impedance. -19- In communication networks it would not usually be the case. A thermal analogue. however. would have to be termina- ted in various impedances depending upon the convection and radiation conditions on the surface. INTRODUCTION OF BITTERENT TERMINAL CONDITIONS General To investigate the problem for different terminal condi- tions a method is proposed where the transfer admittance for the R C cable terminated in a pure resistance is found and compared to the transfer admittance of m-section lumped net- work terminated in the same resistance. If the two transfer admittances could be expressed in essentially the same form. conditions probably could be derived under Which the response of the two systems would not differ by more than the allowable error. The R C lines under transient conditions have been treat- ed originally by Lord Kelvin in order to determine the prac- ticability of a transoceanic cable. Lord Kelvin used the classical methods of solving partial differential equations. A treatment of the same problem by transformation calculus a; given by Carson (6) and Cohen (6). Cohen (a) first gives a solution for an infinite line and a line terminated in an open or short circuit. For the pro- blem of different termination the author states that the pro- blem is by far more difficult and that it is only in special cases that it is at all possible to obtain a completely de- veloped solution. A solution for a special case where a general RLCG line is terminated in a coil with time constant L/R,is given. The solution is in terms of propagation con- stants. Goldman (15) states. however. that there is no dis- tinct velocity of propagation along the R C cable and that the signals are said to be "diffused" rather than "propagated" -21- along the line. It is doubtful. therefore. if the solution for a "propagating" RLCG line could be applied to a R C cable. Carson (6) makes the statement that if the line is clased by arbitrary impedances instead of open or short circuits. the case is quite different. and the location of roots becomes. ex- cept for simple impedances. and then only in the case of non- inductive cable. practically impossible. While. therefore. the expansion theorem solution can be formally written down. its actual numerical evaluation is a practical impossibility. ex- cept in a few cases. For this reason. the althor would not consider it further in his work. From Carson's statement. one could conclude that While a general solution would be impractical. a solution for the spe- cial case - R C cable terminated in a pure resistance could be possible. Unfortunately. such a solution is not given in Car- son's work. SOLUTION FOR THE DISTRIBUTED LINE Investigating further it was found that a solution for the differential equation that describes the problem at hand is given without proof by Carslaw (5. pg. 104) and an indica- tion of the method it is derived by Newman (19). a formal proof is not given there. It is solved by classical methods and it was shown by Dr. Frame (13) that the functions involved result from the boundary conditions of the problem. The differential equation is written for a problem in heat conduction with the same boundary conditions as these in the rod. for which the electrical analogue was drawn. It is obvious. therefore. that the solution applies equally well to the R C cable with terminal resistance and unit step mirrent input. In Newman's (19) work it is given in a dimensionless form and therefore readily rewritten for the R 0 cable. The differential equation. boundary coniitions and solu- is given below. A formal proof. however. shall not be given. because it is not essential for the problem. The Problem: 5!. , p a ,1 _px Fig. 13 The Insulated Rod A rod of homogenous material is subjected to a heat in- put at constant rate into one face I = 0. while the other -23.. face I = a is exposed to a fluid medium. The length of the rod is a. It is assumed that the heat travels only in the x direction. The surface coefficient of heat transfer h. may be considered to be a function of the nature of the fluid and its velocity. but this analysis does not permit considera- tion of its variation with temperature. Find the temperature-time relationship at the surface I = a. The following quantities are used in the develOpment: x : distance in the x direction ?”1 t = time :fir] K : thermal conductivity of the solid Eggfif3%% o : specific heat of the solid 4%;5377] P: density of the solid [—%—1 k : thermal diffusivity )égr -‘€$;—- h = surface coefficient of heat transfer Pigfiggrj q: constant heating rate at x = 0 [7%] 9: temperature [or] From these certain dimensionless quantities are formed: : %2—. a modified Nusselt Number = E; a dimensionless quantity involving time : EEK a dimensionless temperature response The differential equation describing the heat flow is the Fourier's equation in one dimension. beak'aze f 'ax‘ -24- The boundary conditions. 1) The rate of heat flow across the surface x = 0 2569 s-/K’ ____. 9° (8x23 2) The rate of heat flow at the surface x I a 9a=I58q'-K(§f)a 3) The initial temperature of the solid is uniform and e- qual to the temperature of the fluid (assumed zero here as re- ference). when t = 0 e a 0 4) After a long time steady state will be reached and the temperature gradient through the rod will be constant whent=°<> 4, ea‘ 5 and Egg—2— %x K 5) The total gain in heat content by the rod at any time must equal the difference between the total heat input up to faecpdx - crt- J29... 0/9 An equation satisfying all of these conditions is: that time. 94 1 2 g p Z€;,/_H§L-f—£—--£Z figfi:wna¢ _"C°5oanl%)4é16n a Na '3’, [5: o m. (Ii/V9] d 03/ where Bn is defined by #3 cofflarm -25- Since the temperature at x = a is of primary interest. .. 3.1? [ 3 —-—l —2 fl”; * ”“2 “ COSfln ‘6 a Na [8031:301' m. (14 m9] ha] The equation can be rewritten for the R C cable if cor- responding dimensionless quantities are employed. for Nu R1139 = N 31—033 for P HIinzcline = '7%? PT’8 time constant of the cable) for E ifRf?;e : YE} °1Rline (Y : transfer admittance) _- e( I“ TT TT TT 7.0,. Fig. 14 Distributed R 0 Line Rline = total resistance of the line Cline = total capacitance of the line then x 3 f’ I g 7E7beu¥_ fiqu+4Afz ”(g-f%"ir —‘ COS Y), ’Pa" ’30“ 2 fla‘INfiv‘IV] . k -25- The quantity: a? + N“ a.“ [at . ”(1+ny “5/6” is a pure number and function of %%%3% = N only. It can be computed for different values of N up to any desired n if A? tables for the roots of 601288? N are available. Newman (19) and Carslaw (5) give sich tables for Bn (n = 1 to n = 7) and for values of N from 0.001 to 100. If the values for Bn (n1) 7) or different N's are desired they can be found graphically as shown by Cohen (8. pg. 88) (see graph 1) or they could be calculated to a greater accuracy by approximation methods as indicated by Dr. Frame (13). Actually in the regions where N» O. or N-u-oo and the functions: sin x. cos 1. tan 1, cot x. may be assumed e- qual to the arguments. the calculation of Bn and the total numerical coefficient becomes quite simple as it will be shown later. Once the numerical values for c fl”2+/y‘ ~— COS ,2 l4!) 2 flnz[£nz*N(th-7 fl are computed. the dimensionless equation for the transfer ad— mittance assumes the form: 1 I g 2231!- “fl: “lg: ’5" We: ‘1'" X» ”(in Q0“ A’ 6 T +Az€ " A56 4’.- . an I . .4 c v u — o . . c II n o I . _ n u _ . q o . .. p u . . a i . .. . . . L . . . .. .. vI 0.. u o. I I o c e I. I <. I I o .I I . I II II I III II I . I III. III I I. I IIIOIIIIII I I III I v I. II I III I II II I I III. _ a. . I . A . fiv¥ Ia . .. .. i . . . .v . 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I . . . . . I I.II «I .. III. I II. l.|I I I I all. I . III vI . .1 I I I .I. .I .. .L e I III I III. II. III. III I 5 III. I III III .I.I ((((( II II III IIIIII . 9 III . ..I. I II III III TII. III. . . . . .«. o . IH . _ . H .... _ _ H n . H. . . .fl . * .. . h . . . _ g , L . . . H __ . i .. . . ... .III... . I -.. . II. I- III I . >I. . .. . . 4 ~ ... . . . . _ , _ _ . y A _ t _ _ I . .n a . . -I-I.I -- ....I- .. -.Ia; II II. «IiitIIITIIIIIL... H . . T ...- I II-.- . ” m . _ . H . . _ _ . ~ _. .. L A _ , _ «III. I w _ a n _ l i . a _ . a _ _ _ .. . . . . . .. w , . H . . .. r _ I o I . . I I . . . v . I . I I . I I I b . I I ... . ..I II o I IIII I II wIIlo‘IvI III III III II l VI I I I I 4... . III III IL I II . . U . . _ i . . _ . . , _ . _ . _ .. _ i , H _ p _ i ,. _ _ _ ... . — . . _ _ H .h _ i i ._ . . J I .I ...I .. . . I 4 I I*. . . f. I IIIIIIII IITIIIIIIIII . IIIrII . II. .I .IIII- .. IIIJ II. I . . . _ _ . . . i . . . . I l I . . . . . v I . H . . .. . . ‘ . . . ... .c IoIIIIlAIoI‘ . eh _ _ kn . _ _. . . _ . . I 7 . A . H . I _ ..I-I . ILI . IL . ..I k F It a a . .1 q . _ _ i < . . . . , .. . _ i . . . _ i . _ .. . .... I I .. .. . I . . , u . . . . . . . . _ . , . ,. . _ , n _ . . . . . . . . . r . s. ..I I II.......... I .. .I I ..I .. . . I.-- IIoI.I....-I.I,.III.I ».II .IIIII. . ..... . I . _ . . m . . . . . . , . . . .. . . . . . . . . . _ . . i _ __ _. . . , . o. + . . I — I l . . o . . ¢ . ..1III4AI I , _ p _ , . , , ,_ _ _ n w _ . . h . a . . _ _ . ..I. .. .I I .... .....< .... ..r.. . .. « .... ... II.» .I? . I ...« .I JIIIoI-III.o<.. .... I. - ..e . If? I 4I..I.v I It . m _ . _ , . . , _ . . I . u . . n . . v p i _ . i _ . _ i . . . , ‘ . . . - . I _ I I. .. I ~ IIIII v IoII.I -.,0 a o . fl . . _ . . ... _. . . . . . . . . _ . .. _. . _ _ . , . . . . . _. .. L. m . .. _. u . . k i _ . . _ . k . _ v _. u . n a I a c I . I . I. < I . I I. . I I c I I I _ . . . .J I . . IIIIIILII Y Ic I k I . or» I I I I I I I I I . o I I I II; .I I..IIIIIILIIII.I III I y . . I I III . . _ _ . u . A u * o a _ _ _ u ,. . _ A _ . . . . fl . _ . . . . . ‘ . v _. o .. . . . . . .fl . _ I II... I . ...., .. . v II . . . _ . . ,. _ . _ .. _ _ . . . _ . . . .. y . _ i _ _ . . . _ .. . .. . _ , _ . _ i _ . . , . , _ _ . , _ .I «I . . u . ... I. to II.I.II+IL II. ..I_ I- I. . ....Ifl IIII It? I ....I 7IIIII|..IIL...IIIIIII ..I III. I +ILI.II9IIIv IIII. I. It . . . . _ , . _ . . ,. . M . _ . x . ~ . . _ * . _ . . . _ . . v i . . . . . . . . . ... . . . _ _ I, . . _ ._ . fl . _ . y. . . _ . I u _ o . . .fi . . , _ . _ . a . . a i , . _ . . fl . _ . _ __ _ w _ m . . h . . a H . . ..‘ . . . . + — . .. .fl . . . . .. . .. I I. . . . . . . . . .. .. _ . . . _ . I. --. . .. . . . : f . .. ......T..... 1;. 1.5:-.- .»,-I._-,-IVI--.I, 1--.- ......I....:. - -.- e -,. ..-... a _ i _ . ,. . _ _. , A _ 4 . w _ . _ . _ . _, .. . , u , _ , _ _ . w H . . .m w _ , H _ . _ u _ _. . A . . . . . . . . , . _ T _ . l _. . .. . _ i . , , a m . _ , _ . . _ .. _ _ . v i , . . fl . . _ i H . _ I fi _ .. . _ . .. . . . , _ i _ e u l“.¢ OICIIIIQ . o I I I o II I. ¢II a. «I II .II OI... III. III II I...“ II IIIIO II III. III IIIle III I. II II II t I II III. IIIII III A IIII I I I. III. . . _ .u . _ , q . . ~ . .* . .l a . . . . . _ i . ... . _ I , ‘ . i _ i , _ ., . . . “ . N _ . ._ . . o n i , i i . . . I l . _ . u . . , i , _ _. _ I . I . . .. ...... .. I 71.. ..- L . .. .. ,I. ... ... I I; .. g .. . H . . . _ . , u w . . ... . _ . ., _ . . . .. - . . . . . _ .. __ “Farm. The?” f __ . . i. . SOLUTION FOR THE LUMPED LINE The next stqp would be to derive an expression for the lumped line of m elements and to investigate under what conditions it could be considered equivalent to the equation d? the distfl.- buted line, within an allowable error. if ’72 If ’7? ‘77 ’73' L2? 1:? j?- 0" TC TC I Fig. 15 The Lumped R 0 Network Consider a network having n nodes. the mutual resistance be- tween each node being R, and the admittance of each node except the first and last nn The admittances of the first and last nodes would be: : 1 : R R Y1.1 .R. and Yum fi‘é’fiififi The total resistance of the line would be Rline ' R(n-l) and the total capacitance Cline : C(n-Z) The mutual and node admittances can be written in terms of . N. and n only, $.nce N = Rline C “load Rline’ line -28- Replacing R by Rline n- C by Cline (n-Zi and R / Bload by Ni 111-11 R . Rload Rline The determinant for the network can be written: el e2 es I e(n-za) _ e(n-l) . en ! 1 n-l -.2:_ 1 Rline R1133 2 n-l 2 n-l C _ n-l Rline Rline - line I ' x 5 _ n-l 2(n-l)/C¢ ‘ ?2(n-1)/01£ - 2:; n-2 gRline n- line : n -1 1 _ n-l ‘ 9 i: %;£EQLL%%ES%E. %%" n ; - 2:; N / (n-ll line - Rline Dividing each column byn n1 Rline e1 e2 e3 e _ n-2 n-1 en !“‘ r' V ’ f 1 1 i -1 ; -1 ‘ - 2 {(n-2)Tn-l) n 5 -l éfli 3“ ”In-2mm) line . -3 . 2,1 Tp A -1 A (n-2)(h-l) ;_ , . f 2 2-1 g2¥(n-2)In-I) E i. -1 .N 7’ (my ‘1 E‘ i : (n—l) a .i : -50- If the determinant is opened. it would result in a poly- nomial inTp of the (n-2) order, the coefficients consisting of combinations of Rline and N. If it is solved for en the numerator would be (Eris-l) .‘1(-1)(n'l)(-1)(n'l)- = {$13.1 Rlin Eli n it could be cancelled by the factor £2}. in the denominator lin 1 in the denominator. ine n- ].anng fi-f- The coefficient of the highest power in'L'P should be made unity and the polinomial factored in its' roots. If a unit step function is applied. the salution for the dimensionless quantity 1 would be the following: Ytr'Rline -l I I Yogi’s-(”'9 K, :5-“ ...)(5- _9_<_z)(5_ 95,)” .(5_ol__»__.2. Where K l is the coefficient at the highest power of Z79 . From physical reasoning one can conclude that for any finite values of Rload (”ghoqédo) . the roots will be all negative and real and therefore the solution 1‘ -04».1-i: t -,Ae"°";+Ae “ff—... -é Y;P(lll: A0 ”2 The first term “A0' is the steady state term and can be easily determined. h co A —— s W en 2’. Y4; 'plinc A0 Fig. 16 Steady State Circuit - - R e - i R ' a - load : 1 n load ) o H...— - line N The steady state coefficient is the same as that for the distributed line. which should be expected. The determination of the coefficients at rp —-/(,;/|2min general terms is difficult for a determinant of a rank higher than 4. Even after they are found, a factoring in general terms would be impossible for any equation of higher order than two. It was done for a determinant of the 4th rank which results in a quadratic equation. Solving this equation in general terms the square root can not be eliminated. which makes the general solution cumbersome and relationships are not easily recognized. A simple SDIUtiOH in general terms can be obtained, how- ever. for a 3rd rank determinant. SOLUTION VOR ONE LWEPED SECTION .A. Solution by Laplace Transform R - ‘line - N E(add i‘loud Fig. 17 One Lumped Section 1 -l o The determinant: lene _1 2;1u2 _1 2 ...I 702;. .e... C - 2 ‘ 5(51’-———/:2 4mg- -I e2(/Y+2) / a c Q” f 2”“: “ SG+ON+2 .. 2” 1. __l ' =__[_______/__€ 2+” 1- \cV‘7QZbe' ‘ly' [Y CONDITIONS FOR REPLACEMENT BY ONE SECTION The result derived is very useful to determine the con- ditions when a distributed line can be replaced by a single lumped section. Returning to the equation for a distributed line. J _ ,_I___ -—/3, fife—t- fl”? VQ+’13».. ’Y qufa T-+'/qz £5 /9 <5 T+n” H-roc and comparing it with the equation for a single lumped section. 5210’ 1‘ l I / “"'""" ;-______ 21-” T- Y 11>... N ”6 it is recognized that the two equations would be equivalent if ,2“;- cmo/ , =2”, 5“within the allowable error, and that all higher terms in the distributed line equation: ‘AL" fa, H. cwna/ /Q;: #392 -are negligible as compared to A12and El. Al can be calculated for different values of N from the expression: fi’ +‘fyz- (as: 4 Z a [WWW] ”6’ ~ ' -1 1 - ' I' - - The (pantity N(A1 NM??? -J;‘.may be called the 70 error in first coefficient. It has been calculated for different values of N and are given in table 1 and graph 2. For N = 0.01. is less than 0.000l%. K. -34- 24H :00 in first exponent. It. too, has been calculated for different 2N The quantity (Bf—2N)£Lsé;may be called the 95 error I values of N and are given in table 2 and graph 2. It is found that both errors cf,“ and 59‘ decrease for decreasing N. That the higher coefficients will be negligible if the error in the first coefficient. (Sq . is within acceptable limits can be shown if the instant t = O is considered. ”or this condition all the exponentials become unity and the sum of all coefficients must beequal to l. N Alfaz/Ag, ,1 Any, ...-pili- 1r n—a-oc and since the coefficients are periodically changing sign and steadily decreasing in absolute value [All #32! find flan] none of the higher coefficients (n22) can be greater than the difference (Al-%) which was not greater than the allow- able error. The exponents at the higher coefficients would have ne- gligible effect because the coefficients themselves are ne- gligible. From the data obtained the concluSion may be drawn that if the maximum error is to be‘z 1.6%. for instance, any $.3- tributed system having N‘< 0.1 can be represented by a single lumped section if a unit step function is applied. Table l VALUES OF THE % ERRORS IN THE FIRST COEFFICIENTS N <5};7% N iswg 76 100 27.685 0.9 11.066 80 27.340 0.8 10.168 60 27.300 0.7 9.171 40 27.240 0.6 8.121 30 27.079 0.5 7.004 20 27.046 0.4 5.814 15 26.919 0.3 4.496 10 26.162 0.2 3.128 9 25.950 0.1 1.581 8 25.715 0.08 1.305 7 25.331 0.06 0.997 6 24.755 0.04 0.620 5 24.047 0.02 0.246 4 22.867 0.01 0.007 3 20.998 0.008 0.002 2 17.836 0.006 0.0014 1.5 15.387 0.004 0.001 1 11.972 0.002 0.0001 0.001 0.0001 100 zég’n 1.96078 1.87500 1.76470 1.66666 1.42857 1.00000 0.66666 0.40000 0.18181 0.09524 0.05825 0.00995 0.00599 0.00099 Table 2 % ERRORS IN THE FIRST B12 2.4186 2.3110 2.1694 2.0418 1.7261 1.1597 0.7401 0.4269 0.1873 0.0968 0.0588 0.01 0.004 0.001 EXFOICENTS Se, 9: 23.348 23.253 22.933 22.508 20.827 15.970 9.930 ~6.970 -3.019 1.650 0.944 0.502 0.250 0.050 Je, I / .O/ O. 00/ EKAMFLES IN HEAT TRANSFER In a heat conduction problem Nu would be less than 0.1 for a good conductor. with slight thickness and poor convec- tion or radiation from the surface. An example shall illustrate this. Heat is suddenly applied 1_‘_"’-_—\ to one side of 1" thick K-226' )5“ 5 85 infinite copper plate gap/7‘ 3f“ (K I 226). The outer O 1 o ""/"' F ff'o’t’” surface is exposed to air and has a total sur- iface transfer coefficient h = 3.85 (an average value taken from an actual problem). Find the dimensionless number Nu. ‘ hoa - 5085 - Nu . ”K'— - TEE—226' - 0.001415 The number Nu thus found is about six times smaller than 0.01 for which the errors were 6K1. 0.000l% and ($9.: 0.5%. Obviously an analogue having only one lumped section can be used for this case. The conditions are quite different. however. for insulat- ing materials and thick walls. A one foot thick cork wall having the same surface conditions would give a value for Nu : 154 and could not be represented by a single lumped sec- tion. CONDITIONS FOR REPLACEKENT BY "M" SECTIONS General To investigate the cases where more than one section is necessary. the logical method would be to solve the determi- nant for n's higher than three and to compare the coefficients and exponents. As it was indicated previously. such solutions are extremely complicated for higher ranks and would not yield manageable coefficients in general terms. It would be possi- ble. however. to solve for all numerical values of n's and N's of interest. For a wide range of values this would be a tremendous task. Some other properties of the transient solution of the lumped network, however. can help to solve the problem. EQUALITY 0F RESIDUES In the transient solution of the distributed system. I L.._'___ ZOOA €213.11;- ‘Qr'zabad' [Y n n-/ all the «Jefficients An should add up to % fir» A1{A2{As/oeo {An/...9%ifn 0° If the difference at»: 7”;— 24:7;- =41," 02/ is called % error in m coefficients then such a smallest num- ber m could be found for which the error kais equal or less than the allowable error. The coefficients An(n)m) would then have negligible effect for the same reason as in one section solution. and ‘the distributed system could be truly eXpressed by a finite trumber (m) of coefficients. The errorgxncan be computed for any desired m. and if it is prescribed a "necessary number of terms" can be found whidi ‘Would truly represent the distributed line. A transient solution for a lumped network having m 08- pacitors (rank of determinant = m/Z) would have a solution or exactly the same form and with the same number of terms. If the roots 31. 32. S3 ... are arranged in increasing order 81( 82(85 ...(Sn_1{sn, the residues or coefficients Would be decreasing in absolute value and periodicd.ly chang- ing sign, if all the roots are positive (this was assumed by Physical reasoning). -40- The steady state term for both solutions is the same and the m coefficients in both solutions should add up to the same value - the steady state term. Since for the two solutions as»; | fl=ln z .__’_ J . /oo :EE:/Qn "jaf C’"°{ :EE:4/L7 “’ [y 4. In» he] 03/ ... lemme) ...>1A.I it can be said that for each corresponding coefficient An distributed "An lumpedé J": 100 Thereforegkrepresents the maximum error in residues for m In terms. The error (grhas been computed for several values of N In and m and are given in table 3 and graph 3. From these tables a necessary number of sections can be selected for a given maximum error¢£;and N value. a» Table 3 Innmmm % ERRORS IN RESIDUES FOR M SECTIOLJS N/m 100 10 5 l 1 27.685 26.162 24.047 11.972 1.5919 2 14.631 13.153 10.352 3.241 0.3739 3 10.879 7.893 5.522 1.420 0.1288 4 7.800 5.193 3.242 0.748 0.0952 5 6.795 3.627 2.196 0.491 0.0312 6 5.800 2.643 1.459 0.308 0.0200 7 5.200 1.993 1.191 0.249 0.0064 ‘09. till 0'0 .600 - numéer of 5961/0/75 EQUALITY OF ROOTS Even if the coefficients or residues can be considered equivalent for the two systems. it is not obvious that the exponents or roots would be the same. Since the roots 31' 82. 83 ... Sm in the lumped network equation can not be calculated in general terms for higher values of n and a direct comparison is not possible. an al- ternative method of investigation is proposed. In the transient solution for a lumped R-C network hav- ing m different. negative and real roots. certain relations between the roots and residues must hold. The residues for a transform: [4—— K ‘7 .S(S+e‘J(3+aegK3+oLJ).u.(5+1xLQ) are determined as follows: -43- K can be eliminated if it is realized that in the dimension- less system used . - 1 “o R Then the first coefficient Al is given by I. = .... «3°44 «I» ~11. (“I-«1)(dI-d3)... (“I-d”) where 94, [dz/“3 0‘” are the exponents at the coefficients A1. A2. A3...Am. If this same relation would hold for the distributed sys- tem with n = m (m terms considered) to a desired accuracy. then since the coefficients have already been considered equi- valent. it could be said that the exponents are equivalent too. The error "‘ 0‘1“! "‘°<’fi h) _|_ J a me Hi ‘(ewa-olz)Z°‘I’°‘!J"'(°"-°"”) . ”‘— [o/O] rm L A' ' J has beam computed for different values of m and N and are given in table 4 and graph 4. It is recognized that the two errorsgxmand 6;,mare es- sentially the same for the same values of m and N. From the tables the necessary number of sections can be selected if the error is specified. or for a given number of sections the error determined. For cases where N90 or Nooo certain simplifications can be made and an: analytical relation between the number of sections and error found. c>< \IOtUII'RUINI-J Table 4 04.2.“! '7 .1 f AI ..(otp-cta 281’“1)"'6“-d”) H 2 Error (Sr = 100 m A, 100 10 5 1 0.1 27.685 26.162 24.047 11.972 1.581 11.895 10.797 9.818 4.642 .577 8.225 7.218 6.455 2.905 .348 6.311 5.586 4.755 2.111 .246 5.141 4.280 5.761 1.657 .189 4.551 5.545 5.109 1.364 .152 3.782 5.022 2.649 1.161 .127 o . -....- 1) I OI .Oll.- {74101 It A '17.!101 ) I 1 0 . . -.:..L A . o . x v v o . . . . . _ . . . . _ A . A A . o A . . . . . . A f h . . 7 1' v '4! v o 9 124 v 1 ll." ‘0‘ > .I 0 III :9 I o v I. . 1 o! o v o )7 fl 0 10 1411114011. .1! 1 44-)]: I . . A . . . . . I . _ A . . A . A . A A A . A . A . x . _ _ U . . _ A . _ .. , . . . i . . , w , A . . _ . A N IOII‘II‘J III" 0 .l Ill 14).], i I: 0 7|. .6. O I o | I It . 0! I I . VI 0 A 1-11. . o u I I ... c I- v v I l . Y I D I . A . . . _ . _ 1A! A . i . . . . . . . . . . A A . . a _ c n . A . . . . .A . . A . . A . . A . A . A . A A F A A “ _ A u _ . . . A _ 1.17.6: ux-t. In I.-. 5.. n..03-!,I|. . . .41.! ,llvl. 7711.... 6 4741 '1 alt .v 41 0 v0: 0. . I! 0“ 1.0.1: o 1:91 | A >10}. I _ . e _ A . A 6 _ _ A . A A A _ A . . . _ . . A A _ . . . . I -. - h . . . . . _ . . . o A . . A _ n .— A _ . . . . . u A .w , < _ . . A V). . . 0 6| 9 A A I n I VII-VI. . lulu I I r :7 II. I! I .6 9' 7 ll ovlv) ( III, a I I) A +- 0! [I'VQ‘fi-Y. x 1.01.1 a . . . . A . A A . . . . A . . . A A . _ . _ . . A o . . . ,. A - . A A .. . .U I A u e o A . v v v 4 v v _ _ . A _ . A A . A n u . o o - o . A . _ ‘v . A . A . A A 1 . A . 1 u . _ . t L 0 O I- It II...II III a I t It . ,9 III IOI Ill loll! I 0' I I)“ 7.)-) I. '4 :‘i’I celleAL . . A . . _. . . n _ A . A . . A _ -. . A . W _ . A A . A . . , . . . . . p A u A . .fl.-I.Y. . . a . w . . A . . . M A . . m PC. _ A — . _ _ rniro 1| 4 Illivv) )nllnoe [Oil-o'- IO vIVu 3 . _ A . To 1 .4I.( - L - . A . _ e . _ I _ . . A . . , . _ 1.4 A . A . . . . A A A A A . fl . i . .[IIII‘ 7’ ‘ .0", 1110'. '- navlluuo-OII-vov 1 VA .Il‘ :.bt|90|.-l. ... . 4. .:.ul 1):. .v . 0.7 II In 0:6 Ollielltll . n . . , A . . . I: . _ . . A . . . . o . . A _ _ . . A: _ . o _ ... A . . A A _ _ _ _ . 5 . ..--1:11 ....2. _. - .1; _. . _ A c A o A ‘ 5 . . . _ x . . o L I...F a? L1 .IIA O» .11 Iv . . A . N ¢ . ,. . 9 . . a .. . A . A . A A A . . A . . . . . . . m A . . A H O A n e o I: t, )7 ..:I19. I ..l nnnnnn + v)!- v 14.1 :71. .4: ll.||l . . _ . A . . H . A _ . _ . _ o A . r . . n a . _ . H A A o a _ . I I . loill..¢0,|vl 4 . e . _. A _ . u _ .. A . I H A . . _ . . ...A. u 1 o ..o 4.14 o A A . _ o I . A . . A-A4t._-A -.. . o . . . R Il..l1n A s v A . . . 1 4 . . . . q o . . e . A . . . _ . 9 lrlll o v I 1‘11). quV-lnwanifll“. ...-N I? t.’ 5 > 11!!) A III.) nufii-w-QII' O .Llll. ! 00.5 éer 0/ 5e C76 4 ”a”? m.- SIMPLIFIED CASE WHERE N-yO For the case N-t'O in the defining relation for #3 I3 *mfl" ” flA, fem/9 =/9 ,- 507979 3,5; mom) can, - J I—"xvz 69”” The coefficient An now can be expressed in terms of N only. The first coefficient . _ N N2 A1 - ZWFWJl-NZ for N—fi-O the following is true: 1 ,1 N-*1;N2<< N; (1 - N2)—r1 For the limiting case, the expression becomes Al—y fiqfi- as N’O as should be expected. These cases NA-O are the ones where the distributed sys- tem can be represented by a single lumped section. SIMPLIFIED Oil-3E WHERE N—y-oe For the case where N-s>c>othe following assumptions may be made 1'” 25-2- =/y / for ”—v- 6‘9 cof/g —a- (—E-_fl) / The first coefficient Al can now be written in terms of N only I” 17.11. 1 .1"— ZI-r/Y +N ] 2 1+” [ ,‘Z A 2 l/HY) [tin/+17) )2+”(’+I9] forJN-9-=‘= l { N4>N And for the limiting case AI‘J—l as N—, M 1r ll It can be investigated now if the value of Al thus found for the distributed system fulfills the condition for lumped system. A" at“: 84 ... °°? The values ofér have been calculated for m (2 to 15) “hp and are given in table 5 and graph 5. They compare quite closely with those calculated for N 3 100. ‘J- *————"-j Table 5 ‘75 ERROR 3‘" FOR THE CASE WHERE N —’ 00 In 5n... % 27.324 11.643 7.961 6.044 4.869 4.077 3.506 3.076 co 0) -q on (N m. an I» r4 5 2.739 ...: O 2.470 P H 2.249 ...I N 2.064 ...a 03 1.907 H .p. 1.773 H 01 1.656 I u I' ' 1 I. I wQO\\ UN». \0 \NVSVVQ 1&\ O .‘l I I1 IIIlu III.|IIII._1'I. ll 1‘}! Jill i1illIIII~ “l fi 4 j q 4 II -1 I 11'1‘u1 04" j: 4 4 fl . M . l . . _ . . m u . . 3 .. . . . . o . . . . . . . .. . H . _ M. I H . . . ~ . n I . ”- .L1I0153.IH. l m . . _ o . l . . _ _ TI 1 . . l _ . . . . _.. 1 — . . - . . ~ 01 . I111 . I ... vfi'IIIIO.V .. III III. 91 ...II' it. 1 VIII. -11- II III-61.010I1$11.1'I . 1 I... w.1 1 m. 4 n _ u . . _ ... . . — . _ . _ u o . ..... . o o o s ‘ _ . m, u . . . a _ H M l W w W . w . w . _. I p c I? vol. .1 I‘IOOII 'ij‘llt ..... .1 Il-t’uilbv OI OIIQI I_OI| I .II. II¢.I I o Ill-I‘JIALOIJIOI1 “ ¢ . . _ . . Ww . m . _ . . H . . h T. . q . 1 I. . . . 1 . . 1 . ...... . 1%. 11 . .. .. _ fl . . H _ a u M — _ . n . . n .. _ - I. .I.1 wI-IIII: .. -I-.-111‘ 111.19-. 1.1»--- 1 --- L. .. 1. -ellI ..1L1...1111.1II~11|11...111.I__: 1 11- .- I-J.I 111 .11. 1 1111-1111111T. ..I.1.| A . . _ . * o . H _ c. H _ . a . u . . . . o . . a. 1 . . . - m m .1 1.1 H .. n a 1. -.-. m ....m1.-. 2 n m . a . u . _ . . n . U . . . . . m. . . H — . . . p . . . . ~ ~ . I 0 II? t 0 pl 0 I .a .I OIOVOIIIVIIII‘OIIIAI‘InIIOOO'IIIAWIOI OIQIIIIIIIOIOIO.OIOI Oil 1+1 I 16 11 5| .9 I'lIIllIIIIf .IIOOOOIII I Iu-on I . 0011 II. vlt‘l . . _ . . _ . . . . . H n g . . . . _ . . l . - T _ n _. .. . .. ,_ H . . 3 _ . fl . . . . fi . . . . . . . _ . _ . . . . . . . _ m .. . . . _ . . . . . _ _ “I l O OIIIIB ‘I «It II II 1+.1 'IIII 1| I...” O 10 I“ ‘1 II. 1* n I I‘ll! IH1YOIVI'HII I I VOID Oil 1 II. + l ‘1 'Iilo‘clw II, I. V 10,6 l TJIO 1‘09ilf.19010" IIOIOIW 1101011 I I'll . . M _ . a . _ . . _ n . _ ~ . m a . . _ _ . . . . . . . o I _ 6 o N . u ..... 0 s o u u o I I o a m .— . I I 1 III to I. . U . . a . . . . u . v . . _ n . _ m . a ~ II II I II 1 0| 101.91: c I I I. 9 ”cl-1 . I.‘ n I In H I .911] A I I- I .IIIII I’IIII l I! -071- ! I .H I I I”. I - -h 1.. II‘O 'II 0! I I 0 ll 4 7|. A l 10-0giml7blolll . . H . . . . _ . H _ . m . m u _ . — 1 O O 0 . 0 _ _ _ a a u . . . I I. . . — _ . v I . .~ . I o. o I .1 .9, I . . . _ . . . n . . _ . _ . M _ n _ . . .1 H d H < - v — — — . l _ I H — IV I to IIIIII . I «1 .41. 11‘14 I” IV|15 OI w a- 01-0101 ("AI 010' blA III [19-01 vII"O|b1-'I. IIOOIIIIIIV. o 0 .IIII'A 9160]... v- 9 II 11‘! ’1 'II- ‘1’.” .ll - .I‘ldl . . q _ . . . n u . . i . . _ . _ . — . _ . . _ — . . . . - . . o _ o . _ p . M I. 91 o . _ o a. ‘ . w. . . . . . . _ . . _ . p _ . - . fi . o . . — . . . . h . .. . . . o . . . q . . . . _ ~ g . — I. to I nln III .01 10 411‘- I. I v D1Illc110l9~ o 0 4 III? Il'fillllolo‘ III IIIIVI+I IIoIIIIII I .Il O... Q-IIIIIIOIAI tile. Ilb .IIII H{-F‘.Il‘l’ 11-0.1 |||A . _ M . — . _ M . . . . . . u . i g . . . . H H u . u H m — m . ~ .yH. . . . . . . I11 l ..... 1< 1 . q o . .H o -. I v w. I. I o a . _ I... I 14.1.... H _ ¢ _ _ _. .. , . . _ _ fl _ “ _ _ . . _ . . . . n . . h F. .. 'v.61IO I II. I .0 Icicto It‘llI '0‘ h D... '1 'O.~I|IDI'I?I ‘Io'l IIIOV. | I... ‘1‘!ch I0 1‘1! ”I IOIIII 90 -1 'IIIIIII1 III'I .19! IOth AI I IA. fl . . _ . . . o . a “ m l . .. . . . . . a fi * _ m . . fl . l — . . a o . o . . . . . . . . . . . — ~ . . a .31 . ... _ . . .. . _ . . . . ~ . . . . . . L . _ H . . u _ m . . . . m l. _ H . .. . . — . . . _ . . . . . h -. . -. - -. . -1- .- .. - -.. _ 1!; -.--til. 1111-11“: . -----087... l M S. 11.. .-MAQUK ..11Q1mWI 9111mqg . _ . . . . _ . . h .. n N _ W 1... .. .. .... . .. H .. .1 3.3-3---.“ . . .— . _ m _ r ... . . . _ . . _ m H . _ .W.) _ u _ m . ._ It I II ...... OI” III01IIIOIH II 101 ~._ IIIIIOQIIIIIIm 11.0Il0Iov1» II-+|II'19I [+147005'lfivlituilllw. III.QI1.1A I‘lO‘lJIIOIOIIi . H . . N . _ . . _ e . m. . M V . . ... ; IT hm 3- ..chix . w n u . l 0.0- _ . -.-. w .-1. .... 1.- -.1.1111.1l11--1...1-1..--1 -1111} 11-11.. 1. .111111 «111-1..11. . -. . n .141. .. T . H n . _ . _ . _ H . ‘Iillltl'-..ll1.lllai on .-.- _. 4 - 1 a .—..-o 1.Il§llcl.111'IIo '79 I'I. I . W;_ 1‘I‘il’7. 1111!-I1I‘1 ..II...-!1... 1 . . . . w _ m . _ m .01 CONCLUSIONS It was shown that if the number of equal lumped sections representing a distributed line is increased to infinity. the lumped line becomes the distributed line itself. This could be assumed on a heuristic basis from the beginning. It was also shown that if after adding certain numbers of terms in the distributed line equation, the remaining terms have negligible effect upon the total result. the equation for the distributed line with a finite number of terms is emial in form and numerical values to an equation for a lumped line with the same number of terms. . Thus. if the number of significant terms in the distri- buted line equation is found. it can be said that the distri- buted line can be truly represented by such a lumped system that would have the same number of terms in its characteristic equation. In the particular case considered, it means that the num- ber of capacitors to be used is determined by the number of significant terms in the distributed line equation. The number of significant terns in turn depends upon the value of N. The calculation for the error for any number of terms and different values of H is quite simple if Na'O or Nay-.4 It is more elaborate for intermediate values of N, but once it is done and tables or graphs prepared, they can be applied to any case for the given configuration equally well in thermal and electrical systems since all calculations are n 1 .| I .L .L _ ' LufifliA' m d ‘J'W -50- done in a dimensionless form. Something should be said about the time constant‘f'of the system. It may appear that it should be a significant factor in determining the error. It is recognized, however. that the maximum error would occur when t4-0 and as t increases the error would decrease reaching zero at steady state. Now if the response curve is reploted from a dimension- less abscissa “Erin a time abscissa it is obvious that dif- ferent values of’t would only stretch or contract the curve but would not change its general shape. Therefore the value of‘t'would determine the absolute time for which the greatest error would exist but would not have any effect on its value. In some cases, indeed, the error would last for such a short time that it would not even be possible to measure it. In a thermal system, that would be the case for good conductors having high thermal diffusivity such as silver or copper. The parameters K. C. and h in the thermal system have been assumed to be constant to permit an analysis. In prac- tice, however. they vary with temperature. In a control problem, usually the Variation of tempera- ture around the reference point small since it is the purpose of the controller to Keep the temperature constant and the parameters can be assumed constant. If the change of temperature is large and the variation of constants not negligible, an electrical analogue with none linear parameters can be used as it was done by Tribus (26). 6. 10. ll. 12. BIBLIOGRAPFY AhI‘dEIlt, "I'TQRo 3 JoFo Taplin "Automatic Veedback Control". 1951. New York: McGraw-Hill Beuken, C.L. "Economisch Technisch Tijdschrift". 17. January, 1937 Brown, G.H.. 0.3. Hoyler, R.A. Bierwirth "Theory and Application of Radio-Trequency Heating". 1947 . New York: D. VanNostrand Brown, G.S.. D.P. Campbell "Principles of Servomechanisms". 1948 New York: John Wiley and Sons Carslaw, H.S.. J.C. Jaeger "Conduction of Heat in Solids", 1948 Oxford: The Clarendon Press Garson. J.R. "Electric Circuit Theory and the Operational Calculus", 1926 New York: McGraw-Hill Chestnut, H.. R .W. Mayer "Servomechanisms and Regulating System Design", Vol. 1. 1951 New York: John Riley and Sons Cohen, L. "Heaviside's Electrical Circuit Theory", 1928 New York: McGraw-Hill Eckman. D.P. "Principles of Industrial Process Control", 1945 New York, John Wiley and Sons Eckman, D.P. "Electrical-analogy Method for Fundamental Investigations in Automatic Control" Transaction, American Society of Mechanical Engineers. Vol. 67. 1945 Emmons, H.W. "The Numerical Solution of Heat-Conduction Froblems" Transactions, American Society of Mechanical Engineers, Vol. 65. 1943 Eyres. N.R.. D.R. Hartee, J. Ingham "The Calculation of Variable Heat Flow in Solids" Transactions, Royal Society of London, Vol. 240. 1946 Frame. J.S. Oral comiunica tion, Michigan State College 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Gardner, M.F.. J.L. Barnes "Transients in Linear Systems", Vol. 1, 1951. New York: John Niley & Sons Goldman, 3. "Transformation Calculus and Electrical Transients". 1950 New York: Prentice-Hall Guillemin. E.A. "Communication Networks, Vol. II". 1949 New York: John Wiley & Sons Hornfeck, A.J. "Response Characteristics of Thermometer Elements" Transactions. American Society of Mechanical Engineers, Vol. 71. 1949 110081111. GoDo. HOE. CI‘iner "Solving Complex Problems by Electrical Analogy" Machine Design. Vol. 18, 1945 Newman, A.B., L.Green "The Temperature History and Rate of Heat Loss of an Elec- trically Heated Slab" Transactions. Electrochemical Society, Vol. 66. 1954 Paschkis. V.. H.D. Baker "a Method for Determining Unsteady-State Heat Transfer by Means of an Electrical analogy" Transactions, American Society of Mechanical Engineers. Vol. 64. 1942 Paschkis. V. "application of an Electrical Model to the Study of Two Di- mensional Heat Flow" Transactions, American Institute of Chemical Engineers, Vol. 58,.1942 Paschkis. V. "Electrical Analogy Method for the Investigation of Transient Heat Flow Problems" Industrial Heating. 9, 1942 Schack, A.. translated from the German by H. Gohischmidt, E.P. Partridge "Industrial Heat Transfer". 1935 New York: John Wiley and Sons Schmidt, E. "Foeppl's Festschrift", 1952 Eerlin: Springer Smith, E.S. "Automatic Control Engineerhmg", 1944 New York: McGraw-Hill 26. Tribus, Mo "Intermittent Heating for Aircraft Ice Protection with Application to Propellers and Jet Engines" Transactions, American Society of Mechanical Engineers, V01. 73. 1951 P111. 1. TII‘ II In. II '- . l. {'22)}. 1 O" 3, ‘2“: 2.: g- Y. L N E b! S “U M 0 0. R .o\ . ' {.5- ,: . PW” ‘ ~' - 14.. .....an A. ......v. .... 1N. $3.4. o u . . 5...“. . . \V. a. . «I 3me“ Q} . i . . I. r .‘n... . A..0¥W.~.A}m. .32.... .. . . . , .. . i. {stat}. 4 r .... .0.) . .13 F! .c . .v ’. . I n .‘V. . xx. 6 1' uk. .4.“ ..a. C .. . 7.x, z .I \ .fo .. ., .1 .5. , .-.». .... . \w .. . I l a- :HE