-—-————— ._ A POWER SEEREES APPROXIMATION OF 'E'HE DYNAMIC TRANSFER; CURVE OF A VACUUM. TUBE Thai: for the Dost» of M. S. MICHIGAN STATE UNNERSITY‘ John 0. Cheney. 1955 “o. n“ 1.. 1:41:86 This‘ is to certifg that the thesis entitled A FO'vJER SERIES APPROXDIATION OF THE DYNAIVIIC TRANSFER CURVE OF A VACUUM TUBE presented by John O. Cheney has been accepted towards fulfillment of the requirements for M.S. E.E. degree in Major protgser ll Date December 1, 1955 0-169 A POWER SERIES APPROXIMATION OF THE DYNAMIC TRANSFER CURVE OF A VACUUM TUBE 3! John O. Cheney v AN'ABSTRACT Submitted to the College of Engineering of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering Year 1955 Approved‘__ ABSTRACT A POWER SERIES APPROXIMATION OF THE DYNAMIC TRANSFER CURVE OF A VACUUM TUBE One of the major problems in vacuum tube circuits is making accur- ate calculations of the tube's output when the tube is operating in.the non-linear region of the tube. By use of finite power series the dynamic transfer curve is approximated and this power series establishes a relationship between the grid voltage and the plate current. The co- efficients of the corresponding terms for transfer curves of different plate voltages are compared and a means of plate modulation calculations is devised. In addition several examples are worked out including a single-ended smplifier operating class A31, a push—pull amplifier oper. ating class A31, and a plate modulated rf. amplifier operating class 0. Some other approximations are discussed. A POWER SERIES APPROXIMATION OF THE DYNAMIC TRANSFER CURVE OF A VACUUM IUBE By John O. Cheney A THESIS Submitted to the College of Engineering of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1955 IHESIS The Strelzoff 11mm] He I 'h0 provi of the a} lit} POSIible ACKNOWLEDGMENT The author wishes to express his sincere thanks to Dr. Joesph A. Strelsoff, whose guidance and assistance proved to be both timely and invaluable in all phases of this work. He would also like to express his thanks to Mr. James L. Cockrell, who provided much worthwhile advice in the calculation and comparison of the approximations. Without the help of the above two persons it would not have been possible to complete this work. {366676 ”A W lyspfinv'fln Yf \ ostHde- *1 GERBER I SAFER II CREEK 1} BEER “P3521 HR: '31. LEEIEZX “WW? "‘~'h¢. INTRODUCTION . . . . TABLE OF CONTENTS CHAPTER I THE APPROXIMATION . CHAPTER II .REDUCTION OF THE LIMITATIONS . . CHAPTER III APPLICATIONS . ..... APEEIDIX ‘ O O O O O 0 APPENDIX B . . AME-DIX c 0 0 APPENDIX D . BIBLIOGRAPHY . 59 (>1 66 61; 88 FIGURE 1. 2. 3. A h. A 5. .A 6. A 7. A 8. A 9. A 10. A 11. A 12. A 13. A 1A. A 15. A 16. A LIST OF FIGURES Transfer Characteristics of a be . . Basic Tretode Amplifier Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . . . . . Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . . . . Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . . . . . Transfer Characteristic of Approximation . . . . . . Transfer Characteristic of Approximation . . . . . . a 6V6 and 6Y6 and 6'6 and its its its its its its its its its its its its its its First Iirst First Second Second First First First First First First First Second Second PAGE 11+ 17 18 21 23 2n 27 28 29 31 32 33 3h FIGURE 17. A Transfer Characteristic of a b867 and its Four Term Approximation . . . . . . . . . . 15. A Transfer Characteristic of a 6537 and its Four Term Approximation . . . . . . . . . . . . . . . 19. Value of the Coefficient of (ec + 100) vs Plate Voltage for the 6J5 . . . . . . . . . . . . . . . . . . . . 20. Value of the Coefficient of (ec + lOO)3J:lO"3 vs Plate Voltage for the 6J5 . . . . . . . . . . . . . . . . 21. Value of the Coefficient of (ec + lOO)5xlO'6 vs Plate '01 tags for the N5 0 O O O O O O O O 0 O O O O O O 22. value of the Coefficient of (ec + lOO)7xlO~12 vs Plate voltage for the 6J5 . . . . . . . . . . . . . . . . 23. Value of the Coefficient of (ec + 100)9x10'18 vs Plate Yeltage for the 6J5 . . . . . . . . . . . . . . 2“. A Transfer Characteristic of a 6J5 and its Approximation 25. Single-ended Triode Amplifier . . . . . . . . . . . . 26. Push-pull Amplifier . . . . . . . . . . . . . . 27. Single—ended Amplifier . . . . . . . . . . . . . . . . PAGE 37 3s “5 “5 M6 #6 1+7 53 5h 55 - .5 3' "—RW— um I S.“ ' II REE III 3?. I! ‘.%I LIST OF TABLES TABLE PAG~ I suuxnay or COEFFICIEHTS or (a0 + 100) . . . . . . . . M3 II RESULTS or THE APPROXIMATIOI USING ODD POWER SERIES ron PLATE VOLTAGE . . . . . . . . . . . . . . . . . 50 III RESULTS or THE APPROXIMATIOH USING ODD AND svsn POWER SERIES roe PLATE VOLTAGE . . . . . . . . . . 51 IV MORE RESULTS OF THE APPROXIMATION USING ODD AND EVEN POWER SERIES FOR PLATE VOLTAGE . . . . . . . . . . 52 A POWER SERIES APPROXIMA'I‘ION OF THE DYNAMIC TRANSFER CURVE OF A VACUUM TUBE INTRODUCTION Recently there have been papers written by serveral authors per- taining to circuits with non-linear circuit elements.1'2 The method of solution has been to approximate the non-linearity by a power series of few terms and then proceed to solve the non-linear differential equa- tion. This paper will show that vacuum tubes may be handled in a similar manner by approximting the dynamic transfer curve such that ip = {(lbb, ec). In April of 1919 H. J. van der 3131 published a paper which pro- vided a means to determine the plate current of a vacuum tube given th; input voltages and the characteristics of the tube.3 The essentials of his paper follow. 10 8 «(731, + Ec + E + e sin pt)B where e sin pt was the input signal, a: is a constant whose value is a conductance and depends upon the structure of the tube, 8 = 1/“, and £13 a voltage and depends on several things such as the contact potential difference between the cathode and grid, and the power developed in the “9". 0“ O n---m« l. Pipes, L. A.; Forced Oscillations of Non Linear Circuits; flw- ‘ h---“ 2' K11. Y. H.; Circuit with Ron Linear Inductance and Capacitance; Wicatieae am: acetaqnwics: Jammy 1955; pp. b19425. '5. van der 3131, H. J.; Theory and Operating Characteristics of the Thermionic Amplifier; Proceedings o_f_ the institute 03; Radio Engineers; ..-0 Volume VII. Number 2; April 1919; pp. 97-128. V8. filament (which was the cathode in many cases in those days). But 3b = Ebb - Ion where R is the load resistance of the tube and Ebb is the applied plate voltage of the tube. Then 10 = «(x (Ebb -. 103) + 3c + 6 + e sin p173. Also let V= Eb+l°+6. van der 3131 then claimed that experiment showed 3 to be equal to 2 and that 6 was very small compared to the rest of the expression for V. He then solved the expression for Io and obtained the expression below" I 134121110 + esin pt) ab. 111413307 +‘e‘s_ifin pt): ° 20(3232 This paper is still accepted today. However, there have been several modifications that have greatly reduced its value for analytical computation. The first limitation is that the tube must be conducting and that the grid should not be positive. The second and more discour- aging point is that B is not equal to 2 but is approximately 1.6. This makes an analytical solution for ID somewhat difficult and complicated, if not impossible. With the advent of multigrid tubes van der Bijl's equation became less and less useful for the solution of circuit problems. Others1+ tried various means most of which relied on the parameters of the tube. In as much as these parameters varied to some extent with the applied voltages the mathematics became very quickly highly involved. rSe‘eWAppendivafor the steps in the solution of the equation. ’4. Caporale, Peter; A Note on the Mathematical Theory of the Multi- electrode Tube; Proceedings o_{ the Institute 9_f_ w 915.119.113.13 Volume XVIII, Number 2; September 1930; pp. 1593-1599. 3 A popular way to handle these problems is a graphical method of solution. This, however, has several serious drawbacks. With tubes going into cutoff or saturation it becomes difficult to predict the d.c. current drawn if the signal applied to the grid is a.c. .A good way is to assume that the characteristic is linear until the tube cuts off. Using this assumption it is p0ssible to make an approximation of the output current by rourier series. A similar assumption is made at saturation. These methods have the disadvantage that the slightest change in any one of the applied voltages necessitates starting all over again. There is also no way to figure plate modulation. Hence the graphical solution has several drawbacks which become quite serious when an analytical solution is desired. CHAPTER I THE APPROXIMATIOR A solution is to approximate the tube curves with some mathematical expression. This solution poses two questions which must be answered before we can employ it. The first is, mthat curve shall we approximate?” The second is, "Having selected the curve, how shall we approximate it?" Since we are interested primarily in the tube as a circuit element we are most interested in its output for a given input. The transfer characteristic gives us the plate current for a given grid voltage if the plate voltage is held constant. Therefore the dynamic transfer characteristic will give us the curve we want. Figure 1 shows us the curve we have to approximate. The ideal approximtion would be an infinite convergent series. This series would then fit the curve on every point. Unfortunately such series are difficult to find unless the curve fits a known function. In our case this is not generally true. One may point out that the curve is very nearly that of a translated hyperbolic tangent. This is true but the variation from the hyperbolic tangent is on the very part that we are most interested in. If the tube goes both into saturation and cutoff, the hyperbolic tangent may well be the curve to use. This point will be discussed more fully later on. Having ruled out infinite series we are reduced to a finite series of some type. One series'to use would be a Taylor series. The form f a s0..@ ‘ NO 'OHflfiflHQHOdHEO Rousseau. 0H0 \ sezedmvtttrn 9281a a; .v .. .‘ .QQIJCMHnHvJHH. 757...: $33 a . \I]l‘,-’ ’1'“. to be used would be: i =a1(ec1+b) +a3(e°1+b)3+a5(el+b)5+ ... p c The number of terms would depend on the number of points where the curve is to fit exactly and the number of points in turn would depend on the closeness of fit desired. Thus if n points were desired then the curve would have n terms (Note: At the point b 8 '°cl' 11) would automatically equal zero and this point would not be included in the n points). Since the function contains only odd powers the curve would be symmetrical about the point b = "cl' This fact greatly expands the usefulness of the function. To determine the coefficients of the series we substitute the desired points into the equation, thus obtaining n equations with n unknowns. and then solve. Or by matrix notation: w 23-1. hp]. [c1 c13 c15 . . . . . c1 f at " 3 5 2n-l ip3 c3 c3 c3 . . . . . c3 a3 2n-l i . . . . . . . . . . JL Pan-i L c2n-‘l J 1., °2n-ij where ck= 'cl + b at the point k and where 1p]: = the current passing through the tube at that point. Before actually solving a particular problem we shall first go about solving the general case. The method is that of organized sub— stitution (or complete diagonalization of a matrix.). The first step is write the c mtrix and augnent it by adding the 1p column to form an n + 1 by n rectangular “W1" F 3 5 2n~l '— c1 c1 c1 asses Cl 1P1 [ 3 5 2n..l C3 c3 C3 sees. 03 1P3 c 3 c 5 c2n-l i Len-1 c211ml 3 "'°' 2n—l pan-l Divide the first row by c1. Multiply this new first row by c} and subtact it from the second row. This will be the new second row. Re- peat the last step except multiply by c5 and subtract from the third row. Continue this process until the first column reads 1 O O 0 ... For simplicity let us call the number in the second column and second row of our new matrix d3. Divide the second row of the new matrix (which will hereafter be referred to as the second matrix) by d3. This will give us the second row of the third matrix. The first row of the third matrix is the same as the first row of the second matrix. Call the number in the second column and third row of the second matrix d5. Divide and multiply the second row of the third matrix by as and subtract it from the third row of the second matrix. This will be the third row of the third matrix. Repeat the Operation until the second column reads 81 O 0 O ... The process is repeated until principal diagonal of the last matrix (The diagonal that starts from the first term in the first row and continues to the kth term in the kth row is the principal diagonal) consists only of ones and all terms below the principal diagonal are zero, or until the matrix looks as follows: F q 1 '12 XI} 31),; ..... r111 N1 0 1 323 ’2]; ..... gen «3 O O 1 '3“ so... ’3“ “5 b0 0 0 O ..... l “Zn-L1 . .The above matrix is called a half-diagonelized matrix. To com- pletely diagonalize the matrix the same process as half-diagonalizing is used except that you start at the bottom and work up. A completely diagonalized matrix is shown below. [10000 .....051" O l 0 00 .....Oa3 OOlOO.....Oa5 “:3 O O O O .....1 820% The reader will note that the n + 1 column is the term coefficients of the power series that we are looking for. (The Justification for this manner of solution can be easily seen by placing a after every alt-l term in the k column except the n + 1 column and a plus sign between columns except between the last two columns where there should be an equal sign. In this manner every row becomes and equation and the di- aSomalization process becomes merely an organized method of solution.) Now we have decided what curve we are going to approximate and the general manner of way we are going to go about it. In the diagram below we have a single ended tretode amplifier. For the tube we shall use a 6V6 t'. We will let 3 equal #00 volts and 2c be fixed at bb 2 250 volts. The load resistor will be 2,000 ohms. 8p £5 (E ‘ Cc’ ‘6 Etc-‘5f 554* Ecc: * Ecce- Figure 2: Basic tretode amplifier. To obtain the dynamic transfer characteristic, a load line is drawn ontthe average plate characteristics chart (which is available from various tube manuals) and.a curve of plate current vs. grid voltage' is drawn along the load line. For a first approximation we will choose the point ec1 = -lOO volts as our origin. We will use five other points; one at cut-off, one just at saturation, one in the middle of the linear range, one well into 3; The 6V3 is a beam power tretode. It behaves almost identically like a pentode. 10 saturation, and the last one approximately half way between cut-off and -100 volts. The points selected were ec1 = ~b0, -35, 0, +15, +50. For clarity the five equations will be written out in their entirity and solved by diagonalization. Each equation will be of the form: 1p 2 “.61 + 100) + bud + 100)3 + c(ecl + 100)5 + d(ec1 + 100)7 + °(°cl + 100)9 6 12 (1) o = a.uo + b on x 103 + c 102.u x 10 + a 0.153sn x 10 + e 0.000262 x 1018 (2) 0 '3 a 6‘3 4‘ b 27N.625 x 103 + c 1100.3 x 10b + d lL902 x 1012 + e 0.0207119 x 1018 (3) 112.5 =’a 100 + b 1000 x 103 + c 100000 x 106 + d 100 x 1012 + e 1 x 1018 u — 3 . b _ 12 ( ) 177.5 — a 115 + b 1520.875 x 10 +'c 20113.o x 10 + a 266 x 10 + e 3.518 x 1018 <5) 177-5 - . 190 + b 3375 x 103 + c 75937.9 2: 10° + a 1703.5 x 1012 + In the matrix form: Fuo 6h. :103 102.hx106 0.1o38u11012 0.000202x1018 0 '7 65 27h.6251103 1160.31.106 n.902 x1012 0.020711921018 0 (6) 100 1000 :103 10000: :106 100 :1012 1 :1018 112.5 115 1520.875x103 20113.6:106 266 :1012 3.518 :1018 177.5 u H50 3375 110 75937.9x106 1708.6 31012 38.914 x1018 177.5J The first row is divided by M0 and then becomes the new first row for (7). The first row of (7) is multiplied by 65 and subtracted from the second row of (6) giving the second row of (7). The third row of (7) 11 is obtained by multiplying the first row of (7) by 100 and subtracting it from the third row of (b). The fourth row of (7) is obtained by multiplying the first row of (7) by 115 and subtracting it from the fourth row of (b). In a similar manner the fifth row of (7) is found by multiplying the first row of (7) by 150 and subtracting it from the fifth row of (6). E 1.6 x103 2.5bx106 0.00%96n012 0.00000655x1018 0 1 o 170.6251uo3 993.9 x106 1+.6358 1:1012 0.020286 x1018 0 (7) 0 8% x103 97m. 1106 99.590h ::1012 0.999395 :1018 112.5 0 1336.875x103 19818.2 :106 265.529 :1012 3.5172u7 :1018 177.5 L9 3135 x103 75553.9 1:106 1707.986 x1012 38.h39 :1018 ”7'2 '1 1.61.103 2.56 1106 14.090 1:109 6.55 x1012 0 " o 1 5.8251:103 0.02717x109 0.0001189xio15 0 (8) 0 o 11.2551 76.7676 x103 0.899%9 1:109 112.5x10"9 o 0 12.032 229.206 x103 3.358293 x109 177.5x10'9 L0 0 57.292 1622.808 1103 38.06621485x109 177.5:10'9J (8) was obtained from (7) by rewriting the first row of (7) as the first row of (8); by dividing the second row of (7) by l7O.(3251103 for the second row of (8); by multiplying the second row of (8) by 89-01th3 and subtracting from the third row of (7) and then dividing by 109 for the third row; by multiplying the second row of (8) by 1336.875x103 and subtracting it from the fourth row of (7) and then dividing by 109 for the fourth row; and by multiplying the second row of (8) by 31.35x103 and subtracting it from the fifth row of (7) and then dividing by 109 for the fifth row of (3)- 5 ...f 3': “Ah, .J-H_H§.MHM|. ...—gm _ . ....E..n...... 12 To get (9) we will rewrite the first and second rows of (8) as the first and second rows respectively of (9). Then divide the third row of (8) by H.851 for the third row of (9). The fourth row is obtained by multiplying the third row of (9) by 57.292 and subtracting it from the fifth row a: (8) and then dividing by 103. ‘1 1.6::103 2.56 x106 9.0961109 6.55 x1012 0 '1 o 1 5.825;:103 27.17 x106118.9 1109 o (9) o o 1 15.825x103 0.185uxio9 23.191x10"9 0 o o 38.8 1.1276x106 401.53 1:1042 P o 0 716.162 27.111111 x106 4151.6 110-12- The next matrix, (10), is readily found in a similar manner. The first three rows of (10) are identical with those of (9). The fourth row of (10) is the fourth row of (9) divided by 38.8. The fifth row was obtained by multiplying the fourth row of (10) by 710.102 and subtract- ing it from the fifth row of (9) and dividing by 103. r11.6:r103 2.56 1:10‘3 11.096x109 6.55 x1012 0 '- 0 1 5.825x103 27.17 nob 118.9 x109 0 (10) 0 0 1 15.825x103 185.11 x106 23.191x10’9 o o o 1 0.02906x1o6 -2.bl7x10'12 _0 0 o 0 6632 423.09 x10'5_ {’1 1.61103 2.56 x10" ’+.090x109 o -o.7iux10"3" o 1 5.8251103 27.17 x106 0 -12.96 1.10“" (11) o o 1 15.825x103 o 2.982x10'9 0 0 0 1 0 -5.785xi0‘12 9 o 0 o 1 0.109110-15J 13 Matrix (11) was obtained from (10) by dividing the fifth row of (10) by 6632 for the fifth row of (11); by multiplying the fifth row of (11) by 29.06x103 and subtracting it from the fourth row of (10) for the fourth row of (11); by multiplying the fifth row of (11) by 185111106 and subtracting it from the third row of (10) for the third row of (11); 9 by multiplying the fifth row of (11) by 118.9x10 and subtracting it from the second row of (10) for the second row of (11); by multiplying 12 the fifth row of (11) by 6.55x10 and subtracting it from the first row of (10) for the first row of (11). By this time the reader must be fairly familiar with the process. I will. therefore, merely outline the rest of the solution and give the final results. The next step would be to eliminate the terms other than one in the fourth column, and then the third, and lastly the second. The resultant matrix would be: ‘1. 0 0 0 o u27.87 x10'3‘ 0 1 0 o o -l+Ob.‘+2x10"6 (12) 0 o 1 o o 9h.53 x10”9 0 0 0 1 0 -2.617xid'12 o o 0 o 1 0 109:10'12 L ' 41 Our results can now be expressed in the equation that approximates our curve. (13) 1p = o.u28(0c1 + 100) - 0.1496110'3(oc1 + 100)3 + 0.091453x10'6(ecl + 100)5 ~5.785110'12( cc]. ‘0’ 100)7 + l()9xlO-.18(ec + 100)9 l The next step is to plot the expression and compare it with the curve we are acutally trying to approximate. This is done on Figure 3. 1M 9 ... .... ..v. t..- .568 e-odflanag “Ohflh .8.“ .05 ..no» no 1 Q%\ 7 1 9°1°dw91111w 91918 moaaaauoammfl erase 1304 :30 ooom I made» 0mm u... .pa.» co: u ppm . 91w 0 Ho Oaaenhoaodkdno huhmnmka 1 .m 0.3M: 15 In this particular case we note that the approximation is very close to that of the actual curve until saturation is reached. point there is considerable overshoot. than 15 percent (Based on the saturation current). that the curve for the most part is well fitted. Nevertheless the error is less Therefore we note It is now possible to vary the grid voltage over a considerable range without making extensive new calculations. We still, however, are nailed down tight as far as plate voltage. screen voltage, and load resistor are co ncerned. At this Let us next investigate the effects of changing the plate voltage. In order to be consistent we will use the same load resistor value and the same origin. For our first attempt we use a plate voltage of 300 volts and select the same points as we used in the 900 volt curve. Our augmented matrix appears as follows: F110 61+ 11103 65 2714.62511103 (1k) 100 1000 2103 115 1520.875x103 150 3375 x10 b \N 6 6 10000 1106 102.ux10 1160.3110 20113.6:106 75937-9x10b 0.16389x10 b.902 x10 100 266 1708.6 12 12 :1012 11012 :1012 0.000262 :10 18 0.0207119x1018 1 3.518 38.NN :10 18 O 0 112.5 :1018 135 110 18 135 This solves in the same manner as the previous matrix and the resulting equation is as follows:‘.. (15) 1p = 0.60598(ec1 + 100) - 0.58153x10'3(.c1 + 100)3 + 0.19259x10” 6 us 2 u “.1 + 100)5 - 10.136110 1 (ecl + 100)7 + 221110 13m, + 100)9. ... For the rest of the paper merely the first matrix and the solution will be given. See the Appendices for step by step solutions. 16 Before checking this equation let us study the method of solution used in obtaining these last two equations and see if we can notice any similarity. First thing that we note is that the first five columns are correspondingly the same. Only the sixth columns are different. If we were to use the same points in approximating the 200 volt curve, again the first five columns would be identical and only the sixth column would be different. Instead of repeating all the work.all over again for the first five columns the five by five matrix was augmented twice to main a five by seven matrix, and the 500 and 200 curves were solved for simultaneously. The seventh column of the matrix was: 0 0 (16) 82.5 87.5 87.5 The equation of the 200 volt curve was: (17) 11p a 0. 4909(0cl + 100) - 0.117357110'39c1 + 100)3 + 0.1171191110”b (ac1 + 100)5 - 8.5991110"12 ('01 + 100)7 + 1921110480161 + 100)9. Examination of the function plots as compared to the actual curves CFigures u - 5). Shows that the error in saturation is tremendous and renders the curve unfit for use in this area. Also the fit in the linear region could be improved. For these reasons we make a second approxi. mation. In hope that we can save some work we will again solve for the two curves simultaneously. This time we will select our points more in 17 .eoaassauonaaw sauna use and same» no ”weanesaxou :26 #5504 Ilno Doom M .36» 6mm 1 «.6 Q>o d H0 Ofiamahoaudhg houmfiaa 4 .3 ouswuh BOJOdmvttttn 9:914 Inlhvflalfieu-‘Efiwhnii‘ ‘ A234 gist... .....a. ...—11.5.13! 3..“ a I .e .u r . . . w . A . M an . 49.3. if . (11¢ .31 _ e 18 .aoaaesa NOR and «a Huh saw mad mbn 0 Ho Owamnhooowhmfi 0 mowed-fine.— 058 «on» 000m asap» 0mm .oae» HOMmfidkB < BOIGdfl9IIIIW Gusts 19 accordance with the principles we enumerated when approximating the ROD volt curve. The points selected were ’cl = 410, -35, -10, +10, +30. This gave the following matrix. . 2 - no 6% :103 102141106 0.1638111101 0.000262 :1018 o 0 65 2711.62511103 1160.3110b 11.902 11012 0.020711911018 0 0 (1’8 90 729 1:103 59011.91106 447.8297 :1012 0.387112 111018 57.5 50.0 12 10 1331 :103 16105.1:10b 198.8717 x10 2.3579 110181350 87.5 30 2197 x103 37129.3xio6 627.9852 :1012 10.6ou5 11018135.0 87.55 This leaves us with the following results: For the 300 volt curve: ( 19) 1p = 0.3611899(.c1+ 100) - 0.31123148x10'3(ec1+100)3 + 0.078511110'6 -12 -18 (6,:1+100)'3 - l1.563110 (061+ 100)7 + 79.287110 (061+ 100)9 For the 200 volt curve: (20) 1p 2: 0.u57889(.cl+ 100) - 0.ul+u022x10’3(.cl+ 100)3 + 0.111769x10'6 “81* 100)5 - 8.532x10‘12<.c1+ 100)7 + m8.%6x10'18(e c1.1» 100)9. The comparison of these functions to the actual curves is shown in Fig- ures 6 and 7. This set gives us a much more satisfactory approximation. There will be no attempt to compare coefficients of like terms at this time. Instead the next step will be to note the effects of varying the load resistance. After comparing the transfer curves of various applied plate volt. ages with the range of load resistance desired it was decided that the 200 volt set would give us the most desireable information. (This set had all leads going into saturation which was not true with the 300 and L‘00 volt sets.) , . _ ..1. ; . . Prussian“... . .soaael H9394 vacuum spa msoaueaanonamd Us o>o 05 Ho OuaeuhOpouhcno erase danced sane ocom sane» omm asap» can MOHmQflhB 4 .O ORA—MFR eszedmerttrw saetd .‘..Z . ”A “......Ifl. 74112....51...~4§.E a”; .. . . .k r . . 1.. t . E . 7 ..7 . . .1 .m ,4,4 l, , .4 «I41 J 4 4 . 7 11¢ 9t ......6.. satlvot$A49usT8A¥47v o. 7 7bls. s79 O..A7.TO§7o 949 Yt... orfvvfis It‘s .979... ..70767 .It os.o .... ...7+¢... 1... .. .. .7.. a... b... .... .0... 0‘ a... ...t sees 01‘ 96.01007! ...:Avoe7IJ'Yesefissvo ...el ‘Q‘Ov‘tszfiw...vsuusrtsvaVOIIIO.~...7. é... 474:7. 7.. 7... a... 7.7. .. . 7... . .... I... awo. u... ...a .V o... s..¢ v...- ..¢7s vr7vlsvo74. ...7 ..ssile4Vs 40.7. ’... Jsutvbvotw+ssn ...altsltslvstvl77..7777.... ....ifisoo .... ..7.*..t.1 .9. .... . . 7... v... n .. .a..AcsP-1— n. s..¢ ...c s... ..L. ovilve... ..ho ossVlTQ7YO. ..oo7o.¢\. ...?Yjetavos.c ‘ .. .... .. .... .... ...a 0990 > . , 4 4|7 . 4 .0 nvtt sstfiise7vv seas. Qv7v¢ 09.. us.- ssstlvoéo s... «7.77 .o.. ’A707' .10! a ..s. .... ..vVY7777. .v7sviss7vs .. ......... .... .... o7... .... ..o. .-........ .... . 7-<.......1+.1...: . .... . .. ...-.7-.. 7..- --7v0 nevt+07 ...a7u710. ..sv 4... rte. 77.1 ttvswvotn 0.4. a...~...c7fifls. o... c .. ... ..7. .... .7.....07Vl .4... o... 7... .... ..7 ....777... ...o 777..1.v.A o... ..i...i.. ..o....77l4v....7 ... .... .... o.-. 1..9 ..th + 4... .... A... .... .... .... .... .... ...o. o... ..v. v... .... o........ .7... .-.. .... .... ...- .... 7700 ..-..u... .... .... .... .7.. .... .... .... ... .7777? . .... 77.4......_7.... .. . .. .7.. . ...- ... .... .... a-.77.... .... .... a, . .... ...7. .... .... . .... 76.. ....-.o.. . .. .. ...$.9... .77. .... .... .7. 7.$.. .... .... 4... 4.0. 4... ...77.-... ...7 44.. .7.. .. 7..- .9... VI... 9... .... .... ..coT.7so ..-?..o.- p... .... s... .v7o+.-.o. .... ...9 .. .... .... vs..7v... ...+ -... .-.. .... ..4IJ7...- .... .... :...a...7..... .... ...O .. .... ..-. ...? .... .... .... .77. .... .... ...7 .....7..- .... ..777.... .... .... 7. ..0077i..7 ...-wvoo. o.. 1... ..t... 09.. scotiiiot u... 40?. soos t... .... 7.0: .--. .u 4 ..7 .7.. .7.. .. ... ... 7tvfiYs... .... ..47Av.17 0..7 .... .... ...- 7.. .. 7... ...... .... . . ...A 7... .. . .... .... .... .... .... .... ..c .... 6... .... .... .... o... .... .... .... .... .... ..-. .... 77 ...71777.. .... . . . .u ...-.vs 7 .. ....rtvs. ..07 .. , .vslvv.s. .... .7.. 7..d .... . .I.s nos-Avsqé .... A..- .. 7 . . .... ....o w .7 .nvv .7... wet. ... .7. ... .cs. 77.. .. ..7.. ...7717.. .... ..¢4 Q... .... .... .... .. .77. 7... ...alont. .... .7.7 . . ..77 . 7 .. .7 ....r.... .... . . 7... .... 7... .... .... ... ..-. .... .-.. .77. .... ..A. 7 .77. «.07 7.... ..-. 7...7.... . 7. .... .7.-... .3 ......... .... ... .... .... .... .... .... .... ..-- .... 1.7. .... ...- ..-- . «to...o7¢ .... .... ... .. . 7 ..71 .... (7 st 0... ..V. ...7 ... . . v... .. .... ...7 ... .7.. o... I... .~.. . . ..77 .u 7 7. .7.. .97‘7.... 7.7.1... . . 77s. ... .77 7 th 7 s 1 .779. .... .... -... .. .. . .... .. . .... ...- .... .... .... v-.. ... ..A. .. .... .... .... .... ..7. . ..7. . .7.. 77.0 ..-.7 -... ..77 .. .. ... .7.. .... 1. . . a. . . p .. .... ... .... ... . .7 ..4. .0. v|.r...o 7. . . .. .... .. 7 . . . . . .. .. -7.. . .7 ... .ito -.7. 9. . 1 7... ... ... .... 7. . ... .... ... ... .7.. ... .. . s..o .... .... ... .7.. a... .... .. . 7. ..77 ..77 .7. 7.4. 7 . .. 7 .. .. . .... 77-7v. 7O467t. ..V. .... 7... .... .... .... .... ... .... ..-. .... 9... .... 1.. . . ..-. .777 .... ... .... c... ... .... .. s ...o .... ...; .... .t.. 77.. . .707 . -... .... .... . .. .... .. .... .... ... .... ...7..... .... .... . . .......47 ... .... . c 7... .7. .... . .. 7... .... .... .-.. 7... . . 7.... I7... .....-7 . . .. .... ... ..-. ... .... .. . .... .... .... ......v.. ..t- 77 04.5 .... .... ... ...- ... .. . .. 7 .7.. ..77 .... ...7 ....7.7..r7... 7.... . .... .... ... ... .. .... .. O .... .... .... .... .... .7... ..-. 7. ..av 77.. .-...7... ..-. ... .... .... ..-7..... ...A..... .... .7-94... 7.... .... -7ui .... .... .. . . .. .. ... . .. .... .. ... . . o-.. ...7 .. ...- Y.7.v .... .... ...7 . . .... . .7 .... ..-. -..... . . l ...- 7.- 7.7 7..' 7... u... .... 1. . . ... .. . .... 7.. . . .... .... ... ...c 0137 7 o .vol 9... .... n... o7.e ... .v.o ..7 ...-v7. + s..§ .... 7... .... ..u. 7.. . .qu7.s .... .... .... . . .. .... . .. . .. . ... .... .... .... .... . .... .... .... .... 7... ...7 v... .. . .....1. ..7. .. . ..$. -....7... ...7 . .717..- . .. ... .. . .... ... ... .7.. .7.. ...7 .... .... .... 77.... .... .... .... .... .... ... .7. .. .... ..77 .-. 7. . . .-.. . ....... ... . . ..7. .... .... ... ...a -... ..-. ... ...- .7... .... .... .-.. ... ... .7.. .7 .. .. . ... 7. . . . . . - .. . .. .. . Us.» >s . ...: its .7. ...s .. t .-va .‘I‘ Os§‘A7O..v It’s .svfl |twt sell Is.. .- s I.. 71‘. . 7 4st. .0. . 7|!Aw'..- I. u. 7.! .7 vssv Illl . :5 0|?- , , .-.. .. ... . 7. .7.. .... . .. ....A.... .... ..u. .. .. .. . ... .... .. ... .. . . ...7 .... ... .. ... .... .. ..71fi-.. ...1v0.70 -. . . . .... 7. . .....7... .... .... .. . .... .... .... .... ...7.7... .... ..77 .... . . .7 ....7...77.... ....L7..- .... .-.. ..7..77-.. 7 . . .... . .. ..- .... - . .... .... .... .... .... . .. .. .. ... .... .... . .. ..- . .. ...7 . .. . .... .Tva.... ...7 .. . . ... ... .... .... 1 c . 7.. ...“ ..lsr¢.v. .... .... .... .... ..q .... ..u. . . 4... ..77 ... .A.. ... ...A 70.04)... .... .... .ss. .... ...7 i.. .... ... 7..- ...a. . .... .... sss.7o.«s ....76... .... ov.l7.... .... ... .. . ...7l.. ... .. 7.7... .... .... .s . o..s .... .. .wv779 .... 1... .... ... .... v... u... ..v... .... «.oonott ....L...». ....77..- ...7 .... .... ...a -7.. ... .... . 77 n. . ..77-..... v.71 .....o..7l..7s 79 . , , v 4 4 4 o... ...! ...a. .... .... a... 743. . . ..A.L...c ..9. ..4s 0+so7T+v.. ...a .... .... ... .7.. . . ...v .... A... ... .. . . . .... .. 7 .... t... .... .7.. 0.7. ts.AA .... ....i.... .... .... . . .... a... .... .... .... .... ..v. . .. ... .. .. . .. .... ..<. . ... . .... ... A .7.. ... .... .... ...7 .... ...7 .-.. . . .. . . .. .... .... .... h... .. .... ..-. .. . ... . .. .. . . . .. . . .. ... ..77 .... ..7 ....7.... .... 7... . .. . ... .... .... 7... .. . ... ...+ .... .... ......... .. . .... .... ......... ... .. . .... .. . .. 7... 7.7. ....a ... .... . 11 .... .... .... .... .... . ... ... .... ... .... .... .. 4. . ... .. 7.. .. ....+-7.. ...a v. .- . .... .... .... .. 7 7.... ....L.... .... .... .-.. 7... .... ... . .. .... .... .... ... . .. .... .... .... .... . .. .. .. .... .... .7.. 7. . . . .... . .. .. .. . .-. .... .. .... 1... 7... . .. ... .... .. .. .. . .... ...7 .... .... . .. ... .... 77 .. . . .. ... .... . ..-. .... .... .... .... .... .... .... . ... . . .. . _. .... .... .... ... . .... .7.. .... ..... . . ... 7. ..-. ... .... .... .... .... ...L .... a .... o.-. .... .... . ... . . .. .... .... .... . ... .. . _. .... . 7 .. .7. .... . . .. . . . .... .... 7. .... 7.7. .... .... .... .... .... .... . .7. .. . .. .... .... .... . .... . .. .... .. . .. .. 7 .. .... ... ..7. .-.. .... .. . ......... ,.... ..77. .. . .... . .... .. . .. .. . .. -... ...7 ........7..... .. . -... ..-. .... ..-.-7 .... .... .... .. . .. o . 0.. .. ..m .+.. . .... .... .... .... .... .7éélwo... ..o. t .. .... .... . .. .... V0071t-wO7Tv... .. .Oatlsolti‘e‘s 0.7- lit.>l .... t... ... .... .... a +.. ... A... .... v... .... .... .... .... .... .. . .... ... .. ... .... .. ... .... . 4 .... ..7. ..7kir7fvar. vqn7. .... .l7+ .... .... .. . .... . ..- .... . . .... .... ...7 -... .... -........ . .. .... .. .... -... .... .... .... .... .-7. .... .0... .... .. . _.. .... , .. 7... .... ...7 .7.. .... . . .. .... .... o.7 . ... ...-..I7-.. .7.. .11.- .... .... .... .... .... . ... .... ... -... ... .+.- .. . .... .... ....v.... .7.. .-. .. .. . ...A .... ...uv-... .77- .... ...- . ..1771.-.¢..7o..7.l7.777.77.... , a 4 ...o .... .. .... . ... .... ...a .... ...7 .... .... 7... ....i.. . .... .... .. . ...-.4..7 .1. .... #7. . .. -... ....l77.7v.... ....747...477|.7?-. ...v .s.. ..o. .... . 7. . . . . .... ...I. u . .7. .... ..sd 7... .... .... yr. ... .. o .. .. ..n‘lI-.. 7.... ll'- .:..:.7...:..:-. _ .;..:..:..:..; .,..:. ;....-:- :..:, ...:, a :.-...; ..7.7.... .. .... . ..7 .... ...777.7. .... ... ..7 .-.. .... . . .. . .. .... .... 7... .7.. . 7...... .... .... .... .... ....7.... .... .. . 7 .. ... 7.7 . . .. .... .... ...7 ...- .... .... . . .... . .7 -.. .... 7... ... 7. -7.. .... .... . .. . 7 , .. ..77 .... ...77 7. .... .7... .7... 77.77.--. ....77 17. ... ... . . . . .... .... ... o. . .. .... .... .7 . ... 7. . .... ...7 .7.. ...Iov7.... .... .... ... .77. a..7 ..7. . . . . . .. .... ... .... .... ...7. ._ ..7. .. . . . .... 7... ..-. .... .... vol... .... ..101....7 .. . .... ... .... .... .... .... .... .... -.. 7. . .... .... ... .77.. .... .... .7..... To... 771- ...a ...a ... ......... .... .... . .. ...7,..... .... .... -... .7...?... .50 88 -... .--. .....7--. u . .... . .. ... . .7...... .... .... ... .... ... .7777? .... .-.. .. . ..... .7.i¢ ...: ...trltso ... .. . .... ...7. .... . ... .... ..di . .. . . . ...7..... ...11.. ...,.l‘... .... .... 7.... ..7. ..1. .... ..-. .... . . .. .... ..7. .7.. .. .... ..-. .... 9.-..1... . .. . .7 .-.. 7 .7 .77. .. . . .. ... ... .. .... .... .... .7.. . . .... ..su v........ .s.. 7... .... ...... 14.0 9.... 0. . .7. .... o .... .7.. 7 7. ... r. . .... ... .... .7 . . .. 7. ... -7.. .... .... .7.. .... .... ....7.... .... .... . ... . . .--. . .. .... . . .. .. 7. . ..7. .. . .... . 7 4. ..7- .... .17.... .. ... . . ... .... .... .. .... 77.. ...7 ... . ..7. ..... 7... o7..i.v.. ...... .. 7 . . 7... .7. ., 7.. .77 .... ..7.7..... 7.. .... 7... ... 767.‘9 .... ..77 77..7.. .... 7.. .. . ..7. ... 7...7... . 77. ..77 .... 77.0 ....7.... . .7 . .. 4.. -5... I... .... . 7. 7.- . ..o .7.. .... .. 7. 7 _. . .. .7.. ...7..7.- .... . . ....77.-. .... .. .... 7 . . 70.7 . . 7... .7.. .7. .. T ... . 7. .77 77.. 7.... .7.. ... .... .... .743 to..v707.vl(‘7.t. '7.-. . 7AI17.v-#T?.‘JTI.IG7snsO.s "761... 101‘. o 7... c... v... 7... .... .... 7.7.1.7.. o e no oaaouuoooehozo hounneua 4 . oasmam eszsdmettttn azvtd 22 Since the curves were so radically different near saturation it was felt that nothing could be gained by attempting to use the same set of points for both curves. The “000 ohms curve was approximated.using the points ec1 =:-00, -35, -15, ~5, and +20. The 1000 ohms curve was approximated using the points 0 = -hO, -35, 0, +10, +30. Once again cl the initial matrices and the results will be given. rer the first approximation70f the #000 ohms curve: 3 6 0.1038‘711012 0.000202 :1018 12 F370 60 :10 102.17 x10 6- 0.020711921013 (21) 85 011+.125x103 m737.05x10b 32.058 :1012 0.23162 :10“ 3o 05 2717.825x103 1100.29x10" 17.902 no 0 6 95 85703751103 7737.81X10 69.983 11012 0.63025 11018 1+5 920 1728 x103 21.883.21.106 358.3181 1:101‘2 5.515978 x1018 175. ( 22) 11, = 0.u165n(ec1+ 100) .. 0.h0569x10-3(ecl+ 100)3 77 0403113100"6 (ecf 100)5 - 7.95mo-13<.cl+ 100)7 + 177.mo‘15(.d+ 10079. For the approximation of the 1000 ohms curve: [170 bu x103 manna" 0.1638uxio12 0.000202 x1018 0‘ b5 2777.8251003 1160.3:106 77.902 11012 0.02071197no18 0 (23) 100 1000 1.103 10000 1:106 100 7:1012 1 1:1018 107 110 1331 x103 17.105.11.106 1917.421 11012 2.3579 :1018 160 E30 2197 x103 37129.31106 027.N852 x1012 10.80175 11018 162 (2“) 1p = 0.20735(ec1+ 100) - 0.217008x10'3(oc1+ 100)3 + 0.01777J.xio"6 5 ~12 7 -13 9 (ec1+ 100) - 1.250110 (ec1+ 100) - 31.2110 (ec1+ 100) . These results are plotted.along with the actual curves in Figures 8 and 9. Before trying to interpret the results we have obtained let us examine a triode. The method will be the same as we used in investigating .aao» How .. Q»? n .. «caudaanoum I opuso aasao< cane coo: gang» omm camp» com . f u 011‘. 5" 04". .Iowadaanoumgd panda and can o>o on» ma oaaadnopowucnu homuncus 4 .w ouswwh ledathIIIIN azvra 103.3393..." :56 7.30 250 .002 u 33> omm a .09 ‘0 l.“ I a I cv|u O- duf‘ Pl¢4 ‘07:! vbit I... . 33.83893 2:: a: 23 £5 a we 32230230 335:. < .m 933m . “Jed-mun“ new 25 the (Nb. Once again we will use a common tube, the 6J5 or one-half (If a 6817. The latter is a tube that is common to many television sets. ls will first use a 20,000 ohm load resistor with plate voltages of 200, 300, and L$00 volts. Then we will use a plate voltage of 200 volts with load resistors of 10,000 and l$0,000 ohms. As usual we will give the initial matrix and the final result. For the 20,000 ohm set of curves, the points used were e. = ~50, ~20, —2, +10, and +30. Once again ec = -100 was chosen as the origin. For the #00 volt curve: 1P50 125 1103 312.51100 0.7812511012 0.00195311018 0 q 80 512 1103 327b.5x106 20.9715211012 0.13421 x1018 1.2 12 13 (25) 93 9M1.192x103 9039.2x10 86.81255110 0.83375 :10 12.5 l9u.8?l71x1012 2.35795 x1018 19.} 2 110 1331 x103 15105.1:10 O‘O‘O‘ 330 2197 x103 37129.3:10 627.h8517x101 10.oou5 x1018 19'§4 (26) 1p = 0.263(edt 100) - 0.17012x10'3(ec+ 100)3 + 0.03356x10-b(ec+ 100)5 ~18 - 2.1932x10-12(e¢+ 100)? + u7.ux10 (ec+ 100)9. Since the same points were used for the 300 and 200 volt curves the actual solution was preformed by using a five by eight matrix. For this reason for the 300 volt and 200 volt initial matrices only the last column is given. The 300 volt column is the first column and the 200 volt column is the second column. 0 o o o (27) 9.2 5.6 1n.5 9.7 1h.7 9.7 26 The equation for the 300 volt curve is: (28) ip = 0.25Sbl(ec+ 100) - 0.17111110'3(oc+ 100)3 + 0.03.2071110”6 (oc+ 10055 - 2.1197x10'12(oc+ 100)7 + u6.sn0'18(oc+ 100)9. The equation for the 200 volt curve is: (29) ip = 0.00392(ecl+ 100) - 0’07322fl0.3(°cl+ 100)3 + 0.01282xlO”6 (601+ 100)5 - 0.73u5x10'12(oc1+ 100)7 + 13.3n0‘18(ecl+ 100)9. These last three functions are plotted and compared with the actual curves of the tube in Figures 10- 12. For the different load resistors the first approximation used the points so = ~50, ~12, 0, +10, +30. Once again the origin was assumed at °c = -lOO. Both curves were solved for simultaneously. The sixth column is the 10,000 ohm column and the seventh column is the l$0,000 ohm column. p “’50 125 1:103 312.5:10" 0.73125211012 0.001953x1018 0 0 “ as 681.1r12x103 5277.3x10b no.8673 x1012 0.316% 11015 o o (30) 100 1000 x103 10000 1:106 100 :10” 1.0 x1018 10.8 3.3 110 1331 x103 16105.1x106 191M717 x1012 2.35795 x1018 15.6 11.95 _1_30 2197 x103 37129.3:106 627.11651711012 10.00160 x1018 18.6 u.95_ The solution for the LI-0,000 ohm curve comes out as follows: (31) 1p = 0.25609(.c+ 100) - 0.18677x10'3(ec+ 100)3 + 0.03u62x10'°(ec+ 100)5 -18( °c+ 100)9 - 2.l+17x10'12(oc+ 100)7 + 57.5x10 In solving for the 10,000 ohm curve the author must have made a slight error because the equation did not check at the points selected 'here the approximation should agree exactly with the curve. Resolving Yielded the correct equation which very closely resembled the incorrect 0118- This serves as a warning that the coefficients must be calculated mafia-3'1... mfiéiki . r..ll . .. [1 .uofieaauonmnd oak: and use auao> ease numu.. .1 .1 3:93,: .90 .33 04 uses ooo.om u an .26» com u 2n tunl I 9v-‘ 0... n 7 ‘1' 0 m3 e no 03333328 songs? 4 .3 shaman . 9 vii- .--t seaedmsntm equcI as naiadflfidiflésiq ism—1...... 1 . _ . 1 28 If «1.! III! u: L «ago» e11 aoaaeauuennmd chase Heavod .ano ooo.o~ u .116» com u as. .souasa Honnad cough as“ and mum e no owe-«uonodu o sameness 4 .HH ossmah salodmvttttw caste . -.. k If. .13.- undo» dune caucus—«Nouns? I I 0:8 good .eso ooo.o~ u .aap» co: a pan 50318333 can: a: one a no Sauna-A0935 newsman...“ 4 .ma 0.3»; I019d3§ttttfl 01'14 30 to more than three places. In order that the reader might compare the two equations both are given below. Equation (32) is incorrect while (33) is correct. (32) 1D = 0.51965(..f 100) - 0.32788x10'3(oc+ 100)3 + 0.05025x10'b(ed+100)5 _ - 3.u83x10-12(.c+ 100)7.+ 73.2:10‘15(.d+ 100)9 (INCORRECT!) (33) 1,, = 0.51196(ec+ 100) .. 0.321+O0xlO-3(ec+ 100)3 + 0.05597x10'6(.c+1oo>5 - 3.N98x10'12(ed+ 100)7 + 73.7x10'18(eé+ 100)9 (counters) The Justification from this apparent idiosyncracy is that the errors in each term must be cumulative and hence the difficulty. The correct approximations are plotted in Figures 13 and 11}. For a second approximation the same points that were used in the 20,000 ohm set were used in the 10,000 and 1+0,000 ohm set. The initial matrix is shown below with column b as the 10,000 ohm coefficients and column 7 as the 140,000 ohm coefficients. F50 125 :103 312.5x106 0.78125x1012 0.001953x1018 0 <3 _ 80 512 :103 3276.8:106 20.97152x1012 0.13u21 x1018 0 0 (3h) 98 9ui.192x103 9039.2;106 86.81255x1012 0.83375 :1018 8.7 2.8 110 1331 :103 16105.1:106 1914.87171x1012 2.35795 x1018 18.6 n.95 L330 2197 x103 37129.3x106 627.M8517x1012 10.60u5 :1018 19.1 1.95‘ The results were as follows: (35) is the 10,000 ohm curve while (35) is the 140,000 ohm curve. (35) :1 = 0.08257(ec+100) _ 0.0u557x10'3(oc+100)3 + 0.00u71x10'”(.c+100)5 .p "15(oé+100)9. + 0.16N69x10”12(ed+100)7 - 15.99xlO (36) 11):: 0.0oobl(ec+100) - 0.03908x10’3(oc+100)3 + 0.00089x10'b(0¢+100)5 - 0.h020xlO-12(ed+100)7 + 7.47x10'18(e¢+100)9o These approximations are plotted in Figures 15 and 16. 31 mode» cane numwu. noaaalawoammd ergo an 04 sane ooo.oH .30» com oeueno accesses 4 H o soxodmwttttu outta name» duuo 32 7|} 1 4.4 7 d . Ycl..|7vve.. v... .... ..77 .¢.¢ 9009 s... .... otuév 0..- 6.0. 0.17. a..- 77 .... .... ... ... .0.» out. 7.7... ...! 007.... 9.. 77.. .... .... .... a... . .o .e Ilvtéwbc.. .ootwelov .000 '17. ..o v .. .. . ... .. 71.... .... ...7 .7 7... .7.. ..77 .7.. .... .... .... 7... ...07.7¢..7 .7.. ...7 .... ..7...... ......7 ..7. ...7 7 ..07.. 7 ...7. «.17-... c... .7.. ...7 ......7.. .....7 ...$.o..v7n.o.1.... .0... .0070. b.vt 1.... ... .... 4... . 7.. .. v¢.. .7..17... o. r... 7... oc.- otts ...07.v.. ...¢ ..90 ......17. .... ..7- ..1171000 94.. 04¢ o... tt.~ 77.. 7... ..7. 7. 7 .7.. 77.. 77.7 7.77 .7.. 97.. 77.. 77.. ...7 ...7 -.7. ..7. 7.7. 7.77. 77.747... 77.7.... 7... 77.. .. . .777 7.. .... b. 7... .7.. 7... 7... 7... ..7. ...7 ..7. .-..77... 77.7 ...77..777 ....w..7. .70....a777. 7.. ..7. 7.7. .... .7 77.. .7.. .... ..7. .7.. ...7 .77. .77. 777. ..7. .47. .7777.777+.¢.7.~ ... 7.7.9... ..... ... .77. 7... 7.7. 7. 7... .... .... .... 7... ...7 .... ...7 ..7. .7.. 77-8773t. 7Y7o7n.¢7 7.77 7.. 7... 7... .7 7 .7.. .7.. 7. . 7. 77.77.... .... ...7 .7.. ...7 ...7 7... ...7.....7 7.77774.. .77. 77.7 ...71 .. .. . .7 . ._ .. . . .. .7.. .... ...7 .... .7.. 77.. .7. .... .7.. .... 7-717174.-..7....777......71 .. ..7. 7 ..7 7. . .. .... .00. I... .... ...s .7.. 1A§7 ...7 .eetIVt. «say .410 I... ....«qve. 7\ .7. ..7 ..7. . t- .... .o 7 7 . on I90? «I... a... ..7x .eva t... as a so 0.. ..v«Oev.§ .- V I n .4 .s.4 .IQYII‘IO .7 7... .7.. .... ..77 ..7. ...... ... .77..... .7 .7 7 7 7. ...7 7.7. .77. . ...7 .... o7. - 7.. . 7.. .7 .7....... .... ...7. . .7.. 7.7. 7... 77 .777 v. . 7... .7 -7.. . . . .7 7 .- . .. ..7. .77 7 .7 .... 7... . .7 .. 7 7 7 .. .7.. . . .. .7 .. 7 .. ..7. . 7 ... . 7 7. 7 .7 .. .77. 7... . . ..7 .77 7. 7 . .-.7 ...7 ... ....7... . . 7.. .7.. ......7 7.7. . .. . 7. . 7 .... 7 . fie VOIu 7's! . .7 ‘17: l~7 vs. III. e .v r is . -.r~ 1.1- 0... A00. --es s... so . t o . ..7 st . Is. 7 alt. .« 7. 77.7 -07. .7.. . .. .. 7. .... .7.. ...7 7 .7.. 7 .. .. .. .77... ..77 7... ... .... . .7 ..7 7 .7 .. 7-7. 7... 7... .7 ...7 ...7 7.. ... ..7 ...7 t... ..esx..v: vs .vv. . .. . ... .7.. ....7.... a... 7... .. 7... 7 .77 . ta 01.. .1 77 7.7.. ...7 .. . .77 . 7... .7.. 7... w7. ..7. 77.7 . .. ...7 .7.- ..7 .... .7. 7P7. . . 7. .7 .... . 7. . . .7 7. .77 .71. 7.7. 7. . 7. 7 7... 7... 7.. .... .... .. .. 7... .77. 777 ...7 ..l.. 7 7. .7 w. 77 ...7 ..7. . . ...7 7 .7 .... . 7. 7 .7. .... 7 ..7.. .7 77.7 77.-.... . . . 7 ..7. ... .77. ...7 7.7 ...7 .. . .77. ... 7. . 7 .7 .577 ...7 . . ..7. .7 ... 77. . 7 .7 777. 7. .7... 7... .7 .7.. .77. 7... . . 7 7. 7. 97.. 7.77 .. .7 7 . . .. ... . 7 7 7... 7... . 7. .7 777 7 .. 7.7. . . . 7 ..7. .7... .7 7 .77 ...7 . ... 7 .7 . 7. ...7L7.. 7 .7 7. . 7 7. 7... 7-.. .77. . 7.. ... .. ... .. ...7 ..77 . . . . .. . 7. . .7.. . ... . . ... 7 7. 777777.77... ...7 7 . .. .. ..7. 7. ...7 .77. ...7 . 7. . . ..7. ...7 .7.- .7. 7.771...7 . ...7 .7.. . 7 . .7 7 7 .7. .. 7 . 7 7. .. .... 7... .7.. ..7. .7.. .... ... ...7..., ..7 7. 7 ... .. 7 77.. .7 . 7. 7. 7 .7 .. .7 ..7 .. .. .. . 7. .-.7 .7. .77. 7... o .o 7... 7.7. .7.. . . . . 777. ..7. .... ..7. 7577 .77v7u... ..7. . .7 u 7 .. ...7 ..7. ... . .. 7 .. .. .... ..7. ...7 77.. . .7. .. .... 7... 7... . 7. 7... 7. . ...777... 7............ .77. 7... .... .... ... .7 ..7. ..7. 7. .. .7. ..7. 7..- .... 7.. .7.. ... 7... 7... . .7. . ...7 77.. ..7... . .7.. 7. .. .7 .... 777. . .. .... .... .7.. ..7. .... 7... 7..- ...7 . 7 ... .... .... .7.. ...7 .... 7... . .7 7.7. .... .... .... 7 ..4.777 7... .777 ... .. ..7 .... 7 .. .... ..v. .. .... ...v..77 ... .7.. ...71un7a .... ..7. 7.. 7.. 79$. .7. .77 .... .... ...7 . . ...7 ...7 .7 7... 77.: 1'17 77.: 1700 77... 7. 7. .. .7.. .7.. 7 .. .... .... .77. .... .7.. .47. 7... .... .... 7... ...7 7.. ..7. .7 . 7. ..7777... .-.. . . ... 7777 ..7. ..77 77.77-.. .7 ..7 77. 7 .. .. . .... ... .7.. 7. . .. 7... .... 7... 7......7 7... 7... .7 . 7. ..7. 7... .7..... 7 . ... . .7 . -.. . .. 7 7. 7 ..7. ..7. . .. .7 .... .... .77. 77.. .... .... .... .7.. ...7 ...7 ...7 7... ...7. 7 ..7 . .7. . . 7 7...7.7. .7. . 7. .. .7...7 .777 ..7. .7.. .-.. .7-. 77. ... ...7 . . 7. 7 . 7 ... 7 .7.. .7 .... 7... .-7. .7.. .. 7 7... 77.. . 7 ... .. .. .. ..7 7.7. ... .. ... 7... .7.. .... ... . 7 . ..77 .. ... .... . 7 ..7. .. .77-- ...... 7 .. .7 .. . . ... 7 -. 7.. .7.. .7 . .. . 7... .7 .... .. 7 .. . .. .7 .. . ..77 .7 . .7.. .... 7... .7. 7. . ... ..7. .7. 7... .7.. .7.. 77 7 .... .... ..7. 77.7... .... .... ..7. 7.7. 7 7 ...7.77. 7.-. ..7...7. ..7- ..7. ...7 . .. 7 7 ..7 .7.. 7... .. .. 7.77 7.. ..7. .. . ...7 .... ..7. 7. . .... .... .7 7 ...7 .-77 7.. ..7. . .. 77..7-.7. . ...7.71...-....177.7 7... .7771 1.71. 7 .. 7. . 7 .0. . 7. . . .ev . .. .... a.» f... 0... .... ....s ...a .OVo ..d .0. .... v1. .. ‘4 17'. ..7 O . .7 s»... so}: 1". ...... .ne .. ~ . . . 7 . . 7. . .. . 7 . 7 . 7... .7.. 7. . 7...1.... .... .7.. 7-l. ..7. 77.. .... . .7 ... ..7. .. ... 7.7. 7... .77. ..7. .7.. .77. ..7 . 7. .. . . . .. -.. .. . .7.. .7.. 7... .7.. 7 .. .7.. ....7.. .7. 7 . .. . 7 7 .7 . 7 ... ...77.7- ...7 .7.. .7 .. .... 7... .77. .... . .. ... . . .7 . .. 7. .... ...7 .... ...7 . .. .... .7. 7... .... .. 7 ..77 .7.. .. . .-.. .7.. .7.. 777. 77$. ..7. -767. 7..- .. 77.. ... ... .. 7. . . . ._ .7 . . .... .... ... .... . ... ... . .7.7.. .. . ... ..7. . ... ... 17...... .7.. . 7.7... .7 77... ...7 .7. .7 ..7 ... . .7.. .7 . ... .. . .... .... .. . .7 ...7. ..7 ... ...7 ..7. .7. .7.. .. 7... 7. . .7. 77.77-7777 .77. 7.777! .7-. .7 7... .... 7 7.. . . 7 .. . 7. .7.. ...7.. .... 7 7. .-.. .... .. . ..7 .... 7. 7 ..7. ... ... .7 $77.7...7 .7.. a...1 .. ....7... .... 77.. . .7 .. 7. 7 .7 ... .77. . 7 .... .... ..7 .. .7 .. .... .. . .7 . ...... 7. . ..7. .7. .7....77 . .7...7 ..7. .....7... .. .... .. .. . ... ... .. .7 . ._ ... .. .... . .. . 7... .7.. .7. ..7. 7. . 7. . .7.. ..7. . .. .7..... 7...... .7.. 7. ..7 .7. 777.. .. .. . . .. .. 7. . . 7 . .. . . .7 ..7. . 7... 7 ..7. 7... ..7. .... ... 7 ..7 7. ..-.777. .. . ...7 . . 7 .. . .. . .7 7.7 .7 .7. 7. .. . .... .. ..7 .... .7..... .7. ...7 .77. .7.. .. .... . 7 .7.. .7.. .. .. . .... ...7 7. 7 77.. .... .7.. 7.. . .7 7.. 7.. 7... 7... ...7 ... ..7.7777 .7... .7.- ..7 ...7 DCAQVAI .7v .... .7. .7. .7.. ..7. 7.. ...s ...7 . .07 .. q... .... ... .. . s. I... ... e... 7s.- .. I... vise col. ..4. 7.. ten! tun. Isl. . ! n.ll 7. ..7. .... .. . .7..- .7 .7.. .. ... 7... 7 77.. .... .... 7.. ...7 7.7.7...17..7 ..7.771.. 7.7.. $77. 77.. .71.... .70. .... .... .-.7 .. .... ..7 7 , 7.. . v7. ....7 77 7. . ...7 ..7 ..7. . . ..7. ..7. .. .77.. . ..7. ...7 .7. . . .... .7..... ....7 .7. 77.. .7 .... .. ... .7. .7. .7 . .7 . .... ... ... .1 7... .... .... .... .... .... .7.. . .. ... s... .7 .7. ..o~ a... ..7. 7w997§7~7777111|tv..1.77. 7:7 .. .7.. . . .7. 7 7... ...7 .7 ... .... . . . . . .7 .7.. ... .7. .7 7. . .7 7 . ..7.7...7..... ..7 ..7 .7 707.77l.o. .7. .... .. .7.. ... 7 . ...7 t..! s . ...7 .... 7. e .... .7.. 7... ..0. 1... v... ...77 0.4. .v1. .0... .7.. 1... 7.. 7... . .... .... 7... 7. .71v7.7:7xwl..a§lvv¢6sil!7d.t7Inc'ii .. .. 7. . . .. .. ... .... .... ...7 .... 77... ...7. .7.. ... .7. .7... ...7 7.7777..7.7.o. ..7 7.. 77.. ..77 77.7 .777 77.. .97...77- 77.70 .777. . ..7 . .. .7 . .... . . 7.. .. 7 7. . .. .777 .... .7. 7... . . .7 7 . .. 7. 77. . 77 ..7. .7.. 7 . 7. ..7. .77..... .7 . .. .7 . ,7 7.... .. . ... .... ..7. 7... .7 .. . .7.. .7... . . .7 . .7 . 7 ..7 . .. ..7. 7... ..7.1.... 7. .7 . . . . . ... ..7. ...7 .... ... 7.7. .7.. .. . ... .7.. 7.. ...7...7 ..7 77. 7.77 7.77 .77. .7. 7 ...7 7 .7 .... .7.. ..7... .7 .7.. .. . .. .. 7 .7. . .. 7.... . . . ... . .7 .7 7. . 7 .. 7.. 7.77 .7. ... 7..- 7... .-.. 7...1 .. .. 7 . . .. ... .... .... 7 .. .... . .7 .7.. .... .7 .7.. 7... ..7...77.77.7 77.77. 7.171.77.1f7.. 7... 7 .. 7... 7177. 777777.... .7 .. 7.. .7 7.. 7.7 .7 .. 7 . ... .7.. . . .7.7 .. . .-.7 7... .77...... ..17 ...7 ....477.. ..7 . 7 .7 .777r.7.7 .7.. .. ..7. .7 ..7. 7 7. . . ... ... . 7. .. 7.. . . .7 . . 7. .. 7... . . . ...7 ..7 .77. 7.01.117 A... .... ..7. 177.1001.fi14.l 7.77 . 7 7. .... .. 7. 7 . ..7. .7 . ..7- ..7. .... "" .7. ...7 ...7 777. .. . . .... ... .... .7 7.. ... 77 ... . .. ... ..77 ... .... . v79.. 7.... ..7. ..-.1 . 7: :; :: ; :..:..:..:._:..: .:.....:... .,:,7,,,:..:.-:..:._:. doavdlaHOM 4 .:..:-r:--:; .7. .. . 7 7 .. ... . 7. .... v... . .7 .7 . .... ... ..o .... .... ..v!. ...7 ..7. 77.7 ...ncuoh 7... .77 .. 7 .7.. 7... ..7 .7 7.7env..7 .7.. .. 7 .7 7.77 .... ..7 7 . ..77 o-.. v... 7.. .9. ...7 0.0.01.. .7 ..7 7... .7 . .... .... ...7 7.. 77.. ..7. .7.. 7. . ..7. .... 7... ...7. r7t777 ...- 7... ...7 .. . .. .. .. I . . .. .7..;7777. 7 77-. .7... {...-I. 7..- --7..+7.7 . . . ... . . q ..7. 7. n. 7 .. 1.. .... :1 .01. 9..) thl- -. .. . ~.. It). *Dee Duel ls.‘ ..b1 I.-- t". '7‘7‘!"7‘II7I7A. .. .. .. ..., .. . . ... ... ... 7. 7 .. .. .. ... .... ....‘.. ...7 ...7 us. I 7 741.107.. . .7 . .7 . .. 7 .7... .7. . . .. ... .. .... .. ..... .... .... .7. .7... -777 ..7 . 1.3.17.7-.. 7. ..7. . .7 ... .... w. . ..7 ..7 ..7. .0. . . ..7. 77.7.1 7v..17..v ..77 17.1lt7f77 7.57.7.7... 7.-. 0700110.! oitvwat‘t7 O ... .7. ... 7. 7.. .... .... ...7 ... 7 .. ..-...4...... ...7 ..7 . 008 u -...h...7...7t7o77tiio . . 7 7.. .... ... 7.7 7 7. 7.. 7 . ..77 .. . . . .... .. 7 .... ... .... -.. .... 7 V‘C777I7777t177 77.0.70100 7.7.7 . 7 ... 7. . _. .7 77 7 7...1. .7 7... .7.. 7. 7.. 7... 7 . 7 T77.74-77.. .77113710 .. .. 7. .. 7.. .7, . ...7 ... .7.. ..77 .... .... .... . .. ... 7. . .... 7 7 .7 . ° 7177‘.7.. 1777.7Ve.-|1 7 ... . . .. . ... .. .. .. ... . .777..7. 777 77.7.7. .7... .. 77 “ 7.7.777..777 -.7..77-A .. . . . . . . . .7 ._ . . . . 7. ... ...-.Y 7. 77777790717719.7777) ‘7 , .n .777 .71 1457?¢+.0l7. lq...‘ '7‘.‘. r b .. 7. .. . . 7 ... 7 . .... 7 .. .... 7... .- . ....747177179.9777 7 7 777| ..7v9177t.t| 0-77 97t7 6|..- 1... .Ovltlii701'll7vxv777774fi7 .7 7 ..ft 177 . . . 7 . 7 .... 7.. .7 . 7.. 77...! 1..- i .... .7.-. .11. IV.. ’17. .v. 10‘71 .V-Y 7 7 7. 707 lcfittthtlo 7. it ‘Otlxvstsfnfittql 1.1.7.1 Otvllc77761.. . ud.‘ 77'74 .. .. ... 7.7.7 77.. . 7 717‘. ..7 7701779 60.07.. r57t ..77. .070. I... o... 74....l.7..lls(. ...7. T7... ...7: 0|.“ :0. {iti I16! oolthltv.v .....Avltvn. rt..b¢7 7. 747. . , .P77. 7.. ..7rr.77 1. o A... ..-) 107. o..... 17,710 ......a... w.v 7 .117. I... t. a. l 7 777.!!9 it"! slit llvlt. vial: 7.17 07||6|1 IOI. vlttnl Ylvt.t.7...uo1..a7 1777¢ 717 7r .nouuelduehmad «shah new use mac e no oaaewueaosanno nowadays 4 . « eadmam IOJodmvttttn casts 33 .nodudlnnouamd cacao» new and uJ 1 undo» aqua 97? 7 .7 nodudlaxoumn< opuso «1:904 .18 o8.2 .- .3.» 8... u n no own-«hoaodhanu houundua 4 . a ouswwh .919 “VIII?“ OQ'Id 3k cage» draw . . 4 4 4, . 1.1... + o d .¢?,..I§v¢.... ..vq “Vo. T¢I.-a .90.. ..+4 .+...9195.. 0... ..4. ...4 t 09 ..c- v... u».é\ ..- .... ... .-. . . ...- .... A... ... .... . .... ...- -... --.. -... .... .. . . .... . .9 .... ... ... ... ... 11...... -.. . ..A .... .1... .... .... .... ... . . .... .... . ... ... . ... .... .... .... ... -... ...-i-.. .... .... y. . . . .... . .. ... - . .. .. .... .... ...-4.... .... ........¢.-... -11. .- . .... . ... .,. A .. .. ... ... o. - 1.-. .... .... .... . ...-+-116A.-.A.... . .. . 1 .4 . . 7.... . . . . .. . .v .... .. . ... .t.ov.v.. .... . .. .-.01... .-.. ... .- .-. . . ... .. ... . .... ..q. .... ... ...v ...9- L-.. ....-.Y-ol.o .... o... -51 . ... . . . .. ... ... w'vt. --.? t... ... .. . .vkéc..v.+-..i... . -.. .. .. . .1 . .. . .. ... 01.. v1-.v ...: ..-. .... . n ..-...o. wq... .-. .. . -. . u ... ...u.-.. . .... . .. u. . vs v..oAv|«...ul...V.- .... . . . ... $121, ...-1-1 1-1.1-132 . .. 1-13.71... . q . . o . . . . . v . . . a A w T. . A . . 1vL . all-- 1 A . a . . . t. c 9 10 170-11..- I i .L - 111- . a n . b 1 . I t.l.' l6 1‘ .1I I | . 4 'l‘; . . TA..7..- .... ...! ,.A.f¢- c++aL111ql 91.61 p.6. 0....- .... 01.14 1111111-- T o§vO!-16 O‘IAYSI AT1.!-O¢|’4ti .v. . ...LvT.-. . . .... 9.1.? It} +?$ v1.10... .6.019.6. ...-¢4T11§..Yl¢ - fi .1 110 {41" I 0‘ il1.l.+.-t. . .Iogdsnoumn—d vacuum a: 33 3 a no 03020909228 young—aha 4 . H 0.3%?— seladiIIIIN 04911 35 We shall now leave the 6J5 and go on to two more tubes. This pair of tubes are remote cutoff and sharp cutoff pentodes. Only one curve for each tube will be approximated. For the remote cutoff pentode we will use a 6867. The plate voltage will be 200 volts, the screen voltage will be 100 volts and the load resistance '111 be 110,000 ohms. Our origin 1:111 be 'c1 = -30 volts. Since a 6367's grid is not supposed to go positive no points with the grid positive were selected. The points used were e a‘=-l6. cl -9, -5, -2, and 0. The initial matrix and its solution are given in (37) and (38). F111 27.1-1-1102 53.752hx10 21 92.b1x102 hos.u1o1x1o (37) 25 156.251102 976.5625x10 2 28 219.52x10 1721.0 110 0.0206b111012 o _ 18.0109xlO 0.791125 :1012 o 3.8lu70 11012 1.0 12 on 1.05U'110 osos 61.0351110 13h.u929x1o 10.578h6 110 1.0 4: -:* :r or 1: os )0 270.001.10a 21130.0 x10 215.7 110 19.68} x1012 Ind (33) 1p = - 5.023856(scl+30) + 5.078815x10’2(ecl+ 30)3 - 1.6859571104‘ (eclv 30)5 + 22.58N3cld-8(ecl+ 30)7 - 10“.56xlO-12(ec1+ 30). A cursory examination of (38) shows it to be quite obviously a poor approximation (let ec1+30 = l and see what I mean). As an experi- ment we will try using only four points for our approximation. they shall be the same four points as the first four points of our first approximation. Our initial matrix and its solution are given in (39) and (#0). '11- awn-x102 53.752ux101‘ 1.051- nos 0 '2 21 92.611102 now-101110“ 18.01091108 o (39) 25 156.25x102 976.5625me 61.03511108 1.0 L28 219.52::1o2 1721.0 no“ 131-.1-9zgx1o8 “'01 36 (to) 1p = .. 0.02815Mecf30) + 0.03226no’2(o°1+30)3 - 0.0117u2x10'” (ec1+30)5 + 0.1332(ec1+30)7. ' ' This approximation is plotted in Figure 17. The fact that four points give a better approximation than five points may seem somewhat surprising at first. However, in this case it must be realized that any finite power series with positive powers tends to go to infinity as the varible goes to infinity. In the five tern approximation .the curve included a point in saturation and tended to approach a finite value as the varible approached inf inity, while the four term approximation did not include the point of saturation and the last point was in the path of increasing ip with increasing °cl‘ For the sharp cutoff pentode we will use a 6887. The plate voltage will be 200 volts, screen voltage will be 100 volts while the load resistance '111 be 1$0.000 ohms. Our origin .111 be .c1 = ~30 volts. Once again we will use only four points. This time they Will be °cl = -12, -11, -2. and -l. The initial matrix and its solution are shown below. '15 58.32x102 188.9568x10u 6.12221108 o ‘ 26 175.5tnno‘2 1155.137bx10" 80.3181x108 o (‘11) 28 219.52x102 1721.0 non 13I+.u<329x103 2.05 L29 2143459me 2051.11h9x10‘l 1724;985:108 11.51 (“2) 1p = - 0.3b562(ec1+30) + 0.25778110'2(ec1+30)3 - 0.05782110’u (ocl+3o)5 + 0-"11‘10'5(°c1+30)7- This function is plotted in l'igure 15. This concludes our survey of the various tube types and approxima- tions of their transfer curves by power series. So far we have seen u. . .y. a 1 . .. 4 . s, manual , 1.1.11; 143.453 . NJJJ . . . .a. . .»Ho> Hum eauuouaa< 0:8 1304 .enc ooo.o: sumo» com DOHOP CON nu 3 tt?‘ .7 .. . . e o .. ~ .no«»slauoumgfl Ines nook sea was sown o no ouoswhouoouono no manna 4 . H onswah sexedmsttttw etvtd 38 .louaeauuoumas.ahos ouuoauuoanas .pnso aasaoa._nuulunu. .58 08.3 u an .3.» c2 u won node» CON fl .39— 3“ can B3 c .3 03.23326 hotness < .3 933a sexedmettttn OQFIJ 39 that this method offers a reasonably good approximation for a given plate and screen voltage and a given load resistance over a wide range of grid voltages for most of the common types of amplifying tubes. There are several limitations. First the approximation is not particup larly good.at saturation. .Also the tube aust be conducting during sole part of the cycle on the grid voltage for the equation to be valid at all. (If this limitation is not accounted for it is entirely possible to predict a plate current when the tube is entirely in cutoff.) .At this point we still have to start over if we change either the load resistance or the plate voltage. Our next step therefore is to try to further reduce these limitations. CHAPTER II REDUCTION OF THE LIMITATIONS The first limitation that we shall take up is that of the poor approxtmation of the transfer curve when the tube reaches saturation. Ie will sub-divide the problem into three cases. The first case will be when the tube does not reach saturation, the second case will be when the tube reaches saturation but does not cutoff. and the third case will be when the tube reaches saturation and does cutoff. The first case is a trivial case. In this case we simply disregard the fact that the curve is poorly approximated at saturation and use it any“!- The second case presents a far more serious problem. One way of attacking it is to use basically the same method of approximation but choose the origin such that the tube is in saturation and that the var. ible will take the tube out of saturation. In this case the power series will have a constant term that is equal to the saturation cur- rent and the power series except for the constant will be normally negs ative except when the tube is at saturation and then it should be zero. For example if the be tube was to operate from ecl = O to ec1 = +50 volts, one could select the origin at ‘cl = +fl0 volts. One could use the power series of the following form: (#3) 1], = 1' + .1(.c1-uo) + .3(.cl-uo)3 + .5(.c1-uo)5 + where 1. is the saturation current of the tube. In particular if we were to apply “00 volts to the plate and 250 41 volts to the screen with a load resistance of 2,000 ohms, and the above limitations on the grid voltage we might get an initial matrix as shown in (M). .1 F110 - 1x103 - 1x105 - 11:107 0 -20 -. 5x103 - 321105 - 12:51:107 o (“14) ~30 ~27x103 . 2u3x105 ... 2157x107 42.5 :uo -bltx103 ..102ux105 -1b3shx107 4.5.07 In this case the saturation current was 177.5 milliamperes. The reader should note that the last column is not the plate current but the difference between the plate current and the saturation current. In most cases the problem of the tube saturating does not occur unless the tube also cuts off. This case is only included for completeness and (M) will not be solved. The third case prsents a difficult problem. If we wish to use the power series, it is obvious that we would need more terms. This would make our task of solving for the coefficients more laborious and the Whole problem becomes unreasonable. The only other alternative is to find another way of approximating the curve. We had earlier mentioned using the hyperbolic tangent. This curve very closely resembles the transfer curve. Our equation might have the form: (1‘5) 1p = a + btanh c(ecl + d). This seems to offer us considerable latitude as it appears that we “11 be able to pick four arbitrary points that we wish to use. Unfor. t ... .. . unatvly this is not the case. The minute we select the saturation 42 current we have chosen a, b, and (1. One other point on the slope of the curve will give us c. Thus in our previous example we had a satur- ation current of 177.5. The tanh varies from --1 to +1. Therefore our b must equal one half of the saturation current or in this case 88.75. Since this point occurs at °cl + d = O, a must equal 88.75. 6. is also fixed because i = 88.75 for only one ocl‘ c is determined by select- P ing a point on the slope of the curve and putting it into the equation and solving for c. This method is actually easier than that of power series but unless the tube transfer curve is fairly near the shape of the hyperbolic tangent the method will be inexact. (There is also an- other difficulty, but this will be explained later.) In addition, in order to use the quation we should transform it into a power series that is convergent over the range we desire. This appears not to be too dif- ficult. Before we do let us rewrite the equation in light of what we already know, and let x = c(ec1 + d). (’46) i1) = a(l + tanh x). or in power series form 3 2 5 7 (“7)1 =a1+x-£+—£-1-lx—+eee). P < 3 15 315 There is, however, a very serious limitation to the power series In fact x2 form. This series converges only in a very limited range. must be less thanE-a- . Since the tanh does not approach 1 until x is larger than gand since we want to go through zero we cannot use the 1’0"” Series form of the tanh. (There is another form of the power series for the tanh x when x 19 large but this is not valid for small x and 111111.8 we gain nothing.) As a result of this limitation we are al “0“ reduced to Fourier analysis in this condition. q... ...: as Eleni»... 13.4.5 n a- 43 The next two limitations were the fact that the present power series approximations were limited to one plate voltage and load re. sistance. The solution most apparent is to compare the series of the same plate voltage or load resistance and see if some relation can't be worked out with the coefficients of like terms. It is obvious that we*wonfit be able to vary both the load re- sistance and the plate voltage. Our choice is to use a fixed lead re- sistance and vary the plate voltage. Tne reason for this decision is the fact that plate modulation employs a constant load but a varying plate voltage. Actually the method we are going to use would be valid whether we varied plate voltage or load resistance providing the other was held constant. So if the reader desired to keep the plate voltage constant and let the load resistance vary, the same metnou.would work out satisfactorily. The method is one of comparing coefficients of like terms. In order to make the data more convenient we tabulate it. For the be with a load resistance of 20,000 Ohms TABLE I Summary of Coefficients of (ec + 100) M g Plate + - 3 5 - 7 9 Mt. (0c 100) (sci-100) “Heed-100) (oc+100) +<°c+100) 20° 0.11392 0.07322110'3 0.01282x10'b 0.73%:10‘12 13.311045 300 0-25801 0.17111x10‘3 0.03207xlO-6 2.1197310"12 110.3):10'48 n00 -3 A —6 ~12 ~18 \ 0.20300 0.17b12x10 0.03356x10 2.1932x10 147.14x10 Nisan-annex. ...munswr1hniiunnvqgn ..m . ...... - nu l‘rom immediate observation it is obvious that there is no linear relationship among the coefficients. Therefore it is necessary that we find some other method to establish the relationship between them. Let us investigate three methods. First is to plot the points on a graph. The second is to approximate the curve of the graph using the known points and a power series using the odd powers only. The third is a variation of the second. In this case the power series consists of odd and even powers. The graphical method consists of plotting the value of the coef- ficients versus plate voltage. A separate curve is needed for each set of coefficients. To find the coefficients of an intermediate volt- age one need merely pick them off the curves. Figures 19 through 23 give the set of curves for our particular approximations of the 6.75 transfer curves. For a trial a plate voltage of 250 volts chosen and the transfer curve was approximated from the graphs. The equation of the approximation is shown below. (148) 1p = 0.215(.c+100) - 0.138x10“3(ec+100)3 + 0.0270x10‘°(.c+100)5 - l.'IOxlO-]‘2(ec'+100)7 't 37.3110-18(ec+100)9. This equation is plotted along with the actual transfer curve in Figure 214. A more detailed evaluation of the methods will be given later. However, Figure 214 shows quite clearly that (MS) is a very poor approx- imation of that particular curve. 0111‘ second method consists of approximating the curves of the co- efficient; versus the plate voltage using power series of the form: M9) ‘1: = blsbb + 1:231“,3 + b3Ebb5. + 100) vs Plate Voltage 0 C Plate Voltage Value of the Coefficient of (e + 100)3x10"3 vs Plate Voltage. Plate voltage Value of the Coefficient of (e for the 6J5. / 00 figure 19. Figure 20 c acouo «whooo 00 m. as a a n t O w. v v e e n ah a H 1 P s v s v 2 x no 1 o x 7 5 4) m m m efl ... 6c 8 c 80 0 til we... 1 t of If V0 00 Y st to tn on as to 1.1 ad Pm l U Pfi ......“ f a e o o C C o m h t as a .... f o w Av a. m. .uf 0“ Yo 0 / C o 1 2 2 e 2 . e m. r ... m ... n , a , uEOdO HHHOOO of a 6.15. e 4 m ,0 m. = 2 «no / 4‘4247 ‘,'+-lOO)9xlO"18 vs Plate voltage Plate voltage Value of the Coefficient of (e of a 6J5. 1'igure 2}. .noaaoa«Noa gas .»a use «one» ease \ muuv.1 .1 acupeaawoa erase ascend Iago ooo.om u oral? tr ....et Isl, am a no oupeamouomhmnu newsflash 1 . N ohnwwh soxsdmetttiw outta 17.9 In making this approximation we shall‘use only the points we actually know. (Thus in our case we shall use only ’bb = 200, 300 and 1400 volts.) Once again we will use the matrix method of solution. E00 8x106 32::1010 0.11M -0.07322 0.01252 -0.73u,5 13.3- (50) 300 27x106 2h3x101° 0.259 -0.17111 0.03207 -2.1197 ue.s $00 614x106 lOthlO:lo 0.263 -0.l7612 0.03356 -2.l932 347.3111 In this matrix the fourth column is the first term coefficients (.1) of the transfer curve approximations. The fifth column is the third term coefficients (a3) times 103. The sixth is the fifth term coefficients (as) times 106. The seventh is the seventh term coeffic- ients (I7) times 1012. The eighth is the ninth term coefficients (a9) times 1018. (All the even term coefficients are zero.) Solving the matrix gives the following results: (51) a1 = 0.00700x10'21:bb + 0.015u25x10'63bb3 - 0.000735x10‘1ombb5 (52) 03x103 =,_ 0.002u5x10'3nbb - 0.01052x10'bsbb3 + 0.000u95x10‘103bb5 (53) .51106 = - 0.00057hx10'21:bb + 0.0021u2x10'“sbb3 - 0.000099x10'10sbb5 (5’4) a-leO12 = 0.17559Hx10'22bb - O.16‘5807xloubfi.“3 + 0.007539xlodoflbb5 (55) a9x1018 = - 7.14-85x10'21bb + n.309x10'e‘sbb3 - 0.1938x10’losbb5 To test this approxination we will once again select a plate voltage and compare the calculated current with the current calculated from the published curves. Once again our voltage will be 250 volts. This time °m' ‘Pproximation will be (56) 11, = - 0.h59(.c+100) + 0. 312::10"3(.c+100)3 - 0.004x10'b(ec+100)5 + 5.210x10'12(.c+1oo)7 - 1140.66n0’15(.c+100)9. In order to save space merely a table of the evaluation of the plate .ald...hm.n.§ a m agflflmfiim a. gs. 50 current (actual and calculated) and grid voltage will be given. The portion covered is in the linear region of the tube. TABLE II Results of the Approximation Using Odd Power Series for Plate Voltage Plate voltage 250 volts 250 volts 250 volts Grid voltage ~10 volts 0 volts. +10 volts Actual Plate Current 1.7 ms. 8.5 ms. 11.9 ma. Calculated Plate Current -0.02 ms. 0M4 ma. 3.01 ma. Load Resistance 20000 ohms 20000 ohms 20000 ohms This is a better approximation, but it still is too inaccurate to be of much value. Since the third .method is almost identical with the second method only the matrix and its results will be given. In this matrix also the fourth through eighth columns will be modified as in (50). F200 M10“ £53.106 0.1139 -0.07322 0.01282 -0.73u5 13.3" (57) 300 9x10“ 27x106 0.2580 .0.17111 0.03207 -2.1197 110.8 1+ 6 300 16x10 64:10 0.2630 -0.17b12 0.0..50 -2.1932 146.1: The results are given below. (56) a1 = - 0.15005x10‘22bb + 0.15350110'ulbb2 - 0‘02%5fl0.%bb3 (60) asxlob z - 0.02189x10-2Ebb + 0020731104}!be (61) a7x1012 = + 1.80u2110’23bb - 15533110.“:be -1} 2 -b b + 140.70110 Ebb - 0.2ussx10 Ebb}. - 0.00329x10'6'sbb3 — ' 3 + 0.2ussn0 E‘s“ (62) 1:92:10“ - 09.35::10‘22 b 51 Once again a plate voltage was selected and the coefficients were evaluated and an approximation made. This time the approximation was (63) 11, = 0.19u5(.c+100) - 0.1275x10"3(.¢+100)3 + 0.023u3x10'b(.c+100)5 . 1.14970x10-2(age-100)7 + 31.78x10‘15(ec+100)9. Over the linear region the results of this approximation are tab- ulated in Table III. TABLE III Results of the Approximation Using Odd and Even Power series for Plate Voltage Plate Voltage 250 volts 250 volts 250 volts Grid Voltage ~10 volts 0 volts +10 volts Actual Plate Current 0.7 ma. 8.5 ma. 11.9ma. Calculated Plate Current 3.59 ma. 8.27 ma. 12.13 ma. Load Resistance 20000 ohms 20000 ohms 20000 ohms Since this approximation checks fairly well at 250 volts now let us try it at 350 volts. (61+) is the approximation. (61+) 1,, = 0.2877(oc+100) - 0.1915210_3(oc+1oo)3 + 0.03627110"b(ec+100)5 .. 2.u13x10‘12(oc+100)7 + 53.59ux10'15(oc+100)9 Table IV gives the results of this approximation. It is easily seen from Table IV that this is the desired approxi- mation. It is only valid for Ebb from 200 to #00 volts but this range °°u1d Probably be extended by making additional approximtions and in- eluding these coefficients in the comparison. 52 TABLE IV More Results of the Approximation Using Odd and Even Power Series for Plate Voltage Plate Voltage 350 volts 350 volts 350 volts Grid Voltage ~10 volts 0 volts +10 volts Actual Plate Current 14.95 ma. 12.20 ma. 17.25 ma. Calculated Plate Current 5.810 m. 12.26 ma. 17.0“ ma. Load Resistance _ 20000 ohll 20000 ohns 20000 ohms In conclusion it should be added that there well could be other methods of finding the intermediate curves than the ones discussed. Also each of the methods discussed may be of value in a particular case. In our case the third way proved to be the best. If we had investigated the 6V6 we might have found the second way the best, or even the first way. In discussion of the three ways in particular, the second and third ways are essentially the same, and neither offers much advantage over the other except in a particular case. Both of them have the advantage over the graphical method in that they can be calculated to more places. In addition if the plate voltage is continuously varying as in plate modulation the second and third ways can be directly substituted in the equation for the plate current‘while the graph would again have to be approximated. CHAPTER III APPLICATIONS Now that we have made the approximation of the transfer curve and discussed its limitations it is now time to investigate some of the uses it can be put to. Our first problem will be that of a single tube ampli- fier using a resistance load with too much bias for operation in the linear region. The tube will be a 6.75. the plate voltage 300 volts, the load resistance 20,000 ohms, and the bias ~15 volts. We will apply an alternating current voltage of the form lOcos pt. The circuit dia- gram is shown in Figure 25. J ea—‘hL \ " E“ Ec+ E f C ‘6 Figure 25. Single-ended Triode Amplifier. The varible, (ac + 100), now becomes (85 + 10 cos pt). Putting this into (28) and expanding we get. “14‘ haunts—nan... Jana): 1. ...—.3. Snafu: - 5h (b5) 11) = 2.078 + 14.7180” pt 4- 0.97ocos2pt - 0.2Mcos3pt - 0.0577cosnpt + 0.00183cos5pt + 1.153x10'3cosbpt + 10.53xio'5coa7pt 8pl”. + 1H). 81:10-ch s9pt . + 35.80x10.'7cos This function in turn may be expanded by the use of trigonometric identities into the more familiar harmonic functions. (be) 11, a 2.5m + n.53bcos pt + O.‘+59c082pt - 0.056cos3pt - 0.00699coshps + 0.000126cos5pt + 0.03b3x10'3ces6pt + 0.165110'5cos7pt + 0.280x10'7cos8pt + 0.183x10'9cos9pt. To obtain the output voltage it becomes merely necessary to multiply this current by the load resistance. 01’ course if there is a blocking condenser the direct current component of the voltage will be missing. Push-pull circuits can also be easily calculated using this approx- imation. In this case the tubes are figured separately and the plate currents subtracted from each other. 'E ——4 Output Figure 2b. Push-pull Amplifier . 55 For example let us assume that the output transformer appears as a 20,000 ohm load, that Ebb = 300 volts, that Rec = 15 volts and that eg lOcos pt. This once again means that we use (28). For the top tube the plate current will be identical to (65). The variable for the bot- tom tube is (85 - lOcos pt). This gives the following results for the bottom tube. (67) 1p = 2.078 .. “.718cos pt + 0.976cos2pt + 0.2mcos3pt. = 0.0577cosupt - 0.00183cos5pt + 1.153110-3cosbpt - 10.53x10'5cos7pt + 35.8Ox10‘7cos8pt — ‘46.8x10"9cos9pt. Subtracting (67) from (65) gives us the output current (68) (68) io = 5.u3bcos pt - O.M88cos3pt + 0.0036bcos5pt + 21.06x10'5cos7pt + 93.6x10'9cos9pt. Or in harmonic form (69) 1° = 9.072cos pt - 0.llbcos3pt + 0.000252cos5pt + 0.33Ox10'5cos7pt + 0.}6oxlO-9cos9pt. Thus we immediately see that the even harmonics cancel as they should in a balanced push-pull circuit. One of the most difficult problems to handle analytically as far as vacuum tube circuits are concerned is a plate modulated r.f. ampli- fier operating in class 0 conditions. Our approximtion offers a solu- tion. It is not an accurate solution inasmuch as several untrue as- 80111131310113 are made, but the results can be used as a guide on what to expect. Figure 27 shows a single-ended r. f. amplifier. 56 EC. E" Eu" Es 6" Figure 27. Single-ended Amplifier The untrue assumptions that we shall make are (1) if grid current is drawn, that it does not affect the bias of the tube, (2) the neutral- izing circuit can be ignored in our calculations, and (3) the tank cir- cuit presents a resistive load at all frequencies. This last assump- tion is particularly objectionable, but it can be Justified in that we are interested primarily in the voltage output and not the current. This means that we will find the current by making the assumption of a resistive load and then finding the voltage by multiplying the current by the true load. In such a manner we will be able to guess an approx.- imate solution. Let us assume that we have such an amplifier. Let Ebb = 300 + 100cos Int, ec = ‘40 + 6500s ct, and that the tank presents a resistive load of 20,000 ohms at the carrier frequency. The expression for the output current is (70)- 57 .. 2 (70) 1F = L. 0.150o5x10'2(3oo+100cos mt) + 0.15350x10 1‘( 300+100cos mt) - 0.02485x10-b( 300+100cos mt)3J (bro-+05cos ct) +[O.10457x10-2 (300+100cos mt) - 0.10‘4-03x10"u(300+10()cos mt)2 + 0.01t372x10'.b (300+100cos mt)3J (140+b5cos ct)3x10"3 + [- 0.02189x10.2( 300+ lOOcos mt) + 0.02073110-u(300+100cos mt)2 - 0.00329x10'b( 300 +100 cos mt)3] (WbScos ct)5x10'b + [1.80u2x10.2(300+100coa mt) - 1.5833x10-u(300+100cos mt)2 + 0.2%8xlO-b(300+100cos mt)3](1+0+b5cos ct)7x10'12 +[-u9.35x10'2(3oo+100cos mt) + l40.70x10““( 300+100cos mt)2 - 0.21188110~6(300+100cos mt)3] (HO'tbScos ct)9x10']'8 This expression is quite long and its expansion would be of dubious value. Nevertheless we are able to see that the approximation will give us an expression for plate modulation. There are many more problems that can be solved by this method. Indeed other tubes may be used. Also nothing was done involving wave shaping circuits. However, by this time the reader should have a fairly clear idea of the subject and can apply it to his own particular situ- ation. In conclusion once again it should be stated that this is only an approximate method as is the case in most non—linear problems. The method can only be as accurate as the approximation. It was found that the calculations required the use of a calculating machine as a slide rule did not offer sufficient accuracy. It also should be added that the results experimentally may differ from those calculated due to variations from the normal by the tubes themselves. 58 As far as the results are concerned in this paper, no experimental work was performed to prove or disprove them. However, each of the calculations was checked at least once and most of them were performed by machine. It is the author's opinion that this method will be of great value when more than one grid voltage is applied to a tube and when the tube goes either into saturation or cut-off (but not both!) For just one particular case the approximation is too much work for the value received. Nevertheless the reader should not forego his own opinion on the subject. APPENDIX A SOLUTION:YAN DER BIJL'S EQUATION APPENDIX A Solution: van der Bijl's Equation 3 (1) IO = «(ab 4 1c + e + esin pt) where Eb = Ebb — IR and B = 2 (R is the value of the load resistance.) ' 2 (2) Io = «nub, - IR) + no 4»: + esin pg] Let Y= XEb+Ec+£ Since 6 is very small compared to "Eb + no it may be neglected and v = tab + no. Rewriting 2 (3) Io = «(v + esin pt) Expanding the right hand side of (3) 2 (it) I0 = a” + 20‘ Yesin pt + u easinept Substituting the value of Y and further expansion (5) to = «(IaEbe + Ecz + ZIEbEc) + 20‘ (13b +_Ec)esin pt + “easinapt Substituting Ebb — Ion for Eb, expanding and collecting terms (b) “1232102 .. 10(1 + sexenbbs + some + annesin pt) = ..«(v + esin pt)2 Rewriting (7) 102(uxen2) - 100 + 2030 + esin ptj) +x(v + esin pt)2 = 0 Substituting into the quadratic formula (5) IO = 1:12:13" + esin pt) :Jl:h«lll(_!.t£s.i_n. pt) -‘--“...-m- ---- ...-.-- n-'- APPENDIX B b CHARACTEISTICS OF THE TUBES b Generalnll'fectric Company, Electronics Department, Eefihllfirlifi EEG‘SJ Receiving meg; Volume I. «a _ . «nae» opeam Awnvmw 08HO sou“ gage» com a o>o a mo wean-«Aouowusno madam owdao>4 saledmsttttw casts Anny . undo» oaeam nunvmw one u no .o«».«uo»oau¢no oeuam Aunu\ ..agp m.o u uaqaaaau omeuo>4 ausxxno'aastg .uao» oseam QQN 50mm e no noapuwhopoeamno speak omeao>< as» m.@ u asoecaem ..Hos cod u was seieflmsttttn sqetd «use» oeaam “enumw Yved‘fi 11+? 44 feces; a 101 $‘VO Iitlvea sot. oo. 09A. at. Nap-pas» m.e u squeegee galacticlyeesa ed‘Av‘0‘50‘t, vaeve 1" c090+si112sbko+ coy‘ué‘o‘xvttool1¢+o 42v$l .. .4 i 4% <44 «Mi 4 4 44 4 r’ 1 e e ji’rTiL d 1 4 4 4 4 AW. 4 t .010 vr+v4|fléIcl§y \A .10 0‘ o a To! Yvko‘e silOch A AL1 T.lo+AYLIX . . A , , “of? 4 s o t a ‘i o d 0‘ Q it ‘Ith .. cidwilyled c Q A , _W 4 4, H + Q «J 4 tel cot slio9sr 9.4- II s all difi< otbcn 00. o e..‘\ .au Aowk 4 a... $.«v 61‘. wtfiwa4 vet. ‘ $9M]. sun .0- ..Nu ..1 1 4 7.... ..v. .i.. .... ...: v I“. oo‘ot 9.. «..r . .... .... . . . .r.- .40 ... Ttliv .... a... . vu to. . .1 f+ ..- .. rlwfyt c . . . . . v; I . . c . . . . . .1 . 1 ‘Yu . . . . . TV‘Qvoi .‘fowi ..‘VA u... o. . ...a vntc. ... .fl.. a.v.ll¢4$‘o a‘6l4‘¢‘¢4 Y.. . . .... ..fi .. . ...r a... «.o.‘ l .. o ... . .... ...i c+4 .... .c.. .i.. ..«l ... i4v¢u¥+ nil; ... .... .... .... .n.. .o‘ 9 .-.. ... _ .... .... v... .... ... v v.... .. . . t. . y... .... .... ... . ... .. “ .. . .... .... .... ... . .-N nsrw 1 .8. qt. 9T$¢ +v.. .... .... ... q .1.. .. ... c.!. ...9 who: *3... cut. . . . .... ... ... . A— 70.4. o... n.o. ...t o». v; ..iti .l.. .. .-.. ... b... .... ....a oa.so- 1.-.. . \. .‘Y‘v ..ta ...» a... ... A YO..¢ 99.. .. ..o. ,+.. .... .... .... . ..-. ... . II .... ... .... . . s tot. .... ..b» 4 ..ul.‘ . c.90‘ .. . vy- . .... . .. . ..y. -.. ¢.v V I. .. . ..-. v... ... .... . . ... \.¢.. .... .... .... ... . .... v s ... ..s. ...- .... .... ... o .... . . , .... .... «on. .vbéi ... . .... .. . . 90.- luol .... In]. ecu A ‘no. .1 ... .... .... . .. ... . .-.. ... . 0.. .17. .«an . .-+ wt. 0 .... ..s . n 4.. .... .... 1..§ .. . .... .. . . .... .... .... L ...aivu. . .... nt. . c . . .. ... .. .... ...a . . . . ... s... . . .. fl ... .... .. .. . .. . r ..01 ..a6 Y. .. ... ..s|+ 4.; .~. .. .r . .. .... ... . . .I.. ... . ... o. . .v.. .. . i .. .. . . .. . .. .. .... .. r . . .... . .. .r .. ...A . .y ... . . . . ... . .... o... .... ... . .. . . ... ... ... ... A. . . .. . .... .r. ‘... ... . .. ... . .... .. ... .... .... ... ....f4o. . ... ... .. ... .... ....i ~--.s ... . ... ... . .... .. . .r. .... .. . .... .vl‘ . y .... .... ... .... .. . .... r.o ... .... .... y. . A... ... . .... .. . . ... .... .... . . .. 1 .. ... .... .... .r . ..i . .... .. .. .... ... .. . . o .0 .7 . , .. .... .... ..t. ..v . . t ..y. ¢.u y. .... .... ... . y . . . . .. ... .. ... . . .., .r. .... .... . A . ... .. .... .«.. v..t ... .i ... .. . . . r. . .v »» . 0.. . .... .. . .... . . . . . .A ¢._ . Y... . .. .. . v.4 .. . .... ..o . .... ... .Y. o... 1.: . .... .. .. .... .... .. . .L . . .r.. o. 4 . .. .... ... a.. « -... 0.0 I-. .... ... .... .... . . . ... .. .. . .. . ..o an... «I . .,.¢ t» .-. . . .. .. . 1...Ol ... . . v .c .... ..cu .... ...» ... . ...1v :. .q .. .. y. . y ..7 ... .. . .... .. . .. ... ... ... l...i ...? l ..y.. ..c , v D a ... . . ..f .... i 11 1"“.t . . vl.. . ... o .. .... .. .4 ..- .. . .... .. ... . . . t . . . .4 ... .... «o9; vvtlb 79.3 ‘ .i 01'.“ OIL- Iv- ' 1|. I- VA. r to. J 16le. .c T10 ‘Vt' 9 .0A 1“»- \. +¥l his . f. 1 l r. Y 6'5. 7' ...7; VI. .T..J.l .r.f.lTTiT.‘ .00.. . . a ‘7 V . A Ti II --. n.0t Yul... T.l #+ .9.‘ !. IA 9|" ammo a no noaaaasopoaseno oueam owuso>< ssxedmsttttw OQVId :3» 02 u mo APPENDIX C CALCULATIONS FOR THE APPROXIMATIONS APPENDIX C Calculations for the Approximations Since a step by step process was given in the text only the matrices will be given. equation will be omitted. The initial matrix is repeated but the final Also after the matrix has been half diagon- alized Just the complete diagonalisation will follow it. 676 first Approximation Tor Ebb = 300 and 200 volts with BL = 2000 ohms ' no 69 x103 102.11x106 65 2711.625x103 1160. 3:10 (1) 100 1000 :103 10000 x10 100 115 1520.875x103 20113.6:10 266 6 6 6 _150 3375 6 x103 75937.9x10 1708.6 '1 1.6 x103 6 2.56x10 0 170.625x1o3 0 8u0 993.9 x106 6 (2) 1:103 97th x10 0 1336.875x103 19818.2 :106 [o 3135 3 6 b h.09b x109 0.02717x109 ET 1.6x103 2.56 x10 0 1 5.8251103 6 6 o o n.851x10 76.7676 x109 (3) o 0 12.032x10 229.206 x109 57.292x106 1622.808 x109 J<) O 0.1638ux1012 n.902 xlO 0.00u906x10 b.6358 99-590“ 265.529 xlO 75553.9 x10 1707.986 0.000262 :1018 o o ' 0.0207119x1015 o 0 11018 x1015 11015 12 12 110 l 2 1101 3.518 38.hu 112.5 82.5 135 57-5 12 135 87.5‘ 110 12 15 - 0.00000655x10 O 0 18 12 x10 0 0 x10 0.020256 12 x10 0.9993M5 o12 3.5172u7 38 .1439 8 112.5 82.5 135 87-5 135 876‘ 1 110 15 x10 x1018 11 x1012 12 6.55 110 0 0 W 0.0001189x1015 o o 0.899u69 x1015 112.5x1o'3 82.5x10' 3.358293 :1015 135 1:10“3 87.5110~3 38.0662u85x1015 135 2:10“3 87.5xio”' ('4) (5) (6) (7) loooo H1 'oooo ooo '00:) H‘ O '000 1 .6x103 1 0 0 0 1 . 6x103 1.61:103 H 000 6 68 12 2.56 x10 b.096x109 6.55 x10 0 o 5.825x103 27.17 x106 118.9 :109 o o 1 15.835x1o3 0.185ux109 23491::10"9 17.007x10"9 o 38.800x103 1.1276x109 -1hh.03uxio“9 ~117.128x10'9 0 716.162x1o3 27.uuu x109 ~1193.659x10‘9 .886.865x10”?j 2.56 x106 b.096x1o9 6.55 x1012 0 o 7 5.825x103 27.17 :106 118.9 x109 0 0 1 15.525x1o3 185.u x106 23.191x10'9 17.007110"9 0 1 0.02906x106 -3.712210‘12 —3.019x10'12 o o 6.632 :106 1usu.7 110"12 1275.2 1:10"1 2.56 x106 b.096x109 o -1.uu811o”3 -l.258xlO-3q 5.825x103 27.17 x106 0 -26.277x10'6 ~-22.829x10'b 1 15.825x103 0 ~-l7.782x10'9 -18.59Ox10-9 0 1 o --1o.13l+x10'12 --8 .599x10'”12 o 0 1 0.221x10‘15 0.192x10-lé' 0 605.u8 1:10"3 u90.90 x1o'3" 0 -581.53 x10"6 -u73.57 110'6 0 1M2.59 x1d'9 117.u9 110’9 o --10.13I+x10"12 -s.599x10’12 1 0. 221110“15 o.192::10"15 j 69 61b Second Approximation for Ebb = 300 and 200 volts with RI. = 2000 ohms '80 68 x103 102.ux106 0.1638ux1012 0.000262 :1018 0 o ‘- 65 27u.625x103 1160.3:106 n.902 x1012 0.020711911018 0 0 (8) 90 729 :103 590n.0x1ob u7.8297 x1012 0.387u2 :1018 57.5 50 110 1331 x103 16105.1x106 19u.8717 :1012 2.3579 :1018 135 87.5 &30 2197 2103 37129.3:106 627.u852 x1012 10.60n5 :1018 135 87.5“ 11 1.6 x103 2.56:106 0.0014096211012 0.0000065521016 o o ' 0 170.625x103 993.9 x106 b.6358 :1012 0.020286 :1018 0 o (9) o 585 :103 567u.5 x106 M7.u61 x1012 0.38683 :1018 57.5 50. 0 1155 :103 15823.5 x106 19u.u21 :1012 2.35718 :1015 135 87.5 9 1989 x103 36796.5 x106 626.953 x1012 10.6036 11018 135 87.3 Fi.1.6x103 2.56 x106 n.096 #:109 6.55 x1012 0 o ‘1 0 1 5.825 :103 0.02717x1o9 0.0001189x1015 0 (1o) 0 0 2.266875x106 31.567 :109 0.31727 :1015 57.5x10"3 50 x10"3 0 0 9.095625xlo6 163.0uo :109 2.21985 :1015 135 :10"3 87.5x10”3 .9 o 25.2105751106 572.912 x109 10.3671 :1015 135 xl0'3 87.5x10—é' 'i 1.6x103 2.56 x106 M.096 :109 6.55 x1012 0 0 o 1 5.825x103 27.17 :106 118.9 :109 o 0 (11) ()<) 1 13.925 x103 0.139961109 25.365 x10"9 22.057x10‘9 o 0 0 36.383l4x103 0.9M0831109 ..95.7105x10"9 .113.122x10'9 5) 0 0 221.85u7x103 6.83863x109 -5ou.u662x10‘9 -h67.570x10f2j 'fi 1.0x103 0 l (12) O 0 O 0 0 O b '1 1.6x103 0 1 (13) 0 0 0 o £9 0 i o 1 0 (1h) 0 O O O r0 0 0 no 614 65 2711.6253uo3 1160.29x10 (15) 85 611142531103 m37.05x10 95 857.375x103 7737.81x10 120 1728 70 6 2.56 x10 1+.096x109 6.55 x1012 0 o 5.825x106 27.17 :103 118.9 :109 o o 1 13.925x103 139.6 :106 25.365 :10'9 22.057x10'9 o 1 0.02602210b --2.6396x10"12 -3.109x1o'12 0 0 1.06597x106 79.111118x10'12 222.176x10-li 2.56 x106 n.096x1o9 0 -0.u86 110"3 -1.365 :10'3" 5.825x106 27.17 x106 0 ~8.828 xlO'b.~-2u.782 110‘s 1 13.925x103 0 15.000 110'9 -7.o39 1:10"9 0 1 0 -u.563 x10"12 ~8.532 110‘"12 0 0 1 0.07u2u7x10'15 0.208u26x10'1_j 0 o 36u.899 1:10"3 : h57.889 x10”3 ‘ 0 0 -3u2.3h8 x10-6 411111.022 3:10"b o o 78.5u0 :10‘9 111.769 1:10"9 1 0 -h.563 xio“12 -8.532 :10"12 0 1 0.07u2u7x10'15 0.208u26x10'15 d bib First Approximation for Ebb = 200 VOItO with BL = #000 ohms 8 .- 6 0.1638ux1012 0.000262 :101 0 n.902 x1012 0.0207119x1018 o 2 8 1 l 30 0.63025 :1018 M5 1103 102.“ x10 32.058 x10 0.23162 x10 69.983 no12 358.3181 .1012 5.515978 :1018 121 O‘C‘sO‘O‘ x103 2u883.2 xlO (16) (17) (18) (19) H1 000 <3 c: r3 '23 O 'c> hm H3 <3 <3 ca 1.6 1:103 2.56:10" 00011096311012 0000006551110“ (1 170.625x103 993.9 :106 M.b356 x1012 0.020286 :1018 o 1178.125x103 M219.k5x106 31.7098 x1012 0.23106 :1018 3o 705.375x1o3 714914.6121106 69.5939 :1012 0.62963 x1015 u5 1536 x103 214575.80x1o6 357.8266 :1012 5.51519 :1018 85 1.6x103 2.56 :106 n.096 :109 6.55 :1012 o - 1 5.825 :103 0.027172109 0.000118911015 o 0 1.u3hux106 18.7191 :109 0.17u25 :1015 30:10“3 0 3.3858x1o6 5o.u289 x109 0.5u573 :1015 #5110’3 0 15.6286x1o6 316.0935 :109 5.3326 :1015 u5xio'2 1.6:103 2.56 x106 1 5.825x103 "o 1 0 0 o 0 1.6x103 2.56 x106 1 5.825x103 0 1 0 0 0 0 113-1‘Kklo3 3-“3‘403x109 -281.672x10'a 71 9 b 118.9 x109 0 3 0.121148x109 20.915210“ 2 u.096x10 6.55 x101 0 '7 27.17 x10 13.05 x10 9 6.2uux103 0.13u42x109 -25.81hx10'9 n.0962109 6.55 x1012 0 — 27.17 :106 118.9 :109 0 13.05 x103 121.u8 x106 20.915x10”9 1 0.0215x106 .u.13l+x10'12 o ' 1.0230x106 181.71ux10'ti [1 1.6x103 2.56 x106 11.096x109 o 0 1 5.825x1o3 27.17 x106 0 (20) 0 0 1 13.05 x103 0 0 0 0 1 0 £3 (1 o o 1 F1 0 0 0 0 1116.514 3110's" 0 (21) To -'MO 6“ (22) 100 1000 110 1331 .130 2197 [1 1.6 (23) 0 8’40 0 1155 7.0 1989 0 000 0 rd 0 0 1 0 0 0 4105.69 :10" 6 103.113 x10"9 ~12 -7.952 x10 0.1776x10-15J -1.163 x10”3 .- -21.117 :10’6 .0.660 110'9 -7.952 xio"12 -15 0.1776110 J 6Y6 First Approximation for Ebb = 200 volts with BL = 1000 ohms x103 x103 x103 16105.1x106 19u.8717 x10 2.3579 110 1:103 x103 0 170.625x103 11103 971114. x10 102.16.106 65 274.625x103 1160.3:10b u.902 10000 3:106 100 6 37129. 3x10 027.1852 110 2.56m6 993.9 :106 b.6358 6 99.5908 x103 15823.5 x10b 19u.421 :103 36796 :106 626.953 0.16381’11101 0.0011096x1012 0.000006551no1 2 18 0 .000262 :10 O 8 x1012 0.0207119x101 o 1? 18 110 1 - 110 107 12 18 160 12 18 . 8 x1012 0.020286 :10” 12 18 110 10.60u5 x10 160_ 6 o 72 0.999365 x10 107 :1012 2.21985 x1018 160 12 18 110 10.8036 110 160 (2“) (25) (26) (37) F“ O '000 ...TI 'o H1 VO 1.6x103 2.56 x10 1 0 0 0 1 3 . bxlO .01103 6 11.096 1109 5.825 :103 0.027171109 0.851 :106 76.7676 x109 9.095625x106 163.0Ho x109 25.210575x106 572.912 x109 1 0.3671 x1015 160x10'2J 2.56 x106 h.096x1o9 6.55 x1012 0 ‘- 5.825x103 27.17 :106 118.9 :109 o 1 15.825x103 0.185ux109 22.057x10'9 0 19.1022103 0.53352106 ..ho.622x1o‘9 0 173.955x103 5.6931x1o6 -396.070xlO-?J 2.56 x106 b.096x109 6.55 x1012 0 7 5.825x1ob 27.17 x106 118.9 x109 0 1 15.825x103 185.u x106 22.057110"9 0 1 0.027929110b ...2.127x10"12 0 o 0.83u71 x10b -26.068x10'1€j 2.56 x10" 14.0961109 0 0.205 x10'3‘ 5.825x1o3 27.17 x106 0 3.713 :10”6 1 15.825x103 0 27.8u7 x10”9 0 1 o -1.255 :10"12 0 0 1 -0.03123x10‘15_1 73 6.55 x1012 0 ' 0.0001189x1015 0 0.9993u5 :1015 107110'3 2.21985 :1015 160x10'3 7h 267.3u7 x10‘3'7 6 H1 O O O O 0 1 0 0 o -2uo.082 x10- h7.707 x10"9 (28) 0 0 1 0 0 0 o 0 1 0 -1.255 x10"12 . -15 0 0 0 0 1 -0.0312x10 L- .J In order to save space and avoid needless repetition if a matrix, [A], is repeated entirely in another matrix, [B], then F b.“ be m= m- L be where the column of b is the last column( s) of matrix B . Thus for example if abcd andif bed 517:: e f g h [89 = e f g h y ,j k m n J k m n 2 Then ['13] could be written [A] y 2 (Note: This is a particular notation invented by the author for con- venience on his part. There is no justification for this in any laws of matrix algebra but Just is a manner of notation.) (29) (30) (31) (32) 6J5 l‘irst Approximation fer Ebb = 300 volts with B; = 20000 ohms r—50 125 3:103 312.51106 0.78125x1012 0.00195311018 0 .- 80 512 2:103 3276.8x106 20.97152x1o12 0.13821 :1015 o 98 981.192x103 9039.2x106 86.8125511012 0.83375 x1018 9.2 110 1331 :103 16105.1:106 198.87171x1012 2.35795 :1018 18.5 23° 2197 x103 37129.3x10b 627.88517x1o12 10.60850 :1018 18.7i F1 2.5 x103 6.25:106 0.015625x1o12 0.00003906x1018 o _ 0 312 :103 2776.8 :106 19.72152 :1012 0.1310852 :1018 o 0 696.192x1o3 8826.7 :106 85.28130 :1012 0.8299221 :1018 9.2 o 1056 :103 15817.6 :106 193.15295 x1012 2.353653 x1018 18.5 L0 1872 x103 36316.8 x106 625.85392 x1012 10.59982 x1018 18.7 '1 2.5::103 6.25 x106 15.625 x109 39.06 x1012 0 - o 1 8.9 x103 0.06321x1o9 0.000820x1015 o 0 0 2.23059x106 81.2768 x109 0.53752 x1015 9.2110“3 0 0 6.0192 x10b 126.8032 x109 1.91013 x1015 18.5110"3 L9 0 19.6560 no6 507.1288 :109 9.81318 :1015 18.7:10'31 '1 2.5x103 6.25:10b 15.625x109 39.06 x1012 0 - 0 1 8.9 :106 63.21 :106 820.0 x109 0 0 o 1 18.505x103 0.2809sxio9 8.1285x10‘9 0 0 0 15.018x103 0.85962x109 .10.3262::10"9 .9 0 0 183.391x103 5.07688x109 -66.3712xio'9 75 ‘ (33) (3“) (39) (36) 0.75 First Approximation for Ebb = 200 volts and 800 volts with 3L = 20000 onms f 0 O (29) 5.6 9-7 9.7 O ' 0 o l o 1.2 (37» (30) 5.6 12.5 9-7- 19-5 9.7 194:1 '1 2.51103 6.25::10"15.625x109 39.06 1101‘2 0 0 1 8.9 1106 63.21 x106 820 1:109 0 o 0 1 18.505x103 280.98 :106 4.1245x10'9 0 0 0 1 0.03060x106 -0.6876x10'12 0 0 0 o 0.68872x106 32.225 x10~12 '1 2.521103 6.25x106 15.0251109 0 -1.828 1104' 0 1 8.9 .103 63.211106 0 -19.656 x10'6 0 0 1 18.505x103 0 4.1538110"9 0 0 0 1 0 -2.ll97xlO"12 L0 0 o 0 1 0.0408110-15J F1 0 0 0 0 258.612 x10-3‘- 0 1 0 0 0 -171.107 610"6 0 0 1 0 0 32.0710110-9 0 0 0 1 0 -2.1197x10"12 _0 0 0 0 1 0.03468110-15; 76 (35) (NO) (#2) (1‘3) 77 - 0 o - F 0 0 _ 0 0.0038x10"3 0 3.8 110'6 (31) 5.6::10”3 9.8 x10"3 (39) (32) 2.8286x10'9 8.3932169 9.7110"3 15.287 110'3 .8.9182x10"9-11.155x10‘9 L 9.7xlO~J 12.886 xlo.?‘ L ~38.036 x10'9-73.863x10'?‘ r“ 0 0 - ( --0.5195x10"3 -1.851 x10'3'7 (33) 2.8286x10‘9 8.393 x10'9 (81) (38) ..0.7768x10"9 -7.0295x10'9 «0.32751110"12 -0.7828x10'12 “0.7385110"12 -2.1932x10'12 8.9285x10'12 32.687 x10"12 0.013 :10'15 0.0878110"15 L _ L 3 '1 0 0 0 0 113.928 x10'3 263.00 x10-3- 0 1 0 0 0 -73.229 110"6 -176.12 1:10"6 0 0 1 0 0 12.817 x10"9 33.556 1:10"9 0 o 0 1 0 8043853110"12 -2.1932x10”12 L0 0 0 0 1 0.0133x10~15 0.0“?“x10gléi 6J5 First Approximation for Ebb = 200 volts with BL = 10000 and 80000 ohms '50 125 x103 312.5x106 0.78125x1012 0.001953x1018 0 0 ‘ 18 100 1000 110 1331 130 2197 k 12 88 681.872x103 5277.3:106 80.8678 x10 6 12 18 x103 10000 110 100 x10 :103 16105.1:106 198.8717 :10 x103 37129.3x10b 627.88517x1012 10.60850 :1018 19.1 8.95J 12 0.316N8 x10 0 O 1.0 :10 10.8 3.8 . 2.35795 :1018 18.6 8.95 (“8) (“5) (“6) (“7) (’85) [1 O 83 o 1-1 OO '00:: ..J 'c3000 78 2.5 x103 6.25x106 0.015625x1012 0.00003906x1015 0 0 7 861.872x103 8727.3 x106 39.8928 x1012 0.31308272x1018 0 o 750 no3 9375 :106 98.8375 :1012 0.996098 x1016 10.8 3.8 1056 x103 15817.6 no6 193.1530 :1012 2.35365 x1018 18.6 8.95 1872 x103 36316.8 x10b 625.85392 x1012 10.59822 x1018 19.1 8.95l 2.521103 6.25 :106 15.625 :109 39.06 x1012 0 0 ‘ 1 10.288 :103 0.8558x109 0.0006788x1015 0 0 0 1.692 :106 38.2525x109 0.887298 :1015 10.8x10'3 3.8x10'3 0 8.5999x106 102.7895x109 1.637260 :1015 18.6x10'3 8.95110"3 0 17.180 x106 865.2881x109 9.328255 x1015 19.1210"3 1+.95x10'J 2.5x103 6.25 x106 15.625x109 39.06 x1012 0 0 7 1 10.288x103 85.58 :106 678.8 :109 0 0 0 1 20.288x103 0.2880 :109 6.383x10"9 2.286x10'9 0 0 9.66 x103 0.31289x109 40.761110"9 --5.381::10"9 0 0 118.2661103 8.3879 x109 -90.305x10'9 ~33.586xlo‘2) 2.5x103 6.25 x106 15.625x109 39.06 x1012 0 0 ‘ 1 10.288x103 85.58 x106 678.8 x109 0 0 0 1 20.288x103 288 x106 6.383x10‘9 2.286 x10'9 0 0 1 0.03235x10°'-1.118x10'12 ...0.5570::10"12 0 0 0 0.5620 :106 81.883x10~12 32.328 x10'lj 2.5x103 6.25 x106 15.625x109 0 -2.88 1:10"3 2.286 x10'3" 1 10.288x103 85.58 x106 0 -50.00 x10"6 ~39.008 110'“ 0 1 20.288x103 0 -18.883 110"9 -18.318 x10“9 0 0 1 0 -3.898 x10'12 -2.817 x10”12 0 0 0 1 0.0737110“5 0.0575x10'ig WV“) 1 . .- er '-" (“9) (50) (52) (58) .1 r1 0 0 0 0 511.96 x10"3 286.088 x10-3‘— 0 1 0 0 0 -328.00 x10-b ~186.767 1:10“6 0 0 1 0 0 55.970 110"9 38.616 110“9 0 0 0 1 0 -3.898 x10"12 -2.817 x10"12 _0 0 0 0 1 0073721045 0.0575x10'45_ 6J5 Second Approximation for Ebb== 200 volts with BL = 10000 and 80000 ohms ( 0 0 ' f o 0 '- 0 0 0 0 (29) 8.7 2.8 (51) (30) 8.7 2.8 18.6 8.95 18.6 8.95 _ 19.1 8.95_ 19-1 “-957 F 0 0 _ _ o 0 0 0 (32) 8.7x10-3 2.8 x10"3 (53) (33) 3.9001110"9 18.6x10"3 8.95110"3 -8.875x10;9 L 19.1x10”3 8.95x10'?J _ ~57.558x10-9 ' o 0 ‘ 0.625 110"3 o 0 6.716 110"6 (38) 3.900 x1o‘9 1.255 x10'9 (55) (35) 7 753 x10‘9 .0.3286x10"9 -0.1738x10'12 0.16869x10'12 «ll-0137810"12 5.186 x10”12 L. ..0.01599x10"15 79 a?” .r' 0 0 1.255x10"9 ~2. 608x10"9 -19. 718x10"? ~0.2918 x10‘3" ~3.1378 x10”b -0.5851 x10'9 -0.8020 :10"12 0.00787110'124 f" U1 0‘ V O O ...: O O (57) (58) (59) O F‘ O O O 182LMnf 21 92.61x102 28 219.52x10 _30 270.00x10 l. 1.96x102 0 51.85x10 0 107.25x10 o 168.68x10 NMNN 9 211.20x10 r1 1.961102 0 1 0 0 O 0 R: 0 82.5735 110"3 -85.5722 xio’b 8.7058 x10"9 0.16869x10"1 -15 1 -0.01599x10 6807 live Term.Approximation for 60.6088 110'”3 ‘ -39.0827 x10"b 6.8939 110"9 2 ~12 -0.u020 x10 0.00787110’1§J Ibe= 200 volts with BL ='NOOOO ohms 53. 782Nx10 “08.8101x10 2 976 5625200 2 1 u u 8 1721.0 no“ mu 2 2830.0 2 3.8816x10 327.7365x10 1 880.522 x10 5 13 2318.752 x10 21 u u 8 1613.835 x10“ 8 3.8816 x10 7.5 6.3700 x10 0.3 1.973“ x10 28.9 «3: $3M: 5.68678x10 79.8 8 188.9 9.69808x10 1.058 1108 18.01091108 61. 03511106 3“ “9291108 18.700 x10 3.8187 8 0‘ 0.07530x10 6.830 x108 90153 1108 3-7775 2.3885 x108 10.5371 6.881 x108 19.6387 30 x10 18.76 1938x10 038 110 088 x10 O\O‘O‘O\O‘ 968 :10 0.79828 :10 0.001876110 0.02066111012 12 0 x1012 2 10.57886 :101 12 19.6830 x10 12 12 0.76328 xlO x1012 2 1101 2 1101 1:108 0 0.018835x1010 0 2.186786x1o10 1.0 x10 16.505588x1010 8.75x1o 1.0 8.0 8.75; 1.0 8.0 8. 75 8.098666x10lo 8.0 1:10"2 -2 -2 80 .J r1 1.961102 3.88lbx10u 7.530 x106 18.76 x108 0 ' 0 1 6.37 x102 31.938 x10“ 188.35 1106 0 (60) 0 0 1 12.6197xlo2 1.10887x106 0.5067x1o'b 0 0 0 8.5877x102 1.83538x106 1.1388x10"b p 0 0 26.66001102 5.759951106 -0.162 11041) ’1 1.961102 3.8816x1ol‘ 7.530 1:106 18.76 1106 0 ‘ 0 1 6.3700x10" 31.938 :10“ 188.35 1106 0 (61) 0 0 1 12.61971102 110.887 :10“ 0.5067x10'b 0 0 0 1 21.872 x102 0.1332210'5 _0 0 o 0 0.03551210“ -3.713 x10“: ’1 1.9631102 3.8816x10b’ 7.530 x106 0 15.8331110'23 0 1 6.3700x10u 31.938 110“ 0 155.1188x10"” (62) 0 0 1 12.6197x102 0 116.8083x10‘b 0 0 0 1 0 22.5883x10‘5 L0 0 0 0 1 -l.0856xlO-1(:| ’1 0 0 0 0 -502.3856x10‘2‘ 1. 0 1 0 0 0 507.8815x10“ -168.5987x10"(’ 8 (63)0 0 1 0 O 0 0 0 1 0 22.5883x10‘ -10 A O 0 0 0 1 -1.0356x10 6507 Four Term Approximation for Ebb = 200 volts with 11L = 80000 ohms The four term solution is identical with the five term solution 81 from (57) to (60) with the fifth row and the fifth column omitted. (68) is the matrix after (60)- 71 1.961102 3.8816x10u 0 -1.003 x1o'2‘ 0 1 6.37 x102 0 -8.2536x10'“ (68) 0 0 1 0 -1.1782x10"6 ,0 0 0 1 0.1332x10'8J 71 0 0 0 -2.8158x10"2‘ 0 1 0 0 3.2261x10"1+ (65) 0 0 1 0 -1.1782x10'6 -8 0 0 0 1 0.1332x10 .1 6837 First Approximation for Ebb = 200 volts with IL = 80000 ohms 2 188.956.8110,4 6.1222x108 0 - 26 175.56x102 1188.1376x10u 80.3181x108 o 138.8929x108 2.05 18 58.32x10 (66) 28 219.52x102 1721.0 x10“ 2 29 283.89x10 2051.1189x10” 172.8988x106 8.5 2 u 8 3.28x10 10. 8976x10 0. 3801x10 0 o 91.32::102 915.200 :10“ 8 2 pl 8 71.u755x10 0 8 (67) 0 128.80x10 1827.0672x10 128.9701x10 2.05 _9 189.93x102 1786.6885x10u 162.6359x108 8.5_ 71 3.2831102 10.8976 :101‘ 38.01 x106 0 '— 0 1 10.0219 :102 0.7827x106 0 (68) 0 0 1.362865x10“ 28.1583x10b 2.05x10”2 _0 0 2.881010x10“ 85.2857x10b 8.5 x10“€7 (b9) (70) (71) 3.28x10 1 O 0 3.2818102 1 O O O 0 1 0 O O O 1 10.8976x10u 10.0219x102 1 0 10.u976110u 10.0219x102 l O ~36.802x10 ‘ 25.778x10’u -5. 782x10”" 0.8111110”S J 3 7 1 O 8.01x10 0 8.27x10 0 1 . 505110"6 —6 b 8 7.73m2 2 2.01x10 0.826x10 2- h -13.978x10‘ -32.169x10- -5.782:10"" ..J 0. 811x10“ 53 APPENDIX D CALCULATIONS FOR THE COMPARISON OF THE COEFFICIENTS 55 aaoo ohouom 660 mean: oonooz casewowmmooo 05 .«o dogma-So on» no.“ msoaadadoado a “Seams“: 1 e .1 e O u we 03' 3161mm? 6 2-3.5386 6+ 3-38868 6.. 2-28.0868 6+ 2-38.286 o a o g heon" 8?: anoiaommfiéu hiofimimooé 81o? mmoSdu oaofimmzmaé o a o ..NnOHs ama.m mnoaszamaw:.oa mioauzmmaoo.o minds mmeoo.on minds cameo.o o scam: A. “8138368- 8138 a??? one? manned: 813w $610+ one? Blaze: 228% o 9 81282; 812838.61 6122636666 81283386.. 21283805 :2»? a o .mnoaumo.o mnoaumm~o~.on minds 83606.0 wings Hoomo.on minds ~mo.o moaned seas: w fil e e I. e s I. e 11 meoauon om «seas New“ 0 wnoasmmaoo o maoaswhmmo o mioaxmmo o woasoom :oaswo o mundane om mnoaummaao a- maoasmmmao o mioauamaoo o- Nuoaxmmo o moanmma soasma o e \o II o e I 0 " x .muoaumh 6 muoasmmaos o mnoauaeooo o mnoasaoomo o mnoasamo 0 mod OH 608 o A. .r.~o mmma.m- ommmo.o m~o-.o- mhm.o oaoauemoH hoauoe moans. v.0: NmHH.m- Nommo.o Haaaa.ou mmm.o oaoasmzm ooasam moaum .03 $2.? $30.6 mmflodu 2:5 Sedan 6668 modem. a: Amy 2 Adv 86 .6noimm61 8.38386 ..ofimmmood- 21328866 0 c 128M388: a snoawoa.o: :noaummwm.au oaoaunaomo.o anoaxnosoa.oa coioflxommma.c o -msoawmo.mn uneaszmhm.aa macaxammao.o Naoaumemoa.ou mnoasmmoma.o o olefin m or- .suoas scam.” suoHsNMhmo.ou snoaseamma.o snoau wwm~.oa moan» 1:383.» 2103336.. 21606286 1.1283805. 1.38886 mofim .m.o~um6.o «inflammaon.o- ~-oas~:ooo.o mnoauaomno.on micasmmomo.o scan: windsom.om muoasomzm~.ou mnoaxmmaoo.o muoauwommo.on «nods mmmo.o :oaxw: mnoasmw.em muoawmmaao.au muoanznmao.o wnoaummaoo.o- muoaumaawo.o :oaxma -muoasmo.o unoaummaom.oa muoHuazooo.o annasaohmo.on micasmmomo.o Joan: J:.~: mmm~.m- ommno.o whoaa.o- omhm.o hoaxes Joaxoa n.82 amHH.ma aommo.c “Haaa.oa owmm.o eoasam :oaxm rm.ma m2m~.ou mamao.o mmm~o.oa mmaa.o 8688» John: whozom qo>u 688 660.8886: oonaoz oauoaummma.o- canoaummmaoo.o+ oauoasmmocoo.on odioanmmzooo.o+ oaaoawmmaooo.ou olefin mom.: enoawaommha.ou encaumsamoo.o anode mmoao.oa exoaummsmao.o r «scan mw:.~- muoaunmmmaa.o+ wioaw:~mooo.ou muons memoo.o: mnbau mo~oo.o o o a o “my moflm a o m. H o A8 «28 a. mods o modem 0 Aha NOHHN H1 J1 mods: moaxm Asa mocha a o m. o H o “.8 o o H 87 .8638..- 116682.92 Sammie- IN- 613888.01. 812888805- 81.38397 1.138386 maoflmzowJ muoHummHmoé- o s N IOHKN~OHO.O+ o 13838.0: H o c 19332.6- 4.383de 0 H o uoHsameoH.o N 1388816- 0 o H.‘ SC 6. BIBLIOGRAPHY Pipes, L. A.; Forced Oscillations of Hon Linear Circuits; Communi- cations and Electronics; September 1954; pp. 352-358. In, Y. H.; Circuit with Non Linear Inductance and Capacitance; Communications and Electronics; January 1955; pp. 619—626. van der 3131, H. J.; Theory and Operating Characteristics of the Thermionic Amplifier; Proceedings g£_the Institute 22.82912. Engineers; Volume VII, Number 2, April 1919; pp. 97-128 Caporale, Peter; A Nets on the Mathematical Theory of the Multi- 3‘: electrode Tube; Proceedings q£_the Institute g£_nadio Eggineers, Volume XVIII, number 9; September 1930; pp. 1593-1599. Blight, H. B.; Tables 22 Integrals and Other Mathematical Data; The Macmillan Company, 1987. 6 General Electric Company, Electronics Department; Electronic Tubes Receiving Types, Volume I. 6V6 Sept. 1951 6J5 June 1952 6SG7 Oct. 1952 6337 May 1986 ERRATA Page 1 - Line 1 several instead of serveral Page 7 - Line 2 subtract instead of subtact Page 9 - Line 3 tetrode instead of tretode Page 9 - Figure 2 tetrode instead of tretode Page 9 - Footnote ** tetrode instead of tretode Page 10 - Line 3 entirety instead of entirity Page 13 - Line 25 actually instead of acutally Page 36 - Line 7 variable instead of varible Page [.0 - Lines 1.1-12 variable instead of varible Page 42 - Line 12 equation instead of quation R0 1.." int. 0'?" “’5 0m