THE UNIVERSITY OF TEXAS BULLETIN 

No. 3114: April 8, 1931 
A METHOD OF CALCULATING THE 
PERFORMANCE OF 
VACUUM-TUBE CIRCUITS USED FOR THE 
PLATE DETECTION OF RADIO SIGNALS 

By 
J.P. WOODS 
ENGINEERING RESEARCH SERIES No. 28 
Bureau of Engineering Research 
Division of the Conservation and Development of the Natural 
Resources of Texas 


PUBLISHED BY 
THE UNIVERSITY OF TEXAS 
AUSTIN 

Publications of The University of Texas 
Publications Committees : 
GENERAL: 

FREDERIC DUNCALF  MRS. C. M. PERRY  
J. F. DOBIE  C.H. SLOVER  
J. L. HENDERSON  G. W. STUMBERG  
H.J. MULLER  A. P. WINSTON  
OFFICIAL:  
E. J. MATHEWS  KILLIS CAMPBELL  
C. F. ARROWOOD  C. D. SIMMONS  
E. C. H. BANTEL  BRYANT SMITH  

The University publishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue and the last two the position in the yearly series. (For example, No. 3101 is the first bulletin of the year 1931.) These bulletins comprise the official publica­tions of the University, publications on humanistic and scientific subjects, and bulletins issued from time to time by various divisions of the University. The following bureaus and divisions distribute bulletins issued by them; communications concerning bulletins in these fields should be addressed to The University of Texas, Austin, Texas, care of the bureau or division issuing the bulletin: Bureau of Business Research, Bureau of Economic Geology, Bureau of Engineering Research, Interscholastic League Bureau, and Division of Extension. Communications concerning all other publications of the University should be addressed to University Publications, The University of Texas, Austin. 
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THE UNIVERSITY OF TEXAS PRESS 
~ 


THE UNIVERSITY OF TEXAS BULLETIN 

No. 3114: April 8, 1931 
A METHOD OF CALCULATING THE 
PERFORMANCE OF 
VACUUM-TUBE CIRCUITS USED FOR THE 
PLATE DETECTION OF RADIO SIGNALS 

By 
J.P. WOODS 
ENGINEERING RESEARCH SERIES No. 28 
Bureau of Engineering Research 

Division of the Conservation and Development of the Natural 
Resources of Texas 

PUBLISHBD BY THB UNIVERSITY FOUR TIMBS A MONTH, AND BNTERED AS 
SBCOND·CLASS MATTER AT THE POSTOFFICE AT AUSTIN, TEXAS, 
UNDER THB ACT OF AUGUST 24. 1912 

The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preservation of a free govern· ment. 
Sam Houston 
Cultivated mind is the guardian ireniua of Democracy, and while guided and controlled by virtue, the noblest attribute of man. It is the only dictator that freemen acknowledge, and the only security which freemen desire. 
Mirabeau B. Lamar 
TABLE OF CONTENTS 
I. Introduction ----------------------------------------------------------------7 
II. The 	Standard Signal________
____
______________ _
__________________ _ 
8 
III. The Detection of Small Signals_____________________ __ 
_
____ 9 
IV. Ideal Detection ----------------------------------------------------------11 
V. A Limited Theory of Large Signal Detection________ 24 
VI. 	Extension to Include Pure Resistance in Grid and Plate Circuits --------------------------------------37 
VII. Power Series Analysis of the State Characteristic 41 
VIII. Analysis of Detection 	Performance of UX-201A Vacuum Tube ------------------------------------------------------44 
IX. Summary --------------------------------------------------------------------55 
X. Appendix --------------------------------------------------------------------65 
XI. Bibliography --------------------------------------------------------------72 

TABLE OF SYMBOLS 


ea-total voltage in the grid circuit. eo = e. +Ee. e.-signal voltage. Throughout this paper the signal is assumed to be the standard, 
e. = E (1 + m sin wLt) sin <uHt. Ee-de grid bias. E-amplitude of the carrier wave. m-per cent modulation. wL-27rfL, where fL is the audio frequency. wir-27rfH• where fH is the carrier frequency. eg-total voltage grid to filament. 
eg = eo -igr c • ig-total grid current. r e--external grid resistance. ep-total voltage plate to filament. 
eP = Ea -ipra. Ea-plate battery voltage. ip-total plate current. ra--external plate resistance. 

I. INTRODUCTION 
Distortion in the detection circuit.-As the radio art has progressed it has become more and more desirable for the receiving set to give reproduction of good quality and less important for it to pick up distant sending stations. At first, most distortion was thought to be due to amplifica­tion, and little attention was paid to the detector. How­ever, it is now recognized that the detection circuit does produce distortion, and the elimination of most of this dis­tortion is one of the chief advantages claimed for plate detection. 
Difficulty of making calculations for radio circuits.-The 
calculation of the performance of a radio receiving set, or of any portion of it, is immensely complicated because of the large number of variables and because of the lack of known laws to define the characteristics of some of the most important devices included in the set. As an example of the latter, it has been found necessary to express the relations among the grid and plate voltages and currents of a vacuum tube by means of empirical equations in the forms of power series; there are no satisfactory equations based upon theoretical considerations. Carson first em­ployed such empirical equations in setting up amplification and detection theory. Later, Lewellyn and Chaffee used them. 
These equations are usually limited to the square term for two reasons : first, the use of the higher powers makes the calculations very complex; second, for small signals an empirical equation ending with the square term represents the tube characteristic accurately enough. But when plate detection is used, the grid swing covers a large portion of the static characteristic, and the higher powers of the em­pirical equation must be employed. The result is that the 
This paper constitutes a dissertation to be submitted to the grad­uate faculty of The University of Texas in partial fulfilment of the requirements for the degree of Doctor of Philosophy. It is to be used as a doctor's dissertation with the permission of the director of the Bureau of Engineering Research. 
The University of Texas Bulletin 
calculation of the performance of a plate detection circuit becomes very complicated. 
Scope of this bulletin.-This bulletin gives a method of calculating plate detection performance under certain lim­iting conditions. Employing the Carson series method, formulas and tables are developed from which can be cal­culated the harmonics of the audio frequency plate current when the standard signal is impressed on the grid circuit. The fundamental output and the amount of distortion are calculated for a detection circuit employing a UX-201A vacuum tube, and the results obtained are compared quali­tatively with experimental results obtained by Ballantine. 
Acknowledgments.-This work was done in the Bureau of Engineering Research of The University of Texas. 
Professors Brown and Boner of the Physics Department and Professor Correll of the Electrical Engineering Depart­ment of The University of Texas have been kind enough to read this paper and have offered many helpful sugges­tions. 
II. THE STANDARD SIGNAL 
The radio signal emitted by a broadcasting station con­sists of a wave of high and constant frequency upon which are superimposed the comparatively low and varying fre­quencies of voice or music. The audible frequencies are thus transmitted as periodic variations in the amplitude of the high or radio frequency wave. 
Consider a vacuum tube circuit oscillating at some fre­quency wH, and suppose that the oscillating current is a sine wave of constant amplitude, 
Suppose further that I is a linear function of some voltage e. If this voltage e be made to take the form 
then i becomes 
Plate Detection of Radio Signals 
i = C(EK +EL sin wLt)sin wHt EL =CEK (1 +-sinwLt)sinwHt. EK 
Now the variable part of the voltage e (i.e., EL sin wLt) can be made to depend upon microphone current, and the oscil­lator output is then controlled by the microphone. The expression for i given above would be the output of the ideal oscillator when the microphone current is a sine wave. The signal radiated by the oscillator would be of the form: 
e. = E(l + m sin wLt)sin wHt (1) 
where Eis the amplitude of the carrier wave. This ampli­tude would depend upon the efficiency of radiation, distance from the oscillator, etc. The other symbols have the sig­nificance indicated in the preceding Table of Symbols. 
Actually, the waves of radio transmission and reception are far more complex than that represented by the above expression. This expression holds for the undistorted superposition of a single pure audio wave of unvarying in­tensity upon a pure carrier wave, whereas in the transmis­sion of speech or music many varying tones of varying in­tensities are transmitted simultaneously. Further, both sending and receiving apparatus cause distortion. 
However, Eq. (1) lends itself to mathematical manip­ulation. For that reason, in analyzing the performance of any piece of radio apparatus, it is convenient to assume that the input to the apparatus is represented by this equation. From results obtained using this simplified signal as a standard, it is possible to form an idea of what the per­formance will be when the input is more complex. 
III. DETECTION OF SMALL SIGNALS 
Assuming then, that the signal delivered to the grid cir­cuit of the detector tube is the standard signal, it is the function of this tube to demodulate the signal. If there is 
The University of Texas Bulletin 
to be detection, there must flow in the plate circuit of the tube a current of the form 
i = /(E,m)sin wLt. (2) 
There may be also currents of other frequencies, but, if speech or music is transmitted, there must flow current of this original audio frequency. This process of de-mod­ulating the signal may be accomplished in three ways: by grid detection; by plate detection; and by a combination of the two. 
The plate current of a vacuum tube is a complex function of the grid voltage, the plate voltage, and the cathode tem­perature. By direct current methods, it is possible to obtain curves showing the relation between plate current and grid and plate voltages, the cathode temperature being kept con­stant. The performance of the detector tube under sim­plified conditions can then be calculated using these curves as a basis.1 
In such calculation, the following conditions are as­sumed: first, that external impedances in grid and plate circuits are negligible; second, that the variation in grid voltage is small. This last condition means that only a small portion of the plate current vs. grid voltage curve is covered by the signal. Then, over this small portion, the curve can be accurately represented by an empirical equa­tion of the form 
ip = ao +ale. +aze.2 (3) 
where a0, a1, and a2 are constants depending upon the par­ticular tube, upon the grid polarizing voltage, and upon the de plate voltage. The tube input is taken to be the stand­ard signal 
e. = E (1 +m sin wLt) sin wHt, 
and this value of e. is substituted in the empirical equa­tion (3). When the resulting expression is expanded trigo­nometrically, there is obtained: 
1Carson, J. R.: "A Theoretical Study of the Three Element Vacuum Tube," Institute of Radio Engineer.~, Proc., v. 7, p. 187, 1919. 
a2E2 a2E2m2 
ip = a0 + --+ + a2E 2m sin wLt (4) 2 4 
a2E 2m2 m ---COS 2wLt + a1E [sin WHt + -COS ( WH -wL) t 4 2 
a2E 2 m2m 
+ -cos(wH + wdt] +--[-cos 2wHt--cos 2wHt 2 2 2 
m2 m2 
+ -COS 2 ( WH -WL) t + -COS 2 ( WH + WL) t 
4 4 
-m sin(2wH + wL)t-m sin(2wH -wL)t]. 

Eq. ( 4) shows that it is the curvature of the plate cur­rent-grid voltage curve which brings about de-modulation, and that the process of detection not only makes available the fundamental audio frequency fL, but also generates its harmonic frequency, 2/L. 
In practice, if the impressed signal is so large that Eq. 
(3) cannot approximate the portion of the ip -eg charac­teristic covered by the grid swing, there are generated other harmonics of the audio frequency besides the second. These harmonic frequencies cause distorted reproduction and are to be avoided. They vary with varying curve shapes and with varying points of operation on any given curve. This suggests the possible existence of such a characteristic curve as to give "ideal" detection; i.e., detection without the generation of harmonics of the audio frequency. 
IV. IDEAL DETECTION 
The linear form of the static characteristic.-It is gen­erally stated that a vacuum tube circuit having a static characteristic of the form shown in Figure 1 will give ideal detection when the tube is polarized to the break in the curve. 

FIGURE 1.-A linear static characteristic. 
Suppose the input is the standard signal, 
e. = E (1 +m sin wLt) sin wHt. 
This modulated wave is drawn in Fig. 2. The static char­acteristic of Fig. 1 gives linear rectification, and the plate current due to the input voltage e. is of the form shown in Fig. 3. To demonstrate that the tube is an ideal detector, it is necessary to find the Fourier series expression for the plate current wave of Fig. 3, and to show that this series contains terms of frequency h but none of frequency 2fL, 3/L, etc. 

FIGURE 2.-Form of grid voltage wave. 
Derivation of output equation.-If, using 0-0' as an axis, the equation of portion a,-b of the static characteristic shown in Fig. 1 is 
ip =Ke., (5) 
then the plate current is represented by the positive values of the following equation: 
iP = KE (1 + m sin wLt) sin wHt [Positive values only]. ( 6) 
Eq. (6) of course, gives both positive and negative values, and the problem is to obtain from it an equation which will give only positive lobes as indicated in Fig. 3. 

FIGURE 3.-Form of plate current wave resulting from linear rectification. 
Let the required equation be the Fourier series,2 
ip = b0 + a1 sin x + a2 sin 2x + a3 sin 3x + . . (7) + b1 cos x + b2 cos 2x + b3 cos 3x + 
where x = wLt. Rewriting Eq. (6), ip=KE(l +msinx)sinqx [Positive values only] (6') 
WH 
where Q=-. 
To find the coefficient of the zth sine harmonic, multiply Eq. (7) by sin zxdx and integrate. 
f 
21r. • d 
ip sin zx x = a • .,,.. (8) 
0 
Combining Eqs. (8) and (6'), 

a • .,,.= J2,,.JKE(l + m sin x)sin qxl sin zxdx. (9) 
o l Pos. values only J 
The next step is to perform the integration of Eq. (9) taking into account only the positive values of the expres­sion KE(l + m sin x)sin qx. 
WH 
2From this point on, the ratio -is assumed to be an integer. See 
Appendix A. 
The University of Texas Bulletin 
Consider the modulated wave 
E (1 +m sin x) sin qx. 

This wave crosses the axis at the points 
7r 27r 37r (2q-2)7r (2q-1)7r X=O;-;-;-; ., -----;27r. 
q q q q q 
7r 
In the interval 0--, the expression KE (1 +m sin x) sin qx q 
7r 211" is positive in value; and, in the interval ---, it is nega­
q q 
27r 37r tive. In the interval ---it is again positive; in the 
q q 
37r 47r interval ---, again negative; and so on. Therefore, if 
q q 
the integration indicated in Eq. (9) is performed over the 
7r 27r 37r region 0-27r considering only the intervals 0--; ---; 
q q q 
47r 57r 67r 77r ---; ---, etc., the result will be that only the posi­
q q q q 
tive values of the expression KE (1 +m sin x) sin qx will be taken into account. 
Eq. (9) can then be rewritten: 
f 
2ir/ q a,7r = KE (1 +m sin x) sin qx sin zxdx (10) 0 
f 
3ir/ q 
+ KE (1 +m sin x) sin qx sin zxdx 
2ir/ q 
f 
5ir/ q 
+ KE (1 +m sin x) sin qx sin zxdx 
4ir/ q 
+ 
I 
I 
f 
(2q-1)11/q 
+ 	KE(l + m sin x)sin qx sin zxdx 
(2q-2) ,,.;q 
Expanding Eq. (10) trigonometrically and integrating: 
L' f KE
a • .,,. = 	~ -[1 + m sin x] [cos ( q -z) x (11) L 
2 
-cos ( q + z) x] dx 


L'JKE 
m 
= ~ -[cos ( q -z) x + -sin ( q -z + 1) x L 2 2 
m 
--sin ( q -z -1) x -cos ( q + z) x 
2 
m 	m 
--sin (q + z + 1) x + -sin (q + z -1) x] dx 2 2 
L' KE sin (q -z) x m cos(q-(z-l))x 
=~ -[----­
L 2 q-z 2 q-(z-1) 
m cos ( q -(z + 1) ) x sin(q + z)x 
+-------­
2 q-(z +1) q+z 
m cos ( q + ( z + 1) ) x mcos(q+(z-1)) 
+--------------] 
2 q +<z +1) 2 q+(z-1) 
L' 
where ~ indicates the summation of the different values 
L 
resulting from the series of limits in Eq. (10). 
Referring to Eq. (11), there are two typical terms: 
L' 
~sin(q + g)x 	(12) 
L 
and 
L' 
~ cos(q +g)x 	(13) 
L 
where g may have the values z; -z; (z + 1); -(z + 1); 
(z-1); -(z-1) as indicated in the equation. Expand­ing the above typical terms, 
L' L' 
~sin(q + g)x= ~[sin qx cos gx +cos qx sin gx], (14) 
L  L  
and  
L '  L'  

~cos(q + g)x =~[cos qx cos gx-sin qx sin gx]. (15) 
L L 
L' 
Now the symbol ~ represents the summation of the dif­
L 
ferent values of the expression upon which it operates over the series of limits 
7r/ Q 371'/ Q 571'/ Q (2Q-l)7r/ Q ] ; ] ; ] ; ] 0 271'/ Q 471'/ Q (2Q-2)7r/ Q 
When these limits are substituted, any terms containing 
n'1T sin qx disappear because sin qx becomes sin q -, and the 
q 
sine of any integral multiple of '1T is zero. The term cos qx becomes (-1) when the upper limit is used, and ( + 1) when the lower limit is used. 
Substituting the above series of limits, Eqs. (14) and 
(15) become, 
L' 
~ sin ( q + g) x = [ (-sin '1Tg/ q -sin 0) + (16)
L 
(-sin 3'1Tg/q-sin 2'1Tg/q) + (-sin 5'1Tg/ q -sin 4'1Tg/ q) +etc.], 
and 
L' 
~cos ( q + g) x = [ (-cos '1Tg/ q -cos o) + ( 17) 
L 
(-cos 3'1Tg/ q -cos 2'1Tg/ q) etc.] . 
Rewriting Eqs. (16) and (17), 
L' 
~sin ( q + g) x = -[ sin '1Tg/ q + sin 2'1Tg/ q + ( 18) 
L 
(2q-1)7rg 2q'trg .... +sin +sin--], 
q q 
and 
L' 
~cos ( q + g) x = -[cos 1rg/ q + cos 27rg/ q + (19) 
(2q-1)7rg 2q1rg .... + cos-----+ cos--]. 
q q 
The right hand side of Eq. (18) is of the form 
-(sin y +sin 2y +sin 3y + .... +sin ny) 
where y = 7rg/ q, and n = 2q. 
It can be shown that the value of this series is zero pro­vided that ny is an integral multiple of 27r. In the above case, 
ny = 2q ( 'trg/ q) = 27rg 
where u=z; -z; (z+l); -(z + 1); (z-1); -(z-1). 
Now z is an integer; hence g is an integer; ny is an integral multiple of 211"; and the above series has zero value. Hence the typical term (12) is always zero. 
The right-hand side of Eq. (19) is of the form 
-(cosy+ cos 2y +cos 3y + .... +cos ny) 
where as before y = 7rg/ q, 
and n=2q. Since ny is an integral multiple of 211", it can be shown that the value of this series also is zero unless, 
y = 0 ; ± 211"; ± 47r; ± 671'"; etc. 
In these cases, the value of the series is -n. For y to have the above values, 
g = 0 ; + 2q; ± 4q; + 6q; etc. 
Therefore Eq. ( 19), and hence the typical term (13), has no value unless g is a member of the above series, and, in this case, the value is 
-n=-2q. 
Remembering that z, the order of the harmonic being solved for, is a positive integer, and using the relations 
between g and z, and using the above evaluation of the typical terms (12) and (13), Eq. (11) can be evaluated. Now Eq. (11) has no value unless g = 0 ; + 2q; ± 4q; ± 6q; etc. But 
g=z;-z; (z+l);-(z+l); (z-l);-(z-1). There is only one value of z (a positive integer) which will make g zero, and that value is z = 1. 
For g = 0, z = 1 and, from Eq. (11), m cos(q-O)x
a • .,,. =KE ~ [ sin (q -1) x 
2 L q-1 2 q-0 
sin(q + l)x

+ 	
m cos ( q -2) x 
2 q-2 q+l 


+ 	
m cos(q + 2)x m cos(q + O)x] 


2 q+2 2 q+O =KE [O-m (-2q) + _ _ m (-2q)]
0 0 + 0 2 2 (q-0) 2 (q+O) 
=KEM. There are three values of z which will make g = ± 2q. These values are :8 Z=2q-1, For g = + 2q :-z = 2q + 1, Z=2q. When z = 2q-1, Eq. (11) gives a • .,,.= KE [O-O+ m (-2q) -O+ m (-2q) +OJ 2 2 q -2q 2 q + 2q KEm 
3 
3Note that q is here treated as an integer. But the ratio of the two frequencies is not necessarily an integer. See Appendix A. 
When z= 2q +1, KEm 
az7T=---­3 

When z = 2q, az7T = 0. 
In the same way that the case g = ± 2q has been treated, there can be treated the cases g = -+-4q; ± 6q; -+-Sq; etc. 
For g = -+-4q:-
KEm When z = 4q -1, az'Tr = 
3.5 
KEm When z = 4q +1, az7T = ­
3.5 When z = 4q, az7T = 0. For g =-+-6q:-
KEm When z = 6q -1, a ,7T = 
5.7 
KEm When Z= 6q + 1, az7T= ­
5.7 When z = 6q, az7T = 0. But there remains one possibility to be considered. Sup­pose that z = q. Then Eq. (11) becomes 
az7T = ~ KE [sin (q -q) x m cos x (20) L 2 q-q 2 1 
m cos(-x) _s_i_n_2_q_x_ + m _c_o_s_(2_q_+_l_)_x
+----­
2 -1 2q 2 (2q +1) m cos ( 2q -1) x ] . 2 (2q-1) The foregoing evaluation of the terms (12) and (13) shows all the terms of the right-hand member of the above equa­tion are zero except the first which is of the indeterminate 
sin 0 form --. But if the substitution z = q is made before 0 the integration of Eq. (11) is performed, 
L'JKE m m 
a,rr = 	~ --[cos 0 +-sin x--sin(-x) 
L 2 2 2 

m 	m 
-cos 2qx--sin(2q + l)x +-sin(2q-l)x]dx 
2 	2 
L' KE m m sin 2qx 
=~--[x--cosx--cosx----­L 2 2 2 2q 

m (2q+l)x m (2q-1)]
+-cos --cos---­2 2q+l 2 (2q-1) 
KE'Tr 
2 
The sine terms of the Fourier series for the rectified cur­rent are therefore : 
KEm KE sin x +--sin qx (21) 'Tr 2 
KEm 	KEm 
+ 	sin(2q-l)x sin(2q + l)x 
371" 37r 

KEm 	KEm 
+ 	sin(4q-l)x-sin(4q + 1) x 
3.571" 3.571" 

KEm 	KEm 
+ 	sin (6q -1) x -sin(6q + 1) x 
5.771" 5.771" 

+. . 	.. 
To find the coefficients of the cosine terms of the Fourier series for the rectified plate current-see Eq. (7) : 
f 
211" 
b,'Tr = ip cos zxdx (22)
0 

z,,. 
_f 5KE (1 + m sin x) sinqx l cos zxdx -l Pos. values only j 0 
=~ J KE [l+msinx][sin(q+z)x L 2 
+ 	sin(q-z) x]dx 
L'JKE. m 
= 	~ --[sm(q + z)x--cos(q + z + l)x L 2 2 
+ 	m cos ( q + z -1) x + sin (q -z) x 
2 
m 	m 
--cos ( q -z + 1) x + -cos ( q -z -1) x] dx 2 2 
=~KE [-cos(q+z)x _ m sin(q+z+l)x 
L 2 q-f-z 2 q-f-z+l 
+ 	m sin (q + z -1) x cos(q-z)x 
2 q+z-1 q-z 
m sin (q -z + 1) x m sin (q -z -1) x ]
-------+-	. 
2 q-z+l 2 q-z-1 
Again there are two typical terms of the forms (12) and 
(13) by means of which Eq. (22) can be evaluated in the same way that Eq. (11) was evaluated. This time there are two indeterminate cases: z = q + 1 and z = q -1. 
As before, these cases are handled by substituting before performing the integration. 
The cosine series of the equation for the rectified plate current is, therefore: 
KE -­- KEm  cos ( q +  1) x +  KEm  cos ( q ­ 1) x  (23)  
7r  2  2  
2KE --­cos 2qx ­ 2KE  cos 4 qx  
37r  3.57!'  
2KE ---cos6qx ­ ..••  
5.77!'  

Combining Eqs. (21) and (23) the complete Fourier series for the rectified plate current of an ideal detector is: 
KE KEm KE 
ip = --+ ---Sin WLt + --Sin wut (24) 
7r 7r 2 
KEm KEm 
2 2 
KEm 1 1 + ---[ -sin (2wH -wd t --sin (2wH + wL) t 3 3 
1 1 
+-sin(4wH-"'L)t--sin(4wH + wdt 
3.5 3.5 
1 1 + -sin (6wH -wd t --sin (6wH + wd t + etc. ] 
5.7 5.7 
2KE 1 1 
+ [ --COS 2oiHt --COS 4wHt 
7r 
3 3.5 
1 
--cos 6,,,Ht -etc. ] . 
5.7 
As a check on the above, the modulation may be made zero, and the result must be the equation of a rectified sine wave for half wave rectification. If the sine wave, 
ip = KE sin wnt 
is so rectified, the equation for the rectified current is: 
KE KE 2KE 1 ip = --+ --sin wnt---(-cos 2wHt (25) 
7r 2 7r 3 
1 
+ -COS 4wnt + . . ) ; 
3.5 
and, if mis made zero in Eq. (24), Eq. (25) is obtained. Conclusions.-Concerning ideal detection of the standard signal, these conclusions may be drawn: 

(1) 
A characteristic curve of the form of Fig. 1 does produce ideal or distortionless detection for the standard signal. 

(2) 
The rectified modulated wave differs from the recti­fied sine wave by a term of the frequency of modulation fL, and by terms the frequencies of which are the sum and dif­ference frequencies of h and multiples of the carrier fre­quency fH· 

(3) 
The output of the fundamental depends upon E and not upon E2 ; and it depends upon m and not upon m 2• Approximation of ideal detection.-The static character­istic for ideal detection, the broken line curve of Fig. 1, is impossible of attainment in practice. No tube nor circuit has been devised which has a characteristic bending so 


sharply. A typical curve for a modern detector tube is shown in Fig. 4. 
a..,.__ __:.;C;.,:;_-+"'-O-_~,...._----n:"'.' Neg. L..-Ee 0os ' 6rid Volts FIGURE 4.-Typical vacuum tube static characteristic. 
To approximate ideal operation as closely as possible, the tube is polarized to the knee of the curve. The grid voltage then swings about 0-0' as an axis, and, to minimize the effect of the curved portion c-c' of the characteristic, the grid swing is made sufficiently large to include the straight line portions a-c and c'-b. The operating range a-c­c'-b of the characteristic of Fig. 4 is thus made to resemble the characteristic of Fig. 1 except for the fact that the two straight lines a-c and c'-b of Fig. 4 are connected by a curve c-c' instead of intersecting at a point as in Fig. 1. 
If the signal impressed upon the grid of the detector tube were very small, the operating range of the tube would lie wholly within the region o--c' and would not begin to ap.. proximate the broken line of Fig. 1. 
To find how closely large signal detection approximates ideal detection, it is necessary to solve for the plate current of the detector tube when a standard signal of large swing is impressed upon the grid. 
V. A LIMITED THEORY OF LARGE 
SIGNAL DETECTION 

The problem.-The method of solving for the audio fre­quency components of the plate current when the impressed signal is small has been outlined in Section III, the result being summarized in Eq. ( 4). Employing the static char­acteristics ig-eg in addition to the ip-ep characteristics and using the double Taylor series, the above method can be extended to the case where there are external impedances in the grid and plate circuits.4 
However, the limitation of the static characteristics to the square term as in Eq. (3), limits the above theory to small signals since only small portions of these curves can be accurately represented by equations ending with the square term. The entire static characteristic curve, using the axis of grid polarization as the zero axis, can be rep­resented by a power series that is more complete than Eq. (3) : 
(26) 
where the series is to be continued until the desired accu­
racy of representation of the curve is obtained. Taking 
many points on a curve and taking many parameters, it 
is possible to represent an entire tube curve accurately. 
Now, having an empirical equation such as (26), the 
problem is to solve for the direct and audio frequency cur­
rent output of a detector tube when a large signal is im­
pressed on the tube. 
4Chaffee, E. L. and Browning, G. H.: "Theoretical and Experimen­tal Investigations of Detection for Small Signals," Institute of Radio Engineers, Proc., v. 15, p. 113, 1927. 
Pl,ate Detection of Radio Signals 
Method of solution.-For the present, the following limi­tations are imposed: first, that no grid current flows; sec­ond, that the external impedance of the plate circuit is neg­ligible. The equation of the ip-eg curve is taken to be of the form (26), where the equation has been set up with the axis of grid polarization as its zero axis. The input to the tube is taken to be the standard signal. The audio frequency current output is to be expressed as a Fourier series with the audio frequency as the fundamental fre­quency of the series. 
For reference, the two known equations and the desired equation are written below: 
8
ip = ao +ale. +a2e.2+aaes+.... (26) 
e. = E (1 +m sin wLt) sin wHt. (1) ip = Co + b1 sin wLt +b2 sin 2wLt +b3 sin 3wLt + . . . . (27) 
+C1 COS wLt +C2 COS 2wLt +C3 COS 3wLt + .... 
+terms of high frequency. Eq. (26) is the known tube characteristic. Eq. (1) is the known input voltage. Eq. (27) is the desired form of the expression for the out­put current where the coefficients c0, b1, b2, c1, c2, • • etc. are to be determined. The general term of the right-hand member of Eq. (26) is anesn and, from this point on, the general term will be dealt with. If the value of e. as given by Eq. (1) is sub­stituted into Eq. (26) the general term becomes: 
~En(l + m sin WLt)n sinn wHt. (28) 
By means of the binomial theorem, this term (28) may be expanded, and there results the sub-general term 
n! 
(29) 
r!(n-r) ! 
where r has successively all the integral values, zero to n inclusive. 
It can be shown5 that the trigonometric term sinr wLt can be expressed as a finite Fourier series of the form: 
Sinr wd =Cir Sin WLt + C8r Sin 3wLt (30) 
+ C5r Sin 5wLt + . . . . + Crr sin ToiLt when r is an odd integer; and of the form 
Sinr WLt = dor + d2r COS 2wLt (31) 
+ d4r COS 4wLt + . . . . + drr COS TwLt when r is an even integer. Similarly, 
sinn WHt = Cin sin wHt (32) 
+ 
C3n Sin 3wHt + . . . . + Cnn Sin ncvHt when n is odd; 

Sinn WHt =don+ d2nCOS 2wHt (33) 

+ 
d4n COS 4wHt + . . . . + dnn COS nwHt when n is even. 


Substitute expansions (30), (31), (32), and (33) into the sub-general term (29) : If r is odd and n is odd, term (29) becomes n!mr 
(34) r!(n-r) ! 
. . . . + Crr Sin TwLt ] ( Cin Sin wHt 
+ C"n Sin 3wHt + . . . . + Cnu Sin nwHt ] . 
If r is even and n is odd, term (29) becomes n!mr 
(35) r!(n-r)! 
+ drr COS Tw1,t ] ( Cin Sin wHt + Can Sin 3wHt + . . . . + Cnn Sin nwHt ] . If r is odd and n is even, term (29) becomes n!mr anEn ( Cir sin wLt + C3 , Sin 3wLt + (36) r!(n-r) ! 
. . . . + Crr Sin TwLt ] ( don + 
d2n COS 2wHt + • . • • + dnn COS nwHt ] • 
ssee Appendix B. 
Plate Detection of Radio Signals 
If r is even and n is even, 
n!mr 
(37) 
r!(n-r) ! 
+drr COS rwLt ] [ don +dzn COS 2wHt + ....+dnn cos nwHt ]. There are, then, four possible expansions of the sub­genera! form (29) ; these are the expressions (34), (35), (36), and (37). Since n and rare restricted only to posi­tive integers and not to either odd or even integers, all of the above four forms will occur. The following three trigonometric formulas will be used in performing the multiplications indicated in expressions (34), (35), (36), and (37): 
sin x sin y = -112 cos ( x +y) +112 cos ( x -y) ; (38) cos x cosy= 112 cos(x + y) + 112 cos(x-y); (39) sin x cos y = l/2 sin (x +y) + 112 sin (x -y) . (40) Referring now to form (34), the performance of the indi­
cated multiplication will result in terms of the type: 
Cvn Cur Sin UwLt Sin VwHt where u and v designate the general terms of the two series. Eq. (38) shows this product to result in terms of sum and difference frequencies. The summation frequencies are of course higher than the carrier frequency, and the difference frequencies may safely be said to be much higher than the audio frequencies. The lowest difference frequency given by expression ( 34) is cos ( wH -rwd t. Referring to term 
(29), r is limited by n, and the lowest difference frequency will depend upon the highest value of n in the power series for the tube characteristic. This lowest difference fre­quency iS, therefore, COS (WH -nwd t. 
Consider a numerical case. Choosing a high value of audio frequency and a low value of radio frequency we have: 
WH = 500,000 271"; WL = 5,000 271"; cos (<.,H -nwd t = cos ( 500,000 -n 5,000) 27rt =cos (100 -n) 10,0007rt. 
Even in this extreme case, the difference frequency will not lie in the audio frequency range unless n approaches the value 
WH WH 
100; i.e., -. If, then, --n = z, the difference fre­wL WL 
quency is of the value z/L. But this is the frequency of the zth audio harmonic. Therefore, in the solution for the audio frequency and its harmonics, it is assumed that 
where z is the harmonic of the audio frequency being solved for. Since it develops that, even in the above numerical example, which is extreme, the effect of a49e49 upon the 49th harmonic of the audio frequency can be calculated without error, the above restriction is of no consequence. 
Form (34), then, produces no terms of audio frequency. The same is true of form (35). The conclusion may be drawn that the terms of the power series having odd expo­nents (see Eq. (26)) do not affect the audio frequency. 
The general forms (36) and (37) can be treated in the same way, and it will be found that audio frequency results only from the products 
(Cir sin wLt + Car sin 3wLt + . . . . + c,, sin rwLt) don (41) and 
(dor + d2r COS 2wLt + • • • • + dH COS rwLt) don (42) 
where n must be even, where, in expression ( 41), r has all odd integral values from zero to n, and where, in expres­sion ( 42), r has all even integral values from zero to n inclusive. 
From these two products, it is easy to pick out the coeffi­cient of any particular harmonic of the audio frequency; 
e. g., the coefficient of the third sine harmonic is d0n c30 and the coefficient of the zth sine harmonic is d0n Czr· But 
Plate Detection of Radio Signals 
the product (41) is only a typical form. The total coeffi­cient of sin zwLt is given by 
r = (n-1) n!mr ~ anE" don Crz 
r=z r! (n-r) ! 
where the summation is for the odd values of r only, and where c.r is a function of r in the manner shown in the Appendix B. From the above summation the following general formulas can be written. 
General formulas.-The general term ane•n of the power series for the static characteristic makes the following con­tribution to the coefficient of the zth sine harmonic of the audio frequency: 
nlmz2z (l2-z2) anE"f(n) [ z!(~-z) ! J f(z + 1)+ 3! f(z+ 3) (43) 
(l2 -z2) (32 -z2) 
+ f(z+5)
51 
(l2 -z2) (32 -z2) (52 -z2) + 7 ! f(z +7) + 
(l2-z2) (32-z2) .... [(z-2) 2-z2J 
+ z! f(2z)~ 
n!m••22z (l2-z2)f( )
+ (z + 2) ! [n-(z + 2)] ! 1f(z+3) + 3 ! z + 5
(l2 -z2) (32 -z2) 
+ 5 . 1 f(z +7) +.... 
+ (l2-z2)(32-z2) .... [(z-2)2-z2] f(2z+2)~ 
z ! 
n ! mz+•2z (l2 -z2)
+ 
(z + 4) ! [n-(z + 4)] ! 1 f (z + 5) + 3 ! I (z + 7) + .•.. 

(l2-z2) (32 -z2) .... [(z-2)2-z2] f(2z + 4)} 

+ 
z! +. 


+. 
n ! mu-12z (l2-z2) 
+--(n---1-)-,-) f (n) + 3 ! f (n + 2) + .... 

(l2-z2)(32-z2) .... [(z-2)2-z2] /( 1)}]
+ 1 	z+n-. 
z . 
In the above formula : n must be an even integer> z; z must be an odd integer; 

f(y) = 1.3.5.7 .... (y-1>_ 
2.4.6.8 ... . y 
The general term a.,e.0 of the power series for the static characteristic of the tube makes the following contribution to the coefficient of the zth cosine harmonic : 
n !mz (O-z2) a.,E0 /(n) [ z !(n-z) ! 2) /(z) + 2 ! /(z + 2) (44) 
(-z2) (22 -z2)
+ 4 ! f( z +4) 
...L (-z2) (22 -z2) (42 -z2) I ! /(z+6)+.
6 
+ (-z2) (22-z2) .. ·,. [(z-2)2-z2] f(2z)} 
z. 
n ! m zz+2 	(-z2) 
+ (z + 2) ! [n-(z + 2) fT 21f(z+ 2) + 2 ! f (z +4) 
+ 	.... + (-z2) (2• -z2) .... [ (z -2) 2 -z2] f (2z + 2) } z! 
+. 
+. 
n !mn (-z2) 
+ n ! 2) I (n) + ! 	f (n + 2) + ... .
2 
+(-z2) (22 -z2) .... [ (z -2) 2 -z2] f (z + n)}]. 
z ! 
31 
In the above formula: n must be an even integer > z; z must be an even integer; 

f(y) = 	1. 3. 5. 7 .... (y-1) 
2.4.6.8 ... . y 
Finally the contribution of the general term a,,e.0 to direct current is : 
n ! m2 n ! m• a,,Enf(n) [ 1 + 2 ! (n-2) ! f(2) + 4 ! (n-4) ! f(4) + 
(45) 

where n must be an even integer. The coefficients for direct current and for the fundamen­tal audio frequency and its low harmonics.-Using the above general formulas, it is possible to write:­
The direct current output of the tube: 
1 1 	(46)
ao+ a2E2 -[1 + (-)m2] 
2 2 
3 1 3 

+a4E•-[1+ 6(-)m2 + (-)m4 ] 
8 2 8 

5 1 3 5 
a6E 6
+ 	-[1 + 15(-)m2 + 15(-)m• + (-)m6] 
16 2 8 16 

35 1 3 5 35 
+ 	
a8E8 -[1 + 28(-)m2 + 70(-)m4 + 28(-)m6 + (--)m8] 128 2 8 16 128 

63 1 3 5 

+ 	
a,0E 10 -[1 + 45(-)m2 + 210(-)m4+ 210(-)m6 
256 2 8 16 



35 63
+ 45 (--)ms + (--)mio] 
128 256




+. ; 
+ a2E 2 [.5 + .25m2] 
+a,E4 [.375 +1.125m2 +.140625m4] 
+a6E 6 [.3125 +2.34375m2 +l.757812m4 +.90765625m6 ] 
+a8E 8 [.2734375 +3.828125m2 + 7.177734m4 + 2.392578m6 
+.07476807m8] 
+a10E10 
[.2460937 +5.537109m2 + 19.37988m4 + 16.14990m0 
+3.028107m8 + .06056213m10] 
+· 

The fundamental output (i.e., the coefficient of sin wLt) : 
1 1 
a2E 2-[4(-)m] (47)2 2 
3 1 3
+a,E4-[8(-)m + 8(-)m8 ] 
8 2 8 
5 1 3 5
+a6E 6-[12(-)m +40(-)m8 + 12(-)m5] 16 2 8 16 
35 1 3 5 35
+a8E 8-[16(-)m + 112(-)m3 +112(-)m5 + 16(--)m7] 
128 2 8 16 128 63 1 3 5 35 
E 10
+a10-[20(-)m + 240(-)m3 +504(-)m5 +240(--)m7 
256 2 8 16 128 
+20(~)m9]
256 

+ .... 
=~E2 [l.Om] 
+a4E" [1.5m +1.125m8] 
+a6E 6 [1.875m +4.6875m8 +1.171875m5] 
+a8E 8 [2.1875m +11.484375m8 +9.570312m5 + l.196289m1 ] 
+a10E 10 
[2.460937m + 22.14844m8 +38.75987m5 +16.14990m7 
+l.211243m9] 
+................... . 

The second harmonic output (coefficient of cos 2wLt) : 
E2
-a2[.25m2 ] (48) 
a4E4
-[1.125m2 + .1875m4 ] 
a6E6
-[2.34375m2 + 2.34375m4 + .1464844m6] 
a8E8 
a
-[3.828125m2 + 9.570312m4 + 3.588867m6 + .1196289m8] 
10E 10

-[5.537109m2 + 25.83984m4 + 24.22485m6 + 4.84497lm8 
+ .1009369m10] 
The third harmonic output (coefficient of sin 3wLt) : 
E4
-a4[.375m3 ] (49) 
E6
-a6[1.5625m8 + .5859375m~] 
a8E8 
a
-[3.828125m3 + 4.785156m" + .7177734m7] 10E 10
-[7.382812m8 + 19.37988m6 + 9.68994lm7 + .8074949m9 ] 
It will be found that those terms in expressions (46), (47), (48), and (49) which involve the parameter a2, check with the corresponding terms of Eq. ( 4), which was derived from small signal theory where parameters higher than a2 are not included. 
Comparison of effects of first ten parameters of the static characteristic.-lf in expressions (46), (47), (48), and (49), m is allowed to have the value 1.00, these expressions may be written:­
Coefficient of direct current (m = 1) : E 2 E4 E6 E8 E10
a0 + .75a2+ 1.64la4+ 4.512a6+ 13.75a8+ 44.40a10(50)
• 
Coefficient of sin wLt ( m = 1) : 
a2E2 E4 E6 E8 E10•
+ 2.625a4+ 7.734a0+ 24.438a8+ 80.730a10(51) 
Coefficient of cos 2wLt ( m = 1) : 
E2 E4 E8 E10
-.25a2-l.312a4-4.834a0E6 -17.107a8-60.548a10• (52) 
Coefficient of sin 3wLt ( m = 1) : 
(53) 
The coefficients as given by expressions (50), (51), (52), and (53) may be shown in tabular form, where the table holds for the special case m = 1.00. 
TABLE I 
m = 1.00 
Terms 
of Power de Sin wLt Cos 2wLt Sin 3wLt Cos 4wLt Series 
ao ··------·------------------------------1 0 0 0 0 
a.E2 ••...... ------..--.................. •75 1.00 -.25 0 0 a.,.E• ------------------------------------· 1.641 2.625 -1.312 -.3750 .04687 a.E6 -------------------------------------4.512 7.734 -4.834 -2.148 .6445 a,E' . ----------------------------------13.75 24.438 -17.107 -9.338 3.888 
a10E10 
----------·------------------------44.40 80.730 -60.548 -37.260 18.630 
To illustrate the use of the table, the second harmonic in­troduced by the sixth parameter is a,6E 6 (-4.834) cos 2wLt; the third harmonic introduced by the eighth parameter, is a,8E 8 ( -9.338) sin 3wLt; etc. 
It is to be remembered that the current flowing in the plate circuit can be obtained only by summing up the effects of all of the parameters. For example, if the modulation is 1.00, Table I gives the coefficient of the second harmonic as being 
E2
a
2(-.25) +a4E 4 (-1.312) +a,6E6 (-4.834) 
+ asE 8 (-17.107) + a10E10(-60.548) +. Moreover, any of the curve constants a2, a,4, a6 , etc. can be negative in sign so that the actual amount of second har­monic present may be very small or even zero. 
It is interesting to express the above table on a per cent basis; i.e., to express the de, second, third, and fourth har­
E2
monics due to a2in per cent of the fundamental due to the same term; to express the de, the second, third, and 
E
4
fourth harmonics due to a4in per cent of the funda­mental due to a4E 4; etc. 
TABLE II 
m =1.00; Percentage Basis 
Terms 
of de Sin wLt Cos 2wLt Sin3wLt Cos 4wLt 
Power 
Series o/o % % % % 
azE2 ---------------------------------------------· 75.0 100 25.0 0 0 
a,E4 . ---------------------------------------------62.5 100 50.0 14.3 1.8 asE6 ----------------------------------------------58.2 100 62.3 27.8 8.3 asE8 . ·-------------------------------------------56.0 100 71.0 38.2 15.9 
a 10E10 
.... ---------------------------------------55.7 100 75.0 46.0 23.1 
This table shows that the higher parameters introduce harmonics more and more strongly; thus, the second pa­rameter introduces the second harmonic and the fundamen­tal in the ratio 25/ 100, while for the 10th parameter this ratio is 75/ 100. 
This per cent basis table cannot be used to find the amount of the audio frequency plate current or to draw any conclusions as to the amount of distortion. The pre­vious Table I is to be used for such work. Table II is in­cluded solely to indicate the manner in which the effect of each of the higher parameters is distributed among the harmonics. 
Tables can be set up for the case, m = .50, in the same way that Tables I and II were constructed for the case m = 1.00. 
TABLE Ill 
m=.50 
Terms 
of 
Power de Sin wLt Cos 2wLt Sin3wLt Cos 4wLt 
Series 
ao ·--------··------------------1 0 0 0 0 
a2E2 ---------·--·-·---------... .5625 + .5 -.0625 0 0 
+ .8906 -.2824 -.04687 .0007211
~~ ·.::::::::::::::::::::::::::: 1 :~~~o +1.560 -.7348 -.2135 .009236 asE' ·----------------·---------1.7069 + 2 .836 -1.612 -.6335 .04238 
a10E10 
··------·· --· -···-----·· ··· 3.1058 +5.336 -3.396 -1.605 .5636 
TABLE IV 
m=.50; Percentage Basis 

Terms 
of de Sin wLt Cos 2wLt Sin3wLt Cos 4wLt 


Power 
Series o/o % % % % 
a•E" ----·----·----·----------······ 112.5 100 12.5 0 0 
a,E• .......... ·-----··--------. ·----·---------74.6 100 31.7 5.2 .08 
aoE"--·------------------------·-············ 64.6 100 47.0 13.7 .59 

a 5E8 
100 57.0 22.4 aioE10 -------·----· ----------------------------58.2 100 63.7 30.1 10.5 
------------------------------·--·--····· 60.2 1.55 
A comparison of Tables II and IV indicates the manner in which distortion varies with modulation. More complete tables of the above type will be found in Appendix C. 
Effect of grid pol,arization.-The above discussion was based on the condition that the equation of the ip-eg curve had as its zero axis the axis of polarization. Suppose that this empirical equation uses the axis of zero polarization (i.e., the normal zero axis of the ip-eg curve) as its zero 
axis. How is the addition of a polarizing voltage E. to be taken into account? Let the curve equation about the normal zero axis be 
2 3
ip = a0 +a1eg +a2eg+ a3eg+.... (26) 
Then, if the axis is to be shifted the distance E., 
i = a0 +a1 ( e. +E0) +a2 ( e. +E.) 2 +a3 ( e. +E.)3 . . . . ., (26') and from Eq. (26') can be written 
i = a0 ' +a/e. +a/e.2 + a8'e.8 +. (26") where 
2 3 4
a
0' = a0 +a1E 0 +a2E 0 +a8E0 +a4E0 +. a/= a1 +2a2E. +3a3E.2 +4a4E." +5a5E.4 +.. 
3 ! 4 ! 5 ! 
a/= a2 + --a3E c + --a4E c2 + --a5Ec3 + 
1!2! 2!2! 2!3! 4 ! 5 ! 6 ! 
2 3
as'= aa + --a4E c + --a5Ec+ --a6E c+ . 
3!1! 3!2! 3!3! 5 ! 6 ! 7 ! 
a/= a,+ --a5Ec + --a6E.2 + --a1E.3 + . 
4!1! 4!2! 4!3! 
as'= as+. 
aa' =. 
etc. 
In the event that the grid swing is to extend beyond the 
point where ip becomes zero, then the curve equation must 
be so determined as to include this fiat or zero portion of 


FIGURE 5.-Grid swing on zero portion of characteristic. 
the curve. In the extreme case where most of the grid swing is actually on the zero portion of the curve, the equa­tion must approximate the zero line e1-e2 of Fig. 5 and at least the portion of the curve e2-e3 • Exponents of the higher orders are necessary for this. 
Conclusions.-From Eqs. (41), (42), (43), and (44) and their derivations, it is possible to draw some interesting eonclusions: 
1. 	
Only the even exponent members of the power series produce de and the audible frequencies. 

2. 	
The nth term of the power series (i.e., a.ie.n) can introduce no harmonic higher than the nth but af­fects all harmonics lower than this. 

3. 	
The sine harmonics are all odd. 

4. 	
The cosine harmonics are all even. 

5. 	
As the signal strength increases, the importance of the higher terms of the power series increases. 

6. 	
As the percentage modulation increases, importance of the higher terms of power series increases. 

7. 	
As "n" increases, the ratio 
2nd Harmonic due to a.ie,n 



fundamental due to a.ie.n 
increases. 
The same is true when the third, fourth, etc., har­
monics, are compared to the fundamental. 

VI. 	EXTENSION TO INCLUDE PURE RESISTANCE LOADING IN GRID AND PLATE CIRCUITS 
Application of analysis of Section V.-The method of analysis given in Section V has been based upon the condi­tion that the relation between the voltage impressed upon the grid circuit and the current flowing in the plate circuit is defined by a single valued curve the empirical equation of which can be expressed in the form of a power series. This is true when the detector tube alone is considered. It is also true when the impedances combined with the tube are pure resistances. However, when the impedances are functions of frequency, the relation between plate current and varying grid voltage is defined, not by a single valued curve, but by a series of closed curves. 
In order to apply the previous analysis to the case where the tube is used in connection with pure resistances, it is only necessary to use as a basis the static characteristic of the detection circuit rather than the characteristic of the tube alone. The detection circuit may be set up, and its static characteristic can then be taken by the ordinary de methods. Or it is possible to use the static characteristics of the tube alone and, by graphical means, derive from them the static characteristic of the detection circuit for any particular circuit constants which may be chosen. 
r
Method of including external plate resistance.-Consider the vacuum tube circuit shown in Fig. 6 for the case where 0 is zero. 

c.. 
I J 

FIGURE 6.-Vacuum tube with external resistance in grid and plate circuits. 
The voltage on the plate is 
ep =EB -iprB. (54) 
As the grid voltage is varied, the plate current must satisfy the relation 
EB-ep ip=----(55) 
and it must also satisfy the relations defined by the static characteristic curves ip vs. ep. 
Therefore, if the straight line represented by Eq. (55) is plotted on the same graph as the ip-ep characteristics, then, for any given value of eg, both ip and ep are defined. For example, referring to Fig 7, the ip-eP characteristics for a UX-201A tube, if eg has the value of + 8 volts, EB the value of 100 volts, and r6 the value 4,820 ohms, then the tube is operating at the point A; ip is 10.5 milli-amperes; and eP is 49 volts. In this way, given fixed values of rB and EB, it is possible to plot i., as a function of eg. Curve Number II of Fig. 10 was determined in this way. 
But since re is zero, the ip-eg characteristic is the basis of the previous analysis. Therefore, external plate resist­ance has been taken into account, and it is possible to pro­ceed exactly as before. 
Me·thod of including external grid resistance.-Referring again to Fig. 6, if re is not zero and if the grid becomes positive so that grid current flows, then the actual grid voltage eg is not equal to the impressed voltage e0 , but 
(56) 
Further, ig is a function of eg and eP' It is necessary then, to use the ig-eg characteristics of the tube as well as the ip-ep set. The ig-eg characteristics for a UX-201A tube are given in Fig. 8. 
From Eq. (56), 
(57) 
re 
Eq. (57) defines ig as a family of straight lines intersect­ing the vertical axis of Fig. 8 at the points ig = e0/re and the horizontal axis at the points eg = e0• Such a family has been plotted for the values 
re= 21,200 ohms, 
eo = 4, 8, 12, . . . . volts. 
e
Now, for a given constant value of eo, it is possible to plot ep as a function of eg. Fig. 9 shows curves of this type for the above values of eo and re; e.g., the curve marked "2" in Fig. 9 defines eP as a function of eg for a grid resist­ance of 21,200 ohms and a constant impressed voltage of 0 = + 2 volts. 
If there is no resistance in the plate circuit then, 
ep =EB= 100 volts, 
and the operating line of the tube is Curve IV of Fig. 9. 
e
Consider point "16" on this curve. The impressed voltage 0 is 16 volts, the actual grid voltage eg is 6.7 volts, and, from Curve I of Fig. 10, the corresponding plate current ip is 17.2 milli-amperes. Thus point B' on Curve III of Fig. 10 is located. In this way the entire curve is deter­mined, and again there is a characteristic from which analysis can start. 
MethO'd of including external resistance in both grid and plate circuits.-When there is external resistance in both grid and plate circuits, then the operating line for Fig. 9 depends upon both EB and rB. Suppose 
EB= 100 volts, 
rB = 4,820 ohms. 
The previously determined Curve II of Fig. 10 takes into account the effect of rB. The family of curves in Fig. 9 takes into account the effect of re. Therefore, if Curve II of Fig. 10 is superposed on Fig. 9, both rB and re can be taken into account. To do this it is necessary to make the ordinate scales of the curves the same. Curve II of Fig. 10 can readily be plotted as ep vs. eg so as to correspond with the curves of Fig. 9. It then becomes the operating line, Curve I of Fig. 9. For point "16" on this line, 
eo = + 16 volts; 
ep = 53 volts ; 
eg = + 6.7 volts; 
ip = 9.7 milli-amperes. 
Thus point C' is determined (Curve IV, Fig. 10). Again, there has been found a static characteristic, including this time the effects of external resistances in both grid and plate circuits, and from it analysis can start. 
Experimental check.-The same direct current methods by means of which the characteristics of the vacuum tube alone are obtained, can be used to determine the charac­teristics of the vacuum tube circuit. As a check on the above graphical work, such characteristics have been taken for a UX-201A tube under the conditions noted in Fig. 10. The curves of this figure are experimental, and the points indicated by small circles have been determined graphically by the above methods. 
VII. POWER SERIES ANALYSIS OF STATIC 
CHARACTERISTIC 

Method of simultaneous equations.-The previous analy­sis of detector performance has used as a basis Eq. (26), an empirical relation between plate current and signal voltage expressed in the form of a power series. If the static characteristic represented by such a series is simple, or if the range covered is small, the parameters can be solved for by the method of simultaneous equations. For example, if four terms of the series are sufficient, it is possible to choose four points on the curve, solve four simultaneous equations, and obtain the constants a0, a11 a2, and a3 • The graph of the empirical equation passes through the four chosen points. 
However, the characteristics to be analyzed can seldom be represented over any large range by a four-term series. If a close fit is desired, and if the series is extended to say a10e10 in order to obtain this close fit, then it becomes neces­sary to solve eleven simultaneous equations. Further, though the empirical equation so found must pass through the eleven chosen points, in between these points its graph may not fit the original curve at all. 
If the method of least squares be used, this difficulty of divergence between the chosen points will be avoided. But there are still as many simultaneous equations to be solved as there are parameters to be evaluated. More­over, the work of raising a series of many numbers to the higher powers is very tedious. Finally, if the result­ing equation does not fit closely enough, the analysis can­not be extended, but must be repeated. 
Fourier series method.-The following method avoids the above difficulties. Suppose that the region E0-0-Eo' of the curve of Fig. 11 is to be represented by a power series, the zero axis of 
the equation to be the axis of grid swing, which is located at the center of the distance E 0-Eo'. There is to be ob­tained an equation of the form 
3
ip = a0 +a1e. +a2e.2 +a3e.+. where e. = eo-Ec. 
Express the variable e. as the following function of the arbitrary variable x : 
e. = E0 sin x. 	(58) 
Replot the curve of Fig. 11, replacing e. by the new vari­able x. There is obtained a periodic function of x, a single cycle of which is shown in Fig. 12. If this periodic func­tion is analyzed by the Fourier series method, there results the equation, 
ip = c0 +b1 sin x +b3 sin 3x +b0 sin 5x +. . . . (59) 
+c2 cos 2x +C4 cos 4x +C6 cos 6x + . . . . The even sine and the odd cosine terms of the series are 
excluded because the curve of Fig. 12 is symmetrical about a vertical axis through the point x = 90°. 
Now6 	sin 3x = 3 sin x -4 sin3 x ; 
sin 5x = 5 sin x -20 sin3 x +16 sin° x; 
etc.; 

and 
cos 2x = 1-2 sin2 x; 
cos 4x = 1 -8 sin2 x +8 sin4 x; 
etc. But, from Eq. (58), 
e, 
sin X=--. 	(60) 
Eo 
Therefore, 
3e. 4e.3 
sin 3x = ------; 

Eo Eo3 
6See Appendix C for more complete formulas. 
5e. 20e.3 16e. • sin5x=---+--­
JJ}o Jj}o3 Jj}o 5 
2e,2 
cos2x= 1---; 

Jj}o2 
8e.-,, 8e.• 
cos 4x = 1 ----+--­
And so, using the above relations, the Fourier series of Eq. (59) can readily be transformed into a power series expressing ip as a function of e. over the range JJ}o-0-JJ}o' of Fig. 11. It will be noted that, if the Fourier series ends with the angle llx, the power series ends with the term e.11 ; if the Fourier series ends with the angle 12x, the power series ends with the term e.12; and so on. 
This method of analysis has required that the zero axis of the empirical equation be located at the mid-point of the interval over which the equation holds. Of course, after the parameters have been found, the vertical axis of the equation can be shifted to any desired point in the chosen interval. Or, if it is desired to have the axis at either end of the chosen interval, the substitution 
e. = ± lJ}c cos x (61) 
may be used in place of Eq. (58). Since the grid swing of a vacuum tube is to either side of the axis of grid polar­ization, the substitution shown in Eq. (58) is necessary for vacuum tube work. 
Advantages of Fourier series method.-The above method reduces the power series analysis of a given curve to the Fourier series analysis. It thus avoids simultaneous equa­tions and the raising of numbers to the higher powers. Moreover-and here lies the chief advantage-it is not necessary to obtain the power series before the closeness of fit can be tested. When Eq. ( 59) has been obtained, the closeness of fit can easily be tested without any need of raising e. to a series of high exponents only to find that the series is not long enough. If it develops that Eq. (59) does not fit closely enough, the Fourier series can readily be extended, and none of the previous work must be dis­carded. When the Fourier series is long enough to be suffi­ciently accurate, it can be transformed into the power series, and the analysis is complete with no need for testing or extending the power series. 
VIII. 	ANALYSIS OF DETECTION PERFORMANCE OF UX-201A VACUUM TUBE 
In this section, there will be given the analysis of the detection performance of a circuit using a UX-201A vac­uum tube in combination with external resistances. The numerical and graphical work will be given step by step, and all curve sheets will be included in order that anyone who does not care to read the previous derivation can still follow its application. 
Sample calculations.-The static characteristics ip-ep and ig-eg furnish the experimental basis from which starts the analysis for detection performance. These character­istics for a UX-201A tube are given in Figs. 7 and 8. 
The following constants of the detector circuit will be assumed: 
Eb= 100 volts; 
Ee =-10 volts; 
rB = 6,250 ohms; 
re = 25,000 ohms. From the two s~ts of static characteristics and the circuit constants, there will next be graphically determined a curve defining ip as a function of e0 , the total voltage in the grid circuit. 
On Fig. 7, there is drawn from the point 
ii,=0 ep =EB= 100 volts the operating line II, intersecting the ip axis at the point 100 volts ip = ------= 16 milli-amperes. 6,250 ohms 
From this operating line II of Fig. 7, there is found a curve ep vs. eg in this way. Any value of eg is assumed, say +4 volts. This locates the point B on the line II, and the corre­sponding value of ep is 55 volts. Other values of eg are assumed, and there is obtained the curve A of Fig. 14. 
On Fig. 8 there is drawn a family of straight lines inter­secting the eg axis at the points 
eg = 4 ; 8 ; 12 ; . . volts 
and the ig axis at the points 
4 volts 8 volts 12 volts ig=------------; etc. 25,000 ohms 25,000 ohms 25,000 ohms 
From this family of straight lines, there is found a family of curves eP vs. 6g each curve holding for a particular value of eo. To do this, a particular value of eo is assumed, say 20 volts. The straight line for eo = 20 volts intersects the ig-eg characteristics in the points a, b, c, d, e, f, etc., and each of these points locates a point on the corresponding eP vs. eg curve. Thus: 
for a, ep = 0 eg = 3.7; 
for b, eP = 5 eg = 5.8; 
for c, eP = 10 eg = 6.5; 
etc. This determines the curve B on Fig. 14. Other values of eo are assumed, and the remaining curves of the figure are plotted. 
Now, in Fig. 14 the curve A intersects the family of curves of type B in the points a', b', c', d', etc., and each of these intersections defines one value of e0 , one value of eg, and one value of ep. But when values eg and eP are known, the static characteristics of the tube, Fig. 7, give the corresponding value of ip. There can now be plotted the circuit characteristic ip-e0 for the assumed circuit con­stants. Thus : 
for a', eo=2 ep= 65 eg = 1.2 ip = 5.4; for b', eo=4 ep= 61 eg = 2.3 ip = 6.0; for c', eo= 8 ep=55 eg= 3.9 ip = 7.1; etc. 
So are determined the points a', b', c', etc., on the curve ip-e0 of Fig. 11. 
The power series expression for this circuit character­istic is next determined. Since the grid bias E c is to be -10 volts, the axis of the empirical equation is to be the line 0-0' of Fig. 11. Then 
e
0 = E0 + e. = -10 volts+ e. 
where e. is the standard signal, E (1 + m sin wLt) sin wHt. The curve will be analyzed over a region extending a dis­tance of 50 volts to either side of the axis 0-0'. Replace the variable e. by the variable x where 
e. = 50 sin x 
and replot the curve of Fig. 11. There results the periodic curve of Fig. 12. By the ordinary method of Fourier analysis the follow­ing equation is found for the curve of Fig. 12: 
ip = c0 + b1sin x + b3 sin 3x + . . . . (62) 
+ c2 cos 2x + C4 cos 4x + . . . . 
where 

Co = + 4.320 bl = + 6.376 C2 = -1.805 bs =+ .9324 .7348
C4 
b5 =+ .3775 Ca .4722 b1 =+ .1443 Cs .3084 b9 =+ .0451 C10= -.2040 bu=+ .0389 C12= -.1373 bis= -.01511 C14= -.08133 bis= -.004667 C1a= -.05267 
As a check on the closeness of fit of the above equation : 
Value of  Value of  
Value  iP from  iP from  
of"' Curve + 90 --------­--­--------­--­11.40 + 45 ­-­---­----------­-­---­9.68 + 30 -----­--­--------------­8.35 + 15 --­-------­---------­--­5. 75 + 5 -------­---­-­----­----­2.05  --­-------------­-­----­-----------­-­---­-----­--------­---­----------­---------­-------­-----­-­------­------­-----­-- Equation 11.32 9.70 8.32 5.82 2.05  
0 --­---­--­--­-------­--­ .35  -­----­----------------­ .52  
- 5 -­-­-----­---­----­---­- .000 --­--­--­----------­----­ .115  
-15 -----­-­----­----­----­- .000 --------­--------------­ .089  
-30 ---­--------­-­--------­ .000 ­-­---­----------­----­-- .003  
-45 --------­--------------­ .000 ­--­-­--­----­---­-----­ .045  
-90 ----­-----­------------­ .000 ­------­----­--------­-­ .074  

The largest error occurs when x = 0, and is 1.57 per cent of the maximum value of ip. 
This Fourier series is now transformed into a power series by means of Table V of the Appendix C and the rela­tion 
e. 
sin x=-=e1 • 50 
Thus: 
b
1 sin x = b1e1 = 6.376e1 ; b3 sin 3x = b3 (3 sin x -4 sin3 x) = .9324 (3e1 -4e13) 
= 2.797e1 -3.730e13 ; 
c2 cos 2x = c2 (1-2 sin2 x) = -1.805 (1-2e1 2 ) 
2
= -1.805 + 3.610e1 ; 
c4 cos 4x = C4 (1 -8 sin2 x + 8 sin4 x) = 
-. 7348 (1 -8e/ + 8e/) 
2 4
= -.7348 + 5.878e1 -5.878e1 ; 
etc. 
This transformation from the Fourier series in x to the power series in e1 can be best written in tabular form. Then, when all of the Fourier series terms have been trans­formed, the total coefficient of any power of e1 is obtained by summing up all the coefficients in that particular column. 
COEFFICIENTS OF 
Term of Fourier e,' Series 
b1 sin x ----------6.376 bs sin 3x ----------2.797 -3.730 b5 
sin 5x .......... 1.887 -7.550 6.040 b, sin 7x .......... 1.010 -8.081 16.162 -9.235 ho sin 9x ·-·-······ .406 -5.412 19.483 -25.978 11.5411 b11 sin llx·.......... .428 -8.558 47.925 -109.542 109.592 -39.834 b13 sin 13x ---·-·--·· -.196 + 5.500 -44.000 +150.858 -251.430 +201.114 -61.891 bi:; sin 15x ·--------· -.070 + 2.613 -28.226 + 134.410 -328.557 + 430.111 -286.740 +76.464 
17.384 + 140.513 -458.899 591.421 -348.631 +76.464
Sum ·---·------·---· 12.638 -25..218 
COEFFICIENTS OF 
Term of 
Fourier e,• Serles Co--­------···----·-·­--· 4.320 C2 cos 2x -----­--1.805 c, cos 4x -·---·­-.735 Co cos 6x -­----­-.472 cs cos 8x ---­-·· -.308 C10 COS 10x ---···· -.204 C12 COS 12X ------­-.137 cu cos 14x ----­-­-.081 cu cos 16x ·-­---· -.052  e,• 3.610 5.878 8.500 9.869 10.200 9.886 7.970 6.792  e,• 5.878 -22.666 -49.344 -81.600 -115.332 -127.525 -141.577  e,• 15.110 78.950 228.480 492.083 765.153 1132.616  e,• -39.475 -261.120 -949.018 -2186.150 -4449.562  eil• 104.448 843.571 3206.354 9492.398  e 112 -281.190 -2331.894 -11218.289  eiu. 666.255 6903.562  
Sum ---­-----·  .525  62.655  -543.922  2712.392  -7885.325  13646.771  -13831.373  7569.817  
..,,.  

-1725.891 
-1725.891 
From the preceding two tables, the power series for the characteristic of Fig. 11 is: 
3
ip = .525 + 12.638e1 + 62.655e/ -25.218e1 (63) 
4 6 1
-543.922e1 + 17.384e1 5 +2,712.392e1 +140.513e1 
8 9 10
-7,885.325e1 -458.899e1 +13,646. 771e1 
+591.421e/1 -13,831.373e/ 2 -348.631e/8 
14 
+ 7,569.817 e1 +76.464e/5 -l, 725,89le/6 
• 
By substituting for e1 its value e./ 50, Eq. (63) can be expressed as a power series in e.. However, some work is avoided if Eq. (63) is left in the above form. 
The audio frequency output of the tube for various values of signal voltage can now be determined. Suppose that the fundamental output for 100 per cent modulation is de­sired. Refer to Table III of Appendix C. The first sub­table of Table III is headed "For m = 1.00," and one col­umn of this sub-table is headed "sin wLt." This particular eolumn is to be multiplied by the initial column of the sub­table, which contains the terms a0E0, a2E 2, a4E4, etc. This multiplication can be performed only when some particular value of E has been chosen, and the summation of the prod­ucts so obtained is the coefficient of sin wLt for that partic­ular value of E. Since Eq. (63) is in terms of e1 instead of e., E must be changed to E1 where E1 = E/ 50. For ex­ample, suppose that the carrier max, E, is taken to be 5 volts. Then E 1 = 5/ 50 = .1. Multiplying the two columns -0f the sub-table and summing up the products: 
a2E/ X 1=62.655 X (.1) 2 X 1 .626 
a4E 14 X 2.625 = -543.922 X (.1) 4 X 2.625 =-.142 
a0E 1 °X 7.734 = 2,712.39 X (.1) 0 X 7.734 .021 
a8E18 X 24.438 = -7,885.32 X (.1) 8 X 24.438 = -.000 
.505 
Therefore, for a carrier max of 5 volts and 100 per cent modulation, the fundamental of audio plate current is .505 sin wLt. 
In this way the fundamental output can be found for a series of values of E. The work is best tabulated. 
COEFFICIENTS OF SIN wLt FOR VARIOUS VALUES OF E 
AND FOR m=l.00 
E .5 1.0 1.5 2.5 5 10 15 20 25 Et .01 .02 .03 .05 .1 .2 .3 .4 .5 
a2E12 ---------------· .00626 .0251 .0564 .1566 .626 2.51 5.64 10.02 15.66 a,E,4 ••••••••••••••• -.00001 -.0002 -.0012 -.0089 -.142 -2.28 -11.57 -36.65 89.24 "6E1• ···········-··· .0003 .021 1.34 15.29 85.93 327.79 asE1"-··------·----· -.49 -12.64 726.29 -752.75 
10
a10E1-------------.11 6.50 115.52 1075.89 a12E112 --------------.02 -2 .02 -63.80 928.45
au,E114. _____ _______ 
.OS 19.45 442.35 a1eE118 ------·-------.02 -2.51 89.29 
Sum ........ .00625 .0249 .0552 .1480 .505 1.17 1.53 1.77 1.96 

The last line of the above table gives the max. value of the fundamental component of the plate current as a function of the max. value of the carrier wave. This function is plotted in Fig. 15. 
In the same way that the curve for the fundamental was calculated, curves can be calculated for de, and for the 
second, third, and fourth harmonics, using, of course, the 
proper columns in the sub-table. 
Let / 1 be the max. value of the fundamental, 12 be the 
max. value of the second harmonic, etc. Then per cent dis­
tortion is defined as 
Per cent of 


J---------x
100.
distortion = 
For example, reading values from the curves of Fig. 15 for 
a carrier max. of 10 volts: 

(.097) 2 + (.015) 2 + (.001) 2 
Per cent of / 
x 100=19.4.
distortion = \) 
(.505) 2 
The distortion curve has been so determined and is plotted 
on the figure. 
Fig. 15 then, gives the complete story of the performance 
of the detection circuit, for the tube and constants pre­
viously assumed, and for a constant value of modulation of 
100 per cent, and for varying signal strengths. 
Effect of modulation.-By repeating the above calculation 
using, in place of m = 1.00, the values m = .80; .60; .40; .20 
successively, the curves of Figs. 16, 17, and 18 are obtained. 
If 5 per cent be taken as the maximum allowable distortion, 
then the optimum value of the carrier max is 10 volts. 
Moreover, the modulation had best be limited to 80 per cent. 
For faithful reproduction, the relation between fundamen­
tal output and per cent modulation should be linear. Fig. 
18 shows this to be approximately so for E = 10 volts and 
for modulations up to 60 per cent. For higher values of 
the carrier max the linear relation is even more approxi­
mate. Note that for E = 10 volts, the loud sounds are 
softened, while for E = 25 volts, they are intensified. 
Comparison with experimental results.-Ballantine7 has 
obtained experimentally curves of the types shown in Figs. 
15, 16, 17, and 18. It is interesting to compare the two 

7Ballantine, S.: Detection at High Signal Voltages. Part !-"Plate 
Rectification with the High Vacuum Triode," Institute of Radio En­
gineers, Proc., v. 7, p. 1153, July, 1929. 

sets of curves qualitatively. Since different circuit con­stants were used in each case, they cannot be compared quantitatively. The distortion curves all show the saine pronounced dip at some critical value of signal strength. Further, this critical value of E is little affected by the percentage modulation so that the best operating point of the circuit is the same for all modulations. In both the calculated and the experimental curves, the best operating point is reached after grid current begins. Ballantine men­tions that the second harmonic, as obtained experimentally, decreases as E increases becoming zero near that value of E which makes distortion a minimum. At this point the second harmonic reverses phase and then increases as E increases. The second harmonic curve of Fig. 15 is in agreement with Ballantine's results. 
For low modulations, Ballantine obtained curves showing that the fundamental output first increased with E, reached a maximum at some value of E greater than that which gave minimum distortion, and then decreased sharply with a further increase in E. For low modulations, the calcu­lated curves of Fig. 16 show a slight decrease in funda­mental output when E becomes greater than 15 volts, but nothing like the decrease Ballantine obtained. The dif­ference is probably due to the difference in circuit constants. Refer to the curve for m = 1.00, Ee= -5 volts, which is plotted in Fig. 20. This curve has begun to decrease sharply. So has the curve for m = 1.00, Ee= -15 volts. Apparently the dip in the curve occurs when the polariza­tion point is shifted away from the knee of the static char­acteristic. 
Effect of grid bias.-AII of the above discussion has been based, of course, upon the initial assumptions of the circuit constants: 
Ep = 100 volts; 
E c = -10 volts; 
ra = 6,250 ohms; 
r, = 25,000 ohms. 
Suppose that Ee is varied. 
To take this variation into account, it is necessary to shift the axis of the power series of Eq. (63). The zero axis of Eq. (63) was Ee= -10 volts. If Ee= -5 volts, then the axis must be shifted by + 5 volts. 
Using Table IV of Appendix C: 
ao' = 2.340; 
a/= 26.511; 
a/= -176.409; 
ao' = 841.012; 
as' = -2726.284; 
a10 ' = 5815.134; 
a
1/ = -7675.381; a14 ' = 5613.444; a1a' = -1725.891. 
The odd parameters are not calculated because they do not affect detector performance. The arithmetical work of cal­culating the new parameters can be checked by the follow­ing relation. For any particular value of e., say eu, 
112 ip(for eu) + ip(for -eu)= 
a0' + az'eu2 + a/eu4 + a/eu6 +etc. 
Having the new set of power series parameters, the cal­culation of detection performance proceeds as before. 
The curves of Figs. 20 and 21 have been plotted to show the effect of variation of E e. With the exception of Ee, the circuit constants are the same as before 
EB= 100 volts; 
rB = 6,250 ohms; 
re = 25,000 ohms. 
The modulation is taken first as 1.00 then as .20, and Ee is allowed to take the values -15 volts, -10 volts, and -5 volts. 
The principal effect of moving the polarization point away from the knee of the characteristic (see Fig. 11), appears to be to increase the value of E which gives mini­mum distortion. Thus: 
for Ee= -10 volts (knee of the characteristic), mini­mum distortion is obtained for E = 11 volts; 
for E 0 = -5 volts, minimum distortion is obtained for E = 17.5 volts; for Ee= -15 volts, minimum distortion is obtained for E ~17.5 volts. The variation in Ee does not greatly affect the actual value of the minimum distortion; Fig. 21 indicates that, for 100 per cent modulation, it is possible to obtain as low as 8 per cent distortion for all three values of E 00 After the mini­mum point is passed, the curves for E 0 = -5 volts and Ee= -15 volts show the distortion to increase very rap­idly with E, much more rapidly than for the case Ee = -10 volts. 
Graphical determination of audio frequency plate cur­rent.-The following graphical determination of the wave shape of the detection plate current is based upon assump­tions which may not be justified. However, it appears to check with the analytical method, and it may be of some interest. 
Using the static characteristic of Fig. 11 and supposing the signal voltage to be a sine wave, the resulting plate current can be determined point by point. For example, the dotted curve marked (10) in Fig. 19 is the plate cur­rent for a signal voltage e. = 10 sin wHt. Similarly, the dotted curve marked (8) is the plate current for a signal voltage e. = 8 sin wHt ; etc. 
Now suppose that a modulated signal is impressed upon the detector circuit; 
e. = 5 (1 + m sin wLt) sin wHt. Suppose further that m = 1.00. Then Now, when sin wLt = 1, e. is approximately the sine wave 
es= 10 Sin wHt• 
This occurs when wLt = 90°. But the plate current for 
e. = 10 sin wHt is given by curve (10) of Fig. 19. 
In the same way, when wLt = 37°, 
Sin WLt = .60 
and approximately 
es = 5 (1 + .6) sin wHt = 8 sin wHt. 
The corresponding plate current is given by curve (8) of Fig. 19. In this manner, cycles of plate current can be found for various values of wLt. 
Now the detection plate current (de plus the low fre­quency) is the rectified current. Therefore, if the average value over a single high frequency cycle of plate current is plotted against the value of wLt corresponding to that cycle, the result should be a series of points outlining a cycle of the detection current. Fig. 19 shows a cycle of detection current obtained in this way for the above condi­tions: i.e., the static characteristic of Fig. 11, and 
E = 5 volts, Ee= -10 volts, 
m= 1.00. 
If this cycle of detection current is analyzed by the Fourier method, the results should check the values corre­sponding to the point E = 5 volts of Fig. 15, because the circuit constants and the values of E and m are the same in both cases. Comparing the results obtained by the two methods: 
Results from Graphical 
Results from Power Determination of Single 
Series Analysis Low Frequency Cycle 
de --------------------------------------------.92 _
_____
______________ .91 Coeff. sin wLt --------------------------.51 ________________________ .62 Coeff. cos 2wLt -------------------------.10 -------------------------.09 
IX. SUMMARY 
This paper has first reviewed briefly the standard radio signal and the method of calculating detection perform­ance when a standard signal of small amplitude is im­pressed on the grid circuit of the detector tube. The type of wave ordinarily taken to be the standard signal has been discussed, and the mathematical expression for this wave has been given. There has been outlined the manner of deriving the equation for the plate current of the detec­tor tube when the static characteristic of the tube is repre­sented by a power series ending with the square term, and when impedances external to the tube are negligible. 
The subject of ideal or distortionless detection has been taken up, and the detection circuit which has a broken straight line for its input voltage vs. plate current char­acteristic has been discussed. For the case where the cir­cuit constants are not functions of frequency, it has been shown that this type of detection circuit does give detec­tion without distortion and that the audio frequency output is a linear function of the modulation and of the ampli­tude of the carrier wave. 
Limiting the discussion to the case where the circuit con­stants are not functions of frequency, a method has been developed for calculating the direct and audio frequency current output of the detection circuit when the impressed signal is large. The general plan used has been that of Carson where the static characteristic of the circuit, for the particular constants chosen, is expressed in the form of a power series; but the power series has not been lim­ited to the square term. General formulas have been de­rived for calculating the coefficient of any harmonic of the audio frequency plate current, irrespective of the strength of the impressed signal. These formulas give the desired coefficient as the sum of a series of powers of the modula­tion and the maximum amplitude of the carrier wave, the terms of the series being multiplied by the parameters of the power series equation for the static characteristic. 
Certain conclusions have been drawn concerning the rela­tions between the equation for the static characteristic and the form of the audio frequency plate current. 
A graphical method of obtaining the detection circuit static characteristic from the detector tube characteristic has been given. This method holds for external resistances in either grid or plate circuits or in both, but the effect of reactance has not been included. 
The work of analyzing an experimental curve into an extended power series has been found difficult, and, for that reason, a method has been developed which is less laborious than the ordinary method of simultaneous equa­tions. This new method is more easily checked than the old, and the resulting power series does not give a graph oscillating badly about the experimental curve as does the simultaneous equation method. 
Finally, as an example of the application of the forego­ing analytical work, an analysis of the detection perform­ance of a UX-201A vacuum tube has been given. The analysis starts with the static characteristics of the tube, and there is derived the circuit characteristic for the con­stants given, the power series expression for this charac­teristic, and the value of the fundamental audio frequency plate current and the per cent distortion for various values of modulation and carrier amplitude. Each step is carried through numerically so that one can follow the actual cal­culation of the tube performance, whether or not he has followed the derivation of the basic formulas. 
The results obtained from the calculation have been given in a set of curves showing the amount of fundamental audio frequency plate current and the per cent distortion as func­tions of modulation and carrier amplitude. These calcu­lated curves are compared qualitatively with similar curves obtained experimentally by Ballantine. 
An appendix has been included giving in tabular form all information necessary for the calculation of detection performance by the above method. 



Positive ()rid Voltage· e~ Positive 6r!d Voltage -e~ 
E6 • 100 volts for all curves 
Curve I -r, = 4,820 ohms Curve11I-r8 • lf,200ohms 
Fl!J. 9. Curve II -r 8 ; 9,460 ohms CurveIY-r1• O ohms 

20~-----------------~~---~--~ 
---experi ­
mental curves 
o o -Points 

16 16 n .J2 
r-OS 
Fi9. /0. Vo/to9e in arid Circuit -eo 

FIGURE 9.-Voltage relations for UX-201A tube with external re­sistaRces in grid and plate circuits. 
FIGURE 10.-Comparison between static characteristics obtained ex­perimentally and those obtained graphically. 
Plate Detection of Radio Signals 


ng.11. 
-~ 
.:.. 
1:'.

.. 
'} 
~ 8 
~ 

-!: 6 
X in Degrees 

FIGURE 11.-Static characteristic for UX-201A tube with external resistances and its transformation for power series analysis. 
FIGURE 12.-Static characteristic plotted as a periodic function. 

Neg
Fig. /). Vo/taqe in 6rid Circuit -eo 
140 
~ 
•100 
~ ~ 
" 
..... 
~ 60 
~ 
~ ~ 

FIGURE 13.-Static characteristics for various external resistances in plate circuit. 
FIGURE 14.-Relations among plate voltage, grid voltage, and total voltage impressed on grid circuit. 

3.5 
.. 
i:: 
t 3.0 24 
~ 
t<>5 20 
li. 
Fi9. IS. 
Fig. /6. 
FIGURE 15.-Detection performance as a function of the carrier max. 
FIGURE 16.-Fundamental output as a function of carrier max. and of modulation. 


Fi9. l8. Moclulotion 
FIGURE 17.-Per cent distortion as a function of carrier max. and of modulation. 
FIGURE 18.-Fundamental output as a function of modulation. 
Plate Detection of Radio Signals 

FIGURE 19.-Graphical determination of wave form of audio fre­quency plate current. 
The University of Texas Bulletin 

r, ;15,250 ohms 
t 
a
:;: 
~ 
·~ ~ 
" ~ 
~ 

F1<; . 21. Carrier MoK in Volts 
FIGURE 20.-Effect of variation of grid bias upon fundamental output. FIGURE 21.-Effect of variation of grid bias upon per cent distortion. 
X. APPENDIX 
APPENDIX A 
Consider the waves of Fig. 2 and Fig. 3. For these fig­ures the ratio wH/ wL has been chosen to be exactly 5; that is, for every cycle of the audio frequency, there are an even five cycles of radio frequency. The modulated wave of Fig. 2 and the rectified modulated wave of Fig. 3, are there­fore periodic, and the period of the complex wave is the same as the period of the audio wave. This is true when­ever the ratio q = wH/"'L is an integer, and, in these cases, the equation of the complex wave can be assumed to be a Fourier series of fundamental period l/fL. 
When the ratio q is not an integer but is some fraction e/ d where e and dare integers, then the period of the com­plex wave is d times the period of the audio wave. 
The treatment in Section IV can be changed to care for this. In Eq. (7) 
ip = b0 +a1 sin x +a2 sin 2x + . . . . 
+b1 cos x +b2 cos 2x + . . . .' 
WLt 
let x =--. d 
Then Eq. (6') can be written 
ip = KE (1 +m sin dx) sin ex [Pos. values only]. 
And Eq. (10) becomes 
L'" 
a•rr = ~ KE (1 + m sin dx) sin ex sin zxdx, 
L" 
L"' 
where ~ represents the summation of the values of the 
L" 
integral resulting from the series of limits 
1r/d 311"/C 511"/ C (2c-1) 11"/C
] ;] ;] ; ....] . 
0 2Tr/C 411"/C (2c-2)7r/ C 
If the remainder of the analysis is carried on in the same way as in Section IV, the same final question results; i.e., Eq. (24). This means that when the ratio wH/ wL is an integer, Eq. (24) represents a wave which has the period l//L; but, when the ratio wH/wL is a fraction, then Eq. (24) represents a wave having a period d//L. 
APPENDIX B 
Sinr x Expressed as a Finite Fourier Series If r is an odd integer, 
sinr X = C1 r sin X + Cr sin 3X + C5 r sin 5x +. (1)
3 
+ Cpr sin px + . . . . + Crr sin rx, 
where 
(l2 p2) 

Cpr = 2p[f(r + 1) + -f (r + 3)
3 ! 

. (32-p2)(l2-p2)f(r+p-l) 
p. ' ] . 
If r is an even integer, sinr x = dor + d2r cos 2x + d4r cos 4x + . (2) 
+ dpr cos px + . . . . + drr cos rx, 
where 

do= f (r) 
dpr = 2 [ f(r) + <-;~2) f(r + 2) + <22-p~\(-p2 ) f (r + 4) + 
((p2-2)-p2) .... (22 -p2) (-p2)
· .. ·+ p. f(r+p)].
1 
In the above equations, 
f(y) =1X3 X 5 .... x (y-3) X (y-1) 
(3)
2 x 4 x 6 ... . x (y-2) x y 67
Plate Detection of Radio Signals 
APPENDIX C 
TABLE I-Values ef f(y) 
x f(y) 
2 -----------·------·--··----------------·----·-·----·--··--·---···--·-------------·--·----.50000000 4 -------···-----------··--··--------------··--·----·-·-·-·-------------·····-----.37500000 6 --··-·-------------·--·-·--··------------------------------·-------------·-·----·--···--31250000 8 ----·---------------···-·-·------------------·---------------·----------------------···-.27343750 10 ------··--·-·----·-··-··-·-------------·-·-··-·----------·--------···-------·--·--·-·--····-.24609375 12 -------·-·--·-··-·--·-··-··--·-·-·-----------···-------·--···-·-----··---------·-----------22558593 14 ----··-------------------------------·--·-·-----··-------------------------------------20947265 16 -···----·--·--·-·---·--··-·-···----·-·-··-·-·-··-····--·---··---···--·--·-------··----------19638061 18 ----------··------·--·--·-·-·--·-----·----·-------··--·----------------··-·--·---···-.18547057 
1.3.5.... (y-1) 
f(y) = --------­
2.4.6. • •• y 
where y is an even integer. 
TABLE II 
Fundamental Formulas 
Coefficient sin wLt 
m m' m• m' m• mu m" a2E'----·······------·· 
1.000000 a,E" ----·---·------·--·-1.500000 1.125000 ..E• -·-·---------·---· 

1.875000 4.687500 1.171875 asE• ---·--·--·--··-·--··-2.187500 11.48437 9.570312 1.196289 
810£.lO -----···········-· 2 .460937 22.14844 38.75987 16.14990 1.211243 auE12 2.707031 37.22168 111.6650 97.70690 24.42673 1.221336 auE" -··---··--·-·---2.932617 57.18603 262.1027 393.1540 206.4058 34.40097 1.228606 a1s£l6 -----··---------­
3.142090 82.47986 536.1191 1228.606 1105.746 387.0110 46.07273 
Coeffiicent for--cos 2 wLt 
m' m' m' m• mu m" a,E• ---·---------·----··-··-··-·--··--·---3750000 
.5859375 asE8 ------------··--------------------· 
aeE8 -----------------------------·------1.562500 
3 .828125 4.785156 .7177734
aioEIO__________________ ____________________ 
7-382812 19.37988 9.689941 .8074949 a,.Eu ---------------·--·-·-··--·--·--· 12.40723 55.83251 58.62411 16.28448 .8723830 a1,E1" -----------------------------------· 19.06201 131.0513 235.8924 137.6038 24.57212 .9214543 a1e£18 -------------------------------· 27.49329 268.0596 737.1637 
737.1637 276.4363 34.55454 
Coefficient for de 
m• m' m' m• m' ml' m12 mu a2E' .5000000 .2500000 a.E• .3750000 1.125000 .1406250 aoE• -3125000 2.343750 1.757812 .0976562 asE' .2734375 3 .828125 7.177734 2-392578 .0747681 aioEIO .2460937 5.537109 19.37988 16.14990 3 .028107 .0605621 a12E12 .2255859 7.444335 41.87437 65.13794 30.53340 3.664009 .0508890 a14£U .2094726 9 .531005 78.63079 196.5770 172.0049 51.60145 4.300121 .0438788 
•1aE19 -1963806 11.78283 134.0298 491.4425 691.0909 387.0110 80.62728 4.936363 
Coefficient-sin 3wLt 
m' m• m• m• m" m12 m" a2E2 ------------------.2500000 a.E• -·--·-------··------1.125000 .1875000 aoE• -------·-·--·------2-343750 2-343750 .1464844 asE' -··--------------3.828125 9.570312 3.588867 .1196289 a10E10 ··--·--·---···---·-· 5.537109 25.83984 24.22485 4.844971 .1009369 
a,.EU ---·-·-·-----· 7.444335 55.83252 97.70691 48.85345 6.106681 .0872383 auE" --------·----------9.531005 104.8411 294.8654 275.2077 86.00243 7-371636 .0767879 a1oE1• ---------·------· 11.78283 178.7064 737.1637 1105.746 645.0182 138.2180 8 .638638 
ml' 
1.234091 
m" 
.9598481 ml• 
.0385653 
m" 
.0685606 
Coefficient cos 4wLt 
mlO mt' mH m" a.E• --------------------------------­
m• m• m• 
.046875 ,..E• -----------------------------­
.5859375 .0585937 
2.392578 1.435547 .0598144 &10£10 -------------·---------------------6.459961 9.689941 2.422483
asE8 -----------------------------­
.0576782 39.08276 24.42715 3 .489532 .0545239
a,.E12 ----------------------------· 13.95813 a1tE' --------------------------------· 
126.21027 117.9462 137.6038 49.1 4425 4.607271 .0511919 
.047992416 44.67659 294.8655 552.8723 368.5819 86.38634 5.759085
a1aE---------------------------­
Explanation.-Suppose that it is desired to find the 
a8E8
amount of third harmonic contributed by the term 
of the power series. Referring to line three of the third 
harmonic table above, there is obtained : 
-a8E8 (3.828m3 + 4.785m5 + .718m7)sin 3wLt 
where E is the max. value of the carrier wave, and m is the 
modulation. 
TABLE Ill 
Formulas for Particular Values of m 
Terms 
of 
Power de Cos2wLt 
Series 

For m = 1.00 
aoE0 -------------------------------·----------------­1.000000 
.7500000 1.000000 -.2500000 a.E' -----------------------------------------------­
a2E" -----------------------------------------­
1.640625 2.625000 -1.312500 -.3750000 .046875 
,..E• -------------------------------------------------­4.511718 7.734375 -4.833984 -2.148437 .644531 
13.74664 24.43847 -17.10693 -9.331054 3 .887939 810£10 --------------------------------------------------­44.40165 80.73036 -60.54771 -37.26013 18.63006
asE• ----------------------------------------------------­
148.9305 274.9487 -216.0311 -144.0207 81.01209 a1tEtt -----------------------------------------------------­
a12El2 ------------------------------------------------------­
512.8986 957.4107 -777.8960 -549.1031 335.5629 a1aE18 ----------------------------------------------------­
1801.154 3390.411 -2825.341 -2081.831 1353.190 
Form= .80 
0 
1.000000 a2E2 ----------------------·---------------·------------­
aoE--------------------------------------------­
.6600000 .8000000 -.1600000 a.,E• -----------------------------------------------· 
1.152600 1.776000 -.7968000 -.1920000 .0192000 aeE• -------------·------------·--------------------------­2.558100 4.284000 -2.498400 -.9920000 .2553600 asE' ----------------------------------------------­6.303181 11.01688 -7.330870 -3.678528 1.366355 a10E10 ------------------------------------------------­16.47598 29.55903 -21.30184 -12.27091 5.598778 a12E12 -··------·------------------------------------­44.73679 81.68754 -62.10459 -39.20268 20.43920 a1,E1' -·------------------------------------------------· '24.7437 230.6874 -182.2565 -122.8031 70.33639 a1eE16 --------·---------------------------------------354.7244 662.3072 -538.6340 -381.1284 234.1206 
Form= .60 
aoE0 ----------------------------------­1.000000 
a2E' ----------------------------------------------­.5900000 .6000000 -.0900000 
a.,E• --------·----------------------------------­.7982250 1.143000 -.4293000 -.0810000 .0060750 
asE• -------------------------------------------------­1.388619 2.228625 -1.154334 -.3830625 .0786712 
asE' -----------------------------------------------­2 .694681 4.570799 -2.787888 -1.219062 .3780597 
a10ElO -------------------------------------------­5.555846 9.738892 -6.554424 -3.381060 1.330342 
a12£12 ----·--------------------------------------­11.90665 21.33294 -15.33213 -8.829873 4.063920 
a1.,E14 ----------··-------------------------------­26.21311 47.70516 -35.93437 -22.38850 11.51819 
•1eE1e ----··-------------------------------------58.86437 108.3908 -84.57541 -55.89612 31.25469 
TABLE III-(Continued) 
Formulas for Particular Values of m 

Form= .40 
aoE0 --------------------------------------------------------­1.000000 
a.E' -------------------------------------------·---------------· .5400000 .4000000 -.0400000 a.E' -------------------------------------------------------------­
.5586000 .6720000 -.1848000 -.0240000 .0012000 aeE• ------------------------------------------------------------­
.7329000 1.062000 -.4356000 -.1060000 .0152400 asE' ------------------------------------------------------­1.709960 .8722784 .2951760
-----1.079516 --.0671692 a10E10 ----------------------------------------------------------1.696297 2.825554 -1.649848 .6870377
-.2066586 
auE12 ________________________________________ _··· ----------------­
2 .775864 4.774987 -3.053272 -1.466143 .5337864 
4.670770 8.216587 -5.606164 -2.985529 1.249501 a10El6 ---------------------------------------·----------------8.020605 14.34476 -10.27433 -5.917341 2.753933 
814£14 ---------------------------------------------------------­
Form= .20 
aoE0 -----------------------· -------------------------------­1.000000 
.5100000 .2000000 --0100000 a.E' -----------------------·--------------------------------­.4202250 .3090000 -.0453000 -.0030000 .0000750 aoE' ---------------------------------------------------­-.0975094 -.0009412 
a2E2 ---------------·------------------------------------------­
-----.4090687 .4128750 .0126875 asE' ---------------------------------------------------------·· 
.4382002 .5324528 -.1686675 -.0321654 .0039202 
.4996273 .6819854 -.2643909 -.0653885 .0109623
a10ElO ------------------------------------------------------­
.5946067 .8761755 -.3934843 -.1178829 .0248972
a12£12 ----------------------------------------·-----------------· 
iE14 
.7295486 1.133023 -.5685707 -.1975228 .0498423 a1o}E1e --------------------------------------------------------··-.9154029 1.476115 -.8073192 -.3155442 .0918073 
a1.------------· --------------------------------------------­
Explanation.-Suppose that it is desired to find the con­
E8
tribution of the term a8of the power series to the third harmonic for a modulation of 60 per cent. Referring to the table for m = .60, there is obtained: 
-a8E8 (1.219062) sin 3wLt, 
where Eis the max. value of the carrier wave. Note that the extension of the power series to high exponents (e.g. 
E16
a16) necessitates great numerical accuracy in the calcu­lations. This is so because the value of the output cur­rent is obtained from the sums and differences of numbers large in comparison. 
TABLE IV 
Coefficients for Shift of Axis of Power Series 

ao' a,'  a o  a ,  a2 1 2  a3 1 3  a, 1 4  a, 1 5  a o 1 6  a; 1 7  a, 1 8  a, 1 9  a 10 1 10  a u 1 11  a12 12  am 1 13  au 1 14  al~· 1 15  aoo 1 16  
a:i' a 31 a l ai:i' ... a,'  3 1  6 4 1  10 10 5  15 20 15 6 1  21 35 35 21 7 1  28 56 70 56 28 8  36 84 126 126 84 36  45 120 210 252 210 120  55 165 330 462 462 330  66 220 495 792 924 792  78 286 715 1287 1716 1716  91 364 1001 2002 3003 3432  105 455 1365 3003 5005 6435  120 560 1820 4368 8008 11440  
a'!!' •9 aio'  9 1  45 10 1  165 55 11  495 220 66  1287 715 286  3003 2002 1001  6435 5005 3003  12870 11440 8008  
au' a 1::1 ai:~ '  1  12 1  78 13 1  364 91 14  1365 455 105  4368 1820 560  
a11' ai.;:.' air,'  15 1  120 16 1  

Explanation.-Suppose that the power series expression for a curve about a particular axis is 
ip = a0 +a1e +a2e2 +a3e3 +a4e4 +. .' 
and that it is desired to shift the axis of the equation by a distance F, obtaining a new expression 
iP = a0' + a/e +a/e2 + aa'e3 +. 
Line seven of the table gives: 
a/ = a6 + 7a1F + 28a8F2 + 84a9F3 + . . . . etc. 

In the same way : a/ = a5 + 6a6F + 2la1F2 + 56a8F3 + . . . . etc. Note that F occurs successively as F, F2, F3, F4, etc. The table will be recognized as the table of binomial coefficients. 
TABLE v 
Sinmx as a Function of Sinx 
Coefficients of 

Sinx Sin3 x Sin5x Sin7 x Sin9 x Sinu x Sin" x SinlGx Sinx ·------------------·-· ----· 
1 Sin3x .... -------------··----·· 
3 4 Sin5X -------------------------5 -20 16 Sin7X -----------------·------· 
7 -56 112 64 Sin 9x ----------------------·· 
9 -120 432 576 256 Sin 11x ----------------------· 
11 -220 1232 2816 2816 -1024 Sin 13x ----------·----------­
13 -364 2912 9984 16640 -13312 9096 15 -560 6048 -28800 70400 -92160 61440 -16384
Sin 15x --------------------­
Cos mx as a Function of Sin x 
Coefficients of 

Constant Sin2 x Sin4 x Sin6 x Sin6 x Sin1° x Sin12 x SinH x Sin16x Cos 2x ------------------------1 
2 
8 8 Cos 6x -----------------------· 1 
Cos 4x ------------------------1 
18 48 -32 Cos 8x ------------------------1 
32 160 -256 128 Cos 1 Ox ----------------------1 
so 400 -1120 1280 -512 Cos 12x ---------------------· 1 
72 840 -3584 6912 -6144 2048 
Cos 14x .--------------------· 1 

98 1568 -9408 26880 -39424 28672 -8192 
Cos 16x ----------------------1 

-128 2688 -21504 84480 -180224 212992 -131072 32768 
Explanation.-Suppose it is desired to express sin 5x in terms of sin x. From line three of the Table A, 
sin 5x = 5 sin x -20 sin3 x +16 sin5 x. 
TABLE VI 
Powers of E 
El 2 3 4 5 6 7 8 9 E' 4 9 16 25 36 49 64 81 E' 16 81 256 625 1296 2401 4096 6561 E• 64 729 4096 15625 46656 117649 262144 531441 E• 256 6561 65536 390625 1679616 5764801 16777216 43046721 ElO 1024 104857.6 9765625 60466176 2.8247525 1.0737418 3.4867844 59049 (8) (9) (9)El> 4096 16777216 2.4414062 2.1767823 1.3841287 6.8719475 2.8242954 531441 (8) (9) (10) (10) (11) E" 16384 2.6843546 6.1035156 7.8364163 6.7822307 4 .3980464 2.2876793 4782969 (8) (9) (10) (11) (12) (13)ElO 65536 4.2949674 1.5258789 2.8211099 3.3232930 2.8147497 1.8530202 43046721 (9) ( 1.1) (12) (13) (14) (15) 
Explanation.-The table gives, correct to eight places, the even powers of the digits from two to nine inclusive. The numbers in parenthese are exponents of ten. Thus: 
510 
= 9,765,625 
51 2 
= 2.4414062 X 108 
514 
= 6.1035156 x 109 etc. 
XI. BIBLIOGRAPHY 
Ballantine, S.: Detection at High Signal Voltages. Part I-"Plate Rectification with the High Vacuum Triode," Institute of Radw Engineers, Proc., v. 17, p. 1153, July, 1929. 
Beers, G. L.: "Power Detectors for Broadcast Reception," Electrical Journal, v. 26, p. 413, Sept., 1929. 
Breit, G.: "The Calculation of Detecting and Amplifying Properties of an Electron Tube from Its Static Characteristics," Physical R eview, v. 13, p. 387, 1920. 
Carson, J. R.: "A Theoretical Study of the Three Element Vacuum Tube," Institute of Radio Engineers, Proc., v. 7, p. 187, 1919. 
Chaffee, E. L. and Browning, G. H.: "A Theoretical and Experimental Investigation of Detection for Small Signals," Institute of Radio Engineers, Proc., v. 15, p. 113, 1927. Lewellyn, F. B.: "Operation of Thermionic Vacuum Tube Circuits," Bell System Technical Journal, v. 5, p. 432, July, 1926. Medlam, W. B.: "Improving Detector Efficiency," Wireless World, 
v. 24, p. 524, 1929. Page, W. I. G.: "The Valve as an Anode Bend Detector," Wireless World, v. 24, p. 279, 1929. Terman, F. E. and Morgan, N. R.: "Some Properties of Grid Leak 
Power Detection," Institute of Radio Engineers, Proc., v. 18, p. 2160, Dec., 1930.