1 I B ^AHY 
 
 OF THE 
 UN IVER.SITY 
 Of ILLINOIS 
 
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Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/tunneldiodecircu155moto 
 
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 ceo. 2, 
 
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 DIGITAL COMPUTER LABORATORY 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
 REPORT NO. 155 
 
 TUNNEL DIODE CIRCUITS COUPLED INTO TRANSMISSION LINES 
 
 by 
 Tohru Moto-Oka 
 
 August 16, 1963 
 
 This work was supported in part "by the Office of Naval Research 
 under Contract No. Nonr-l83Ml5) 
 

 ro , 
 
 ACKNOWLEDGMENT 
 
 The author wishes to express his appreciation to 
 Professor W. J. Poppelbaum, Professor T. A. Murrell, and 
 Mr, Henry Guckel for their invaluable help and encouragement 
 
1. Introduction 
 
 All interconnecting wires should be considered as transmission lines 
 in very high speed circuitry. Therefore, there seem to be two groups of methods 
 which are used in the design of very high speed tunnel diode switching and logic 
 circuits. In the first method it is assumed that all constants are ideal lumped 
 constants and that they are interconnected ideally at first. They are then replaced 
 by distributed circuitry. In the second method tunnel diodes are connected to 
 distributed circuitry from the beginning. In the latter- method it may be possible 
 to discover the circuitry in which the characteristic of the distributed circuit 
 is used in our favor. It is impossible to derive this type of circuit from the 
 lumped constant circuit. 
 
 The character of the tunnel diode circuits coupled into distributed circuits 
 is not understood well enough to design very high speed switching and logic circuits „ 
 In this report, typical tunnel diode circuits coupled into transmission lines are 
 investigated as the preparatory stage of latter design methods. 
 
 2 , Amplifiers 
 
 The circuits which are shown in Figure 1 are supposed to be the typical 
 basic tunnel diode circuits which are coupled into transmission lines. Therefore,, 
 familiarization with the characteristics of these basic circuits is very useful 
 in order to design more general tunnel diode circuits which are coupled to trans- 
 mission lines. In addition, these circuits are expected to have the characteristics 
 of voltage or current amplification from the investigations of tunnel diode lumpea 
 circuits, which are mentioned in an earlier report. 
 
 -1- 
 
r 
 
 °i' x i 
 
 I- 
 
 v i ° 
 
 Transmission 
 Line #1 
 
 M 
 
 V T 2 
 
 Transmission 
 Line #2 
 
 -o v 
 
 -o v 
 
 (a) 
 
 V l °" 
 
 -o- 
 
 B 
 
 -o v. 
 
 V \ 
 
 °2> T 2 
 
 V l ° 
 
 ^Transmission 
 Line #1 
 
 -O 
 
 Transmission_ 
 Line #2 
 
 -G V. 
 
 M 
 
 Figure 1. Basic Tunnel Diode Circuits Coupled Into Transmission Lines 
 
 -2- 
 
The equations of the circuit which is shown in Figure l(a) may be 
 written as: 
 
 1 -s(V T ?) i -b(t -T ). 
 
 |^(G 1+ G 2 -Y) e _(- Gi+ G 2 -Y) e 
 
 (1) 
 
 vl "S(T -T ) -S(T +T ) 
 
 ^(-G 1+ G 2+ Y) 6 ^W^ G 
 
 where the transmis ion lines are assumed to ha.ve ideal characteristics and G. and t. 
 
 i i 
 
 are respectively the intrinsic admittance and the transmission time of the section 
 "i" of the transmission lines. The tunnel diode equivalent impedance Z is assumed 
 
 to be r + sL + —= = , that is, 
 
 -Lr + SO 
 
 _ sC - G / v 
 
 1 + (slH-r)(sC-G) K ' 
 
 If the terminal 1-1' is terminated by the series circuit of a signal 
 voltage source "E" and a resistance R and the terminal 2-2' by a. load resistance R , 
 
 V and V become E and zero respectively, because no reflection occurs at these 
 terminals. Therefore, the output voltage at the terminal 2-2' may be written as: 
 
 ,7 \ _ " s(T i +T 2 ) s 2 LC! - s(GL - rC) + (l - G ! r) 
 
 V = V = E£ 
 
 ' + ^ s2l ° " ^ GL " C ^ r + l^G-J H 1 " ^ + V G 2 
 
 (3) 
 
 ■3- 
 
The following equations are the necessary and. sufficient conditions of circuit stability; 
 
 1 - Gf r + 
 
 > 
 
 G 1+ G 2 ) - 
 
 -GL + C/'r + —^~\ > 
 G 1 +G W " 
 
 w 
 
 (5) 
 
 The following conditions are the conditions of sinusoidal oscillation: 
 
 GL > C ( r + 
 
 G l +G 2 
 
 G 1+ G 2 
 
 4LC <[ 1-G ( r - "L H > -I GL-C f r + 
 
 G-^+G^ 
 
 (6) 
 
 (7) 
 
 i. 
 
 ULC > 4 GL+C ( r + 
 
 
 G l +G 2 
 
 (7') 
 
 And the conditions of relaxation oscillation or switching may be written as follows: 
 
 V V G 2 
 
 G(r + G^) > 1 
 
 or 
 
 GL > C r + 
 
 G l +G 2 
 
 (10) 
 
 (9) 
 
 L gl+c ^ + g7g 2 
 
 > hue (11) 
 
 -h- 
 
If the intrinsic admittances of transmission lines satisfy the following condition; 
 
 - = r + - (12) 
 
 G G ! + G 2 
 
 the output voltage may "be rewritten as: 
 
 v G i E c ' s(T i +T 2 ) (s-oQ(s-p) 
 
 V2 = v ^°i + ° 2 Ee s .| s yo.i^ (13) 
 
 V c ig . 
 
 where 
 
 "'p-Kt'-DWKs-O'-a (1U) 
 
 These equations mean that this basic circuit is considered as the basic amplifier 
 which has an infinite voltage gain at zero frequency in the ideal case. 
 
 Also, the backward input voltage may be written as: 
 
 -21 s 
 *" G " G " Y L 
 
 V = — e - (15) 
 
 1 G + G + Y 2 V±p; 
 
Since the input impedance may be mismatched, it becomes necessary to know 
 how much of the available power has been amplified. For this purpose, power gain 
 will be defined in terms of transducer gain. The available power from the source is 
 
 P. = G.-ET/h 
 
 in 1 ' 
 
 (16) 
 
 The output power is: 
 
 (s-qQ(s-P) 
 
 (17) 
 
 The transducer gain is 
 
 G, 
 
 UG 1 G 2 
 
 Tr P. /„ _ n2 
 
 in ^ G ± 2 
 
 (s 
 
 -a)(s 
 
 -3) 
 
 sj s 
 
 ■d 
 
 ■fc)} 
 
 (18) 
 
 The input impedance is 
 
 in 
 
 , G coshT s .+ . (Gp+Y)sinhT s 
 G G sinhT s + (G„+Y)coshr s 
 
 (19) 
 
The cascade circuit of the basic amplifier which is shown in Figure 2 is expected 
 to have the characteristics of an amplifier. In this circuit, the number of tunnel 
 diodes can be selected arbitrarily. The constant current supply should keep all 
 tunnel diodes in the negative resistance regions. The negative resistances of the 
 tunnel diodes, the characteristic impedances of the transmission lines, and the load 
 and source resistance must satisfy some conditions which are needed so that this 
 circuit will have the characteristics of an amplifier. Otherwise, it may work as 
 an oscillator or a binary cell. 
 
 C 
 
 
 
 1 
 
 <Vo> 
 
 I 
 
 \h TD#1 O TD#2 
 
 (V ) 
 
 n-1 
 
 I 
 
 Q 
 
 \7 TD#n-l WTD#n 
 
 ( G i T i ) 
 n-1 n-1 
 
 Z. 
 
 (GT ) 
 v n n 
 
 Figure 2. Transmission Line Type Amplifier 
 
 The transmission line type amplifier which has two tunnel diodes is shown 
 in Figure 3- The circuit equations of this circuit are as follows: 
 
 ~ sT o ST o 
 
 /gi-(G 2+ G 3 -Y 2 ) £ sgT<-WV e 
 
 ■ST 
 
 \ 1 
 
 (-G +G^+Y ) G 
 
 2G v ~g 3 ^ 2 . 
 
 3 
 
 2G^ < WV 6 
 
 -7- 
 
v. 
 
 'aC (VW sfc (-WV 
 
 -ST. 
 
 v o e 
 
 (21) 
 
 *i V 55" (-WV 2r ( WV 
 
 ST. 
 
 v o e 
 
 When it is assumed that the input and output terminals are terminated by matched 
 resistors, the values of which are respectively l/G, and l/G_, the voltage gain may 
 
 be written as 
 
 G = 
 v 
 
 V, 
 
 V. 
 
 
 -s(t i +t 3 ) 
 
 -ST. p ST 
 
 (-G 2 +G 3 +Y 2 )(-G 1+ G 2 -Y 2 ) e ^+ (G^+Y^Gg+G^) € 
 
 (22) 
 
 If it is assumed that tunnel diode admittances Y and Y are respectively sC - G^-, 
 and sC - G , the voltage gain at a frequency oo iss 
 
 G' = 
 
 2G.G e 
 
 ■Jco ( T 1 +T 3) 
 
 r"G 2 (G 1 +G 3 -G T1 -G T2 )cos ojt 2 + -joX^G^-Gy + ^^Tl" 6 !^ J 81 * 1 ^2 
 
 j|^a/C 1+ C 2 )G, 
 
 G 2 + (G 1 -G T1 )(G 3 -G T2 ) - a^^Cg 
 
 (23) 
 
 ] 
 
 
 -8- 
 
If we set G = G and G = G , the voltage gain may be rewritten as 
 
 -jco(t +t ) 
 2G G e X J 
 
 g ; - .- — — -2 — §: 
 
 j"o^C 1 +C 2 )G 2 cos oju 2 + (G 2 - co C^ ) sin aO 
 
 Also, when section 2 of the transmission line is chosen to satisfy the condition that 
 t G = (fl/2) n/C C , the voltage gain becomes infinity at co = and co , where co is 
 
 given by (n/2T ) or (G„/ vC C p ). If the time constants of two tunnel diodes have the 
 
 same value, that is, C /G_, = C p /G,p_ = T , this condition may be rewritten in the 
 following form which has clearer physical meaning; 
 
 TO = § t^O (25) 
 
 Equation (2k) may be rewritten as: 
 
 G 1 2 
 I G" I « =i F(x) ; F(x) = ■ — - (26) 
 
 2 kx cos pX + (,1-x ;sin — x 
 
 where 
 
 co 3 , 12 
 
 x = 7— and k = ■ > 2 
 
 3 1 C 2 
 
 % ^ a 
 
 F(x) is shown in Figure 4. 
 
 -9- 
 
V. 
 
 v. 
 
 V, 
 
 o- 
 
 Input (Vl^ 
 
 1 
 
 •- 
 
 V, 
 
 TD#1 
 
 1 
 
 1 
 
 (G £ T 2 ) 
 
 2 
 
 M 
 
 -*- 
 
 V 
 
 TD#2 
 
 Y (G_T ) Output 
 
 Figure 3- Transmission Line Type Amplifier with Two Tunnel Diodes 
 
 -10- 
 
3|3 
 
 I 
 X 
 
 u 
 
 0) 
 Oh 
 
 *-» o 
 
 On 
 
 CO 
 
 VO 
 
 ITS 
 
 OJ 
 
 ■11- 
 
 IJNIVERSITY Of 
 ILLINOIS LIBRAE 
 
From the investigation of the two tunnel diode transmission line 
 amplifiers, it appears to follow that amplifiers with more tunnel diodes have more 
 peaks in the gain characteristics. It also seems that optimum gain for pulse 
 amplification would be found with an infinite number of tunnel diodes. The ideal 
 case is given by the strip line type tunnel diode which is shown in Figure 5« 
 
 //////////////////////// ///////////TT77 ^-v 
 
 (a) 
 
 x * L 
 
 $ i-q > C <> 
 
 o- 
 
 - I 
 
 U 
 
 n 
 
 1 
 
 -o 
 
 00 
 
 Figure 5. Strip Line Type Tunnel Diode 
 
 ■12 • 
 
The characteristic equations of the strip line tunnel diode may be 
 written as: 
 
 -yx -yx 
 
 V = Ae" /A + Be f (27) 
 
 I = Y Q (Ae~* X - B£~ fx (28) 
 
 where 
 
 Y o =vf (1 - J f )>*•«*»- *»» h (1+j f ) • <* - § 
 
 and L, C, -G are the inductance, capacitance and negative conductance per unit 
 length of the tunnel diode respectively. 
 
 If the input terminal (x = 0) is terminated by a series circuit of a 
 i R and a voltage source E and the output terminal (x = 
 a resistance R, ; then A and B are given by the following equation; 
 
 resistance R and a voltage source E and the output terminal (x = H) is terminated by 
 
 1 + R Y_ 1-EY. 
 s s 
 
 k-VoK* 1 (1+ Vc>^ 
 
 (29) 
 
 -13- 
 
The coefficients A and B are derived from this equation: 
 
 (1+R-Y )e* £ 
 a= -. £-2 « | (30) 
 
 (R +R*)Y_ cosh 7^ + (l+R R.YT)sinh 7-0 
 
 S * (J S JO U 
 
 (R Y -±W H 
 B = -. ^ — — (3D 
 
 (R +R„)Y- cosh 7 I + (l+R R fl Y^ )sinh yi 
 s I' s £ 
 
 The output voltage which is the voltage at x = I may be written as: 
 
 v» = 
 
 m i y o 
 
 (R +R„)Y A cosh yl + (l+R R„Y^ )sinh yS> 
 s i> s i! 
 
 The voltage gain which is defined "by the ratio of the output voltage and the source 
 voltage is derived from the above equation. 
 
 vER+R/i _ _ _ .. 2 
 
 s l rl+R R/jY» 
 
 cosh yl + \ — ' > sinh yl 
 
 <VVV 
 
 and 
 
 G I = ^V 1 ' (33) 
 
 v ' R + R 
 
 ^ ^(l+^+M 2 ) 2 -^ Jcosh(2ai+cp i )-cos(2^-cp 2 ) 
 
 0A- 
 
where 
 
 M + jN = 
 
 + \*Az) G 
 
 +;-5e 
 
 l + R R fl Y n 1 + RR„ 7/ I 1+i -f- 
 
 '«.*»,) 1/(1) (w ? 
 
 if 
 
 , -1 2M 
 
 cp s tanh — 75— 5- , and 
 
 1 + M" + IT 
 
 -1 2N 
 
 r.an 
 
 92 ,,2 A _2 . 
 
 ^ M + jNi - 1 
 
 The approximated equation of the gain at low frequencies is derived from 
 Eq. (32) as: 
 
 V * R s + "*£ l + R R,Y? 
 
 1+ (nr + . ° (H) 
 
 <V H J> T 
 
 h 
 
 
 1 
 
 
 
 R s + R l 
 
 
 | of LCi 2 
 
 i /l+R R ( 5 
 
 G^ 
 2 
 
 (3V) 
 
 -15- 
 
When QiH = (l/R ) + (l/R,,), the voltage gain becomes infinity at zero frequency. If 
 
 1 + ViL 
 
 then the voltage gain in the low frequency region may be rewritten as: 
 
 *i 
 
 (35) 
 
 V R + R« RR/7 TOO 3 
 
 1 - R + R GZ - - uWJ + jorTPCa)) 
 s i 
 
 where P(cd) is a polynomial in jco. If these two conditions are satisfied 
 concurrently, that is: 
 
 «-t*ij~3"i,-*(jf-£-l <36) 
 
 ;he frequency characteristic of the voltage gain at low frequencies becomes 
 
 2* £ x 
 
 G v 7 " R + R. 2/2 ^' 
 
 s I to LCiJ 
 
 ■16- 
 
When it is assumed that R = R» = 2/(Gi) = ^Jh/C, which satisfies Eq. (36). 
 the absolute value of the voltage gain is derived from Eq. 33 as: 
 
 where 
 
 y—J N/cosh(20{£+cp 1 ) - cos(2p^-cp 2 ) 
 
 1 fik\ 1+ V + \c 
 
 
 V 2w 1 M + (J) , :i 
 
 ^{\R5) 2 -4=^ to i 
 
 6 = ^^il {f + * + (?) J = "^ tanh "2 
 
 -17- 
 
3 
 
 Oscillator 
 
 It is apparent from the stability criterion of the tunnel diode amplifier 
 which was mentioned in Eq. (k) and Eq. (5) that the tunnel diode circuit coupled into 
 a transmission line may oscillate under some conditions. The behavior of this type 
 of the tunnel diode oscillator which is shown in Figure 6 is analyzed from two 
 different standpoints, that is, the small signal linear analysis and the large 
 signal analysis which uses the broken line approximation of the tunnel diode 
 characteristic . 
 
 (hi) 
 
 
 
 -*- 
 
 V, 
 
 Tunnel Diode 
 "C 
 
 S R 
 
 (7,Z Q ) 
 
 -Ri = v 
 
 (a) Equivalent Circuit 
 
 (b) Broken Line Approximation of 
 the Tunnel Diode Characteristic 
 
 Figure 6. The Tunnel Diode Oscillator 
 
 -18- 
 
The equivalent circuit and the broken line approximation of the tunnel 
 
 diode characteristic which are shown in Figure 6 are used in the large signal 
 
 analysis of the circuit behavior of the tunnel diode oscillator „ The circuit 
 equations of the oscillator may be written as: 
 
 V = V + v o 
 
 (39) 
 
 R 
 
 V = 
 
 R + Z„ 
 
 -2st -> -2st -, 
 V Q = pe V 
 
 
 
 (to) 
 
 v = -E 
 
 dv 
 
 JT- + (l-GZjv = 2V_ 
 dT v 0' 
 
 v = E 
 
 in the region I 
 
 in the region II 
 in the region III 
 
 (te) 
 
 where T = — m,t Q = \fjieil . 
 
 If it is assumed that the switching time of the tunnel diode is 
 negligible compared to the transmission time t <, the following points become apparent i 
 
 1) It is assumed that the tunnel diode is switched from v = -E to v = E at t = 0, 
 
 2) During < t < 2t , V is assumed to be V } which should be larger 
 than (l/2)(l-GZ )E in order that the tunnel diode remains in region 
 III, that is, v = E. 
 
 -19- 
 
3) During 2t < t < kt , V = p(E-V ) and the tunnel diode remains in 
 region III if 2p(E-V ) + (GZ -l)E > 0. Otherwise, the tunnel diode 
 is switched back to region I at t = 2t . 
 
 k) During kt Q < t < 6t Q , 
 
 a) if the tunnel diode remained in region III during 
 
 f T 
 
 2t < t < kt , V = p ^E - p(E-V ) >■ then the diode 
 
 remains in the region III if 2p ^E - p(E-V )Y + (GZ -l)E > 0, 
 Otherwise, it is switched back to region I at t = ^-t • 
 
 b) if the tunnel diode is switched back to region I at t = 2t 
 
 V = p -s - E - p(E-V ) f the tunnel diode will remain in the 
 
 i 
 
 region I if 2p A - E - p(E-V S)T - (GZ -l)E < 0. Otherwise, it 
 
 is again switched' back to region III at t = kt . 
 
 In general, if the tunnel diode stays in region III during < t < 2(m-l)t , 
 the backward voltage may be written as : 
 
 m-1 
 
 V = (-p) n - 1 V 1 -Z l-p)^ (k2) 
 
 n=l 
 
 during 2(m-l)t < t < 2mt . The tunnel diode will remain in region III if the 
 following condition is satisfied: 
 
 2V Q + (GZ Q -1)E > (U 3 ) 
 
 Otherwise it is switched back to the region I at t = 2(m-l)t . 
 
 -20- 
 
In other words, if the following two conditions are satisfied, 
 
 m-1 
 
 m-1 Tr pE 
 
 -ee (- P ) n + (-prs. = 
 
 _=1 
 
 1 + P 
 
 {l -(-P) m_1 } + ("P) 111 " 1 V r >^| (GZ Q -1) (kk) 
 
 -E E (-p) n + (-p) m V 1 - j-f-jl -<-p)»} + (-p) m V < q (GZ -1) 
 n=l ^ J 
 
 (W 
 
 the tunnel diode is switched from E to -E at t = 2mt . This behavior is illustrated 
 in Figure 7. 
 
 ^►t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 1 
 
 1 
 
 I 
 
 1 
 
 1 
 1 
 
 1 
 1 
 
 
 
 
 " r L 
 
 
 
 
 
 
 ~ 2t o^ 
 
 
 ! ! 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 I i 
 
 
 
 
 
 
 
 
 u 
 
 
 
 
 
 
 
 
 
 
 
 
 •" L 
 
 
 
 
 
 L , 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 Figure 7. The Waveform of Ideal Oscillation 
 
 -21- 
 
The steady state oscillation condition may be written as: 
 
 m 
 
 in / \m 
 
 -v 1= -eZ (- P ) n + (- P ) m v 1 (k6) 
 
 n=l 
 
 or 
 
 = ^g_ i - (-P) m {k6 <y 
 
 1 1+p i + (- P ) m 
 
 It is worthwhile to mention that in the above discussion V- must be larger than 
 (l/2)(GZ -l)E in order to switch the tunnel diode from -E to E. And also, GZ 
 must be larger than 1 for switching behavior. This condition is apparent from 
 Eq. (^-l). It is apparent from Eq. (k-6 1 ) that p should be negative, that is, 
 R should be smaller than Z „ 
 
 "m" must be one in order to get oscillation of maximum frequency. This 
 condition may be written as: 
 
 (W 
 
 or 
 
 i -~>gz - l>§-- l (hi") 
 
 z o ° z o 
 
 -22- 
 
This condition means that p should he negative, hut GZ may be larger or smaller 
 
 than one. The latter part means that even if switching behavior is not assumed, 
 oscillation of a tunnel diode is possible. Z should be in the following region 
 for the relaxation oscillation of frequency of approximately (l/^t )i 
 
 1 < GZ Q < 1 + -Jl - GR ' (h&) 
 
 If V is given by -pE/(l-p), the switching time of the tunnel diode is 
 derived from the following equation: 
 
 dv 2pE /,-x 
 
 — = av - r— - — (49) 
 
 dT 1 - p v ■ /J 
 
 where a = GZ - 1, 1 = (t/Z C). 
 
 At T = 0, v = -E, then 
 
 *- fc^ a *-^ aT } B < 50 > 
 
 At t = t , v = E, then 
 
 , -2p + a(l-p) / C1 v 
 
 aT l = lQ g - 2 p - a(l-p) (51) 
 
 -23- 
 
Then the maximum frequency of relaxation oscillation is expected to be in the 
 following region: 
 
 2Z„Ct. > f m > hZ^Cn, 
 
 1 
 
 1 
 
 (52) 
 
 or 
 
 -2p + a(l-"pT f m : 
 
 2Z.C log -tr }. r { 
 
 & -2p - a(l-p) 
 
 kzn log i P + a T?^ 
 -2p - a(l-p) 
 
 (52') 
 
 Equation (5l) may be rewritten as: 
 
 n- -> X + 1 
 
 ^1 " ^ FT! " ' 
 
 111 1/^A 
 
 (51') 
 
 where 
 
 Therefore, 
 
 f < 
 
 ~m 2Z Q CT 1 
 
 1 / R\ 1 
 
 (5.3) 
 
 -24- 
 
The above discussion gives some ideas on the oscillation conditions of 
 
 the tunnel diode coupled into a transmission line. Another method of approach to 
 
 the oscillation conditions is a small signal linear analysis. The equivalent circuit 
 
 for small signal operation of the tunnel diode oscillator shown in Figure 8(a) is 
 given in Figure 8(b). 
 
 
 
 (Z) 
 
 h!) 
 
 E 
 
 pA 
 
 -nm^ 
 
 fc 
 
 -G < 
 
 U-J 
 
 (z) 
 
 > 
 
 (a.) 
 
 00 
 
 Figure 8. The Equivalent Circuit of the Tunnel Diode Oscillator 
 
 for Small Signal Operation 
 
 The oscillating conditions of this equivalent circuit may be written as: 
 
 G = Re 
 
 R+jcoL+Z 
 
 l+pe-J £a *0 
 l-pe-J 2a *0 
 
 (5*0 
 
 -cjoC = Im 
 
 R+,ia)L+Z 
 
 l + p€-J ?aJt 
 
 (55) 
 
 where p = 
 
 r - Z 
 r + Z 
 
 and t_ = V(iei 
 
 -25- 
 
These equations may be rewritten as follows: 
 
 or 
 
 G = 
 
 R(l+p 2 -2p cos 2cob ) + (l-p^)Z 
 
 (R 2 +oo 2 L 2 )(l+p 2 -2p cos 2out )+ 2 -JR(l-p 2 )- 2pooL sin ^ojt Q V Z 
 
 2 2 
 
 (l+p +2p cos 2a3fc )Z 
 
 I 
 
 ooC = 
 
 cuL(l+p -2p cos 2ast ) -2p sin 2oab Z 
 
 (R +cd L^)(l+p -2p cos 2cot )+ 2 -JR(l-p^) -2pcoL sin 2a5b Q \- Z 
 
 (5^) 
 
 l 
 
 2 "2 
 
 (l+p +2p cos 2aJb )Z 
 
 (55') 
 
 TG |(R 2 + a) 2 L 2 +Z 2 )(l+p 2 )+ 2ZR(l-p 2 )| - R(l+p 2 ) - (l-p 2 )Z~| 
 -2p cos 2cot Jg(R 2 +oo 2 L 2 -Z 2 )- R^ -2p sin 2cut (2ooLGZ) = (5V) 
 
 clC |(R 2 +a) 2 L 2 - £ + Z 2 )(l+p 2 )+ 2ZR(l-p 2 )| 
 ^ -2p cos 2aat JaC(R 2 +a> 2 L 2 - ~ - Z 2 )\- -2p sin 2aat ('2a) 2 LC-l)Z = (55 n ) 
 
 -26- 
 
or 
 
 (l+p 2 ) |g(R 2 +o) 2 L 2 +Z 2 )- R +( i: % J(2GR-l)z| 
 
 where 
 
 (R 2 +CD 2 L 2 -Z 2 > r R^ ■ + l+(coLGZ) 2 sin(2cct -hx) = ( 5*^' " ) 
 
 poo p 
 G(R +a> L -Z )-R 
 
 a = Arc tan ^^ 
 
 (l+p 2 )a£! |r 2 + a) 2 L 2 - | + Z 2 + ( il %) (2ZR) 
 -2p ||/-{aC(R 2 +co 2 L 2 - ^ -Z 2 H 2 + (2oo 2 LC-l) 2 Z 2 sin(2act +p) = (55"') 
 
 where 
 
 a£ ( OTL +R - - - Z 
 
 P = Arc tan - 
 
 (2a) 2 LC-l)Z 
 
 The relation between oscillation frequency and the length of the trans- 
 mission line can be derived from Eq„ (55)j> if the other circuit constants are given,, 
 because this equation does not include G. The negative conductance value is given 
 by Eq. (5*0 after inserting the oscillating frequency and the traveling time of the 
 transmission line, which in turn are derived from Eq. (55) a nd Eq. ( 5*0 • This 
 negative conductance seems to be the average negative conductance occurring in the 
 large signal operation of the device. Some information about the oscillation 
 amplitude can be obtained in this way. 
 
 -27- 
 
Examples of the relationship "between cd and t and the relationship between 
 G and t are given in Figure 9* The dependence of G on cd, which is derived from 
 Figure 9, is shown also in Figure 10. In this case, it is assumed that C = UpF, 
 R = 1.5fl, L = 0.6nH and Z = 50ft. It is worthwhile to mention that the difference 
 nit/cD of the transmission time does not change anything and that the relation of 
 Figure 9 corresponds to the case for n = 0. 
 
 The upper frequency limit of oscillation is derived easily from Eq. (55' ,? ) 
 If the coefficient of the sine function is larger than the first term of Eq. (55' ?, )j> 
 any t does not satisfy this equation, that is, the upper frequency limit is 
 derived from the following equation: 
 
 2 _2 L2 2 T 2 L 
 
 ooCiR + cd L - — 
 
 m m C 
 
 2ZR 
 
 1+p 
 
 (1+P") 
 
 kp f 2 „2 f _2 2 T 2 L v 2 
 
 o Q i cd C (R+co L - - - Z 
 2 S 2 I m V m C 
 
 + (2cd LC-l) Z 
 
 m 
 
 (56) 
 
 At the upper limit of the oscillation frequency, the length of the transmission 
 line must be 
 
 t. 
 
 
 
 m V jj.e 
 
 to = cd r 
 si 
 
 CD=CD 
 
 m 
 
 r 2cd |2 
 vjae m 
 
 ^ it + 2nit 
 
 -Arc tan 
 
 (2cd LC-l)Z 
 
 m 
 
 The negative conductance may be written as: 
 
 R(l+p 2 +2p sin p ) + (l-p 2 )Z 
 
 CD = CD 
 
 m 
 
 (R 2 +cd 2 L 2 )(l+p 2 +2p sin p )+ 2 ^R(l-p 2 )+ 2pCD L cos p fZ +(l+p 2 -2p sin p )Z 
 m ' N K nr v ' m m m 
 
 where 
 
 P = Arc tan 
 
 m 
 
 cd c(cd 2 L 2 +R 2 - £ - Z 2 ) 
 m - m C 
 
 (2cd 2 LC-l)Z 
 m 
 
 (58) 
 
 -28- 
 
in 5 
 
 cu 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V G " 
 
 2t o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 >I] 
 
 
 
 
 
 
 
 
 
 
 
 in* 
 
 
 
 
 -q J" "-■ 
 
 
 
 10 
 
 m|x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 mi 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 ! 
 
 \i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 !/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V CD - 
 
 2t 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ,«3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 3 
 
 0.1 
 
 1 nsec 
 
 1/G 
 (0) 
 
 100 
 
 min 
 
 10 
 10 
 2t Q (nsec) 
 
 Figure 9« The Relation Between t~ and u> or G of the Oscillator 
 
 
 -29- 
 
lr ,l 
 
 Q 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1/G 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 100 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 io- 
 
 10 
 
 10" 
 
 Figure 10. The Relation Between Oscillation Frequency and 
 Negative Conductance of the Oscillator 
 
 -30- 
 
It is evident from Figure 9 or Figure 10 that the negative conductance 
 at the upper frequency limit of oscillation becomes very large, so that the maximum 
 oscillation frequency seems to "be limited, in fact, by the maximum value of the 
 negative conductance of the tunnel diode . Since Figure 9 shows that two principal 
 oscillation modes correspond to a given length of transmission line, the maximum 
 oscillation frequency seems to correspond to the minimum length of a transmission 
 line when the oscillation mode which corresponds to the smaller value of G is 
 predominant. These values are shown in Figure 9- 
 
 If p is assumed to be -1, the relations mentioned above are simplified. 
 The upper limit of the oscillation frequency and the negative conductance may be 
 rewritten as: 
 
 oo' = -=- (59) 
 
 m 2RC yjyj 
 
 G 
 
 R(l-sin p» ) 
 
 m 
 
 oo = oo' ,J2. .2^2 „2 
 
 m 
 
 m - m mm m 
 
 (R 2 +o^ 2 L 2 +Z 2 ) - (R 2 +ay 2 L 2 -Z 2 )sin p^ - 2oo^LZ cos ^ 
 
 (60) 
 
 From Eq. (59) and Eq. (60) 
 
 CO 
 
 60=00' G^ 
 
 m < 7T- (61) 
 
 m C C 
 
 where G is the maximum absolute value of the negative conductance of the tunnel 
 diode . 
 
 -31- 
 
Therefore, the upper frequency limit in this case is given by 
 
 o£ = Min.(G /C, 1/2RC) 
 
 (62) 
 
 If it is assumed that the series Impedance of the tunnel diode, that is, 
 R + sL can be neglected, the oscillation conditions may be rewritten as follows: 
 
 G = 
 
 (1-P 2 ) 
 
 (l+p +2p cos 2cofc )Z 
 
 (63) 
 
 -2P sin 2oot 
 
 cue 
 
 
 
 (l+p +2p cos 2cot )Z 
 
 (64) 
 
 These conditions are illustrated more explicitly by Figure 11 where GZ 
 
 2 2 
 
 is given by (a/b) and ooCZ by c/b . The maximum value of ouCZ may be written as: 
 
 ooCZ 
 
 -2p 
 
 Max ' . 2 
 1 - p 
 
 (65) 
 
 In this case, 
 
 r, * "2p 
 
 cos 2o> t = ~t 
 
 m 2 
 1+P 
 
 (66) 
 
 ZG 
 
 l+p 
 
 : CD ' 2 
 
 m 1 - p 
 
 (67) 
 
 -32- 
 
The upper limit of the oscillation frequency is derived from Eq. (65) and Eq. (67) as: 
 
 CD = 
 
 m 
 
 -2p G 
 2 C 
 
 1 + P 
 
 (68) 
 
 Also, Eq. (53) may be rewritten as: 
 
 "m < 2 Z Q C < 2 C 
 
 (53 ! ) 
 
 Therefore, it is concluded from a comparison of Eq. (53* )> (62), and (68) that the 
 upper limit of the oscillation frequency is given as G n /(2nC) in an ordinary tunnel 
 diode in which R is smaller than l/G„. 
 
 Figure 11. Illustration of the Oscillation Condition 
 
 -33- 
 
k. Monostable Multivibrator 
 
 The circuit which is shown in Figure 12 works as a monostable multi- 
 vibrator when some conditions are satisfied. The optimum design conditions based 
 on the assumption that the characteristic curve of the tunnel diode is approximated 
 by the idealized N-shape split line, which is shown in Figure 13, are mentioned 
 below. The other assumptions are that the equivalent circuit of the tunnel diode 
 is given by the parallel circuit of a constant capacitance and the nonlinear 
 conductance. The conductance is expressed as i = f(v). Furthermore, it is assumed 
 that the trigger input is the constant current pulse I , its width is longer than 
 the rise time (t ) of the output waveform and shorter than twice the transmission 
 time (t ) of the transmission line, that is: 
 
 -st. 
 
 I. = I.(l-€ X ) , t < t. < 2t_ 
 
 1 l ' r — i — 
 
 (69) 
 
 ©■ 
 
 
 M 
 
 % R 
 
 Figure 12. Monostable Multivibrator 
 
 Figure 13. The N-Shape Approximation 
 of the Tunnel Diode Characteristic 
 
 -3fc- 
 
Under these assumptions, the switching "behaviors are given "by the 
 following equations: 
 
 i) During t. > t > 0, if it is assumed that v = at t = 0, 
 
 I + f(v) ■ x o + h - \ ' v " v > v " ° 
 
 (70) 
 
 where V and V are voltages of the forward and backward 
 traveling wave of the transmission line at x = 0, and the 
 characteristic impedance of the transmission line is Z . 
 
 ii) During 2t > t > t., 
 
 ° % + 'M 
 
 = I 
 
 v_ 
 " Z, 
 
 V = v, V = 
 
 (71) 
 
 iii) During t > 2t , 
 
 dv 
 
 /e- 
 
 "i^W^o-^^^^V 
 
 •R - Z -> 
 
 V(t) = v(t) - V(t) = v(t) h R + z ° V(t-2t Q ) y 
 
 ^'■RT^^- PV(t-2t Q ) 
 
 (72) 
 
 where it is assumed that the transmission line is ideal. Even 
 if the transmission line has some attenuation, but no phase 
 distortion, the attenuation can he included in the reflection 
 coefficient. 
 
 -35- 
 
These equations, with the assumption of Figure 13, become in the region 
 of E > v > 0: 
 
 i) During t > t > 
 
 i r _ 
 
 % + (l-GZ )v = (I 0+ I.-I)Z (73) 
 
 ■where t = t/CZ . If it is assumed that v = at t = 
 
 and I. + I. + I > 
 l 
 
 JZ -arr 
 
 v = -^(1-e aT ) (74) 
 
 where a = 1 - GZ^, J = I_ + I. - I > 0. J should be 
 i 
 
 positive for triggering. The rise time t may be written as 
 
 l Jz o 
 
 \ - k £ « JZ^il ™ 
 
 And for a = 
 
 \ - m w> 
 
 The necessary and sufficient condition for complete switching is 
 
 JZ > aE and J > (77) 
 
 because it is assumed that the rise time t is shorter than 
 
 r 
 
 the input pulse width t.„ The output voltage is kept at E 
 
 after t = t . 
 r 
 
 -36- 
 
ii) During 2t > t > t. 
 
 |^+ av = (l„-l)Z n 
 
 dT v ' 
 
 (78) 
 
 a) If (I -l)Z - aE = I Z - E > 0, the output voltage 
 stays at E. 
 
 b) If Z I - E < 0, the circuit flies hack, that is, 
 
 v = 
 
 Z ° VE \ - € _a(T ' 
 
 a 
 
 "] 
 
 (79) 
 
 where T . = t . /CZ^ . 
 i l' 
 
 iii) During t > 2t 
 
 a) In the case of Z I - E > 0, the value of the 
 output voltage is kept at E until t = t' + 2t 
 which is given by 
 
 
 
 -2pJZ 
 
 Vo " E - — 
 
 p.. -at ■ -i 
 [l-C ] 
 
 (80) 
 
 During t. + 2t > t > t' + 2t 
 
 - + av= Z (I -I) + -j- 1 
 
 ] 
 
 (81) 
 
 Therefore 
 
 av = 
 
 z oV E+2pJ V - 
 
 a 
 
 1 - -\l + a(T-T» -2t ) \- £ 
 
 -a(T-T"-2T n ) n 
 
 aE (82) 
 
 -37- 
 
If, in the above equation, v is less than zero at 
 
 t = t + 2t_, the fall time t„ - CZ^ of the output 
 r f 
 
 voltage becomes 
 
 l X- 
 
 Z I -E+2pJZ \p -3.1 -, 
 
 + aE = 
 
 (83) 
 
 The output voltage is usually kept at zero after 
 t = t + t' + 2t . If v exceeds zero by v at 
 
 t = t + 2t_, then 
 r 0' 
 
 f + av . Z (I -I) ♦ 2pE 
 
 (81t) 
 
 and 
 
 av = 
 
 -3,(1-1 -2T ) 
 ;i -l)Z Q + 2pE ■ av> (1-e ° ) + av Q 
 
 (85) 
 
 In this case, the fall time may be written as 
 
 T f - - S fa 
 
 ZqCIq-^)* 2pE 
 Z (I -I 1 ) + 2PE - a VQ 
 
 + T 
 
 (86) 
 
 b)« For Z I - E < 0, Eq. (8l) is applied during 
 
 t + 2t_ > t > 2t^ with the initial condition that 
 r — 
 
 Vo - E r -*& -\h 
 
 V ' ^=2T a 
 
 D 
 
 + E 
 
 (87) 
 
 -38- 
 
The procedure after that is the same as for a) . The results of the 
 
 rise time calculations t , the dead time t' and the fall time t„ of the output 
 
 r' f 
 
 waveforms are shown in Figure Ik, where I = la, E = lv, G= l(ft)~ , I Q = 0.8a., I. = O.Ua, 
 p = -1 and Z is changed from 1ft to 3 • 7 5*1 • From this calculation, it is concluded 
 that the "best value of the characteristic impedance of the transmission line, where 
 the rise time becomes equal to the fall time, is approximately 2.8ft. In order that 
 the circuit is switched off by the backward wave of the transmission line, the 
 characteristic impedance of the transmission line should be kept between 1,25ft and 
 3.75ft. 
 
 If it is less than 1.25ft, the circuit flies back by itself after termination 
 of the triggering pulse. If it is larger than 3.75ft; it does not fly back in this 
 ideal case. The waveforms are shown in Figure 15. 
 
 5. Binary Counter 
 
 The circuit which is composed of a tunnel diode and an open-ended 
 transmission line as shown in Figure l6(a) has the characteristic of a binary 
 counter for a bipolar signal which is shown in Figure 16(b). The most important 
 merits of this counter are its simplicity and very high speed behavior. The demerits 
 are the necessity of an uncommon input signal and the complicated interstage 
 circuitry. 
 
 It is assumed that each input signal consists of a positive pulse followed 
 by a delayed negative pulse. (A negative pulse and a delayed positive pulse can also 
 be considered. ) When the tunnel diode is in the low voltage state, the first positive 
 pulse sets the tunnel diode into the high voltage state and sends a positive forward 
 wave to the transmission line. The delayed negative input pulse would reset the 
 tunnel diode to the low voltage state, but the positive backward wave of a trans- 
 mission line generated by a positive input pulse cancels the negative input pulse 
 and inhibits the device from returning to the low voltage state, that is, the input 
 signal sets the circuit which is in a low voltage state to a high voltage state. 
 When the tunnel diode is in a high voltage state, the first positive input pulse 
 does not change the state of the tunnel diode and sends only small positive forward 
 waves into the transmission line. But now the negative pulse of the input signal 
 resets the circuit to a low voltage state. 
 
 -39- 
 
■p 
 
 II 
 
 I- 
 c 
 
 (T r Z Q = 1.7918) 
 
 Figure ik. 1 , t' and t„ of the Monostable Multivibrator 
 r f 
 
 -ho- 
 
(a) Typical Wa,veform 
 
 (b) Ideal Waveform t = 2t_ = t n = t- 
 
 r 1 f 
 
 Figure 15 . Waveforms of the Monostable Multivibrator 
 
 41- 
 
Input 
 O 
 
 T 
 
 — * 
 
 1 
 
 
 
 *1 
 
 n 
 
 -1 
 
 k- 
 
 
 
 (i) 
 
 11 
 
 (a) Counter Circuit 
 
 (b) Input Waveforms 
 
 Figure l6. Tunnel Diode Counter 
 
 This circuit can be analyzed in nearly the same way as the mono stable 
 multivibrator o The delay time of the second input pulse (t ) should be longer than 
 twice the transmission time of the transmission line (2t _) plus the switching time 
 (t ), but it should be shorter than four times the transmission time, that is, 
 
 2t + t < t < kt 
 
 s - d 
 
 Of course, it is preferable that the input pulse width is equal to or longer than 
 the switching time^ 
 
 t. > t 
 l — s 
 
 The maximum repetition frequency of the input signals depends on the 
 attenuation factor of the transmission line and the reflection coefficient at the 
 tunnel diode side, and so on. 
 
 -k2- 
 
6, Conclusion 
 
 As the results of the analyses of the typical tunnel diode circuits 
 coupled into transmission lines, the many design conditions of these circuits 
 became evident. 
 
 It is difficult to derive the characteristics of the combined circuits 
 of these standard circuits directly from these results. As the next stage, it is 
 necessary to investigate the interconnecting circuits. These circuits should have 
 the characteristic of directivity and they should also have the characteristic of 
 the input -output separation in order to make the designs of systems easy. 
 
 43-