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0.84- 
 
 -^^^ DIGITAL COMPUTER LABORATORY 
 
 o UNIVERSITY OF ILLINOIS 
 
 OP- ^ 
 
 URBANA^ ILLINOIS 
 
 REPORT NO. Ik6 
 CABLE DRIVER AND TERMINATOR DESIGN 
 Shlgeharu Yamada 
 
 I 
 
 August 21, 1963 
 
 This work was supported in part by 
 Atomic Energy Commission Contract AT(11-1)-1018 
 
T-U, b ,^ 
 
 ^ ^ P • -^ 1 = INTRODUCT ION 
 
 This paper gives a theoretical treatment of a cable driver and cable 
 teiTuinator design of very general applicability. In general^ problems of cable 
 driver and terminator design separate into two parts: 
 
 (1) the noise problem^, which includes both inter-channel 
 crosstalk and the ground noise problem of coupling between 
 different machines at different ground references; 
 
 (2) performance of the driver and terminator network^ subject 
 to prescribed level shifting^ to give sufficient termina- 
 tion--including an allowable limit for reflection under 
 the worst-case operation. 
 
 The first problem^ ground noise^ sometimes becomes considerable in 
 magnitude;, especially if a high-level system and a low-level system are connected „ 
 Obviously in AC systems ground noise can be suppressed by using a transfonner 
 to separate the longitudinal circuit coupling the two systems. In a DC system^ 
 a linear differential amplifier or a balanced transmission line can efficiently 
 reject ground noise. On the other hand^ from standpoint of economy a differential 
 amplifier terminator is not preferable if ground noise can be held within allow- 
 able limits. 
 
 The second problem mentioned above can be summarized as the matching and 
 level shifting problem. Cable impedance matching is not necessarily complete 
 because of the nonlinear amplification of the terminator. (This situation will 
 be discussed later.) A linear emitter follower driver and terminator are con- 
 sidered to give a better m.atch at the sending and receiving ends^ but level 
 shifting is rather poor. 
 
 From the above circumstances^ a driver and a terminator which give 
 reasonable matching and complete level shifting are preferable in most digital 
 applications. Such designs impose no additional level shifting circuitry at 
 the stage preceding the driver and the stage succeeding the terminator. 
 Accordingly^ the driver and terminator can be connected directly to logic 
 circuits without these intermediate stages. 
 
 These considerations suggest a suitably designed on-off driver and 
 on-off terminator. Table 1 shows possible combinations of an on-off driver and 
 
 -1- 
 
Driver 
 
 Terminator 
 
 \ — O 
 
 (a) 
 
 A/ — o 
 
 '+v. 
 
 1 
 
 \ / 
 
 (a) -> (a' ) ^ , , 
 
 0/ \-v 
 
 r v-^2' 
 
 (a) -> (b-) 
 
 (b) -> (c') 
 
 \ / +V^ 
 
 ■V / V 
 
 ''+VA / 
 
 (b) - (d') 
 
 .-v^y V / V 
 
 (d') 
 
 Table lo Possible Combinations of Cable Driver and Terminator 
 
an on-off terminator which give both noninverted information transmission and 
 arbitrary level shift. These combinations work as a regenerative OR circuit 
 and AND circuit in cooperation with the terminator. 
 
 Consider allowable limitation of the reflection for the driver- 
 terminator pair. The limits on the reflected wave depends upon the delay 
 time of the reflected wave from the main signal and upon the turn-on time of the 
 terminator transistor. When the reflected wave comes after the completion of 
 turn-on^ the limits on the reflected wave will be less severe: once the tran- 
 sistor comes to a stable state^ the reflected wave does not change the operating 
 state unless 
 
 min S(t) - |R(t) + N(t)| > critical signal (l.l) 
 
 Here^ critical signal means that signal level which breaks the saturated 
 operation. Signal min S(t) means that signal level which satisfies the lower- 
 most operating condition. If a represents the ratio of the critical signal to 
 the minimum signal^, then the above equation gives 
 
 ^ . JjKtWMt)! ^ „ (1.2) 
 
 mm S(tj ^ 
 
 Therefore, if a is designed to be 1/2, this gives 
 
 |R(t) + N(t)| < 1/2 min S(t) = l/k ts{t) (I.3) 
 
 For the most part, min S(t) is close to the 1/2 of the signal level difference 
 of the two stable states. On the other hand, if the reflected wave arrives 
 before the completion of switching, the allowable limitation on the maximum 
 reflected wave becomes more severe. 
 
 When the switching speed of the terminator is less than 2i/v (where 
 V is the group velocity in the cable and & is the length of the cable), the 
 condition 
 
 max |5S(t)| > |R(t) + N(t)| (l.^) 
 
must be satisfied. Here max |&S(t)| is the allowable signal level variation 
 which guarantees worst-case operation. This case is much more severe than the 
 former case^ but if the switching delay caused by reflection is allowed to a 
 certain extent^ the limitation of the reflection constraint is relaxed o 
 
 The operational equation which defines this delay is 
 
 1 - e"^ = (1 - e"^)F(p), 
 
 where 
 
 1 R '^^'^ 
 
 Here R is the ratio of the reflected wave to the signal and T is the relative build- 
 up time (or fall time) without the presence of the reflected wave. T is the cable 
 delay. The delay depends both upon the reflected level and the cable delay 
 time. Figure 1 shows the relative build-up delay caused by the reflection as 
 calculated from Eq, (1.5 ) = 
 
•H 
 -P 
 
 ft 
 pi 
 I 
 
 H 
 •H 
 
 2.0 
 
 o 
 
 •H 
 -P 
 
 O 
 
 D 
 H 
 Ch 
 
 (D 
 
 -P 
 
 o 
 -p 
 
 •H 
 
 oj 1.8 
 
 •H 
 
 -p 
 
 ^ 1.6 
 
 H 
 •H 
 
 f^ 1.1+ 
 
 1.2 
 
 1.0 
 
 
 
 
 
 
 k 
 
 
 
 
 
 q/ 
 
 
 
 
 
 / 
 
 
 / 
 
 
 
 / 
 
 / 
 
 ^ 
 
 
 
 y 
 
 /. 
 
 ^ 
 
 
 
 X 
 
 ^ 
 
 
 
 
 
 (1) Reflected wave delay 
 is 0.5 T 
 
 \2J Reflected wave delay 
 is 0.8 T 
 
 0.1 0.2 0.3 0.4 0.5 R 
 
 Figure 1. Ratio of Reflected Wave to Signal (r) 
 
2c EQUATIONS AT EQUILIBRIUM 
 
 Consider the circuit shown in Fig. 2. This circuit can be expressed 
 by the following equation (especially convenient for noise estimation): 
 
 C) = [A][V] + [B](p 
 
 (2.1) 
 
 (p-iBH-^) 
 
 Figure 2 
 
 » 
 
 (v) does not depend upon the termination condition. [A] and [b] depend upon 
 the circuit configuration o If the network is divided into cascaded subcircuits^ 
 each has an [A.]^ [B.] respectively. [A] and [b] can be expressed^ as in the 
 ordinary chain matrix, by the matrix of a subcircuit as follows : 
 
 [A] = ... [A^^] + [B2][A2] + [B^][B2][A^; 
 
 [b] = ... [b^JLb^JIb^] 
 
 (2.2) 
 
 Equation (2.l) should be solved for given input/output voltages and current 
 conditions (these are given by the speed requirements of the regenerating 
 transistor) and for given condition of input impedance, say, 
 
 -6- 
 
Z . - R^ ; 
 
 m 0' 
 
 ^. = QRn (2.3) 
 
 m \ ~j / 
 
 for one of two equilibrium conditions „ 
 
 Network N is a passive linear circuit; therefore^ the solution of 
 Eq. (2.1) under the condition of Eq. (2.3) a-nd given voltage and current 
 conditions^ must be positive. 
 
 Figure 3 shows a simplified structure. In this case the input circuit 
 gives only the termination condition and there are no energy sources in the 
 input circuit except for noise and the reflected wave. The input terminating 
 condition gives the change in the output voltage and current. This structure 
 is suitable for a cable drivers--terminator pair^ because the cable terminator 
 need not serve as an energy source for the driver; it sometimes causes trouble 
 with the preceding stage of the driver. 
 
 Figure k is the simplest circuit topology of the form of Fig. 3 and 
 also is a necessary and sufficient circuit for a level-shifting terminator. 
 (.) can be eliminated from Eq. (2.l) in this case: 
 
 Q) = [A](V) . [Z] (y (2.U) 
 
 Here [Z] is the output impedance matrix which corresponds to the 
 input condition. The problem is to find the network from Eq. (2.l) or Eq. (2,^) 
 subject to the condition of Eq. (2.3). 
 
 For the circuit of Fig. h^ the steady-state equation for the two input/ 
 output states, as indicated in Eq. (2.^);, is 
 
 :;) ■ G:; S) © • C" -l) (':) --- 
 
 Here (e ;, j ) corresponds to the input switch open; {e , j^) corresponds to the 
 input switch closed. 
 
% 
 
 Figure 3 
 
 Figure h 
 
11 Rq + R^ + Rg 
 
 R^Rg 
 
 21 R2(Rq + R^) + R^(Rq + R-j_) + RqR^ 
 
 A 
 
 ^o"\ 
 
 12 Rq + R^ + Rg 
 
 22 R2(Rq + R^) + R^(Rq + R^) + R^R^ 
 
 \i Ro -^ Ri - \ 
 
 ^ \^\^\ + R^) ^ RqR'] 
 
 ■'22 r^CRq + Rq) + RqCRq + R^) + RqRi 
 
 The impedance condition for shorted T gives 
 
 ^0^0 
 
 R. ^ ^0' 
 
 (2.7) 
 
 The 
 
 se equations determine R^ R^ R for the given (e^ j )^ (e^ j ) and V^ V , 
 
 Generally, to get the value of R , R , R , we must solve a set of higher order 
 
 equations. When we limit the input impedance variation AZ. , one of V., or V^ 
 
 m ' 1 2 
 
 cannot he given arbitrarily and the equation to he solved becomes of ths sixth 
 order. For simplicity, in our case we omit the input impedance variation con- 
 dition; therefore, V and V can be given arbitrarily within the limitation of 
 the passive network approximation, and the equation reduces to a third order 
 equation--or even a second order equation. 
 
 Equation (2.8) gives R and R as the function of R , e , e , j , j , 
 
 V^ and V^; 
 
 ^0 = 
 
 ';-^o-^iQ 
 
 1 + 
 
 (e^ - e^) - R2(j^ + jg)" 
 
 V. 
 
 \ ^ Jl^2 
 
 1 V^ - e, + j R^ " 
 
 2 
 
 1 
 
 '1 2 
 
 (2.8) 
 
 These two equations and Eq. (2.7) give an equation for R , If another impedance 
 relation, which specifies the impedance condition where T is open, is added, 
 the equation for R becomes much more complicated. From Eq. (2,7) we get 
 
10 
 
 AX + AX + A = 
 
 (2.9) 
 
 Another added impedance relation 
 
 R„(R, + Rp) 
 
 R ,-^0 
 
 «0*\ 
 
 2 
 
 Q > 1 
 
 gives a similar equation: 
 
 BqX + B^X + B^ = 
 
 (2.10) 
 
 where 
 
 X = Vg - e_|_ + j^Rg, 
 
 Ao = Aq(R2) 
 
 Bo(R^) 
 
 ^1 " ^1^^2^ 
 
 B^(R2) 
 
 Ag = A2(R2) 
 
 B^ = 
 
 B2(R2) 
 
 Equation (2.9) and Eq. (2.10) must have a common root; therefore^ the condition 
 (necessary and sufficient) is described by: 
 
 A, 
 
 
 
 =0 
 
 
 
 *0 
 
 ^1 
 
 =0 
 
 *1 
 
 \ 
 
 =1 
 
 ^s 
 
 
 
 \ 
 
 = 
 
 (2.11) 
 
 This gives a sixth order equation for R„. 
 
 By solving Eq. (2.11) and substituting the root into X^ V can be 
 determined. But this procedure is rather laborious; therefore we employ a 
 
11 
 
 practical method which checks the input impedance variation only after getting 
 a trial result^ and if 
 For example^ if we put 
 
 a trial result^ and if not satisfactory, revise the value of V and recalculate 
 
 V, = -6 volts e = -O.5 volts j = 0.1 ma 
 
 V = +6 volts e = +0.7 volts j = 1 ma 
 
 then the equation for R becomes 
 
 r| + 63.78 Rg - 77.98 Rg + 22.35 = (2.12) 
 
 This equation involves a trivial root; therefore, after factorization this 
 "becomes a second order equation, and gives the roots 
 
 R^ = 0.7^13 (KH) 
 
 R-L = O.i+995 (Kfi) 
 
 Rq = 0.1207 (Kn) (2.13) 
 
 These values show about 20 percent impedance variation. However, if we put 
 V = +12 volts, and the other conditions held to the same values as before, we 
 get 
 
 Rg - 2.676 (Kfi) 
 
 R = 1.05^ (Kn) 
 
 Rq = 0.09907 (Kfi) (2.14) 
 
 In the latter case [i^l is less than six percent for the extreme case; this will 
 
 
 be satisfactory. 
 
 So far, only mathematically accurate elements have been considered. 
 In practice every element has tolerance variations. Consider now any circuit 
 with all components having small variations but such that the behavior of the 
 
 UNIVERSITY OF 
 ILLINni.<; I iDDAnu 
 
12 
 
 circuit can still t>e described completely by the analysis above. The worst 
 case of the circuit behavior can then be described completely by an analytic 
 method. On the other hand^ if the circuit equation itself can be described only 
 by statistical methods, even in a macroscopic view, we cannot describe the 
 worst-case behavior by an analytic method. Another design philosophy of net- 
 works will be most economical: essentially a statistical design of circuits. 
 Worst-case design sometimes gives uneconomical results. But such discussions 
 are beyond the confines of this report. Therefore we will describe here only 
 the determination of the worst-case equation. Assume, for convenience, that each 
 element has small finite variations. Cross-terms of the variations can then be 
 neglected when estimating the incremental change. 
 
 A worst-case equation corresponding to Eq. (2.5) becomes 
 
 5) = l^lie)HV(i,,)] - t^(ii5)V3 (^-^5) 
 
 Here [A/ s], [V/ -pn]^ and [Z/ n] represent matrices, the elements of which 
 are multiplies by (l + e), (l - £), etc. The signs of e, 6^ T are determined by 
 the sign of the partial derivative, -v^ is always positive. Therefore, to 
 determine the appropriate sign of a variation, it is sufficient to check the 
 sign of the partial derivative of e. Equation (2.5) and Eq. (2.6) give 
 
 de de ^e ^e 
 
 5Ar>°' 3v->°^ 5F>°^ 5r^>° 
 1 2 1 
 
 ^e ^e 
 
 5F<o^ 3^<o» (2.16) 
 
 2 
 
 The circuit parameters for the equation e , therefore, should be 
 incremented as follows : 
 
 V 
 
 ^ - v^(i - e) Rq -* Rq(i + n) R^ - RgCi - n) 
 
 Vg - VgCl + e) R^ - R^(l + T]) (2.17) 
 
13 
 
 Correspondingly we have for the equation e 
 
 2 
 
 \ " \^^ "" ^^ 
 
 Ro-Ro(i- n) 
 
 R^ - R^{1 + n) 
 
 V. 
 
 V2(l - e) 
 
 R^ - R^(l - T]) 
 
 % 
 
 Ro(l + t) 
 
 (2.18) 
 
 Now, if we give (e , j ) for the condition (2.17) and give (e , j ) 
 for the condition (2.l8), we find we cannot assign arbitrarily (e , j ) for 
 condition (2.17) ai^d (o-,^ j-,) for condition (2.18), because there are more con- 
 ditions than degrees of freedom of the circuit. (s-,^ j-, ) under the condition of 
 (2ol7) determines the maximum level of the cutoff. (e , j ) for the condition 
 (2.18) determines the minimum level of the saturation. On the other hand (e , j ) 
 for the condition (2.I8) gives the minimum level of the cutoff, and (e , j ) for 
 the condition (2.I7) gives the maximum level of the saturation. These latter two 
 values should be deduced from the former two values, but the interrelation of 
 these values are not defined until the equation is solved. Therefore, the 
 equation for worst- case operation for specified input impedance can be given, 
 not by a straightforward method, but only by an iterative procedure. 
 
 Equations (2,5)^ (2.6) and (2.7); modified by the condition (2.I7) and 
 (2.18), give a third order equation of R . R cannot always be realizable for 
 preselected voltage and current levels, given percentage variation of circuit 
 elements, and given impedance condition. If we ignore the impedance condition, 
 as for the case of ordinary logical circuits, the worst case equation is deduced 
 much more easily. In our case, in addition to the passive condition, another 
 realizability condition (a sufficient condition) m.ust be met: 
 
 and 
 
 \< 
 
 for the root of the equation. 
 
 SQ^Rg + 2Q2R2 + Q^ - 
 
 A 
 
 Q^Rg + 2Q^R2 + 3Qq < S 
 
 J 
 
 (2.19) 
 
Ik 
 
 12 3 
 where Q , Ql , Q^, Q^ are the corresponding coefficients of R , R , R , R . 
 
 If Eq. (2.19) cannot be satisfied for given initial conditions^ these 
 conditions must be changed until they satisfy Eq. (2.I9). The following is an 
 example : 
 
 V = -6 volts + 5/0 
 
 e^ = -O.3 volts 
 1 
 
 j^ = 20 laa 
 
 V = +12 volts + 5/0 
 
 e = +0.8 volts 
 
 j^ = 1 ma 
 
 R^, Rq, R^, Rg vary + 5f° 
 
 input impedance for saturation termination = 91^ + 5^ 
 
 For these conditions^ equation of R becomes 
 
 r| + 579.7955 Rg - 1658.870 Rg + 1049.5538 = 
 
 (2.20) 
 
 and satisfies the condition (2.I9). The solutions are: 
 
 Rg = 1.90^3 (Kfi) 
 
 Rq = 0,1059 (Kfi) 
 
 R^ = 0.61+93 (Kn) 
 
 These values give about '^'p impedance variation when T is in either the short 
 or open condition. 
 
 Cable transmission loss enters through the variation of the resistance 
 
 ^Q. \, Rg 
 
 R'. The practical values of R^^ R., ^ R^ will be 
 
 Rg = 1.8 Kfi, 
 
 R - lOOfi, 5/0 
 
 R = 650fi, 3/0 (will be selected from 68on, 5fo) 
 
15 
 
 Practical configurations and data are reported in a separate paper. 
 
 S. Yamada and D. Foster, Data on Cable Drivers and Terminators for ILLIAC III_, 
 Digital Computer Laboratory, File No. 565, August, I963, 
 
3. TRANSIENT CONSIDERATIONS 
 
 The aim of this section is to investigate the nonlinear influence on 
 the wave form at the input of the terminator. 
 
 Consider the circuit of Fig. h. This is equivalent to the circuit of 
 Fig. 5 if the ground, potential problem, is neglected. Z and V„ are the 
 equivalent impedance and supply voltages respectively. V and Z vary accordingly 
 as the input switch changes and can "be represented as a function of time t and e . 
 Assume now that these changes are rather slow compared to the operation of the 
 switch. 
 
 For the first case^ consider the switch open at t < 0^ and closed at 
 t = 0. e equals e'(t) + V (t); e' is the voltage change at the terminal 
 (2, 2') which is caused by the voltage change at (l^ I')- Consider now a small 
 time interval T. For this interval, the circuit is assumed to be linear. 
 
 1' 
 
 € 
 
 2' 
 
 
 
 Figure 5 
 
 The equation of the circuit, neglecting cable loss, becomes 
 
 'cosh pT 
 
 -1 
 
 R sinh pT 
 
 R sinh pT cosh pT 
 
 (3.1) 
 
 ^ = -VqW - i^Ro 
 
 For a first approximation, V (t) is taken as a staircase function, that is 
 
 -16- 
 
17 
 
 t-t 
 
 i+l 
 
 VQ(t) = Z 
 
 i 
 
 V(t. )6(t - t. )dt 
 
 (3.2) 
 
 t-t 
 
 The successive values of V(t.) converge to the equilibrium value. From 
 Eqs . (3.l) and (3-2)^ we obtain 
 
 e^{t) - 
 
 V^(t - T) 
 
 -pT 
 :::: e 
 
 
 + V 
 
 (^ - ^) ' ^ ^ ^o(^ - ^^'^-'^ 
 
 for 
 
 t < T + T 
 
 where 
 
 I = R(i) + pi(i) 
 
 ^0 = ^0 ^ ^ 
 
 (3.3) 
 
 For T > T the equivalent source voltage V_(t) increases successively^ 
 and Z_(t) decreases. These changes add together. The circuit design in the 
 preceding section satisfies 
 
 5^0 
 
 for the first approximation. Therefore^ the voltage change at the input of the 
 terminator is equal to 
 
 •5Z, 
 
 6Z, 
 
 V^ + 6V, 
 
 V. 
 
 2Rq "0 2Rq 
 
 This change shows the increase of voltage for a given magnitude of impedance 
 variation. 
 
18 
 
 By the same method as ahove^ e (t)^ which corresponds to the switch 
 suddenly opened, can be calculated. In this case, instead of inserting a voltage 
 source, we insert a current source: 
 
 Vo(t) 
 
 ° - I(t) 
 
 ^o-^^o 
 
 at the place of switch. Therefore, e (t) becomes 
 
 1 , Mi) 1 +^ 
 
 Vo(t - T) - Rq V^Ct - T) - Rq ^-PT 
 
 e^(t) = VQ(t - T) - — - — R(i) + — 2 . , R(i) Z7wir7r~7^] 
 
 - 2Rq - 2r/ - 2Rq ^ ^ 
 
 (3.^) 
 
 for 
 
 t < T + T 
 
 V^(t) is again the staircase function for t > T + T and is also determined by 
 the resistance characteristics of the transistor and e . For t > T, by the same 
 observation as before, e (t) decreases to the equilibrium voltage. These 
 results are shown in Fig. 6. 
 
 Equation (2.3) represents multiple reflections; the first reflected 
 wave, delayed by 2T from the main signal, is 
 
 ,Mi)v(t).iiil^!°^ 
 
 - 2Rq ^0^^^ - 2Rq dt 
 
 In both cases, if the imaginary part of I (which is essentially wiring inductance 
 or parasitic capacitance) is small, the wave form distortion depends mainly upon 
 the characteristics of the terminating transistor. 
 
 The above analysis shows the equilibrium, level is higher in absolute 
 magnitude both in the switch-on and switch-off cases; it is better to take into 
 account these situations for a worst-case design. 
 
19 
 
 
 
 r 
 
 © 
 
 Driver on 
 
 Driver off 
 
 (T) Be^ caused by bV^it) 
 
 (2) 6e caused by gZ^ 
 
 © ©-© 
 
 (k) Beg caused by bV^ + bZq 
 
 Figure 6. Dependence of e„(t) on Variations of V^^ and Z_ 
 
h, NOISE PROBLEMS 
 
 Noise problems which arise when two computers are connected are 
 sometimes very complicated o Noise is composed of two parts « One part is noise 
 from adjacent channels through direct coupling between wires; the other part is 
 noise from the ground circuit. The noise ratio of the former can be expressed 
 
 by 
 
 ,/|^..„,,^) 
 
 ^ dx + Z 
 n^ 
 
 £ 
 
 (^^^^n(x)R^e" 
 
 2JCUX _ jud 
 
 V V 
 
 dx 
 
 (^^1) 
 
 The first term represents the direct noise to the receiAz-ing end^ and 
 the second term represents the reflected noise at the sending end. Ordinary 
 Zmn's; Ymn's are reactancesj^ therefore^ the wave form of the noise has a pulse 
 shape. This can be easily shown by the Laplace transformation of Eq. (^.l). 
 
 The second term of Eq. (^.l) shows the distributed pulse train along 
 the time axis, which length corresponds to the pass length of the noise. For 
 coaxial cable, Zmn, Ymn are small. This can be checked by experiment. 
 
 The second noise contribution arises from the mechanism shown in 
 Fig. 7- When we use a transformer, longitudinal pass can be shut off; of if we 
 use a balanced system it can be eliminated. Otherwise this kind of noise cannot 
 be eliminated. 
 
 side pass 
 
 longitudinal pass 
 
 hypothetical ground potential 
 
 Figure 7 
 
 Estimation of the noise is as follows. Let V , V , I , I be the 
 potential differences and the currents referred to the hypothetical ground 
 potential. Translate these potential differences and currents by matrix [S], 
 
 -20- 
 
21 
 
 [S] = 
 
 1 
 2 
 
 1 
 2 
 
 where now 
 
 [e] = [S] • [V] 
 [1] = [S] . [I] 
 
 (^.2) 
 
 The equation of the transmission line is 
 
 (1,) 
 
 cosh [r]-^ 
 w~^ sinh [r]^ 
 
 w sinh [r]^"" 
 cosh [r]i 
 
 
 The matrix [r] may he considered diagonal^ hut w is not a diagonal matrix, 
 
 [V] = [Z] • [I] 
 
 [z]^ = [z] 
 
 [z] = S[Z]S" 
 
 Wq = [r]"^ • [z] 
 
 w = 
 
 
 hi ^ ^22 7 A 1_ 
 2 " 12^ 7^ 
 
 h ^hl ' ^22^ 7^ 
 
 ^22^ 7 
 
 22 
 
 1 
 
 + Z 
 
 12/ 7 
 
 (^A) 
 
 Therefore^ if Z = ^oo-? "'^■'^^ longitudinal pass and the side pass are independent 
 
 of each other^ and of course no noise occurs. But for an unbalanced cahle this 
 
 cannot be satisfied^ while, in the balanced cable Z is equal to Z . Therefore 
 in this latter case there exists very little longitudinal noise. 
 
 Equation (^.2) also illustrates this situation when V = -V (i.e.,, 
 balanced transmission); there exists no longitudinal crosstalk. In any case 
 
22 
 
 from the above considerations a pair cable will give better isolation from 
 longitudinal noise. 
 
 Figure 8 shows the measurement of ground noise. 
 
 €[ 
 
 'V 
 
 Figure 8. Measurement of Ground Noise 
 
m ? 01969