appendix a: resume of practical line characteristics978-1-4615-9707... · 2017-08-24 · appendix...

Appendix A: resume of practical line characteristics

In Chapter 2 we defined the parameters and properties of an ideal lossless line, and subsequent analyses and discussion were based on that ideal concept.

Figure A.I summarizes the structure, characteristic impedance and application potentialities of some practical lines. Some points of general interest are:

• Cases (a) and (e), only, are 'balanced' lines. • Zo is only weakly dependent on the accuracy of mechanical dimensions

because of the logarithmic dependence. Thus, if (d/a) increases by an order of magnitude, loge(d/a) increases only by a factor of 2.3.

• Zo, tu , for any of the lines shown, can be found quickly using a TDR technique. For the determination of Zo low-inductance metal film resistors can be used as test terminations.

Attenuation has been ignored, since lossless lines have been assumed. However, attenuation though often small, is finite in practice. Thus for the cable type UR-67 (50 n, polyethylene) one manufacturer quotes an attenuation of some 2 dB per 100 ft at an operating frequency of 100 MHz. This means a reduction in signal amplitude of about 20070 at that frequency, over a distance of 100 ft. The effects of attenuation in fast pulse work are minimized by choosing a low-loss cable and interposing 'repeaters', if the signal level has become degraded, at appropriate points when long cable runs are used.

For analogue signals a repeater must be a low-noise, wide-band amplifier: for a digital logic signal a repeater is a 'reshaping' or 'regenerator' circuit that restores pulse rise and fall times and logic levels (see Chapter 7).

90

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Appendix B: laboratory demonstration work

This appendix is written for those readers who wish to acquaint themselves, in the laboratory, with the practical aspects of transmission-line pulseelectronics but who do not have ready access to very wide-band oscilloscopes (either real-time or sampling) and TDRs.

GENERAL CONSIDERATIONS

Consider the set-up in Fig. B.1. PO is a pulse generator, with output impedance Zo, that supplies either a triggered or a free-running rectangular waveform to a transmission line in the form of a coaxial cable of characteristic impedance Ro and one-way delay-time td' PO also supplies a trigger pulse to the cathode ray oscilloscope (CRO). Waveforms vI(t), vT(t) are displayed on the Y l' Y 2 channels respectively.

Let the 10070 -90% rise-times of the pulse generator, cable, and oscilloscope

Trigger eRo

J PG

Fig. B.l Set-up for basic practical work.

92

CIRCUIT DESCRIPTIONS AND OBSERVED WAVEFORMS 93

be tpro ter, tor respectively. If the oscilloscope is to exhibit distinctive horizontal sections of waveform on the Y1 channel, and thus illustrate the distributed nature of a transmission line as compared with a lumped circuit, for ZT * Ro it is necessary that:

(B.1)

Note that ter has been ignored in all our discussions because we considered only lossless lines. However transmission lines do have a finite, though small, ter; the interested reader is referred to the book by Matick listed in the References. The 50 {} cable type RG-58U has ter< 1 ns per 10 ft length.

As yet, tpr is unspecified but we can make an estimate of a possible value, e.g. EeL, which employs the long-tailed pair, switches well within 4 ns.

Laboratory oscilloscopes with a bandwidth in excess of 30 MHz (tor"'" 12 ns) tend to become expensive. It is thus evident that tor has a dominant effect in the determination of the minimum acceptable td.

Using the figures quoted, we require:

td> 6.5 ns (B.2)

Allowing a cable delay of 5 ns/m, inequality eqn (B.2) is easily met with, say, 2.5 m of cable which is not an impractical length.

It should be appreciated that the demands on tor are most easily satisfied by using, say, 50 yards of cable on a reel. However, long lengths such as this introduce the problem of interpreting waveforms distorted by cable losses, which is best left until the case of the 'loss-free' cable has been understood.

CIRCUIT DESCRIPTIONS AND OBSERVED WAVEFORMS

A block schematic for the cable drive circuit is shown, between the dotted lines, in Fig. B.2. A pulse from virtually any low-cost, rectangular-waveform generator (including one based on a simple '555 timer' astable circuit), is squared-up by passing it through two TTL stages, TTL!, TTL2 connected in series. (If the waveform generator has a TTL output, TTL1, TTL2 are unnecessary but if a sinusoidal signal source, only, is available its output can be squared-up by using a TTL Schmitt trigger.)

Fig. B.2 Block diagram of cable drive arrangement.

94 APPENDIX B: LABORATORY DEMONSTRATION WORK

A

FromTTL2

RT 1 :~cable --------,---~--

Ground plane

Fig. B.3 Simple form for S of Fig. B.2. 011 0 4 = Schottky diodes; 0 21 0 3 = IN4148; R = 1 kO (metal film); Ro = 50 0 BNC termination;

C1 = Ceramic capacitor (e.g. 1000 pF) for rail supply decoupling.

The output of TTL2 is not suitable for driving a cable directly for two reasons. First, pulse rise-time is relatively large ( :::::; 10 ns); second, the output impedance of a TTL stage is not known with certainty and varies over the output voltage range. Pulse shaper S overcomes these difficulties. Rise-time is reduced by 'slicing' a section from TTL2's output. A known source impedance is achieved by arranging for S to drive a known current into a cable with a specified resistor connected across its output terminals.

One form for S is shown in Fig. B.3. The 'earth' should be a printed circuit board copper ground plane. The p.d. across RT is zero when A is at logic '0' . When A is at logic level '1' , point C assumes a potential vo, where

(B.3)

~VD is the sum of the voltage drops across conducting diodes D2, D3, D4 •

Variation of Vee gives amplitude control. Equation (B.3) is applicable provided, under all conditions,

+ 1 V> vo> -0.5 V (B.4)

Figure B.4 shows the delay-line 'differentiation' that results when a length of cable (RG213U), shorted at the far end, is connected to C. (For this particular test a wide-band oscilloscope is necessary.)

In cases where it is desirable to use a fixed voltage supply Vee' the resistor R of Fig. B.3 may be replaced by a variable current source such as a suitably biased common-base transistor stage with a potentiometer in its emitter circuit. Thus, in Fig. B.5, which shows an improved version of the shaper S of Fig. B.3, Vee and RB are chosen so that zener diode Dz operates in the breakdown region. Q, operating in the common-base mode, supplies a collector current I from a high incremental source impedance and adjustment of Rv permits variation in the magnitude of I, which is given by

I:::::; Vz/(Rv + RE) (B.5)

where Vz in the p.d. across Dz in the breakdown region.


Fig. B.4 Waveform at C of Fig. B.3 with a line shorted at the far end connected to C. Vertical scale 100 mV/cm; horizontal scale 20 ns/cm.

c,

FFromTTL2 D,

Fig. B.S Alternative form for S of Fig. B.2. O2, 0 3, Os = IN4148; 0 11 0 4

= Schottky diodes; Oz = IN7S1A; Q = Any pnp with Cob < 3 pf; C1 = 1000 pF Ceramicon; Rr = SO n BNC termination; Rv = 1 kn; RE = 220 n;

RB = 1 kn; Vee ~ 10 V.

/ is sensibly constant even when Vee is not well stabilized. When the output of TTL2 is at '0' it acts as a current sink for /; when its output is at '1', / divides between Ro and the cable. As with the simple circuit of Fig. B.3, the sending-end impedance for any pulses reflected from the far end of the cable (Zr;f::.Ro) is Ro provided the condition (B.4) is satisfied.


Fiq. B. 6 View of practical cable drive unit.

Note that Q in Fig. B.5 need not be a high-speed transistor because it never switches, but it must have a low Cob in order not to degrade, unduly, the output pulse rise-time. For operation with a 5 V rail, Dz can be replaced by a band-gap voltage reference source (e.g. Intersil ICL 8069, which has a breakdown voltage of 1.23V approx.), provided R y , RE are reduced in value.

A practical version of that section of Fig. B.2, between the dotted lines, that includes the shaper S of Fig. B.5 is shown in Fig. B.6. For mechanical rigidity and protection of components from dust and damage the assembly consists basically of a box approx. 3 in x 1 in x 1 in (76 x 25 x 25 mm), one side of which is a p.c.b. with copper covering its inside surface to form a ground plane . Aluminium brackets at right angles to the board constitute the ends and in these are mounted BNC 50 (1 panel sockets for input and output cables. Holes in the board permit the outer cases of low-inductance ceramicon decoupling capacitors to be soldered directly to the ground plane. Another hole accommodates a DIL socket for removal of the (TTL) IC package.

The internal construction and wiring are visible if the remaining sides are made of transparent plastic pieces screwed together. Holes allow Ry to be adjusted externally and accommodate small sockets for power leads. An (optional) plinth ensures that the unit stands sufficiently far above a bench to allow the cable sockets to be easily accessible.


Fig. B.7 Waveforms for Vl(t), upper trace, and VT(t), lower trace, for the scheme of Fig. B.l using the unit in Fig. B.6. ZG = Ra = 50 0; cable hasRa = 500; ZT = RT = 75 O. Horizontal scale: 40 ns/major division; vertical

scale: 50 m V Imajor division.

--------~-.----~----1_----5V

1.2V

Re.

1.2V

To cable

Fig. B.a Further, improved, form for S of Fig . B.2.

Figure B.7 shows waveforms obtained using the unit of Fig. B.6. Many improved designs for the circuit of Fig. B.S exist. Thus, Fig. B.S shows, in outline only, a scheme that offers much greater isolation between the input and output circuits . It should, like the circuit of Fig. B.S, use a p.c.b. with a copper ground plane.

In Fig. B.S, Q3 gives a collector current I that is switched by the long-tailed


pair comprising the high-speed transistors Q •• Q2 (e.g. 2N4260). The conducting state ofQ •• Q2 depends on the logic level at the base ofQ •. D. prevents the possibility of reverse breakdown in Q. should large amplitude inputs be applied.

Appendix C: general notes on waveform observation

The observation of waveforms with 'fast' edges, e.g. ECL signals, can present problems to the less experienced engineer involved with pulse and digital system hardware. The aim of this appendix is to offer some practical hints to help the reader avoid some of the more obvious pitfalls.

In order to ease the transition from the familiar to the less familiar, we review first the observation of waveforms in lumped systems. We then consider the special precautions necessary for waveform monitoring with distributed systems.

LUMPED SYSTEMS

Most basic laboratory oscilloscopes (CROs) with a bandwidth in the region of 30 MHz (and a corresponding rise-time of about 12 ns), have BNC input connectors for use with coaxial leads, the outer-conducting sheaths of which are earthed when connected to the instruments. These CRO leads thus offer good screening against unwanted, but often unavoidable, electromagnetic interference that can be a problem particularly with low-level signals.

Suppose it is required to monitor a waveform having a minimum transition time, te , of, say, 200 ns appearing across a resistor Ra, one side of which is earthed in the circuit under test. The typical set-up shown in Fig. C.l(a) could be used. The CRO lead is assumed to have a length of approximately 1 mcorresponding to a two-way delay time 2td of about 10 ns - and a total capacitance, CT , of the order of 90 pF. The CRO input resistance, R1, and input capacitance, CI> tend to be standardized by equipment manufacturers at 1 MO and about 30 pF, respectively.

The measurement system can, in this case, be regarded as lumped (see

99

100 APPENDIX C: GENERAL NOTES ON WAVEFORM OBSERVATION

I _ ~ m_---r-y eRO front panel

~ C,

connector ......,p-<>--...... Probe clip, hook, or 'prod'

(a) (b)

Fig. C.l (a) Acceptable measurement scheme for tc» 2td ; (b) Section to right of broken line shows equivalent loading on RG • RI = 1 MO,

C1==30 pF.

Chapter 1), as 2td« te , and the equivalent loading across Ro presented by the lead and oscilloscope, is shown to the right of the broken line in Fig. C.l(b). If the magnitude of either the resistive or capacitive loading across Ro is unacceptable, then we can make use of the long-established and popular 10 x ('10 times') probe, one version of which is shown schematically in Fig. C.2(a). The insulated probe-housing provides a facility allowing mechanical adjustment of a small compensation capacitor Cp, which is connected in parallel with a 9 MO resistor. The outer conducting sheath of the cable has an earth clip connected to it, usually close to the point where the cable enters the probe-housing.

The probe is correctly adjusted when V2 is a scaled replica of VI. For this to occur it is necessary that CpRp = (CT + C1)R1. This condition is achieved, in practice, by observing the CRO display, when the probe is used to monitor a

r= t,=200ns

-I l-

V

(a)

CP(C,+CT)

(CP+C,+CT)

(b)

= O.9CP

Fig. C.2 (a) Use of 10 x probe (signal attenuated by factor of 10); (b) Load across RG for correct probe adjustment, CpRp = (Cr + C1)R1.

DISTRIBUTED SYSTEMS 101

square-wave, voltage-calibration signal normally available at a front panel connection of the CRO, and making a mechanical adjustment so that there is no overshoot or undershoot on the edges of the observed waveform. In one form of mechanical adjustment a screwdriver (preferably non-metallic) is used to set the position of a screw, and thus effectively alter the dimensional separation of the plates of the small trimmer capacitor Cp , via a hole in the probe-housing provided for that purpose.

After the probe has been adjusted correctly the attenuation factor A ( = v/v2) = 10 and the equivalent load presented across RG is 10 MO in parallel with a capacitance slightly less than Cp , as shown in Fig. C.2(b) for the normal case (CT + CI»> Cp: typically, Cp::= a few pF.

Active probes offering a high input resistance and low input capacitance without significant attenuation are available as CRO accessories from some manufacturers (e.g. Hewlett-Packard). They can also be constructed in the laboratory using field-effect and bipolar junction transistors, but they generally require auxiliary power supplies which may not be available from the circuit under test.

DISTRIBUTED SYSTEMS

Suppose, in Fig. C.I, that te = 2 ns instead of 200 ns as considered above. The condition 2td« te no longer holds, the system must obviously be regarded as distributed, and the set-up of Fig. C.I(a), without modification, would be unsuitable for waveform monitoring. There would generally be reflections at the ends of the oscilloscope leads which could interfere with, and possibly cause malfunction of, a circuit under test - particularly a logic gate. The waveform observed will have transition times limited, of course, by the oscilloscope rise-time. The arrangement of Fig. C.2(a) is also unsatisfactory because the condition CpRp = (CT + CI)RI depends on the assumption of a lumped system which is not true in this case.

To minimize, if not eliminate completely, reflections at a CRO input terminal in standard run-of-the-mill laboratory CROs, real-time wide-band CROs and sampling CROs, it is obligatory to use correctly terminated CRO leads. For a low-cost CRO a standard, commercially available, 500 termination can be connected via a BNC 'T-piece' as shown in Fig. C.3(a).

For a sampling oscilloscope, with which we will be concerned for the remainder of this appendix, the 50 0 is a built-in feature. Input connections can be of the BNC or the 'hermaphrodite' General Radio (GR) variety. There are no real problems here since GR/BNC (male) and GR/BNC (female) connections are readily available. For simplicity in drawing we will show, schematically in section form, BNC connectors only.

However, the arrangement in Fig. C.3(b) is still not suitable as it stands


I

__ J I

GR connec::.J

:1,-= nl~~+ ~===~

oon"o' - n \ jJ;~ , 50 n (built-in)

Standard 50 n termination (a) (b)

Fig. C.3 Oscilloscope connections for fast-edge monitoring. (a) Basic laboratory CRO; (b) High-speed CRO, e.g. Sampling CRO. A GR input connector is often used but BNC - GR and GR - BNC adaptors are

available.

~CRocase

Fig. CA Passive probe for observing waveform across RG •

(R+Ro)>>RG·

because it presents to the test circuit, at the input to the cable, an impedance of 50 n which will cause problems if it is required to monitor the waveform across a resistor, Ro, having a value comparable with 50 n. Clearly, a series resistor, R, must be incorporated to give, with the cable input impedance, Ro, a potential divider that gives a known, convenient, attenuation factor and does not load, significantly, the circuit under test (see Fig. CA). Ro is not heavily loaded if R is chosen so that (R + Ro)>> Ro, i.e. (R + Ro) ~ lORo. This is the principle of the wide-band passive probe.

Table C.llists some useful values of R and associated attenuation factor A = (v/v2) = [(R + Ro)/ RoJ, for the case Ro = 50 n, Ro = 50 n. For Ro = 50 n

Table C.I Convenient values of R and A for passive probe of Fig. CA.

R (!l)

1200

950

450

R+Ro (!l)

1250

1000

500

A = (v,/v,)

25

20

10


Dielectric

PVC cover

Outer braid of coax. cable

Fig. C.5 A practical realization of probe tip in Fig. C.4.

and R = 450 n the loading effect of the probe is such as to give a voltage reflection coefficient of about -0.05 for a 50 n test circuit cable that RG is intended to match. This is often acceptable but lower values of Rare generally not advisable.

A passive probe is a purchasable item. However, the reader may wish to construct a simple low-cost version, as in Fig. C.5, for himself. A lowinductance resistor, R is soldered to the inner conductor of the coaxial cable, and the outer braid is twisted to form a lead that is soldered to an earth clip, 'prod', or hook. All wires must, of course, be kept as short as possible. In a more elaborate form of the probe, R can be surrounded by a cylindrical metal sheath soldered to the cable braid. In this design the sheath can surround all but the probe-tip end of R and be separated from it by an insulating sleeve. An earth clip is soldered to the metal cylinder. For a well-made probe the signal displayed on the CRO screen is merely an attenuated and delayed version of the actual circuit waveform.

It is important to remember that the set-up of Fig. C.4 is not suitable if the signal being monitored does not have an earth rest level. Such is the case with the signal from an ECL gate. Direct connection of the CRO passive probe, just described, might upset the d.c. conditions in the circuit under test. However, this difficulty is overcome by using a series coupling capacitor, shown as Cc in Fig. C.6. The magnitude of Cc depends on the maximum pulse width, (p' that it is required to display without appreciable 'sag' or 'droop' on the waveform. To avoid significant sag it is necessary that Cc(R + Ro) » tp. Typically, Cc is in the range 1 to 100 nF.

It may be possible, in certain circumstances, to obtain a waveform for observation without any loading at all on the circuit under test. This is the

R.

-o.an - 1.7SV I--t

t.

c, R ~=J~3 -+-= 1 \ J;

Fig. C.6 Use of coupling capacitor, Ce , to observe fast signals with significant d.c. bias levels.


'Primary' cable

! I

Vx[ foo Y,RX I I

- I --~RO t Vv i l

~. - - I (Rx+ Rol n 'Secondary' cables

Fig. C.7 A 'signal-splitter' circuit can be used for waveform observation: Rx=nRo; (vx/Vy) =n + 1.

case for a signal-splitting technique exemplified in Fig. C.7. The 'Primary' cable in the test circuit and each of the n 'Secondary' cables are all assumed to have the same characteristic resistance Ro. One or more of the secondary cables can be used for waveform monitoring. It is easily shown that all ofthe cables are correctly terminated if Rx = nRo: thus, if Ro = 50 {} and n = 2, then Rx = 100 {} and A = (vxlvy ) = 3.

Occasionally it is necessary to employ an attenuator to reduce the magnitude of a fast pulse signal which it is required to observe. This is the case when the pulse source is the output from a mercury-wetted relay pulse generator and an 'avalanche' pulse generator. Connection, even via a 10 X passive probe, to the input circuit of a sampling oscilloscope could well destroy the input stage of the instrument since it is normally capable of accepting a signal in the range of only approximately ±2 V about earth.

Switched attenuator boxes are available but fixed attenuator pads are easily constructed. These can be of the 'T' or'1/"' variety. A symmetrical-T type attenuator is shown in Fig. C.S. This is required to operate between two sections of cable each with characteristic resistance Ro. It can be shown that Zab' the impedance seen looking to the right between input terminals a and b, is equal to Ro, the load across the output terminals c and d, if Rl and R2 are chosen so that the following relationship holds .

.J(RI (R 1 + 2R2)] = Ro

Then the attenuation factor A = (v/v2) is given by,

A = (Ro + Rl + RJI R2

Fig. C.B Symmetrical-T attenuator pad.

(C.l)

(C.2)


Equations (C. 1) and (C.2) are useful for checking a given design, such as might be specified on a data sheet. But what if it is required to design an attenuator for a given Ro and A?

The relevant equations are,

Rl = Ro(A - l)/(A + 1)

and, R2 = 2RoA/(A2 - 1)

where, as before,A = (v/v2)

(C.3)

(CA)

(C.5)

In any problem involving attenuator pad design the requirements will generally be met only by using non-preferred resistor values. The choice then is to use selected, measured, components chosen from large batches or to choose close-tolerance nearest-preferred values and sacrifice some accuracy. The course selected depends on the problem.

Thus, if we require A = 2 when Ro = 50 n, a calculation gives Rl = 16.7 n, R2 = 66.7 n. These are not preferred in any range. Suppose, however, we choose Rl = 16 nand R2 = 68 n. From eqn (C.1), -J[RI (Rl + 2R2)] = 49.3 n, and from eqn (C.2), A = 1.96 which is close toA = 2. Two other useful sets of data using preferred values based on Ro = 50 n are: A = 5 for Rl = 33 nand R2 = 20 n; A = 10 for Rl = 39 nand R2 = 10 n. 'T' stages can be cascaded, in which case their attenuation factors are multiplied.

An alternative to the symmetrical-Tis the symmetrical-1I" pad of Fig. C.9. This may give more convenient resistor values in a particular application.

Z"-D-~~}+ b d

Fig. C.9 Symmetrical-1I" attenuator pad.

It can be shown that Zab = Ro if,

RA/-J[1 + (2RA/ R B)] = Ro

in which case,

Alternatively, we can write,

RA = Ro(A + l)/(A - 1)

and, RB = Ro(A2 - 1)/2A

(C.6)

(C.7)

(C.8)

(C.9)

Answers (including worked solutions to problems)

CHAPTER 1

Pl.l From the data given, td = l/u = 3/(3 X 108) S = 10 ns Thus,2td = 20 ns. ' a conventional circuit theory is applicable for tc» 2td. A safe choice

is tc = 1O(2td), i.e. tc ~ 0.2 J.ts. b conventional circuit theory is definitely not applicable if tc« 2td • A

safe choice is tc = (2td/1O) = 2 ns. Pl.2 tc = 2ltu when 1= t/2tu

Now tu = 5 ns/m '"" 5 ns/39 in :. I '"" (1.5 ns x 39 in)/(2 x 5) ns '"" 6 inches.

Pl.3 In Fig. S1.3, straight line OC corresponds to 2td = tc; horizontally shaded region gives 2td ~ lOtc; vertically shaded region gives 2td :s:;;(t/1O)·

10

Fig. S1.3 2td = 2ltu; for tu = 5 ns/m, 2td = 10 ns when 1 = 1 m.

106

CHAPTER 2

CHAPTER 2

P2.t [L] = [V] [T]/[/] [C] = [I] [T]/[V] Hence, a [LIC] = [V] [V]/[/][/]

and [../(LIC)] = [V]/[/] = [R] Similarly, b [../(LC)] = [T]/unit-length, because L, Care per-unit-Iength.

P2.2 a Zo = ../(LIC) as the line is lossless

Z(O) ='(73.75 x 10- 9) ='(73750) = 500

o ~ 29.5 X 10- 12 ~ 29.5 b tu = ../(LC) = .J(73.75 x 10- 9 x 29.5 x 10- 12) sift

= (10- 9) ../(0.7375 x 2.95) sift = 1.475 ns/ft

c tu = 1.475 x 3.281, as 1 m::::: 3.281 ft ::::: 4.84ns/m

P2.3 Zo = ../(LIC) and tu = ../(LC) = .JC(Zo.JC) = ZoC = 72 0 x 69 pF/m

:. tu ::::: 5 nslm

P2.4 a v = f(t-xv'(LC») = f(y),say,wherey = (t-xv'(LC»)

(:;) = (:.:;) (:~) = f'(y) ( - ../(LC») , where f'(y) = d~~)

Extending this argument,

where

Also,

and

Thus,

= f"(y) (../(LC») (../(LC») = f"(y) (LC)

f"(y) = d2f(y) dy2

( ~~) = (:.:;) ( ~ ) (a 2v)

at2 = f"(y)

(a 2v) ax2

= LC (~) at2

= f'(y), as (~) +1

or, L1C e;~) = (~:~) which is eqn (2.8) in the text.

107

108 ANSWERS (INCLUDING WORKED SOLUTIONS TO PROBLEMS)

band c follow by a similar argument, i.e. puttingy = 1+ :>N(LC) in b and y = (I + :>N(LC) + XJ(LC)] in c

d Let v = f(y) + g(w),wherey = I-:>N(LC) and W = I + :>N (LC)

f'(y) ( - -J(LC)] + g'(w)( - -J(LC)]

( - -J(LC)] (f'(y) + g'(w)]

Similarly,

(a2v) = LC (f"(y) + g"(w)] ax2

Also,

(~;) = (f'(y) + g'(w)] as ( ~ )

( a2v) and al2 = (f"(y) + g"(w)]

Thus (~) = LC (~) , ax2 al2

(~~ )

or, _1_(a 2v) = (a 2v) which, again, is eqn (2.8) LC ax2 al2

P2.S a When the step is at a point x' on the line the energy stored in the electric field is equal to (Total capacitance to point x') P/2.

:. WE = (Cx') P/2

b The energy stored in the magnetic field is equal to (Total inductance to point x') 12/2

:. WM = (Lx')J2/2

C WE = (CX')V2/2 = ( RLo2) x' (IR2 0 ) 2

and V = loR.

:. WE = (Lx')J2/2 = WM

-JL as-JC =-

Ro

d Total energy supplied to line when voltage step has reached point x' along it is W.

W = WE + WM = 2WE, from P2.5 c, above.

Now, the rate at which energy is increasing is the power, P, continuously supplied to the line, as the wavefront moves from x' to (x' + ox') in a time 01.

CHAPTER 2 109

p = (dd~) = edd~E) = e~E) ( ~,) = CV2u

But, U = Ih/(LC) and Ro = ~ ~

:. P = CV2IV'(LC) = V2 ~ ~ = V2IRo = VI

P2.6 By analogy with the argument leading to eqn (2.1),

v-(v+ov) = [(LI2)ox+(LI2)ox] (oi/Of) + [(RI2)ox + (RI2)ox]i :. - OV = [L(ox) (oi/M)] + R(ox)i

In the limit,

Similarly,

i - (i + Oi) = [(Cox) (ovIOf)] + (Gox)v :. - Oi = [C(ox) (ovIM)] + (Gox)v

In the limit,

( ai) (av) - ax = Gv + C at

P2.7 From problem (P2.6),

(a2v) = -R (~) _ L~(~) ax2 ax ax at

Also - ~(~) = G(~) + C(~) , at ax at at2

:. -L:t (:~) = LG (~~) + LC (~;~) From eqns (P2.7.1) and (P2.7.3)

(:;~) = -R (:~) + LG (~~) + LC (~;~)

But, - (:~) = Gv + C ( ~~ )

( ai ) (av) -R ax = RGv + RC at

(P2.7.1)

(P2.7.2)

(P2.7.3)

(P2.7.4)

(P2. 7 .5)

Hence, from eqns (P2.7.4) and (P2.7.5), after some re-arrangement of terms,


(::~) = LC (~;n + (RC + LG) (~~) + RGv

P2.8 Assume v = exp( - ax)f (I - ~)

where, a = R ~ ~ and u = lI.J(LC)

(P2.8.I)

then,

(:;) = -aeXp(-ax)f(/- ~) - ~ exp(-ax)f' (/- ~)

= exp(-ax)[ -af(/- ~) - ~f'(/- ~)] (P2.8.2)

(::~) = -aexp(-ax)[ -af(/- ~) - ~f'(/- ~)] + exp( - ax) [: f' (I - ~) + :2fll (I - ~)] (P2.8.3)

Also, ( ~~) = exp( - ax)f' (I -~ ) (P2.8.4)

( ~;~) = exp( - ax)f" (I -~ ) (P2.8.5)

For the case RC = GL the final expression in problem P2. 7 reduces to

(::~) = LC (~;~) + 2RC (~~) + RGv (P2.8.6)

Using eqns (P2.8.2) to (P2.8.5) we have to show that eqn (P2.8.I) satisfies eqn (P2.8.6). Now eqn (P2.8.3) can be rewritten,

(::~) = exp(-ax) [a2f(/- ~) + 2:f'(/_~) + :2 fll(1 - ~)] (P2.8.7)

(P2.8.8)

Eqn (P2.8.8) makes use of eqns (P .2.8.4) and (P2.8.5)

Nowa 2 = R2CIL = RG; 2alu = 2R· f C . .J(LC) = 2RC;_I- = LC ~ L u2

:. (::~) = LC ( ~;~) + 2RC ( ~~) + RGv, which is eqn

(P2.8.6)

CHAPTER 3 111

CHAPTER 3

P3.1 The 'boundary conditions' at x = 0 and x = I give:

PVG = 0; PVT = -1

Hence from eqns (2.33), (2.37) of Chapter 2,

PIG = 0; PIT = + 1

x=O x=/ PVG=O

@] PVT= -1

".JT" I 0.5V I 10 (VI t 20ns

20 VT

(bl PIG=O

PIT= + 1 20

10

l10mAI i(xl

(mAl

15ns -11= 15 nsf

20 x=O x=1

(el

(al

Fig. S3.1.

Figure S3.1(a), (b), (c) illustrate the solutions to parts (a), (b), (c), respectively, of the problem.

112

PVG=O

20

ANSWERS (INCLUDING WORKED SOLUTIONS TO PROBLEMS)

V,

PVT= + 1 (V)

20

tIns)

VJ I -1V

P,G = o~----,---==----, PIT= -1 ~

20

P3.2 See Fig. S3.2.

10 (b)

10 (~d n lO

~t(ns) 5 15

'---____ -----'1

(a) (e)

Fig. S3.2.

CHAPTER 3 113

P3.3 See Fig. S3.3.

PVG = - ~ ~-""-------:[I]=o=----'

v,tti 10

-t----.----r- tIns)

! tIns)

20 40

20

25

tIns) 10 30

40 (a)

1V-(b)

vIti x=114

tIns) 2.5 17.5 35

1~V-1; V

(e) v(x) t= 25 ns

v

t x=1 x=o x=//2

Fig. S3.3.


P3.4 The reflection chart is shown in Fig. S3.4(a). At t = 10 ns, Sw opens so the net input current is zero. However, the existing current is 20/3 rnA. Hence a forward current of - 20/3 rnA, corresponding to a forward voltage of - 1/3 V, must traverse the line. Waveforms are shown in Fig. S3.4(b).

Sw closes

Sw opens

tins)

(a)

P3.S See Fig. S3.5.

x=!

PVT=O v,(t) L ~V ( ) (V)~ ~tns

;: ~~):-'-t \ ,.;- ~(ns) ~t(ns) ~t ~ 10 2025

Fig. S3.4

PVT = + ~ PVG=O

--~----~--~=---~ o

10

20

30 I I

viti

I I tins)

(a)

Fig. S3.5

(b)

(b)

CHAPTER 3 115

P3.6 Application ofthe Thevenin-Norton transformation in Fig. S3.6(a) gives Va in (b). This, as shown in (c), can be resolved into positiveand negative-going step-edges. The reflection chart is shown in (d) and from this we obtain the required vI(t) shown in (e). For 40 ns> t> 20 ns the line voltage is zero and the energy stored in the electric field is also zero. However, energy is stored in the magnetic field and is dissipated in the source resistance for the period 60 ns~t~40 ns.

( 1

:lrn~~' l' J--r ----~ 50n

~ 10ns la)

50nO\--===::J VGcL?T.-. xL

Ib)

x=O x=1

PVG=O toc:"-----, PVT = -1

10

20 I I : 1 tins)

Id)

1V~ VG o -tins) t

40

Ie)

v,It) 1 -rrn-_+().5V _tins)

t ---0.5V 20

40 60

Ie)

Fig. 53.6

l1S ANSWERS (INCLUDING WORKED SOLUTIONS TO PROBLEMS)

Pl.7 Referring to Fig. 3.6 of the text: For 40 ns> t> 20 ns the line current is 40 rnA. Opening Sw imposes the boundary condition that the net line current at x = 0 is zero. This is achieved if a -40 rnA current step starts down the line. On a voltage reflection chart, which we plot because we are interested in VI(t) , this corresponds to a voltage step = - (40 rnA x 500) =

-2 V. The voltage reflection chart is shown in Fig. S3.7(a). For com

pleteness, behaviour is considered from t=O. Figs S3.7(b), S3.7(c) show vl(t), vT(t), respectively.

x=O x=1 (b)

PVG=O -I~------~=3=V::-{PVT= +~ ~ V,(t)

-5V _3~V

20

40

60

80

(a)

10

,l tIns)

I

65

70

Fig. S3.7

-+-r-----r-r-- tIns)

sb t 65

40

-+-r--r---- tIns)

40 50

(e)

CHAPTER 3 117

P3.8 The voltage change Ll VI is the sum of incident and reflected waveforms. In Fig. S3.8: a = [Ro/(Ro + Ro»); PyO = [(Ro - Ro)/(Ro + Ro») a For r = 1, 2, 3, ... n.

LlvI(2td) = aV(1 +Pyo) = [Ro/(Ro + Ro») [1 +Pyo) V

b Ll VI [2(2td») = a V(1 + Pyo)PyO Ll v I [3(2td») = a V(1 + PYo)p~o

Fig. 83.8

It follows that,

[LlVI [r(2td))] [LlvI[(r - 1) (2td)1] = PyO


P3.9 For Ra» Ro,

vI(t) = ~r-n .6vI[r(2td)J r~ I

vI(t) = ex V(l + pya> [1 + PYa + P~a . ... Pyan-1J and, PyaVI(t) = ex V(1 + Pya) (Pva + P~a· ... p.qaJ :. vI(t) [1 - PYa)J = ex V(I + pya> [1 - p.qaJ :. vI(t) = vI[n(2td)J = ex V(l + Pya) [1 - p.qaJ/(l + Pya) Nowex(l + pya)/(l - Pya) = [Ro/(Ro + Ra)J [Ra/ RoJ = Ra/(Ra + Ro)

"'" 1, as Ra » Ro. :. v.(t) "'" V(l - p.qa)

P3.10 a p.qa = exp( - tIT), given.

:. n log. PYa = - tIT

But n log. [~:: ~:] = n[log.[1 - (Ro/ Ra)J -log.[1 + (Ro/ Ram

"'" n [ - (Ro/ Ra) - (Ro/ Ra) + negligible higher-order terms]

"'" - 2nRo/Ra

. 2nRo t --""'-Ra T

However, t = 2ntd

• Z1iRo = ZJ'1td

Ra T

:. T = tiRa/ Ro)

If CT is the total line capacitance, td = (CT/C) .J(LC) Substituting this and Ro = .J(L/C) in the expression for T,

T "'" RaCT

b It follows from problem P3.9 and part a, above, that

vI(t) "'" V[1 - exp( - t/CTRa)]

The interpretation of this result is that, for an open-circuited line and Ra» Ro, td« CTRa and the incremental voltage steps at the

V,(t) t ~ v,(t)

~ ~--------~ (a) (b) (e)

Fig. 53.10

CHAPTER 3 119

input are very small in comparison with V. The charging of the line, actually of the form indicated in Fig. S3.1O(a), approaches the form of Fig. S3.1O(b), corresponding to an effective equivalent circuit shown in Fig. S3.1O(c).

P3.ll See Fig. S3.11(a) to (d).

la) Ib)

rv ~+1 VI

IV) tins) 0,06 2026

~ IV)

tins)

0,0 1016

Ie) Id)

tins)

t=-VT=O

V,

VT

V,

t 10

I --1V

I

tins)

110

V, tins)

__ -+-----k:- + 0.4 V

I -0.4 V~----+---;~

t t t 20 50 70 100

Fig. S3.11


CHAPTER 4

P4.1 Figure S4.1(a) gives the graphical construction for finding vl(t), vo(t); Fig. S4.1(b) shows the waveforms obtained; Fig. S4.1(c) is the reflection chart.

x=O x=1

2

l t Source characteristic

10ns v=2-(100 !l)i

(c)

10 20 30

~ ~ tIns) i(mA)

v,ttI VT(t) (V)

-1 -1 Charged line characteristic v= -1 +50i

(b) (a)

Fig. 84.1

P4.2 It is instructive to regard Vy as the input voltage to the line. Fig. S4.2(a) and (b) are solutions to parts a and b respectively, of the problem. The graphs are scaled so that the Bergeron lines are at right angles. If Vx is regarded as the input to the line - the usual convention - then the graphical solution is in the right half plane i> 0 (see Example 4.2 of text).

CHAPTER 4

-40

2

Line voltage before Sw closes_ 1 V

v

~O.1 V -+-~~---+---

i(mA) 0,0

- - -l- - t!-------I t= 10

Characteristic for 2 V in series with 100 ()

(a)

Fig. S4.2(a)

v(V)

i(mA) o 10

-tins)

Characteristic for 2 V in series with 100 ()

Fig. S4.2(b)

121


Characteristic for 2 V in series with 100 D, Sw open

Line voltage, Swclosed -

-40 -20

Fig. S4.3(a)

v,(O +) ~ 1.33 V

i(mA)

2

vx(t)

____ L..-+--,

tins)

-40 -20

Fig. S4.3(b)

P4.3 Fig. S4.3(a) and (b) are solutions to a and b respectively. (See comment in problem P4.2, above, regarding the location of the plot in the v, i plane.)

P4.4 1 m = 3.281 ft. tu = 1.475 ns/ft (given.)

:. td = 3.281 x 1.475 = 4.84 ns.

When Sw closes the initial circuit is shown in S4.4(a). A-50 V step travels back along the line towards the 47 kO resistor. The reflection coefficient Pva is

Pva = [(~7 kkOO: :00 ~] = [1 - (500/47 kO)]/[1 + (500/47 kO)]

r "'" [1 _ (~)] - [1 _ (_2 )] o ,Pva 50 kO - 1000

CHAPTER 4

X~O x~/ PVG .------""PVT = 0

50n

~~ 4.84-

vzlt)

I tins)

t ~ ~+50 V

100V 0 1 V,IOT) ! vz(O T ) ~ 50 V tins)

~ 0.1 V (Exaggerated, I~ 'forelarity)

-9.68 0 ~ r T - tlnsl

9.68 la) Ib) (e)

Fig. S4.4

123

There is no reflection at t = 9.68 ns because the line is correctly terminated at the switch end with Sw closed.

vz(9.68 ns) = 50(1 - PVG) = (50 x 2/1000) V = 0.1 V

Figure S4.4(c) shows how the waveform vz(t) is obtained from the reflection chart of Fig. S4.4(b).

P4.S Figure S4.5 is self-explanatory,

vz(9.68 ns) = (50 - 49.9 + 16.63) V = 16.73 V

4.84

I I _ ,

1-===--------:::>1 PVT - -"3

tlnsl

150V I 1 ~50V

vz(t1 l -16.73V

t t -tlnsl

9.68 12

116.73VI 1-49.9Ixl-~)~+16.63V

(al Ib)

Fig. S4.5

P4.6 As the RG 8A/U is 1 m long it has a total line capacitance CT = (29.5 pF/ft) x (3.281 ft) = 100 pF

When Sw opens, RG = 47 kO» 50 0, and the circuit for estimating vx(t) is shown in Fig. S4.6(a). The time constant RG CT = 4.7 p,s » td·


t Swopens

(a) (b)

Fig. 54.6

As in the solution to problem P3.1O this means that the incremental voltage steps in Vx are minute in comparison with 100 V and the waveform for vx(t) resembles an exponential with time-constant 4.7 p,s, as shown in Fig. S4.6(b). Obviously the line is not fully charged until an interval of "" 5 x 4.7 p,s, say 25 p,s, has passed after Sw opens. To obtain 50 V pulses across RT on a repetitive basis means that Sw must operate at a pulse repetition frequency < (1125 p,s), i.e. 40 kHz.

P4.7 Refer to Fig. S4.7(a). 'a' the point of intersection of the output characteristic of gate A, in Fig. 4.18, and the input characteristic of gate B, gives the initial condition prior to the '0' to '1' transition.

The Bergeron lines describe the '0' to '1' transition: the resulting waveforms for v,(t), vT(t) are shown in Fig. S4.7(b).

-1

(a) (b)

Fig. 54.7

CHAPTER 4 125

Ii)

v

~VT(t)

- ...--------- ------ -------(ii) " , state

------------ o t

td 3td -2V (iii) '0' state

(a) (b)

v

V~T(t) ----------

Iii) '" state - ----------

t t t

liii) '0' state td 3td

RT =25{J

(e) (d)

Fig. S4.8

P4.8 For the EeL gate set-up for Fig. 4.22(a) with RT = 25 0, Fig. S4.8(a) shows the construction for a '0' to '1' transition. Bergeron lines correspond to slopes of ± Ro. Fig. S4.8(b) for vT(I) is obtained from (a) by cross-projection. Similarly, Fig. S4.8(c), (d) show the constructions for 4td> t > 0 + , for a '1' to '0' transition.

P4.9 At X the impedance seen looking to the right is (150 0 II 1500) =

75 O. This matches the 75 0 line to the left of X. Looking to the right at Y we see a correctly terminated 150 0 line in parallel with a correctly terminated 75 0 line. The impedance seen looking to the right at Y is thus (75 0111500) = 50 O. This matches the 500 line to the left of Y.

P4.10 Figure S4.1O explains, graphically, series matching using the SLL approach. In (a) the load Rp of Fig. 4.26 cuts the output characteristics in the '0' and' l' states at P and Q respectively. The addition of Rs (Fig. 4.26), if appropriately chosen, modifies the output characteristics from (ii) and (iii) to (ii)' and (iii)', respectively. The slopes of (ii)' and (iii)' are each, ideally, - Ro.

The arrows in (b) show the state of affairs for a '0' to '1' transition. The Bergeron lines are either coincident with or perpendicular to (ii)' and (iii)'. The resultant waveforms for a '0' to '1' transition are shown in (c) and those for a '1' to '0' transition in (d).


v Q

(i)

v

(i)

(b)

CHAPTER 5

(iii)'

(ii)

-RGl

(a)

Fig. 84.10

•

(e) (d)

PS.l Figure S5.1 shows vI(t): this is a redrawn version of Fig. 5.3(£). The exponential rise is given by:

VI(t - 2td) = V[l - exp[ - (t - 2td)/ RoCLl]

But, tk = (t - 2td)

Thus, VI(tk) = V[l - exp[ - tk/ RoCLl] VI(tk) = V /2 when

V/2 = V[l - exp[ - tk / RoCLJ]

Hence, tk = CLRo loge2. For CL = 100 pF, Ro = 500,

tk "" 3.5 ns

Vlt~ V/2 __ RoCL

-t 2td

Fig. 85.1

CHAPTER 5

PS.2 From eqn (2.11) of Chapter 2, vf(x', t) = U(t - t')f(t - f'), where t' = (x' /u) From a Table of Laplace Transforms: vf(x', s) = f(s) exp( - sf') = f(s)exp( - sx' /u) = f(s)exp( - sx' tu)

PS.3 PVT(S) = [ZT(S) - ROJ/[ZT(S) + RoJ

But, ZT(S) = lIsCL

:. PVT(S) = [(lIsCL ) - Ro]/[(lIsCL) + RoJ = - 1 + [(2/RoCL)/[s + (lIRoCdll

127

A step input starts down the line and is characterized by Vl2s. However, when it reaches the end of the line at t = td it is U(t - td) V 12, in the time domain. In Laplace Transform symbolism this is (Ve- S1d)l2s The reflected waveform is [PVT(S) Ve - SldJI2S

However, when this arrives at the beginning of the line, after a further delay td , it becomes [PVT(S) Ve - 2s/d]/2s. Substituting for PVT(S) gives the expression

[-[Ve- 2s/dJl2s] + Ve-2s/d[lIRoCLS[S+(lIRoCL)Jl

Adding the component Vl2s gives,

v[(s) = Vl2s + [-[Ve-2s/d]/2s] + Ve-2s/d[lIRoCLS[S+(1/RoCL)Jl

Taking the inverse transform, and remembering that

5£- l e- bsp(s) = f(t-b)U(t-b)

gives,

V[(t) = [VU(t)I2] + U(t - 2td ) [V[1 - exp - (t - 2td)/ RoCd - VI2]

This is eqn (5.8) of the text.

v o~~~----------~

Fig. S5.3

v _", () 2se PVTS


IAI

z,~1R' IItIU VI2

lal Ro Ro Q

V~CL ~CL Ibl I I I V I 2RoCL

lei t v,ltl I

I td

Idl ~V12 v,(t I 2RoCL

td

lei ~V/2 I 2RoCL gltl

I 2RoCL

V

I If) t I- V/2

v,ltl

0 t

2td

Fig. SS.4A

IBI

Fig. SS.4B.

PS.4 (A) See Fig. S5.4A (a) Vf = f(t); (b) Equivalent circuit for t = td; (c) vT(t); (d) vr(t); (e) g(t); (f) v,(t). (B) See Fig. S5.4B (a) vf = f(t); (b) Equivalent circuit for t = td; (c) vT(t); (d) vr(t); (e) g(t); (f) v,(t). (C) See Fig. S5.4C

CHAPTER 5

(e) (a) fit) f r--- VI2

~ ~ Ro

'"' i vfT} VT LLrV /' LT/2Ro

(e) t 1 VI2 vT(t) 1

Ltd (d) t 1 ~,/LT/2Ro V,(t)~ '---

V/~Ltd Wit I LT/2Ro (e) 0

1

~v LT/2Ro

VI2

(f) VI!tlt 1

2t d

Fig. 85.4C. (a) Vf = f(t); (b) Equivalent circuit for t = td; (c) vT(t); (d) vR(t); (e) g(t); (f) vJ(t).

PS.S From problem P5.4: (A)

vJ(t) = [VU(t)I2} + U(t - 2td) V[l - exp[ - (t - 2td)I2RoCL}]12 vT(t) = U(t - td) (VI2) [1 + 1 - exp[ - (t - td)I2RoCdl or, vT(t) = U(t - td) (VI2) [2 - exp[ - (t - td)I2RoCdl

(B) vJ(t) = [VU(t)I2} - [U(t - 2td) VI2} [1 - exp[ - (t - 2td)RoI2LT}] vT(I) = U(t - Id) (VI2) exp[ - (I - td)RoI2LT}

(C)

vJ(I) = [VU(t)I2} + [U(I - 21d) VI2} [exp[ - (I - 2Id)2RoI2LT}] vT(t) = [U(I - Id) VI2} [2 - exp [ - (I - Id)2Rol L T}]

129

PS.6 Consider the ramp waveform of Fig. S5.6(a), for which vp = (Vtl2lc) = Kt (where K = Vl2lc), applied to the circuit of Fig. S5.6(b).

Vp

1 ;t-(V/2)

v~_i.-o--It,

,17 r-'" -4-t

(a) (bl (e)

Fig. 85.6.


( 2VR) (dVp) Hence -- = C --'Ro L dt

Substituting for (dvp/dt) gives,

2VR = C K _ C (dVR) Ro L L dt

or, dVR + ~ = K dt CLRO

The solution of this is,

But, Vc = vp - vR

:. Vc = Kt - vR

If the incident waveform is Vp , the reflected waveform is Vr •

Vr = Ve - Vp = - KCLROI2 + K(CLROI2) [exp[ - 2tICLRoJ]

For t» (ROCL I2)

Ivrl max = KCLR OI2 = (VI4te)RoCL

If the input is not a ramp but a truncated-ramp the same argument holds providing te » R OCLI2. The duration of I Vr I max is of the order of te·

CHAPTER 6

P6.1 From eqn (6.9),

Kef ~ [(Cm Zol2) - (LmI2Zo)]

Since Land C are measured per-unit-Iength, [CmZO] = [C] [viLIC) = [viLC] = [l1/unit-Iength Similarly [Lml Zo] = [vi LC] = [l1/unit-Iength From eqn (6.21),

Ker ~ [(CmZo) + (Lm/ZO)]/4tu

We have already shown that the two quantities in the square brackets have the dimensions of time per-unit-Iength. As tu is defined as time per-unit-Iength, Ker is dimensionless.

CHAPTER 6

la)

"~.v o t tins)

9 Ib)

V"lo,t)1~15mv 1 v"lo",t)~7.5mV

- ,/ t ""---o t tins) 0 t 9 t tins) 9 18 4.5 13.5

Ie) Id)

Fig. 86.2.

P6.2 In Fig. S6.2 (a) and (b) are self-explanatory; (c) gives ver(O, t) for te = 2td (compare with Fig. 6.11 of text); (d) gives ver(O, t) for te = 4td.

131

P6.3 Ker x 1 V = 15 mY: 1/2 corresponds to td12, i.e. 15 ns. See Fig. 6.6, of text, for explanation of waveform duration.

r---------~·--15mV

-tins) 15 24 45 54

Fig. 86.3.


P6.4 In Fig. S6.4, (a) shows vcr(O, t) if line B correctly terminated. (b) shows v~/, t): this is double the amplitude of the correctly

terminated case because of the open circuit (PVT = + 1). (c) vcr(O, t) is the sum of (a) and a time-displaced version of (b).

(a)

(b)

(e)

V (0 t) 1 15mV '" y ........ t-------------t"t- - tIns)

9 60 69 v,,(/,t) t

:...---------, ,--------- - tIns)

W-__ "mv 30 39 60ns t 15mV , 69"s

V"(O't)~-:jO-----------;u _tIns) o t

9

(15 - 67) '" -52 mV

-67mV

Fig. S6.4.

CHAPTER 6 133

P6.S Vcr(O, t) = Kcr[vI(t) - VI(t - 2td)]

Figure S6.5(d) shows vcr(O, t) as comprising constituent parts (a) and (c).

dV1 1 - = +-exp(-th) dt T

1 _-----------1 V V,(t)V1ns

o t(ns)-

v,(t- 2t,) l:....-_______ ~ __ ---1 V

_tIns)

18

- v,(t- 2t,) "'-__ -_ tIns)

1 _-----'~v v,,(O.t)V ____ _

t tIns) t 18 v,,(/.t) tc--------tIns)

t--'"-O.18V

9

Fig. S6.S.

(a)

(b)

(e)

(d)

(e)

P6.6 Vo(t) = V[l + tanh [a(t - tbm This exhibits symmetry about point t = tb at which vo(t) = V. Note that for x->oo, tanhx->1 but for X= 2, tanhx> 0.999. Similarly, for x< -2, tanh x< -0.999. We will assume tanh a(t - tb) = 1 for a(t - tb) = 2, i.e. t = t2 = tb + (2/a) in Fig. 6.6(a). We also assume [1 + tanh a(t - tb)] = 0 for a(t - tb) = -2, i.e. t = t1 = tb - (2/ a). As tb = 3 ns and a = 2 ns -1 (given) the curve for vI(t) = vo(t)12 is shown in Fig. S6.6(b).


v--- - I VG(t)-~-- 2 V

I I (a) 0 t t r-

(b)

(e)

(d)

V'l 1V-~'t(ns)

t, ~ 2 th ~ 3 t, ~ 4

15 mV - - - __ ------......

I t 3

I t tIns)

12 21

~ -"\J.L" tIns)

I IKo• :t [v,(t- td)) ~ - 360 mV

Fig. 56.6

d Now, dt [tanh a(1 - Ih») = a sech2 a(1 - Ih)

= a[l - tanh2 a(1 - Ih)]

The maximum value of a[1 - tanh2 a(1 - th)] occurs at t = th and has the value a. vcr(O, t) and vcr(l, t) are thus shown in Fig. S6.6(c) and (d) respectively.

CHAPTER 7 135

P6.7 In Fig. S6. 7, (a) shows vJ(t); (b) shows - vJ(t - 2td ).

(c) which is the required ver(O, t), is the algebraic sum of (a) and (b) multiplied by K er .

(d) shows vef(l, t). This is a differentiated, scaled and time-displaced version of (a). KeflV

-----'''--- = - 0.01 V. 2te

V,(t) I + 9

-1V

- v,(t- 2td ) f------'+,

-15mV

13.5

1 45 Vcf(/,t)~ .

-10mV--'--· --------1

13.5

+ 18

+

--1 V

27

~

1-+10mv

(a)

----tins)

tins) (b)

tins) (e)

t tins) (d)

22.5

Fig. 86.7.

CHAPTER 7

P7.1 For Cable 2 to be matched, R3 must be 50 n. The impedance to earth at X is R J in parallel with (R2 + 50 n). Given that (vy/vx) = 0.5, it follows that R2 = 50 n. Hence, R J = 100 n.

P7.2 The problem requires that, [R JR2/(R J + R2)] = 100 nand [5R/(R J + R2)] > 3 V. If we replace , >' by '=', we can substitute into the first equation to obtain R2 = 168 n. Let R2 = 150 n, a preferred value. Then it follows that R2 = 390 n, also a preferred value, is suitable. The parallel combination of R J and R2 is then 108 n. In practice this would be an acceptable termination for a 100 n line.

In the '0' condition the line drive transistor must have a collector-emitter voltage not exceeding 0.4 V. It must thus be able to


sink [(5 - 0.4)/0.15J - [0.4/0.39]] rnA in this condition, i.e. approx. 30 rnA.

P7.3 Desirable device parameters of Q are: high common-emitter direct current gain, at the maximum operating current J, to reduce loading on the TTL drive gate; a high transition frequency iT (e.g. > 500 MHz) for collector current~ in the range 0 to J; a low collector-base capacitance, to reduce capacitive loading on the drive gate.

Ifthe '1' level output of the gate is assumed to be 3.7 V (a typical value) and we allow a base-emitter voltage drop of 0.7 V, nominal, this gives a 3 V signal sent down the 50 0 line. Thus J:=::: 60 rnA.

The circuit would not be suitable as a high-impedance probe for the detailed observation of fast digital signals because the base-emitter junction of Q does not conduct until VBE >0.6 V. Hence, digital signals having an amplitude less than 0.6 V would not be observed. (Waveforms negative-going with respect to earth would also be ignored.)

There is a problem with this circuit as a line driver. If the terminating resistor were inadvertently shorted, during circuit tests, Q might be destroyed because the collector current would be of the order of the short-circuit output current of the gate multiplied by the transistor current gain. This would cause an excessive power dissipation in Q. This condition can be mitigated by the use of a current-limiting resistor in the collector lead of Q. However, the resistor must be small in value if Q is not to saturate when the gate output produces a '1' on its base terminal.

P7.4 The use of a further double-pole-double-throw switch SWT in Fig. S7.4 (compare with Fig. 7.5(a) of the text), allows tri-state operation.

Fig. S7.4.

CHAPTER 7 137

P7.5 Let Ro( = 10 n) be the output impedance at x and x'. Matched operation requires that 2(R + Ro) = 100 n. Hence, R = 40 n. The practical choice R = 39 n is appropriate.

P7.6 For balanced drive operation in Fig. P7.6(a),

Imax = (2.8 - 0.4) V /100 n = 24 rnA.

For the termination in Fig. P7.6(b),

Imax = 2.8 V /50 n = 56 rnA.

The termination in (b) would be superior to that of (a) with respect to the common-mode noise-rejection requirements of a receiver connected there. This is because the common-mode impedance to earth at each of the receiver's inputs is less in (b) than in (a), and thus the magnitude of any induced noise spikes would be less.

P7.7 A sketch of a bidirectional repeater is shown in Fig. S7. 7. RT = Ro R. and D. and R2 are operative while D2 are inoperative, and vice versa.

Main line side

1 Line-extended

side

Fig. 87.7.

P7.8 Imagine the line disconnected just to the right of points A, B. Then the conditions which would exist in one state of the current-mode driver switch are shown in Fig. S7 .8(a), and the conditions which would exist in the other are depicted in (b). The open-circuit voltage change at A is (V2' - v2) = IR and this is 'seen' from a source resistance R. Similarly

~ M1:'l (al

R

IR~A IRl

'L.c=r-s R

(cl

F;~ R v,t _ v, , '. I ,

(bl

OR A

2R = matched 2lR S line impedance

R

(dl

Fig. 87.B.


the open-circuit voltage change at B is (VI' - VI) = - JR ,again 'seen' via a source resistance R. Fig. 7.8(c) shows that the open-circuit voltage change between A and B is 2IR seen via a source resistance 2R. The voltage change propagated down the matched line, when it is connected, can be determined from the equivalent circuit of Fig. S7.8(d). It is JR. (The magnitude ofthe signal swing on each wire is JR12.) But, R = 50 {}; hence for a 1 V change, J = 20 rnA.

References

The literature on lines and cables is extensive, being scattered through textbooks, learned papers, and handbooks. The following lists make no attempt at being exhaustive; they represent a selection appropriate to the material covered in the text.

BOOKS

1chnson, W.C. (1950). Transmission Lines and Networks. McGraw-Hill, mainly Chapter 1.

2 Millman, 1.M. and Taub, H. (1965). Pulse, digital and switching waveforms. McGraw-Hill, Chapter 3.

3 Matick, R.E. (1965). Transmission lines for digital and communication networks. McGraw-Hill, Chapter 5.

4 Oliver, B.M. and Cage, 1.M. (1971). Electronic Measurements and Instrumentation. Inter University Electronic Series 12, 61-4, especially.

5 Coombs, C.F. 1r. (1979). Printed Circuits Handbook. McGraw-Hill, New York, 1-29.

6 Harper, C.A. (ed.) (1977). Handbook of Wiring, Cabling and Interconnecting for Electronics. McGraw-Hill.

7 Lewin, D. (1985). Design of Logic Systems. Van Nostrand Reinhold (UK),30.

139

140 REFERENCES

REPORTS/PAPERS CHAPTER 4

Latif, M.A. and Strutt, M.l.0. (1968). Simple graphical method to determine line reflections between high-speed logic circuits. Electronics Letters 4 (23), 496-7.

2 Crowther, G.O. (1970). Reflection phenomena when TTL gates are connected to long lines. Electronic Equipment News.

3 Hart, B.L. (1972). Graphical analysis of pulses on lines. Wireless World, 427-31.

4 Garrett, Lane S. (1970). Integrated-Circuit digital logic families II -TTL devices. I.E.E.E. Spectrum, 63-71.

5 Schottky TTL. (Undated). Application Report B93. Texas Instruments. 6 High Speed Comparator Applications. (1979). Publication PS 1652, The

Plessey Co. Ltd, 10-19. 7 ECL High-Speed Logic (400 MHz 100 K Range). (1985). Technical

Publication 62545003, Mullard Ltd.

CHAPTERS

Time Domain Reflectometry. (1964). Application Note AN62, HewlettPackard.

2 Cable Testing with Time Domain Reflectometry. (1965). Application NoteAN67, Hewlett-Packard.

3 Hart, B.L. (April 1971}. Demonstrating line pulse reflections. Electronic Components, 333-5.

CHAPTER 6

larvis, D.B. (Oct. 1963). The effects of interconnections on high-speed logic circuits. IEEE Transactions on Electronic Computers, 476-87.

2 Feller, A., Kaupp, H.R. and Digiacomo, 1.1. (1965). Crosstalk and reflections in high-speed digital systems. Proc. Fall Joint Computer Conference, 511-25.

3 DeFalco, 1.A. (1970). Reflection and crosstalk in logic circuit interconnections. IEEE Spectrum, 44-50.

REFERENCES 141

CHAPTER 7

Hart, B.L. (1984). A Schmitt trigger design technique for 'logic-noisespike' elimination. IJEEE 21, 353-6.

2 Industry Standard Line Circuits. (UK 1985). Booklet LL8E. Texas Instruments.

Index

Page references in italics indicate information contained in either a table or a figure.

Active probes, 101 Amplitude,

and ideal lines , 5 Attenuation, 90-1

passive probes, 102-3 Attenuators, 104-5

Balanced systems, 77-9 noise, 83-5 types of lines, 90

Bergeron lines, 43-4 Bidirectional bussing, 74-5, 79-80

drivers, 86-7 Boxed voltage, 24 Bussing of data, 74-87

Cable bundles, and crosstalk, 62, 72

Cable drive circuit, laboratory demonstration, 93-8

Capacitance, coupled lines, 63 distributed gate loading, 51

Capacitors, 56-61 linearity of, 37 series coupling, 103

Cathode ray oscilloscope, 99, 10 1, 102 Central processor unit (CPU), 74 Characteristic impedance, 5, 9 Circuit interconnections,

crosstalk, 72 reflections with, 47-52

Clamping diodes, 49 Clock frequencies,

and TLL gates, 49 Closed-loop systems, 54

Coaxial cable, passive probes, 103 transmission lines, 47, 72, 91

Common-mode noise, 75, 81 balanced systems, 83-4

Conductance leakage, ideal lines , 6

Conductor positions, and sign convention, 12

Control signal, Enable (En), 80

Coombs, C.F., 139 Coupling,

capacitative, 63 inductive, 63-4

Coupling coefficients, 70 forward crosstalk, 65 reverse crosstalk, 67

CRO, 99, 101, 102 Crosstalk, 62-72, 81 Crowther, G.O., 140 Current,

ideal lines, 5 induced, 63 reflection coefficients, 15 reverse waves, 14, 15

Data transfer, 74-87 DeFalco, J.A., 140 Delayed signals,

classifying systems, 2 ideal lines, 5 step function, 6

Differential-mode noise, 85 Differential systems, 77-9

noise in, 83-5 types of lines, 90

143

144

Digital systems, elementary, 74-5

Diodes, clamping, 49 semiconductor, 38,46-7

Distributed circuits, defined, 1-4 waveform observation, 101-5

Double-throw switch, 77-9 Enable control signal, 80

Drivers, multipoint bidirectional, 86-7 single-ended point to point, 85-6

EeL gates, 49-52 EIA (Electronic Industries Association),

75 Electronic Industries Association (EIA),

75 Emitter-coupled logic gates (ECL),

49-52 Enable control signal (En), 80

Feller, A., Kaupp, H.R. and Digiacomo, J.J., 140

Flat cable, 72 Forward moving waveforms, 13-14

crosstalk, 65-6, 69 reflection charts, 24

Garrett, L.S., 140 Gates,

ECL,49-52 TLL, 47-9, 77, 79, 93-4

Generator, phantom, 19-21,31 Generator impedance,

and reflected waves, 15

Harper, c.A., 139 Hart, B.L.,

(1971), 140 (1972), 140 (1984), 141

Heaviside unit step function, 6 Heaviside shifting operator, 65 Hysteresis voltage, 83

Ideal lines, 5 equations for, 6-13

Impedance, and reflected waves, 13-15 and resistance, 9 characteristic, 5, 9

determination of, 55-61 non-linear, 38 surge, 9

Inductive coupling, 63-4 Interconnections, logic circuit,

crosstalk, 72 reflections, 47-52

Internal noise, 81

Jarvis, D.B., 140 Johnson, W.C., 139

INDEX

Laboratory demonstrations, 93-8 Laplace transform variable, 58 Latif, M.A. and Strutt, M.J.O., 140 Lattice diagrams,

see Reflection charts Law of conservation of energy,

and reflected waves, 14 Leakage conductance,

ideal lines, 6 Lenz's law, 64 Lewin, D., 79, 139 Line voltage,

ideal lines, 5 positive, 12 rectangular pulses, 28-9, 30 reflection charts, 24 reverse waves, 15, 41

Linearity, and SLL, 45-6 defined,37

Loading, distributed along line, 51-2 non-linear, 46-7

Logic circuit interconnections, crosstalk, 72 reflections, 47-52

Logic signal transmission, 74-87 Long line systems,

defined, 2, 3 Loss-free lines, 6-9

shock-excitation of, 9-11 Lossless lines, 6-9

shock-excitation of, 9-11 Lumped circuits, 33-4

defined, 1-4 waveform observation, 99-101

Matching, 14, 16-17 parallel, 50 series, 52

Matick, R.E., 139 Maxwell, J.C., 1

INDEX

Microstrip lines, 91 and crosstalk, 71

Millman, 1.M. and Taub, H., 139 Mixed data transfer, 76 Modems, 75 Modulator-demodulators, 75 Multipoint bidirectional drivers, 86-7

Noise, blindness to, 82 crosstalk, 62-72, 81 in balanced systems, 83-5 in bussing, 75 in unbalanced systems, 80-3

Non-linear loads, 38, 46-7

Oliver, 8.M. and Cage, J.M., 139 Oscilloscopes,

laboratory demonstrations, 92-3 TDR, 55-6, 58 waveform observation, 99, 101, 102

Overshoots, 33 prevention of, 49

Parallel data transfer, defined 76,

Parallel matching, defined,50 ECL gates, 50-2

Parallel wire transmission lines, 91 and crosstalk, 62-71 classifying systems, 1-4 ideal lines, 6-9

Passive probes, 102-3 Peripheral units, 74, 75 Phantom generator, 19-21,31 Positive line voltage, 12 Principle of Superposition,

linear systems, 37 Printed circuit boards (p.c.b.), 91

and strip lines, 47, 62-71 reducing crosstalk, 71

Probes, 100 active, 101 passive, 102-3

Propagation velocity, and transmission time, 2

Ramp voltage waveforms, see Truncated-ramp voltage wave

forms Rectangular pulses, 28-9, 30 Reflected waveforms,

see Reverse moving waveforms,

Reflection charts, 23-34 compared to SLL, 38 three-dimensional, 28, 29

Reflection coefficients, current, 15 voltage, 14, 16-17

Repeaters, 90 Resistance,

and impedance, 9 at terminations, 38-46 ideal lines, 6 matching, 14, 16-17,50,52

Resistors, linearity of, 37 matching, 14, 16-17,50,52

Reverse moving waveforms, 14-15 crosstalk, 63, 64, 66-8 ECL gate with series matching, 52 noise, 85 phantom generator, 19-21,31 reflection charts, 24-34 SLL,41-2 with interconnections, 47-52

Ribbon cables, 72 Ringing waveforms, 33 RS232C, 85-6 RS423,86 RS485,86-7

Schmitt trigger circuit, 83 Semiconductor diodes, 38, 46-7 Serial data transfer,

defined,76 Series matching,

defined,50 ECL gate, 52

Short line systems, defined, 2, 3

Signal delay, classifying systems, 2 ideal lines, 5

Signal splitting circuits, 103-4 Signal transition time,

and distributed circuits, 2 Signal transmission, logic, 74-87 Signals, unwanted,

see Noise Single-ended systems, 77

driver, 85-6 noise in, 80-3 types of line, 90

Sliding-load-line (SLL), 37-52 SLL (Sliding-load-line), 37-52 Step functions, 6, 9

145

146

Step voltage waveforms, and ideal lines , 9-11 finite rise and fall times, 31-3 reflection charts, 23-8 TDR,56-8

Strip lines, 47 and crosstalk, 62-71

Stubs, 51 noise due to, 75, 85

Surge impedance, 9 Switches,

double-throw, 77-9, 80 Symmetrical-7r attenuator pads, 105 Symmetrical-T attenuator pads, 104-5

TDR (Time Domain Reflectometry), 55-61

10 times probes, 100 Termination, parallel,

EeL gates, 50-2 Testing,

transmission lines, 55-61 Thevenin equivalent circuit,

sliding, 40, 50, 58-9 Time-base sweep circuits,

waveforms, 3-4 Time domain reflectometer (TDR), 55 Time domain reflectometry (TDR),

55-61 TLLgates, 47-9, 77, 79

laboratory demonstrations, 93-4 Transceivers, 86-7 Transistor-transistor logic gates (TTL),

47-9,77,79 laboratory demonstrations, 93-4

Transmission, logic signal, 74-87 Transmission lines,

defined, 2 ideal, 5-13 reducing crosstalk, 71-2 testing, 55-61 types of, 90-1

Transmission paths, parallel operation, 76 series operation, 76

Transmission time, calculated, 2

Tri-state operations, 79-80 RS485,86-7

INDEX

Truncated-ramp voltage waveforms, and ideal lines, 11-12 classifying systems, 2, 3 crosstalk, 68-70 finite rise and fall times, 29-34

TDR,59-60 Twisted pair transmission lines, 72, 77,

91 interconnections, 47 parallel terminations, 52

Two-way bussing, 74-5, 49-80 drivers, 86-7

Two-way transmission times, classifying systems, 2

Unbalanced systems, 77 driver, 85-6 noise in, 80-3 types of line, 90

UR-67,90

Voltage, boxed, 24 crosstalk, 63, 64 hysteresis, 83 ideal lines , 5 induced, 64, 65 positive line, 12 reflection charts, 24 reflection coefficients, 14, 16 reverse waves, 14, 15,41

Waveforms, and system classification, 2, 3 crosstalk's' shaped, 64 ideal lines , 5 in balanced systems, 78 in time-base sweep circuits, 3-4 in unbalanced systems, 77 observing, 99-105 ringing, 33 with finite rise and fall times, 29-34 see also under names of individual

waveforms Wire-over-ground lines, 91

appendix a: resume of practical line characteristics978-1-4615-9707... · 2017-08-24 · appendix...

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