5.3.1 Cable Terminated In Zn.

For alternating current, the transmission line equation^

when the line is terminated in its characteristic impedance. Power transmission as a function of frequency is, then,

(5-Ю)

where о is a function of frequency. The term о is given by the formidable expression

(5-11)

However, an examination of this equation for the case under consideration often leads to con­siderable simplification. For example, a stainless-steel quartz-fibre cable. used for Campbell operation in the frequency range 0. 3 to 0. 6 Me /sec has the following parameters:

c =	30 x 10’12	farad/ft
Zo =	75 ohms ■
( =	Z02c = 1	.69 x 10′
г =	1. 9 ohms/ft
g = •	Q Щ-10 3 x 10 m	ho/ft

h/ft •

An examination of the terms in Equation (5-11). using these values, results in

Г2 : ■ u2 (2 " — » — ‘ .,2 24, 2

a.’ c ■ g

(wre — u.’2 Г c) . /:> rg

so that for this case, Equation (5-11) may be written as

(5-13)

2

Hence, if the power spectral density of current from the detector is Gj(w) amp sec and the frequency response function of the amplifier fed by the cable is Y (ju>), then the power spectral density of voltage at the output of the cable is

2

volts sec, in the range of u> over which the approximation (12) is valid. The power spectral density of voltage at the output of the amplifier is

? — x }ТГш7с~ІЛуТ, ,2

G3(w) = Z^GjHe 1 I Y(jw) I (5-і!

o

volts sec, and the mean-square voltage at the amplifier output is

No range of validity of и was considered in Equations (5 = 15) and (Б-16) because it is assumed that! Y(J u.)!1 falls off rapidly, in this case, below 0. 3 Me/sec and above 0. 6 Me/sec.

= pulse rate,

voltage at amplifier output, and

at t = o, at amplifier

So the problem is to determine the pulse shape, v (t). at the amplifier output.

The conditions under which an improperly terminated cable will be used are: •

a. Relatively lossy cable with characteristic impedance of about 75 ohms, .

. b. Terminating resistance of 5000 ohms, and. .

c. Amplifier with bandpass of 8 kc/sec to 60 kc/sec.

Photographs were taken at the input and output of an amplifier having an input impedance of 5000 ohms and a bandpass of 10 kc/sec to 60 kc/sec. It was driven by either 200 feet of RG-59/U or 100 feet of. nickel-clad copper, quartz insulated. cable; the cable, in turn, was driven by 5 x 10 sec current pulses. The results of these tests are shown in Figures 5-7 through 5-10.

Figure 5-7* shows the amplifier input when driven by the low-loss RG-59/U cable. It

has two distinguishing features or components: (a) the original pulse and its reflections, and

(b) an exponentially decaying tail. The first component contains a considerable amount of energy,

but this energy is nearly uniformly distributed among frequencies from zero to 1/T, where T

is the width of one pulse, so very little energy lies in the narrow pass band of the amplifier. The

second. component decays with a time-constant of RC. where R is the input impedance of the

amplifier and C is the total capacitance of the cable: furthermore, if it is extrapolated back to

zero its amplitude is Q /С. where Q is the charge in the pulse. Hence, for this case, the pulse • ’ P — P • ‘

at the amplifier output is. essentially due to. an input pulse of the form

%e-t/RC.

C ’

♦See footnote at the bottom of Page 5^19.

5-16 . .

Figure 5-7. Amplifier Input Driven by Unterminated Low-Loss Cable. 5 psec and 0. 05 volt per division.

Figure 5-8. Amplifier Output Driven by Unterminated Low-Loss Cable. 10 psec and 1. 0 volt per division.

Figure 5-9. Amplifier Input Driven by Unterminated High-Loss Cable. 2 psec and 0. 05 volt per division.

Figure 5-10. Amplifier Output Driven by Unterminated High-Loss Cable. 20 psec and 1. 0 volt per division.

The amplifier output is shown in Figure 5-8. Note that the original pulse and its reflections are highly suppressed and that the’energy lies mainly in the slower components, as predicted in the above paragraph. ■

Figure 5-9* shows the amplifier input when driven by the high-loss nickel-clad copper, quartz insulated cable. The original pulse and its reflections contain very little energy, again distributed over a wide frequency band, so a negligible amount lies in the narrow pass band of the amplifier. The slow component decays with a time-constant of RC, and, extrapolated back to zero, has an amplitude of Qp/C. Hence, for this case, the assumption that the pulse at the amplifier output is essentially due to an input pulse of the form

S e — t/RC

C. .

is an even better approximation than for the case of low-loss cable drive. The amplifier output is shown in Figure 5-10. Note that it has practically no fine structure due to the original pulse and its reflections. . . ‘