Stevenson J. Power system analysis

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228

CHAPTER 6 CURRENT AND VOLTAGE RELATIONS ON A TRASMISSION LINE

Zc = 30 Q

(a)

120

120 V

-60

120 V

-30

: 180 V

90 V

-------+----

120 V

135 V

(b)

180

------ -------

90,

4.25T

135 V

(

GU 6.14

Circuit dia

ram, lattice dia

ram, and plot of volta

versus time for Example 6.8, where the receiving-end

resistance is 90 D.

Example 6.8. A dc source of 120 V with negligible resistance is connected through

a switch S to a lossless transmission line having

c = 30 . The line is terminated

in a resistance of 90 . If the switch closes at I = 0, plot VR versus time until

t = ST, where T is the time for a voltage wave to tra\·el the length of the line. The

circuit is shown in Fig. 6.14(a).

Solution. When switch S is closed, the incident wave of voltage starts to travel

along the line and is expressed as

v = 120U(vt - x)

where U(v t

x) is the unit step function, which equals zero when (v t

x) is

negative and equals unity when (vt - x) is positive. There can be no reected

wave until the incident wave reaches the end of the line. With impedance to the

incident wave of

= 30 , resistance of the source zero and v

= 120 V, the

reection coecient becomes

90 - 30

PI<

90 + 30

6.12 TRANSIENT ANALYS IS: REFLEIONS 229

When v+ reaches the end of the line, a reected wave originates of value

and so

v -=

(�) 120 = 60 V

= 120 + 60 = 180 V

When

I =

2T, the reected wave arrives at the sending end where the sending-end

reection coecient

s is calculated by Eq. (6.90). The line termination for the

reected wave is

s' the impedance in series with the source, or zero in this case.

So,

0-30

-1

ancl a rc l1eeteu wave of - 60 V starts toward the receiving end to keep the

sending-e ncl vo ltage equal to 120 V. This new wave reaches the receiving end at

t = 3T and reccts toward the sending end

a wave of

( - 60)

and the receiving-end voltage becomes

-30 V

180 - 60 - 30 = 90 V

An excellent method of keeping track of the various reections as they occur

is the lattice diagram shown in Fig. 6.14(b). Here time is measured along the

vertical axis in intervals of T. On the slant lines there are recorded the values of

the incident and reected waves. In the space between the slant lines there are

shown the sum of all the waves above and the current or \'oltage fo r a point in that

area of the chart. For instance, at x equal to three-fourths of the line length and

4.2ST the intersection of the dashed lines through these points is within the

area which indicates the voltage is 90 V.

Figure 6.14(c) shows the receiving-end voltage plotted against time. The

voltage is a pproach ing its steady-sta te value of 120 V.

Lattice diagrams for current may also be drawn. We must remember,

however, that the reection coecient for current is always the negative of the