D A153 611 THE K-PULSE AND RESPONSE WAVEFORMS FOR NON-UNIFORM

1/1TRANSMISSION LINES(U) OHIO STATE UNIV COLUMBUSELECTROSCIENCE LAB E M KENNEIUGH ET AL. OCT 84

UNCL SIFIE ESL-7269i- NB 4-7-C-F/G 2 NL

mA15h7 6E-1hTImEEhhhhhmhhhhhhhhhhhhu

imo'Ummo m

~12.4..

*1.2II

-- 'll-I~IIIIII 1.8-

MICROCOPY RESOLUTION TEST CHART

NAly NAL IPURE AL AOf *

0%

W, -"

.O.

,* *,.

-= - r : . . _ . . =, 4G ; .,- - - -. .. r-

IrffPRnl ICr) AT GOVR: NMI N I F XPFNSF

The Ohio State University

THE K-PULSE AND RESPONSE WAVEFORMS FORNON-UNIFORM TRANSMISSION LINES

by

E. M. Kennaugh, D. L. Moffatt, and N. Wang(0

'n

I The Ohio State University

ElectroScience LaboratoryDepartment ot Electrical Engineering

Columbus, Ohio 43212

,. 'DTICTechnical Report 712691-1 ELECTEContract N00014-78-C-0049 MAY 1 93

October 1984A "

JDepartment of the NavyOffice of Naval ResearchL800 North Quincy Street* Arlington, Virginia 22217

"..2

This document has been approvedfor public release and sale- its

di .ributin is unlintited.

. .. . .i

IUPt(*)R t Al (OVFINMtNT F XPVNSF

NOTICES

When Government drawings, specifications, or other data areused for any purpose other than in connection with a definitelyrelated Government procurement operation, the United StatesGovernment thereby incurs no responsibility nor any obligationwhatsoever, and the fact that the Government may have formulated,furnished, or in any way supplied the said drawings, specifications,or other data, is not to be regarded by implication or otherwise asin any manner licensing the holder or any other person or corporation,or conveying any rights or permission to manufacture, use, or sellany patented invention that may in any way be related thereto.

41

0-.--.a,

S%

.'

0

- -

3 0272-101

REPORT DOCUMENTATION 1. REPORT NO. 3 2. 1. ReCIpent S Access,on NoPAGE .... L

4. Title and Subtitle 5. Report Date

THE K-PULSE AND RESPONSE WAVEFORMS FOR NON-UNIFORM October 1984TRANSMISSION LINES s

7. Author(s) S. Performing Organization Rept. No

E.M. Kennaugh, D.L. Moffatt and N. Wang_ ..- 712691-1HaOt Name and Address 10. Project/Task/Work Unit No

.. . ..The Ohio State University ElectroScience LaboratoryDepartment of Electrical Engineering I11. Con,,,c,(C) or Gnm) No1320 Kinnear Road (c) N00014-78-C-0049

a Columbus, Ohio 43212 (G)12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered

* Department of the Navy TechnicalOffice of Naval Research800 North Quincy Street 14.

Arlington, Virginia 22217 .... ..IS. Supplementary Notes

16. Abstract (Limit: 200 words)

• Application of the K-pulse concept to a class of distributed-parameter systems whichcan be modelled by finite lengths of non-uniform transmission lines Is demonstrated in this

'- paper. The K-pulse of such a system is the excitation (input) waveform of finite durationwhich yields response waveforms of finite duration at all points of the system. Numericaltechniques using a finite element method are developed to derive accurate approximation ofthe K-pulse and response waveforms for uniform and non-uniform transmission lines.Comparison is made with exact results to illustrate the accuracy and utility of themethod. _

-a

17. Oocument Anatysis a. Destriptors

Ib. Identiflers/Ope-nEnded Terms

o| I c. COSATI Field/Group

Il, Availability Statement 19. Security Class (This Report) 21. Nn of Page%

-UNCLASSIFIED-20Seur las ThsPage) 22. Price

(Sea ANSI-fl9.IS) See V',t..'' - y. ?verse OPTIONAL FORM 272 (4-71)(oformerly NTIS-3%1Department of Commerce

I

ACKNOWLEDGMENT

p2

The research results given in this report represent, in large part,

the last major work of Professor Emeritus Edward M. Kennaugh before his

death. The report was completed by the coauthors who accept full

responsibility for any misinterpretations or errors.

I

|K

3 1-Y

I I-

QUALITY

iii

TABLE OF CONTENTSIPage

ACKNOWLEDGMENT iii

LIST OF FIGURES v

Chapter

I. INTRODUCTION 1

II. THE R-MATRIX OF A TWO-PORT LINEAR SYSTEM 2

Ill. FINITE ELEMENT ANALYSIS OF LINE WITH DISTRIBUTEDSHUNT CONDUCTANCE 5

IV. FINITE ELEMENT ANALYSIS OF LINE WITH NON-UNIFORMCHARACTERISTIC IMPEDANCE 11

V. TERMINATION EFFECTS 15

VI. EXAMPLES 19

A. UNIFORM CONDUCTANCE G 19

B. LINEARLY VARYING CONDUCTANCE G 29

C. LINEAR VARIATION OF CHARACTERISTIC IMPEDANCE 33

VII. CONCLUSIONS 36

APPENDIX I 37

.. APPENDIX II. ELEMENT VALUES FOR DISTRIBUTED SHUNTCONDUCTANCE LINES 45

APPENDIX III. ELEMENT VALUES FOR VARIABLE CHARACTERIST"IMPEDANCE LINES 47

VIII. REFERENCES 48

iv

t.! ~ -.- 7 * . * .

*

LIST OF FIGURES

Figure Page

• 1. A linear two-port system. 1

* 2. N-section model of finite transmission line. 6

3. Lines with non-uniform characteristic impedance. 12

4. A finite line inserted between entering line with Zs and 17exiting line with Zd.

5. Lossless grounded slab. 21

6. Shorted uniform lossy slab. 22

7a. K-pulse and reflected waveforms for uniform lossy slab 24(ZS = Zd = 2.0, k = 1)

7b. K-pulse and reflected waveforms for uniform lossy slab 25(Z S = Zd = 2.0, k = 1)

8a. K-pulse and reflected waveforms for uniform lossy slab 26(ZS = 2.0, Zd = 0.5, k = 1)

8b. K-p,' se and reflected waveforms for uniform lossy slab 27(z s = 2.0, Zd = 0.5, k = 1)

8c. K-pulse and reflected waveforms for uniform lossy slah 28I(Zs = 2.0, Zd = 0.5, k = 1)

9. K-pulse and reflected waveforms for shorted tapered line 30(ZS = 2.0, Zd = 0.0, k = 1)

10. K-pulse and reflected waveforms for tapered lossy slab 31(Zs = Zd = 2.0, k = 1)

11. K-pulse and reflected waveforms for tapered lossy slab 32(Zs = 0.5, Zd = 2.0, k = 1)

12. Profile of the dielectric constant for a planar 33dielectric slab

13. K-pulse for line with continuously varying Er 35

A-I. Lossy line configuration 37

~v

FI

I, INTRODUCTIONIThe purpose of this report is to illustrate the application of the

K-pulse concept to a class of distributed-parameter systems which can be

I modelled by finite lengths of non-uniform transmission lines. The

K-pulse of such a system is the excitation (input) waveform of finite

duration which yields response waveforms of finite duration at all

points of the system.

Numerical techniques using finite element methods are developed to

derive accurate approximations of the K-pulse and response waveforms for

r: uniform and non-uniform transmission lines. Comparison is made with

exact results, where these can be obtained using other methods, to

illustrate the accuracy and utility of the method.

, II. THE R-MATRIX OF A TWO-PORT LINEAR SYSTEM

3I Let a two-port be represented symbolically as in Fig. 1, denoting

traveling wave amplitudes (voltages) at ports I and 2 by ai (inwari

traveling wave) and bi (outward traveling wave). Then the scattering

matrix (S) is defined:

. h = (S) a , (l(a))

b2 s21 S22 a2

II

! I 1

!U

02

LINEAR0. o SYSTEM o0

b2

Figure 1. A linear two-port system.

£When cascading two-ports, a more useful relation is given hy

alnI1 r12 (2)

\\bl \r2l r22 a?

a matrix which relates the right- and left-traveling wave amplitudes at

port 1 to the corresponding right- and left-traveling amplitudes at portF-

2. If we denote this matrix by (RI2), we see that the matrix (Rln)

relating the wave amplitudes at the first port to those at the nth port

is just the product of n-i matrices:

(Rln) = (RI2) (R23) . . . (Rn- ,n) . (3)

To use the finite element method to analyze a non-uniform line, we first

find the (R) matrix for a representative element, and then use matrix

multiplication to find the resultant n-element approximation for the (R)

matrix of the continuous line.

We note in passing that Dicke [11 defined the matrix in Eq. 2 as

the T-matrix, but in order to avoid confusion with later usage of the

term "T-matrix" to represent the perturbation matrix (S-I) [21, we shall

adopt the notation of Kearns and Beatty [3], calling it the R-matrix.

The relation between (S) and (R) is

LS

(R) = 1 ( (4)Sili(s12s2l

- s11s2?)

and

r~l(rjjr22 -rl 2r2l)- (S) 1 r. (5)rlI /

1 -r12

When entering and exiting lines in Fig. I (at ports 1 and 2) are

assigned the same characteristic impedance, the scattering matrix (S) is

symmetrical (s12 = s21) and it follows that det (R) = 1, r11s12 = 1. In

such a case, if one "flips" ends of the two-port, exchanging right and

left ends, the new (R) is given by

3

(ril -l b,

bl -r1 r22 a

4

III. FINITE ELEMENT ANALYSIS OF LINE WITH DISTRIBUTED SHUNT CONDUCTANCE

It is customary to analyze distributed parameter networks by

utilizing lumped constant elements, such as breaking a transmission line

into lumped element T- or Pi-sections. However, we shall use delay

elements corresponding to infinitesimal lengths of transmission line

with fixed delay, which permits the final R-matrix of the system to be

expressed in terms of polynomials in the variable z = exp(-2sT), where T

is the element delay.

Referring to Fig. 2, let the continuous shunt loading be modeled by

N sections of loss-less line, each of length AL = L/N, with a lumped

shunt conductance Gn for the nth section. We shall adopt the symbol dn

to indicate the amplitude (voltage) of the wave traveling to the right

(dextra) at the left end of the nth section, and the symbol sn to denote

the amplitude of the wave traveling to the left (sinistra) at the same

ref 2nce plane. For a typical section:

= (Rn) ( ) (7)

Sn Sn+l

(Rn) = :((8)

0 e-S T _wn 1_Wn

5!

where wn Gn/2Yo, one-half the normalized conductance of the nth shunt

U load. Using the z variable, we can define the matrix (Rn):

(Rn) = 4 2 (1+wn) (Rn) ,(9)

where

(1 Pn(Rn) -P ny (1 2P)z ),(10)

TN-N

b 0 bNL%,NAL

Figure 2. N-section model of finite transmission line.

and Pn Gn(Gn + 2Yo)-1. The relation between input and output planes

of Fig. 2 now becomes

do( dN,.= (R) (I1)

so SN+1

(R) a z-N/ 2 (R) , (12)-po (1"2po)

Nwhere a TT (1+wn), and

n =0

(R11(z) R12(z)(R) (R1 ) (R2 ) . . . (RN) : , (13)zR21(z) zR22(z)

and the Rij(z) are polynomials in z of order N-I.

7

i- 7

--

NowU

(R) = c z-'~/ (P 11(z) P12(z)P21 (z) P22(z)

(14)

where

P11 (z) = R11 (z) + pozR 21 (z)

P12 (z) = R12 (z) + pozR 22(z) (15)

P21 (z) = -poR 11 (z) + (1-2po)R 21(z)

P22() = -poR1 2(z) + (1-2po)zR 22 (z)

Hence the Pij(z) are all polynomials in z of order N. Each represents

the Laplace transform of a train of N + 1 equally-spaced pulses with a

fixed net duration T = 2N T = 2L/C, independent of N; where C is the

wave velocity on the unloaded line.

Let us interpret these 4 finite duration waveforms for the

distributed parameter system, in the limit as N +

From Eqs. 10 and 13, it is easily shown that if SN+1 = 0, or the

exiting line in Fig. 2 is terminated in the characteristic impedance Zo

of the unloaded line, P11(z) is the transform of that special input

waveform P11(t) of finite duration applied at the left end of the

8

loaded line which produces (a) a single attenuated (by a factor 1/a) and

delayed (by T/2 = L/c) impulse at the exiting terminal, and (b) a left

reflected wave P2 1 (t) with finite duration. If P1 1(t) is applied at the

right end of the line of Fig. 2 with the left end terminated in Zo, then

(a) the exiting wave at the left is the same attenuated and delayed

impulse as above, while (b) the right reflected wave produced is

-P1 2 (t), also of finite duration. We thus define:

P11(t) = P11(z)} K-pulse of network

P2 1(t) = {P21(z)} = rL pulse of network-1 (16)

-P12 (t) {P1 2(Z) = rR pulse of network

P2?(t) = P2 2(z)} = -ptjlse (K-pulse undertime reversal)

The property suggested for P22(t) means that the time-reversed from or

P22(T-t) is the K-pulse for the same line with all dissipative elements

replaced by negative equivalents, i.e., exchanging -Gn for Gn. In a

lossless system, we shall find that Pil(t) = P22(T-t).

While the properties above hold for the Pij(t) of the line-lumped

approximate of any order N, we are interested in the limiting form of

these as N * -, and how accurately this limit may be extrapolated from a

finite element model with N less than 50. These questions will be

9_

addressed in Section 6, where computational results for various N are

compared with exact waveforms, derived for the uniformly loaded line in

Appendix II.

Values of the shunt loads Gn(N) must be determined before

calculations for the finite element model are initiated. These values

are derived in Appendix I for the line with uniform shunt conductance as

well as the line with a linear taper of shunt conductance. For

simplicity, we have to this point assumed entering and exiting lines in

Fig. 2 have the same characteristic impedance as the unloaded line; the

case where these differ can easily be accommodated by adding at most two

simple R-matrices for junctions between dissimilar lines to the chain

product in Eq. 12. This case will be discussed fully in Section 5.

in'10I

• * *-..2 2 : . , - - ..-. .-. ""''' -.. '''''""" - ' --.- ' " - ' . '. T . : " " • , -

IV. FINITE ELEMENT ANALYSIS OF LINE WITH NON-UNIFORM CHARACTERISTICIMPEDANCE

A line with continuously varying characteristic impedance is

modeled by N uniform line segments in cascade, the characteristic

impedances of the discrete elements matching that of the line as a

staircase function. If the phase velocity is non-uniform as well, the N

sections will be of dissimilar lengths, but constant phase delay. We

shall consider here the special case arising when a non-uniform line is

used to model transmission and reflection by a dielectric slab with a

varying dielectric constant. Referring to Fig. 3, the non-uniform line

is represented by N uniform sections with Zn, an the characteritic

impedance and phase constant of the nth section. The lengths (AL)n of

the sections are chosen such that an (&L)n = BL = constant. Thus, the

propagation time T/2 through the non-uniform slab is composed of N equal

delays T for each section where T = 2NT.

The R matrix of the nth element of Fig. 3 is given by

dn d n+1/= (Rn) (17)

Sn Sn+1

where

(Rn) (1)0 e- s T 1-kn kn

11

." . " " -- -- - " '' ' *-." -j *S h.., - i . - : r- ..'-* . , -. --. .. .

Using z =exp(-2ST), and

(Rn) =kn Z-1/2 (Rn)

(Rn) ( zn n)(19)

d dN

0 NN

0 --

_ __ SN

LN pAL

Figure 3. Lines with non-uniform characteristic impedance.

12

where rn =(Zn+l Zn)/(Zn+i + Zn), kn =(1 + Zn/Zn+i)/2. As before

(see Eqs. 10-14),

do(~ )= (R) ( ) , (20)

so SN+1

where

(R) a z-N/2 1 )o (R) , (21)

Nand =II k,,

n =0

(R) = (Rj) (R2) . . . (RN) =(R 1 z 1 ()(22)zR21(z) zR22(z)

Since the Rij(z) are polynomials in z of order N-1,

13

"' (P11(z) P12(z)

(R) = z-N/2 ,((23)

P2 1(z) P22 (z)

* 'where

P11(z) = R11(z) + rozR 2 1(z)

P12(z) = R12(z) + rozR 22 (z)"( =R'z ((24)P21(z) = rOR 11(z) + zR21(z)

. P22(z) =rOR12(z) + zR22(z )

q As in the previous section, the Pij(z) are all polynomials in z of order

N; the interpretation of the 4 finite duration waveforms, each of fixed

net duration 2NT = T, which are the inverse Laplace transforms of the

UI Pij(z) follows as in Section 3, preceding.

The values of rn(N) and kn(N) must be determined from the

particular dielectric constant variation of interest before calculations

U mof the Nth order approximants are initiated. These values are derivedin Appendix II for the line representing linear variation of cn in a

* dielectric slab. In Section 6, the convergence of the waveforms as N

increases will be examined.

-14

V. TERMINATION EFFECTSwIn Section 3 and 4, the discrete element model was employed to

approximate continuous variation of conductivity or impedance of a

non-uniform line. Discontinuous changes in line parameters such may

arise at exiting and entering ports were not considered. In Section 3,

it was assumed that the line with distributed conductance was inserted

between lines of the same characteristic impedance with unloaded line.

The resulting R matrix of the system has unit determinant. If entering

and exiting lines have characteristic impedance Zs and Zd respectively,

the R matrix for the sysem is given by (RT).

(RT) = kd kd (R) , (25)rs 1r d 1

where

ks : (1 + Zo/Zs)/2, rs : (Z0 Zs)/(Z 0 + Zs )

kd = (1 + Zd/Zo)/2 and

rd= (Zd - Zo)/(Zd + Zo )

When Ls = Zd, the determinant of the matrix (RT) has unit determinant, a

property which holds whenever entering and exiting lines have a common

characteristic impedance. In Section 4, the characteristic impedance

15

~ ~ K .1 . ~"C

of entering and exiting lines were not generally the same, so that the

resultant R matrix of Eq. 2 does not have unit determinant, but a value

of ZN+1 . If the limiting input and exiting impedances of the line of

*11 Fig. 3 are Zo and Zn+l, respectively, then we can obtain the RT for new

entering and exiting impedances Zs, Zd, respectively, by Eq. 24 where

ks = (1 + Zo/Zs)/2, rs = (Zo -Zs)/(Z o + Zs) ,

kd = (1 + Zd/ZN+1)/2, and rd (Zd - ZN+1)/(Zd + ZN+1)o

The resultant (RT) will have a determinant of Zd/Zs. If Zd =Zs,

S. entering and exiting lines have common characteristic impedance, then

the resulting (RT) will have unit determinant.

With reference to Fig. 4, let the R-matrix of the distributed

parameter network referred to entering and exiting line characteristic

impedance Zs and Zd be derived as outlined in Eq. 24. Then the K-pulse

of the system with load Zd and generator impedance Zs is given as

S_

I'

r ,* _ - . . , . , ., '- - " " . --i . .--

za Z nZN Zd

exiting line with Zd.

17

-,,,- . - .

. . ..

K(t) {P1i(z) + rdP12(z) + rsP21(z) + rdflsP22(z)1 where rd

RZd -ZL)/(Zd + ZL) and rs (ZL -Zs)/(ZL 1 Zs). Similar expressions can

*be obtained for other response waveforms. Numerical results of K-pulse

and reflected pulses for a finite transmission lines inserted between

lines with various Zs and Zd will be illustrated in Sec. 6.

I

I-fj

UC ~

|'° "':18

'1"

VI. EXAMPLES

To illustrate simple examples of the K-pulse in a distributed

parameter system, we model the reflection of a plane wave at normal

, incidence to a planar dielectric slab by a network in which a length L

of line with conductance G is inserted between two semi-infinite lines

. - with characteristic impedance Zs and Zd. The K-pulse and reflected

waveforms are then calculated using a N-section line-lumped

approximation to the finite transmission line. In the following,

various numerical results obtained for the approximations of the K-pulse

r and response waveforms for uniform and nonuniform lines are presented.

Comparison is made with exact results, where these can he obtained using

other methods, to illustrate the accuracy and utility of the methods.

A. UNIFORM CONDUCTANCE G

Consider first a lossless line with length L shorted at one end

and appended to a semi-infinite line of twice the intrinsic impedance,

* .corresponding to a slab with dielectric constant Er = 4. Thus, G = 0,

Zs = 2Zo and Zd = 0, where Zo is the intrinsic impedance of the lossless

slab.

In Fig. 5, the K-pulse consists of a unit impulse followed by a

second impulse of 1/3 with a delay of 2L/C, corresponding to the transit

time down and back the shorted line. The reflected waveform is the time

reversed negative of the K-pulse. If uniform shunt conductance loading

19

G is now introduced along with shorted line, such that the total

conductance in the finite line equals the surge admittance, the K-pulse

and reflected pulse are shown in Fig. 6. As in the lossless case, a

unit impulse begins the input. Thereafter, signals returning to the

Un input junction and reflected down the line segment are cancelled or

"killed" by subsequent input contributions until the final signal

reflected from the shorted end is returned. Since no further signal

travels down the line segment, the shorted section is at rest after

At = 2L/C. The K-pulse or "kill-pulse," has a duration equal to the

round trip time.

The waveforms of Fig. 6 are calculated using a N-section

line-lumped approximation to the uniform line. For comparison, the

exact results are also presented in Fig. 6. It is ohserved thal *"P

K-pulse and reflected waveform converge rapidly to the exact res, s

N increases from N = 10 to N = 40.

20

*~ M4

K M

0.2 0.4 0.6 0.8

(t)/

I -8 0t-I)

Figure 5. Lossless grounded slab.

21

0.2 04 0.6 0.8

-0.3-

-0.4-

° - - /

81t40

-0.46

----- I

2LC

0.2 0.4 0.6 0.8 1.0

-0.2-

3

3

Figure 6. Shorted uniform lossy slab.

22

Fig. 7 and Fig. 8 present the K-pulse and reflected pulse for a uniform

lossy line inserted between two semi-infinite lines of different

characteristic impedance, corresponding to a planar slab separating two

half spaces with characteristic impedances Zs and Zd respectively. Fig.

' 7 shows the results for the symmetrical case (Zs = Zd = 2Zo), where Zo

is the intrinsic impedance of the unloaded line. It is again observed

that as N increases from 10 to 40, the N-section line-lumped

approximation rapidly converges to the exact results. Note that for the

symmetrical case, the left-reflected and right-reflected pulse are

identical. For the asymmetrical case shown in Fig. B (Zs = 2Zo, Zd =

Zo), though, the left and right-reflected waveforms are not the same.

In both cases, the K-pulse and response waveforms have a duration time

equal to the round-trip transit time.

2

- 23

S - -. .. -- " - ." "" -".-. ..-. ... . .. ..- " " .. ... " ' " " - - ".-.-,

EXACT SOLUTION

-~ 0.05-

0.2 0.4 0.6 018

t /ZL-0.04088 1 -)

Figure 7a. K-pulse and reflected waveforms for uniform lossy slab(Zs Zd 2.0, k 1)

24

0.1226 Ct-11)

S 2LC

00.2 0.4 0.6 0.8

-0.3338(t)f

-0.2-

-0.3-

UN-10 X

- Nt~.EX ACT SOLUTION

-0.A 1 N w40

Figure 7b. K-pulse and reflected waveforms for uniform lossy slab(Zs Zd =2.0, k =1)

25

, -. --- . , . ,..- .-- ----- -- • - -. • -- z ~ - - -w - -• - . . . - - . - . o -, ' T ' ~ . - ,- - ,

r.

8(t)

0.0408 8(t -I)

C0.2 0.4 0.6 0.8

-0.1

Ki(t) N 1 0

-0.2 /

EXACT SOLUTION

-0.3

Figure 8a. K-pulse and reflected waveforms for uniform lossy slabIb (Zs = 2.0, Zd = 0.5, k = 1)

2-

- -- - -

- - -. . .. . . . . . . . "U bf , . . ~.- - .- -

k7

0.333 8 (t)

S2L 0.12268 t-I)

0.2 0.4 0.6 0.8

-0.2-

R(t) N 0

N =40

-0.3 *

//7 " EXACT SOLUTION

-0.4

-0.5

Figure 8b. K-pulse and reflected waveforms for uniform lossy slab(Zs = 2.0, Zd = 0.5, k 1)

27

1/ 2LC

0.2 0.4 0.6 018

-0.333()-012 (-I

-0.1

N z40 e

EACT SOLUTION

-0.4-

Figure 8c. K-pulse and reflected waveforms for uniform lossy slab(Zs 2.0, Zd 0.5, k =1)

29I

* .* *. V- .-

B. LINEARLY VARYING CONDUCTANCE G IConsider a finite line with linearly tapered shunt conductance,

with the total shunt conductance equal to the intrinsic impedance of the

unloaded line. Again, the continuous shunt loading is modeled by N

sections of lossless line, each of length AL = L/N, with a lumped shunt

conductance Gn for the nth section. Values of the shunt loads Gn are

determined in Appendix II. Figs. 9-11 present the K-pulse and response

waveforms for a linearly tapered line with various terminations. It is

observed that convergence to the K-pulse and response waveforms are

rapid and simple, even for lines with continuously varying parameters.

', 9

p.

-0.20

-0.4-

K (t) N4

-0.6-

N=400.2-

FC010.4 0.6 0.8 1.0

P(t)

-0338t)-0.3677 (t- 1)

Figure 9. K-pulse and reflected waveforms for shorted tapered line(Zs 2.0, Zd =0.0, k =1)

* 30

0.2 0.4 0.6 0.8

-0.2

-0.3-N1

;LMt-0.4L

-.0.5

t L0121801

0.2 0.4 0.6 0.8

-0.333 (t

-0.2

12R M

-0.6 /

Figure 10. K-pulse and reflected waveforms for tapered lossy slab(Zs Zd =2.0, k =1)

31

L i

0/2LC 0.04087 8(t

0.2 0.4 0.6 0.8

K(t)-0.1- -_

Ia 0.333IL tL 0.122614 8t-1C

0.2 0.4 0.6 0.8

-0.2-

-0.4t2L

C

0.2 0.4 0.6 /

-0.333 81)I I | ,, " -0333 ( t1 ///J / 0.1226(4 t ( - I)

• -0.2 -

RM--N z40F- /

-0.8 L

Figure 11. K-pulse and reflected waveforms for tapered lossy slah(Z 0.5, Zd 2.0, k 1

32

p U

_: 7-

C. LINEAR VARIATION OF CHARACTERISTIC IMPEDANCE

Numerical results of K-pulse and reflected waveforms for a line

with continuously varying characteristic impedance are presented in this

section. Waveforms are obtained using a model with N uniform line

segments in cascade, the characteristic impedances of the discrete

elements matching that of the line as a staircase function. If the 1"1phase velocity is non-uniform as well, the N sections will be of

dissimilar lengths, but constant phase delay. We shall consider here

the special case arising when a non-uniform line is used to model

transmission and reflection by a planar dielectric slab with a linearly

varying dielectric constant. The profile of the dielectric constant Er

is shown in Fig. 12. The two half spaces separated by the slab have a

dielectric constant cs and Ed respectively.

-0*I I Ed

LMEDIUM I SLAB I MEDIUM I[I I

Figure 12. Profile of the dielectric constant for a planar dielectricslab

33

., . .

Fig. 13 shows the K-pulse and reflected pulse for a planar slab

with a dielectric constant Er, linearly tapered from c. =E to

Ed = 2Eo. The waveforms shown in Fig. 13 are calculated using a

20-section line-lumped approximation to the planar slab. Calculations

*m using N = 40 shows that convergence to the K-pulse and reflected

waveform is rapid and simple.

4

34

.: ,-".

0.03

0.02-

K (t)

0.01

0.2 0.4 0.6 0.8 1.0t/ 2L

-0.2 0.4 0.6 0.8 1.0

-0.1

-0.2

SE

-0.3- N -20 en

r 2E1 (n-0)+ e

Figure 13. K-pulse for line with continuously varying cr

35

VII. CONCLUSIONS

In this report we illustrate the application of the K-pulse concept

to a class of distributed-parameter systems which can be modelled by

finite lengths of non-uniform transmission lines. The K-pulse of such

a system is the excitation (input) waveform of finite duration which

yields response waveforms of finite duration at all points of the

system. Numerical techniques using finite element methods are developed

to derive accurate approximations of the K-pulse and response waveforms

for uniform and non-uniform transmission lines. Comparison is made with

exact results derived for the uniformly loaded line, to illustrate the

accuracy and utility of the method.

The next logical step in this analysis is to address the inverse

problem, i.e., given the K-pulse and response waveforms, what are the

electrical parameters of the line. Kennaugh had claimed in an earlier

report [41 that synthesizing the parameters of the non-uniform line from

measured K-pulse and response waveforms was as equally tractable as the

direct problem. The key role of the K-pulse in factoring the system

before attempting synthesis has been clearly established in this

approach, which differs from the one-dimensional inversion techniques.

It is intended to investigate this problem as time and funds permit.

36

APPENDIX I

K-pulse for Uniform Lossy Line with Short-Circuit Termination.

Reflection by a lossy distributed-parameter network furnishes a

useful example for application of K-pulse concepts. As shown in Fig.

A-i, a length 1 of lossy line with characteristic impedance Z1 is

shorted at the far end. The reflection coefficient (voltage) at the

input terminal when connected to a uniform line of characteristic

impedance Zo is of interest.

SHORT£i CIRCUIT

3/

REFERENCEPLANE

Figure A-I. Lossy line configuration

37

ap~.. **

We assume that line loss is modeled by a uniformly distributed

shunt conductance, such that the total shunt conductance over the length

1 is a specified fraction of the surge admittance of the line without

loss, i.e.

total Gshunt K

The ratio of the surge admittance of the entire line (to the left

of reference plane) to that of the line is also a specified parameter:

j=Yo

m The expression for the voltage reflection coefficient can he derived

as:

K +S'T JK +S Tnp

v = tanh (ST S ) - S

K + ST K + STv tanh (ST ST ) + St

where S = jT, x = T- and C1 is the velocity of a wave traveling the

finite line section in the absence of loss.

38

Recognizing that

eZ - e-ztanh (z) - ez + e-z

the voltage reflection coefficient rv can be rewritten as

I S+K [ 1ST - ST ST W sT +1 -ST

STv +1 -

S -+S 1 T+KSr Se ST S ST

' 1 ST+--K +1] e- S +--K e.

" ' Multiply numerator and denominator by

-STe , one obtains

US

7v (ST+K; e ST(ST+K) -St ST(ST+K) )eSK =S t(St+K -S t(V) /ST(ST+K)-ST

T( K) eiS ( T - ST ST+K) e

with a change of variable, " = ST + a, where a = K/2, rv can be written

as:

39

L'.

rv = V 2-a2 ; e ;?a2

( s-a + Vs2a2 ( 2.a2 esI2

It can be seen that the expressions in the numerator and denominator of

the expression given for rv are entire functions of S. The inverse

Lapalace transform of the denominator gives the K-pulse K(t) while the

reflected pulse r(t) is the inverse transform of the numerator, i.e.

;-a -(s- V/s2.a?) . s-aeatK(t) K(s) ea L 2a 1 e - a )- =a1=+

s22 -]e e

and

[ s-a " (S 52)s-a

atT(t) 4(s) ea ee LK7 2a2 /) s_ s2a2

e e

Ry using the tables for inverse transforms:

40

S -(S.-;s2-a2) a~t-l)-e ++ 6(t) + 1(a t?.-2t) IJ(t)I~s2 a2 V 7- -2t

e .. V+.a 6(t-1) at 11I (a t- 1) it)

Vs2-a

(-sa 2)t-1

e ++ 1( la-'?2t -t) U

e-(S- s2-a2) 5(t)-

e \b22

41

Vas2) a(a t1) U(t-1) + 6(t-1)e Svr a t-

S we obtain

K~)=-K( 2T 2 )[~+~) 6t 6(t-2T)L

A - 2JO(KV2 ))LUt) -U(t-2T)I

42

and

R(t) =e ''(1- 6(t) - (t-2T)

:T(w) L2 rQ

noJ (K (-2) ) 2 )j U(t) ~- 2).

If we set 2T 1, the total duration of R(t) =1 and K(t) =1, then

the normalized K-pulse is given by

3 K(t) =6(t) - 6(4~(t-1)

+ +K ek (2~t-1)- J1 (K- t-t2)

2-J0 Ktt [U~t) -U(t-1)]

43

and the reflected pulse is

-kt1+ e K (u 2t-1 + ui Ji(K t-t2)

tt2

-2J 0 (K- tt2)] [U(t) -U(t1l)]

44

C.7 Z

APPENDIX II

ELEMENT VALUES FOR DISTRIBUTED SHUNT CONDUCTANCE LINES

The relation between the R matrix and S matrix has been indicated

in Sec. 2. The elements of the S matrix for the finite element model of

the distributed line is given by

Yc - Yin

11= Yc + Yin

L-Z

SI S$21 1+S11

where Yc is the characteristic admittance of the line. For a typical

section of the line with a shunt conductance load Gn, Yin = Yc +Gn,

thus

S11 = -Gn = -Gn/Yc2 + Gn/Yc

and

2

1 + Gn

4C

45

In the case of a line with distributed shunt conductance, we assume that

the total shunt conductance Gsh equals to a specified fraction of the

characteristic admittance, i.e., Gsh = kYc. Then, the element value Gn

for a N section model is simply:

Gn kYc for uniform lossy line,N

and

n 2kN Yc for linearly tapered line.

With the (in given as above, it is straightforward to calculate the

R-matrix elements described in Sec. 3.

46

-.-

APPENDIX III

ELEMENT VALUES FOR VARIABLE CHARACTERISTIC IMPEDANCE LINES

The approximate K-pulse and response waveforms for a line with

continuously varying characteristic impedance are derived by using a

model with N uniform line segment in cascade, the characteristic

impedances of the discrete elements matching that of the line as a

staircase function. For the special case arising when a non-uniform

line is used to model transmission and reflection by a planar dielectric

slab with a linearly varying dielectric constant er, the dielectric

constant cn for the nth section of the N-section model is simply

En CE2 - El) (2n-1) + El

where el is the dielectric constant corresponding to the entering line,

* and E2 is that corresponding to the exiting line. With En given as

above, the element for the R matrix can be readily determined as

described in Sec. 4.

47p.

REFERENCES

(1) "Principles of Microwave Circuits", Chap. V, Sec. 5.18, RadiationLaboratory Series, 8, Montgomery, Dicke and Purcell, Ed.,McGraw-Hill Book Company, 1948.

(2) "Acoustic, Electromagnetic and Elastic Wave Scattering - Focus onthe T-matrix Approach", Varadan and Varadan, Ed., Pergamon Press,1980.

(3) "Basic Theory of Waveguide Junctions and Introductory MicrowaveNetwork Analysis", Kerns and Beatty, Ed., Pergamon Press.

(4) "Fifth Annual Report 710816-12", December 1982, The Ohio StateUniversity ElectroScience Laboratory, Department of ElectricalEngineering; prepared under Contract No. N00014-78-C-0049 for theDepartment of the Navy, Office of Naval Research, Arlington,Virginia 22217.

48

FILMED

6-85

DTIC6 i