I realize that, no matter how careful I have endeavored to be, occasional errors may still exist. I should be grateful if you would be kind enough to notify me a3 you discover them either in the book or in this manual.
Sincerely, David K. Cheng Electrical and Computer Engineering Department Syracuse University
Syracuse, NY
(For the use of instructors only.)
13210
-roduced by Addison-Wesley from camera-ready copy supplied by the lhor.
I'yright 0 1983 by ·AddiRon-Wealey Publishing Company. Inc. I rights reserved. No part of this publication may be reproduced. Ired in a retrieval sY8tem~ran.mitlted, in any form or by any Ins, electronic, mechanical, photocopying, recording, or otherwise, lhout the prior written permi8sion of the publisher. Printed in . United States of America. Published simultaneously in Canada. N 0-201.10129.1
FGHIJ·AL-89818
..
A note to instructors using FIELD AND WAVE ELECTROHAGNETICS. Dear Colleague: As teachers of introductory electromagnetics, we are all aware of two facts: that most students consider the subject matter difficult and that there are numerous books on the market dealing with this subject. It is understandable that s~udents find electromagnetics difficult. First of all, the subject matter is buil t upon abstract models that demand a good mathematical background. Second, before the course on electromagnetics, students who have studied circuit theory normally encounter functions of only one independent variable, namely, time' whereas 1n electromagnetics they are suddenly required to deal'with functions of four variables (space and time). This is a big transition, and visualization problems associated with solid geometry add to the difficulty. Finally, students are often confused about the way the subject matter Is developed, even after they have completed the course, mainly because most books do not provide a unified and comprehensible approach. As I point out in the Preface of the book, the inductive approach of beginning with the various experimental laws tends to be fragmented and lacks cohesiveness, whereas the practice of writing the four general Maxwell's equations at the outset without discussing their necessity and sufficiency presents a major stumbling block for learning. Students are often puzzled about the structure of the electromagnetic model. I sincerely believe that the gradual axiomatic approach based on Helmholtz's theorem used in this book provides unity in the gradational development of the electromagnetic model from the very simple model for electrostatics. Although a rigorous mathematical proof of Helmholtz's theorem is relatively involved (not included in the book), the physical concept of specifying both the flow source and the vortex (circulation) source in order to define a vector field is Quite simple. Many review ~Destlons are provided at the end of each chapter. They are deSigned to review and reinforce the essential material 1n the chapter without the need for a calculator. You may wish to use them as a vehicle for discussion in class. I have tried to.m~ke the problems in each chapter meaningfUl and to avoid trlvlal number-plu~1ng types. This solutions manual gives the solutions and answers to all the problems'in the book. I hope it proves to be a useful aid in teaching from the book. Answers to odd-numbered problems are included in the back of the book.
Cho p #-. " 2 { CIt J
4-
a:
_A_._
i ...
T
J
A
A
+:i =:r2
11 ..
A·a·
(<
8 = cos-I(A-iIAs) Aa
(t)
-
=-(A"4
.z-~3)
y
-'41 = 1,&."1,,,,1 -Iii
(.6) IA-il-li" "'4,'ee)
I rs:
- Q&J
.z'.. (-3)'" , Iii •
0+ J.(-4) + (-3)- -It c:
ctll-'(-I'//ifIFi):'la~.s·
A-ac = A· eC - A -,,',h 2.. (~s -4..1) .... lv,,'
A JI C- -4.. of - Ay IJ - ~/O C,) A. (i.e) =(X-i)·l a -4-2 (I,) (A.1)- C. = A (AeC)-A (E-i) sti,.2 - Gy4 0 .,. 4. f «)
A.(,-c) =
1. 2 "2 r6s;#ion n
E(~·
_r 'm---rnrc~ ,« 1'_ C4rna,..$: or
V«C/DTS
oP, - ay - Zi.s 1,
ts- ca..n - ti;,IJ "'iIy 4 4 -s;«
~
s
ii.. 4 - 2fT - if.",. ~
""a... , ~;;y2 + 0.; s
Vee fDr.. rtlp,cs.nl'n.!J -//,. H,,~. Sid.S of HJ. 'iridlt!Jl.
.... I: ~ - o~ - o~ ~
~-
-D• • -
~ ·~ ~ 0
(a)
-.. tiz J ~~
cra.Z. ~
. .. A P, P, ~ is
.
(b)Ar.a 01 1"'iAn~Jc
4
4.', ~~ ~-Q.'-Qy- Q; ,-/# +,.,i.ft.f/Ie . t'
= r' In , J x ul a J ~ /7•f.
s, = 8" -r ,
~2-J
:
..--
~ ·b~ =
1l -"8-A
(s.A,.(i-A)
==8-i-;--.4-0 · -. ~Z·4(I)
~rDItt
DI
J.
D.J,
A·i =A·C. w.hllv. A,,(B-C)=O. co F,o," A ai = A-C. W& 1t4". A Jt (i-c) ~ o. (.2) ;",,(1., A..1. (i-c) and (J) Imp"·.' A OCB- c). S/"Ctl A i:
flot
CI
u"I...
",1.111
I
v,c.,~~ (I)
r.- C) iJ
A
",,1/
AnJ (2) cannot Itold v..c""'r. TJ,u~, 8
~ . ~. 4. - CoS(.,-~)
4'" the same lim':
-c - 0
- c.out
1
C4SJ
4-
or
silt"S;',".
BeC.
}i
A
Jt.
.~~ l::~A
It
CJlA==:j .. c
C1l.:
BJI:
BJlC=A"3rB s'-" '?A. - CA J"" tlA -
ro/,,#;ons : A8
;i·: ....'n;-#uJ.
A
8
=
Sin ',(.
s,."
s
8(,4
ri"
;'. -r. r ' = r (c-r')-(c-r). -.
E~p=tnd
(c - t').1.
s..,,~c
f,C
C eAt
(c.;)·(c-r)
= o. cc: -r).
AJl(AJtX) a=A(A-X)-X(A-A) . A • P =- ~A -A',i
0,.
X • ;, (1':4 oS PleA). 2-fO
AP, = -ifY ~ ~
-- 4".z Z
. oP.I =
I
-41 2. .,. aa 3
it)t(~CO$t#) + ~(,.r,'"+)#
-
ci.- iii'l--iiyf+ii"
1« I
~ ~ == O~ - o~ = a.1 + tiy:' - Qzl, 1i A--P · a-/1.1. -A-P. - M -~==I.f.2 , , a ,I~ ~ I 2. 2.- f f (a.)
x. .. r e.cs
+ =- 4-
Y - r ~i,.+
== .3 (b) R == (,.l.-f
-
c:
ff
(1.:rrh) - - 24 J;n (2~h) - 2.fj Cos
Z,
2·12
(4)
%~)'/1 - {4 ~ .,. .J&)'1'2 -
f} ..
t-" -I ("/z) -
~
~7T/J:= 12.0·.
ICr
{::I 4 II
r.,,-I (4/J) -
lS • (·J)I.+ 4 a", (-sr
SSJ.I·
ii L -.2
(E p )x - -I(- .r!.::. -)i1.· 2/Z, ~ , 5.\0 (P) 0:.. -
do (~;;;,
J + cij.4
9 - c,.s -I (a. · Q.) -
-
- a. $") a:. - :
C.,-I (-
I
nn-)-
= f(~2 -i,1 + i.J
(S 4-
~
r
P. 2-14
£0 til
~
CoJ
X
-2.y l
-1" (yJx .,. xJ,). ~ d" -4ydYi
,
~
(b)
)t.
"J.r; .I.'
-'y-4, J,.
J
j"-CoJI=!(4y IJy ." y IJy) = 14 .,. r: tiL _jJ['ytly
,
I
I
E9uo! It". I""ply
itrN!lrah aJo~ I -
c.a.,.. hcctJ,us.
!!:k.!l. .
i
.Ip-ci(lc pa.H"
E
do CDltluvc+I.,.. (itlIJ.
i~
41
eV(~y.,C).
CO' ~ "·,,tI] [e_]= [e.s~ [ £" ]. [-1;,,'1 c..JSfJ 1, ~
E-
Jo ,,01 ".c~"o,.~'Iy
CDttSllrvMi". r,.c/J In --IJ",
-I;"~
0,. r II·nl~
of
~
r COS
Sllt~]
['1;,.4]
c..sl
~~Ol#
~~
j. I/Z" r S"n ltl Jr + "~~O$ ~~
JtI.
JJ(1,4,-I)-~(S,SJ./~-f); J;(",,-J.,-f)-~(r,-J'.9~-I). 7h~,.c i, 110 ChAn".. in r (- s) /r.m ~ J~ L (-J'.,.
· ··
I",.,.
E· Ji •
1
cos lftl~
S')
:a
-24.
SI.'·
P. 1.-f& (.)
V V =[ ~ (1eel T"XJilt Jy},.~{1""l-1&XI(01 ¥yJ + ~ {s,'" !jlt)($i" fy}].-". (VV), - -(iI,
(f,J
f ~ ;;.1)c·J --(iiy o. 0 21.I ifs o .0 4 J) .
PO = - a.- ayz - if& 3; ~ :-.,:. (a +~Z .,. '4,.3)" lt
.'. (v V), iI~ 0
=~ (f - "f)e-
J
=o.041S.
1.1-/7
J {~JI""o)·(a.It~Js"nll)J9JI"lJ;s S"l9d.tI/> =7S1r.
P. 2-/9
f
~
AoJi" - (I,." 1.#.
+
.1,.._ +J 1'6W6.
",-II,
3
~
)AoJF.
J
70' -lAC. (~ .. 4): A- = q,. r J4-a..
1I-t>p A· cit ~ 1t.,. Mea
1
= "(?rs
l )
= 2 0 0 7T.
I~.
-
r.. e« (a-.o):
801111,.
Ids
cls-~iillds.
I,
A:: Q, r t
J$ =
~
- J S.
- ds
A·J,:; O.
D."~
~.-
(,. -$): A- a,. 25 + ~2Z
W~lls
J
A·JT
~/'I
~
Js:zqrJs.
= 2S/tN.,11
J A-J; == 2-00fr+()" IOD01T "1.~OO1t. V. A :::::J r .,. 2. jV.,14".cJfl7rf'V.A i ' NJnltl J -I.l~tr. &
0
f
V·'j
P.1·Z0
~~ (rF,,)
= :
+
•
~~ ~ - le,. j.·FJ.,=Z4,,1c,.
/) i vcr'~KC thCtll-... -h.i/s HeauoS. j= It., A $'it.!fu/QrHy 41 "-0. P. Z·U
III.}
J D· dS- • j (u~:t)Ra .si"lJ c1(Jd~ =fir( f -t) ec'&f' J-Jf :::: - rr . $,"n4
•
j))
v· i
0
=- (c.osa~)11l4. J9.$Jvm19.iJJl."II;'.JAfI~ o
Pl-22. .
v.~
- .tL
J
&)
LC"'"F) =.1. 2..r"I.c/~\l=O ~Il.. I ( ~ .c a dA, L~ 7 l.'VJ .
·'. III (It) z: C-fH)$ia"" ~ c; i... f .1
P.l-:14
"-T(.
J
flt)= : ' ·
0.) A.JI. .. J;1t'y'J* -/rly~Jy
,
~A'JL'" z t » ~6 -7 - 9{ -11-}' b) V -A ~ - 4& (2""y' J (iJ(~). Js- '" JJ rl.r-~fl.¥Y').f4\d,. .I.,).. 31 .s.. I
e)
No, • bcC4U~"
~
J.
J
V J( A rJ= 0 .
p. z-z~ V Jc A -
~ ~"'. (iia. c.ru. s/" t -i, .si"_ r,·" .;) h l(v .. A}. J; -j.. ~r "1fhcAJ.~ ·(,:, j,~3"".J.Jtl). 4-1..
...
Jt
4
: =IO.S
5
til
-7
~-----~
-..
~
---..
~;I.
•
£. Q.-I- H,. cenlfr 01 -I",,~. woulJ bIZ ~rt' i-f all -/4,.e« '/nc c.J,AI'ftU w .. r. 01' +h~ .$.",. 4u'~y. The, probletrJ /, "9uJva/."f 10.".,. Di Q. ~I·""(" ("I?~ c/,~';'oJ q....".s,.~y ~1/2. By $ymmed-ry lit.,. will only 1M Q .1e() ""po"."f.
ch.,.,.
r:
I
a., £..y =- - f
F _ s: -
D¥
L/i. (IJ FII I J ) J.L~) =ii 4.,,£ Il' Jt ~
.!!..
.,~
(J
3~,
- 4_
T 4'77\£
&r
G
¥
lJi I 1. "'C.L
JE-J."
Ii TIl nI
J.J-'
~ha,.t.,.,J/& -l/s '1r~~(4 ./ .
Usc <7aussi law: Q./~O. a.) £ ;s normal /0 11M fwo r~s o.l f O.DS(rtt)J wi).".. £ :a r ~ G anJ a;. -i 4x re$pflclive7·
'.:-!.
,,&:
Q .. 2 ~ (I .0./&) :. D.I-£. =r b)
l. - ii, (I(J(J",) ~II> - ... (too,,) s,.,. +-ii,.(foo,cJ~}-.".,s;,;'~. A -
-
. (Itr
'yS-aftJs -./.. (J
(IDDxo.aSC4Sa.~)(O./J(O.OSJ.)= ~oJ..S"'.
Q .. O.D7~~£(I == &'.,4-
U
o
I. r+"lo'-/J(C).
Spl,e.rlu.J 0
I:R $: h.
ItIO-fJ
(c).
;)''"'"~: £-,\£. Apply Ga&lSS~ law. .,,,~£~ .1lJ~/C')~tr.b'J.a _ .....Ji (~' _ Jl~ 1, ~, I ~~. ~" CO J S ~i/
£~,. ~ R(j -:,~.
,
e-
4::At•
£
4-).4t > Il..
ss«,
JIll
•
I'c~.t
Cy"'''4,,''~.1 sY'"'"~"'ry:
P.3·/0
4)
£,.-0
/'*<.4,
,. > ", b) J,/ A P.)."
••
0
:I
~E,.. Apply ~Clnr law. tl~"
j
s.: (Ai;. -
-I
£-
b~.)/4.'.
J:-/J'.j .
1J..t..,. f(J Ef:.
CtI..,,.,,.,,,, iF
(~-4cV IJI.,.J
t:"i.1. ~-/.f. £
~ eel. - ~ "'/"_',
=ff.,,"-
8
d"J
"",ill h_ no %0'- 2 Casjtl- 5/"-8
IlJ.J"
V=
P(II, •• N
-1- (..!. + ..!..
~7C'.R
~,
~
-ZJ
A.,'" Il' I-(:Y- Atlc.os_. ~
Il, -
[ /-1 '4)'. ,l-~ 'Ei -:i: c.'''J
a I *' Lee.... .!!! ~
~
a.)
v-
*,
~~.,&.-,
~~
"If' ~ _..!L c"r. ~ d& Je.'·.-I ~Jt '-A'.t
. .
~'.»J~
'J,(dht{JCGlatl_f) 4."...... A
£ • - V V - - 4:" ~A. 3V - G~ ~. ':ftf:!./~ 1{JC.,',-I' .. A~. I. ~ r./ "'~l$,.,.~"l·
6) ~1"a./-ion -h,. c,'&I~~",'&J surFp,e,u: R
::a
C.. {J CO$'. - f)'/~
£1U4H." ill' ..rf~""(''''cl:
!!.J!. _ E~
11.#1.
or
= ·.-f "(Ii,,' _--!!1!cil(
£.
Jc••
J It.. _ AI
-
J
.z
$;'..
N.fI.
S;II~.
$,;'1._
A - t:~ (SI,.I/~.I{j~.,)"
7
( AlsII.I.."er
·12.
*'
f "'"Iler 'YprDII.cJ, (nl:i-l #t• .141-.& cI.~r«c Q;' iA,prokil'l14//on) is h:J Ct'Jn$,·J..,. 11Hz ,roil/eM ,.J a.pq,'r 01 d/,fGctld d"po/_.~ •• cI, wl,1t fA Mom_,.1 ,c7h .
A
f-
e,.
,c~ (~-f.#) I polflhltC/ clue 10 ; I; ~
_
,{d/~c.",
P.-/cniY',J t/U4 10 - , ~
__
~.. t(d/J)c•• "
., ..
-
41Y~R!
(-' ,( - 4r) e; .
4ft'''.
1('" ~ ~-~(I - .1~ C'lI$') ·
R.;a ~ Il.~J {! + ~ CMS) ...
= ,.("c.,.)'/J
V
a)
4"~Ai
IS
,,(cI/~)ca• .
-
v _~
.
~""~R~
I-
.
i :- Vv • ~:;;'~ ("Jc.sa'~"""It:l'). h) Et'lQlio/J
IfI,.,cJI'tllcnlials&lr!Oce.: II.-C;(coJey/J
c,,,,a"l.n 10,. sln."li••, :
Il ~
c; (s,·".)~&
-/J
'(0,1)
£-4 £
Y ."
c) ~ 1-/4
&: -
=2
n -'~,. II
7
b) F,.o", Cou(_~ I.",,: J,r
4"~~'
• ii Y
~
I.h
Jry l(lIJ)&~r
.
V V .!II~«" N." Sa... "itS tlWr •
S"rI'"c;c ~•. J.n'ily ~ •
-tr.
Us« fh. rcsulh 01 p,.oJ,lam P. J-fJ -lor fire. c••,.J",,~1a lyJhm Chosen ;,.HM /-I!JLtre "'J~e"c1C."'P4fC' llap/4C4 Ii. /,y J:Jy tJ-nJ y AYJya.,,,_,
r
P. J-IS A r~cn,.c- fItc c;rell14,. /u... ,;.,., 0/It.,1t« %)'-,/4n. wId, lis aX/$- clJ;nc;J/"9 t.A/,'fJ, #'.. z- 4 ",.. ThtL $U,.fAUl. C.AA"!j'" 0,. -In. +...,btL Wdfl is 1; - Q/~1tblt . .,1:;;" f -I;,.,J +I.. fd-#«"It'A1 Q/,,~ .JI". AX;. 4f".z JIJ. h .... t:""C4J/~,L,'r,. cJ,""Jd "f 4"ns,'fy -" ~"III."'J ~ Z~
V
00:
~ _.I
f
41r"~
j".
~ 6 it
~ J,
r
"'''*.'"Sl-fit.1/1 - Z• .'b'*(~.2'}·
-,
·
4) Th., .~/r ••• /olt above. I~ fI,~ ~rJlIfri~ufI'OI) Jv c/tJ" fo a circular I,,.. c/,o."9" J .. ,../I-y ~ = ,~ Jz'.
'*
dv _
J; ~J2'
2*'.1 ~"'(Z"2" Af q ftJ 't-,f OuH:qc. fh~ /-elk ': J_ r::---: V = ~'·" Jv :s ".1: (It % .., fPt!' 1
1
.-:0
I
=
.2'a,
-i\7i Jv - i ~" [ • 2\
(.-1,)+/ ,'-#-(.-1.)'
I
-
I
1
:)~"4(Z.b}' J6'f~JJ
.
b) SA""._ Il'J/P,..,. i O I1S .,... oAt4/;"J Fo,.. V«"d E aIQ.. poi"f lit ,jd~ 1/', 11.14•.
A'f'iki ~ FDrce
CACl6CS
tH sop4.ri4Ht:J1't
I<.tls/(),,'~ I.rca
~ .. I£.I = 4"'~,,'
A) Alo'!!J /+,C ~4r.J,o/A
b)
=
t E.t
Dr ' ; -
I
--,I£'Ji ... -1f(yJx .,..cJy).
w
)-'1
J/~plu~.nf ~.
(a.Hnu:h·""): 'J£•.
~NI.'
A f ~1.u'li~",um:
a
AIo,.s
"h.
-tt
JL -
2)'J.
j
dJ( =4y J .r
--14'·
Z
'y'Jy 2.'f"7). sl,.4!,,1rI lin. 1l = 'y -"; q)l - 'ely W·
w --1f~(f2.Y-4)Jy--141·J.~~r). I
3-(r a)
l
- P. ~ .,,, 14'/ sitt -IAU' ot fh.. culJ..
... p" if
"
It'.
'1' •-v· b) QS -
ji == - J ~ ·
'L 1 1 ' . & II
J
Q::I ., L
J1. L'.,
-
TtlfAI ~unJ cI,Q,.,_ - Q. + Qv 1-1'1
A
S.J:"'. if =a. P.
's"4rr..c..
= O.
J,..' II -
-JP. L'
d.14 rs.. J.,.iI~ 1f,s:= p. ii.,. -/aa P ) 14_ )
7h e x- C#»5YJII"C.rtf-
oF NJe ".cJr:c.+'~J
a -
p (,oS #I•
,;'1"'.;7
du. 10 A ~/'''!J 01 ~. C4"hI'·I)U In wlclH, Il..J. ~ II 1$ cfE
=-
%,
-
A+ f/,. cUrh,. tit Me c v i4y g
:
E: t: •
PeAl. /. . .f",\R I%"~ ,;n_){RJ*)CilS. ' ' : . CtH'.
s;II1141••
- L 1" , ., "aEs .~ 2~ .. C's , s;".J.· 3~. ·
10
I! t-so ~
VJ,.w
£•• J
= .J
.) ",. £'1 c/-
It
50
I~() (ltV)
a
20 11 SO -
1,000
,) ~ _£ lJ-J)'" 1..£ ~.
"
!U:E.. A,.
~
E" .0,,,
(2-0):6
V J ::sJ(4t1-1.l J - /D) - f JO (k )
,. ~
3
lit. %=0 plc,,"_:
£,
0=
.
~ 2Y -~J" + a:. s.
':L ·
~,(~.(J) .. s, 1.1" iiy
(z.eO) IS J?;" (~o(»)- 2l",fi - 0 ) =3 £M ( i -O) ~ 1p (2.0).1..(. J . o-s.• !!!. 3 .
· ~(%-o)a;;1I.ly-d1]""" ':
Po )-1.1
[leV)
J
~ (%-0) - (4w'y-iiy ' • .,. .. fo)~. ~! - ~,,. (4 -I) 1/, .. ....., F - £.'it ----. "'1'1., -l.-P.If =--!-I!. ~,..,-I af. ~'=~I? ~r, _,,,. ~,. -, JIt.
e" lIf'" 4: l--. lIS rj
A ,su,"« £.-
&.C.: 4,.
s, r.
-I
,,.
s; e T
l'
X£:= Jt l ---- E.1t1= , 4It:a.
,
For ~ J Q"J ".nc~ ~ 1-0 be pA,.all.. , ID ffJ.. x-OtJ(i$ J
E1I:ll - E~r ~ E,Jr- J . &. c.:
dIf · Ii, • e.
P. )-,15
"
=Ii · D
'£rJ. -
"
L
s/s .
---+0
~
=J4
IZ
I)
Chapler 4
v'v. 0
P.4-f
P. 4-~ A f
Q
-
Vel -
e,y .. C,
VA
(J)'
-
,
+~.. '
~"'-D1C., • DJa:-i","'o(r C,
£4=-.,'J'
.
i5.. ·-dY~OCJ'
V J$ a ftfA,)til7lUM ("';"''''''.11''') .;I,c $fCond
poinf wl, .. r-.
D-f. V wI+!, r .. r~c:" fa )t.1 Y anJ % wDulJ 0./16e nCjerl-ivc (po.illv«); tIt.lr SUM coulcJ nat v4nish.. as
pQrftQI drlr;vDf/vlI.J
tf7ulrcd hy '-ClpIQC~
P 4-'
~
j
V V
.Ie -
J. e'luG/;oll.:
.L ..L (I,. ~V) = _L
L
e»
r
V =- : 1 + ~ /." r ., :! ~ • --fa + c, J." q. ... ~ { ()- _..t!.j, + Co l,.,J, + ~
~r
~
dr
c=
r
l('·tI.)~V,I
, •
LIt 0/4)
, C ""
v.l.. J, * 1{"~~-~l"tL) .
~~: ~_ iI : -4i, ~~UJ;h;"~~~~;;rr:~'H,})J.· =- . 1
F
----
.Q
6)
j"~ , l'1rrJr
r
~
(r)
4}~.Q .~ll IS
=- Q . 14-
'r
:
,-0
. -,I" Qd (d'" ,..)v.
cJ,Q,.,.
1'.4-1 tA) 0"l9;"'/ point Q. a-l y -= d/J Q"cI /1'Yt~!Jes Q &t Y= (1+''')dlJ J ?t-:tl, t.J 1 • • • •
-Q. .fy-.(~+'n)J/3
, "-O~ :ttl~l~···
b) 0",'9;_11,',.. cJ.ar"!Jt ~ tAl 8 - JO· QhJ """fI&~" /,'nc. Cha.r,.s ~ QI- B - I.O·GnJ -'/0·; -~ QI 9 .'O~ -~D:QnJ-150°
•
.v•
...---- ... ---........
...---da---'" toeo----
D -----
,
h'
J. = _. d
Q·--dQ ,
h)
-'---. 4 ) V - Q. - - (4"~ At dR; b)
I --•.",'' -.J
fa _ •
_
-£ "V
•
~A
131'
t.l (J,a-d
.
,
a _
1'4-{3
') 4-14
Tj,~ solu#ion V()r.,Y)= VO sal'-s/lel
bound4'-Y
V(x~)')
• LJnirLi.
Cl1nt;/llioltl.
I
Vi-.."')- V, ~ A:
A" -
7ntl.
0
It
rtoJJ "ct V.I1.
Is,." '/,..[A"Sl,J, ':a,ri +8..cos/' ":"] ~ 1'·"1. n", «e COJIt ~ _{ "'II' ItHJ 4
"
0I
4
J ntr ,,~I, ~li (v, -~ cosh It:"). l·o~
I
t
It
n
0. 4-IJ
Sf;'/'
,.
At Y-b,
all
SD!u"'''Dn.
:"'1 "A,.CoS/' :"Y]. [5 vc~o,-~ - r B S"n r;:)C, -... ' n- It"
~ s'-', :."~[A"
...
AI y-o
V"v=o ana
n even.
"he".
n ocJJ
.
J'olul-ion /r fhq, sup.. rposif;ot) 01 Mol ~or
4-'
£')C~"'pl. Qrui //,aI -lor F,'9- 4-12 ro/-qf~J ~o· ;n thQ c./ocl<\IV"~" dl,..cJ"on. (In btlM CG~.S ~ $houlJ h. r 4 pla c «J ~.Y ~/.2.J 1 nsiJ.: V lr, ~) .. '~.L f [Sin"+ + si" n("~I)J, r« 6
" ,,=.u
OulsiJ.: l
4
,n
_-_"J
V(r,+)·~
r
1t. .~tI
VCr,,~)
A+
(il
~(#[s;nnl +I;""(~~:~l. r>6.
t! uo~ ...l (Af ,.», ) =-E.,.c.tI~+ ,.~ _ _ ,..1 B, cos "r-". E-ca.£., V--l;," c•• ~
,-h.
6
It 0
\
V('~#)·"'!,l-I;4I~ '~I, ,"'c.~".L flfJ
r
"
---. ~ ·~h·; B,." 0 /Or n/a'. Oullicl. Nt. cy!ittJ.,.: r >b )4-10
o-
.. - ~,. ~) c• .I +' r ,. ,. (I; ~) - - iV - i,.4 {,.. ·Oc.'II.,E.(fI·I,;,,~·
V (r, 4-)
E-
For r~1>~ ~('i~J .. -t.~c.J~"i,,.r·"coln1,
.. It-'
n.'
",h~ ~C~;).r A",,"cos"f.
Af r~b: V.('6.)-~.("+) --..-
- ~, L.. ·-(r~ L..-
,,,J,.... A,.~,., n~l. r."8,"·'_-~rA,; ftl"l;"'·'L-f."A"b":/~. -t,II +',li'=A,S,
'II,
j
C h o.pf.,.
S-
R ~!
p.r u,,;"'tM~" of CtJDh~
SJ
('1.) /l.,== R J
:a
~ -Z/2
:
l
r~Aa
•• tr.sl-
) .
em -I)IA == 2. JZ 1, - ~I £,. i- z;•.,.
b-
---...
Q, •
a
'
-.!:• lorr.· ~, ~ J~
J .a I
= 1r(2alJ -Ih
1Tl4+bJ'-na&
os
..J.:.-
-
L tll b =- Thick"•• s ~ e.tJa.fln,.
b) I J -
'!s,
=R.'/111l11U1 p.r unillon.!J""" 01 co,.. -
Il,
J
£ • .h- • -L-..£ . :1
l"",
0."
I
J,-- 0.7 CA> J ~,- O./'J ~w);
; -O./f.O(A)~ ~... 0.J91. c.w);
IJ=O. 0'll (A), ~, ~ O. 2.~"(w) ;
:1.,,-0.113 (A), ~. - D.... J' (W);
It'-o.41J1CA), ".,- 2.11r (w).
f.,. a)
n
'"
c
f'" ~ b
.. (r/c)t
~~4
, r;.,.
~
.c:
R.. :> h
-
10·'
Q.
,~- G",~)ItP" J
(41'1!.J)R ,
4-71". Ifa
_
C",,,IJ)lt>.J)J - D.ZJ9 (el",,).
~ ~ S- xlt) 'A. e-'.• ~.IO'"
-
.-wcn
~~ .~.J« ~
l V/m)i
-..!l.' ( ,~
Q.
E. - a.. ... ".~~a - ca.. 11:'". tD v, Itt). b) h ,. _I~./IIW'" / a « ~ J,.-rC,·aQA '.~J110 RC lA/Ma),• J
'> h. .~
R.
ID
O.
, 4.) t!.-(r/~)t = ~ h)
e: J,,' = v' •
W - Lj 1.
l.
.'.
•
-O.Of
-
"
t - ~~~ -4.".'D-/~Cs)
--4-..,
2rJr,~· ct--1(~/~~t -('.£11 4" -c.
.,.'"
,Co
~. - [ e-erICl']' a D. 0 I' _ ;0-4 E"ergy aissipaJ_J (WJ, · QS h.,.,./osS.
c.) f lee..froJhJ.lic ~ntl,.gy S/oraJ oul-~i". +/,.. Spl,..re
? ~..c.,' 4'FrR'JA... ,.:i., ·-loS" (/'3). (;",sl.",': a.) R.- ~= -f tr §t "'J.SJ1J110' (S/m). b) L = -f- = '11 (V/wJ t:» : .. ~ · ~
'J
(;1).
[-(rf4HJ' J.
I::
&I
0
(0
J
'
c.) p,. V I - , tw), Ie
1M"
c,
+
zrrtr:. l,,{b},) .
Z "1;
I',j
=r
J
f9
=tr £.
.
P. 5-14
A.11U",« ChQrgtl.l
- '1
"1 ant:!
10 ~ohc.n"rat..
In., CAnl-.rs I
d»b
'-"
l·
Yz ~ 0_ ---L--
t ... v, - VJ I
n
~
=I'
a
(t
4'Wi ~J
.!£
l
(clf_);~
4,",~ f
--r t , b; .. -s; - CI=b;" - ~ 4
o-f spit .res 2 ""p~cl"v.ly.
"a...1...-(_f v, 4-rr-E, b __d-f_), ("
d'-~b
"
Ortc/
t
f
-
i)'
r
c
~,
_
,
- -Itr '
f ) __t_(1. . . 1- _ .!-'.
T - d.-Iit - d-b. . ~
20
4""
eLf
h,
b.•
d J
Tn_ curr.nl ,flo"';
~S-IS
pa. N&rn of fit.. lower ~
hall
oj,
F'·9.
(1,);1
- the. ((J"cJucfbr
hoH,
o.ncJ "11
irt1age are. fed wi-lh -ih. same c.u,.ren'" is ~)(4e"',>, #ttl Sam~
fhAl 0{ F/~. (Q). All
(b)
(0.)
QS"
hou"cJo.ry CQncJilion$ Ore Sall,I,cc:J.
The sfrCQmJ;nc~ are slmi/Qr .Jo fh~ eleerri« lielcJ It"ne.s 01a. concJuc.lor Qnd i+s iMa..!JaJboth CAr'yin9 Q c/'af'"gfZ +Q, In Ihe el.. cfrosfG'''' eels_. ~ S-/~· Ac.c...orcl;"!J
10 P.S-IS, th« woulJ /)a +he Salft& as .,lta./" 0;
curran/- .f/ow fa~'Iern a. whole sphere in
unhOund.J. ~arfh m"diu",. lienee Me f:..urnftf liner are radial. A.uumc Q C-urr.,nf I . -
-.1
J = Q~ 21TR' V,=
1
y
2nrh
I I , 2",,.. . . 2.". (IO-6)(J,./o.J) ='.36-/0 (n)
r:
X l7tJ - sin/, (
b x).
V(¥,y) =
s.c, at X=a.: ~
..
I
elll.:zI
Ita
o.f 0 Q.rtJ y=b NUju/ra (1) .... cos ( ": y); fl... J.ounJary C6nc/i/;on o.t .%=0
indicaf«.s filA!
4)
~Tf'''
C o n c:/, l- ,'o '1$
Tilt, bounJa..y
thor
1
I_j'
_}II [cJR. ..
R=-2L:a
~
--
£ -'\ 27Trll" ,
J
f
TJh/s.
en s/,.h(¥,,)CDJ( h,;'" y).
"-0 • V(ca.. y)-" - I
....
=0
r-
~
CIt Sinh
(.!lfa)cos{;" y)
8" Cos(';,Try) .
A,.tI) -
~[ )ctI'(!f-y)Jy : 0 = 8,.-0 IDr For "-0, " - ~ --... c. - s;nh~It"A/b) . ll"l {n"x,I .\1 = ~ V(x~y ) ~ l{, [ ~,." lit".~1.,) (01('!!tl TYIJ~ A. x . It-O
21
n,.
~\ - 6'~ b) j"'-E- -rvV -- ~ ~ .
p
v('; ~) =- ~.. (A"'''.",, r"') (c" n; ., 1',.1,,, ,,~) . It-. B. C.: VC,.,., - VC,.I-.) ---. 1>.--0.
s-"
CDt
r.,cos' ---.. A,.-c,.-O(Drn,ll. V(,.~) = (1<,1'-1' ~&)CD.J +. k;-A,C,..I k,-I,C,.
r--..o-, W,.ilc
I-
s. C.: ~V 6" ,., . ..
V(r6
~)c
- :.
k.--'!'" , r k - .!i. 0
0 -..
I
6'
k•• j,JJ(,. •
{!-b"
V (t; 1)- - ~ ( r » 'r6&) GiJs~. ~ (f'
j .. -6VV --,.(ti,.~~ "*4~
::/») 1,'
~,.l,('-;r)col~-a4"(1 + -;:r)l;"~ t
s
6
/"
= i.cs, C(Js~-~s'·nl)_1,~'(G,Col'l+~S;n+) r ~ Q• .r. - .7;~ (4,. ~ iii., .5,.,,~ ) > a. . I
CDJ
J z
R
0 I r « b.
,.
J
,.
~ -
'-2 4)
~ii .. _
.l!.! ( i .,. U M i)
at
t:«
•
1ft
B=aJ£B
£II ,
.,A
-
au.. = 0 ~t
auy = ~t
3 us
~ _
-.!!La
u "', s
lei -
'"
t.J
'1
u. _D. II
~(I
.. ,
My
:,,1- (:,.r
u s, Is.-". £I.- uJ-O, I
I
i
a
3-"1 .l t - -
au a 1&-
t.)
0 U.
u,,-u.
j
--.
c - u -~ .
·
8,
~ .%.0, Y .&I,t.
-~8...
a. u.. = a t , .l!l , ""·7
au", ..
.
..
(y- f:~j I> (Z +
£ --as £.
So
y"~s"n~t <#.!.L t , Za-~(I-CI)$l4J{Jt).
Xc:rO,
£,uGlion 01 mol-ion:
h)
£ + .=t.
)
If .#,c uc,clron iz i"J·.. ,I..cI at fhe or,:'in (.JC.-Y-2-0) a.t t-O, Wc/,c.v.
£) COl wi =(u.~.. -~ 'II
\l..}
•. ¥
l.t)
Circ.u/Q,. Ibotion
(.I••
I' I £ fit •.
23
P. ,_I) ·
}
H.I'-cal ,"o"'~rt ~ wH}, CD"slatrl' accill.ra/ion '.1£0 /,.. ;'1 z, dircc;l-;on.
1
"
2.4
y
~6-11
l5
b) For
very long +wo-w;,.. Irdnlm"s;on /i"a L"": (,..~ - ALII ( +,/)•.,JlI - - n .:.L. - 4 ~ n A- -4- AIL
t1
l
Ii
• 2.,"
c}
B-
fi k
(-y)'+Jt'
• .. "
A _a
IA. _ - aA, x ay Q,~)C,
"-~l t+y "x Ztr (f+y)a.,~a
J-l ]
( _ljl.,~
-iiY~-:[(f~l+~ - (1~,Y~x·J-1: fa,,:, -~~l P.6-/4
IJ,&D~"," fr, (FMC):
Apply e!i v Ar 9 «n ce
J.l
1~· CFXC)Jv v
Now
(jfJt.C)·Jr.
v. (f•. c) =- c· (9 ~F) - F· (vJtc) -
c· (~J(F)
c)· Jj' a - (j JC cJ;)·l . Thus, C. !
J
V
-
j
-J FMdi h.. CDun l is ana,.,,;,.,..,., ConnGlt' ".chH': J£' " la
(vJtF)Jvc:
"
. ".,I
ii-G_"l -
P'-fS a) ,.<4:
, ,,-IJJ
8 =4./A »i t
I
~
>
!4r::L_ H · . (.t!,. t)nI • P,
JA,
8 :a ii... fI_"l M-a
I I
JIIV- Mila" • (cIs. a,)(~ -~"l.a"(~-~"l. P 6... (1
(1)
I
I'
Y,.. -
I
J --r" 4".0.,
4i1
di.dc
«r. a.-(cOJfI/)'d,d~ 2
•
b)
8-
-A.f /..
wltiCh
V~ 1ft
--i
A.l ,~ _ -
./-.
;s M• .III,.,.
,u, 1 J,J. , "a Z (%... b')Jh.
~z
II'
£,. ('-J,).
l'
~
i Itt =a V • M
A)
::II
i"" - M" all -
O.
4. AI• • ~ (ii.~ '4,y"", z)
- -T(-a.y + iy)l)-ii,~J1tI"l i~ M,Jj", .
::a
-
, : Dr J. -(a COI-~.rin.)M • • aA A ,
, ' lit.
:=
ii~ M.li"S ;
b)A,ply £,.(I.-JI)
It> c/oop 01
rdt:/iu, 1J lIn. C4''Y''''' " CU,.,..nl
J.,~J.:
~
8 - J J4 -ii~/lilt'.~ z 2. ~I -
2
- a. T .fl. At•
-Ji".M.
Co
u., -Li -i ~, ~
$.O,
Ii
a~
,
-
"c =~o,u,. Be.
1:1
(H-').
-ii,.l.OI1 JtIO J { T)
10'" I 71• O. OJ J
Y'
.1.2f"ID6 (H-') _ ,. 7~ w ID4
.2".. D.o. - D.OQJ ,]t»o-(4"./D-'.".D.Olr
8'-1 -a... , C
b)
J-/D-'
4"'.'li"."" QDJS'
~.s
tI. -
- iiZ ~ Si,.'. . l
.L-
-
a)Cf - ~_
.,
_ _ A(J",~J.)lAs;n~)J cI~.4s ~(&~VJ
T'
lt/O-'.A...4.0.s.'O'{AI,.,)
41'1 J1 lo· Y
"
Q,
10s-/o' iooo =
e) NI-1(4_,,«.), ";I
f.JS (A/",).
:1-.1- 1 (11 all
f'tj )_,tJ'.I.J1'Sk 1tJ' N ~ 'I SliD 0.02$6 (A)-=.lf".6 (JrJ~.
F! '-20 M":j"lli;c c,ircuif: S
t
::I
_1_=
~S
10
-J
f
L",J)
«zss»
(4""o..,).,~J .
N.gl.c"'''",,, l.akagfJ Ilux anJ allum/n.9
l,· •• /z
CDns/Q.nf
over SI ~
f
o -AI
Ut
/
(O.~Q2 -
AZ4-0.0J)
Jo.o
-l'oIC/o
,(
H
tl - _,_(,..I4~~.A2_\ • 0 lo.l ~ 10'(H"') .. 'AI ~I)OO 1 . 27
-I,
J.
'''.1)( J~n$il/
l
NL
-
Gt. l' d,1l
(J -
~, .. ~....
0.{112).,0'
-1 l.JJtIO· 4 (Wh).
·
f.' Z _ 10-4 (Wb).
h) HI - L API'
(I-IJ-
2DO-J
(I. 60 #
- A"u,. Ie $ - (1.9S"'0" I.n.,o-~ 2'• f . ~ SOOD
!l - -'-1. • (,. 9S.
,-'. I".s ..
10
fiJI)- 3. 'JJC 10- -1 •.
in
(A/m)
'Q"" -/0·
(AI"'),
gap.
(H"Jc'" (HIJ),/SDOO - ~1.6 (~/'")~ in Core . . 6-21
a.)
W ar. k rtl1u·''''&4 per I.t "if le"~ ,;, /-i,."e cit::
,.
B
':dt-
---
Wo~k
rtlcJ1..
p_r u,,"" vDlunt. itt Jt-:
Jw-l
H
Thus,
tv,
p,Jt-nIJ8cHtJ,.
-1• H JB . •
'I
b) Wo,k don. p.r un"" vo/~",. l" Cht:ln~ing I,.,.. 0 10 ~ die,!, P4'" AI, ;3 w" w"ic/, is ,..P"~S."'.d IJ? AreGJ' I .n
110"",,,
Alon~ path b~ IBis J~u-.lVeJ/ ;ne/ucln~ Q +hof ~."J1 hJ M~/'n·fa,n #r.. cu".,.nf. WCJrk /s Janr ~Qin.rf -H,• .source.. /h. WGrlc ptt. r un//- volu",e. ~ ;.1 r.pr..s .. by -(IJ"CA~) . .I" ~o"n!J {riNf'l c IDJ, lit. cJ/"~c/"on 01' cu,.,.cn# i$ ,..".".... J allc/ /~. vvorlc Jon. ~ 1$ r.prescnl-ui J,>'.. Qrea ~. Scvn. G"'Dunf- of worlc. i.s dDne In c..ha", 9 A 4o/o"JI Ma pal-" fro", cI 10 • iAl)J I>tu.k 10 teL (loS +hAf NVJulreJ ill !JDI·n~ -from tl -10 b -II,NJU,1t
".,..tI
""!l
c 1-0
a.
. '. Work clona ptt..ru",·fvolume ;ntJne. el~/.:I l(W,~Wa"'WJ) -2 1t Ar«4S[(Z.,Il J- Jl+J1l.] =- ArC.A of ",he hysleres;s IQlJp. 6-1J
H, --v V. ,
J
~.-VV"'J .
J,ounclary Con J i liol1l :
~,
H
'':' -p~~"
- - + ~I
dV..,
~
~n Ie}', fin •
H" . - fl.,. 1:, ----... ~ -~ ." dt at -:ls,,·
S, - a" 0 , "
4.)
8~-~
- Qy 10 (mT)
8,. + 4
.J.a-
H
la=- SDOOA -
y Bay.
1-1
o.s ,,,,--p;
---... 8~JI,-J~SOO ("'T)·
8 - 8 - -/0 ("'7) 11 '1 · .. SllS 4",2fDO - QyfO r"'tr).
b) II ~-~fO"~O.s ("'1), H
~
S'y.
~,~ o.s.
la·
~, -
~
H
lie - ~a
ol,.f_"-' ~,. ~ 07
·"
S,-a.8'.1C"'~B,y'
--.
a,- ~
D;"OOZ -s
~..,
8
to
Il
~ra ..... ~ -QDo2.
I. -
C).04'.2 I- ily
tJ."
oa).
-O."D4(roJ)-O.~J~
~ 0) Con si'Je r 11N0 ,illiaf/ons: (f) L .. nJ ~. boN. in air; anJ CD 1 AttJ -L hot-I. in mA~n.""c ",. ."/wm w/-t/, r./Afr'vc pflrMetJ/,i/ily 1
fA,.· 1
1
0
--(Z)
~"nJ
- - --IJ
"
p
•
Ii
~.
-I'
cD
~r
B,y and H,. a.t P (yaD).
F,.nJ Bly And HJ • of Pl,s oJ.
81V - •~. {l.r,,)c"fI.~..!:J .. -¥(~-r.IJ(.,.-..&f!.!...-~I .., ",. ~ Jr~"') ,., -;; ~ .w,. • "(P,:~/) r .
St.· ~~~(Z-~~;""- "qr,#IJ;'Z; Jl :a.As. -~_.!!. -
I. ,."
8,'1 .
,,~.,(}
"'I
b) For fir» I,
,.a ,
If.nJ
Hilt -
'
s.- {tt(J-ZJf.it-e. ~) ~ I . II _ ~ _
'111
,u~
.J:.."~.JJ
d .
rs
#/,. ( /-,.Jar>, e-J,.,'-t, .JoJ,',r;-.J)
I.;" ~;:~ Z .. L.
A.e/.r fo .follow"~ li"u,e .
3D
I-....&~--.......¥ - -. . .
,
L -
, L' L." •
.I
Po '-31 Sin,.. h, »ha +hc rrt~n~h~ .fl&l~ d&lc 10 ""e/o,,!} loop "~Inking wl"'" tIJ.. .,j",,,llloop C41n 6. approJ(;",al.J lIy #'lAf dv e: io fwo iJ'n"'.7 fo".!J wire' carryln.9 efua/ ~"d ~,~P09,"t1 I
cur~."f
I.
A -~1~(~-
,a '''"
L • ~• 1
Po
d""
.&-"" 1.W"
f
W,+11*.
)J~ .AIl ~,~d 2"" (~ --,r- °w,·w,*dJ
(~,.cI)l,,+.,)
-,
·
d (w, .. Wi'" cI)
'-n £',. lj,-f40J: ~ - fL,I/ + La,Z,I.. .,. fL,X:. ~) W,1- i;[L, fiJ' +lL*I-1;) + L,]
-t
(L, z·.,.JL"x rL~) J II
%- : : ·
°
~
6-JJ
I , ~1~ -:rJ -2)
d - O./~
'A' l. I.J
eM'
A" a 2 8 CaDS Jo e .. 6 & • II • ii P.~n- .
1.
Lt!J_J Ii:
(
Z 7tdl 2
:1&
~
l1Td
Fot-c. per un'" !Q",1h
Oft WI;'.
2.:
A I~IJJ fj- --4 ,IA ---it .a
"Y
~
'Y ZTtd
... - 'S'15o/'• .. -a, I. 44
1f 10 '
) (Alk).
Forc.es on all fhre .. wires are 01 C-'1C/4/ "'A~nihiJe ana /owQrd fh .. ~4n/4r of I-h~ fr/t:l"9/~.
Jl
- ._ - P,ItZ, b
F Oft loop
Tr
41.
• a,.",utl ;~, p.
'-3'
j" JCOs./lUI. tI, D
I ,,'-;6/dJ' t 1 (Ilepulll've ford) r
IJ.fl-solvc fhe circular (l,Q
eft vJ;fh
Q.
.
wi
loop
lnfD
m.ny smAil
loopj~
mA!Jn.lic c/'-polc ",oment dm -.I~ Js.
- J d T- == I, }til ~ B-
clf-dm.S·
T"
- - i. I, $;I'I.,j8 Js ='-QJlJr,ll';~(d-Jd·-Ji)sin-. ,
:J
/n flt4 ellr~c/;on 01 a,I',l1ln!l Ihe cllr~cf"6n 0; the. +1..,)/ ~)' I .. "" 11."/00' 10 fho.f ~ j du~ '10 in .,It_ .sf,,~"f wir•. B .13 0 ve» Ihe c;re.ulo.,./oop I,AI h•• n FD~nd in
prob l .
e..l::E.
.r,
,.,
Po
'-JD
ca. A/~. )
M4~".""C .(('-1. cI"'~i7 aJ- th~ CA-nh, ~ H.fl IQ~9(l ~"rcL.lI4" Il.Irn J 'N,ra CArryi"9 Currll.nt I~ l.s
8~ -s:I I &L.. 2. 'i
o« z.O ,',,£,.('-J')
Tor7 u• on the smAIl c.ireulqr w/r« : -7 - -m,. B- ~ (Q.,l, /'- "t;J) • (-..8!l\ ~- -) )J,I,l,fr r/ Q 2 Ii ~- 4 a,-a.s J. 'i sa
1
--.Mt:AAI1iIuJ. - J",li.l,,,1f rl". in Q c/l,..c.filJll fo a.1/on J 2r.' ~ fftc, ltt4!JnQfic. .fluxu Jy I, 4.Jtd Zz .
produc."
JJ
/
t:z, fhfln
F
P 6-39
-'-
fl,. h4r M4,,,.f is O,./cMoJ SU~J, /--hGo! 8",.L 8. ,
+9. If- < S~. r~
0( -
~a..
(N~)a
- At - ;4-V(:':~ -
\
J
4..$)'
(N.lJiu:lS
_
(2J.,- ~(j
F .... 10011.'.1_ 9.0 (N)~ S -0.01
.l; =- J
l,. -Za,o·'e,.,).
(m'J,
;.',. -4000 .
(m),
.solvl"g: IntrW == NI - f.JZ'Jf/D' (A. t)
HPH'clv
WIn -
P6-40
ASJ'umc. a v'-rl-ual c/"spIQc.",.nf'" AJt~
Wrn (X +.4,,)
w,.,(X)'"
ala
oF -lite
iron core.
if (.fI-MJ) HaJv l~".
.. W,..(x)
(F~) -' ~~ x ax
+ fAlp,.-lJnaZ1JAJe
_,u" r..u
z v:»
- f)n" I'S ;n th .. J,'r-.c.licm til
c Ju:~efcr p 7-2
-1s
7-
P.7-f
#11
x:
7
~J.Js --I~(vJ(A).Ji .. -.i~A.Ji
~f
r at
's ~,
K -~ J US ($1r,o"-t - }""')'fO-'
J-.f i· cli
i,,~r•• s""1
J
J'4. ,
3 CD$(S1r/o"t -
·
(7)
J?1X-)fO·'.(~ 0.1 J,,)
- - :.~ ~,n(s",o'-t - O~f..".)-s;n$".".,o"t}/o~ (W£)
ii--d~J8.Ji - 45 [COS(T7'fO"t-Oo.tr)-C.OS.s1rfO't.] (v) I-
_..::l!:..... I.~ II rcol(S1rlo"t-(J.4,,)-co~S7Tlo'tl 2R. :IS -
4)
B
-
0. 211)
$"" {-a. 2..,,)
- f. ,'s""(~"'D"t-D.271)
.- -a,
P 7)
3 sin (S1J IO"t -
&!sinlAJt
%
r- .-.l 8·d$
(A)
-
a,
.o 1-.J]" &I;/nu!j'l'.tI( t : !-JJr_{f,Pjl:''''~!JJ,-J/,,(-!J.~#tl» J=
f
2.,.,.r·
~
Z.
I
..,-- U- •- R 1;"" V",
,.s.,t
34
J
cI$DQ,Z%dr., Z"T1"•
{If - fln (.Ii -IIJ)cos ~t
[Ji -In en 41)]
~,_. _.if.. Z I v:.M 1- n&IlIJb fZ." .. O.047ZI'l.ZWJ, (V).
Z-.Jlf r'4ff)- rj .
b)
I("AI
r &IJ"iltllJt. 11 Js-erflw lrl i (J<#r;)r·~U't 4,
(/·itJ
- J-I,u11Nlt. r1."4-~)l 1.l!li)_{ib J V3~ lz I' "Jj; n'-'''n!
J'
r
v,.,-; ~~.., 1/(f'~)/fl(:4~)-JiI • o. 04' 9P. I Wb (v).
From probl.m P. b-30: ~J-
-V ..
1'0I (sin c.Jt)(cI-}c/C h0
-!if - -1"0 1 wt.os t.oJtJ(d-1d'-b') ~
i
CQ~
wt
..... _ Iv...
!'!.oIfIJ(cI..,J;jqi) Jl. a - /iR. -
_
I b)
./i.R."3J&IO-l
/"ow{J-Ida-b-)
.,~ , .,(0.2.) Q:T -
n.
..OJ
41.2· .
7-' --
P.
JS
_
3./ilfto-'
"""i'(~""o).D.DU.z
-0.234 (;.4).
1-7 T4f«,
+/'11. J"vII';9cnuz 0& £,. (7-l1a.): fj·(VJcE)
----.. f?
--k{v.K)- 0 IromE,.f2-IJ1) 8 == f (X 1J z) Which /s 10 hold -
J
'" wlt.""~r IJ fZ~;$fs or not; Itcnc. f(%ly,~)""ul,.veAnj.shanc/ veK-O. S;m,i/arly ~ f4k. fh. c!ivllrgenea 01 £". (7-.J7h): af all
f,mu
tlvtl'ywhere
v- (VIC H)~ v-j ., tr (V-D) == 0
- - ~f
+k(i-iiJ ,lro_£9' (7-J1)
-----... v- D .. .f. Eo,.
7-1'
(1-4-":
V -A .,..,-Ale ~ at - o. elv V ~1T'" L 4-1'( v: L A t' ' ~ v'"
A-· e...l
_LJ
d v.'
f# {~~[v'(f) +~ M-}Jv'1- o . v·(f) - ~ v- f
Now.
+
- -r- v'(f)·
(;)
r- v(~) == 1- 6ff) (,.;,
v~(-l-)-~ V:T+j-V'(:J. oni" em, v· (f) - ~ 6'. 7 - v:ff)·
A/IO, SuAsfilu#ln1
SubJflluH"1 0,,),',,(1),
) 7-9
£9.f7-J1b):
From
£1"
livJ
Jv' ,(.J.(v~J + 1l.)Jv'_jv' v(L)cJv' ~t
- - -0. L 0-1 R. -P"', S '-....,., 3·d~
.'. 6: j
CJii)
~
• ~,. ~ I·,,;,. Il.
~ : : =0 (E"J,oIconfinuify).
V";:; - j
VJt(!J-J '*~~~ ,
+H -
-u- -) • (1-J.9)a"JI7-4Ii:
K-VaA. l-- vv-
;X -f'~ .(~V) flVIC P ~"A -pJ -I-'~~ v :sf .
M- ·
Use !)D.Llge conc/,.,..lDn for polc.n"'aJ~ ,e" Qn;""'o~""ouJ meJium: V'{EA'+I-'~~ ~ WAve .,/ua.litwl.for weierpolen";.':
IF-O
-pv.l; 6IlAJI-~£ V[JI~I V-(4;A)}-PE.!ls--pJ-. ~ro", Elf ("'-J1c)~ 6·6-, ". (~ V vJ I-J,. v· (~A).-" Wave ~9(JQ/,M for .r~"I",.poI~"';41:
f
U·(E vVJ-}l~
fl.- -1'.
As shown in prolJl-m ~ 7-1: 4) £". (7-J7J) ~"btl clcrittc.cI.from (7-37,q). Hen U. fhc boundary C4JnJ/+irHU lor rh« norm.1 Ctlmponcnf~ 01 s, whicJ, arG o!JfalncJ-rrDWt v-a-o" ar e not: indtlpcndenf of -/-h'" bounJary ConJ,./-ionJ I'or fha. f-an!}tlnfiQI COtnponcnl. of E ~ which a,." obf~/n.cI
P.1-f:l
£,.
I'rom V J( £ == -
b) S/m;/Clr~
I
.lL. ~t
fhe hound,;,,)'
CornpoTMn/:s of
D
COnd//-r"ons
ar« nof indtLptlnJll.nf 01 -Ihosa
for t-h. +Qn.,cnlr~1 CtJrnpon_rt/s 01 VAryin, CA$tl.
~ Jv!....clium 1.: Fr"" spac .. j H~ mas} be zero SD f/,al
- Qn.z It H,· 0
"".. Jiuw,2:
8" is nof
if
/n
fh"
ft'me,-
/"-l-oo.
in+inif-•.
8ft , = Bn l
an~.It (~- q)-.J; 1.1-14
-kJr .,I,e normol
j
£,,- £'1·
l,(z.t)-~o.OJ.Ii" fO'n-{f- f).ii.4ID.f)Ji"bi/~(f-%'t)J
E,. (Z,t) - ~D. D4 COl [t06,,{t - t )-11-Ile[O.•fi~.i"',,(~-z"Jl Pheasors: £" -t, .,l... ~ [D.OJ-e-i ."n + O.04 e-;,,/J j - ii. [-jf). f)J., (tltl2 -)..0.025))
=s, (0. Eo P.
7-"
II:
0.0&. ,
•
AJff! • - L
V' V - r:
~t
C
0"
e-":")
= ~ E. ei~
= -1.17 rlAJ.
or - 1l. ,~
IN,·,1t II(.Q t) =L ~
vJV_L~_-.L. &A'
We
heed
,
~ta
/
4"..}u'
lit ··.Uu) do • ~
v.
V
e-
ijlff)- ~ V'I +,V l (f ) +2(fJ/). v{f ) . ,
»«
{Fo r mu I4 : Vz Cf9J- fl • g}-, Va, "fV} "2(VfJ-lv9~ Let J. t -1jU1l. -t -Il/u~. r(~-I{f-Il/u.) V'/f.) ) _.L u' J', d'J - L!1l. UR- d~ J 37
-'f') r R • -4wQ{A ).
V
P, 7-11
V
It
l
=:
-u ~H
r:
~t
V · -;i :. 1.. c
(l)
V· H = 0
:t
:!)·
V w Q): V K V £ - -j-A (1 + € - F;~ ~ _ ~"£ v- (Ve£)v £ - -P~tj+f'£ ~tl· I(
Wavc c9uQhiJn lor £:
VI( (V:
v·£ -p£ ;:~ =p
;r
+~
VJr V1tii- VI('j +-€cf~ (VNE). -(- - I"iJR - v-r -/'l~"Hi. ~lR V V·H)- v
Wave 1l9 1J t:if i tJn lo,. H :
VJH-P'~ --Vl(y.
For S'-nLiSOidQI Ii",. clepcncl«nc . : ~ --.j"-,,
He!m""lfz's e1u a -l"" ns : ( for
phQlor~)
fP jj +
U'J'E jj- -
r-
H=- -'- V
jlAl"" '"
-*.
£ -
/is ~ (-w'J.
V IE + i4J'p~ £ == jUjJT +
iiy 0.' .sin (ton») cos(,.,,/Ott-~%) Us. pJ,GIOI'I: i' ti a-
P. 7-/1
vr,
,
i-
- - J'wp. ~ O
fi}f
tv~
7.
(vi",)
1 L
11 n£y
0
[jj.jo.~ Sin (/0" It) + ii~ O.f(/~")COI(I(J1TXJ] .-J,l.Z(l)
i =J!;;~ fiJlH =4y Cd~fo {('07f)'-t~lls;n{fo",,) ii,~ JI
(!)
PhasiIr
lor" fDr given £ ; K =. Q'y 0./ sin (fO"X) ~-;}z. (IO.")ao#,,a,.tAJjJ.'o-4007l~
lfjuallng (j)allJ(j):
.'. P -·/Joo n
-/o/j." - $4.4 (rae/1m).
Re (H ..i Ctl t )
From (i)" H(--J;ljt) -
• -a.1. 30.... 10.4 S;,,(IO"x.> c.os{,,,,o',-r4.4i) - d z 1.JJ k 10"" coS(fO"~) t"" (6" lo't -54.4.1), (AIM) (AIm).
HC'JI."Zjt) - ay2 CQs(IS"")si,,(6,,,O't- '1.)
P.7-f9
PhQsor: H - Gy 2 Cos {IS1T¥)e:i l
" f,J I " .,. (/~Trj =- W:;.u.f" - '''''0 J (-).--,-0')-, fl~ -= 400 1fl- 2.2S 11& - f1S r a
I
{
~ =/3.111-41.'
£ == .L: v- H Ic.1c.
t:,} W4..
:=J - .
(-Q•
(ralll",)·
~H¥ + 4- ~H,K) ~z • ~x,
- -j' [-~jl} Cos (ISnz,)-a. J01l s/" (fn,x)] illS =
i
[-a. IS"", ClJS(f~"'1I.)+ a..i'''07f.sin('~'''''>Je-j,..%
(-'%jt) -
"'-'
- ii",
(£ tl~)
4"
ells (1$1r%')s/" (6",,0 9t
- ~1.6%)
+ ~ $'& S;" (IS"") c.s ('.",D't - 4/.' 2) P. 7-20
(V/Ih).
i =' ii. ~ .si". (4S (wt-IcJl).
a., Ii
-.1-
_"'Ai ~s,'n. cu(ec#-(JZ.). ~: -i;vali --d,~ k(Jl.J.I~)
VJt 11-1,;: - - f £-=._. (ompa,.;ng (j)a."J
,
A/~o.
{•"'t-fo ·
From @,
H-al
-
Q•
ff/F. S;n~ 39
r_4' ~.
~
..
~~ S,n.s",(I.J'f-JUl.)dt
..,;:,~ ",." •
CM
.
@
.fIn (.f-
wit -./tJ,\R.).
U>.
~ 7-21 MA)twcl1~ From
V
V· i -
FrDm (!)
0,1
A'•.
d.iinc
.
H - 6JJA .L v. r
I
~ X ~ ... -)?~
Curl 4,&J,,""ons:
IC
H
sa
j
tAJ-E
(j)
£ - VitA.
such thai
-= -Jl:....
CD
£
V• V-A.
Wf!- .
- -dp{V{fi.X)-v"A.l.
@
(!t f/x(H-j4J4.A.)-o, /.1 ii-jfA}.~--Vv,.,. (!) SUDf"QUi"9 @) ;ro", (!): lU4:A.- ~ff1(v·A.)-'~.l-jf1v .... FrD/It
-
L C"DOle
-
V • A. -
4) £,.(1) 1,«'0",_$ h) £ '/. @ h«c.o",.,s ~ .
i-WP V", · ii - iw~:;'."'* fi(fi.XJ. v£,4'. + C&I~4: A.-o, 4 J,.",~."UCIS H./"./,.Its.t .".
Fi - jw~. V x w. V • ~ - -jltl~R. ,.,~c. V It ~ V" (£ - -.: iifl.)- 0 - - . L.f l - i.,= V II, V)( R- jwi- jW(~.l4p) ·,jw~ (1., L).
p. 7-21.
@
I.:
CD GnJ @ in (!): 0 .~. iw~ VwV- W. -).w~.("ri. + VV. + f.) .4) •• ( (1- V.- ii. - vJ ~).
SululH-uling
.
I:: )
Choo.rc b)
V· ff. - v.. £'/. -J-
1-
V"• ., ~ 11. •
@) hc.comc.s
P ---i E.
(7-,S)
&1.) £9- @ becomes
1 - ': We .,. fi V • i.
- -It: i, .. (,' i.
-I
C' • t'. 1f.) ·
WmblnQ..fi'on of £,L ('I-fS) anJ (!) 9/v.s
l. = V. u. w. -.l · C'. P 7...2J 4) --
b)
I/Jil l . ' '''''' cu,.,.", , _ CtJnJut:.fion cur"'"
V.s-«, VIe ij Jc H-
wE _ (Z1f'ODJlIO').J6-/tJ-·. tr
s: 10 '0"
V~£· -J'WJAH f1 (C; ·11) - H - r V Jl 1.
va
40
Ie
~
II- 7~1Jl'D.
V·jj::lo.
Suf
v' H + IT V ,,£ = 0, 0': Vi ij - j~trii- o ,
-------5) (8--
f)
jj - is 4.'0-' C()S (10".".1 --';,y+
~.J
4) k. =
(A/lit) .
W~.'. -= ;::;, - ~ - o.tos
At t
-J.,O-1 ( 6) , w~
CrQd/m}.
,..,ul,..
IO?1T(J.'O-J)-.J~Y+: -tn7l+JjI n:O.I.1..':: -1~olt-1.S (hi'.
y
81.1t h)
(rJ; 4
~ -
A.
£ - ? it #I
Let
Clf-
~~ ~
&
Jt
'0
.:
y - 22. S
:t nA/Z
(M).
4" :II -:-If .. l.s(), ./0-' COI(to'"t - ~ r + .. 1) (V~,' ,,_/
wi -lea.
e; · £,. = s,,,.
(m).
£-
ii.,£",sinfl(
+ .,£.Sin(tlH'/r)
-4xE".+ ~Ey.
!.x S' I £ . " mCr4+YJ-s/"eccDJf+cos4s/nyr Ie -
; : Cos '"
+/ ' -
rt.f r . Sill
.
'?.-" ~ . v"0l~ CDS t \1. (, .s;\ sin'", ( .!x.. e: - .s: £,. I £;j
I.. £r _\1+( £. _,J_2~~_f, l e:. . Sift r) £. 4/" vI E,.E.. Sllt',vvhich is the ~fuQliDn of an 4/lijJse. I,. Ord., 10 "'."J Ih", /,orllmakN 01 fhtz (lIllI's e , rofaie fhtZ coorcl/nah 4W'el x-y Cou"hrc/oc.'wlsfl hy t:tn DI19'. 8 .}" x~y~ Auume /oj, e «'Iuo/ion t>f
fhe ellipse in
Y
.jfbrMl
of lit.,
hew cODrJ"nd/.~ 1-0 h.
(-!if + (-:fJ -I.
0
whera
"
E. ' == E GIIf. + I: ~"n. (!) "ox£ sin_ -IlvCMII.@ ~ £ y " ", ...s~h$l"'k""n!J @ oncJ (j) in @ Q"d NZ,Qrran9;"9 : 1 __
£
'f
:"c
Co'-' + Q!
s""'._' .,. £ 2(r;n'fI .,. co,.. ,_Z£ £ :sin,,£.,S(.L - .!.\_, h*"7
Q~
Y
II-
CD'-. slot. .,
J
1
@
£.1. sin''''
co,·. __./__ ..
£:lit& sl" . .,. .. ,
b'
1
.r;nBCO$iJ(...L_L)_ j,a
(Z) tbtJ @ Cdn 2.
Q'
COJt
E,.~li"l",
•
h. so/v.J -lor fIt,.~ unknowns:
8 = .1.,..
a. ..
4
sin
+
a..
£'p. CD.
b*
Y
ohM;"
CO"'PQrln9 (!) dnd @.
4
7
w. I. __,__ 1· A'
-'(2£teE M '.1 y=,\ ~; _~.a )
LQn
j e ('''sacZIt) 'foE! (I-~.Cl') · s/" '" L
,
Ie
..
In parHculQ.t> i, £,,,= £,.:
-t. ' 4Z
IJ 4
4S~
-JI. £.Ut!f: •
"-Ji
E. $;n
1- ·
·
-P. '-H
LeI an 411/l'f;c,ally po/arizecl pla.n. wave h. r e-« p,..stlnl_d ~Y fh. p/'asor (wi~h p"'pagalion lac lor e-JJ,~ omiH.J):
s = s;e, % 41 £J ei~
0.) wber« £I
J
£i » and ~
are
arb,.Ir41'Y
co"'fanls.
Ri~J,t-honJ circula,.,y PO'tlriz~dWtJ"":£"c-Cre(a..-iay).
L.Ft-hllnJ circultlr/y polarlzeJ WII".: ~ • ~, (;Il"'i iiy ) .
I
i-,
£ '1:, . . z' ( £, t ) ·£, e '" £.1.,i j£.ej.A), .
rr
f{£,
£ ... £,." -+ ~ .
fhen
b) LeI a.. r"ght-J,anJ C;rc.uJa~/ypD14 rt'z_J wav4Z. he
E" e cs, - ia~) :a
- l. -
(a:.f -a,jZ) +£
~#
4£._ ,
(a.; + Qyj)
wAe,... E•., and E._ cue I"'9/'I-hA"J and 1..lf-hand ell'-pflc.dlly pola";ZI.J waves respflclivttl),.
SImIlarly, a 1...ft-JuanJ cJrculor'Y pola.rizec:J wave can h. written as
~~ • £ lA,c ., j iiy ) - £ (ii~f +~jJ)+£ (ti.1 = ('c- ~ £'•.,. .
!-?f-'
For
COil du c li"E In II
kc1=:s
.I
W~€,-W~E.(I- ; , : )
ecJ .
Ice. .. ~ -jo(, ~.-
J"a
- ~j)
Icc' ..".- etl. - 2.j
R. (k~l) - W~£ ~l., -<1_ 1-'": 1_ (,Jju(II'--~-(-~-~-)'.
Fro", ([) and 0
0(' -
W4I
OJ,lain
4J
'-7
-11TJjJ",
do
a) f =
(:"Y»,.
a,.c gOOd cl1ncludo,r ~
A /I
'0
$-
~
7~ -(1 01j)
I
: ·
(Hz)
J
Co"a,. 2.D2('''jJ.1fJ O.II1-1t/ /.01..'"
I.s)-'..J
$; Iv. r
Z.II' tl?)."" 0.12/alD' I. oJ'" ttl '.29 _III"
8".$'
J."~.
".1tJ
.,0'
o.•
,.s)
c..",er
4.'''-10'
4."1110
S/Iv.r
.. "./0' 4.J.P/O" 2.0)M'''·'
Btal/fJ.s
I..S'~/O' J.I'kffJ ,. "9)11.-4
b) f
- I (Mil..)
c) f
.:a
I/O
16.) 11/0·'
I (G,Ha)
J
1.09 X
'0-'
_'D"l . wJ7! [114 (:cJ- ,]'/~ .. wpj£ ,,: (-~ )1 -E.,-1.~1
-=.3"10' (Jlz).,
0(-
4
L t 4 n tfc :: Cd4
- 0.49 7 (N~/".) .
~
h)
.,"',
... z --...
x -
.L II(
ln2
-I.J9S (~.
'?t. -kit (,.,j 2':.C) -23' (t*JO.()D~) (.4) II -w./jU [ 1-# f(-!i;Jl-3/.,,, (T4J/,.,) A - .,. -
u,. u..-
O.
()~J
.(hI)
61
T-1.t''1JJlfO. ("'Iv
y-n-, [,4 f{:C~1·1.
f91S 1110· (11th).
J'--'is' .1 c/
At
(~
. £- = lAy - S o. i-b
x ~ 0,
Ii· ~ G,,1tl- a.. O'z/Qiff-....u"J
· .ii- a:z 0 • 210 ~'-9 4)
e-···'·
s"/t ('"ID't-3 1. ,"~~r-1100/'7f).
(AA,,)
roC - 0.1.
e»
I/J,
d.=fJjiijlll+(i!iV-I} - 14- (Nfl"') ~ -wJiii [JI+{-4i!.flj'/&- 300" (raJlm) ~.. IRO" c1c....c,/ttIC)_41 . '.•• 'ft) 'e =,~ a',. {f (:crJ"· . . ....
n
J6J1I
It
"""A. -
- 33 . .3 1110" lm/$)
;
-IJ--
1-
~
-11.9
J,) c-"y - /~ •
e)
fl. 6 7 • 10·' em) 11 / 0 .
' em).
Y -!fln /0 -17.4 a/o·' (m).
At
y - 0. f (m), HCy.tJ-i,.o.,.-,·ao. r S/n(lo'·"t-Joo".,.c --f) - ii",S.7S a /o· · sin (I()-""'tAlm) '
y)
l
(10 *)-.Jm['Ie. Ucy). i" j.i", - ..l.fl.,i.".Iin(M",·f••.•~~ ,
p. '-/0 a)
(V/_).
,f- '1I'f/:,r
tr- ""f~fIJl 20
t.d
%-
~ './1 II)
~/.
C·
-
(SI",).
'
-10 (JB)
., ;.(,.. • I. 71
.'" 4) From £,_ (~-~.2) cIA
21r
11 / 0
"'-J'"fpr.I.,'lllo" C~/,.,)
b)AI f-fOfCHa). 1I2
-0.99
..
II
'0'"
em>.
u,. :; -!--(!'II,)- ~.,.,~"IJ. 2,..
A.
~
""¥---"--7· flu d u, 4A..\ u.~ • ...., +~ ( dA. 9 1 >"7il. ·
>"-7
1
u,-
4S
'}
~
'-'2 a.)
h)
~
10.J (W~em")
2
tr:
:: -
1£1
-
2 '1.
a"
t£t .. Jo.oz 7. 2.7S (VIc",) .. IH I = I - 1.J' It lo-J (Ale.,.,.) 1&
;.'£ 1
"- -
1£1
I. 'J J(
-
'D
J
2?S (VIm) 0.126 (AIm).
(wl,",a)
~ 7)I. 4 1£1- 9qtJ (V/~m) - 9.90-/ 0 (VI",)
."
IHI.- ~.'J , I'-/J
(Alent)- 2'3 (AI",).
AS.luma /-J,af
r"pres«nf.d
Q.
cireuJarly piJ/GI';z.J plane "¥t2v « k
by
£(z.f) =- Qxl; c.ol(lJt-1c1l +4')+ a;,~ ~;,. (Iii-It" +'1)
Ii Lz, tJ .. 4y Tho. Poynli"!J
~. cos (""- k%"~)- Q,. ~ s/,. (wi-'"'' /». v~c~o,. ts
~ - f ~- liz =iia '7E,. , a It
0'-/4
if
[c4S'(61I-lczof,#)+ S;"~(,,,,-* s»
unJlGnt jnJ.~"J~~ of r:
l- s.e, + 4,£" Ii - :, ~Jr £" = ,; (~£. - Ii. E,,) ~,,=; ~ (E~jf')-a.2~('£l+I£ ..O.
P'l-f5 FrDm Ga U$s'1 /Dw ,
£-a.P
r 2'11".. r '
+J]
t onJ s:
(WI"..)
wl..".'''r+!,.
/,one C},a"!Jc J~',,~ify Dn lhe inn..,. ~nJucfor.
~D" -1° l· dF :II:
. £- - r . -= "
-
..L in (.!.) 4
2,,-£
~
,. l"lj/.)
From Amp~rc'S c.;reuJIrJ IA~
PD)'nf,'''' v.e/o,tJ. - £ Ie
Power P
ir4nlmif'"J
2;r · H- a. v.:
H'" ~
ot CF'O,,-Sfl.CflDnal
2
",. In lIJla)
QIea .-
=1(/· ds - 2"''?n~,,/., 1''''[''( :.)rd,. cI+ -It.1. D _ ~
.
l;-£.(4~-i4¥)('-IJ~. AS..Iume. 14e. ,,~"'.,h.J e/.c,.,.,.,f,.'J ,,,,".,.~'Y 106e E,.lS) .fa. E,.. .,. a,. £,.,.) .;Jf6.
P.t-"
c, .·en
4)
8tJunJary CtJndi";lIn tal'
2- 0 :
2" = 0
[ ~(.) ., t: (a)] r
£
(a) -
e
£.f-~ +j iiy)~iJ: Q./~/t-/'anJc:ircu"rly
""-3
poJ.,iz..,d wave,
r
d,·recbeon.
b)
ci;,a M (JI,-HJil(D)-.!. s
H,(o)= U,(o) -
It
~(.) -
..!. (iii
*,4.)
i (-4.'>"£'(" - t (iii..'" a '1. ~
I
~ ----+- -Q.a{i{lD)-Ijj,.toJ] =~
~
' . ' . -T
ii· 0 ~
;"p.rf.,~
CMtduc#ol
y)
H;Co)" H,ID) -
24: lja.. .'-iiy )
~ - - ~ It H,(O) - ',- (G. -j.,) , f, (%~t) -lR.. [Et (%).,. £,.{~)j .J.,t = lR.. ~[(a.-J4j,)c·i".+ (-.. 4ja,) ~~zJ(illlt - 01.... £.[-2j (.", -j~s,·nlJ.lci•• ,
c)
- 2£. Si,,~.(. . ~;,.tJt-aycoltJt).
Given
P.'-17
£; lx,z)" Q1 fO e-i t ,1T. ., ~~
(v/",).
~-"L"')....-~ kx~'. Jc&==~ -Jc.-~ -'/Jr:.~. 10 (rQd/ltJ) A -
2" T1" - tii -
0.
'2 r (",)
I - f: - 4.1'.,0· (Hz.);
h)
W-J.."'f .Jc~ -JJI f0 9 ("tAcI/s)
E; (x,z ;t)- ~fO CO'(J.(O't - '.-I~)
(VI_)
fi,.l~Z). ,;. iilM·xl,. ci",-f-~o.'+a.4' - 11D1( L I 14 0.'''' Ii 41'- d f(J .-Jfb."., '· 2 v -y -(-4 L -f s.L).,-j{'a,. .2) ~ IS" ~ZOJl
c)
ii,(...~z;tJ- (-i. ,;'" +~ J~,,)(,O.(J./O't -(ur.-tl%) (A/1ft). Cos.,. - (1.,,;" 0. - (f)-a.-fi;.o.,+i.ac).i... D.'. 47
1=
-
c.,
cJ)
("1 0 ) '" £,(-,,0) ..
ii,. ex-, s) -
;f aM
II
-j('.-,.) (v~).
"
0 - . . . I:.,.(X".) -= -ii., fOC
aM' .. i;, D.' .,. ~ D.,·
i,(~~)
- _.(5. ;f.i - ... J:,,) c-jl,,,-f:4)
(AI".).
l, (1t,,2) - t; ex"s) "f,Ca"s) -a,ID(e-i,S. e11S) c-j ""
c.)
• - 5yilO e-i ' " ~,." IS
("Im).
~(.,S) • ~(""I.) .,ii,C",I). . --(i'./~",COI'. -i". ,f-or si"ts)':I'. (AI",). '-II
(i.ivc"
a.)
1; (y,d- S(ay.., a.Jj)Ci.{Jiy;j)
kY=-'fl, 1.-"
Jc .l~;.,Jt; =12. (rlAcJlm).
kz . ' .IJr.
..
T .. ,; - : - O.SI4 f - :.. - 1.1J ./0' eN.);
A •
b)
(vllft):
(,..).
IAJ-Z,,! -Jt~ -J.,./o't,...,,,).
l,. tv, 2j t) • S" {~"A,IiJ C./U (.J.''''0,':1' " Uiy-'s) (vI,..). Hi (1. 2 ) - a,.;-¥; • IJ~" (-iT"/-. ~~:).r("""/j),i4llf14
t
= 4 '_..L) .i4 Clly · J)
•(-ii. I'" (-,1.,,) {J. .· 'fjy- ,~) (AI,.,.). ii"l · a. - : . ~ -cos·'ff>·."'-.
#;(y,2it) -
c:.) cJ)
COJ
6; -
US
' lI I D' of .,.
A_JCi;~%)=o a.nd E.ycy,O)+E.,y(y,oJ-O laoJ 10:
£, (Y,S'). -
$(-~ of . . n) ci'llJ,••)
if a,.,. • E,. e~ x) . -/;~" (-iy"f-a.fJ-s(-iy""OJe i' 14y# S)
H,. (Y. %) -
- a;. (- ,:,,) .-;'(6'1#%)
e)
(vI",),
I -
£, (~z) -~. (y, s) + £,.ly,,;)
(A/",).
- (4y (-I0j) I"" 's"'l; ,on cos ,y"iUfy. ii, lYJ z) = i,. (y" z) + ii,. (,/,,2) =
a. (- ,~)c:os 'z
-e ;Ufy
P. 1-/9 4) frDm £". (.-,0A) and (I-'O"):
£, (:c., z;t) ii, lx,s;t)
iiy~£;.s;" fJ,a cos .;J.rin(wt -~,1t s,h9,).
= "flt)C"4.C.~SC4lr4')~S("'t"AXS,it9;)
.,. i.. (2
b) ~ '-11
it).,."..·s'·"f),zOI6I;)Si"(61t-J,1tc",,
-ttl. (f. Ji") -all ~:;• ~/,,_,.#;nl{J/1f.
~" ..
(Ji)'
C#I.lli ) .
4) FrDm £'1. (,.•, ..) anJ (I-,'II): l, (_.. a;tJ· -Z£.. [lI_CMlIls;nt,A%~.;)ClUr.,t.J;tS'·"It) ~ 0'. sill -I (;4$fJ,aCOS(}I)Sin(",t -AUl"S,.)j,
ii, (x"z; t)- "0,
2.~"
C4'l!,% c.s 'IJs/n(lJf- J, 1C slnBi)- . J
i!a." - 2.1.. ~ (£ -ii·)- a..
b) ~ 1'-1/
t +
r-
~~
tt"! "
J~J -Irl --+
-
'7,. Jill
S -
-
Z;/tsin.; c.SllJ,.~~S••). -',
f.
r< 0 - . ?,- '1.- '-'1a· Jrl-:·
:::~: .. J
StI.=20/tJg,.J - 9.54 (dB).
P. '-21 4) L" -lit.. lossy tn.Jlum (",.Jiu",Z): r _ - £. .~z -iJaZ t=t - ~ to e
t,
•
-/Till ~ (~J-I)~ /1-Ji¥[/I*(!:;J+ I]t.
w},.rc
.,
I" air.
f',.'
(raJ/",),
w
-ftc. = I.s» /0' ("4e/ls)
~"&: - :~J - ~.5 ~ ~
I~O"
fA,-l.JD
"fJJ·
72 -oJ", - ~(/~t-14)'4 -125 ~ l t-iie s, . ,,-,.J•• e."j"~'8 ii -4 • !J. _- --!JL i"b.J·c·J.J%.-":"z. y f'
It
'11
a
12 f
49
(o/...~ A-fl. 7'(fDtI ·
ii - I, -j'l "i - 4, 7'iOW L ·
Wei also hova £-r
K
a
K
-
£., . ~ ~'s - - - - . . . HI'
80 u nJ Qry eona;";o"J
.f." l
- --f.u... i's a.r IJ01r (.. .
-=r -
Ii o! ~
ana
0:
£'. [,.Ii"" (I * t..' J.)". ~-j IJ.J.£ = 7. SJ .-)"",. £ .. 2 71· J 1sr l,(z,t) - ci. 2.71 GOlft.,./o', '6% ""~1·) (vi,.,) 10 .,. £,..{ /0 - e; -
I. " ..
...
"
""~1·)
H,. (s,d--a, a .0 1J CoS fl.•• ,O·t ., U
(A/",)
it (a,t) - AIL 7.IJ e.-J.'u e.6 (J.,./ort - ,.,~~-/7.z1 H, (s.t) ':'" ~ O.DJJ il. H • G4U (/. ~.lOft -9."% #11$'
-
- (10' Z ~ IZ01T -
2. 77
Ai ) ( u;.v "1 - C1s
P '-23
F,.Q", and
)
-
-0" 0.122
(.,
W/m'h
t
..5J ·4.1.' ~ 2;? Z2~ (Cos 13. Je)e
a. O. 122 ~-4 ,'z
==
(A/Ito).
1
Z./~."
h) (<1.J, - Os
(V/"')
.(W/mt).
£,~. (~-Ia'a)~ lr-lo,6). (r-iD'a)~(5'-/ot;j)~ (~~ lo,a)
ce- fD'Ih)
\Ala
hC7.""
£.I = a. ,(E.•-jAr"l I~s) . ,.
" J
HJ -4Y .1 '1. tcIefl·1~~_£." rz2"1.%)
il-. '1 -e:.. -4 i. e
i-a (.c.",,-11-2,£-Ci /.SJ' a , L l
".1
t
M
E= a e
JovndQ,.v ,
J
H-
e-i,Az
'II'
I
Cond/fionf QI
of z-J:
(EI>f,"~1
l
J~2)
,'.10 e-ip ·
z-o; E:, (oj-£'• Co) . 'H.(o)-R.(o\ , I 'J.
l;UJJ- ~CJ) ..
ii,(J)-ii,cJ~
Four rzl1UQ-hGns fo .$0/".. -four unl"owIJS ~.,. £- E I ~~ 1. "0. Q nJ £", i" -f.rMJ 01 E,.. : a) E .. _ Je,,;-,,;) tlu.AJ ,. ,. ?.& of j(,t .",·)'t.,.Affl £".
£ 4-_ z.
.".(70+
zJ..*;1..1
'I. 'I.. eMAJ*'i('.~f:J~;n,.J £'.. · so
"l
,. it. -
£_ _
d
'I.'7a u,}.J ..J. (,:..
J
£,.-
PJ_
,:> SiltJ,J s.
eila
~
.2'_" .l}.J '"eo'At/1-J l,:.,:>s;" ~d
£;e •
w/,.,.. " •• {IleI~.-I1D"'.
!,.- w/c. , b) 11 J=
AJ/
.·. r
...
'7a-/iJ,fE,. w~.c. ·
}, -
.
"J.-,.L .,. 0
11: -
LIlt/US
1oa .,,, I
'7 1
E,.- - ,:.: :'Jl E-,.•.
,,/z.
J-
"
~~
7J _ JJ • (J
-'.
-,~
:' r .. 0
when d • "~J/2
,
, ,,- f~
,
~, J, . · .. ;
r- 10: ?..-Jil, 'I, ~"-~"'.(.rJ = 2. W4v.l.nglh 01 reJ '(jhl in J,;"I.cl-,ic COAl;,.!]:
p'I-1.4 ti)From £xo",,.te
A~ -
O.7S
-i"- - ~ = d'
>..; -
- ~·.l97 ~M).
~
J
.
>{ - 0.4-11- AtIFrDwt
£,. (I'-"') and u6'""9
wi.,.!, ruptlcf '10 ',. '1.
Z
(D)"
1
rz '1J I
=-
o. '94".
impcc/onc,dS normal/zQc/
:
·i,. t."Ad t-h
tl _.L
"I. *";'1. (). S - ilJ.J41
Y'''')
YtntJ·
J - A a/4 - D./JJ lJ) For vio/.t ti9ltt.
-O.f'JO
If'
f +iJft." tat! if.,. JftA..
",d
•
f - ~·o.Jf-1
r=
2.(0)-1
Z, (0)11
_
;ncit/"", rall.cfcel = Ir)1 x-fOO%
p.,ec,,,la~. 4 I:
p. ' .. ZS
r = o
r:a ,J
D.J''- .i'f'~ (o. )/')) X 100"/0 == (O~.
1,(0)-'1•. Z (ol-? %1(0)+'1,
~.1
't·,. ----. '1,
1
'z. +i'b,t-."AJ
'II. + j'l, ~"AJ
!!L ='1J
+'1
t,
I - 1;, f ... a
r;
·
S2
-; 1A.(f,-n,-J-O. (ti_~ - o
d) (t1_~
eo) I) L.i £, .--C. ---.. d-"A,/2~ n=~~2. . ··· . . . ---- t.n~J-O p
A - ;'01, -
ka -
Po f-l7
"
4)
• (Hj)
F,.om
t•
,
PrDbl.". I! '-2J,
[.fl.) •..,d .iAdEI. j
I..!AJ
-
t; .-7f"H,·--J\TJ
) e
f.- .,.H...•
£
JO·
J)
; -J"'f/lat{.
QI IO(MIIa) .
E~# - '1. H: • -i (-'I. b)
.... A-
(t - j ) ; '
£. ,.
"lit
.
CIa-i",)d
c·"afl.-/Afi£;. . ,lill(,.-J-.Jd
.(-~) I.«J~flEIe -J 'I. silt (1,-j-'J)t/ · _
~£_l4,,--
t co'~.j~t/ - (,.,;!h. I*i-AIIWeI,J)t. '1. ~AtJ-JW'SJ ••. f-j
a
((,,), ={ cfa [ lE;.lI ii,:) -
•a. .,;;,. (A
'I a) ,
'I.....
,.d u,It,J
l$/nA.~u.ketaJ)'I-(c.."J
,
SI;'h-f.J)&
,I.' (Y (1,·It"JC416"~J)·+(~J.J',"'''~dY 2'1. r... 'f I
(#.J - ' I .. J -
il:'J)
~;It ~.tI C.O'A~ ~ .8;,.1.
0(
.." I ·
It
f +J,t."htlf'''''''.,. _ A";B. t." AJ - }.t."J. «a ~
wher«
((1.)
a;
4E;.
rJ
.
(,..), • .!...(er_\ (t1..),
(~")i -
'.
r
, . 7 s""I.dc.'}.J+s""",,.J c.J,."d
z;. ~.(:.!U
C/I'..J, _ ..L
ell•.,); 'I:
1
,
r ) (.I1'n},JCMJ,.t.J)l-#(c.Atls,j,A-.J)1
AI/-to" lila). r-r.,oa/o"(S/",),
(".,J,
(IJ..,); • f.
'J,
-I, It If)
•
tJ
.
""-}.-",'S1f'"'' c1.i.~
k
P.. '-21
lc J
.a
~¥
Ja
."
-
J_J. K.l -
1
•
W~t"E~ - J
"'Jle'; .
(onl"nu;Iy conC/,.fiDnS Af Z.o for
Gil
([) a"c:ly rtl9 u'-",:
X
kl X- It,ll. - t.lJA*. S;,.'" =- ~.-2.D'JlI()"" ® le,... ~~I"J.~. · (j) COl'hh'-"ittg ({), @)ttntJ(j)~
01
W, )JOI~&' Ii
J
and
'u
ot~QnJ
We. CQnsolve
A.
1;'''''''''$
Juf" .sl/tc,.
~1C.' « "'ju.~1 ' Wca
have d~-elzx al A.
-/wIP." - D.I,74
:Ii ;
(IF;').
a.) 9 - (a,,-I A ~ t#n-I .l:!!!LJelo# a s.s« ,1/0-4- (r-J) t
--,;.
_ D.D.J~
4J974
-t
b) F: =-
•
?,
z2. etl.... ~
(o.J
tI
().O,9J(I"J)" )77
= t.
c.,'"
O.DISI (l'Ii) - 4112'. £~ -:!L..
c) (f1.. '. £
I
1 ••. O"J(f~i)
_
c~.
'20 - ~ (I • ;.)-o.099J (1*j).
,.c.I.,;
Cool . , . c..1 D. oJ·" .~
~.~
t't.
,. 'I.
(~..~
(#_), -
cI) 20 IDg,
:-
'.Ii
~ ...... I
e-~·
'6! j -2
H a IF•._ t. '1.
• 2£. ..L..
e
- ·
- -3D. -
P '-19
\1....
(I."
'I
·1 II
%4)
ID
.,
- • .,....
e
-'c - A
=d
L, n
s;n-l(';;' .1,,, 8.).
co'e,-I,-f-!;s;"./j.
t.n. - II .f'-o., _
~'c.$.,
c) ).,- /JD - Ae
(lit).
Jaw:
e;
If.
,
.
f S' ~ - ,."
St·ne, _
6)
(,lw, il"\~ 'I, ~
., Of.
Sncl's Silt
~
t.
tI'i,. ••·
.
J ". - Ii ..• ••. ~i"(';-4)- ("~'e(S,".".c.l.,-c.".s"tt41
S,''''i
[I -
J,J!'1;'t..;J.
S4
P. '-JD
a)
FrD,"
$i,.
'e. -If -
J/"'t
-If..
eo, "
--;.J tt)'ill·....'
Ejl.
s,;"i
(f-/UA) 41ft! (,-/JsJ,j: £t C-..~ ., - , .£ . c·j 1,.- , ~ - a; · ) H~ (x,s) • (~ ;'''a + S,1t ",.
au
c·_'Z
r
wlt.rw
> I 1;,,. ~:>
an'i:
•
.
--,s e"i"~", &
II -As,'". _ j!E.jin •. r,· r, • file; ,, A I(tJs,.·.. -/ ~ fA,-
Z7a CH ' : EM · 'lcoslI;-;',I(t)~""""-1
£ t. -
6) ('..,)u P. '-31 4)
~- s;"JJ~"I/~, -
b) 9.- ·zo· >
r. J.
c)
't _ L
J)
1tL. (E,y N,~J "c ·
Ji" S,
.(,.dM£,.(r-IJ'1~
O.
$;"-'JI/.,
"'.JI·.
-I{ 6'""'i -J.o,
CD.~ ••; / ('t)Sitt'.,.-1 --j:z.ql ,fi;, eose,- - c,., I, -• .. iJ'·. cio:" ~ c.••,. ., C4AI .,
R"
_I.',
Z& CO".
c,,~ '," .~., ~
Tit. iransmi,,... J
e cf,
eJi'·_1.'9,}~JJ
w~v. i" air vor/Ils as --\-
~
-j~. I
-"I(t)I;,,·.. -I •
Affw,,,ucl-/on i" Q/r +Or cad,
~(I.'1I).
wOe".I.".,.,/,
- 20101,. c..~.l. - Is" (dB)
ss
lolql ",.'!.cl-ions
bD'" s
DccUr ins/elft Jhc lo"I-,·n!J Su,.l'.ce. II.clAuse
9,· •
4C-·
>
'e =
On e.Jli.,. {,.om ",h. ,,.ism,
=
J
Sln
"u -
4+
(f) cJO·.
.Ii,. ..,
1:
P'-;I",
"2Z'l.. . '1. ·
(i. (~) ·
Llncl.,. COndi/i.n 01" no ,.~{lc~li.,.:
~
US
-/'-71,,:1 $;,.'. ..
•
Silt
I
-u
'U
I +
I
f ft;J
~." ~ - W/2.
,-
h) For parallel pola,lzlI./-illn (A.nJ tE,"'~.I: ,.
~ COJ
JI1
"' ••,
~
(
II -# (f) · :a
4 -/,- ~"ina" -/1 -#'(E J
- Sin
p. ~-J" (l)
Sin
IIi
9 e
"
T,
~, J4:Ti. Ii)
L.t 4,/c.-x.
~.
e.. Sin
-
~
••
-
" I
" '. -nI2. I
+
f-l:-)
t·"',,-If· Slit ... -
to,.
9...
('e) ',.)
JJI
2.•
JJj I.i
&2
,"0
'0
.-
~
,.. ,...
aa 0
~
Cl2
,0-
0
I!
'-J'
4)
~.
r:• _ ~'_ ,
W.
I,.. c..., £.. (l')-l- £,. c.... . fl,),."
£,. CU'; _
(£/~...·~.
(£I)'M..
have
wJt;~h
.w
•_ "ce.,,· ,. (.'4 '1/CO'~.
.l.. r:
?,e.,.;
of
E,. c•• ., _ t (~\_
'. ..
I ., ,..' -
~'
CO""pa,.., w/Ilt c,. (f-IS/):
S1
C•• '/
J
J , .. CO",
Jll
co'4.'1, COl';
p.
9-1
Q,..
iaiia:
La ~x
H.
., a. »
ay -j} 0
0
t'
• iy}tAI€ I , ~ ~=o ay ·
P. 9-9
~+~,.".
Z, -
(i-4>JIIC
J- ( Z )-O•
F,.om {"'I'. (9- #,.)" (9-""). Gn"(f-~'oJ: Giver): Z•• SO"jO (Jl) 8( -
tI.Of (J'I..)-D.OOI/r
tf -=- 10' 0.'." (rarl/,.,) CHa) A. a (ia
..
Z.- O.l)S."
-f -
21.
~.r/.),
A 9-10 4) FDr ft",o-wi,..
L.
(A/".),
(NjJ,.,)
l!j - AZO (}lNIM) c.. fr • '0 (1Ft",).
,/
L.
•
fJ.""",• .,./,',,...-
-If: - :"Wus''-'(kJ-.J(t,,[/a ./IkJ'-,]-JtJO.
'0
.Jl... 24 b)
21
·
17 --.-. D=
z~n-l"(~)·HL,(l).7S
b 4"='.$1.
'-It
« lO·J (JrJl.
Fo~ cD(J~;lJllranJ~,$.,Online:
Z.-
P.
zs.s
• />-3.9/·,0
·)
t t4[v,. I;J
(PaJL = (~v)L -
z,
lti - z,
IV," Ri
(PQ v
,
)"
, _--Yaz, +Z,.
s cf }
At
V(D)-~-
%-0,·
f
---+
b)
V.· -f (V,. .,. ~.Z.) ,
1 (z) -1" P.f-/J
tol\ r;J
-
')%
-liZ. SinJ. )s
COJI,")'z. -
~
Sl"I.
"'1..
-~ GZ - wV
Jv -R,Z , dz
..b.._/J.GV dz·
{ dj~ -JlCi~ za J,) V(z' - ~+e-"z + ,,-
:z {z, -
"ea
z-
y. -~~ ~.. . , . t wla , ~.,
-A-_ 10
W..
JL
I.·
rtr: ., -vR.&
- Il _fL. • {~ 61
"r Z. az,·,
,,- - : (~. -.Ii Z.).
flV; 4I/Z,,),;'" + rev,. -liz.)e"'· I .')1 , ..... (zJ .. zz" (v.-.,..l;Z.> ~ - Z%.t~ -I;.z.). ·
V (z) - ~. co.."
Il..• -/Jf tX. ' --x,
ICQ\.:l.• I~"'Y-.1..(,,·-\!-) ~,..£o Z, , 'I
\l"'.,.IJ'0 y. ~
V(2:)·
I
~z'i Y:9
I.
.. (Jl~*~)&f (X, ,.x;1
To Wto.ll ;,.,,;z.
)
(1ft.
J)
For
CI
.finite/:". 01 f.. n~M
I.. -= Il. ,
" .R, t.,J, elt
D
11.'" R filii" ~l
J. J.u-m,·"af~J,."IlL: ·
L
i/~;J J)"$forHo,,(e.ss I,n.: A.o - If - so (.Il), ~/ itA,. (~) -t4"(...1~-D.oo,, -+
Jl. -
-!t- _ I. 79999 11 /tJ- J (,., ...y J",.-I,S tAl C. """#/Ifi" _~CU
O.
s (.fl.I".)
"cc~sso.ry fo f.r
r-G.I. ".,.,""".,.
t L II( -
a) V(z) _ 1.(2) -
phASe. ,I.,-/'I ;If
1000" (I.
Q~C :
.
....
7#1"'''. Jt)-'J - es. 24 - f
-0."05
0.01
(HI,.,),
(N1/m )
C-
,.,.f ~)
~I =-I:~1.1 (pF/,..)
f -UJ.{LC. = s. SUS,,,) (rwJ,4)
I
.
Jhl. r:ela .-i}z _ -.!!!...... i"··'· e-i}a
~ "zJ
VCZ)/SD
'tlJJ
.,S .sin (IODD1I'· s. ff'~4J% - 0./014.,,)M
V(s .,) - S.17 ,,-•. ~
I (s~ t) h)
At z -
e) (~y )L
-
- 0./0$
~ J( 104 (1ft) ~
i··o'~ ..s,',.
(.-QJO
,·t-.I->f".f"~J%-4fQ1.4"')~
w« O~4"" V(.r"/O~I') 4",} I(~.'IJ;tJ.
i I.ta. [v, ~t] -
V.'y v.ry S ",.11.
(No/e: Tk« ~iv .. n line 1.."".1.
SO (k.)
"nl),~J'roblc",
is Q mlsp,.;nl. II J},ouIJ hAve .H.~n SO (",l• . WJ.icJ, wou/cJ mo.1:e -lite. n",,"be,~ m~. ",et:AI);"!J.~/. \ ."
6z
~'1-'S
£,
a) From £,. (11-97): Z,.-Z.t.nJ,,,,l 1It>.J.. F,..m s. (9-J7) QnJ ('I-ill): .., -/l-(R-.J-·w-L-~-G-.-j-tII-C) tlll z• -=/ t;./t"'i £ .,. jwc
.
z.. .. (R., iWL)J. I.)
D
p. 94&
From E"
!J. -
Z,'. - Z• co tlt~,l."IV -
(9- ,,):
2.:' J. -
raft JJ. a t'a.n
': - 41tJ~
4 WD- - -I. '1J2 ,
Z, - Z %, +;..%.. t.,.AL _ SO I
Z.
•
*i Z£
t.:j.t.
- 26.3 -),.9." P.9-11
Gi"a,,:
Z.I . -
- lWC [GGa.,(w c1r
Z.
,>,1.. -
(4D"j.JO)~;S-o(-f.7,JZ)
S(J"Jl4~;JtJJ~1.
732)
(A).
z· cdh>" .. 250{-50·
c.. a.) Zo··/Z,.Z... tAIl" ").t -it •
z. =- z to" *>/.
~
(.4)
- 360/20-
CA).
=Joo!-fS·=Z".'-j11.6 (.4).
ll. h
,I, • 4 (m) ---...
~
Z
.-
-
"j,., f-f!iJc A.+,j. .L G
Obi-Gin
D,'
rJ
of
D./J' (N,lm) D.l J.s .. (,MInt) .
W
(n - 0 tn orr/.,. 10 en.ur. "R, ',6 rr, C)
. . ,..-----
z,; .
G + ;CtJC -
..,. ·
Z.
-!'c = O.2J~1I3J(/o·=·a70.5 ~/O· (raJ/".). It - &."
(Jl) I
G- - 0.24' (".SI.).1
P. '-/~ 4)
+;. "r - t."J, (fII) illJ
~ '>-{fA*jIJlL)('*;.,C)
Il toj.",L .. ,
ca We
.. ~-
Je -
J. = o,rlz y.J#/trt) C - 12."
(,Fltn) .
Inpul imp.JanG& Z; ? arms ot
4:'
•
_
Z.,
i Il ..
e"prc.sfcJ if')
rtlJ".s/t:t."Ul Ili and 4 co.paci-f,'va ra.ac:llAnu
eecsllsl:
X. in
he
Can also
e. ~l
X; 1/.,.
J x, /I
-#
D~
v£
-#-
" i ~."'J
-
ti
~t X,
Comb;".;"" ~s. (f), ®. anc/ Q>, w. f,-,.,f:J J
fl.' I=~ CL
R1
X. --~x
I,
Q"J
'-12.:
~.a
a a...a ~;, ...~.
;X~'. •
of.
'
bofh of whie.h ore ,.",lnlsc.nf oi Ef. (9-114).
h) Fro," £,. (9-IOG)~
A1
fI.,~ input,
Af f},.,
V(~) -lL (%1. C~}.' + A. .sinjS').
Z'- ~/4,
,z:-",/z,
IN.
},ava
~ • V~/4) - I,.Il.•. .z'.IJ, ~~'- 0.1 .."J ~ - V(D)-ZL%I.·
lOAd,
I~I
~
..
Iv.. I • IZJ - J A"I*X"a
It-II
S-I
P'9-'9 4)
-
+~.
JCt:.-f
trt»
--s;r -It .•'
___
wit.,.. r:. .. RI,/%. ~... ~!1..WJ·(r£+'f-(~-'lj'~
(",."')'+1\& •
''''I
x L - tlflor" .Jr;,J.J)/J .
£) S - J and 'L .. /S(J/7'
t
XL .. ",. Z. -
• 2.
~
.¥, -
... ~t
r: _ (/"'''1. 1 .,.
a)~~""'(I-~-'i-a.-.. -.-)a_-4--~-fa
...
'"
I
• .1 or .: ~ ~ D,
&-
:1;/ s/J .
9'.1 (11).
c) From £,. (q - , '4) : '&. ..).~. ,I". ..~. t
A 1.1
~L-X&/z..
I _ (~\a
When S--3,
----+
GrtJ
f"~ t'
•
Q.ItJ
t - t.,,}l....
'i
-
."J
x..~ - S/J. , k "1 t ·1....r.:l'·r.:J~(f-r,:j-4",lr,:J
-fo, Ii -z.
(1- r. f Jt
, • • r.... '-%
For r -: . t _ {J/Jlf - . I . -i)."6SA Dr Jj; --.... I,.. • D.JO,'A U.$. .I". - 0.1 0 114 I. UIoi" v..1o. ,.••,.11 I• ."..
,::J::'::U
,.. . +Jt
.
I +;j.r;".t - O.• "';.D.'.
.
A,
L
t -
VO
--La [(1- r:
Z ~':..
I-LrAg'" """'mum • • ~
t • .,
(Use
tAnJJ,.,
a) • J('.t':..)~41(~1':.') 'V ". , -.
:' r«e,roJ/.","', 'P. 9_1'. ,.'
lit
r",
= TI - Tt
·
n_,""·v" ~,·,n.)
• 1'".,.(l."J
IOt /
A)'"
J. - : .
-
,
VOlf~9C,/1I",,'irtum/JllarcJf foflt .. /~eI is (~_ A) . or JAI, Irom ~h. loAd.
·
~
I! 9-21 4) Fro", £". ('-IDOa) ~nc1 (9-/01) :
v(%,· ; wAc,.
(ZL .zJe~lli I
""rl ..-l~ll~ifJ.
Z r - Ztz, .,~., _/rl"i-,.'
~=IJ.-lAz· r .
M-.Jt• IVrz',I-tt l:" .Z.,..../(f-# trt c·Z
I Vtz') 1-/;f"·lJi d ll '-Ir,.-Id)
S (z'J-
M
IVld
7' IV,."
•
,",,,,
•
.for
ti'-1r.
, 1., \
, + Irlc- J· ' " {Tit.,..;, .. 1 1 I _ I ~, • oWo' I, __ ._1... • • z·.,_ "'~•• :
"';'-7;tt fl' ."",.,c.,,1 ..L
,
flPPl.." , .
\ INA.,. ac IS ~,.,.. II.
},\ From
£.".
(fI- fU):
Zifa') . '-flrli"'·~;' Z I
e)
v-.,J.I..,. MAlt., ~
-0, A+ a.. volf~ .. 1ft""" ~ ."',
At
.:I
-trt f--rw;n ,
Z,.l,1-
S(z.')z, ,
Z. (z') -
or :;, ·
~
I
'
~9~22
• -/}'.,.. Zt . .~ £
..
W"'h Z; A
ro (.4) J)'
A·
.,.,.,*1';10 - ,.
z,z;.-j~t. .. R. R' - "·Z.t
~
Z =1l.,z,"iR;t
•
And ZL" 4O+jID(.4),
S(J-/·~' --...
'0
A.' - ;sot
t .... tan})· - 0.774' a.)
I rJ
From £91.(
•
W ..
I f.DA; + SIDt -
Il; - JI.'J (.4). ~ 9-13
,
A.V.
SO~
R ~ - ZHot - -Il" t
•
•
J' - 0./049 A.
- .L:l- _ .kL.. _ L. s .., .
2.
'*,
J
9-/004) 4"J (c/-/ol).-
;c. (ZL .. Z.)
V (z').
f. ~.'[ I of ,,.,
.i"]
J
i r - Z~-Z, Z,"z. -Irl. '" ' +••r -lAa' r ·
V (Z') ;$ Q "';ni"'l.Im wit.,. f/I- -I: " - ',. - 2rZ:).o.3>..-7J . r - jI • .~.," - o. .z n, .. . h) Z, - Z. ( e)
P 'I 24
. -
::~) - . " "';2'"
(.4).
1'-- 3;0
Ter"'i"qf;~ ra"·,h,,u. l.,.-ISO (.a), l.•• ~ - z,:. .: (O.S-O.J)~ - O.2~. E ( ) · R" ., II. t""lJ.. ". 9-114 : 11; + JX.. -
Lef Ii
at, ~- ~', ~-t.
r.. • ' •• ~
• J
~"ill... ":JI..
R.
I
W. hAve
,.."J t-tM,4.
'e~j,t ----. {,..('+~t).,.r . 4'.~ , ;J '" t CI-r...r,) • ]Ii
,
c- Z.~i [ll.nj'4"i )1/(I*r;.......1-4 ,:1. t
2~i. {-{'-(~.,U;")J:tI[I-(',a""'t>t+4.; J. ~ t _n t • ,I -,.. Z1T t •
.1
"
,. r
ZL -
Z,,.-.:r
r-
/rl. l .,.
/rl- ~:~ ,
.
'r· ~2;
!-71.
1
(S,.I)_(S_I)Cilw
~-
u.oij~·u-O~~~
- Z,
J6~+(I. ')~
(S.,) ..
• Z I-iS t.,.(J• .:./Al .
• s -;
tell ('''~/A)
I! 9-2' 4) G; van: \l, -
z·
Vi -
%,. ; Zl
z•• z,
D. II£. (V).
a
S-o (Jl), RI.· 2~ (
«as ;
~
I; • z • : z., ~
V,.
where Z.Z '.62,.J-Z.tMJ' - z ,*,·zte,.~l. , • Z.+J.·~Z."tM11 "Z.i-o.,.p
·
_
~. ''''il''~J ~ , J (1-1- t.,.).cJ' 1. _ 2 • j t.. 1.1.
L( I.)· l~il",.~J..' ""'P .I
· '"lJ'.Ji;a''';J,v,-T, (
Z,- Z.
Setli,., w. Aave
(v)
III
c )
a ..i= t..11. ) '''it-l1.
,"A.
~- 0
,en £'6. ('-IJD.) and (f-/ZOh), It _ V(.-.'J. :,z, .·i~~(I.rJ tr =- ~'-Zz' .4ncl
... "2,
1
I
'l.,* •
•l
• iO .-il (vJ 1 -Ila"O). 2i- .·ip(,-r).~c·iJ' (mA) z.'~
&.
II)
I
.
S - I-Irl '."'_ 2 ·
fA (l{z".). :
c.) (J~r)1. -
(J~){4-1I'(f').2.Z2"O·J{W) cr'.0112 (MW).
I.f i L - SD CA)I\{. ~ .";"1, 1 - ~.·i," • Z I. . , .. ~
•
/)
IYI. . . .
~9 17" _._-'J =- 0.
l.. -AI.,
r
•
Vi")
..
~a tl; • Z. GOll'O -r (w).
~-t -J-, · - j:tr
. %-/'*11',
)-c.J!U
tI·i~ _ c·;!(A./".-sIJ - c-~(""-!")_-j,,;~&'
r'."'£fI' r''''111Ja) 0)
~ z, .,11.
~" I.
And ('-/~(JA):
_.;:It ci/a'{I";e:Jlla). rS(c-jla'''i~!a) "
(v)
I(~')-
-jIt ..;P(l.j~·i'l&j_.,.I("I"P·+i~~)
(A).
£. [v(a) ~fIt} - S~ [S,.,.(tJt-IJI')-t:•• (ui..~D i (.z~ f) - -1.1 [s"" (ftlt-1"')'" c.S{fttlt,,}&'J] CAl c) AI- Me/GAd" .'-0, ~ in - v(~t);l~t)
h) v-(z:t) -
I.
_
(v)
'().S"(~.,."'t-$i"'lt#t) -'o.S COI(ZQt) (w).
\{ - -10-IJ,
It·
-it (14;).
(Pa,,),.-; A(~Z:)- ..1.,(Zj>- o. P. 9-2.1
a)
f -
2
.,tJ' CU.).
•
.1\.-
0,.". clrcu"+cJ ',n., ,l- I S",,;Ht clt.rl: 51.", I,..,
t- -/. s
eM).
J:c
Oft
Cfrt)
J/A. - ('.667. lite .Jlf,... ,.~ltl, '0101.
e/ocltwi,. M.~" r:• •1tIIieIt ( ••'. A/I) A'tcl cowli,.". Ot +'or • It IItMII"l."./ • ,. 7 A I. 1.A. on r/,., IIwo".I.,.,,- -/r,w.,J ,.".".,.,. ".Ie.!• . A....., )1,- -Jo.nf ----. Z; -.,•• (-i&17S) --;~J.' (4). D,..w. rlrlliglrll,... f,..,..1/te (O-,·tI.I,,,,oittl-lJ.,..u.JIt .,.he, canf., .IHI ,"'''''••e# eI (o"'jI.1f) 011 #ta .j"J• .., ",I,_ cJ..rl. ---.. ~ • .L. (.'.'4)- jaOlJJ ($).
«",
o,,..iM
b) Sho,.l- circulHJ 'l,.., L
, ,s '
.1/.1\- •..s'J._
I." ,.i'" ,,.1-.,.. -Iw-".'(M)•
S",."II,... I-J,. ...1,.•• ~ Dn. ,.tv.I.,N." .,,~ e,.,.I.illu. on
eotrt"I.,..
c/ocltw/,. an *'tIi,.ieMj
'.'JJA to r.~J 1t-i"J./---'Zi.7S.J·tA~I-,J·'S.' ta). D,aw. slr.{,Itt /,.". ~,.. #M (_~j •.II)I'.j",. IIt,..ujJ/I th. ."d l,.l.".el aI .if..~f)'"fIM. elPs"~ SlJ.. tJ f4c u,.,./: ....... t;= • 1~ "(-j,.7~) .';0.04. (S),
,1,,,1.,
.'-l,
P
AI.,
o."p.~
.I~
fI
s,. s., ('O-#j.IO) -0.' .jfl.l
~1.-0"+itl.2 .n S,..,-,,. cit." ('.,itt ~).
a) I.t.cole J .•
W,,, eW.r aI (}
.,GW A
111-
eire/. "',..~ ~~ i"hrleclw.,
.'. "'.".--.. S ·'.77.
"
h)
r-
G)
I.
'.77-1 1.77'"
.iH'-_ O.J' Cil.~·
D,..w I'·"e O~, in,.,.,cc,.,it, I/,. ,.ripltvy G/- If~ A.•• J D.04-(, Oft ·WaW/.,.,a.6 -I.w..,JgCltCItltIor ''scA'e. 2. MDV. c1"cltWt'/~·. O.I.'~ 10 D.'47 (PolJ If). J. 3.,,, 0 .ltd ':'. ;",..,.•• ~I,;" lit. Irl-clrcJ• .J '1.
"y
4.
A.•• J
a,·.'. io.s, .f~.
~. • SD.,. - So "i~'.1 CA). J)
Exh.,.J I;ItIj'fIl D 10
'1. 1l••
~. D.11-),0.4J.
J
to ~ • D.OII-JII."., (S).
~.
is no voNas-
""·"';"11'" %L·
:0
011
"'e/;,.., In.J1 ~ c ~..
(10 -J·to) -
4) LoccJe
tJ.'-;D.2
.,-'.'-jO.2
Cho,.f (P.;"t ~).
ce"I..,. «10
S.,w, W,·n, On
c/rQWa.
IrJ-c/rc/c
H.rou.!J/-' ': , I."I.,rccn", "·nc O':c .4.1 I. "7. - . . S -1.71. J,)
c.)
f. Draw
II".
R..AfI
r. O.~' .iJ'~·
OJ:. ,."I-.,•• c/-;,., Jltcp."'Ps..,.,
olIJ ~
Q.4ff. lit· wAv./e,y4, -h,tNOrJgane,.-ID,."
scola.
s. tvllV& cJDcI&wls. 6y fl. 101 ~ fD o.os," (P";,,I P,'). 3. .J".ln 0 .",i Jf ~ l,.le.,.nc,",-", M. Ir/-circ/& oJ P, . 4.R•• J %;-o."4>J.0.U cJp,. Zi- ~o.;. 3fJ.S '# j. I'.S (11). J) £Jt'.ni li,,_ Ie',- 0 ,. P, · 11.....,/ ~ -1.4-2 -j•. S"'. ~ - .t~ Y,- - (J.o~lf -~·o.O/tJ' (S).
~ Tit.,.
I!.!!::l!
~/~
is
A
- 2~.
volle!J& ".l"IMurF) till
<-
tJ.04'A..
~ - ~tJ (em)
':"NI- voff-.aa ~
n,i",;."u""
.
"
oc.e.u,.~ at-
s'» '- L so
• (). , ,..
"-0.
Jk etna Tofal.
4) I. Sleu1 Ir~",
~unl.,.eloc,kwl"" 0./0 ~
f'owa,J tlac lead 10 1;'. 2. Draw f/'., Irl-c"rcl.~;nN"$.c.I in, li,,& 0J;.c .1 z (S-Z).
J.lo/n DI;', lnla"acfi,., Htc Irl(.,I·,c./. .1 J: .
f.
IJ)
J.••J
z..-SD\--
r _sa: ~r • .1. .il l l . ' 2'" J ·
,) 11 %1.
#If;
1
s..=ao."r-jo.41S. JJ.'S·;~J.1f (Jl).
O~
'2-If1t -J.../J.,
fl.. I,·r.l vo~t.,c "';n;fPtII'" wo.,IJ I>- J .. lS (eM) ,Jr.," +h. ~"al"'t-c"rLu;I.
P.9-J2.
4) Z;- ,:. (40-;'1.0)
=-
D.fo-jl.IO.
t. £n'., .,. on .$",;", clt-" (Poinl I' J. i. 7oi,. 0 II"J P, and •• Iend f. P: J. A.,.II
,J.-
O. ,~"
Of
0;;- - O.9'S ----.
011 IIW4I v.(.,.,M.
S.It.'.""'·.
,
1. -loS (M)
CI('. .
,
-
I
}
Zl in i:9ii -
iowarJ
cal.: D.JOJ. -I.
tJ.Ol97
u, (,..JI-l (,,1,.).
6) I. EnI ..r %"... ~.jl., on S.ifl. chQr# ('oi,.1 J:). 2.
',n.
D,... w I,.,. 0 tit,• • J: 10 ~: I.. QJ .......".'u.fl'. Iow&rJ ". Oft.
9.""".""-leal. :
J. Move c/.e-IIw1le J y #0 41'/ (1'.;." p.).
A'" 70
Cl '.J~
f.
7D'-" d
0': l"I.,-,."I';,., M. Irt- C'-re/. IArGuglt ~
,,:
S. Mark poi,., P ." Ii". Op' sucb -I_I ~ - D.9fS. ,. Il..&d AI P : .,. tJ.'Jr-j. O.'f ---".r-jI9. D (ll)
Z.-
c) I. Move eloele.,.,;,. fro",
g..,..,o,h,-Ic.a/.
"0
J:c
Olt
·wav.I_,.,M.
foworJ
o.,~ ~ s.~ P.'
,.,.,.t ,.'4K
Z. 30'-" oP: J. AII.rlc 4.
P.9·))
P on "'fte o/,'&Uu. fItaI
0} .. 6;' • o., • ., ~: R••J p: s; ••. o'r.j/.J'-----%,·6.r ..; .IJIC.n).
A'
A.,.sC")-l~; -tJ.l"S(m).
f-J.'tJ'(H&)J
'.0 - 1
Z. - 7J. If CAl. For fwo·w;,... Ir&Mrrtin;"'/,.,..: Z. -/u cod." (,-'J D -2 (c".) ---.. • - o.s# (em). ad1 (,,0..'I'i
-. lfialltl,
r---.. , (.,."
--.~ :)
~
\\
~- AI "}O-'
._;~
-. -?- ---J..--~".
1&,. t - j.
A)
IJ.
._! .;
/ ( ~(
J, f C.,.nrucl,"n. P, : Z&- f.r .,~.o.~ ~
r;
: >&-'-i'· •
•
l---+tJ.-O
'. : YJ -
' "
J'
........ ~ •••"'~ .(.. , .... ",).4.
~.: •
·'.JI•
.A
jl·;,.....,.·(;.'...nIJ.A • •• J'S~
1 : ~._;, -'J,.r""'·"'S)A • •• /~S.A.
, T.,. ~."s.tJ'z., ~ -•. "'~. . 11.. r.,,,,,,.,.tI".,.,.."••4,,,,, atIrreJHeIIe.,.,.., (--(- •.L -jt. '
,
)
7
"'...r.;..' '..- , ·JI
..,. '.'''fA,4·""rA
'It
II;• '."fA.l;"".~
Aq-JS
.11 - O.S .,. ;'0.1
USC SMi lit chari «r tilt ''''peJ.,,~. cltart. Slime CII/lj f'-lIcltM" as thai in prolJem ~ q-U atl,ep' ~c wou/~ IJ.. 011 Jolt. tlltl~.m. 1.lt (M4,.lteJ ~1 A . ) ,OItJ •• / C,·rc/. ~&c • •. , ,;,~Ic.
,.1
Jl: 'l, ..
~:
:Z'L • D.S
~; •. t;
~:
Z;J- 1- j I vv i'llt ~. (D.JI'-"O") At. O.~C'J\.
:a ,,,;.,
tAll,A d, -f•.i,,-tAtJu)A. •
A:. HR" -;' :: - 1"0'" J".,.,,1. w. o. JICtI~ hAvlnJ
70 Acni",,,.
G nt4lclt w/H,
we. nc.~
IID,,,.Q/lJ,U $flJJ, SUIC.p'~.
Q
Cl. , .. ,-l.6
.fo,. so/",I,·o" cor,.'pon,J,,,, to ~ . Dbro; I) +II.. re1uJr.J sfub /.",11, I J • G
I.,
Which ~
~·1.4J
chorl
J4 7 .It. ·
1.
~o/&l"".n COUU/H'mllit,-Jo A. ~.Ju~ n . o ,,.a/iz&J ~f,1IC""'IJ"C. "J.- I. 4J is Itced.d I 4 sluh '.Ir#J ~- (J.ISJA.
Sl".;!arly
w,..,j,
O.074~.
,..,uir••
9-J' 2L -
~: ~ -
'.JI ~ j,.JJ I.SO
-itS. (4Jo'A."/f)
~: ~,- /.s.-)/." (aJe4""~) ~: ~.I.S.-jo.'4 (D.l."A. .I~)
11: ~.'.H.j.I.'O (.."tAo"')
t: ~,. t".;'4~(Q'4'A.")
-
'AI
':' CI) SIt.,I-c;rcuilcJ ') Oput-c;r~tlJ'.J II,,"
.Iu"
(y~. ~,- ~
( 1"")1- Y...- ~ -
(yr,),· -jf.
"AI·
- -;o.JO
'0
42'1.-.
JAa~ •• J'fA-
J,.".
(~,),. -~o.fO
1,.
J,,- •.••,~
J.,. AJI, >-..
I.,. ,.'"A.
~ a& o. 4J9 ""' .'J .
JOO
YL •
P. q- 37
loo*JSO
=Z.4 -J· I. 2
!oi,,1 J:. on jm;th c#to.,.t. (O.2fO>" 41' ~')
fh .. re/&1at:J g-I.e clre/. ;, ia",CM i\ ,.he , 9 -Z.O C"r~/. ClIJ addt-c S,-nee.
I
moving (,.om
I in c, 1""91" elL ," n«cJc.c to Convert 91. l'.4) 10 2.0, ~ Q/on, fh. I rl- atrel« 10 ~, (nt1#Jhown
It;>.
on fit. 9=2.0 e'-re/. (D.~'I A Gf Nofc. H.af ~, ;J cJ,·(-(.,cnf IroM ~ ~ H,& po'-,,'" til Idn~cncy 6.lw&"" fh«
,-2.' 4"J ro/.lae/ 1.4,.".
A) II)
0.J9IA-0.2'0>... - tI.DI/A...
elL -
YA-J·;,
'.
1,-''';' lD."2" 01':>. -
YA -
Y£I
Let
,-
(D.l'7A
at ':.').
= (2-)·')- (Z-jf.J,,-jO·J,---....J - O' J D4;
)s.-=-,jl--..
9-"
c.""'e/.~.
PA ~A
Po
g.'.o
A
J.-(J./~SA..
JJo - 1" d. ·
Ilc,u;,.: ~ ~
J.
sJ"..
•
(Anofylic ~Glvlion~) g.. ~
"'If'
22.5-
"-/1 :A/4
4S· 90·
)~/r
IJ r e
" 2.0
117.5-
E '~.J
7A
/'6
'.t'J
~
2.0
~
,.0
t Sec. D.k. CI.."9 QnJ C. H. Lilil"9. "Co",pul-",SDlu/;Ofl 01 Douhle-flub Zrnp.tlo"ce.-MG:Jchinlj Problem.," IE££ Tr4nlAcf,.0IU 011 EJuco"'·o"#.,tJ.£...2~1'. fJ~-11J I
Nov_mb.,.
fq.~.
7J
Ch ~/O-I
'0
ptc r
A
F,,,," O.UailAJ~f
Fro," vM£--jwflR
+' ,.
E:. r~.
E-
~,,:
H~T- ' -i ItH l,· ->H-- ~< ='iUff':r ~r
· ». }1AJf4'H,
,.11
.t ".,
-....c- u: =-iw~H· r
~C"EiJ_ IE: --,'lJJIu'" ~
r sr
Co",6ini,.,.-
., £, - -
1
r sr
•
(~t· , ~) ,~ + J "JUr'l
•
•
•
l'
H, - -
•
~~
,,14.
-
, (H1 ., 'If
~ H~ - - hi
E&·
rJI'
t lEI • IIH.) ".I ,-Y ,,+ - 1CM~ ~,. • • I ( ~U~. ~£, )
• £,.•
,
~lrHP_ ~H:. ,oU4£:'
,., n.
hi
.,. ,
£: .-'11, wh.,. £:
,.11
~£:) , h J _
·
+ J"'C ~,
")1+ wlJ.lf.
r
Sali,liu fh. 10/1.""';'9 },"",,,,,cncOUl He/mho/I•• g'l.Ia/-ion: ,,; 4-h aE:. 0, or
£:
~.~_.
"..
.!..::a. + L ~ +
"
~,a
S"m,'/4''Y .for Hz ~ !D-2
~r
H: c·~·
..
~
.
I ~ f
.1'
-;J
hi
e:_ •
(J
'
t7 1& £ = -j?H V
If
Fro", 0,
•
Raj w~l: ® (~+ a. :z)1I(Ey+i\£.)--i?(R,,,,Q.H.)
'7
It
(~Iz)
V, E. 11 Ciz ,
t
Simi larly
(f)
®
.f,-o,"
1',
IT)( 4'1.=
.,..,. ET
-0'
W4r
It
4«" - JIJ,Jj-4fi,.
PT· (~E2)-
VT Hz a 4. + rr HT X
S/,..:Iarly.
-
~£aJf ii. :
Qz - }' W'€
w'jJc.£y-' [vT £&'-4• .,.') £7. Q&]
ET "
,
ohfQl"
Combining (j)Gnd @, we hove
---.
-iwp Fir
I(
ET
G>
a.-ji4f' ~Hax if.
-b ('l fi, E'I. - aa;'~x ~ liz>. J,&.')ol."it~
ii.,- - It- (~~ H% + i1xjCu~ x ~ £z).
Fro," E,.
(/0-3J):
(fJ + (fJ=1 From
ffft + (fY- f.
80th tire c9 ua:J"ons 01
o
Lln;f circle.
6)
o
£,. (1"-J1):
I
o
1S
41-
P.IO-~
flc,IJ &~p,.sl;on~ 10, 7M,. trloJe" IrOM£,s.(I(J-~4"4'~:
£.. (1) -.Aft s,',. (t"'ylh) H: (1)·
le;: A..
£; (y) • - ~
CtU {rt.". ylbJ
A"
COl
("frY/6) .
S &.4r(Ace charSa den.,"/e,:
1.~ = a" · D1~o - of ED (0) 1 ~K - 4,,· D
-
~ A n
ft
I,. - -cE-(6}. (.1)" ~AII
S..,,./ACe, c.ur,ant clan.;";.,;
.1" - 4". ii tr..- 41 1r Rto) - - 4. i.:< All .r.~ s:.. -;;I,.. --4,-H(").4&(.,r~A ...{~' I... ,,-u
-
~/D-'
F,..IJ ••p,•••ionl
'6~ F., • _"'I. lor TE,.",.oJ••• lromE,.. (I,-" •. ,#ac):
Hz· ty) • 4" COl ("wy 16) ~ Cy)- B"Si"C"try/6)
r £: sze
~.. - i"
(1)-
Alt St",. ("trY/h)
.. H(D) -
cia ' ..
.\.." -tiy. iiU)- a.i-I)"·",. _{ ~f r.r n oJJ -1.. .,.,. n cv.n .
P'1'-7 A) S.,,. n.~ in the fielJ ••""';Onl in pro"."" P. lO-S.
"
- - EI.c'r;~ I,·.IJ I,'... , • • M4,,..cHc 1i.1J /,."••
~
F'I u o l lOll (",. aI& cIrIc f,eJ,J /ina: c..~..- cwflr:;:t>.
r
b) Sa' n -a
~"(J.y~)
;" Ht.~•• 'd • ."...~,..".;" prolllcwt ~ 10-6. • • [Ieel-,/c (i.IJ /,.,,&, M_,.eI,e /,..1.1 Ii"., E'" .Ii•• I., "..,,,eI;~-Il.JJ
-
'l,.&,:
7'
t.,,~
r
•• "?,(JrrJt). I,,. (ny/,,)
P."-'
Give,.: '; -~.fD"'O"(sI,..). ~,..l.ZS. }-Jr-'
----
tr
= /0- 10 (sa''''), 1J -
S" to-! (m),
f
ID
-/0
(Hr.)
4) T£M moJe
fJ - IAJII" - J /4.1 at."•. : Cltc •
~
/li - f.U1·
I "If -
u - u - u -
"
(,.ad/~) t O· '
1..07'
(1/fJ1".)
.'0·' (""1,,,)
b - 2- 10' (1ft")
9'C
2.'0.1 (In).
~ • A - ..!!- -
'f It) TM ,"ode ,
(L)
Tc 7M,
, ==.,'.fII' ..
c:; -
9
2- 10 (H.) <
I.
~.J I-(fel,f -0.979'. '.~.F, -J'7.' (raJ/,.) -ld:ll
"c·
~'1
.~
Ja
I. ~'3 ./0·' (N~/",)
;4 - .~ J!!{!.. -
~. u
IF, •
u,·.·u-,:;
/Wi(
4.ZJlI: 10·' (NflI",)
2.041 ./0' (mlr)
-'."0./0' (",/.)
A,. A/~ - ~_'4' •
to" a (M).
e) TM, moJ.. - -
(fc).,~. 6~ - 4- a/o' IN.) < ,.
i; -/I-llcl/)'
Z',.,
""- r, 1'·'.J1I-'O·' 1:.*- -
. 1-w/jii·~ I
-'c· b~I
u,- "IF" •
-O.,'6S'_
(".~/m)
(N1/"')
4.SJo a,o·'l.',..)
&
J .',2 • ID- (""~) -1.'J)1IIO·
u,. U.-fi ("'/~) A,. .2.1'1. w/o·! A./~
(Itt).
P. 10-9 4) TE,- Mod. - - (fc.)'6, - If.)''''I - ! • If)· lHa)
All ".,/uiraJ lb.
,u&nfl"'''~.1 Gr-
tIr. 8Am. AI' 1-/'01 •
7"', MoJ. in pro'I.", P. 10-' ell), _wee"
£ ,_ (10"'))., we. hAve
-'c -
~.
b~ I""~· {tl-f."~1f 104 (J/I#lmJ 11
.for
U.,·",
b) T£2- ,"oJ. - - (~J ., Ie. T~,
(oL ) _ 4A /0· (Hz) Tc FA\
.
All ~c,u",..J 9",4"';";&' 0,.. Ih. 8.,"& A' I/,os. 10, fha 1M~ ,"oJ.. in pro/,I_", 1.10-' ee) ~ 11Ic-pf .Ie .
(fl-
1:. - ,~/"!r P.lo-/O
FDr TMn mod". DC 1 e - ,.
_ ..L
7·J4·'· 10·· (IIt'J",).
,n oporaJ/.I-p/Gfc. wa.,.,uiJe
jffji;£ ~ J(#cll)(I- (1.'1)'1
l
I
j-6ft
"
.i.: ,
'i
.fFi;)
whe,.. F(x.) - ,,- ,,' "-/&1,.. Do) To lina miniMum aI~ , set j
.
~~~ Fed
At l c /f - llfi., DnJ
c)
I - J Xl - 0 ----.. JL -
I-Ii Ie.
• e.
6)
:I
_.
".In.
01 _
c
~.
~J
b.. '.',1., ".
.,~(If)
J.l1+)""'" . , ~
For I; -S.'OJlfO"(S}m). 6-S.'O-J{"'))'1.IZD"(.A.)~ lAnd p.- ."1110·' (HIIft).
(le)D(.-
2~ - .I • to'
(H..)
In;". Cltc - 1 . .,4# ./0'" (Nllm).
P.lo-II Ptar4JI.J-pltAl. WAv_,uida: 6-tJ.DJl,..\ a)
f- 10 ' · {H6. ).
TEM rnoJ. I'rom £".
(9·14)
dnd (9-lb):
£./ . E. {HO=-.!1 "
ft
'1.
wJ·.~.H. J
'41" == T
•
-"1
&
1·
wil E -rn ... I. 1
Dle.l.cf,.ic. tl,..n"th 01 oi,. .. Mtn. ~. J ./0' (VI,.,).
M_. (~.,) =/, (J.,.~t - .lSI'''' (WI,..) -.lSI (Afw""-l
?
.
,.
P. J()-'l
f-~, A
,
:a
A2 ,
£,..
';:&,/-/OnJ
£:(1C.. y) -
£;
£:
(x.y)=(II..Y)
-
H: ex,y) ~
Co
Ac
A ~ - -::=======~ J I-(f.lll J I-(A/Act , I I
---.
P 10-'J
J.-~.
~
(x.y)=
=
A a -~. c
(~0-9f4) fhro~ 00-'1#-4) 4r TMffhrt>cI~:
-'t"(:)£. COJ(¥).r,n(~)
7,£: (:}£"Sin (-':.a)COS(¥J E. "'''('!:)Sin (1!fj i':a€ (7;)£. Si" (¥)c:os('i;') -{~(';')£.COS(":)Si,,(~l). 79
.
4) Sur/ace curren' Jcnsili,,:
J.,
(y - 0) ::a 4tt Ie =a -
(x-o)
c:
=
X
[if" H.oC-. 0) + 4¥H;{1I..tJ)}
~ H:C-.o)--a. ~(f)E.J;,,('r:Jci}'. (1-6).
cs .;;
{
iii,.,-7i,
a-.... RL- ii.- [iiJ' H: (O,y)
a. H;CO,y) -
- 4s
#
l1y
H;t ,l >] o
1!!t:)£.$~·It{T);i).,a
.. ~ ( ..-4). x.
y
a.
h..-------.. .
,
\
.
-/•
I
\ 0
) 10-14
a)
~ 4# y.b
(fc l. - lJ~«..;(.JIJ~(7Ja o."~b,
F(m.n)=J ,.,a.4-n'
Moa., T£o,~ T£J. 7 £",
rItA"
TE.J .,M,j TMJ 1
- lA/,;;f F("'."). b) A-b, Fl"",tt) Modu
~(,".,.)
7£10
lIO-IS
o
2
.,
=J,..•.,.a.
F(m.,,)
I
T£,o, T£.,
I
2
7£". TM"
n
IS
T £.1 t TEl.
Z
4
1M,,,
Ii
Ji1
TM,j
Z/i
ffo
f-3Jl/O'(Hs)~ A .. elf-D.I(",).
L .. t a.-Jcb. l
lle>".,.- ~1(~o'/,"a.Ic." •.
".) (1'),0 = I.~/O' lor fh. d,,,,,inllnl T£,.,",uJe . For
f > I. 2 (f.),. :
TJ,e
,,_--f },i,hu-orJe r trJl>tIc is T£", WI;''' {jc).,_I.~·hJ.
For f c We
D. i
chOOJC
(f.)., :
4
> D. 06 (1ft'.
b < D.04 (1ft).
a.. - ~.$ (em' Qnd
10
b
ZI
l.S (c.m).
h)
e
_ 4.70 Jf /0" (m/.)
A..
"O.fS7(",):rsIS.1 (CI'n)
'tAl. • r
"-(~.).
A _ g
J I-(A/~A)"
2.?f - 40.1 (rod/".)
A a=
r
.At
11. _ ( ZT£~. - 11-(AlltU~
-2.5"'0-' (1ft).
P'lfJ-" G-lvalJ: Q 4) A. - ..L JIr
.J.. ,0'
_ 0. of
,-S-/O'
f
1; -11-(A./1.At AJ - A/F, -
·
J
j,-/'S6fo· e",,) .
f-7.~XfO'(H%)
(m)
(,.,) - ,. "
(em)
94.2 (,..dim)
u,· elF, - S· lD' (M/S) ~ • C • F, lD' (M/$)
I.'.
(Z',). - ,.IF; -
-/r
x- ;
.
• 0.'0
iJ. D"7
~ .. l,fJ/A.,.
b)
S90 (11)
2DDff ..
'z,
(.11).
-D.DJlJ (Itt)
F, ·II-(A'lza'f -D.ru .A' - ~I'.e= -0.OJ4J
J
I"
.l-
a
u;-
Ztr/>.;
c
,,,.)-3.4. (em)
'.1.1. (r.J/,..)
ulF;. -2..S1./tJ' (m/s)
U;. U.1;. f.1S .'D' (Ift/,)
(%TE),,,·
-it:-
JZJ (A).
.I.
~tD·/7 Giv.n: 4 - '!-J.OJ(ItiI{trt>. b-J.fDalO·'(tft),f -JaIO. (Hz.). 4) A.c · l 4 . 14.40Jr'O-~(M)
f c·':" b)
.A -
- 2.0'. lo9lHa).
f -o. ,
(M),
Ir-I_-(~A-)a - IJ.1lD
A.
"
~ IO-/~
c;. ;v.,,:
a) A
z:r
a. -2 .2~ a 10·'(",>. b -
t
..A.,. 24 •
3"'0·&(110),
z
II-flelf)' =jf-(A/A;f
-D.74S
E,. (to-I',): ("c~ - .i:
• ?b
7r'P.
'C ['-(1.//)'1
-'.2 9S
b) FrDm £9&. (ID-'044).
J _10 10 (Ha). 4.so-/o·' (III)
tDO _/D-Jtwt) ~
M
[,.,
JI.{'Af J\&] A
10·a ("1/".).
(flJ-/0 4 Al, AnJ (IO-I"J):
E; -E.sin{':..X)
.
H; - - ~ /f-ffrs;n(¥) liD == 'la\.iL z 1(-, J "I.
cos(trX) A
.. ." /£/1 -
~v- ffj#lr-Ey"H;>JxJy • ~;:hll-(tl. ,For
P -
10J (w) Q'I
fh. 10AtI (1:1"",.,.4) J tlS$umi,."
under mq/ch.c1 c.ondi+ionl:
E" - 9 f ., oo (VArt>. JH,,·':&/~7.f(AI... ),IH'-"7.'lAl.J.
The wGveguiJe, is 1(".) I.",. ----.. Th• ./i.JJ ,'nh,.,;I"., an h'9hcr of th. sanJ,'n, .."J 6y • .f.. e:lor tH .-"'.1.11'.
Mo.JI• I £;1-
'0. 11~
MIA-./ H;I- 21 J. J
Mil". J~/- I,D·7· c) .J(".0) za a~)t
( z;
(VI,.,) CAl",) (AIM)
H: . 0, H:)J •-ii H'll z ,...
1
y) -
I
-~). I J.\£ ·7 tTl '1
I j(x-D)/. }H:I-"7.' (AI,..)
J ly-O)
:;2
.:i,K
•
(~H: +&\.H: )~-o- -AzH:(X.D)"4,.I(·(~D)
li(Y-o)~· [(H!)'~ (~)'
fli' {rtf..(t-z{f)~ilt·fl.'Jl~1 I
Whlt:.},'$",A",,-,,,.,m'" ",,-all.·
A:&~~~~ end: M•• ./7/:or d) Total amourd: 01
t1/-(1)' -'.'16
tlVC,.lJgc
.
-21J.J (AI,..).
power d,:Uip.fcJ"" f
. .!Juidc :
( ) / \lVQv ..
fd
~ 1000
('Olf'-1/1\ (tz 1 .~ -I) -1000.
,z
.2'.2 (w).
(ht)
P.IO-2/
F,-.lcI
.1tpressiDns lor TM"MtJJ• .frDME110 (10-92) and
l'D-94):
£:(",y) a - ~(f)£" UJ1(";)t,;.(tr,fJ
£; (~y) 1:- (-ZJ £. Stn(l!f)ctls (-7f) £: E. (~")s,;, ("!-) - -
H:
sin
(x.y)=
(It,y)-
~~(l)£o$t;'(:&)CDs{:!!f)
U; l~y) i;(-:)E. c.os(": =a -
Cq/c.u/Qlfl etc Irom ED (ID-'7); 7
J _
0
e,
p(%)=Lf6/4.[rlJ,~, £"H~lcJ 2
,,1.
)Sif)
c x", -
~
y
FrDm pr"hl.M P, IO-/J.·
(:'l) .
/liz).
2 P(z)
-l
)lay -
idt~£~14h
8 [(:')'~(
.
~ (y-o) •
.i;ty-b) - - 4. ' ' : r:)£osin{1lf).-J!"Z
1, (~=o)-
ilJCaA)-- 4 ;""
.
'" (%) '=
,,".0
't
(")£ s;n(JrS)4-iJ,,% A' a. .
2 [p,. (Z»)
y=o
f IIJ,(".oJ/'R1ay-
a; J4'1
yao
hi
&
2 [It (z)]
[l',.lZ!}hOc {II (z)}
•
tJ
~
f)7"
(y::lD),JD
1
"I
13
.
fiX:' f:J£/lJ
J)l. (~YA, 4 ,.,..
{.!!.)£I a b D
•
p
L(z)-
2
(til) e
-? /0"12 FrOM
(teIE,a14 Eal.r( .". -,2h
{(:t+l1!tl
2& (14. * 41/1&) 1
_
(11)')
+ b
a
.
frG611-(le/f)' (IlIA'" 1/6'
T"'II
E,s.
LI '"iLJ
(ID-I3.4)
(ID-/Z'):
Gltt!
InsiJ. Ih. sIal,: ~'=- W:.u~fcl-k; e 6JflIlC., O",I-lit:!. M. slaJ,: ~... w~.~, .. tl a. > I.i~••,
("Jlp.~. «
'<
~ .> Ie
tlncl
.,P . -I. ~ 10-13
Fro,"
£".
(IO-fJ/4)
t.Jlpd
_ t.J .) _ '_ .
~~~
'.'"
and (10-/10):
f~dt + (1cIJ!· (¥l("P~ -~
Q)
-t (~J ~ - (- Icr'J 1"0" ( ¥)
(i)
Lei X=lcyJh. Y-IJIJh. A-"'/~fbG,.,JIl.¥/1t:-f. £,s. (j)G~J (j) b.~om& x 2 .,. y' • R,.I Q)
I
a)
f = 21(/0' leo d/2
(Hz)
7
Y- AX t"" X . >.. - e/f - I.S em).
.. lIJ/A - tJ.lJlof ,
y
@
A - 4./~fI· O.JlJ' J I. -D.DJ/4.
, GraphIcal solulion: X.-o. OJ /4 ,., y'.l.IIJ'.'O~~
x F~Drh
.-( "2~/d - 0.0"
(~/",)
Ity - ZX.IJ-'.2.
£1. (fo-IU):}-.ji,~fJ-~
(rlJdllft)
·"."r,.."",,)
C./f - 0.'0(,..). Ic.J/z.-o.f)sz" A - 0.30' , R- O.07"$".
b) f -S-.JfIO'(Hz).
A.-
X.- O.D1~.
We
ob-fG/"
0( -
O.J.'"
~ - ID.4'
'f
~-,. ,o/·/o(N1/m) (radj,..).
j
R 10-24 FrDm E,. (ID-fJ~): (~d)
__ :: (kr/~c",(ltlJ~
U~""9 H.c nofafrDfI& in prob/£m ~ 1D-1.3 , we ohfAln
e
a,uo:lions (rt'tn CD i" ~ ID-J3 And yl. R'
l
x: ".
y- -AX
above:
CDfX
4) f:ll2 1(10' (Hal, A-I.S(m) I bJf-r./fJ'(Hz),
I
A=-O.JDf
R. -
rIJ.,..,
A-O.60(m)
A-o.Jor
I I
D.OJ/4
Avo
R. -
O.D7S
Qrc no lnfars.c"''-1Ht1 .fQr curves repr.~cn.j.,"', E,~. 0 dnd ®; h.l1ce «van 7Af c/o not exist at
""OJ.,
flte !J1\lcn f,..,u.ncic,.
P./~-lS Usc
"r £"
~
(1D-1.Jd)
_-M At
and (lo-ZJa,J ..
~E;. ~y
N". iwc ~. ~
1
h"
Ely, zit) -1I..[i.cy) .ir"" -} 2' ]
iI(y,z;,) -
A [ii"I.'I) .. JC6lt-,IZ) 1.
Iytl: J/L: £:C1)-£. Cos lryY
-
£.Iy,%;t,- £.co'kyyc.os{wt-}z.)
'I:.7 r.SinkyY.J·;' (IJt-}~)
s;(1)"'1:. t, sinlyY -
£,('/,z;tJ--
II: (y)-- ~Z:~£.s"nJr"y -
H1I (r,2;tJ-1;I£. s.itllyY~""(6It-Jz)
y
Y~d/l. :
£:(y).E. COl (¥)ifC(J-I)-- £.cY.Z;f)-E. cos(¥Ji"(Y- tJCo, (ilJt-p.,
l;(1)C-.~£.C"{¥)i~-~ - £y(1,Z;t).IC.,.,{¥Jc-Cy-f JSilt (flJt-p.) U:(y)-l!fliE. ,0'(¥Ji"- J -It. (r,&;I). - !J?£,c.,{¥Ji4 f1-f J.,."(utjl ~ Y~-dlz :
,ut¥J•
-£i'/,Z; t) -E. .,ty·tl,OI (",t~ s) (Y)·lI C. t.s(¥J."i¥~~J [y('" ;t'·-IE.cos(+"J,,·~iJI iI) (tJt -}Z, Cyf I(Cy). - i:~ £.'01 (¥)e" tJ- 11"(,,.j t). ~E.CO. {¥),"ey. f~ in (wti'z
t;CY) a £.'01(!f).Oftr1 J
£;
".10-1' a.) ~,.om TaJ,l. 10-2. On
lor TE, mode
~ It
aC ..
Neg/eel-in, 1J,(l
fNhi~h
J
F1"0". £". (l0-14Z):
.a..
1_
• CIC -
D
2PII
/ca
{or kyJ«f.
1 •
E1- (11}-/2'): ~ ~ tAI)P.E.... Jc., .
0 -
..
f k"J ( e; - I ) .
~-.c(y-1):g 0.3'1' ~
(Nf>lm).
(J. i97
fIl(y-f) - f ..
(y - t>::or r,o,~ (".).
UJ&
Ie.
clom,'nanf "'DJ•.
~ ~ P.tl 2.
fhJ
a 1~ (.(.~ -Ie;" -(.I -3 •• ~ JlJ-p" f =JxID' eUs), k.~2". at
p. (0-27
;S &CII"
k;- IoIjucrEcI-"· ~ Ie; -Jc;
From £9, (10-114):
h) d-s_/o·
if
is fb.
fdn
rd Y
I-rrrt in
e(1
p CoIf.'., E. J ( ",. "
/J.4'"
£,s. (fo-f.lh)eu.J (fo-,Zc):
II. • • y
HCy.ls;
iA. ~N: h ;)1 ' I):.
I
r' _ -
~
~ ~· h' ~1
a. [ji.'yJ c.i(..f~2)]
l. (y,2:jt)-tR... [£'Cl).J.'''''-)Z)J. ~~/l;
Iy/
H; (y)- II.c.o,1ey y
H;
(1)·
-H.Cy.z;t)-U.uslt,ycOJ(rJtjz)
1;He Si,.11, y
- - Hy(.,.jf).-~S;.J.,.yj';,,("",..)
s; (y)'" JW/iJ II. .,;.II,y y
~
£".c.....;I)·-
_
~C~.;f)-~CfJ (¥J,,·~"fJc"("'-I.)
ell1- :
H;(y). U.co, (¥)i (r-t
H; (y).-ijH.c..{¥)
c:
-
(y) -
J
ty-I) J- ~H. t:os(¥J•
y ~ - cll1.
..c,.t
I, Jl.li"9 s;n(fl f j zJ
~~c.,.(¥)i4f1-IJ.,;.("'i~) C" (1..;1)- "'!'''-.c.'(¥)a-#lfrf's,',. (.n~) UyCY,Sjf)-
:
H:(1).I4,os(¥)...(~tJ H' (y). fH.ctl(¥)c~"·~) -
~ (y)-¥H.tor(¥Jt"(r+IJ -
• N.(y,s;f)-II".,O.(¥).lfey·l)eos(wI-}rJ
"yCY.I;O--&H.ce'(¥)e.,r.,.f>.s;"{"'"}JJ
~b~ll;f)--~U.C .. (¥>&.fJ.s""("';h).
"
.Ittllt'", y_ d/~ i"
i~JIt H.slnkyY tlJrtl • "'y £:(,/) __ ~H., (¥)~·fI(y·t)
£-(y) -
••
ire and .9 U G '
''' ' ,
l,.\1'-
I
~Il - ~
Ie
Y
wa oIJI.""
s· (l.
II
". In
.!!Jc~
~ lO-Z,
'
II(
If.
J
=_ b. cot( lert/. \ . zI
P4
4) Odd TM and even T£ modes are fAa prDpG!J41in~ mocJcl. Lh,e n, JJ lor d i" the lormu/Gs"""
'.4".
7Q~/. 10-2 ,
Ic. = f.ceo6)
J\ __ ~II Cos(~)
TJ
For
n-
Aav.
F.,.
2 ./ ,,-, eI j-IdcfI-P.••
t
2d Jp4l~d-,u.4.
oad 1M moJa. -
Jyll!d/tJ.
woe
otld 1M m.J••
10, ."." FE mfHI.s.
Fro", £"s.(fO./J.U 4ItJc.):
.
£YCY)---'v • .i.A £.COJ~Y
.
~
H. ('f). ~wJc/ E. c.os Icyy. S U'/. CMrr.n/- d. "sily on C"M/CAe"'" 1.•
~
S..,,.I. clt.,.,a
• - 4s
I:
4"
U:<,)c·aa it;e, ·
l'
if
I
yao
J..".;ly 0" c."J~~/or1; • 4'"· D fya.
I, • ~cI E"{fJ)- - ~..... £. · y
Fo,.
~v.n T£ ,"GJ.t -
..
Iy 4 J.
H;(y) •
E.
(y) -
Fro," probl_". ~ 10-21 :
.if ~ J""~Y
..~~ H • J,·n!t,Y *t-
Ha Cy) & H.
.r. - i
y 11
.,
(.J
Je,y .
14y H;'IJ' +.. Ii:(tliJ • 4. H.
~-Q1·~dl(o)-o. P'IO-11 FrD",
£" (fD-'~O):
1""'1"" ;
f".", -IS It/D'· F(m,,",I), '7
j (':J~ (ff+(.y·
F(,...",,>-I(fJ&"'{-n~ma.
MoJ.,
F l,",,,,,~)
.(~>.-, itt(JI.)
TM".
0.11>13
3.'lS.'O'
7£'0.
0.21S'
l.SJI./O'
T E."
0-1'DJ
J.'10$'/o'
7E"/~TMH'
O.J",
4.3J1-/0'
TM"o
O.Joo,
4.!07ICttl
T£~.f
'.Jl01
4..0~ ~'o·
7M,UJ
D.
J r'o
S.J~O ./0'
T£I.,,,T~1f
TEo"
TE,a"
s. 4.14
O.
s.• J f ./0'
I",
(J.4o'J
TM",
P 10-JO a.) S l"c« d » A> b )
-10'
D.)'09
Me
'.12~~'ot
1(JlAlcsf-tJrJ. r ruo~f
mode. is TEl", meel•. L =: .J,o,
h) {rom
£,-
0.,.,-
T~
I' + ,
A d& - ",.,OJ, • 10 "
At
(Hs ) .
(Io-I"): l
7f/",p.,a6d (a + Jl) Rs [ 2 j, ( QJ+ d ' H AJ(tIIl+d1lj
.. JJr4 ,,P. . . .'J(AJ.,.Ja)
~h(A'+d')"Ad(oa"d&)
"
Il
\.
-/Jrl'i'
-,,,., .
FroM E, s. (fD-ISl.tI.) And (ID-,,'6):
~/D-Jt
--
We - :
".P: a hd f;' U: - D.071l' "(J·n (1)
WIll os
a.6d
*'
J
(fa +I) H: - O.(J172' ./O·,J(:J) -
Wa .
4)(.l .L. - ' (f;Ie, )c-] T,., )C'~ - .!!elL". 2 4" d& • Ji; . · OJ1./o' (II:) & •
6) (Q,.,\, -
(~:)". (Q'M)*. - $"'2.
c) (\>\'e)*.t- {W.}~. tJ.D771,dO·lI l ]J -D.D71U (,,:1)
- (W.~J.
,.
~ ;
P'IO-32 ---
4) Q.
_
; 711,..&
.- aAJ (4'", d'J
26 (a'.d J)
,.,
o.cJ( a ' ., d 'J
1.I.' - Z'"u.'_ ~/L+.!. at d
For' A-J-f.,b. Q.,..~
.,
'
... ,.1791t'O'{fb')
lo.az/rb .
b) fo, Q;',~ I.ZDQ,.,
J,'.,.JQJj, -1.440.
t
R ItJ-JJ (1) FrD"t t/'. ,,..IJ Cllnli!JUTAfillnl I·n M. cAvify we s•• 'IItAf .,.11. 1M /l'toJ. with r.'pcc.f fo % is II,. ". Mod. wi/"h r.sp.c.+ I.y. 11.1.11 SAme 4.1 +It~ T£ (Q,'I.~ CQn he ohlo.in.J ,,.om (Q,.,),1. in E". ((0-//,1) 6y chan,'-"g b 1-0 d G'JJ d 10 b. or, (~) Q for I-h. 1M". ,"ode C.An h. J"r;v.J Ir~m +~
,.,
~
field I.prescio"$ in 4 s- (l.o-/ffA,d. onJ.) by sclli", ",. PI: I J ont:l us,·ng £9. (ID-IS~).
W -ZW",·
~
i'-{';,f!)a.J,d£,'
at ~,•.
=i /1 i,rll• cis =J1 tNtI'll, tis
.".£1.
. /Hycs.o>r]
.. ~ {lj'tHyta-o)/J
[IH.C~=O)r+
c1xdy}
- t (";;1)£: { ~a (:.r"" + ~a r-:Ja,.d+:faJ,} , h'-(-ff+(})',..
~".-
~/(J-J4
~,.w 'L
--;;-- -
c- Qd
n
11{".j,I, 4J,d (4'...I) r. ., L Ll' 1. 1 d (tl. "b ).,. G., (Q' .../II)J
ft$
•. C7f41 d
4)
L"1~ln(:) f'
it)
A-
,
• z,,/LC • """F/~ln(t)
f/pc - J~lnifaJ' "'11
"
Il,.
1"41'& . It
a:
C h a.pf.,r
Po f1-1 Mo.)tweIJ~
If
10,... ~i,"ple m.~J;a.: VX -p 'IN _ V IC ii = f of ~ :~ tJ1L1Gl-ions
e-
~t
ve£_L E V·H==O 4)
VIC ([): VJt fhl.l-
-u.,7
-Ll.£
~'E
®
r: ~t r: ~t' fi X V 1( E ~ ~(v·l) - ~~i
Buf
~ ~ uf-V l £
(""",-tti"9 ® and ® ~ we.
® ® (9
-}-t; (filcH)
c::
CD
@
OM4"n
VI.E.-~€ ~E .Lf'jp.,..n ~I_ ,. ~t· • .J r: ~t
b) S;/ftiltvly,we"Avc vlR-}J~ :~4 =-Vxl. P'11-2
£,
r
(ff-2):
E..
III -
III -
VV-jwA' -a,,~ +~E.""~~.
:~ jwA"
The cwp,"cm;lns J AA,A,.
-
£ __ ..lY.. _ iwA
•
and A... ar«
,
R~'"
£ • _ IV _ }t.)A ~ Rj;ft'~ '.~. 1· -,• ..-
"..
.gIven in £~_
y
(11-14 Q , II.. and el . .
Q. [e-j}4 c·jPJ v-4"'.'~ R_
R.,. a R. - 21. til. (OJ' R_a R .. fdleor.
Q-
V
-j;;; ,
(111.)'
-< ( fl.' .
~
.: (,..;~"
_
~c"i}A ells,.-.llcUeor.l"Z(tfI.c.,.leos"ac.",7 4"'~.iIAlIl'l~l J, "\7 ~ 1 .J 7 (~ 2 1J f11 e z f,-.JI- {z ( I ••• ,,. J,J ,.s.l 4"~.)·AJ/lll·)R r Z 'I :J
:!
-
"'*follA)
-4 L'(II + ,,/c.,vl)(dltM~(A_rc.,,)iJ1filr.'NJl ~
A.
r
e
~~';'7JolR ~ k)e-J1
A
90
•
%
J. ~ T~,
X
J «r-».
_jB. #of ~F Il, c.' M
/l.'., ,.' - 2 i · r
III _ I
#oj ~~ ~f 1/ ~
r
DA
.
I
~
K := Qa !l$;rte CGstI·-f iyRJin.6in4 ..,. 4z R C.OJ' F l' c1). X·;: -Il Ie l;n.Ci)s."R.?S;"~Sill+ i; a (I ~ f Ii,.. t:.J~'" kSilleSin~)
-7. i-[, ,. 'ill- f
a."
s.~·Jp(l#j.lf.)fcli'.ci~ijJl(14~.) 1./ (1-1 -fSilf.~S;I~s"IJI>J"IJ;)"~ 4" I' ~a R. • .,.. "7" R " ~ .l.~
-.,
'..12
-ii~~f,-;~''';~'').!..(-L - ':I:z.sin.Ji,,;' 4" r: Il. ll.. 'J )I
1" f),. so".. manner, wc/tG"4 tM ii!'h.J'}I.)'J(!Ii: - Q.N .i}I(I-#~)~Id L - ~ S,it, s;n~) .,,, R,"'" ~ .
~.
. ".
II
~;c·iP(14i)~1. ~--~ *'~ i J}A( I#j ) II.)'-xLy l '-" , s'-" i -.c.
and
1·b'!If-
~ c-i'''lt?/d LeI-
m::a
1.L.Ly
•
Q
,
{'~a i'!AO-#jJA)/.}ILy Si ll lJ
COli
IS.
A = i:r~ c·~A('+i,A~.J;n.(-a.,s"n~+a,.C4I.)
c) h)
-...&.!!!... -;,A. · = 4_ 4trl! £ (I"J'A.)~"" , . if- ~ v. A aA H.. + 4. "-" E~pr.sl;"n.slor 1ft. H••
l -
j:"" V
a=.
JI
H-
4~ £. .
o.nd Etl
-.
!Jiven /n
9/
IhOJfl ~.{lf-l.'a,~"cJ
So",,, dS
IR..
.In !he fa" %on& »f, I/f;)/l.t a.ncl IJ{J)V I f.,.",. can he ntlg'a clad. We ha...ve fhe .follow/it!! "nJftflnl.nMlul exprcsslons j Cl-ss um i l1' i (-t)-L ~os l.t>t: I
A lll•• ;t)- -4,~ }Sin.sinl,.,t""jl.) £. lA.,";t>- "'J;; ~s"n" co. (tdt-}A.)
a,
H(R,.;f) --a.
4";,A
p"s;". C~J(lJt-111.).
S
21. (,)•
sin'
JAs i n A {J._/2/)e~cos·d% r
-It
_
-
2. (c.s(~Ac.s.J-C4$1hJ
F)
sill.
»
•
-a) For hQ/+-WI!J.vc c/ipDI., iJ.. A/4, "11-"/2. Z h (,) _ 1 CIIJ(l~GS') · C " 6'~.• b) For ma1l.imu", 2"'.C,). s.f 1.J..CI).O-S-90·...1;
·.,. 17D ' - ~ -(:)~-O.Ul(1JJ
Max. 2hc(') -U.itlO Wr
P.1I-1
-
:0
flL }27 £, H,· AJsin'd'cI~ • • (;r:J·'l'4A{[fp.+~lf);A+lj;R1-t;~~l)
f
~~ Ji ..
71
.j7l'sinJ •
J.
.. f [10'" (!Jtl) , same J
q P'/I-,
'
4U
£,. (II-J7).
From £,s. (0-274) and (II-1.7b):
[~ -
cr,t=';'" (f:~) )sin fJ
, ~ 4-'. .
(C-iJI''!SinfJ Z
H- -
"Y,. -
f fAa 1'1
- n •• ~r -
71 ( - E_
H:) R'sin, tI. tI~ •
({1J11J11' (:..y
~ J J.D1r 4( J\& .1)1 • (Zill)·
a) Circular 1001
of rtlJlu&6: 1l,.J2DlI'(-1)~2D71'f~
. It) l.~cMnsula, I••p 01sill. ~ antlly: Ill' - J2D 11" (L~~lJ' ,_ -119 __
F(,''J _ ~hc.J.')-~"h .3,,,.
4) 21t-1.1.$A. IF(rl1-1C'./l.l$.C.,.)-coJtt.~-illb)Z.It-~A..JF(')J-J ....
I
Si",
I
I
I I
I I
r
93
COSll~'OI')-~
S.n'
~[
a) £• -£., .,E.~-
.P.II-IO
tss,l.r'J,it.t iP 4."A
• ( , fo .~fl e••')
= 2· 60JIt
R
~
Z, e
J -i)(A-TU'·)!If)
I
'
\Nb~
F(.) -
slr,.
CO,
(If cos .).
P'II-II
r
fJl
-t -
~ I
")I.
----.y
•
£..,- E"" + £~ . jhJ!.~ C-i)~ J.s, it 9J co••),,;,r
£tf' =- >'::'( --::'-)7"JSi"9J CM .>/I-S "",'+,in" I
. (, Iii-cos",.,.,#" ii, . E..,. a.£, ., ii £.£,.. - - "/~'" _.I,,," $',.'. -,.C4S#)
a) I" fh.. xy- p/4n. ~ e - 90· ~
'..y C., ~J 1:#0.
b) 1" the %% - plo..ne, ~:fA 0·" £~- -£tI.1lF;a(.)/·'~;nlJJclU')' c) 1" tit. y%-plo.n., ~.9o~ E",--£.,IFyll(,)/_/C.,h,itf}Jco,'J/. d) cJ::J\/4, ~J.a 12:
1
,
,;
P'II-/J
h)
4) 1l.la"''va
tlJlc.ilwafl,," a"'pi.~IuJu: 1:4:' :4: I.
Array lac#-.,.: IA(~)/"'/cos{:c...~Jr c.) cos(fc•• ~)
-tar:..
----. ~ -74.1'~
Hall-pow., J,clI",wiJI!J
= 2 (9D-- 74. r,·) == 30.l.~
'or.1./ ,;,,( arrar,.,,.om--..J._, ". f a unil",,,,
S
£,.(II-'Z):
·eM I .. f
SI"
e••,)
T
,-
HAIl - power bc,,"'wieilit I., 5-.1.,..",. uni".", .",~ .,.,ill, ~ SpGCi".J
=1, (90·· 7'. "~e) - 20. 7'~
P. 11-/4 4) Fr.,..£,. (11-'1)""" J/.,l: IAt~ I .
91
fA".· 111 t=C~~dJ.
V _,Jco.J +.
b) 8rfJaJsiJ., Opcra#ion.
lAc.,'· ~ I ~~~.t1f1 i5
I S;X I I",. t~d.
""/,.,-c X ~ N1/1/2.
AI hall-power pDlnls: For
/siXI -h -
X .'.391
( -ior J,ol-" "rotk.J.jJ.l,cnJI;..,~,..,...,..,) brotLJslJc ope.ro/-itJlt M. J,ell-power IItLAlrtwid'llt l
l1A~)'h -
iJ
For N- f1..
D. I"
(~)
(,..J.)
- $0. 7S
(:d)
(degJ
(2ij~)I/~ DI -I. ~J (~) (Je,.)
Fro..'" £,.(11-'$):
(2A~).Jf:& 9. rS (':-J Cdc,.)
Yr-!'J(co'f-I)
c,,)£nJ+ire Opcrolion.
(ZA~),/,-I.'U~ (,.oJ.) For /01-/2.
./01-,j:c,
ltJa,.)
(2..d~)'IJ=JI.IJf5 (Je,J
",.0,." E,. (If."):
(l~I). =4'.7'/5-
(de,.)
IAlttl/- L/,,·,.(Ntl 1.) JS'I sinX I where X • -Nt . N Si,,("'/~) X 1 2 As.lUlnc brOGdJiJe operation: ~ .~CM'.
P. /I-/S
•
•
.c:::L.
•
D
4" /Al!tl".../"
-f'J'rlsinX/~';"'J'J+ • • ., I X - I A ctt)",." J.. "
!"/~"X/~""'J' a N~JI7S;XIJX-N~ (:~. ~ . . ~NJ · • /) =- -X-
p. /f."
COlUlr'-lcfion fol/olAls on
. : I
"TF/J.
: t
0
'h-.Il
air
lL I , A ' w".... L- ar,ay ,an.,jIJ..
fh.. shps oul'/ncJ On pp. S2.5 -S21.. . P.0 is i' _ -.!!.. . ..J 2
"0'"
4'
"aJ/u301 Cl,.c/. il ~c/= ~ (-7-)- ~
.
~ 11·17 ('Dm
·
if
(11-4J):
•
wh.re
-Jp['·'(fc··'>]IAJ"'M(1#;>1 •
l -=· ;60z..N,~ R -
y
C
SI'"
•
fA ,_ ..LI sln(1df,!J) 1.6 ~
&
"'-. -
lin(~,/J)
~
4c1, Sin. co,#;
~, -#J..Sin' '0$+. IA1 ,- .!..1"·(~*II)I' ~ SinC1Jy/J) I III
I fe.sllt.,,)]/ s;n(¥)~ilt(¥)f· IF(... ~)I- N,~ Sitt. SIItC"')"i,,($)
~ F".o," £,.(11-'7):
.
PL -A.t'.v.
(j)
Co"..;d.~ 4n .I.",."lal .I.cl,.ic (Hcffai.,.)clipo/. oI/.,.,M ell. l,. ,.". {/c/J 01 4n incid.nI plAne \MQV. wifA can .I.c.t,.ic In"~lIsill E.. '- _ /(,J.. ® ,
M d-xi",um pow.r
1 'I.
a"
i. 4/'Jo,.lted
61 11..looJ il' Z£-
P,. - : /1/,. -t(li.,~t/It.
,,
z;.
(£j:rI
Q)
Co",6inj"g (I). @.1 dncl (i), we ltave
., J.. , J_ A -...LA. A. - _It: 44 t~ - T Ctll ) . Fro,., E,. (11-Jf).· Jl·".·{~l • ,,,, · From !J.S"I D-Gall·I)-I- ~&. 4>;' ·
P'ff-f9
~
Fr'""',.{If-'f):
-f:"rtG.,G••..
4) For hGII-wGv& dipole.: (;.,-G.. -,.,. f -3-/0' CHa), A..- elf -/ (lit), r-1.11I10'(".).
~- 1.f7.'O·' P,
-
1. S7./0·' (W) -fJ.1n
h) for JI.,.I%;an dipole.: C;.,-G.. ~. D.'JJ ~W).
4)
97
-'.S
~W).
A4CJ,'A·/-iOI) pAttern lor
L =>"/2:
~----''"----... ~ PI of DI
- - - - - . . i......
91
IF(.Js;n 'I.
~ f I-Z2
From
£,. (11- 91 II) : F(',~)=JJ fc..:y')i)~'it.(a'c.'~"Y'linl'J}#'Jy'.
.
-,.,..
(1.) .In fhe. xz"- pIGn&,
F
K.
•
I·~ «., I (X') c~"ll $1". QX'
(e) = "./_
~A/2
=2'f~(I- ~ 1l.')C/I'l!"'sinll)4)1.'
= a.~
rJlZ (,). h)
Sef
+~ 0" :
(~
1- CIII (
L.t '" - ~Ii" •• ~QSI;'~
I/Sh.') . )1
24 .$,-" ,
"
1.1' [ ~in("12JJ2. 1 (P-Il)
.f.. -t.DOS.
[sin(~/dJ" _ .L (';/l)
J
Ji
l50
Holl-power b.4",WI"dIJ. (1.(9)1/, -2 si,,·'(O.'" ~). For
.A/a.« I. (2JjIJ)I/~
a'.l'D:
CrQcI)
- 73.l~ Cdeg). CIL
c) Set
Tit =- 'It
- - . . 8.~, -
&;,.-'(~\ ('tAd' 4 1 .a ~ A • flf. ,~ Cd.,).
J) Flrsf-s/Je/oll.l.v./: L,'" - 201.1,. ( U"i(o,m
Pol,.,." luncH.,.
lIall.,.., H..w/dllt J.o~oIiu.
lin' ItUII F,., .1-";.1.-
I... ,.".,
D/.,,..
abe;·) A fOG (Ii_,)
A S1.Jer
(J_,>
13 . .J (tI,)
99
:"llY- .u., US).
Tr;II"~ul.,. Dislr.
~' (.!!f--J 13.J ~ (de ),.
114.' A ide
2'.'
CcI')
