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Anhang I
(a) Die Transfonnierten des Speisepunktes
1 Z=R
2 z = L8
3 Z=_!_ Ca
4 Z=R+LB
--
5 Z= RCa +I
Ca
6 Z= LCa1 +I
Ca
7 R
z = RCa +I
8 Z= RL8 LB+ R
9 L8
z =LOB'+ I
IO z = (R1 + R 1 )LB + R 1R 1
L8 + R,
11 z = (Rl + R1 ) + R 1R 1Ca R 1Ca +I
~JIJ I
---L
--tt-
~ I L
~
~
--o--[J-
L
-CJ--c
CJ---'tN•' R,
L
CJ--~/IN
R, c
270 Anhang I
I2 z = RLCa1 + La + R LOa1 +I -rl::J-
c
13 Z= RLa1 +La+ R -t~ a(La + R) R
14 z = R(01 + 0 1)8 + 1 a(R010aB + 0 1) ~H:J-c,
c,
15 z = L(01 + 0 1)a1 + 1 a(L010.S1 + 0 1) -i~ c,
c,
16 z = LROa1 + La + R ROa + 1 ~
c
17 z = a(L1LaB + R[L1 + L 1]) ~ L.a + R L,
18 z = 8[L1L10a1 + (L1 + L 1)]
L 10a' +I ~ L>
19 Z= La+R LCa1 + ROa +I --c::J-
c
20 Z = a(RLOa + L) LCa1 + ROa + 1 --c::::J-
L
2I z = R1R10 10.a• + (R10 1 + R 10 1 + R 10 1)a + 1 a(R10 10.a + 0 1) ;-cJ---~
c,
22 z = R1Ra010sB1 + (R10 1 + R 10 1)a + I 8((R1 + R 1)010.a + (01 + 0 1)] -c ::r
c, R,
(b)
über
trag
ungs
funk
tion
en
Net
zwet
k
YN
t
lc
R
1 I
2 ::
:I
3 3
' :::
E
5 ~
Ic
6 3
--
Zr(
e) =
E
,(e)
; T
=
R
C;
T1
= R
P1
; et
c.
Ex.
(•)
1 z,
.(e)
=
RC
B +
1
La
Zr(
•) =
La
+ R
R
Zr(
B)
=
La.
+ R
RC
a z,
(e)
=
RC
B +
1
1 Z
r(e)
=
LC
e 1 +
1
w.•
Zr(
•) =
L
Ce1
+
1 --------~~------~~----
I N
-l
......
Net
zwer
k
7 t~cT
8 ~
Tc
I'
9 ~
t I'
t
10 ~
t t
t t
11
~'L\ifTiiTitT'fi'•' I
t t
t t
t
12
~fc ~
R
E (
s)
O.
Zp(s
) =
-
0-; T
= R
O;
T1
= R
1 1
• et
c.
Em
(s)
L1 C
Jis'
+ L
os•
Z:r
(s)
= L
1 0"s
' +
3L
0s1
+
1
1 Zp
(s)
= R
•o•a
• +
3RO
s +
1
1 Z
r(B
) =
T•r
+ 5
T1 s
1 +
6T
s +
1
1 Z
r(s)
=
T's
' +
7T•s
• +
15
T1 s
1 +
lOT
s +
1
1 Z
:r(s
) =
T&s&
+ 9
T's
' +
28
T8 s
1 +
35
T1 s
1 +
15T
s +
1
Ta
Zp(s
) =
T1 s
1 +
3T
s +
1
N
....:1
N f
13
-"«~
'R
14
-ritfc
f2R
--
Ein
Abs
chni
tt is
olie
rt (
eine
r von
bei
den)
15
'(:fr'(~·1,;···--.rrr
t t
t K
atho
denv
erst
ärke
r zw
isch
en A
und
B
-- 16
"t;•'!c
~R2
17
~±c
IR
18
~fc !R
Ts
Z2'
(a)
= T
2 a• +
37's
+ 1
(die
selb
e Ü
bert
ragu
ngsf
unkt
ion
wie
in
Nr.
12)
4RC
a Z
2'(a
) =
4R
1 C1 s
1 +
8RC
s +
1
1 Z
2'(s
) =
(Ta
+ 1
)(T
1 sl +
3T
s +
1)
R•
Z2'
(s)
= R
R
' R
R
1
1Cs+
1
+
•
Z2'
(a)
= lc
( s1
1 )
81+
RC
+ L
C
Z2'(B
) =
1 L
•c•a
• 2L
B
L"C
"a' +
--
+ 3L
Ca1
+ -
+ 1
R
R
f N
-.J w
19
20
21
22
23
24,
Net
zwer
k
.....
''/(
'ii:r·
;r Tc
' Tc
a
Tir
r
___L
_.i
-f~
t I
R,
lla
~~CIT
R
R
R
I
~r.l
fcT-
cT
:TI
Ee(
•)
Zr(
B)
= E
m(B
);
T =
RO
; T
1 =
R10
1;
etc
.
1 Z
r(•)
= T
1Tr
8• +
(T1 +
T1 +
R10
1)B
+ 1
1 Z
r(B
) =
2TB
+ 3
TlT
rB'
Zr(
B)
= T
1Tr
B' +
(T1 +
T1 +
R10
1)B
+ 1
T'B
' Z
r(•)
= T
IB• +
6T
··· +
5TB
+ 1
T's4
Z
r(•)
= T
'•' +
IOT
'•1 +
15
T1 B
1 +
7TB
+ 1
z 1
8)-
TB
f'\
-T
1 B1 +
3TB
+ 1
(d
iese
lbe
For
m w
ie i
n N
r. 12
und
Nr.
13)
IV
-...)
.j:
>. ~ ~
25
-1:~ f·
-
t 26
Ic
-
-cJ
27
lc
c I
28
-or
29
-L?T
Ta
Zf'(
B)
=
3TB
+ 1
TB
+ 1
Z
f'(s)
=
2T
s+
1
TB
+ 1
Z
f'(B
) =
2T
B +
1
(die
selb
e F
orm
wie
in
Nr.
26)
(TB
+ 1
)1
Zf'(B
) =
T1 s
1 +
3T
s +
1
z~) =
TB
+ 1
8
TB
+ 2
~ Jl N
-.J
VI
Net
zwer
k
--
30
=2
I 31
er
--
~'
32
I --
33
~'
----
---
E (
s)
R C
·
t Z
(s)=
-0--· T
=R
G·
T1
= 1
10ec.
T E
IN(s
)'
'
k(T
s +
I)
ZT(s
) =
k'l'
s +
(1
+ k)
Ts
+ 1
ZT(s
) =
Ts +
2 (d
iese
lbe
For
m w
ie N
r. 2
9)
Ts
+I
Zr(
s) =
1•s
+
3
(Ts
+ 1)
2
Zr(
s) =
T
•s•
+ 5
Ts+
2
----
--~-------
N
-.1
0\ ~ § ~
(0
I 0 rr
0 0 ;<"
34
35
37
--
38
39
TI=
R
v-
y ~
c_, _
_
c,_R
,
~t ~~
3Ta
+ 1
Z;r(
s) =
T•a
• +
5T
a +
2
6T•a
s +
5T
s +
1
Z;r(
s) =
T•a
• +
6T
1 s1 +
5T
s +
1
·-··
T3 s
• +
5TB
s1 +
6T
s Z;
r(s)
= T
•s• +
5T
•s• +
6T
s +
1
---·-
R1(T
1s +
1)
Z,.(
s) =
(RtT~
+ R
aTt)
s +
(Rt +
R,)
k(T
s +
3)
Zr(
B)
= 2
kTs +
3(1
+ k)
~ ~ N
-..J
-.
.J
Net
zwer
k
40
~t f
R
41
~'fc,
!R•
42
~wf7
'fx
!R
43
~v:F~'·f·'~]c
!R,
--
44
-G3
:J-
~ ~R,
~
Zr(
s) =
Eo(
s) ;
T
= R
O;
T1
= R
101
; et
c.
Em
(s)
2Ta
Zr(a
) =
T•a•
+ 4T
a +
1
R10
1a
Zr(
B)
= R
1R
1010
.a• +
(R10
1 +
R10
1 +
RsG
t)B -
Ir 1
1 Z
r(s)
= 2
T•a•
+ 6T
a +
3
1 Z
r(B
) =
RSO
•a•
4R
1 Ga
3R
R
•o•8
• +
--
+ 5
R10
1 a2 +
~ +
6RG
a +
R +
1 R
1 1
1
R1R
101 a
1 +
2R
1Ga +
1 Zr
(a)
= R
1R
1G•a
• +
(2R
1 +
R1)G
a +
1
N
-.)
0
0 ~ ~
(nor
mal
e S
erie
n)
'-.'J
ll
I, 4
5
I
I
46
"1iT
47
11
Tf
48 ~
49 ~
I I
Z,.(
«) =
« +
I
!
I Z
,.(.!)
= «
1 +
( y2
)&+
I
I Z
,.(«)
= ~ +
28
1 +
28 +
I
I Z
,.(«)
= «
' +
2.6I3I~
+ 3.
4I42
81 +
2.6
I3b
+
I
I Z
,.(«)
= ~ +
3.
23&
' + 5.
23~
+ 5.
23&
1 +
3.23
& +
I
I N
-.l
\0
Net
zwer
k
50
~1-!1
~
51
,.
:117
11
56
52 ~
~
1!151 1
53 ~~
. 141
55
--
I M
~ J H
Ü~ >5iö
~:04i
i~~ 1
1 I
I ..
55
Hl7
~2
92
~1'6
26
IHS6
4
--
Eo(
8 )
RG
T
-R
G ·
tc
z.
,(8)
= -
-; T
=
; 1
-,
,, e
· E
IN(8
)
I Z
r(8 )
= 8
• +
3.86
378•
+ 7.
4641
8' +
9.l
4lf
lal +
7.4
64la
1 +
3.86
378
+ l
l Z
,.(8 )
= 8
1 +
4.49
398'
+ 1
0.09
7886
+
14.
5918
8'
+ l4
.59
I8a'
+ I0
.097
881 +
4.49
398
+ l
l Z
,.(8 )
= 8
• +
5.I2
6a7
+ 1
3.14
88 +
21.8
586 +
25.
69a'
+
21.8
58• +
I3.I
481
+
5.1
2&
+ I
I Z
,.(8 )
= 8
• +
5.75
988 +
16.5
887 +
3l.
I6a8
+
41.9
986
+ 4
1.99
8' +
3l.
l6a3
+
I6.5
881 +
5.75
98 +
I
I Z
,.(8 )
= 8
•• +
6.39
2a' +
20.4
3s• +
42.8
087 +
64.8
88' +
74.
238'
+
64.8
88' +
42.8
081 +
20.4
381 +
6.39
28 +
l
N
00
0 ~ ~
(c)
über
trag
ungs
funk
tion
enak
tive
r N
etzw
erke
-E>-
or
~A•n
Ver
stär
ker
sind
dur
ch d
ie S
ymbo
le g
eken
nzei
chne
t, w
obei
en
die
Ein
gang
sspa
nnun
g am
Ver
stär
ker
bede
utet
--
---1
1
qJ
c 'R
1
Zp(
B)
=
8
RC
•I
--
VJI
I
8EJ c
f R
1
2 Z
p(B
) =
8
RC
•I
T ~ 'R
RC
B
3 Z
p(s)
=
(1
-A)R101
~1 +
3R
Cs +
l
--
---il
ElJ
c
lc R
CB
4
I Z
p(B
) -=
R10
1 s1 +
3RG
's +
1 -
A
~ ~ N
00
D4
J
5 C
C
A
Q=J
--
YIN
lc R
6
I
~
7 Tc
8 ·'·'{{
·' L&
Jc
9 'ii
' 'Ii
' 1
'""L;
J .... ,
2 r
I (R
Ca
+ 1)
2
ZT
(a)
= R
•C•a
• +
(3 -
A)R
Ca
+ l
1 Z
T(a
) =
R•c
•a•
+ (3
-A
)RC
a +
1
1 Z
T(a
l =
(Res
+ 1)
1
1 Z
,(a
) =
1 -
RC
a
2 Z
T!a
) =
R
•c•a
• +
1
N
00
N
f
10
1
IV
e-IN
•o
M
o
11 I
'• :::
::t:[ '•
-1
----
121 ~
~Mo
~Beo I
131
J~J~!·
14
z z
V
~y
c.Q
zy =
Z
(s)Y
(s) 1
ZT
(8)
= (1
_
A)Z
Y +
1
1 Z
T(s
) =
(1
-B
)Z•Y
• +
(3 -
4B
-A
)ZY
+ 1
-2
B
1 Z
T(s
) =
R
•c•s
• +
(5 -
B)R
•c•s
• +
(6 -
2B
-A
)RC
s +
1
1 Z
T(s
) =
~·"·
. ,.
~·
~·"·
•A
~~
.. ~" ••
ZT
(s)
= -·
···
-· -·
·-·-
. -~··
-··-
__
__
.. -·
·
f N
00
w
284 Anhang I
Aktive Butterworth-Filter- Gleichung in 1-b
15 n=2
•
16 n•3
• •
1·0824 2·6131
17 I
10"92388 o-38268I n•4
• •
18
•
• 19
I·OFI~ +I >-'--N~I\...J.~..,.....-4
• o-~93I
Anhang II
Operationen mit Laplace-Transformierten
No.J f(t) F(s)
1 af(t) aF(s)
2 E"'.f(t) F(s- a)
3 -tf(t) dF(B) ----;{8
4 f (~) aF(as)
5 Lt f(t)dt F(s)
8
6 f"(t) B"F(s) - B"-1f(0) - s"-1f'(0) - · · · - j"-1(0)
7 (t - a)U(t - a) e-••F(s)
8 f(t) = f(t - a) 1 J0
11 f( t )e-.e dt 1- "-••
!) limj(t) limsF(s) t--+0 ,_,. 00
10 1imf(t) lim sF(s) t--+00 1-+0
11 [t j 1(t - T)j.(T)dT F 1(s) · F 1(s) -0
12 f'(t) sF(s) -f(O)
-
No.,
F(8
)
1 1
2 8
3 1 - 8
4 1 Bi
5 1 ;,;(
n =
I,
2, 3
·
· ·)
6 I ;.;
(n >
0)
I 7
81
/0
8 1
8"'
"
Anh
angi
ii
Tab
elle
der
Lap
lace
-Tra
nsfo
rmie
rten
f(t)
U (
)
!im
U(t
) -
U(t
-a)
E
inh
·t .
1
1 t
= ,
et S
lffip
u s
a-+
0
a
U (
)
-!i
m U
(t)
-2U
(t -
a) +
U(t
-2a
) E
inh
"t .
1 d
b1 t
t 1
t -
1 ,
et S
IIDpU
S U
e
a-+
0
a
U(t
), E
inhe
itss
prun
gfun
ktio
n
t
en
-1
(n
-I)
!
en
-1
r(n)
1 y(
7rt)
J(~)
---
---
--
~ ~ s
1 91
8
+IX
- 1
10 I
(8
+IX
)"
-- 1
111
(8 +
IX
}(8 +
{J
)
1 12
I
8(8
+ IX
}(8
+ {J
)
1 13
I
(8 +
IX
}(8
+ {1
)(8 +
v)
1 14
I
8(8
+ IX
)(B +
{1
}(8 +
v)
1 15
I
81(8
+
IX)
---
1 16
I
8°(8
+
IX)(8
+
{J)
1 17
I
8'(
8 +
IX
}(8
+ {J
)(B +
v)
1 18
I
8• +
IX
a - 1
19 I
s2
-
cxt
e-a
l
t"-l
e-o
<l
(n
-1
)!
e-a
1 -
e-P
1
~-
1 {Je
-<1.
1 -
IXB
-pt
IX{J
+ '-tX
-:{J"'"
( IX--
-{J"
"')-
e-<~
.1
e-P
I e-
•1
({J -
IX}(
V-
IX)
+ (IX
-{J
}(v
-{J
) +
(IX -
v}({
J -
V)
1 e-
•1
rPt
e-<~
.1
IX{J
V V(
IX -
V}({
J -
V) -
{J(IX
-{J
)(v
-{J
) -
IX({J
-
IX)(
V -
IX)
1 äi (e
-<~.
1 +
IXt
-1)
1 [
1 IX
2{11
(IX
-{J
) (1
X•e
-ß1
-fl2
e-<1
.l) +
IX
{Jt
-IX
-{J J
_IX{l,_
_('---v_
t ----:1
--'l..".
---=-
IX-v_
-___
,_fl_v
+
e-<~.
1 +
rP1
+ e-
•1
(1X{
Jv)2
1X
2 (v
-IX
)({J
-IX
) {J
2 (v
-{J
)(IX
-{J
) -v2
-({J _
__
v_)(_
IX _
__
v)
Bin
IXt
IX
sin
h IX
t IX
~ ~ - - - N
00
-.
J
No.
I F
(8)
20 I
l 8(
8° +
cx2 )
21
I l
s2(8
2 +
cx2 )
22
I l
(8 +
cx){
82 +
ß2 )
--
23
l 8(
8 +
cx){
82 +
ß2 )
-
24
l 8°
(8 +
cx){
82 +
ß")
---
25
I l
(8 +
cx)(s
+ ß
)(s•
+ v1
)
-
26 I
l 8(
8 +
cx)(a
+ ß
)(81
+ v2
)
-27
I
l s2
(8 +
cx){
8 +
ß)(
8° +
v2)
j(t)
l -
cos
cxt
cx•
t si
n c
xt ;o
---o
ca
cx• ~
ß• (
e-cx
• +
lJ si
n ß
t -
cos
ßt)
l l
{sin
ßt
cx c
os ß
t e-
<'~ '
\ cx
ß• -
cx• +
ß•
-ß-
+ -p
+ a
}
t l
r"''
cos
(ßt +
</>)
cxß•
-cx
•ß• +
cx•(c
x2 +
ß')
+ ß•
v(cx
• +
ß•)
' "'=
tan-1
(i)
..,...e-,
--"'1 ;;-
:---
-;;-
c +
e-ß
t +
sin
( vt
-</>
) (ß
-cx
)(cx2
+ v•
) (cx
-ß)
(ß' +
v•)
vv {v
1 (cx
+ ß
)1 +
(cxß
-r)
'}'
<f> =
ta
n-1
{~)
+ ta
n-1
(~)
l e-
cxt
e-ß
t co
s (v
t +
</>)
cxßv
• +
cx(cx
-ß)
(cx•
+ v2
) +
ß(ß
-cx
)(/JI
+ r
) +
rv' {
(cxß
-v•
) +
r(cx
+ ß)
"}
4> =
tan
-1 (~
+ ta
n-1
(~)
l (
l l)
e-
cxt
e-ß
t cx
ßv•
t -
~ -
ß +
cx2 (
ß -
cx)(c
x2 +
r) +
ß"(a
. -
ß)({
l" +
v•)
CO
S (
vt +
</>)
-1 (ß
\ -1
(V)
+ v
"y'{
(a.ß
-v•
)• +
(a. +
ß)•
v•}'
"'=
tan
v} -
tan
~
N
00
0
0 ~ § O
Q - - -
28
l
(a0 +
a.2)(
s2 +
ß2
)
29
l
(8 +
a.)•
+ ß
•
30
l
s2[(
a +
a.)2
+
ß2 ]
31
l (s
+ v)
[(s +
a.)•
+ ß"
]
32
l
s(s +
v)[(
s +
a.)2
+
ß"]
33
l s2
(a +
v)[(
s +
a.)2
+
ß"]
34
l (s
+ v)
(s +
6)[(
s +
a.)2
+
ß"]
35
l (s
2 +
v2)[
(s +
a.)•
+ ß"
]
l (s
in a
.t si
n ß
t)
ß• -
a.•
-a
.--
-ß
-
e-«
' sin
ßt
ß
l (
2a.
) e-
cx•
sin
(ßt
+ </>
) a.•
+ ß
• t
-a.
• +
ß"
+
ß(a.
• +
ß")
• .p
= 2
tan
-1
(~)
e-~•
e-«
' sin
(ßt
-</>
) "'
= ta
n-1
(-ß
) ß•
+ (v
-a.
)• +
ßv {(
v -
a.)•
+ ß
•}'
v-a
.
l c''
e-
«' s
in (
ßt +
</>)
-+
.
v(a.
• +
ß")
v[(a
. -
v)• +
ß"]
ßV
(a."
+ ß
2)[
(a.
-v)
2 +
ß2
]
.p =
tan
-1
(~
+ ta
n-1
(a
. ~
v)
l (
l 2a
. )
e-••
e-
«' s
in (
ßt +
</>)
t------
+
+
. v(
a.• +
ß")
v a.•
+ ß•
v•
[(v
-a.
)• +
ß"]
ß(a.
• +
ß"lv
{(v
-a.
)• +
ß"}
.p =
2 ta
n-1
(~ -
tan
-1
(v ~ a.
)
c••
e-6•
e-
cx•
cos
(ßt +
</>)
(<5 -
v)[(
a. -
v)2 +
ß"] +
(v -
6)[(
a. -
6)2 +
ß"] +
ßV
[(a.
-6)
2 +
ß2][
(a.
-v)
• +
ß•]
'
.p =
tan
-1
(a
. ~
6) -
tan
-1
(a
. p 11
)
ß si
n (v
t +
</>1
) +
ve-c
x• s
in (
ßt +
</>2
) .p
_ _ 1
(
2a.v
)
ßvv {4
a.•v
• +
(a.•
+ ß"
-v2
)2} .
1-t,a
n
v•-a
.•-ß
•
"'
--1
[
2a.ß
]
• -
tan
(a
.• -
ß" +
v•)
--
6'" §" OQ
.... .... .... N
00
\0
No.j
F(8
)
36
8(81
+
vl)[
(8 +
cx
)1 +
fJ"]
37 I
1 (8
+
v}(8
° +
<51
)[(8
+ cx
)1 +
fJ"]
l 38
I
8(8
+
cx)•
-- 1
39 I
8•(
8 +
cx
)• -
l 40
I
(8 +
cx){8
+
fJ)"
l 41
I
8(8
+
cx)(8
+ fl)
" !
1
I 42
I
81 (
8 +
cx
)(8 +
flj
•
j(t)
_l _
_ +
B
in (
vt -
cf>tl
_ B
in (
vt +
</> 2)
v•(c
x• +
f11
) {J
V(c
x1 +
{11)[4
cx1v1
+
(cx1
+ {1
1 -
v1)1
] ~v•::
-:v'147 cx
""•:=v1 :==
+'=(:=
cx~•=+
~f11
-v2
)2
<P
t= t
an
-t (~
) +
tan
-t (
cx
'-!cx
~ +
v"),
cf>2
=
tan
-t (-
cx1 _+_ 2!
~=V:)
e-•1
e
-C<
t B
in (
{Jt +
cf>t)
(v'
+ d1
)[(cx
-v)
• +
{12 ]
+
{JV
[(v -
cx)1
+
f11][4
cx1 <
51 +
(cx2
+ {1
2 -
<52
)1]
Bin
(&
+ </>
2)
-d
v' (
v' +
d1)[4
cx2 d
1 +
(cx2
+ fJ"
-<5
1)2
]
cf>t
=
tan
-t (V
ß CX)
-ta
n-t
(cx
" -!cx
~ + d
}
cp 2 =
ta
n-t
m
+ ta
n-t
(cx
l + :~-
d")
1 -
1r·«
• -
cxte
-«1
cx•
t 2
te-«
1 2e
-cc•
---+
-+
-cx
• cx
• cx1
cx8
e-«
' [ (
cx -
{J)t
-l ]
e-fJ
' (cx
-{1
)2 +
(cx -
{1)1
l e-
cc•
r-t
cx -
2{1 J
cxß•
-cx
( cx -
{1)1
-LfJ
( cx -
{1)
+ {1
1(
cx -
{1)2
e-
fJ'
e-cc
t l
( l
2)
[ t
2( cx
-{J
) -
fl]
_ ,
cx1({1
-cx
)• +
cx{12
t
-ä
-ß
+
{12
( cx -
{1)
+ {18
({1 -
cx)2
e
fJ
N
\0
0 ~ § ()Q
.....
.....
.....
43 I
I
(8 +
ß}(
8 +
v)(8
+ cx
)• --
44 I
I
8(8
+ ß
)(s
+ v
)(s
+ cx
)•
--
45 I
I s•
(s +
ß)(
s +
v)(8
+ cx
)2
--
46
I (8
1 +
cx2)(
s +
ß)2
--
47
I s(
s2 +
cx2)(
s +
ß)•
---
48 I
I
82(s
2 +
cx2)(
s +
ß)2
---
49
I (s
+ v
)(s2
+
cx2)(
s +
ß)'
50 I
I (8
+ v)
2 [(s
+ ß
)' +
<X2
]
[ t
2cx
-ß
-v
J e-
P•
e-••
-;--=
-;-----: +
6-a
t +
_
+ -;
-n--
-'-;
-;--
--:-
;;
(<X
-ß)
(cx
-v)
(cx
-ß)
'(cx
-v)
• (v
-ß)
(cx
-ß
)2
(ß -
v)(c
x -
v)•
I e-
P•
c••
-ßv
-cx•
+ ß(
ß -
v)(c
x -
ß)1
+
-v(:-v
----;ß;
;-)(,---
cx -
v)•
+ [
t (cx
-
v)(c
x -
ß) +
cx(
2cx
-v
-ß
)J
cx(cx
-v)
(ß -
cx)
-cx
2(cx
-v)
2(cx
-ß)
' e-
cx•
[ t
+ 2(
cx -
v)(c
x -
ß) +
cx(2
cx -
v -
ß)J
6_
"'' +
e-
P•
cx•(c
x -
v)(c
x -
ß)
cx3
(cx
-v)
2(cx
-ß)
• ß"
(v -
ß)(c
x -
ß)2
I (
2 I
I)
e-••
+ ß
vcx•
t
-a -
ß -
v +
v"(
ß -
v)(c
x -
v)2
Bin
(cxt
+ </>
) [
t 2ß
J -
p•
cx(cx
2 +
ß")
+
cx2 +
ß• +
(cx2
+
ß2
)2
6 '
.p =
2 ta
n-1
(~)
I B
in (c
xt +
</>)
f _
t 3ß
2 +
cx2 J _
1
cx•ß
• -
cx2
(cx2 +
ß2
) -
Lß(c
x2 +
ß2
) +
fJ2
(cx2 +
ß2
)2
6 p
' .p
=
tan
-1 (~
) -
tan
-1 (~
)
Bin
(cx
t +
</>)
te-P
' 2(
cx2 +
2ß'
)cP
' t
2 cx
3(cx
2 +
ß")
+
ß'(c
x2 +
ß')
+
ß•(c
x2 +
ß2
)2
+ cx
•ß•
-cx
2ß
3'
</> =
2
tan
-1 (~
)
Bin
(cxt
-</>
) tc
ß'
[2(v
-ß)
ß -
(cx2 +
ß')]
cx
(cx• +
ß"J
v'(c
x2 +
v2
) +
(v -
ß)(c
x2 +
ß2
) +
(v
-ß)
2(cx
2 +
ß2
)2
e-ß•
e-••
(cx)
(c
x)
+ (c
x• +
v')
(ß -
v)•'
</>
=
2 ta
n-1
ß
+ ta
n-1
v
e-ß•
Bin
(cxt
-</>
) 2(
v -
ß)c
"'
te-•
• cx
[cx2 +
(ß -
v)2
] +
[(ß
-v)
' +
cx•]•
+ cx
• +
(ß -
-v)2
' .p
=
2 ta
n-1
(;--~
p)
~ ~ - - - N
\0
No.,
F(s
)
51
1
8(8 +
v)"[
(a +
ß)" +
a.•]
52
1
(8 +
cx)1
(8 +
ß)1
53
1
8(8 +
cx)1
(8 +
ß)1
54
1
81(a
+ a.
)2(8
+
ß)2
55
1
(8 +
v)(8
+ a.
)2(8
+
ß)'
--
56
1
(81 +
v")(8
+ a.
)2(8
+
ß)1
~I
1
[a.•
+ (8
+
ßl"J"
j(t)
e-ß•
sin
(cx
t +
</>)
([cx
2 +
(ß -
v)1 ]
-2v
(ß -
v))
_ 1
ee-•
• cx
[cx•
+ (ß
-v)
2 h/(
cx
1 +
ß")
-v•
[a.•
+ (ß
-v)
1]1
e
' -
v[a.
• +
(ß-
v)3 ]
1
+ v1
(cx•
+ ß
•)'
q, =
2 ta
n-1
(v
: ~
+ ta
n-1
(~)
ee-a
;• 2e
-a•
te-ß
• 2e
-ß1
-------+
-
(ß -
cx)•
(ß -
cx)3
(c
x -
ßj•
(c
x -
ß)3
(3cx
-ß)
e-a•
ee
-a•
(3ß
-cx
)e-ß
' te
-ß'
1
a.•(
ß -
cx)•
-
cx(ß
-
cx)•
+ ß
'(cx
-ß
)"
-ß(
cx
-ß
)• +
a.•ß
•
2(ß
-2c
x)e-
a•
ee-a
• 2(
a. -
2ß)e
-ß'
ee-ß
' t
2(cx
+ ß)
a.'
(ß -
a.)•
+
a.'(ß
-a.
)• +
ß'
(a.
-ß
)•
+ ß"
(cx
-ß
)" +
a.•ß
• -
a.•ß
•
"-••
[3
ß -
(a. +
2v)]
e-ß'
te
-ß'
[3cx
-
(ß -
2v)]
e-<
X'
(a.
-v)
'(ß
-v)
2 +
(a
. -
ß)8
(v -
ß)2
+
(v -
ß)(a
. -
ß)1
+
(v
-a.
)2(ß
-cx
)3
te-a
• e-
" +
+
(v
-a.
)(ß
-a.
)1
(cx
-v)
1(ß
-v)
1
sin
( vt
+ 4>
) 2
[( cx
• +
v")
-a.
(ß -
a.)
J -<
X<
te-a
• v(
a.• +
v')(
ß• +
v")
-(c
x• +
v•)(
ß -
a.)a
e
+ (a
.• +
v")(
ß -
a.)•
[ß• +
v• -
ß(a.
-ß
)J
_ 1
te-ß
' -
2 (ß
" +
v')(
a. -
ß)•
e
ß +
(ß• +
v•)(a
. -
ß)•
' q,
= 2
tan
-1 (~
) +
2 ta
n-•
(;)
e-ßl
(B
in a
.t -
a.t C
OB a
.t)
2a.8
N
\0
N ~ ~ ......
......
......
1\J
0 I 0 CT
0 0 ><"
58
59
60
81
82
83
84
85
1 (aA
+ ot
1)1
1
B(aA
+ «"
)"
1 aA
(aA +
ot1
)1
1
<• +
{J)(a
A +
ot1
)1
1
•(• +
,8)(
•1 +
cx•)
•
1
aA(•
+ P
H•" +
«"l"
1
•<• +
at)•
1 aA
(aA +
ot1
)
Bin
aJ
-aJ
00
8 a
J
2ot1
(I -
00
8 a
J)
lsin
aJ
«' -~
3si
na.
t. +
loo
saJ +
_!_
2ot'
2ot'
«'
trfl
' I
sin
(aJ
+ .;
1)
(v' fJ
1 +
4«"l
co
s (a
.t. +
.;.>
--
(ot•
+ (J
I)•
2ot1
y(o
t1 +
(JI)
2o
t1( f
X1 +
(JI)
•1
= t
an
-1
(ID·
." =
tan
-1
[ß
(3o
t1 +
fii>J
1
2cx1
'C0
8 (
aJ +
.;1)
(V9c
x1 +
4{11
) co
s (
aJ +
.;,)
e-Il•
1
2cx1
y(o
t1 +
(JI)
-2a
t'(at
1 +
(JI)
-
,8(a
t1 +
{11)
1 +
«'fJ
.;1
= t
an
-1 (~.
.;. =
tan
-1
(~~) +
2 ta
n-1
(~)
I sin
(aJ
+ of>
1)
_ (I
6at1
+
9(JI
)111
sin
(aJ
+ .;
.) +
e-
Il•
+ _
, __
_ 1_
2ot'v
'(ot
1 +
(JI)
4c
x1(a
t1 +
(JI)
(J
I(cx1
+
{11)
1 «'f
J at
',81
.;1
= t
an
-1 (~.
.;, =
tan
-1
(!~) -
2 ta
n-1
(~)
--
-+
-+
-a-
«'
I e•
t I)
at1
2at
at1
at1
t•
(co
s a
J -
1)
-+
2c
x1
cx'
~ ~ - :::: N
\0
w
294 Anhang III
·i"tl > -IN i; ~ ~
-l"tt I 't:: ·S ~1-
Cl)
I ;:::;- >
~'"' ilr > -.,
i i i·tl I "t~ ~ I I ~
i"' 'i ;; 'i;"tf
j;IN 't:: .::._.:; -., Cl)
.<:: 1 =I~ -., I ~
~I~ ~
I~ • b ~
" 1;" ;:::;- "tt "t! ;:::;-
tl
~I~ '> + +
~\? "tt
I I + ~ + ~'I. I ~+ ~ ~ ~ ~~ ~ . 'lo
.. > .. " .. > ~ '> I~ ~
" 1;' .. I~
+ '> '> + ~ ~ ..
'>
<C I .... I ao I Cl> I 0 I ~ I IN I ..,
I "" I IQ <C <C <C <C .... .... .... .... .... ....
I J,
(joJ
) 76
I ,, ..
' •'I
J,::)
I
77
8 +
y(s
2 +
oc1
)
78
Vs' +
OC2
(8 +
V1 8
1 +
oc1)"
79
s&J•
--~-
80
va&'
' 1
---
81
e-••
8
--
82
8
s• +
oc1
83
8
a•
oc•
84
I 8
(81 +
oc1)(
81 +
ß1
)
---
85 I
[8' +
(oc +
ß)•i
8• +
(oc
-ß)
'] -
86 I
8
(s1 +
oc1
)1
J,.(
a.t)
oc
"
J,(
2V
a.t)
cos
2y
(a.t
) y
(1rt
)
U(t-
a)
cos
oct
cosh
a.t
CO
B r
J.t -
CO
B ßt
{J
' -
oc•
'
2ocp
t Bin
(1
j
2cX
oc• *
{J'
?,>
::r
§ ~ - - - N
\0
VI
No.
F
(8)
87
82
(8• +
a.')
'
88
8
(8 +
a.)2
89
8
•' -
cx:'
90
8•
B'
-et'
91
,,. 8
• -
ot•
92
8
8• +
4ot
' --
93
8
(8
-et
}(8
-ß)
94
8ot8
82
(8• +
ots)
a
95
8
(8 -
at)"
2
96
y'e
8 -
a.•
I --
------
----
··-
-----
-...
j(t)
Bin
Cd
-j-
cxl
CO
B a
.t 2c
x
f:-«
'( l
-a.
t)
co
sb
a.t
-C
OB
a.t
2a.•
sin
h a
.t +
sin
a.t
2a.
cosh
a.t
-j-
CO
B C
tt 2
sin
h C
tt •
Bin
a.t
2ot1
Ctf:
-«1
-ßE
.-fJ'
a.-
ß .
a.#
ß
( 1 -
j-et
1 t1
) B
in o
tt -
a.t C
OB
Ott
E."'1
(1 +
2a.t)
y
(".t
)
1 ••
rf
y(
..C
) +
!XE.
"' e
(oty
t)
N
\0
0\ ~ ~ .... .... ....
97
8 +
a 0
8(8
+ et
)
98
8 +
a 0
(8 +
oc)(
8 +
ß)
-
99
8 +
a 0
8(8
+ oc
)(8 +
ß)
IOO
8 +
a 0
82(8
+ et
){8
+ ß
)
IOI
8 +
a 0
(8 +
et){
8 +
ß)(8
+ v)
-
I02
8 +
a 0
8(8
+ oc
)(s +
ß)(
8 +
v)
103
8 +
a 0
81(8
+ et
}(8
+ ß}
(8 +
v)
I04
8 +
a 0
81 +
oc•
I05
s +
a 0
8(s•
+ oc
1 )
I06
8·+
a0
82{s
• +
oc2 )
I07
8 +
a 0
(8 +
cx)(
s2 +
ß")
a 0 -
(a0
-o
c)c
'"
Gt
(a0
-oc
)e-«
1 +
(ß -
a 0)e
-ß'
ß-et
a 0
(oc
-a
0)e
-«'
(ß -
a 0)e
-ß'
-+
+
oc
ß oc
(ß
-oc
) ß(
oc
-ß
)
(I +
a 0t)
_
(oc +
ß)a
0 +
_I_
[(a
0 -
oc)e
-o:•
_
(a0
-ß)
e-ß'
J oc
ß oc
•ß•
ß -
oc oc
1 ß•
(a0
-oc
)e-«
' +
(a
0 -
ß)e-
ß'
+
(a0
-v
)e-•
•
(1•
-oc
)(ß
-oc
) (o
c -
ß)(v
-ß
l (o
c -
v)(
ß -
v)
~ +
(o
c -
a0)E
-«'
+
(ß -
a 0)e
-ß'
+
(v -
a0)c
••
ocßv
oc
(ß -
oc)(
v -
oc)
ß(oc
-
ß)(
v-
ß)
v(oc
-
v)(ß
-v)
l +
a 0t
a 0(e
tß +
ßv
+ oc
v)
(a0
-oc
)e-«
' (a
0 -
ß)e-
ß'
(a0
-v)r
••
---
+-----+
+
oc
ßv
oc•ß
•v•
et2(ß
-oc
)(v
-oc
) ß'
(oc
-ß)
(v -
ß)
v1(c
t -
v)(ß
-v)
V (a
01 +
oc2
) si
n (
oct +
t/>)
4> =
ta
n-1
(~)
Gt
. a0
j c
a02 ) •
-+
-
+ -
sm (
oct
-,P
), ~z
az
~·
4> =
ta
n-1
(~)
[I
a0t
Je
a0')
.
J -
+ -
--
+ -
· sm
(oc
e +
t/>)
, cx.
z a2
cx
t a•
4>
=
tan
-' (Y
.)
(a0
-oc
)e-«
' j (
a 02 +
ß•
) .
cx• +
ß•
+
cx•ß
• +
ß'
. sm
(ßt
+ t/>
), 4>
=
tan
-1 (~
) -
tan
-1 (7
J)
~ § 0<1 .... .... .... N
'D
-..J
112
113
f(t)
~ -~=-
a)e
-«'
J( ao
• +
ß•)
aß
• a(
a• +
ß")
-
a•ß•
+ ß
• · c
os (
ßt +
</>)
, </>
=
tan-
1 '~)
-ta
n-1
(71)
--~
I cu
;--«
' l
J(a•"
-+-ß"
) . c
os (
ßt +
</>)
----
_I _
~
(.! _ e
) +
"( a~• +
ß") +
7fi
o:• +
ß•
_ 1 (~)
+ ta
n-1
(.f!_
\ o:
ß•
o:ß•
o:
o: .p
=
tan
ß
a"
J
8 +
a0
a"(8
+ a
)(8
+ ß
}(81
+ 111
)
8 +
ao
(81 +
o:1
}(8
1 +
ß")
_(a
0--=-a)e~
+
(a0
-ß
)e-ß
' I(
ao" +
v•
) .
(ß -
a)(o
:• +
v•)
(a
-ß)
(v•
-+-
ß")
+ .;
\~• +
v•)(
v• +
ß")
. s
m (v
t +
</>)
rf> =
ta
n-1
(~)
-ta
n-1 (~)
_ ta
n-1
(~)
a0
(o:
-a
0)e
--«'
(ß
-a
0)e
-fJ'
I J(
v•
+ a 0
2 )
• -+
+
+
-·s
mv
t aß
v•
o:(ß
-a)
(o:•
-+-
v•)
ß(a
-ß)
(a• +
v•)
v• ·
v•(a
+ ßl
" +
(aß
-v1
) (
+ </>
)
</> =
ta
n-1
!. +
tan
-1!.
+ ta
n-1
~
(X
ß
"
I ao
(t -~ -
~) (a
0 -
a)e-
-«'
(ao
-ß
)e-f
J'
-aß-
"• +
aß
v•
+ a
'(ß
-a)
(a•
+ 11
2 ) +
"'ß•~(
a-'--"
-~ß).!
..(v-'
-o•-+-
ß"J
+ ~ J
(v•(a
+ ;j.
•: (~ß
_ r)•
) · cos
(vt
+ </>
), ß
-
V
-II
</>
=
tan
-1 -
-ta
n 1
-+
tan
l-
V
0:
a 0
a• ~ ß"
[ J (
I -1-;~"
) 'CO
B (
ßt -
r/>1
) -J (I
-t-~:)
• cos
(at
-rf>
1)]
.1.
-ta
n-1
ao
'1'1
-ß'
"'·
= t
an-1
~
a:
N
\Cl
00
~ ::r
§ 0<> - - -
114
s +
a 0
(s +
cx)2
+
p•
115
s +
a 0
a[(s
+-;)2
+ ß'
]
116
8 +
a 9
a'[(
a +
cx)2
+
ß']
117
8 +
a0
(s +
v)[(
s +
cx)•
+ ß
']
118
8 +
a 0
8(8
+ v)
[(8
+ cx
)2 +
ß']
--
119
8 +
a 0
s2(8
+ v)
[(s +
cx)2
+ ß
']
120
"+ ao
(8
+
IS}(
s +
v)[(
s +
cx)•
+ ß
']
J ( l +
(ao
Po cx
)') ·
5--
«1
• si
n (ß
t +
f>),
"= te.n
-1 (-
P-)
a
0-
cx
a0
J (ß' +
(a0
-cx
)")
--«
• •
cx• +
ß'
-ß'
(cx'
+ ß"
) .
6 •
sm (ß
t + +>
. +
= te,n
-1 {!
_ +
te,n
-1 _
{J_
cx
a,-
cx
l +
a0t
2cxa
0 V
{ß' +
(a0
-cx
)1}
1 •
+ =
tan
-1_
ß_
+ 2
te,n
-1 /!_
cx
' + ß
' -
(cx'
+ ß
')' +
ß(
cx' +
ß•)
. e-«
. s
m (ß
t +
+J,
a 0-a
. cx
(a0
-v)
e-•
• l J (ß
' + (a
0 -
cx)"
) --
«'
. (v
-cx
)' +
p• +
ß
ß' +
(v -
cx)'
. E
• sm
(ßt +
+>,
+ =
te.n
-• (-ß
) _
tan
-1 (-
P )
a 0-a
t •-cx
a0
(v -
a0)
c.,
l
j (ß'
+ (a
0 -
at)'
) 1
•
v(at
' +
ß')
+ v[
ß' +
(cx
-v)
'] +
ßy
'(cx
' +
ß')
p• +
(cx
-v
)'
• e-«
. s
m (ß
t + +>
.,. =
ta.n-1
{!_ +
tan
-1 _
ß_
+ ta
n-1
_ß
_
cx a 0
-cx
cx
-•
a0
[ l
l 2c
x J
(a0
-v
)r"'
v(
cx1 +
ß')
t + a.
-; -
(cx1 +
ß')
+
r[ß
' + (v
-cx
)1]
5--
«'
J (ß' +
(a0
-cx
)')
. +
ß(cx
' +
-ßi)
p• +
(v -
cx)'
. sm
(ßt +
.,.),
+ =
ta.n
-1 _
ß_
-ta
n-1
_ß
_ +
2 ta
n-1
/!_
a 0-c
x
v-c
x
a.
(v -
~;[(~
~):~:·
+ ß']
+ (<5
-:~[
(~ ~):
~:1 + ß'
]
-t j (
ß'
+ (a
0 -
cx)'
0 --<
XI
(ß
.,.)
-ß'
[ß' +
(cx
_ v
)'][
ß' +
(cx _
<5
)' • 6
·
cos
t +
,
.,. =
tan-
·1 _
/!___
-te
.n-1
cx
-"
+ te
.n-1
_{J
_
a0
-cx
ß
a
.-ß
~ § Oq
- - - N
\0
\0
I N
o. I
I2I
I22
I23
I25
I26
I27
I28
I30
I3I
---
F(8
)
8 +
a0
(8+~
8 +
a0
8(8 +
Q:)1
-s +
a0
82(8
+ 1X
)2
8 +
a0
(8 +
ß)(
s +
1X)1
8 +
a0
8(8
+ ß
)(8
+ Q
:)0
8 +
a 0
82 (
8 +
ß)(
s +
Q:)1
8 +
a 0
(e +
ß}(8
+ v)
(8 +
Q:)1
s +
a0
8(B
+ ß}
(8 +
v)(s
+ Q
:)l
8 +
a 0
(al +
ß')
(8 +
Q:)l
j(t)
[I +
(a0
-Q:
)t)E
--<~
<'
a0
[(
a0
\ a
0 ]
~ +
I
-~J t
-;ä
e-
-«1
I (
2a
0 )
I (
2ao
) 1
-I
+ a
"t -
--
-I
--
+ [Q
: -
a 0]t
e--«
(X
I IX
IX
t (X
(a0
-ß
)rfJ
' +
[a
• -
a; +
..!!_
-a
0 J
e--«
' (ß
-(X
)' ß
-(X
(ß -
Q:)•
~ +
(ß -
a0)r
fJ' +
[(a
0 -
Q:)t
_
Q:1
-a
0(2
1X -
ß)J
e--
«'
(XIß
ß
( IX
-
ß)l
Q:
( 0:
-ß
) (X
I( (X
-ß
)1
~ (.!
.. _ .!_
_!
+ t}
_ (ß
-a0
)rfJ
' _
[ (X
-a0
+
(2a
0-
Q:)
(2Q
:-ß
)--•] e-
-<~<'
Q:1 ß
ao
ß
(X pt
(Q:
-ß
)1
Q:1(ß
-IX
) Q:
8(Q
: -
ß)1
(a0
-ß)
e-fJ
' (a
0 -
11
)r••
[
a0
-(X
+ a 0
(2Q
: -
ß -
11)
-Q:
1 +
ß"]
-«
I
+
-+
E
(v -
ß)(Q
: -
ß)'
(ß
-11
)(Q:
-V)
1 (Q
: -
v)(Q
: -
ß)1
(Q
: -
ß)1
(Q:
-11
)1
ao
(a0
-1
1)r
.,
(a0
-ß
)rfJ
' -
(X•ß
v -
,.(p
-11
)((%
-v)l
ß(
v -
ß)((
X
-ß
)'
+ [
(Q
: -
a 0)t
_ a
0(Q
: -
,.)(Q
: -
ß) +
a;(a
0 -
Q:)(
2Q:
-P
-")
] e--
«' Q:
((X -
v)(Q
: -
ß>
Q:1
(Q:
-v)
1(Q
: -
ß)1
[2a•
Q:
-(X
I +
ß'
-((X
-
ao
)J t
--«'
+ y
(pt +
ao
l).
sin
(ßt
+ f)
((X
I +
pt)>
(X
I +
pt
ß( (X
I +
pt)
'
• =
tan
-1 .!!
... -
tan
-1 p_
a,
Cl
t
....,
0 0 ~ ., ::s "" - -
13
2
8 +
a0
8(8•
+ {J
I)(8
+ IX
)•
13
3
8 +
a 0
8•(
•• +
{JI)
(• +
IX)•
13
4
8 +
a0
(8 +
v)(8
1 +
{11 )
(8 +
1X)1
135
8 +
a 0
(8 +
")1 [
(8 +
a)1
+
{JI]
13
6
8 +
a 0
8(8
+ v)
1[(
8 +
1X)1
+
{JI]
--~
---·· -
-------------
~-
+ [2
1X1
(1X -
ao)
-ao
(1X1
+ {J
I) +
(IX
-a
,)t J e-
«l
_ 00
8 ({
Jt +
~) .
y(
I +
{JI)
IX1 {
JI
1X1
(1X1 +
{JI)
1 1X
(IX1 +
{JI)
1 {J
I(1X
1 +
{JI)
a.
' ~
= ta
n-1!
!..
-2
tan
-1 !!.
a"
IX
1 +
aot
-2a
, +
[<IX
' +
{JI)
(2a
. -
IX) +
21X1
(a,
-IX
) +
(a
. -
IX)t J e-
«1
IX1 {
JI
IX1{J
I 1X
1( IX
1 +
{JI)
1 1X
1( IX
1 +
{JI)
+ v
' (a 0
1 +
{JI)
Bin
({J
t +
~)
{JI(
IX• +
{JI)
• ~
= ta
n-1
.!!.. +
2 t
an-1
~
a,
{J
(a0
-")E
_,.1
+ [1
X1
(1X -
a0
) +
{11
(11 -
a0
) +
IX(v
-IX
)(2a
0 -
IX) +
(a
0-
IX)t
J e--«
•
({JI
+ "S
)(IX
-")
1 (1X
1 +
{JI)
1(v
-1X
)1
(v -
IX)(I
X1 +
{JI)
+ J
(a01 +
~ .
Bin
({J
t +
~)
{JI +
".
{J( I
X• +
{JI)
• ~
= ta
n-1
!. -
tan
-1 ~ -
2 ta
n-1
!!. {J
{J
ot
(a0
-")
t +
(1X1 +
{JI +
2a 0
(v -
IX)
-v•]
r•• +
v' {J
I +
(a,
-ot
)1•
e--
«'.
sin
({Jt
+ ~)
{J
I +
(a -
11)1
[{
JI +
(a -
v)1
] {J
({JI
+ (IX
-v)
1 ]
'
' =
tan
-1 (-
{J-)
-2 t
an-1
(-{J )
a0-a
"_
IX
a0
e-«
1 J (<
IX -
a0
)1 +
{JI)
.
"S(a
• +
{JI)
+ {J
[(v
-a
)• +
{JI]
IX• +
{JI
. Bm
({J
t +
f) +
...
+
[2v(a
0 -
v)(a
-v)
-
a.[
(v -
1X)1
+
{JI]
(" -
a,)
l ~
1 . .
. +
e
-"'
"s((
v -
a)1
+
{JI]
11((1
1 -
1X)1
+
{JI
' =
tan
-1 {!_
-2
tan
-1 _
{J_
-ta
n-1
_{J_
IX
v-a
IX
-a
0
~ ::r
§ 0<1 .....
. .....
. .....
.
VJ
0
No.
F
(8)
137
8 +
Go
8
1 (8
+
v)1 (
(8 +
ot)
1 +
ßl]
138
8 +
Go
(8
+
.5){8
+
v)1 [
(8 +
ot)
1 +
ßl]
139
8 +
a 0
(8 +
ot)1(8
+
ß)1
1•o
8
+G
o
8(8
+
ot)1(8
+
ß)•
141
8 +
G 0
81 (
8 +
ot)1
(8 +
ß)
•
f(t)
G 0t
(v -
2a0
){ot
1 +
ß1)
-2a
0ot
v V
{(ot
-G 0
)1 +
ßl}e
-«1
.
r(ot
l +
pt)
+
r(o
t' +
ßl
)l
+ ß(
otl
+ ßl
)((o
t -
v)l
+ ßl
] B
ill
(ßt +
4-
) +
...
... +
[<
2a0
-v)
[(v
-ot)
1 +
ßl]
+ 2v
(G0
-v)
{ot
-v)
+
(a0
-v)
t J r
"'
r[(v
-ot
)1 +
ßl]
. r[
(v -
ot)1
+ ßl
]
4> =
tan
-1 p_
_ t
an
-1 ~ +
ta
n-1
(v
-ot
) _
tan
-1 (
-P
-) +
ta
n-1
(-P
-)
ot P
P
V
-ot
G 0-o
t
(Go
-.5J
e--.!
1 e-
<~1 si
n <P
t +
4-l j (
<ot
-G
o)•
+ ~
(.5 -
v)1 [
(.5 -
ot)1
+ ßl
] +
ß[{o
t -
v)•
+ ßl
] {o
t -
.5)1
+ pa
+
...
... +
[<
" -
G 0)(
(v -
ot)1
+ ß
l] -
2(at
-v)
(.5 -
v){G
0 -
v) +
(a
, -
v)t
J r.,
(v
-.5)
1 [ß
l +
(v -
ot)1
) (.5
-v)
[(v
-ot
)1 +
pa
4> =
ta
n-1
(-P
-) _
2
tan
(-P
) _
tan
-1 (-P
) ~-ot
v-o
t .5
-ot
[<G
o -
ß)t +
ot
+ p
-2a
0 ]
rfJI
+ [<
Go
-ot
)t +
ot +
P -
2a•J
e-<~l
{ot
-PJ
• {o
t -
Pl'
<P
-ot
)• <P
-ot
l'
~ +
[J
P-
G0)t
3a0
ß -
G0ot
-2~
-fJI
[
(ot
-G0
)t 3
a0ot
-Goß
-2o
t1 e
-<~l
ot•ß
i {o
t -
PJ•
+ ßl
<ot
-P
l'
e +
ot(ß
-ot
l• +
ot•<P
-ot
)•
[1 +
GJ
_ 2
a0(o
t +
Pl]
+
[ (G
0 -
ot)t
+ 2a
0(ß
-2o
t) +
ot{3
ot -
Pl]
e-«
' at1
ßl
ot1ß
l ot1
{ß -
ot)1
ot1
(ß -
ot)1
+ ~Go -
ß)t
+ 2a
0(o
t-2ß
) +
ß(3ß
-ot
)J r
fJI
(ot
-P
l'
ßl<a
t -
Pl'
w
0 10
~ ~ .... .... ....
14
2
8 +
a0
(a0
-v
)e-P
I [<<
X -
2a0)
(v-
at)
+ (v
{J -
at•)
-a
0({J
-at
) (a
0 -
at)t
J
1 (8
+ v
)(8
+ at
)1(8
+
{J)1
+
+ ~:--<~+
...
(v -
at)1(v
-{J
)1
(<X -
v)1
(at-
{J)1
(v
-at
}(at
-{1
)1
+ [<P
-2a
0)(v
-{J
) +
(vat
-
{11
) -
a0(a
t -
{J)
+ (a
0 -
{J)t
J ~:-/fl
({J
-v
)1(a
t -
{J)1
(v
-
{J)(
at -
{J)1
14
3
8 +
a0
[ (a
0-
at)t
2a
t(a 0
-<X)
<X
+ fJ
-2
a 0 J -
ou
V(a
01 +
v')
Bin
(vt +
~)
(81
+ v'
)(B
+ a
t)1(8
+
{J)1
(<X
-{J
)1(a
t• +
v1
) +
({J -
at)1
(at1
+
v')
1 +
({J-
at)1
(at1
+ v
1)
E: +
v({J
" +
v")(
at1
+ v')
[ (a
0 -
{J)t
2{J(
a 0
-{J
) <X
+ {J
-2a
0 J
1 +
...
+ (<X
-{J
)'({J
" +
".,
+ (<X
-{J
)"({
J" +
v")
' +
(<X -
{J)'(
{J"
+ v')
cfl
~ =
2 ta
n-1 ~
+ 2
tan-
1!!.
+ t
an-1
.!.
v v
a•
14
4
8 +
a 0
e-«
1
[(8
+ <X
)1 +
{1"]
1 [(
a 0
-<X
+ {J
"t) B
in {
Jt +
(<X
-a
0){J
t co
s {J
t] 2{
1"
~ ::r
14
5
8 +
a0
(a0 +
at1 t
) B
in r
d -
a 0rd
COB
rd
(81 +
at1
)1
2at1
~ - -.... 1
46
8
+ a
0 ~ +
( 1
-a"
t) B
in r
d _
(2a 0
+
at1 t
) CO
B rd
8
(81
+ at
1)1
at
' 2a
t1
2«
'
14
7
8 +
a0
!_+
a",_
V{4
at1
+ 9a
01)s
in(a
tt +
<{> 1
) +
(Va 0
1 +
at')
tco
B(a
tt +
</> 1
)
81(8
1 +
at1
)1
at'
2at5
2a
t'
." -
tan
-1 2
at
at•
"'· =
tan
-1_
1
-3a
,' a
•
14
8
8 +
a0
(a.
-{J)
E: ~~
-_
t_ Je
·· + a
t')
Bin
(at
t +
"' ) +
V {
(at1
-{J
a.)
' +
4{J"
a.1 }
COB
(att
+ ."
) (8
+ {1
)(81
+ a
t1)1
( a
t1 +
{J")
l 2a
t1
at1
+ {J1
1
2at1
( at1
+
{J")
I
"'1 =
ta
n-1
~ +
tan
-1 !!.,
."
= ta
n-1
(at
• -{J
a.) -
2 ta
n-1
~
a0
at
1 2a
ta0
{J
w 8
No.j
F(8
) f(
t)
w ~
149
8 +
a 0
a0
(ß -
a0)e
-ll'
t j (
'x' +
a01
)
8(8
+ ß
)(8
1 +
1X1
)1
ßiX'
+ ß
(IXI +
ß')
l +
21XI
lX
I +
ß'
COB
(Qtt +
</>1)
v' {1X
1(3
a0 +
ß)1
+
4(ß
a 0
-1X
1)1
} co
s (a
tt +
</>1
) -
21X'
(IX1 +
ß1
)
""1
= ta
n-1
~ +
tan
-1
~.
a0
IX
</>
= tan
-1
1X(3
ao +
ß)
_ 2
tan
-1
~
1 2(
ßa0
-1X
1)
ß
15
0
8 +
a 0
1 -
a 0fß
+ a
J
(a0
-ß)
s-11
1 t
j (IX
1 +
a0')
•
81(8
+ ß
)(81
+ 1
X1)1
IX
'ß
+ ß'
(1X
1 +
pt)1
+
21X'
1X
1 +
ß•
sm (a
tt +
</>1
) +
· · ·
..
. +
y{4
1X
1(2
a0 +
ß) +
9(a
0ß -
1X1
)1}
• (a
tt +
</>
)
41XI
( IX• +
pt)
sm
•
• ~
"'1
=
ta.n
-1 ~ +
tan
-1
~.
</>
= ta
n-1
[2
1X(2
ao +
ß)J
-
2 ta
n-1
~
a0
IX
1 3(
a 0ß
-1X
1)
ß
::r
§ "" - - -15
1 8
+ a 0
[
1 +
e• ; IX
) t]
e-1
(8
+ 1X
)1
152
8 +
a 0
~ +
[<IX
-a
0)t
1 _
2a0
_ a0
t] ~-
· 8(
8 +
1X)1
~
2 ~
IX
15
3
8 +
a 0
1 +
aJ
3a 0
[
1 (a
0 -
1X)t1
(2
a 0
-IX
)t]
-•
81(8
+ 1X
)1
-1X
-.-
--;x
< +
~ +
21
X 1
+
1X1
e
15
4
8 +
a 0
(a0
-ß)
s-11
1 +
[ a
0 -
ß +
(ß -
a0)t
+ (a
0 -
1X)t
1 e-
«'
(8 +
ß)(
8 +
1X)1
(IX
-ß
)•
<ß -
1X)1
(IX
-
ß)•
2(
ß -
IX)
155
81 +
a18
+ a
0 a
11X
-a
0 +
a 0att
+ (1X
1 -
1Xa 1
+
a 0)e
-«1
81(8
+ IX
) IX
1 IX
1
156
a' +
a1a
+ a
0
I 8(
8+ <
X)(
8 +
-ß)
-
157
a• +
a1s
+ a
0
s'(a
+ oc
)(s +
ß)
158
a2 +
a1s
+ a
0
(a +
oc)(
s +
ß)(
s +
v)
--
--
159
s• +
a1s
+ a
0
a(s +
oc)(s
+ ß
)(a
+ v
)
160
s• +
a1s
+ a
0
s'(s
+ oc
)(a +
ß)(
s +
v)
161
a• +
a1s
+ a
0
a(s'
+ oc
1 )
162
a• +
a1a
+ a
0
a1(a
1 +
oc1 )
163
a1 +
a1a
+ a
0 (s
+ ß
)(a1
+ a;
2 )
164
a• +
a1a
+ a
0
•(• +
ß)(
a' +
oc1 )
a 0
(a1
oc -
a 0
-oc
')E-1
1 ' (ß
' -
a,ß
+ a
0)e
-P•
-+
+
oc
ß oc
(ß -
oc)
ß(ß
-oc
)
a1 +
a0t
_ a
0 (oc
+ ß
) +
__ 1
_ [
(1
+ ~ _
_ ~) E~
' _
(1 +
~ _
~) d'
] oc
ß oc
•ß•
ß -
oc oc
·' oc
ß•
ß
(oc1
-a 1
oc +
a0)
e-"'
+ (v
' -
a 1v
+ a
0)t
-"' +
(ß•
-a 1
ß +
a0)t
-P'
(v -
a.)(
ß -
oc)
(oc
-v)
(ß -
v)
(a.
-ß)
(v -
ß)
a 0
(a.1
-a
1oc
+ a
0)E
-att
(ß
' -
a 1ß
+ a
0)e
-P'
(v'
-a
1v +
a0)c
"'
-+
------+
+
a.
ßv
oc(o
c -
ß)(v
-a.)
ß(
a. -
ß)(ß
-v)
v(
oc -
v)(v
-ß)
a1 +
aJ
a 0(a
.ß +
ßv
+ a.
v)
(oc2
-a
1oc
+ a
0)e
-"'
---
+
a.ßv
a.
•ß•v
• a.
'(oc
-ß)
(oc
-v)
(ß
1 -
a,ß
+ ao
)e-P
' (v
1 -
a 1v
+ a
0)r
01
+
ß'(ß
-a.
)(ß
-v)
+
v'
(v -
a.)(v
-ß)
~ +
J{(~r +
(~ _
In cos
(<XI
+ q,)
. q,
= -
tan
-1 (~)
az-
ao
a, +
a.e
ain
(oct
+ r/>
) )(
• (
a·n
_
a._
•_ -
a.•
a1
+
oc -
oc •
+ =
tan
-1 ---
( a,
a.
) a 0
-
cx'
(ß' -
a1ß
+ a 0
)E
P•
ain
(<XI
+ 4>
) J {(
a.1 -
a0)
2 +
a 11 a
.'}
a.'
+ß
' +
<X
a
.•+
ß'
• q,
= ta
n-1
{~)
-ta
n-1
(a• a~a.
a.")
---
~ _
(ß
' -
a 1ß
+ a
0)e-
P'
_ co
a (a
.t +
rf>) )
{(o
c1 -
a0)
1 +
a11
a.1 }
a.'ß
ß(
a.' +
ß')
oc
• a:
• +
ß'
'
"'= tan
-1 (~
_ ta
n-1
e· :.a. a
.') ·--~-·-·-·--·-··-·-
~ § (JQ
- - - w
0 Vl
No.,
F(•
)
165
8• +
a 18 +
a 0 8'
(8 +
ß)(
•• +
a.•)
166
81 +
a 18 +
a 0
(8 +
ß)(i
t + 1>
)(81 +
cx1 )
167
81 +
a 18
+ a 0
8(
8 +
ß)(8
+ tt)
(81 +
a.1
)
168
111 +
a 111 +
a 0
s•(s
+ ß)
(a +
v)(a
1 +
cx1 )
169
a• +
a 1a +
a 0
(a1 +
cx1 )
(a1 +
ß')
--
J(t
)
_1_
[ _ ~ +
J +
(fJ'
-a1
ß +
a")r
l' +
coe
(cxl
+ <(>
) J{
(a.1
-a0
)1 +
Gt1
a.1 }
a.•p
a1
ß
ae
t ß
'(a.
' + ß
')
a.&
a.• +
ß'
.,. =
t.
n-1
(ID +
t.n
-1 (~)
a. a
,-a
.•
(fJ' -
a 1ß
+ a 0
)e 1
11
("S
-a
11> +
a 0)e
-1'1
B
in (
eil +
<(>) J {(t
:r.1 -
a.)'
+ a 1
1 a.'}
(1>
-ß)
(a.•
+ ß
')
+ (
ß -
tt)(c
x1 +
tt1)
+
a. (a
.• +
pt)(
a.1 +
"S)
•
+ =
t.n
-1 (~)-
t.n
-1 (!
)-t.
n-1
(!)
a 0
-cx1
"
P
a.
(ß-
a1 +
7]) e
-ßt
(~~- a
1 +
~) e-
~'' si
n (
cxl-
<(>) J{
(c
x'-
a")'
+ a,
•a.•
} a.
•p" +
(ß
-tt)
(a.•
+ {J
') +
(tt
-ß)
(cx•
+ "S
) +
a.•
a.
'(/1 +
•)' +
(a.•
-fl•
)'
<(> =
te
.n-1
(~)
+ ta
n-1
(;)
+ ta
n-1
("•
a~a. a
.i
a1 +
a,
(t -~ -
~) +
(fJ'
-a 1
ß +
a 0)e
-/1'
+ ("
0 -
a 11>
+ a,
)e_,
.' a.
•ßv
ß'(
v -
ß)(a
.' +
p•)
tt"(ß
-tt)
(cx•
+ "S
)
+ co
a (c
xt +
<(>) J{
(a
.1
-a
0 )1 +
a11
or;1
} cx
• rx.
"({J +
P)'
+ (a
.• -
p,.)•
+ =
tan
-1 (~
) +
tan
-1 (
a. a~
cx cx•
) -
tan
-1 ~)
CO
S (
tXt +
r/>1
) J{a
t + (
ao
-czt
n + CO
S (
ßt +
t/>,)
J{a
1 +
("•
-p•n
ß"
-cx
• 1
cx cx
• -
p• 1
fJ .,.
, =
ta
n-1
e~·
-a·).
a 1a.
+• =
tan
-1 (fl
' a~{J a
•)
---
w
0 0\ ~ ~ ""
170
s• +
a1.«
+ a
0
s[(s
+ oc
)2 +
ß']
171
s2 +
a1s
+ a
0
s2[(
s +
oc)2
+ ß
2 J
172
s2 +
a1s
+ a
0
(s +
v)[
(s +
oc)
2 +
ß']
173
s• +
a1s
+ a
0 s(
s +
v)[(
s +
oc)•
+ ß'
]
s2 +
a1a
+ a 0
17
4 s•
(s +
v)[(
s +
oc)'
+ ß
']
~ _
e-'"
sin
(ßt
+ </>
) J(
ß"(
2o
c -
a1)
2 +
(.oc2
-ß'
+ a
0 -
a1o
c)'}
oc• +
ß'
ß rx
• +
ß'
'
q, =
ta
n-1
(~)
+ ta
n-1
(
~(a1
-2 o
c) )
rx ~2
-
fJZ +
ao
-al
rx
a1 +
a0t
2
a0oc
s--«
' sin
(ßt
+ </>
) 1
(ß'(
2 ) 2
(
2 ß'
2
tX2 +
ß2
-(;
2 +
ß')
' +
ß(tx'-t{J~ \-
oc -
a1
+ o
c -
+ a
0 --
a1 o
c) ),
q, =
2 ta
n-1
(~) -
tan
-1 (
ß(
2oc
-a
.)
) oc
oc2
-ß
2 +
a0
-a
1oc
(v2
-a
1v +
a0)
c••
+ t
:_~"'
sin
(ßt
+ </>
) J(
ß'(
2o
c -
a,)2
+ (
oc2
-ß'
+ a
0 -
a,oc
)'}
(o
c-v
)'+
ß2
ß
(oc-v
)2+f
J"
' q,
=
tan
-1 [
(a
1 -
2 oc)
ß J -
tan
-1 (-
ß-)
oc2
-ß'
+ a
0 -
a1o
c v
-ot
ao
(alv
-flo
-
v2)E
-vt
---
+ -------
(ot2
+ ß
')v
v[(o
c -
1•)'
+ fl'
J ,_
c"' s
m (
ßt +
</>)
J[ß2
(2o
c-
a1)
2 +
(oc2
-ß2
-a
1oc +
a0)
]
; ß
(o
c• +
ß')
{(ot
-v
)' +
ß')
'
</> =
ta
n-1
(~)
+ t
an
-1 (
-ß
-) -
-ta
n-t
(
ß(2o
c -
a1)
) ,I
X
rx -
l'
cx.2
-ß2
+
a0
-a
1rx
( I
a1 )
a
t--+
-0
v a
0 2o
ca0
(v2
-a
1v +
a0)c
"'
-----
-----
+-
v(oc
2 +
ß')
v(
oc2
+ ß2 )
2 v2
[(oc
-v)
2 +
ß2 ]
+ c
"' si
n (
ßt +
</>)
J(ß
'(2
oc
-a
1)1 +
(oc2
-ß'
-a
1oc
+ a
0)2 }
ß(oc
2 +
ß')
(o
c -
v)2
+ ß'
"' =
tan
-1 [
2
ß~~l
-2 oc
) J -
tan
-1 (
-ß
-) +
2 t
an
-1 m
!X
--
-a
1 oc
-f-a
0 v
-!X
oc
> g. ~ ~ .....
.....
.....
w
0 --..1
No.
I F
(8)
175
81 +
a
18 +
a 0
(8
2 +
v1 )[(
8 +
1X)1
+
ß2 ]
-
176
81 +
a
18 +
a 0
8(
81 +
v0 )
[(8
+ 1X
)1 +
ß"
]
177
s• +
a
18 +
a
0 8(
8 +
ct)1
178
81 +
a18
+
a 0
s1(8
+
1X)1
-
179
s• +
a
1s +
a
0 (a
+
ß)(8
+
1X)1
--------
--
--
J(t)
e-«' si
n (
ßt +
r/>
1) J
tx•
-ß•
+
a 0 -
a11
X)1
+ (2
1X
-a
1)1 ß
*}
ß (1X
1 +
ß2
-v2
)1 +
(2
1Xv)
1
sin
(vt
+
r/>1)
j{
(v
1 -
a0 )
1 +
a11
v1
} +
V
(1X1
+ ß
1 -
v,l)1
+
(21X
V)1
'
.p -
ta.n
-1 (
21
Xß
) +
ta.n
-1 [
ß(
a1 -
21X)
J
~ -
IX1
+ v•
-ß"
IX
1 -
ß" +
a 0
-a
11X
'
.p _
_ 1 (
a
1 v )
_ 1
(
21Xv
)
2 -
ta.n
---.
-ta
.n
I ß
' I
' a 0
-v
IX+
-v
a
0 e-a
.• sin
(ßt
+
r/>1)
j(<
IX'
-ß•
+
a0
-a
11X)
1 +
ß*(2
1X -
a1)
'}
v2(1X
2 +
ß')
-ß
(1X1
+ ß
1 )[(1
X1
+ ß"
-v1
)1 +
(2
;tv)
1 ]
_ si
n (
vt +
r/>
1) j
{
(v1
-a
0 )1
+ a
12v"
}•
v•
(1X1
+ ß•
-v"
)1 +
(2
1Xv)
1
.p =
ta
.n-1
(~
_ ta
.n-1
(IX
" -
ß• +
v')
+
ta.n
-1 (I
X1
-ß*
+
a0 -
a1 1
X).
1 IX
21X
ß ß(
21X
-a
1)
tf. =
ta
.n-1
(IX
" +
ß* -
v") +
ta
n-1
(~)
• 21
Xv
~ -
v"
~ +
[<
a11X
-a
0 -
1X2 )
t +
IX2
-a
0 ]
e-«'
IX1
IX IX
1
---
--
+ -
--
a1
+ (1X
1 +
a0
-a
11X)
s-
«
a1
+ a0
t 2
a0
1 [2
a0
J 1
IX1
IX3
IX1
IX
(ß1
+ a
0 -
a 1ß)
e-fJ
' +
[(1X
1 -
21X
ß -
a 0 +
a 1
ß) +
(1X
1 +
a0
-a
11X
)t] s
-«'
(1X
-ß)"
(1
X-ß
)"
ß-IX
--
w
0 00
~ ~ "" .... .... ....
"-l :r
0 0:
0 0 "
180
181
182
183
184
a• +
a1a
+ a 0
8(
8 +
ß)(a
+ a:
)•
a• +
a1a
+ a 0
a1(a
+ ß)
(a +
a;)1
a• +
a1a
+ a 0
(8 +
ß)(8
+ v)
(a +
a:)1
a• +
a1a
+ a 0
(a1 +
ßi)(
a +
a:)•
a• +
a 1a +
a 0
a(a'
+ ßl
)(8
+
a:)1
~ +
(a1 ß
-a0
-ßi
)e-f
l' +
e-
«'
[<a:
• _
a a;
+ a
)t +
a:1(ß
-a
1 ) +
a0(2
a; -
ß]
a:•p
ß(
a; -
ß)•
a.
(a;
-ß)
1
0 a;
(a;
-ß
)
~ c~ -~ -~+ e]
a:
•p
ao
ß a;
+
e-<X
I [<
a:'
_ a
a; +
a )t
+ 2a
oß -
a;(3
ao +
a1ßl
+ a:
'(2a1
-a:
)J +
0 0
0
a:'(
ß -
a;)
1 0
a:(ß
-a;
)
+ (p
t -
a 1ß
+ a 0
)n''
{J''(
ß -
a:)•
(v;p
-_
a;~(; ~0
~~~" +
(ß:v
~ ap
~:-~·
~~~~~·
+
e-
a.•
[<a.
' _
a a;
+ a
)t +
(v -
ß)(a
:1
-a
0) +
vß(
a1 -
2a:)
-a;
(a1a
; -
2a 0
)]
(a:
-v)
(a;
-ß
) 1
o (a
; v)
(a:
-ß
) ~ ~
sin
(ßt
+ </>
) V
{(ß"
-a
)' +
a •p•
} ß(
a:' +
ß•)
0
1 + ~ [
<a:•
-a
a; +
a )t
+ 2a
:(ß"
-ao
)" +
a1((J
• -
a:')
] a:
• +
p•
1 0
(a:•
+ ß"
)
(Jq
- - -"'=
tan
-1 (~) -
2 ta
.n-1
f!. ao
-p•
a;
ao
sin
(ßt
+ t/>
)v {ß
2 a1•
+ (ß
• -
ao)2
}
a:•p
s -
ß'(a
:• +
ß•)
_ [a
:'[(a
;2
-ß
1) +
a 0(a
;1 +
ß") +
2(a
0 -
a1a
;)J +
(a;1
-
a1a
; +
a 0)t
] 0
-«C
a:•(
a:• +
ß')
' a:
(a:•
+ ß"
) e
"'= tan
-1 (~
)0 -ta
n-1
(~
+ ta
n-1
(~)
ß a;
a.-
ß•
w ~
No.
I 18
5
186
187
188
189
F(s
)
s• +
a1s
+ a
0
s•(s
' +
ß•)
(s +
ot)2
s• +
a1s +
a0
(s +
ß)1
[(s
+ ot
)1 +
v2
)
s• +
a1s
+ a
0
(s +
ot)1
(s +
ß)1
s• +
a1s
+ a
0
s(s
+ o
t)'(
s +
ß)'
s• +
a1s
+ a
0
s2 (
s +
ot)1
(s +
ß)'
j(t)
a 0t +
a1
-2a
0fot
[(
ot1
+
ß')
(2a 0
-
a1ot
) +
2ot1
(ot1
-
a1ot
+ a
0) +
(ot3
-
a,a:
+ a
0)~ e
-a•
a:•p
• +
ot
3(o
t1 +
ß1
)1
ot1(o
t1 +
ß">
-J V
(ß1
-a
0)1
+
a,•
ß• s
in (
ßt +
<f>)
+
ß'( o
t• +
ß')
"' =
ta
.n-1
(~) +
2 t
a.n-
1 (~
) a
.-ß
•
ß
ca.•
sin
(vt
+ <
f>h/
{v1(a
1 -
2ot)
1 +
(ot1
-v•-
a1ot
+ a
0)1
}
v[v•
+ (o
t -
ß)1
]
+ [
(ß•
-a 1
ß +
a0)t
+ (a
1 -
2ß)[
v1 +
(ot
-ß
)1]
-2(
ot -
ß)(ß
• -
a,ß
+ a
o)J
1,-{
J•
". +
(o
t-ß)
• "'
=
ta.n
-1 [
v(
a, -
2ot)
_l
-2
ta.n
-1 r _
____.!.
_] ot
1 -
v1
-a
1ot
+ a;
;J
Lß -
ot
[a
1(ot
+ ß
) -
2(ot
ß +
a0
) +
(ot1
-
a,a:
+ a
0)t]
e-«'
(ß
-ot
)•
(ot
-ß>
'
ao
a.'ß
' +
+ [<
ß" -
a1ß
+ a
0)t
_
a1(o
t +
ß)
-2(
a.ß +
a0)] e-
fJ'
(ot
-ß)
• (ß
-a.
)•
[(a
,a.
-a 0
-
a.')t
+ 2o
t{ot
1 -
a 1a.
+ a
0) +
(a.1
-
a0)(
ß -
a.)J
e-«
' a.(
a. -
ßl'
a.•(
ß -
ot)1
+ [
(a1ß
-a
0 -
ß")t
+ 2
ß(ß
" -
a1ß
+ a
0) +
(ß1
-ao
)(a.
-ß
)J t
rfl'
ß<
ß -
a.)•
ß•
(ot
-ß)
•
a1 +
a0t
_
2a 0
{ot +
ß) +
[(o
t1
-a
1ot
+ a
0)t
+ 2
ot(a
1ot-
a 0-
ot1
) +
(2a 0
-
a 1a:
)(ß -
ot)J
e-«
' ot
•ß•
a.•ß
• a.
•(a.
-ß)
• a.
'(ß -
ot)•
+ [
<ß'
-a
,ß +
a0)t
+ 2
ß(a 1
ß -
a0
-ß"
) +
(2a0
-
a1ß
){ot
-ß
)J e
-fJ'
ß'(
a. -
ß)•
ß"
(a.
-ß>
•
w -0 f .... ::::
I 8
1 +
a1s
+ a 0
e-
«•
190
1 [(
s +
o:)'
+ ß
']'
{[(<
X1 +
ß')
+ (
a 0
-a 1
o:) +
(a1
-2o
:)ß'
t] s
in ß
t +
[(a 1
o: -
a 0)
-(o
:' -
ß')
] ßt
cos
ßt}
2{
P --~
191
a• +
a1s +
a 0
2~3 {
[a0 +
<X1(1
+ a
1t)]
sin
o:t
+ (<X
2 --
a 0)or
;t co
s o:
t} (a
' +
or;')
' --
192
a• +
a1a
+ a 0
ao
t s
in (<
Xt +
~1h/ {(
o:1
-a 0
)1 +
a12
o:1 }
-
CO
B (<
Xt -
j-</J
2)y
'(a 1
2<X
2 -j
-4
a 01
)
a(s2
+
o:1
)1
;t-
2o:•
2o:'
. ~ =tan-1(~)
1 a
o-c
x•"
.P.
= t
an-1
(~:)
--
193
81 +
a1a
+ a 0
~+ a
0t +
t co
s (o
:t +
</J1)v
' {(o
:1
-a 0
)1 +
a 12 o
:2}
_ si
n (<
Xt +
.p,)
·l/(o
:2
-3
a 0)2
+
4a 1
1or;
1
8'(
•' +
o:')"
~
~
~
. ~
= ta
n-1 (~).
~ =
tan-
1 (
2a,o
r; )
1 ao
-cx
2 •
3ao
-or;
• I -
194
82 +
a1s +
a 0
[2 +
(a1
-2o
r;)t +
(o:' +
a,;
+ ao
)t~
6-«
t (8
+ o
:)•
- -19
5 a1
+
a18
+
a 0
~ _
e-
«'
[2a
0 +
(a0 +
<X2)t
+ (<X
1 -
a1
o: +
a0)t~
8(8
+ or
;)1
o:•
<X
o:•
<X
2
196
a• +
a18
+
a0
a 1or;
-3
a 0 +
a 0t +
[3
a0
-a
1<X
+
(2a 0
-
a1o
r;)t +
(o:•
-a
1or; +
a 0)t
'] e
-«t
81(8
+ <X
)1
o:•
o:•
o:'
o:•
2o:'
197
a• +
a.,s
• +
a1s
+ a 0
a
1 +
a0 t
_
a0(<
X +
ß) +
_I_
[ (<
X +
~ _
a _ ~) e-
<xt
_ (ß
+ ~ _
a _ ~) e-~
•J
81 (
8 +
o:)(
a +
ß)
o:ß
o:•ß
• <X
-
ß <X
2
"<X
2 p
2 p•
198
83 +
a.,s
• +
a1a
+ a
0 ~ +
(<X1
-a,
tX +
a 1
-a 0
/<X
)e-«
1 +
(ß'
-a
,ß +
a 1
-a
0/ß
)e-f
J1 +
(v'
-a
,v +
a 1
-a 0
fv)e
_"'
8(8
-j
-<X
){B
-j
-ß)
{B -
j-V
) rx
ßv
(v
-rx
)(ß
-rx}
(v
-ß)
(rx
-ß)
(ß
-v
)(rx
-v)
w
---------------------
--------
--
-----
---------------------~
312
.. I
J
t! + ~ ... 1:1 IS
++ }t + '1.
. .,_ + "as '!.+ .,,. +;.~ ., + +~ '1.
Anhang Ili
.. I
J ....
+
1.::""1 ~ I
} + IS ,; I
!. +
llas + ,; I
~ tl"as +
ll~ I
;l"as
t! + .. .. tfti' ++ .. .. "_ .,,. + '1.
I ~
20
5
s1 +
Gsß1
+
a18
+ a 0
8(8
+ ß
)(8
+ 1X
)1
20
6
s• +
Gsß2
+
a18
+ a 0
s2 (
s +
ß)(
s +
1X)2
20
7
s• +
a2s
2 +
a1s
+ a 0
(s1 +
ß2 )
(8 +
1X)1
20
8
s1 +
asß
1 +
a 1
8 +
a0
s(8
2 +
ß1 )
(8 +
1X)1
20
9
88 +
a.a•
+ a
18 +
a
0
(s +
1X)1
(s +
ß)1
~ +
(ß•
-a,
ß• +
a,ß
-a
0)e
-111
IX1 ß
ß(
IX
-ß>
' +
[IX
' -
2ß1X
1 +
IX1(a
1ß
-a
1) +
a0(
21X
-ß
) +
(1X1
-
G11
X1 +
G11
X -
ae)'] s
-«l
1X2 (
1X -
ß>'
IX(ß
-IX
)
~ c~ -~
-.!.
+ e]
+ (a
. -
a,ß
+ a
,ß•
-ß
')e-/
11
IX'ß
ao
IX
ß ß
'( IX
-ß)
" +
[IX
"(ß
-a
1) +
a0(ß
-IX
) -
(a11
X -
a0)(
ß -
21X)
+ (a
0 -
a11
X +
a11
X1
-1X1)~ e
-«1
1X1 (
IX -
ß)1
1X
1(ß
-IX
)
sin
(ß
t +
t/>
h/{
ß'(
a1
-ß
1 )1 +
(a0
-a
2ß')
1 }
ß(IX
• +
ß">
+ ~ [<
_ +
1
_ ')
t +
(I
X'+
3ß
'IX1
-a
1 1X1
+
aiß'
+
2a0 1
X-
2a1
ctß
')J
IX• +
ß"
a0
a1 1
X a
1 1X
IX
(IX" +
ß">
t/> =
ta
n-•
[i•
ß'
-ao
J
_ t.
an
-1 (~ +
t.
an
-1 (~)
(a1
-ß
')
IX
ß
a0
+
e--«
1 [<
1 1
+
)t +
21X
'{a1
-ß
') -
IX1 (
2a0
+
a11
X1 -a
1ß')
-a
0(c
t1 +
ß
')]
IX•ß
• IX
( IX• +
ß")
IX
-a,I
X a
,IX
-ao
IX
( IX•
+ ß'
)
sin
(ßt +
t/
>h
/ {ß
'(ß
'-a
1)1
+
(a1ß
' -a
.)1 }
+
ß'(
IXI +
ß")
•
t/> =
ta
n-1
[ß
(ß'
-a
,)J
_ t.
an
-1 (~ +
t.
an
-1 ~)
a 1ß
'-a
0 IX
[(a
0 -
a11
X +
a21
X1
-ct
1 )t
+
1X1 (
3ß
-IX
) -
2a
11Xß
+
a1(
1X +
ß>
-2
a 0]
e-«
l (IX
-ß>
' <ß
-IX
)'
+ [
(a0
-a 1
ß +
a 1
ß'
-ß'
)t +
ß
'(31X
-ß
) -
2a
11Xß
+ a
1(1X
+
ß)
-2
ae
J rl'
(IX
-ß
)'
(IX -
ß>'
f (JQ - - - w -w
No.
I F
(s)
j(t)
210
2ll
212
213
214
215
216
-~--
s-3 +
a2s
2 +
a1s
+
a0
a0
[(cx
3 -
a2
cx2 +
a1
oc ::_
a0)t
cx
'[a2
(cx +
ß)
-2(
a 1 +
cxß)
J +
a0(3
cx
-ß
)J
-at
--------
-+
+
f
8(8
+
cx)2
(s +
ß)2
cx2 ß
2 oc
(ß -
cx)'
cx'(ß
-cx
)3
+ [
(ß3
-a
,ß• +
a1ß
-a
0)t
+ ß
'[a,
(oc
+ ß
) -
2(a
1 +
cxß)
J +
a0(3
ß -
cx)J
f-{
J•
ß(cx
-ß
)'
ß'(o
c -
ß)'
s• +
a s
2 +
a s
+ a
f-
cxt
sin
ßt
[(s ~ c
x)' +'
ß'J
' 0 ~-
[(a
, -
cx)(c
x2 +
ß')
-cx
(a1 +
2ß
2) +
(3cx
2 -
ß2
-2
a2cx
+ a
1)ß
2 t +
a0
)
f-C
ll c
os ß
t +
-2-
-;p-[
2ß3
-
(a0
-a
1cx
-1-a
2[cx2
-
ß2J +
cx[3
ß2
-oc
'J)ß
t]
---------
83
+
a2s
2 +
a1s
+ a
0 a
1 +
a0t
sin
(cx
t +
</>1
) '{
4 2
2 +
( 2
3 )'
} --s'(
s' +
cx
')-,
----~--
2cx•
V
a
, cx
<X a
,-
ao
+ t
cos
(cxt
+ </>
2)
• 1
{ '(
2
_ )' +
( 2
_ )'
}
2cx•
v
cx cx
a 1
cx a2
ao
4>1 =
ta
n-1
(3
2cxa
1 2
_\,
q,, ~
= ta
n-1
(-c
x(_a
,_-_2
cx_2
)) a 0
--
cx a
J
a 0
-cx
a2
!----------------1
----------
---------------------
83
+
a2s
2 +
a1s
+
a0
~ +
[(c
x3
-a
2cx2
+
a1c
x -
a 0)t
2 _
(a0
-a
2cx2
+
2cx3
)t _
2(
a 0
-cx
')] f
-at
s(s +
cx)3
cx3
2c
x cx
2 cx
3
---------------
-----
83
+
a2s
2 -1-
a,s
+ a
0 a
1 +
a0t
3a 0
[(
6a
0 -
2cxa
1) +
(2a
0 -
a1
cx +
cx3)t
(a
0 -
a,cx
+ a
2cx
2 -
cx3)t
2 ]
-cx•
s
2(s
+ cx
) 3
--cx
.--
7 +
cx
4 ----cx
-.---
+
2cx2
E
1---------
t.ß
'-e«
' v
(s-
cx
)-vi(
s-
ß)
·2y
(rrt
3)
w l
f--
+ f
2(s2
+
w')
Js
in w
tl
w -~ ~ ~ - - -
.. 21
7 e
' - 8
.. 21
8 e
' y'8
.. 21
9 e'
y'8
220
tn(~)
B+P
221
lnB
- 8
222
lnB
81 +
1
223
8ln
8
81 +
1
224
ln (
~
a)
225
tan-
1 m
22
6 "-
...
J0(2
y'(
at))
cos
2y
'(at
) y'
(?rt
)
cosh
2y
'(at
) v'
( ?rt)
e-fJ
' -
e-«'
t
-0.5
77
2 -
ln t
Si(
t) c
os t
-O
i(t)
sin
t
-S
i(t)
Bin
t -
Oi(
t) C
OB t
2(1
-co
sat)
t
sin
at
t
d(t
-a)
f - - - w -VI
No.
F
(•)
227
E_
..
- • 22
8 E
-.. ...
229
E_
_.
(8 +
IX)
230
E_
..
<• +
IX
)•
231
E-.
.
<• +
IX
)(•
+
ß)
232
1-
E-.
. --- •
233
(• +
a
)r ..
a.
•
23
. E
_ ..
8(8
+
IX
)(B +
ß
)
I /(
I)
U(t-
a)
(t -
a)U
(I -
a)
E--
«11-
•lU(t
-a
)
(t -
a)E
--«
U-.
lU(t
-a
)
[E--
«(1-
•l -
E-t
JU
-•]
ß _
IX
V(t
-a
)
U(t
) -
U(t
-a
)
(t + ~
-a)
U(t
-a
)
[ I
E--«
1 ·-
·)
E-t
iU-•
l J
--
-U
(t-
a) IX
ß 1X
(ß -
IX)
ß(IX
-ß)
w -0\ ~ ~ - - -
Sachwortverzeichnis
Ähnlichkeitssatz 103 aktive Tiefpaßfilter n-ter
Ordnung 221 Algebra I Anfangsbedingung 65, 153 Anfangsstrom in einer Spule
ISS Anfangswertsatz 89 angepaßte Tiefpaßfilter 200 Anregungsfunktion 67 aperiodische Funktion 39 -- Schwingung 43 Ausschnitte von Funktions-
teilen 97
Bandfilter, geometrische Symmetrie von 197
-, Konstruktion von 198 -, maximal flache 197 Bandsperre 200 Beschleunigungsfunktion 166 Beschleunigungsmesser 166 Beseitigen einer Polstelle 28,
75 Besselsche Polynome 213 Binomialkoeffizienten 151 Blindwiderstand 107 Borelsches Theorem I 00 Brückenoszillator, Wienscher
113 Butterworth-Filter 186 Bu tterworth- Funktionen,
Polstellenlage von I 9 I
Cauchy-Riemannsche Kriterien 20
Dämpfungsfaktor 47 Determinanten, Lösung mit
Hilfe von 72 Differentialionssatz t'ür die
Bildfunktion 93 differenzierende Schaltungen,
aktive 131 Diracsche Deltafunktion 168 direkte Laplace-Transformierte
5 I, 52 direktes Fourier-Integral 44 Durchlaßbereich 187
Eingangsimpedanz 108 Einheitssprung 45
Fourier-Transformierte des 47
Einheitsvektor 2 einseitige Fourier-Transfor-
mation 49 Endwertsatz 86 Erregerfunktion 67 Eulersche Formeln 4, 36, 37
Faltungsintegral97, 100 Faltungssatz 97, 100 Filter, die Legendreschen
Polynomen entsprechen 217 die Tschebryscheffschen Polynomen entsprechen 217 Grenzfrequenz eines 186 Ordnung des 186
Filtergrenze, Schärfe der 187 Flächenelement 18 Flußverkettung 156, 161 Fourier-Analyse eines Rechl-
eckimpulses 44 Integral 42 Integral, direktes 44 Integral, inverses 44 Koeffizienten 172 Reihe 35 -, komplexe Form der
36 Transformation, einseitige 49 -, komplexe 49 Transformierte des Einheitssprunges 47
direkte 39 -, - einseitige 49 -, inverse 39
Frequenznormierung 185 Frequenzspektrum 39, 40 -, reelles diskretes 42 Funktion, analytische 20
aperiodische 39 -, maximal flache 187, 188 -, nichtanalytische 20 -, periodische 33 Funktion von Y s 227
Gegen-EMK 159 Gegeninduktivität I 53, I 58 geographische Ebene 6 Gleichrichtung, Halbweg- 178 Grenzfrequenz 186 Grundfrequenz 33 Grundschwingung, Kreis-
frequenz der 38
Halbweg-Gieichrichtung 178 Harmonische als Verzerrung
119 harmonische Schwingungen
35 Herationsschaltungen 233 Hochpaßfilter 196
Integrationssatz für die Bildfunktion 95
Integrationsweg 2 0 -, geschlossene Schleife als
20 integrierende Schaltungen,
aktive 131 inverse Laplace-Transformierte
5 I, 52 inverses Fourier-Integral 44 imaginäre Größen 2 Impedanz 109 Impedanzpegel 183 lmpulsfolgen, Laplace-Trans-
formierte von 17 4 Induktivität eines L-C-Reso
nanzkreises 50 induzierte Rückwirkungs
spannung 159 Integral, reelles 18 Integration als Summierung
der Residuen 31 in der komplexen Ebene 18 ff. längs einer geschlossenen Kurve 21 um eine Polstelle 21
zwei oder mehr Polstellen 29
Kapazität eines Koaxialkabels 11
- - Plattenkondensators 13
Kathodenverstärker 121 Kern 78 Kettenleiter 1 51, 241 -, Formeln Itir die Koeffi-
zienten von 247 Kirchhoffsche Gesetze 108 Kirchhoffsches Gesetz 69 Knotengleichungen 154 komplexe Ebene 3, 4
Integration in der 18 ff.
318
Fourier-Transformation 49 Funktion 13 ff.
Definition der 5 -, Nullstelle einer 16 -, Pol einer 15 Variable 13 -, Funktion einer 13 ff. Zahl 3, 4, 5
algebraische Form einer 3 Exponentialform einer 4
Imaginärteil einer 5 Polarform einer 4
, Realteil einer 5 Konvergenzfaktor 48 Kopplungsimpedanz 108
Ladungsverstärker 135 -, Analyse des 137 Laplace-Integral 51, 52 Laplacesches Grundintegral
53 Laplace-Transformation 49
einer·Stammfunktion 63
, Symbol der 54 Transformierte 50, 52 - der Ableitung 61
Exponentialfunktion 54 Exponentialfunktion von imaginärem Argument 55 - von komplexen
Argument 57 Hyperbelfunktionen 57 trigonometrischen Ausdrücke 55
des verschobenen Einheitssprunges 164 direkte 51, 52 eines einzelnen Sägezahnimpulses 178 gepulster periodischer Funktionen 179 inverse 51, 52 von Impulsfolgen 174
Leitwert, transformierter 154
lineare Phase, Approximation einer 209
Linienintegral 32
Maschengleichungen 154 Mehrfachschleifenschaltun
gen, Netzwerkgleichungen für 108
Merkatorprojektion 5
negative Zahlen 1 Netzwerksynthese 188 Nichtlinearität 119 Normierung der Frequenz 185 - - Übungsfunktion 183 Nullstelle einer komplexen
Funktion 16
Oberschwingung 33 Operator j 2
parametrische Variable 89, 99
Partialbruchzerlegung 79 Partialbrüche 77 Pascalsches Dreieck 1 51,
241 periodische Funktion 33
Funktionen, gepulste 179 -, Satz für 178
Phasenschieberoszillator 116 Phasenwinkel 10 Pole, Gruppen von 15 - höherer Ordnung,
Residuen von 79 Pol einer komplexen Funk
tion 15 - Nullstellen-Diagramm
17 f. Polstelle außerhalb des
Integrationsweges 22 -, Beseitigen einer 28, 75 -, Integration um eine 21 Polstellen, radiale Kontrak
tion der 105 Potenzreihe 248
radialer Fluß im Koaxialkabel 13
Radiusvektor 21, 2 2 R-C-Aufspannetzwerke 126
Filter ftir die Hochfrequenzstromversorgung 223 Kathodenverstärker 121 Oszillator mit Drehkondensator 128
Sachwortverzeichnis
Tiefpaß, dreiteiliger nicht verjüngter 149 -, einteiliger 146 -, zweiteiliger nicht
verjüngter 14 7 Rechenverstärker 134 Rechteckimpuls 165
Fourier-Analyse eines 44 Gleichstromterm des 41 Grundschwingung des 41
, Oberschwingungen des 41 reelle Achse, Translation der
90 reelles Integral 18 reelle Variable 13 - -, Funktion einer 13 Relaisdämpfung 110 Residuen von Polen höherer
Ordnung 79 Residuum 24 -, Definition des 27 Rückkopplungsverstärker,
Analyse des 3-stufigen induktiven 143
Rückwirkungsspannung, induzierte 1 59
Sägezahnimpuls, LaplaceTransformierte eines einzelnen 178
Schaltungsmodelle , verlustfreie 259
Schaltungsparameter, Anfangsbedingungen fUr 153
Schwingung, allgemeine 175 -, aperiodische 43 Schwingungssystem,
mechanisches 166 S-E bene 6 ff. -, Translation in der 83 Selektive Variable 99 Siebschaltungen 131
Spannung, transformierte 154
Spannungsverstärker, induktiv rückgekoppelte 139
Spektrum, diskontinuier-liches 40
-, kontinuierliches 40 Sprungfunktion 45 -, verschobene 46 Spule, gegenseitig gekoppelte
159
Stammfunktion, LaplaceTransformation einer 63
Stoßcharakteristiken 16 5 Stoßspektrum-Computer 165
Sachwortverzeichnis
Strom, Definition des 65, 159
-, fiktiver 72 -, transformierter I 54 Summotionsindex 25 I
TabeHiermethoden 238 Tiefpaßfilter 186 -, angepaßte 200 Trägerwelle 179 Transformation eines
Gebietes II punktweise 7 zwischen z- und s-Ebene 7 ff.
transformierte Batterie I 56, 161
transformierter Leitwert 154
- Strom I 54 transformierte Spannung I 54 Translation der reellen
Achse 90 - in der komplexen
Ebene 83 transzendente Funktionen
77
Übergangsvorgang I 57 Übertragungsfunktion 67
, Normierung der 183 -, ungerade und gerade
Teile der unendliche Reihen in ge
schlossene Form 250 Unterdeterminante I 09
319
Variable, parametrische 89, 99
-, selektive 99 Vektor 2 -, Drehung eines 2 verlustfreie Schaltungs-
modelle 259 Verschiebungssatz 175
W-Ebene 6
Z-Ebene 5 ff. Zeitebene I 5 Zeitverschiebung 91 Zeitverzögerungsfilter,
maximal geebnete 205