Electromanetic Formula Sp2011
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Transcript of Electromanetic Formula Sp2011
Formula Sheet for Comprehensive Exam, Electromagnetic SystemsSp 2011
EE540 Microwave Devices and Systems
1. Maxwells EquationsDifferential Form Integral Form
t
= BE SBE dt
dLS
.. =t
+= DJH SDH d
tIdL
S
.. +=v= D. = dvd vvol
S
SD.
0. = B 0. =S
dSB
(Notation and symbols follow the convention in Elements of Electromagnetics, 5th ed., M. N. O. Sadiku)
2. Plane Waves
==o
o
HE
1
=v
HES = je=
+
=12
12 +==+
= 12
12
2
je
3. For Lossy Media Permittivity )"'("'and rro jj
===
Loss Tangent'
tan
=
Propagation constant
+2
'811'
'2
and
4. Propagation in Good Conductor
Propagation Constant pi fjj )1( +=+= and Skin Depth pi f1
=
5. Transmission Line Theory
Telegraphers Eqns. s
s Vdz
Vd 22
2
= ))(( CjGLjR ++=
)()(
0 CjGLjRZ
+
+=
+
+=
lZZlZZ
ZZL
Lin
tanhtanh
0
00
+
+=
ljZZljZZ
ZZL
Lin
tantan
0
00
0
0
ZZZZ
L
LL
+
=
6. Coaxial Line )/ln(2)/ln(
'2io
io
rrLandrr
Cpi
pi ==
1
7. Open-Wire Line
+==
)2/(cosh41
4)2/(cosh' 1
1 rdLandrdC pi
(Notation and symbols follow the convention in Microwave Engineering, 3rd ed., D. M. Pozar, John Wiley & Sons)
8. Smith Chart Equations
22
2
11
1
+=+
+
L
i
L
Lr rr
r and ( )
222 111
=
+LL
ir xx 9. Microstrip Line
Wd
rre /121
12
12
1+
++
=
Z 0 =
+
dW
Wd
e 48ln60
for W/d 1
Z 0 = ( )[ ]444.1/ln667.0393.1/120
+++ dWdWepi
for W/d 1
=dW
282
A
A
ee
for W/d < 2
=dW }
+
+rr
r BBB
pi
61.039.0)1ln(2
1)12ln(12 for W/d > 2
where, A =
++
++
rr
rrZ
11.023.011
21
600 and B =
rZ pi
02377
( )
( ) ,/12tan10 mNpk
re
erd
=
and mNp
WZRs
c /0
= where, R s = 2/010. Waveguides
2
11. Z-Parameters
NV
VV
.
.
.2
1
=
NNNN
N
N
ZZZ
ZZZZZZ
.............................
......................................................
21
22221
11211
NI
II
.
.
.2
1
or [ ]V = [ ]Z [ ]I
12. Y-Parameters
NI
II
.
.
.2
1
=
NNN
N
YY
YYYY
..............................
.......................
1
21
11211
NV
VV
.
.
.2
1
or [ ]I = [ ]Y [ ]V
3
13. S-Parameters
NV
VV
.
.
.2
1
=
NNN
N
SS
SSSS
......................1
21
11211
+
+
+
NV
VV
.
.
.2
1
or [ ]V = [ ]S [ ]+V
forVVVS K
j
iij 0==
++
k j
14. ABCD-Parameters
15. Impedance Matching with Lumped Elements
4
X = L
o
L
oL
BRZ
RZX
B+
1 X = ( )LoL RZR LX
B = 22
22/
LL
LoLLoLL
XRRZXRZRX
+
+ B =
( )o
LLo
ZRRZ /
16. Single Shunt Stub Tuning Y = G + jB
22
2
)()1(
tZXRtR
GoLL
L
++
+=
22
2
)(())((
tZXRZtZXtXZtRB
oLLo
oLLoL
++
+=
For RL Zo
oL
oLLoLL
ZRZXRZRX
t
+=
/])[( 22
For RL = Zo o
L
ZXt
2
=
For t 0 td 1tan21
=
pi For t < 0 )tan(
21 1 td += pipi
Open Circuited Stub: )(tan2
1 1o
o
YBl
=
pi Short Circuited Stub: )(tan
21 1
BYl os
=
pi 17. Single Series Stub Tuning Z = R + jX
22
2
)()1(
tYBGtG
RoLL
L
++
+=
22
2
)(())((
tYBGYtYBtBYtGX
oLLo
oLLoL
++
+=
For GL Yo
oL
oLLoLL
YGYBGYGB
t
+=
/])[( 22
For GL = Yo
o
L
YBt2
=
For t 0 td 1tan21
=
pi For t < 0 )tan(
21 1 td += pipi
Open Circuited Stub: )(tan21 1
XZl oo
=
pi Short Circuited Stub: )(tan
21 1
o
s
ZXl
=
pi 18. Double Stub Tuning
5
B 1 = B L + ( )
ttGYGtY LL
220
20 1 +
B 2 = ( )
tGYGtGtGYY
L
LLL 0222
00 1 ++
The open circuited stub length is found as: 0l =
pi21
tan 1
0YB
The short circuited stub length is found as: sl =
pi21
tan 1
BY0 where B= B 1 or B 2
19. Quarter-Wave Matching Transformer
Z = LZZ0 and
= m
pi2
2
pi
mmmff
fff
ff 4222)(2
00
0
0
==
=
= 2
0
0
2
1 2
1cos4
ZZZZ
L
L
m
m
pi
20. Three Port and Four Port Microwave Components
Three Port Network (such as T-Junctions) [S] =
333231
332221
131211
SSSSSSSSS
oZZZ111
21
=+
N-way Wilkinson Power Dividers
3
2
031
KKZZ o
+=
)1( 20032
02 KKZZKZ +==
)1(0 KKZR +=
Four Port Network (such as Directional Couplers) [S] =
44434241
34333231
24232221
14131211
SSSSSSSSSSSSSSSS
Coupling dBPPC
3
1log10= Directivity dBPPD
4
3log10= Isolation dBPPI
4
1log10=
6
EE541 Electro-Optics
Formula adopted from the textbook, Optoelectronics and Photonics, Principles and Practices, Prentice Hall, S. O. Kasap, (2001).
E = hE(r, t) = Eo cos (t k r + o) v = = (r o o)- 0.5 n = c/ v vg = d/ dk I = v r o Eo2/ 2 tan p = n2 / n1 R = (n2 - n1)2 / (n2 + n1)2T = 4 n2 n1 / (n2 + n1)2 t = 1 sin = 1.22 / D d sin = m , m = 0, 1, 2, ---- (4n1a cosm) / m = m No sinM = (n12 n22)0.5V = 2a (n12 n22)0.5/ M = 1 + Int(2V / ) for small VM V2/2 for large V/ L (n1 n2) / c dB = 10 log (Pin/ Pout) / L external = Pout(optical) / IV int = [Po(int)/ h]/ (I / e) N2 / N1 = exp[- (E2 E1) / kBT] = 2 o (1.386 kBT / Mc2) 0.5 gth = - 0.5 ln(R1R2) / L L = mm / 2n, m = 1, 2, 3 ---- =(Iph / e)/ (Po/ h)R = Iph / Po = en (e + h) in = [2e(Id + Iph) B] 0.5 SNR = Signal Power / Noise Power Iph = eGoA (ln + W + Le) I = -Iph + Io [exp(eV/ nkBT) 1] FF = ImVm / IscVoc ne() -2 = cos2 ()/ no2 + sin2 ()/ ne2
7
= 2L (ne no)/ I = Io sin2 (0.5 V/ V /2)
EE 641: RF Wireless Communication SystemsList of Commonly Needed Expressions and Relationships
I. Fundamentals of RF Wireless Communication Systems1. Spectral Efficiency spec = Data rate Rb (bits/sec) / Transmission bandwidth B (Hz)
2. Power Efficiency pow = Radiated power Prad (Watts) / Power drawn from source PDC (watts)
3. Shannons channel capacity
+=
NSBC 1log 2 bits/sec
II. Electromagnetic Waves and Radiators
1. Maxwells Equations( ) ( ), , . , . 0H EE H J E H
t t
= =+ = =
2. Wave Equation2 2 2 20, 0E E H H + = + =
3. Wave impedance of the medium/ 120 /rel rel pi = =
4. Propagation constant of the medium = + j = j ( 2 ) ,if = r + j i, then = + j = j [ r(1 j i/ r ) ]
5. Phase velocity of the electromagnetic waves in the medium (ohms) = ( /) = 120pi = 377
6. Poynting VectorS = E H
7. Radiation Intensity due to a source at the originU(r, , ) = r2 . S (r, , )
8. Power radiated from a source at the origin
8
= =
=
pi
pi
0
2
0
sin),( ddUPrad9. Far Field condition
Conditions of Far Field : R 2D2 / R >> DR >>
10. Fields due to a Hertzian Dipole of length z and current I, placed at origin along z axis.E(r, , ) = Er(r, , ) ar + E (r, , ) a + 0 a
2
2 3
cos cos( , , )2 ( ) ( )
j rr
I zE r e jr r
pi
=
2
2 3
sin sin sin( , , )4 ( ) ( )
j rj I z jE r er r r
pi