Formula Sheet for Comprehensive Exam, Electromagnetic SystemsSp 2011
EE540 Microwave Devices and Systems
1. Maxwell’s EquationsDifferential Form Integral Form
t∂∂−=×∇ B
E SB
E dt
dLS
..∫ ∫ ∂∂−=
t∂∂+=×∇ D
JH SD
H dt
IdLS
..∫ ∫ ∂∂+=
vρ=∇ D. ∫∫ = dvd vvol
S
ρSD.
0. =∇ B 0. =∫S
dSB
(Notation and symbols follow the convention in Elements of Electromagnetics, 5th ed., M. N. O. Sadiku)
2. Plane Waves
εµη ==
o
o
H
E
µ ε1=v
HES ×= φ
ηηηη jeΓ=
+−=Γ
12
12 Γ+==+
= 12
12
2 φτηη
ητ je
3. For Lossy Media Permittivity )"'("'and rro jj εεεεεεωσε −=−==
Loss Tangent'
tanω εσθ =
Propagation constant
+≈≈
2
'8
11'
'2 ω εσµ εωβ
εµσα and
4. Propagation in Good Conductor
Propagation Constant µ σπβαγ fjj )1( +=+= and Skin Depth µ σπ
δf
1=
5. Transmission Line Theory
Telegraphers’ Eqns. s
s Vdz
Vd 22
2
γ= ))(( CjGLjR ωωγ ++=
)(
)(0 CjG
LjRZ
ωω
++=
++
=lZZ
lZZZZ
L
Lin γ
γtanh
tanh
0
00
++
=ljZZ
ljZZZZ
L
Lin β
βtan
tan
0
00
0
0
ZZ
ZZ
L
LL +
−=Γ
6. Coaxial Line )/ln(2)/ln(
'2io
io
rrLandrr
Cπµπ ε ==
1
7. Open-Wire Line
+== −
− )2/(cosh4
1
4)2/(cosh
' 11
rdLandrd
Cµπ ε
(Notation and symbols follow the convention in Microwave Engineering, 3rd ed., D. M. Pozar, John Wiley & Sons)
8. Smith Chart Equations
2
2
2
1
1
1
+
=Γ+
+
−ΓL
i
L
Lr rr
r and ( )
22
2 111
=
−Γ+−Γ
LLir xx
9. Microstrip Line
Wd
rre
/121
12
12
1
+−++= εεε
Z 0 =
+
d
W
W
d
e 4
8ln
60
ε for W/d ≤ 1
Z 0 = ( )[ ]444.1/ln667.0393.1/
120
+++ dWdWeεπ
for W/d ≥ 1
=d
W
2
82 −A
A
e
e for W/d < 2
=d
W
−+−−+−−−rr
r BBBεε
επ
61.039.0)1ln(
2
1)12ln(1
2 for W/d > 2
where, A =
+
+−++
rr
rrZεε
εε 11.023.0
11
21
600 and B =
rZ επ
02
377
( )
( ) ,/12
tan10 mNpk
re
erd −
−=εε
δεεα and mNpWZ
Rsc /
0
=α where, R s = σω µ 2/0
10. Waveguides
2
11. Z-Parameters
NV
V
V
.
.
.2
1
=
NNNN
N
N
ZZZ
ZZZ
ZZZ
..........................
.
.
.
...........................
...........................
21
22221
11211
NI
I
I
.
.
.2
1
or [ ]V = [ ]Z [ ]I
12. Y-Parameters
NI
I
I
.
.
.2
1
=
NNN
N
YY
Y
YYY
...........................
.
.
.
.......................
1
21
11211
NV
V
V
.
.
.2
1
or [ ]I = [ ]Y [ ]V
3
13. S-Parameters
−
−
−
NV
V
V
.
.
.2
1
=
⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
NNN
N
SS
S
SSS
......................1
21
11211
+
+
+
NV
V
V
.
.
.2
1
or [ ]−V = [ ]S [ ]+V
forVV
VS K
j
iij 0== +
+
−
k ≠ j
14. ABCD-Parameters
15. Impedance Matching with Lumped Elements
4
X =
L
o
L
oL
BR
Z
R
ZX
B−+1
X = ± ( )LoL RZR − LX−
B = 22
22/
LL
LoLLoLL
XR
RZXRZRX
+−+±
B = ( )
o
LLo
Z
RRZ /−±
16. Single Shunt Stub Tuning Y = G + jB
22
2
)(
)1(
tZXR
tRG
oLL
L
+++
=
22
2
)((
))((
tZXRZ
tZXtXZtRB
oLLo
oLLoL
+++−−=
For RL ≠ Zo
oL
oLLoLL
ZR
ZXRZRXt
−+−±
=/])[( 22
For RL = Zo
o
L
Z
Xt
2
−=
For t ≥ 0 td 1tan
2
1 −=πλ
For t < 0 )tan(2
1 1 td −+= π
πλ
Open Circuited Stub: )(tan2
1 1
o
o
Y
Bl −−=πλ
Short Circuited Stub: )(tan21 1
BYl os −=
πλ
17. Single Series Stub Tuning Z = R + jX
22
2
)(
)1(
tYBG
tGR
oLL
L
+++
=
22
2
)((
))((
tYBGY
tYBtBYtGX
oLLo
oLLoL
+++−−=
For GL ≠ Yo
oL
oLLoLL
YG
YBGYGBt
−+−±
=/])[( 22
For GL = Yo
o
L
Y
Bt
2=
For t ≥ 0 td 1tan
2
1 −=πλ
For t < 0 )tan(2
1 1 td −+= π
πλ
Open Circuited Stub: )(tan21 1
XZl oo −=
πλ Short Circuited Stub: )(tan
2
1 1
o
s
Z
Xl −−=πλ
18. Double Stub Tuning
5
B1
= − BL
+ ( )
t
tGYGtY LL22
02
0 1 −+±
B2
= ( )
tG
YGtGtGYY
L
LLL 0222
00 1 +−+±
The open circuited stub length is found as: λ0l =
π2
1tan 1−
0Y
B
The short circuited stub length is found as: λsl =
π2
1−tan 1−
B
Y0 where B= B
1 or B
2
19. Quarter-Wave Matching Transformer
Z = LZZ0
and
−=∆ mθπθ
22
πθmmm
f
f
f
ff
f
f 42
22
)(2
00
0
0
−=−=−=∆
= 2−
−Γ−Γ−
0
0
2
1 2
1cos
4
ZZ
ZZ
L
L
m
m
π
20. Three Port and Four Port Microwave Components
Three Port Network (such as T-Junctions) [S] =
333231
332221
131211
SSS
SSS
SSS
oZZZ
111
21
=+
N-way Wilkinson Power Dividers
3
2
03
1
K
KZZ o
+=
)1( 2003
202 KKZZKZ +==
)1
(0 KKZR +=
Four Port Network (such as Directional Couplers) [S] =
44434241
34333231
24232221
14131211
SSSS
SSSS
SSSS
SSSS
Coupling dB
P
PC
3
1log10= Directivity
dBP
PD
4
3log10= Isolation
dBP
PI
4
1log10=
6
EE541 Electro-Optics
Formula adopted from the textbook, “Optoelectronics and Photonics, Principles and Practices,” Prentice Hall, S. O. Kasap, (2001).
E = hνE(r, t) = Eo cos (ωt – k ∙ r + ϕo) v = νλ = (εr εo μo)- 0.5
n = c/ v vg = dω/ dk I = v εr εo Eo
2/ 2 tan θp = n2 / n1
R = (n2 - n1)2 / (n2 + n1)2
T = 4 n2 n1 / (n2 + n1)2
Δν ∙ Δt = 1 sin θ = 1.22 λ/ D d sin θ = m λ, m = 0, ±1, ±2, ---- (4πn1a cosθm) / λ – ϕm = mπ No sinαM = (n1
2 – n22)0.5
V = 2πa (n12 – n2
2)0.5/ λ M = 1 + Int(2V / π) for small VM ≈ V2/2 for large VΔτ/ L ≈ (n1 – n2) / c αdB = 10 log (Pin/ Pout) / L ηexternal = Pout(optical) / IV ηint = [Po(int)/ hν]/ (I / e) N2 / N1 = exp[- (E2 – E1) / kBT] Δν½ = 2 νo (1.386 kBT / Mc
2) 0.5
gth = γ - 0.5 ln(R1R2) / L L = mλm / 2n, m = 1, 2, 3 ---- η =(Iph / e)/ (Po/ hν)R = Iph / Po
Δσ = eΔn (μe + μh) 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(θ) -2 = cos2
(θ)/ no2 + sin2
(θ)/ ne2
7
ϕ = 2πL (ne – no)/ λ I = Io sin2
(0.5 πV/ V λ/2)
EE 641: RF Wireless Communication SystemsList of Commonly Needed Expressions and Relationships
I. Fundamentals of RF Wireless Communication Systems1. Spectral Efficiency η spec = Data rate Rb (bits/sec) / Transmission bandwidth B (Hz)
2. Power Efficiency ηpow = Radiated power Prad (Watts) / Power drawn from source PDC (watts)
3. Shannon’s channel capacity
+=
N
SBC 1log 2 bits/sec
II. Electromagnetic Waves and Radiators
1. Maxwell’s Equations( ) ( )
, , . , . 0H E
E H J E Ht t
µ µ ρε
=− =+ = =
2. Wave Equation2 2 2 20, 0E E H Hω εµ ω εµ + = + =
3. Wave impedance of the medium
/ 120 /rel relη µ ε π µ ε= =
4. Propagation constant of the mediumγ = α + j β = j √ (ω 2ε µ ) ,
if ε = ε r + j ε i, then γ = α + j β = jω [ µ ε r(1 – j ε i/ε r ) ]½
5. Phase velocity of the electromagnetic waves in the mediumη (ohms) = √ (µ /ε) = 120π Ω = 377 Ω
6. Poynting VectorS = E × H
7. Radiation Intensity due to a source at the originU(r, θ , φ) = r2 . S (r, θ , φ)
8. Power radiated from a source at the origin
8
∫ ∫= =
=π
θ
π
φ
φθθφθ0
2
0
sin),( ddUPrad
9. Far Field condition
Conditions of Far Field : R ≥ 2D2 / λR >> DR >> λ
10. Fields due to a Hertzian Dipole of length ∆ z and current I, placed at origin along z axis.E(r, θ , φ) = Er(r, θ , φ) ar + Eθ (r, θ , φ) aθ + 0 aφ
2
2 3
cos cos( , , )
2 ( ) ( )j r
r
I zE r e j
r rββ θ θθ φ η
π β β− ∆= −
2
2 3
sin sin sin( , , )
4 ( ) ( )j rj I z j
E r er r r
βθ
β θ θ θθ φ ηπ β β β
− ∆ −= + +
2
2
sin sin( , , )
4 ( )j rj I z
H r er r
βφ
β θ θθ φπ β β
− ∆= −
Directivity 23( , ) sin
2D θ φ θ=
Radiation resistance2
280rad
zR π
λ∆ =
11. Half-wave Dipole Rrad = 73 Ω D = 1.76 = 2.15 dB
III. Receiving Antenna Characteristics
1. Friis Equation2
2 24 4 4t t t t
r eff r
PG PGP A G
d d
λπ π π
= =
2. Effective Area ( , )( , )av
effinc
PA
Sθ φ
θ φ@
3. Reciprocity Theorem 2
4effG A
πλ
=
4. Noise power available from a resistor Pn,av = kTB
5. Noise temperature of antenna2
0 0
1sin ( , ) ( , )
4ant BT d d D Tπ π
θ φ
θ θ φ θ φ θ φπ = =
=
6. G/T Ratio G/T = [ 10 log10
Gant
] / Tant
dB/K
9
IV. Physical Model of Wave Propagation1. Reflection Coefficient
For E field parallel to ground 2 1
2 1
cos cos
cos cosref trans inc
inc trans inc
E
E
η θ η θη θ η θ
−Γ =
+P
PP
For E field in plane of incidence 2 1
2 1
cos cos
cos cosref inc trans
inc inc trans
E
E
η θ η θη θ η θ
⊥⊥
⊥
−Γ =
+
2. Transmission Coefficient
For E field parallel to ground 2
2 1
2 cos
cos costrans trans
inc trans inc
ET
E
η θη θ η θ
=+
PP
P
For E field in plane of incidence 2
2 1
2 cos
cos costrans inc
inc inc trans
ET
E
η θη θ η θ
⊥⊥
⊥
=+
3. Power received under free-space propagation (Friis equation) : 2
4rec
t rtr
PG G
P R
λπ
= 4. Power received due to perfectly reflecting ground with antennas at heights h t and hr :
22
44
=
λππλ
R
hh
RGG
P
P rtrt
tr
rec
5. Normalized diffraction parameter
21
21 )(2
dd
ddh
λν +
=
6. Excess Path Loss due to diffraction from single knife edge
V. Empirical Models of Wave propagation1. Delisle model of path loss in urban environment
10
4 217
2
4 216
2
( /1 )4.27 10 10
( /1 )4.27 10 10
mobbs mob
mobbs mob
r f MHzfor h m
h hL
r f MHzfor h m
h h
−
−
<
= >
2. Ikegami’s model of excess path loss between two edges separated by ds :2( /1 ) ( ) /1
186 ( /1 )o mob
s
f MHz h h mL
d m
−=
3. Okamura – Hata model for VHF/UHF (150 MHz to 1 GHz)
10 10 10 10( ) 69.55 26.16 log ( /1 ) 13.82log ( ) (44.9 6.55log ) logc bs mob bsL dB f MHz h a h h r C= + − − + − −Where
10
10
10 10
8.29 [log (1.54 /1 ) 1.1 arg 300
( ) 3.2 [log (11.75 /1 ) 4.97 arg 300
1.1 log [( /1 ) 0.7] ( /1 ) 1.56 log [( /1 ) 0.8]
mob c
mob mob c
c mob c
h m for l e city and f MHz
a h h m for l e city and f MHz
f MHz h m f MHz for small city
− = − > − − −
and 210
210 10
0
5.4 2 [log ( / 28 )]
40.94 4.78 [log ( /1 )] 18.33 log ( /1 )c
c c
for Urban area
C f MHz for Sunurban area
f MHz f MHz for Open area
= + + −
VI. Statistical Model of Wave Propagation1. Rayleigh density function for received signal amplitude
>
−<
=0
2exp
00
)(2
2
2y
yyy
yfY
σσ
2. Exponential density function for received signal power
( ) )(2/exp2
1)( 2
2puppf P σ
σ−=
3. Rician distribution in the presence of a strong signal
fX(x) = ( x / σ R2 ) exp [ - (x2 + A2)/ 2σ R
2 ] . Io ( x A / σ R2 )
4. Error Probability in the absence of Fading
1 1
2 2S
EO
ESP erfc erfc
N N
= =
5. Error Probability in the presence of fading for Rayleigh-distributed signal
+
−=)/(1
)/(1
2
1
NS
NSPE
11
6. Diversity GainchannelgleforNS
systemdiversityforNSG
sin)/(
)/(=
7. Doppler Shift Frequencyc
vff carrD
ϕcos=
VII. Channel Characterization
1. Given the power delay profile P(τ ),
Average delay 0
0
( )
( )D
PT d
P d
ττ ττ τ
< >=
Power delay spread
[ ] 2
0
0
( ) ( ) ( )
( )
( )
D
D
t T t P d
t
P d
τ τ τσ
τ τ
−=
2. R.M.S. Delay spread rms Dτ σ= or, Multipath spread 2MUL DT σ=
3. Coherence time – Doppler spread relationship: Tcoh ≈ 1 / 2fD
4. Coherence Bandwidth – r.m.s. Delay Spread Relationship1
2corrms
Bτ
VIII. Multiple Access and Cellular Systems
1. Minimum signal-to-interference Power Ratio (for hexagonal cells with ν -th power law)
[ ] 2/
min
3)1(
νν
−−
−=
RNN
R
I
Sc
c
2. Erlang’s B formula for the probability of call blocking with N available duplex channels, as a
function of total caller traffic intensity of U erlangs:
0
[ ]
!!
N
mN
m
UP Blocking
UN
m=
=
12
3. Adjacent channel interference ratio :
2
2
( ) | ( ) |
( ) | ( ) |
sig BP
sig BP
S f H f f df
ACI
S f H f df
−
−
− ∆=
IX. Noise and Interference
1. Power spectral density of noise at the output of a noiseless linear filter with frequency response H(jf) excited at its input with a random signal of power spectral density Sx(jf) :
Sy(f) = |H(jf)|2 Sx(f)
2. Noise bandwidth (or noise-equivalent bandwidth) of a filter with frequency response H(jf):
2| ( ) | o
eqo
H jf N df
BN
−=
2. Noise figure of a linear system:,
/
/n source ref
in in
out out T T
S NF
S N=
=
3. Equivalent noise temperature Teq = (F – 1) Tref or, Noise figure F = 1 + (Teq/Tref)
4. Noise figure of a passive filter at temperature Tp: 1
1 1 p
av ref
TF
G T
= + −
5. Combined noise temperature of a cascade of n linear systems, 32
1,1 ,1 ,2
....totav av av
TTT T
G G G= + + +
Or, combined noise figure, 32
1,1 ,1 ,2
11....tot
av av av
FFF F
G G G
−−= + + +
X. Nonlinear and Intermodulation Distortion
For a memory-less nonlinear system with transfer characteristic
y = a0 + a1 x + a2 x2 + a3 x3 + ignorable higher-order terms
1. When excited with a harmonic signal of amplitude Vamp,
Gain compression33
1 34
1
ampnonlin
lin
a a VG
G a
+=
13
Second-harmonic generation Vamp|@2f = ½ a1 Vamp2
2. When excited with two harmonic signals of frequencies f1 and f2, with equal amplitudes Vamp
Amplitude of intermodulation signal (at each of 2f2 ± f1 and 2f1 ± f2) : VIMD = ¾ a3 Vamp3
Intermodulation power ratio: IMPR ≡ VIMD2 / (a1Vamp)2 = ¾ (a3 / a1
2) Vamp4
Third-order intercept (TOI) point, referred to the input:
PTOI = Input Power (½ a1Vamp)2 |@IMPR=1 = 2a13 / 3a3
3. Dynamic range DR = [ PTOI / No ]2/3
EE645 Antennas and Propagation
(Notation and symbols follow the convention in Antenna Theory and Design, 3rd ed., C. A. Balanis, Wiley)
Notation and symbols follow the convention in Elements of Electromagnetics, 5th ed., M. N. O. Sadiku) 1. Antenna Fundamental Parameters
A e = Dπ
λ4
2
Gd (θ , φ) = 4radP
U ),( φθπ D = (4π Umax/Prad)
W 0 = 24 r
Pt
π Prad= ∫∫ U dΩ = ∫0
2π∫0π U sinθ dθ dϕ
= ecdt ecdr (1- | Γt |2 ) (1- | Γr |2 ) Dt(θt,ϕt) Dr(θr,ϕr) | ρt . ρr |2
where ρt is the unit vector of the incoming wave ρr is the unit vector of the antenna ( polarization vector)
= ecdt ecdr (1- | Γt |2 ) (1- | Γr |2 ) σ | ρw . ρr |2
where ρw = polarization unit vector of the scattered wavesρr = polarization unit vector of the receiving antenna
2. Hertzian Dipole AntennaAlong z -axis
rjors e
r
j
r
dlIE β
βθ
πη −
−=
32
1cos
2 rjo
s er
j
rr
jdlIE β
θ ββθ
πη −
−+=
32
1sin
4
rjos e
rr
jdlIH β
φβθ
π−
+=
2
1sin
4
2280
=
λπ dl
Rrad
3. Half-Wave Dipole AntennaAlong z- axis
( )θβπ
θπµ β
2sin2cos)2/(cos
reI
Arj
ozs
−
=θπ
θπβ
φ sin2
)cos)2/cos((
r
ejIH
rjo
s
−
=
14
ss HE φθ η=
4. Small Loop Antenna
Along xy-plane (S=Nπ ρ o2) with N-turns
rjos e
rr
jSIjE β
φβθ
πω −
+−=
2
1sin
4
rjors e
r
j
r
SIjH β
βθ
π ηω −
−=
32
1cos
2
rjos e
r
j
rr
jSIjH β
θ ββθ
π ηω −
−+=
32
1sin
4
424 /)320( λπ SRrad =
5. Array Antennas
AF= 2 cos ( ) 2/cos2
1 ααθβ jed
+ and | AF| =
2sin
2sin
ψ
ψN
where αθβψ += cosd
15
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