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Page 1: Electromanetic Formula Sp2011

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

Page 2: Electromanetic Formula Sp2011

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

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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

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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

Page 5: Electromanetic Formula Sp2011

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

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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

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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

Page 8: Electromanetic Formula Sp2011

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

Page 9: Electromanetic Formula Sp2011

∫ ∫= =

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

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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

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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

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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

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

Page 13: Electromanetic Formula Sp2011

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

Page 14: Electromanetic Formula Sp2011

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

Page 15: Electromanetic Formula Sp2011

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