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