Microwaves

Chapter 1 ‐ Propagation on a transmission line

Đại Học Bách Khoa Hà Nội / 2012-2013

Propagation of a wave

2

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −×=−×=

λπω z

TtEkztEE 2coscos 00

Electromagnetic field 

( ) ( )zzkttkzt δδωω +−+=−

fTkt

zvP ×==== λλωδδ

fvP ×= λ

During propagation the phase remains constant 

Phase velocity 

Microwave electronics concerns high frequency or small wavelength signals

The size of circuits as the same order of magnitude as the wavelength

Microwave domain

3

AM Radio 30 kHz – 30 MHz (LW, MW, SW)

FM Radio 88 – 105 MHz

DAB (Digital Audio Broadcasting) VHF III band: 174 – 240 MHzL band: 1452 – 1492 MHz

TV VHF 54 – 72 MHz / 76 – 88 MHz / 174 – 216 MHz

TV UHF 470 – 608 MHz / 614 – 890 MHz

Cellular phones GSM 900 MHz and 1.8 GHz

Bluetooth 2.4 GHz

WiFi (802.11 a/b/g) 2.4 – 2.4835 GHz / 5 GHz

Wimax 2.5 GHz / 3.5 GHz / 5.8 GHz

Hyperlan 5 GHz

Satellite5.925 – 6.425 GHz / 3.7 – 4.2 GHzIntelsat V 14 – 14.5 GHz / 10.95 – 11.2 GHz

/ 11.45 – 11.70 GHz

Radar FMCW 77 GHz

Wavelength and frequency

4

Examples of transmission lines

5

Circular waveguideRectangular waveguide

Twisted or untwisted pairs line Coaxial cable

Transmission lines on a Printed Circuit Board (PCB)

6

Microstrip line Coplanar waveguide

Materials

7

Conductors are characterized by their conductivity Copper (Cu): σ = 5.88 107 S.m‐1

Gold (Au): σ = 4.55 107 S.m‐1 

Aluminium (Al): σ = 3.65 107 S.m‐1

Silver (Ag): σ = 6.21 107 S.m‐1

Insulators, between the conductors, are of dielectric materials; they are characterized by their low conductivity, the dielectric constant and the permeability (usually µr =1).

Epoxy (FR4): εr ≈ 4Polyethylene: εr ≈ 2.25 Duroid: εr ≈ 6 or 10Alumina: εr ≈ 8.4Teflon: εr ≈ 2.1; σ = 3.3 10‐14 S.m‐1

Losses are expressed by tanδ (see later)

Hypothesis for this lecture

8

• 2 wires conductors waveguide, quasi TEM mode

voltage and intensity are defined on the circuit

• The length of the transmission lines has the same magnitude order as the 

wavelength: the propagation effect has to be taken into account

• Components have a small size QSA is valid (surface mount devices)

‐ Passive elements: R, L, C; diodes

‐ Active elements: transistors (BJT, J‐FET, MOS‐FET)

‐Integrated circuits: MMIC (monolithic microwave integrated circuits)

• Small bandwidth signal (the signal is centered on the frequency carrier, 

case of a modulated signal, with a small occupied channel) 

Coaxial cable

9

211

4dhRa

πσ=

( )eedhRg +

=22πσ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

20 Ln

21

ddhL rμμ

π

( )12

0/Ln

2ddhC rεπε

=

( )12 /Ln2

ddhG iπσ

=

Resistance of the core

Resistance of the envelop

Inductance of the core placed inside the envelop

Capacitor made of the core and the envelop

Conductance of the insulator

Only in case of a coaxial cable the wave is really TEM!

Distributed elements modelling

10

C1dzG1dz

z z+dz

i(z, t)

v(z, t)

i(z+dz, t)

v(z+dz, t)

R1dz L1dz

tiLiR

zv

∂∂

+=∂∂

− 11

tvCvG

zi

∂∂

+=∂∂

− 11

Kirchhoff laws:

Telegraphers equation

11

Telegraphers Equations

( ) vGRtvGLCR

tvCL

zv

1111112

2112

2+

∂∂

++∂

∂=

∂

∂

( ) iGRtiGLCR

tiCL

zi

1111112

2112

2+

∂∂

++∂

∂=

∂

∂

Harmonic Solution

ωjt↔

∂∂

( ) )()()()(11111111

22

2zvGRzvGLCRjzvCL

zzv

+++−=∂

∂ ωω

( ) )()()()(11111111

22

2ziGRziGLCRjziCL

zzi

+++−=∂

∂ ωω

tjezvtzv ω)(),( =tjezitzi ω)(),( =

22

2ω−↔

∂

∂

t

General harmonic solution

12

( )( )vjCGjLRz

v ωω 11112

2++=

∂

∂

( )( )ijCGjLRz

i ωω 11112

2++=

∂

∂

( )( )ωωγ 11112 jCGjLR ++= βαγ j+= 0and0 >> βα

Incident wave Reflected wave

VzV 22

2γ=

∂

∂

Iz

I 22

2γ=

∂

∂

)()()()( ztjzztjz eeVeeVv βωαβωα +−−−+ +=

)()()()( ztjzztjz eeIeeIi βωαβωα +−−−+ +=βω

=Pv

Propagation velocity

= Phase velocity

Characteristic impedance of a transmission line

13

tiLiR

xv

∂∂

+=∂∂

− 11

ωω

11

11jCGjLRZ c +

+=

γω11

)(

)( jLRIV +

=+

+

cZIV

+=+

+

)(

)(

γω11

)(

)( jLRIV +

−=−

−

cZIV

−=−

−

)(

)(

Case of a lossless line1

1CLRZ cc ==

11

1CL

vp =01 =R01 =G

Zc is called the characteristic

impedance

Example for a coaxial cable (RG‐58U)

14

-117 .m 1088.5 −Ω×=Cuσ

mm406.01 =d mm418.12 =d mm25.0=e

25.2r =ε 1=rμ-1114 .m 103.3 −− Ω×=iσ

180

0011m.s10211 −×====

rrP

cCL

vεεμε

Ω=⎟⎟⎠

⎞⎜⎜⎝

⎛== 50Ln

21

0

0

1

2

1

1

rc d

dCL

Rεεμ

π

,,

,

11 pF.m100 −=C 1

1 μH.m25.0 −=L

.

This is the same value as for a plane wave in the dielectric material.

Dielectric material polyethylene is such as:

Conductors: copper, conductivity

Geometrical characteristics

Characteristic impedance of a microstrip line

15

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ++⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ +

++

++

= 222

02/11

'4

11/814

'4

11/814

'41Ln

122πεεε

επη rrr

rc w

hwh

whZ

Ω==≡ 377π1200

00 ε

μη is the characteristic impedance of free space

www Δ+='

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

++⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ +

=Δ22

1.1//1

4Ln2/11

twht

etw r

π

επ

Reflection coefficient

16

)(

)(

−

+=

L

LL

V

VΓ

L

LL Γ

Γz−+

=11

11

+−

=L

LL z

zΓ

c

LL Z

Zz =Normalized impedance:

L

Lc

L

LL Γ

ΓZ

IV

Z−+

==11

cL

cLL ZZ

ZZΓ

+

−=

ΓL is called thereflection coefficient

Important cases for the reflection coefficient

17

cL ZZ = 0=LΓ

=∝LZ 1+=LΓ

0=LZ 1−=LΓ

If a transmission line has a load equal to the characteristic impedance, there is no reflection.

In this case the load is said to be matched to the transmission line.

If a transmission line is terminated by an open circuit the reflection coefficient is equal to +1.

If a transmission line is terminated by a short circuit thereflection coefficient is equal to -1.

1=Lz

Reflection coefficient at a distance d from a load

18

djdLdL eeΓΓ βα 22

,−−=

For a lossless transmission line: djLdL eΓΓ β2

,−=

LΓ

LZ

dLΓ ,

0

d

zd−

Moving from the load toward generator = clockwise rotation on a constant reflection coefficient modulus circle

Standing wave ratio

19

( )dLddd ΓVVVdV ,)()()( 1)( +=+= +−+

L

LΓΓ

VVSWR

−+

==11

min

max

-1 0

ΓL,d

1+ΓL,d

dLΓVdV ,)( 1)( += +

LΓ

LZ

dLΓ ,

0

d

zd−

Standing wave ratio

20

Standing waves in case of a reflection coefficient equal to 0,6 (λ = 1 cm)SWR = 4

Smith Chart

21

jqpΓ +=

ϕjeΓΓ =

jxrz +=11

+−

=zzΓ

jxrjxrjqp

+++−

=+11

( )22

2

11

1 rq

rrp

+=+⎟

⎠⎞

⎜⎝⎛

+− ⎟

⎠⎞

⎜⎝⎛

+= 0,

1rrC

r+=

11R

( ) 2

22 111

xxqp =⎟

⎠⎞

⎜⎝⎛ −+− ⎟

⎠⎞

⎜⎝⎛=

x1,1C x

1=R

The Smith chart is a representation of the reflection coefficient  in polar coordinates

Smith

 chart

22

Microwaves

Chapter 2 – Impedance matching

Đại Học Bách Khoa Hà Nội / 2012-2013

Power received by a load

24

*21

LLL IV=PZLVL

IL

( )LLP PRe=

( ) ⎟⎠⎞⎜

⎝⎛ −==

+2

2)(

121Re L

cLL Γ

R

VP P

v

i

v(t)i(t)p(t) =

)cos(21)( ϕLLIVtp >=<

( )( ) ( ) ( )**)()(*)()()()(* 1121

21

21

LLLLL ΓIΓVIIVVIV −+=++== ++−+−+P

)()( ++ = IRV c)()( −− −= IRV c ( )( )**)()( 11

21

LLL ΓΓIV −+= ++P

Incident and reflected power

25

⎟⎠⎞

⎜⎝⎛= +++ *)()()(

21Re IVP 2)(

2)()(

21

21 +

++ == IR

R

VP c

c

⎟⎠⎞

⎜⎝⎛= −−− *)()()(

21Re IVP

2)(

2)()(

21

21 −

−− −=−= IR

R

VP c

c

ccL R

V

R

VP

2)(2)(

21

21

−+

−= )()( −+ −= PPPL

)()( ++ = IRV c The incident wave propagates seeing an impedance Rc

The incident wave propagates seeing an impedance ‐Rc)()( −− −= IRV c

incident power  

reflected power  

The power transmitted to the load is the sum of the incident power and of the reflected power   

Normalized waves

26

21 )(+

=V

Ra

c 21 )(−

=V

Rb

c

abΓ L =

⎟⎠⎞⎜

⎝⎛ −=−= 2222 1 LL ΓabaP

ZLV (+) V (-)

ΓL

ZL

b

a

ΓL

Unit of power

27

⎟⎟⎠

⎞⎜⎜⎝

⎛=

W1)W(log10dB

PP

⎟⎟⎠

⎞⎜⎜⎝

⎛=

mW1mW)(log10dBm

PP

30dBdBm += PP

in

outPP

G =

⎟⎟⎠

⎞⎜⎜⎝

⎛=

in

outPP

G log10dB

Decibel

Gain

Power transfer from a source to a load

28

ZL

ΓL

a

b

ΓS

ZS

eS

*SL ΓΓ = 2

20

1

1

SSav

ΓaP

−=PL is maximum if: Available 

power

aΓΓabΓaa LSSSS +=+= 00

LS

SΓΓ

aa−

=1

0

Sc

cS

cS ZR

RE

Ra

+=

21

0

2

22

022

1

1

LS

LSL

ΓΓ

ΓabaP

−

−=−=

)cos(21

1

1

122

22

02)(

22

0LSLSLS

LS

jLS

LSL

ΓΓΓΓ

Γa

eΓΓ

ΓaP

LS ϕϕϕϕ +−+

−=

−

−=

+

Impedance matching

29

ZL

ΓLΓin

(C)

a1

b1 a2

b2

ΓoutΓS

ZS

eS

Matching network (matching to the source and to the load)

*Lout ΓΓ =

*Sin ΓΓ =

For a given source and a given load:matching is obtained by a matching cell placed between source and load

Matching with discrete reactive elements

30

ZL

ΓLΓS

jX

jBpRc

(a)

ZL

ΓLΓS

jXs

jBRc

(a)

)(1

)/(11

LLpLLpc jBGjB

jXjXRjB

jXR++

+=++

+=)(11

LsLc XXjRjB

Z +++=

Thanks to the Smith Chart it is possible to avoid calculation

Serial association on the Smith chart: the point is moving on a constant real part circle of the impedance

Parallel association on the Smith chart: the point is moving on a constant real part circle of the admittance

)(' sLL xxjrz ++=

)(' pLL bbjgy ++=

Matching with a single stub placed in parallel

31

ZL

d

l

Rc

)tan()(1)tan(

)tan(11

djbgjdjjbg

lj LL

LLββ

β ++++

+=

)tan(1)tan(1

djydjy

zy

L

L

LdLd β

β++

==

Ldstub yy +=1

Two solutions for d then two solutions for the length l

)tan()(1)tan(

)tan(11

djbgjdjjbg

lj LL

LLββ

β ++++

=−

Thanks to the Smith Chart it is possible to avoid calculation

Matching with two stubs

32

ZL

d1

l1

Rc

l2

d2

Two distances d1 and d2 are fixed, the length l1 and l2 have to be determined

Matching with a quarter wavelength line

33

RL

d = λ/4

RxRc

Lcx RRR =

11

+

−= x

L

xLx

LrrΓ

x

LxL R

Rr =11

11

114,

+

−

=+

−=−=== −−

xL

xL

xL

xLx

Ljx

Ljx

Lx

dL

r

rrrΓeΓeΓΓ π

λβ

xx

dLdL RrR ,, = xL

xcdL R

RR

RR ==,

This method concerns only real loads

L

xxL

xdL R

R

rr ==

1,

Microwaves

Chapter 3 – The S parameters

Đại Học Bách Khoa Hà Nội / 2012-2013

System with n connections

35

"Connection" equal to "port"  (input or output)

Scattering parameters

36

For a linear system with n ports (1, 2, … i, …j, …n)The normalized waves at the output can be expressed as a linear combination 

of the normalized waves at the input.

∑=j

jiji aSb ( ) [ ]( )aSb =

Reflection coefficient at port "i" when all the other ports are matched to their corresponding transmission line.

Transmission coefficient from port "j" to port "i" when ports "k≠j" are matched to their corresponding transmission line.

0=≠

=ijai

iii a

bS

0=≠

=

jkaj

iij a

bS

[ ]S Smatrix

ijS Scattering  or S parameters

Case of a quadripole

37

2221212

2121111aSaSb

aSaSb+=+=

01

111

2=⎟⎟⎠

⎞⎜⎜⎝

⎛=

aabS

01

221

2=⎟⎟⎠

⎞⎜⎜⎝

⎛=

aabS

02

222

1=⎟⎟⎠

⎞⎜⎜⎝

⎛=

aabS

02

112

1=⎟⎟⎠

⎞⎜⎜⎝

⎛=

aabS

Q(1) (2)

a2a1

b1 b2

S Parameters: 2 ports devices

38

Attenuator: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

0'0

ββ

SR3

R2

R1

R3

R2

R1

Amplifier: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

000

21SSPort-1 Port-2

Filter: low-pass, band-pass, high-pass or band-reject ⎟⎟⎠

⎞⎜⎜⎝

⎛=

0)()(0

21

12fS

fSS

0=iiSIdeal devices are matched at the different ports:

Three ports devices

39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

333231

232221

131211

SSSSSSSSS

S

jiij SS =

0=iiS

A lossless reciprocal three ports device cannot have all ports simultaneously matched.

Reciprocal device, the S matrix is symmetric:

All ports matched:

Lossless device, conservation of energy:

∑∑==

=3

1

23

1

2

ii

ii ab

Power splitters (dividers)

40

cR2

cR2

cR

cR

cR

Port 1

Port 2

cR2

Port 3

4/λ

4/λ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

011101110

21SPort 1

Port 2

Port 3

3/cR3/cR

3/cR

inout PP41

2 =

inout PP41

3 =

Resistive divider:

Wilkinson divider:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

=001001110

2jS

Circulators

41

Circulator: Source 1       23

Matched  load⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=0000

00

θ

θ

θ

j

j

j

ee

eS

31 aeb jθ= 12 aeb jθ= 23 aeb jθ=

03 =a 01 =b

The power returned at Port‐2 is totally absorbed by the matched load at Port‐3.

The source placed at Port‐1 is protected against any kind of change at Port‐2.

Directionnel couplers

42

(3)

(2)

(4)

(1)in

isolated coupled

trough

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

000000

00

αβαββα

βα

ϕ

θ

ϕ

θ

j

j

j

j

ee

ee

S

αlog20log20log10 212

1 −=−== SPPIL

βlog20log20log10 313

1 −=−== SPPC

4131

41

4

3 log20log20log10SS

SPPD β

=−==

414

1 log20log10 SPPI −==

Insertion loss: 

Coupling factor: 

Directivity: 

Isolation: 

(1) (2)

(3)

(4)

dBdBdB CDI +=

Port‐4 internally matched 

Some examples of circuits: filters

43

Bandpass filters

44

Couplers

45

Measurement planes

46

a1a’1 a’2a2

b’1 b1 b2 b’2

d1 d2

Q

(S)1 2

P’1 P1 P2 P’2

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= −

−

−

−

2

1

2

1

00

00' dj

dj

dj

dj

eeS

eeS β

β

β

β

11 1' beb dγ−=

11 1' aea dγ=

22 2' beb dγ−=

22 2' aea dγ=

Measurement of the S parameters

47

RBSij = R

AS jj =

Att

DUTAtt (i)(j)

S

A B

R

Rc

Phas

The measurement instrument is called a network analyzer

The inputs at A, B and R are matched

Network analyzer: basic structure

48

(1)(2)

(3)

(1)(2)

(3)

DUT

(4)

(4)

PowerSplitter

RFSource

Att. Phas

R B

A

Rc

Rc

ForwardPort-2

Port-1

Att.

Calibration process = measurement with standard loadsopen circuit, short circuit and characteristic impedance load

Reverse

Two ports measurement

49

Network analyzer

Measuremen

t of the

 4 S param

eters

50

Network analyzer structure with local oscillator

51

Heterodyne technique

Examples of network analyzers

52Rhode & Schwarz (Germany)

Anritsu (Japan)Agilent Technologies (USA)

Microwaves

Chapter 4 – Amplification

Đại Học Bách Khoa Hà Nội / 2012-2013

Amplification basics

54

LO

IFA

Γ = ‐1Γ’ =1

λ/4

Γ = ‐1

λ/4

Γ’ =1

(DC)(DC & AC)

Decoupling for biasing the amplifier

Amplifier (LNA) in a receiver stage

Operating point

Amplification is based on transistors

55

RD

λ/4

λ/4

In HF

-VGS

Out HF

+VDD

C1

C2

C1

C2

RC

λ/4

λ/4

In HF

Out HF

+VCC

C1

C2

C1

C2

R1

+VCC

R1

T1

T2

J‐FET transistors BJT transistors

Reflection coefficients on a quadripole

56

22211211 1 SΓ

ΓSSSΓL

Lin −

+=11

122122 1 SΓΓSSSΓ

S

Sout −

+=

a1 b2

ZL

ΓLΓin

(S)b1

a2Q

ΓoutΓS

ZS

eS

Insertion gain

57

av

Li P

PG ==sourcetheatpoweravailable

loadthetopower

( )( ) 221122211

222

2111

11

SSΓΓSΓSΓ

ΓΓSG

LSLS

LSi

−−−

⎟⎠⎞⎜

⎝⎛ −⎟⎠⎞⎜

⎝⎛ −

=

22

01

1

SSav

ΓaP

−=

⎟⎠⎞⎜

⎝⎛ −=−= 22

22

22

2 1 LL ΓbabP

SinS

L ΓΓa

SΓSb

−−=

11

1 022

212

22211211 1 SΓ

ΓSSSΓL

Lin −

+=

Conjugate matching

58

ZL

ΓmL

Q

Γms

ZS

eS C1 C2

ΓLΓs Γin Γout

The problem is to find matching networks (C1) and (C2) to be placedsimultaneously at the input and at the output and leading to a maximumvalue of the insertion gain, remembering that the output has an influence onthe input and the input has an influence on the output.

Unilateral quadripole

59

012 =SProperty of an unilateral quadripole

222

211

2221

2

11

11

SΓSΓ

ΓSΓGG

LS

LSiui

−−

⎟⎠⎞⎜

⎝⎛ −⎟

⎠⎞⎜

⎝⎛ −

==

*22SΓΓ mLL == *

11SΓΓ mSS ==

Conjugate matching is easy:

⎟⎠⎞⎜

⎝⎛ −⎟

⎠⎞⎜

⎝⎛ −

=2

22

2212

11max,

1

1

1

1

SS

SGiu

There is no feedback between the output (Port‐2) to theinput (Port‐1)

Unilateral practical condition

60

⎟⎠⎞⎜

⎝⎛ −⎟⎠⎞⎜

⎝⎛ −

=2

222

11

22122111

11 SS

SSSSuUnilateral figure of merit

( ) ( )22 11 u

GG

u

G iui

iu

−≤≤

+

1<<u ( ) ( )uGGuG iuiiu 2121 +≤≤−

iui GG ≈12 <<u

Examples

‐ Transistor 2N3970 at f = 750 MHz

‐ Transistor FPD4000AF at f = 3 GHz

151.011 =S 38.621 =S 061.012 =S 63.022 =S 2106.122 −=u

8.011 =S 926.121 =S 062.012 =S 488.022 =S 210342 −=u

Designing amplifiers

61

Stability

Noise figure

Gain

Bandwidth

Trade‐off: