Fundamentals of Microwave Engineering RF coupling...

Fundamentals of Microwave Engineering &

RF coupling issues

Luigi Celona

Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Sud

Via S. Sofia 64 – 95123 Catania Italy

To: Agatino, Giacomo, Nicolò, Andrea and …last arrived Emanuele

CERN Accelerator School and Slovak University of Technology, Senec, June 2012 L. Celona, Fundamentals of Microwave Engineering & RF Coupling issues

2

Contents

Introduction

Equivalent circuit of a TRL

Derivation of wave equation and its solution

The propagation constant

Characteristic Impedance

Reflection Coefficient

Standing Waves, VSWR

Impedance and Reflection

Incident and Reflected Power

Smith Charts

Load Matching

Single Stub Tuners

Impedance and admittance matrices

S matrix

Waveguides Rectangular and circular

WG

Propagation and cutoff

Attenuation

Field lines

RLC model

Resonator


Frequency spectrum

3

103

3×105 3×106 3×107 3×108 3×109 3×1010 3×1011 3×1012 3×1013 3×1014

102 10 1 10-1 10-2 10-3 10-4 10-5 10-6

Frequency (Hz)

Wavelength (m)

Microwaves

AM

bro

adca

st

radio

Long w

ave

radio

Short

wav

e

radio

VH

F T

V

FM

ra

dio

Far

infr

ared

Infr

ared

Vis

ible

lig

ht

Approximate band designation

L-band 1-2 GHz

S-band 2-4 GHz

C-band 4-8 GHz

X-band 8-12 GHz

Ku-band 12-18 GHz

K-band 18-26 GHz

Ka-band 26-40 GHz

U-band 40-60 GHz

Low frequencies: so large that the phase variation across the dimensions is little.

Optical frequencies: much shorter than the component dimensions.

Geometrical optics regime

Phase of the voltage and currents changes significantly over the physical lenght of the device, microwave components are distributed, the lumped circuit element approximation are not valid.


4

Electrical Model of a Transmission Line

R= series resistance per unit lenght, for both conductors, in /m. L= series inductance per unit lenght, for both conductors, in H/m. G= shunt conductance per unit lenght in S/m. C= shunt capacitance per unit lenght in F/m.

i(z,t)

v(z,t) +

-

z

z

•

• •

• i(z,t)

v(z,t)

+

-

Rz Lz

Gz Cz v(z+z,t)

+

-

i(z+z,t)


5

Wave equation – Propagation constant

wave travelling along +z direction wave travelling along -z direction

0 0

0 0

+ -

+ -

jβαCjGLjRγ is called attenuation constant [dB/m]

is called phase constant [radians/m]

β

2πλ2πλβ

)(

0

)(

0

)(

0

)(

0

)(

)(

ztjzztjz

ztjzztjz

eeIeeIzI

eeVeeVzV


6

Characteristic Impedance

has negative sign +

+

-

- - 0

0 0

0

0

0

0 0

0 +

+

-

-

)(

00

)(

00

)(

0

)(

0

)/()/()(

)(

ztjzztjz

ztjzztjz

eeZVeeZVzI

eeVeeVzV


7

Terminated lossless transmission line

i(z)

v(z)

z

IL

ZL Z0 ,

0

VL

V0

V0

-d

zjzj

zjzj

eZVeZVzI

eVeVzV

)/()/()(

)(

0000

00

0

0

000

00

00

)0(

)0(0@ V

ZZ

ZZVZ

VV

VV

I

VZz

L

LL

0

0

0

0

ZZ

ZZ

V

V

L

L

][)(

][)(

0

0

0

zjzj

zjzj

eeZ

VzI

eeVzV

+

-

The amplitude of the reflected wave normalized to the amplitude of the incident wave is known as reflection coefficient

ZL = Zo = 0 NO REFLECTIONS (maximum power delivered to the load)

MATCHED LINE


8

Standing Waves

LZoZ

0z

z

d

At z=0

oL

oL

ZZ

ZZVV

00

Along the transmission line:

zVeVV

eVeVV

zj

zjzj

cos21 00

00

traveling wave standing wave

If ZL Zo

(by adding and substracting: ) zjeV 0

zjzj

zjzj

eZVeZVzI

eVeVzV

)/()/()(

)(

0000

00


9

Voltage Standing Wave Ratio (VSWR)

1

1

1

1

0

0

min

max

V

V

V

VVSWR

The VSWR is always greater than 1

2

1

Incident wave

Reflected wave

Standing wave

][log20 dBRL

zjzj eVeVV 00


10

Input Impedance of a Transmission Line

LZoZ

0z

z

d

oL

oLL

ZZ

ZZ

INZ

towards load

towards generator

djZZ

djZZZ

e

eZ

dI

dVZ

L

L

dj

dj

IN

tan

tan

1

1

)(

)(

0

002

2

0

][)(

][)(

0

0

0

zjzj

zjzj

eeZ

VzI

eeVzV


11

Reflection Coefficient Along a Transmission Line

LZoZ

0x

x

d

oL

oLL

ZZ

ZZ

G

towards load

towards generator

dj

Ldj

dj

L

genforward

reverseG e

eV

eV

V

V

2

)(

0

)(

0

Wave has to travel down and back

][)( 0

zjzj eeVzV


12

Impedance and Reflection

G

L

Re

Im

d2

There is a one-to-one correspondence between G and ZL

oG

oGG

ZZ

ZZ

G

GoG

1

1ZZ

d2jL

d2jL

oGe1

e1ZZ


Focus on “special lenghts”

13

For a general transmission line with lenght:

•l=n/2

meaning that a half-wavelenght line does not alter or transform the load impedance

•l= /4 + n/2

•Such lines are called quarter wave transformers since they have the effect of trasforming the load impedance in an inverse manner depending on the characteristic impedance of the line.

ZIN = ZL

L

INZ

ZZ

2

0


14


LZoZ

0zd

dz

The rate of energy flowing through the plane at z=-d

o

L

o

zjzj

Z

V

Z

V

Z

VP

zVeeZ

VzIzVP

2

02

2

02

0

2

0

222*

0

2

0*

2

1

2

1)1(

2

1

)(1Re2

1)()(Re

2

1

forward power reflected power


15


Power does not flow! Energy flows.

The net rate of energy transfer is equal to the

difference in power of the individual waves

To maximize the power transferred to the load we want:

0L

which implies:

oL ZZ

When ZL = Zo, the load is matched to the transmission line


16

dB and dBm

A dB is defined as a POWER ratio. For example:

log20

log10

P

Plog10

2

for

revdB

A dBm is defined as log unit of power referenced to 1mW:

mW1

Plog10PdBm


17

Load Matching

What if the load cannot be made equal to Zo for some other reasons? Then, we need to build a matching network so that the source effectively sees a match load.

0

LZsP 0Z M

Typically we only want to use lossless devices such as capacitors, inductors, transmission lines, in our matching network so that we do not dissipate any power in the network and deliver all the available power to the load.


18

Normalized Impedance

LL

o

LL jxr

Z

Zz

It will be easier if we normalize the load impedance to the characteristic impedance of the transmission line attached to the load.

1

1Lz

Since the impedance is a complex number, the reflection coefficient will be a complex number

ir j

22

22

1

1

ir

irLr

221

2

ir

iLx


19

Smith Charts

The impedance as a function of reflection coefficient can be re-written in the form:

2

2

2

1

1

1L

i

L

Lr

rr

r

2

2

2 111

LL

irxx

22

22

1

1

ir

irLr

221

2

ir

iLx

L

L

L

rradius

r

rC

1

1

0,1

L

L

xradius

xC

1

1,1

These are equations for circles on the (r,i) plane


20

Smith Chart – Real Circles

1 0.5 0 0.5 1

1

0.5

0.5

1

Re

Im

r=0 r=1/3

r=1 r=2.5


21

Smith Chart – Imaginary Circles

1 0.5 0 0.5 1

1

0.5

0.5

1

Re

Im

x=1/3 x=1 x=2.5

x=-1/3 x=-1 x=-2.5


22

Smith Chart


23

Smith Chart Example 1

Given:

50Zo

455.0L

What is ZL?

5.67j5.67

35.1j35.150ZL

circleL 5.0||


24

Smith Chart Example 2

Given:

GHzfZ ro 356.275

755.37 jZL

0.15.0 jzL

cm25.656.2103

1039

8

cml 2

?? SWRZin

32.025.6

0.2l

455.0135.032.0

circleSWR 3.4

Ref. position of the load=0.135 We need to move toward to the gen. an electrical distance equal to the line length

0.2175.1828.025.075 jjZin


25

Admittance

A matching network is going to be a combination of elements connected in series AND parallel.

Impedance is NOT well suited when working with parallel configurations.

21L ZZZ

2Z1Z

2Z

1Z

21

21L

ZZ

ZZZ

ZIV

For parallel loads it is better to work with admittance.

YVI 2Y1Y

21L YYY 11

Z

1Y

Impedance is well suited when working with series configurations. For example:


26

Impedance and Admittance Smith Charts

For a matching network that contains elements connected in series and parallel, we will need two types of Smith charts impedance Smith chart

admittance Smith Chart

The admittance Smith chart is the impedance Smith chart rotated 180 degrees. We could use one Smith chart and flip the reflection

coefficient vector 180 degrees when switching between a series configuration to a parallel configuration.


27

1 0.5 0 0.5 1

1

0.5

0.5

1

1 0.5 0 0.5 1

1

0.5

0.5

1

Admittance Smith Chart

Re

Im

g=1/3

b=-1 b=-1/3

g=1 g=2.5 g=0

b=2.5 b=1/3

b=1

b=-2.5

Im

Re

jbgYZY

Yy o

o

1

1y


28

• Procedure:

• Plot 1+j1 on chart

• vector =

• Flip vector 180 degrees

Admittance Smith Chart Example 1

Given:

64445.0

What is ?

1j1y

116445.0

Plot y

Flip 180 degrees

Read


29

• Procedure:

• Plot

• Flip vector by 180 degrees

• Read coordinate

Admittance Smith Chart Example 2

Given:

What is Y?

455.0 50Zo

Plot

Flip 180 degrees

Read y

36.0j38.0y

mhos10x2.7j6.7Y

36.0j38.050

1Y

3


30

Matching Example

0

100sP 50Z0 M

Match 100 load to a 50 system at 100MHz

A 100 resistor in parallel would do the trick but ½ of the power would be dissipated in the matching network. We want to use only lossless elements such as inductors and capacitors so we don’t dissipate any power in the matching network


31

Matching Example

We need to go from z=2+j0 to z=1+j0 on the Smith chart

We won’t get any closer by adding series impedance so we will need to add something in parallel.

We need to flip over to the admittance chart

Impedance Chart


32

Matching Example

y=0.5+j0

Before we add the admittance, add a mirror of the r=1 circle as a guide.

Admittance Chart


33

Matching Example

y=0.5+j0

Before we add the admittance, add a mirror of the r=1 circle as a guide

Now add positive imaginary admittance.

Admittance Chart


34

Matching Example

y=0.5+j0

Before we add the admittance, add a mirror of the r=1 circle as a guide

Now add positive imaginary admittance jb = j0.5

Admittance Chart

pF16C

CMHz1002j50

5.0j

5.0jjb

pF16 100


35

Matching Example

We will now add series impedance

Flip to the impedance Smith Chart

We land at on the r=1 circle at x=-1

Impedance Chart


36

Matching Example

Add positive imaginary admittance to get to z=1+j0

Impedance Chart

pF16100

nH80L

LMHz1002j500.1j

0.1jjx

nH80


37

Matching Example

This solution would have also worked

Impedance Chart

pF32

100nH160


38

50 60 70 80 90 100 110 120 130 140 15040

35

30

25

20

15

10

5

0

Frequency (MHz)

Re

fle

ctio

n C

oe

ffic

ien

t (d

B)

Matching Bandwidth

1

1

vi

Im gammai

11 ui Re gammai

50 MHz

150 MHz

Because the inductor and capacitor impedances change with frequency, the match works over a narrow frequency range

pF16100

nH80

Impedance Chart


Matching with lumped elements

39

The simplest type of matching network is therefore the L section which uses two reactive elements to match an arbitrary load impedance to a transmission line.

If zL=ZL/Z0 is inside the 1+jx circle on the Smith chart the solution (a) is used.

If zL=ZL/Z0 is outside the 1+jx circle on the Smith chart the solution (b) is used.

If the frequency is low enough (frequencies up to 1 GHz) lumped capacitance and inductance can be used. There is, however, a large range of frequencies where this solution may be not realizable. This is the main limitation of the L matching section technique.


Analytical solution

40

If ZL>Z0 (implies that zL is inside the 1+jx circle on the Smith chart)

BR

Z

R

ZX

BX

XR

ZRXRZRXB

LL

L

LL

LLLLL

00

22

0

22

0

1

/

If ZL<Z0 (implies that zL is outside the 1+jx circle on the Smith chart)

0

0

0

/)(

)(

Z

RRZB

XRZRX

LL

LLL

Positive X=inductor Negative X=capacitor

Positive B=capacitor Negative B=inductor

Two solutions, but…one may result in a significant smaller value for the reactive components and may be preferred due to the matching bandwidth or VSWR on the line.


Example

41

Match a series RC load with impedance 200+j100 to a 100 line at 500 MHz

C1=0.92 pF L1=38.8H

C2=2.61 pF L2=46.1H

Solution 1 is slightly better than solution 2.


Single stub tuning

42

An important matching technique uses an open-circuited or a short-circuited transmission line (a “stub”) either in parallel or in series with the transmission line at a certain distance from the load.

Two adjustable parameters: distance (d) from the load, susceptance or reactance provided by the shunt or series stub.

Shunt: select d so that the susceptance seen is Y0+jB and add the stub with susceptance –jB.

Series: select d so that the impedance seen is Z0+jX and add the stub with impedance –jX.

Tuning circuit very easy to implement


All together!

43

Given: 1015 jZL

2.03.0 jzL

Assuming load consisting in a RL series matched at 2 Ghz, calculate the shunt stubs to have the match.

GHzfZo 250

zL

yL

y2

y1

d1 d2

0.284

0.328

0.171

33.11

33.11

2

1

jy

jy

387.0171.0)284.05.0(

044.0284.0328.0

2

1

d

d

J1.33=0.147

J1.33=0.353

Add a lenght of stub with susceptance –jB: Solution1) l=0.147 Solution2) l=0.353


Frequency response

44

Two possibilities: it is normally desired to keep the matching stub as close as possible to the load to reduce the losses caused by a possibly large standing wave ratio on the line between the stub and the load.

Solution 1 has a significantly better bandwidht than solution 2.


Double stub tuning

45

The single stub tuning is able to match any load impedance to a transmission line, but suffer of the disadvantage of requiring a varianle length of line between the load and the stub.

Design of matching with double stub tuner, or multiple stub tuner is far away from the scope of this course.


The Quarter wave transformer

46

Simple and useful method for matching a real load impedance to a transmission line. It can be extended to multisection design for a broader bandwidth.

Drawback: QWT can only match a real load impedance. A complex load impedance can always be transformed to a real impedance by using an appropriate series or shunt reactive stub.

LZZZ 01

Z0 Z1 ZL

l

At f0 l=\4 At f0 +Δf l \4

ljZZ

ljZZZZ

L

LIN

tan

tan

1

11

𝛽𝑙 =2𝜋

𝜆 𝜆

4=

𝜋

2

L

INZ

ZZ

2

1

To get =0 0ZZ IN

Geometric mean of load and source imp.


Mismatch vs. frequency

47

LZZZ 01Z0 Z1 ZL

l

00

0

2cos

2 f

fwith

ZZ

ZZ

L

L

For near /2

If we set the max value of Reflection Coefficient mthat can be tolerated, the bandwidth of the matching transformer is:

0

0

20

0

0

0

2

1cos

42

42

)(2

2

ZZ

ZZ

f

ff

f

f

ff

L

L

m

mmm

mm


Example

48

36.2201 LZZZ

Z0=50 Z1 ZL=10

L, f0=3GHz

Determine the percent bandwith for which the SWR<1.5

For this value of Z1 L=/4 at f0=3GHz. A SWR=1.5 correspond to:

2.01

1

SWR

SWRm

%2929.02

1cos

42

0

0

20

ZZ

ZZ

f

f

L

L

m

m

MHzf

f870

0


Multisection transformer

49

Z0 ZN ZL

Z1 Z2 Z3

0 1 2 3 N

N equal sections of transmission lines ZL real Z monotically decreasing: Z0> Z1> Z2> Z3>… > ZN

jN

N

jj eee 24

2

2

10 ....)(

We can synthesize any desired reflection coefficient response as a function of frequency (), by properly choosing the N and using enough sections. Commonly used:

Binomial transformer (maximally flat design) Chebyshev (equal ripple) transformer


Binomial Transformers

50

For a given number of sections, the passband response is as flat as possible near the design frequency (maximally flat design).

This kind of response is designed, for a N sections transformer, by setting the first N-1 derivatives of the reflection coefficient to zero.

Z0 ZN ZL Z1 Z2 Z3

0 1 2 3 N

jnN

n

N

n

Nj eCAeA 2

0

2 )1()(

!)!(

!2

0

0

nnN

NC

ZZ

ZZA N

n

L

LN

where:


Binomial Transformers Bandwidth

51

If we set the max value of Reflection Coefficient mthat can be tolerated, the bandwidth of the matching transformer is:

N

mmm

NN

mA

aA

/1

2

1coscos2

N

mmm

Af

ff

f

f/1

0

0

0 2

142

42

)(2


Tables

52


Example (analytical solution)

53

Z0=100 Z3 ZL=50 Z1 Z2

0 1 2 3

Design the 3 sections transformer, calculate BW per m=0.05

3!1!2

!33

!1!2

!31

!0!2

!3

71.02

1cos

420417.02

3

2

3

1

3

/1

00

0

CCC

Aa

f

f

ZZ

ZZA

o

N

m

L

LN

5.54;00.4100

50ln)1(27.70lnln2lnln;2

7.70;26.4100

50ln)3(27.01lnln2lnln;1

7.91;518.4100

50ln)1(2100lnln2lnln;0

1

33

023

2

33

112

1

33

001

ZZ

ZCZZn

ZZ

ZCZZn

ZZ

ZCZZn

o

LN

o

LN

o

LN

Greater BW!!!


Example with tables

54

Z0=100 Z3 ZL=50 Z1 Z2

0 1 2 3

Design the 3 sections transformer, calculate BW per m=0.05

53.548337.1

71.704142.1

68.910907.1

03

02

01

ZZ

ZZ

ZZ

Greater BW!!!


Chebyschev Multisection matching transformers

55

In contrast with binomial transformers, the chebyschev ones optimize bandwidth at expenses of passband ripple.

)()(2)(

.................................

188)(

34)(

12)(

)(

21

24

4

3

3

2

2

1

xTxxTxT

xxxT

xxxT

xxT

xxT

nnn

For -1<x<1, |Tn(x)|<1. In this range the polynomials oscillate between ±1. This is the equal ripple property and this region will be mapped to the passband of the matching transformer. For |x|>1, |Tn(x)|>1. |Tn(x)| increases faster as n increases. This region will map to the frequency range outside the passband.

Design procedure based on Chebyshev polynomials:



56

We need to map: m to x=-1 and - m to x=1

)()(2)(

.................................

1)12(cossec4)32cos44(cossec)cos(sec

cossec3)cos33(cossec)cos(sec

1)2cos1(sec)cos(sec

cossec)cos(sec

21

24

4

3

3

2

2

1

xTxxTxT

T

T

T

T

nnn

mm

mmm

mm

mm

Often used for D.C., filters…



57

)(sec

1)cos(sec)(

0

0

mNL

LmN

Nj

TZZ

ZZAwhereTAe

Now if m is tha maximum allowable reflection coefficient in the passband then A=m, since the maximum value in the passband of TN(secmcos ) is the unity. Then m and fractional bandwidth are given by:

m

L

L

m

m

f

f

ZZ

ZZa

N

42

1cosh

1coshsec

0

0

0


Tables

58


Example with tables

59

Z0=50 Z3 ZL=100 Z1 Z2

0 1 2 3

Design a transformer with maximum reflection coefficient equal to 0.05 (A=m=0.05).

%10202.14

2

2.44395.11

cosh1

coshsec

0

0

0

m

m

L

L

m

m

f

f

ZZ

ZZa

N

15.877429.1

71.704142.1

37.571475.1

03

02

01

ZZ

ZZ

ZZ


Tapered Lines

60

As the number of the discrete sections increases, the step changes in the characteristic impedance become smaller. Thus, in the limit of an infinite number of sections, we approach a continuosly tapered line.

By changing the type of taper, we can obtain different passband characteristics.

)(zZz

Lzj dz

Z

Z

dz

de

00

2 ln2

1)(

Three special cases analyzed in the following: •Exponential taper •Triangular taper •Klopfenstein taper

For a given taper lenght the klpofstein impedance taper is the optimum in the sense that the reflection coefficient is minimum over the passband.

It is derived from a stepped Chebyshev transformer as the number of sections increas to infinity.


61

Impedance (Z) and Admittance (Y) Matrices

This linear two port “black box” can be fully described if we know how the voltage and the current at port 1 relate to the voltage and the current at port 2.

1V 2V

2I1I

2221212

2121111

IzIzV

IzIzV

zij and yij characterize the properties of the black box.

IZV

2221212

2121111

VzVyI

VyVyI

VYI


62

Impedance (Z) and Admittance (Y) Matrices

We need 4 measurements to determine [Y] or [Z]:

Open output (I2=0)

1V 2V

2I1I

2221212

2121111

IzIzV

IzIzV

2221212

2121111

VzVyI

VyVyI

1

111

I

Vz

1

221

I

Vz

Open input (I1=0)

2

112

I

Vz

2

222

I

Vz

Short output (V2=0)

1

111

V

Iy

1

221

V

Iy

Short input (V1=0)

2

112

V

Iy

2

222

V

Iz


S matrix

63

It is helpful to think about travelling waves along a TRL!

gV

2V

2221212

2121111

VsVsV

VsVsVgZ

LZ

2V

1V

1V

j

iij

V

Vs

Means we are sitting and measuring the receding voltage wave at port I, while the driving source is at port j.

02

1

111

VwhenV

Vs

01

2

112

VwhenV

Vs

02

1

221

VwhenV

Vs

01

2

222

VwhenV

Vs

the input reflection coefficient

the reverse transmission coefficient the reverse reflection coefficient

the input transmission coefficient


64

Normalized Scattering (S) Parameters

Since measuring voltages and currents at microwave frequencies becomes unpratical, the S- parameters are related to the incident and reflected power.

The S matrix defined previously is called the un-normalized scattering matrix. For convenience, define normalized waves:

ii o

ii

o

ii

Z

Vb

Z

Va

22

Where Zoi is the characteristic impedance of the transmission line connecting port (i)

|ai|2 is the forward power into port (i)

|bi|2 is the reverse power from port (i)


65

Normalized Scattering (S) Parameters

The normalized scattering matrix is:

Where:

2221212

2121111

asasb

asasb

j,i

io

jo

j,i SZ

Zs

If the characteristic impedance on both ports is the same then the normalized and un-normalized S parameters are the same.

Normalized S parameters are the most commonly used.

Energy conservation: for a loss less network SUM(Sij)=1


66

Normalized S Parameters

The s parameters can be drawn pictorially

s11 and s22 can be thought of as reflection coefficients

s21 and s12 can be thought of as transmission coefficients

s parameters are complex numbers where the angle corresponds to a phase shift between the forward and reverse waves

s11 s22

s21

s12

a1

a2 b1

b2


Why S-parameters?

67

• S-parameters are a powerful way to describe an electrical network • The ease with which scattering parameters can be measured makes them

especially well suited to describe transistor or other active devices. • S-parameters change with freqency / load impedance / source impedance /

network • The most important advantage of S-parameters stems from the fact that

travelling waves, unlike terminal voltage and currents, do not vary in magnitude along a transmission line.

• S11 is the reflection coefficient • S21 describes the forward transmission coefficient (responding port 1st!) • S-parameters have both magnitude and phase information • Sometimes the gain (or loss) is more important than the phase shift and the phase

information may be ignored • S-parameters may describe large and complex networks


68

Examples of S parameters

2 1

10

01s

1 2

0G

00sG

Short

Amplifier

1

2

01

00s

Zo

Isolator


69

Lorentz Reciprocity

If the device is made out of linear isotropic materials (resistors, capacitors, inductors, metal, etc..) then:

ssT

j,ii,j ss ji

or

for

This is equivalent to saying that the transmitting pattern of an antenna is the same as the receiving pattern

reciprocal devices: transmission line

short

directional coupler non-reciprocal devices: amplifier

isolator

circulator


70

Lossless Devices

The s matrix of a lossless device is unitary:

1ssT*

j

2

j,i

i

2

j,i

s1

s1for all j

for all i

Lossless devices: transmission line

short

circulator

Non-lossless devices: amplifier

isolator


71

Network Analyzers

Network analyzers measure S parameters as a function of frequency

At a single frequency, network analyzers send out forward waves a1 and a2 and measure the phase and amplitude of the reflected waves b1 and b2 with respect to the forward waves.

a1 a2

b1 b2

02a1

111

a

bs

02a1

221

a

bs

01a2

112

a

bs

01a2

222

a

bs


72

Network Analyzer Calibration

To measure the pure S parameters of a device, we need to eliminate the effects of cables, connectors, etc. attaching the device to the network analyzer

s11 s22

s21

s12

x11 x22

x21

x12

y11 y22

y21

y12

yx21

yx12

Connector Y Connector X

We want to know the S parameters at these reference planes

We measure the S parameters at these reference planes


73


There are 10 unknowns in the connectors

We need 10 independent measurements to eliminate these unknowns Develop calibration standards

Place the standards in place of the Device Under Test (DUT) and measure the S- parameters of the standards and the connectors

Because the S parameters of the calibration standards are known (theoretically), the S parameters of the connectors can be determined and can be mathematically eliminated once the DUT is placed back in the measuring fixtures.


74


Since we measure four S parameters for each calibration standard, we need at least three independent standards.

One possible set is:

10

01s

0e

e0s

j

j

01

10sThru

Short

Delay* *~90degrees


Waveguides

75

The mathematical solution to the waveguide problem is tedious. In order to find the various field components in a rectangular or circular guide six differential equations have to be solved.

EjH

HjE

zj

z

zj

z

eyxhzyxhzyxH

eyxezyxezyxE

)],(),([),,(

)],(),([),,(

^

^

Assuming that: a)waveguide region is source-free, b)z direction propagation (inifitely long structure), c) time harmonic fields with exp(jwt) dependence, d) perfectly conducting wall, the Maxwell equation can be rewritten as:

TEM waves Ez=Hz=0

TE waves Ez=0 Hz0

TM waves Ez0 Hz=0

TEM wave has no cutoff and can exist when two or more conducors are present.

Cutoff!


Rectangular Waveguide

76

No TEM waves can propagate!

EjH

HjE

Boundary conditions

0,0

0,0

zy

zy

EEbyy

EEaxx

TEmn waves Ez=0 Hz0

zj

mnzz

zj

mn

c

y

zj

mn

c

y

zj

mn

c

x

zj

mn

c

x

eb

yn

a

xmAHE

eb

yn

a

xmA

bk

njHe

b

yn

a

xmA

ak

mjE

eb

yn

a

xmA

ak

mjHe

b

yn

a

xmA

bk

njE

coscos0

sincoscossin

cossinsincos

22

22

22

222

b

n

a

mkkk c

22

b

n

a

mkk c

>0 for propagation !!!

numberwavek

2


Propagation constant and cutoff frequency

77

22

2

1

2

b

n

a

mkf c

Cmn

Each mode (combination of m and n) has a cutoff frequency given by:

At a given frequency f only those modes having f>fc will propagate! The contrary will lead to an imaginary meaning that all field components will decay exponentially from the source excitation (evanescent mode)!

The mode with lowest cutoff is called dominant mode, if a>b the lowest fc occur for the TE10 mode:

zj

y

zj

x

zj

z

yzx

ea

xA

ajE

ea

xA

ajH

ea

xAH

HEE

sin

sin

cos

0

10

10

10

afC

2

110

mNpkabkba

R

aabARP

bAaP

Sc

sl

/2

22

]Re[4

232

3

2

322

10

2

2

10

3

10


Rectangular Waveguide

78

EjH

HjE

Boundary conditions

0,0

0,0

zy

zy

EEbyy

EEaxx

TMmn waves Ez 0 Hz=0

0sinsin

sincoscossin

cossinsincos

22

22

z

zj

mnz

zj

mn

c

y

zj

mn

c

y

zj

mn

c

x

zj

mn

c

x

Heb

yn

a

xmBE

eb

yn

a

xmA

ak

mjHe

b

yn

a

xmB

bk

njE

eb

yn

a

xmB

bk

njHe

b

yn

a

xmB

ak

mjE

22

222

b

n

a

mkkk c

>0 for propagation !!!

Fields are zero if the indexes m or n are zero! The lowes TM mode that can propagate into the waveguide is the TM11.

22

2

1

b

n

a

mfCmn


Field lines for some low order modes

79


Other parameters

80

wave impedance

k

H

E

H

EZ

y

x

x

y

TE

/

medium intrinsic impedance

kg

22 The guided wavelenght is the distance between two equal phase planes inside the waveguide

kH

E

H

EZ

y

x

x

y

TM

1

kvp Phase velocity


Table of rectangular waveguide

81

WG EIA-WR IEC-R Fc1, GHz Flo, GHz Fhi, GHz Fc2, GHz A, in B, in A, mm B, mm

00 2300 3 .257 0.32 0.49 .513 23.000 11.500 584.200 292.100

0 2100 4 .281 0.35 0.53 .562 21.000 10.500 533.400 266.700

1 1800 5 .328 0.41 0.625 .656 18.000 9.000 457.200 228.600

2 1500 6 .393 0.49 0.75 .787 15.000 7.500 381.000 190.500

3 1150 8 .513 0.64 0.96 1.026 11.500 5.750 292.100 146.050

4 975 9 .605 0.75 1.12 1.211 9.750 4.875 247.650 123.825

5 770 12 .766 0.96 1.45 1.533 7.700 3.850 195.580 97.790

6 650 14 .908 1.12 1.70 1.816 6.500 3.250 165.100 82.550

7 510 18 1.157 1.45 2.20 2.314 5.100 2.550 129.540 64.770

8 430 22 1.372 1.70 2.60 2.745 4.300 2.150 109.220 54.610

9A 340 26 1.736 2.20 3.30 3.471 3.400 1.700 86.360 43.180

10 284 32 2.078 2.60 3.95 4.156 2.840 1.340 72.136 34.036

11A 229 40 2.577 3.30 4.90 5.154 2.290 1.145 58.166 29.083

12 187 48 3.152 3.95 5.85 6.305 1.872 .872 47.549 22.149

13 159 58 3.712 4.90 7.05 7.423 1.590 .795 40.386 20.193

14 137 70 4.301 5.85 8.20 8.603 1.372 .622 34.849 15.799

15 112 84 5.260 7.05 10.0 10.519 1.122 .497 28.499 12.624

16 90 100 6.557 8.20 12.4 13.114 .900 .400 22.860 10.160

17 75 120 7.869 10.0 15.0 15.737 .750 .375 19.050 9.525

18 62 140 9.488 12.4 18.0 18.976 .622 .311 15.799 7.899

19 51 180 11.571 15.0 22.0 23.143 .510 .255 12.954 6.477

20 42 220 14.051 18.0 26.5 28.102 .420 .170 10.668 4.318

21 34 260 17.357 22.0 33.0 34.714 .340 .170 8.636 4.318

22 28 320 21.077 26.5 40.0 42.153 .280 .140 7.112 3.556

23 22 400 26.346 33.0 50.0 52.691 .224 .112 5.690 2.845

24 19 500 31.391 40.0 60.0 62.781 .188 .094 4.775 2.388

25 15 620 39.875 50.0 75.0 79.749 .148 .074 3.759 1.880

26 12 740 48.372 60.0 90.0 96.745 .122 .061 3.099 1.549

27 10 900 59.014 75.0 110 118.029 .100 .050 2.540 1.270

28 8 1200 73.768 90.0 140 147.536 .080 .040 2.032 1.016

29 7 1400 90.791 110 170 181.582 .065 .033 1.651 .826

30 5 1800 115.714 140 220 231.428 .051 .026 1.295 .648

31 4 2200 137.242 170 260 274.485 .043 .022 1.092 .546

32 3 2600 173.571 220 325 347.143 .034 .017 .864 .432

- 2 - 295.071 325 500 590.143 .020 .010 .508 .254


Circular waveguide

82

TEnm modes

0

)()cossin(

)()sincos(

'

2

z

zj

cn

c

zj

cn

c

E

ekJnBnAk

jE

ekJnBnAk

njE

zj

cnz

zj

cn

c

zj

cn

c

ekJnBnAH

ekJnBnAk

njH

ekJnBnAk

jH

)()cossin(

)()sincos(

)()cossin(

2

'

Jn(x) is the n-order Bessel function of first kind

J’n(x) is the n-order Bessel function derivative of first kind

2'

222

a

pkkk nm

c >0 for propagation !!!

a

pkf nmc

Cmn22

'

Cutoff frequencies

where p’nm is the mth root of J’n(x)

n=0 n=1 n=2 n=3 n=4 n=5 n=6

m=1

m=2

m=3


Circular waveguide

83

TMnm modes

zj

cnz

zj

cn

c

zj

cn

c

ekJnBnAE

ekJnBnAk

njE

ekJnBnAk

jE

)()cossin(

)()sincos(

)()cossin(

2

'

0

)()cossin(

)()sincos(

'

2

z

zj

cn

c

zj

cn

c

H

ekJnBnAk

jH

ekJnBnAk

njH



2

222

a

pkkk nm

c >0 for propagation !!!

a

pkf nmc

Cmn22

Cutoff frequencies

where pnm is the mth root of Jn(x)

n=0 n=1 n=2 n=3 n=4 n=5 n=6

m=1

m=2

m=3


Field lines for some low order modes

84

0 1 2 3

11TE 21TE 01TE31TE

41TE

12TE

01TM 11TM 21TM 02TM

)( 11TEf

f

c

c


Circular waveguides table

85


Microwave coupling to plasma source

86

Water-cooled Bend

Microwave pressure window Placed behind the bend in order to avoid any damage due to the back-streaming electrons

Matching Transformer It realizes a progressive match between the waveguide impedance and the plasma chamber impedance, thus concentrating the electric field at the center of the plasma chamber

DC- Break • Low microwave losses - Good voltage insulation up to 100 kV

Automatic Tuning Unit

Adjusts the modulus/phase of the incoming wave to match the

plasma impedance Dual-arm Directional

Coupler

It is used to measure the forward and the reflected power

Water- cooled copper plasma chamber

Waveguide /Coax Adapter Excytes the TE10 dominant mode in WR340 waveguide


Multisection matching transformers in WG

87

TRIPS ELECTROMAGNETIC

FIELD @ 2.45 GHz, 500 W

VIS ELECTROMAGNETIC

FIELD @ 2.45 GHz, 500 W

10 % ENHANCEMENT WITH VIS TRANSFORMER

VIS TRANSFORMER INSERTION

LOSS 0.0085 dB @ 2.45 GHz

S21 Vs FREQUENCY FOR THE TWO STRUCTURES

ELECTROMAGNETIC FIELD @ 2.45 GHz,

500 W

Four step double ridges

Matching transformer coupled to the plasma chamber


Microwave resonators

88

Used in a wide variety of applications: filters, oscillators, frequency meters, tuned amplifiers, etc… The behaviour near resonance is very similar to the lumped element resonators.

EmlossIN

ININ

IN

WWjPP

CjLjRIIZVIP

CjLjRZ

2

1

2

1

2

1

2

1

1

22

When resonance occurs the average storage electric and magnetic energies are equal

LEm PWWondlossenergy

storedenergyaverageQ /

sec/

At resonance: R

L

RCQ 0

0

1

LC

10

Fractional BW: Q

BW1


Rectangular waveguide cavities

89

Usually short circuited at both ends forming a closed box or a cavity. Power dissipated on metallic walls as well as in the dielctric filling the cavity.

Cutoff wavenumber and resonant frequencies

222

222

22

d

l

b

n

a

mcckf

d

l

b

n

a

mk

rrrr

mnlmnl

mnl

Q for TE10l mode

8

22

1

2

2

0

3323322

3

EabdP

addalbdbalR

bkadQ

d

S


Circular waveguide cavities

90

Resonant frequencies TEnml mode

22

22'

2

2

d

l

a

pcf

d

l

a

pcf

nm

rr

nml

nm

rr

nml



where p’nm is the mth root of J’n(x) and pnm is the mth root of Jn(x),

Resonant frequencies TMnml mode

Q evaluation

Fundamentals of Microwave Engineering RF coupling...

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