Lecture 32017/2018

2C/1L, MDCR Attendance at minimum 7 sessions (course + laboratory) Lectures- assistant professor Radu Damian Monday 16-18, P2 E – 50% final grade problems + (? 1 topic teory) + (2p atten. lect.) + (3 tests) +

(bonus activity)▪ 3p=+0.5p

all materials/equipments authorized Laboratory – assistant professor Radu Damian Monday 18-20 II.12 even weeks Thursday 8-14 odd weeks II.12 ? L – 25% final grade P – 25% final grade

RF-OPTO

http://rf-opto.etti.tuiasi.ro

David Pozar, “Microwave Engineering”, Wiley; 4th edition , 2011

1 exam problem Pozar

Photos

sent by email: rdamian@etti.tuiasi.ro

used at lectures/laboratory

Not customized

ADS 2016 EmPro 2015 based on IP from outside

university or campus

“Engineering” Sinapses

> 2010 < 1950

Complex numbers arithmetic!!!! z = a + j · b ; j2 = -1

standard unit for angles – radians microwaves traditional unit for angles –

degrees in decimal form (55.89°)

rad 180

180

rad

0,2

,2

0,0,arctan

0,0,arctan

0,arctan

arg

anedefinit

baa

b

baa

b

aa

b

z

Attention to angle numerical values!! math software – work in standard unit: radians

▪ a conversion is necessary before and after using a trigonometric function (sin, cos, tan, atan, tanh)

scientific calculators have the built-in option of choosing the angle unit▪ always double check current working unit

rad 180

180

rad

0 dBm = 1 mW

3 dBm = 2 mW5 dBm = 3 mW10 dBm = 10 mW20 dBm = 100 mW

-3 dBm = 0.5 mW-10 dBm = 100 W-30 dBm = 1 W-60 dBm = 1 nW

0 dB = 1

+ 0.1 dB = 1.023 (+2.3%)+ 3 dB = 2+ 5 dB = 3+ 10 dB = 10

-3 dB = 0.5-10 dB = 0.1-20 dB = 0.01-30 dB = 0.001

dB = 10 • log10 (P2 / P1) dBm = 10 • log10 (P / 1 mW)

[dBm] + [dB] = [dBm]

[dBm/Hz] + [dB] = [dBm/Hz]

[x] + [dB] = [x]

Behavior (and description) of any circuit depends on his electrical length at the particular frequency of interest E≈0 Kirchhoff

E>0 wave propagation

lllE 2

2

Source matched to load ?

Ei

Zi

ZL

I

V

impedance values ? existence of

reflections ?

*iL ZZ

*iL

The source has the ability to sent to the load a certain maximum power (available power) Pa

For a particular load the power sent to the load is less than the maximum (mismatch) PL < Pa

The phenomenon is “as if” (model) some of the power is reflected Pr = Pa – PL

The power is a scalar !

Ei

ZiPa

aL

iL

PP

ZZ

*

Ei

Zi ZL

PL

Ei

Zi

ZL

Pa PL

Pr

+

TEM wave propagation, at least two conductorsI(z,t)

V(z,t)

z

I(z,t)

V(z,t)

Δz

I(z+Δz,t)

V(z+Δz,t)

L·ΔzR·Δz

G·Δz C·Δz

TEM wave propagation, at least two conductors

time domain

harmonic signals

t

tziLtziR

z

tzv

,,

,

t

tzvCtzvG

z

tzi

,,

,

zILjR

dz

zdV

zVCjG

dz

zdI

02

2

2

zVdz

zVd

02

2

2

zIdz

zId

CjGLjRj

022 EE

022 HH

j 22

zz

y eEeEE

Characteristic impedance of the line

zz eVeVzV 00

zz eIeIzI 00

zILjR

dz

zdV

zz eVeVzV 00

zz eVeVLjR

zI

00

CjG

LjRLjRZ

0

CjGLjRj

2 fv f

0

00

0

0

I

VZ

I

V

R=G=0

CLjCjGLjRj

CL ;0

C

L

CjG

LjRZ

0 Z0 is real

zjzj eVeVzV 00

zjzj eZ

Ve

Z

VzI

0

0

0

0LC

2

LCv f

1

voltage reflection coefficient

ΓL

Z0 ZL

0

0

0

0

ZZ

ZZ

V

V

L

L

l

0

0

0

I

VZL 0

00

00 ZVV

VVZL

zjzj eVeVzV 00

zjzj eZ

Ve

Z

VzI

0

0

0

0

Z0 real

voltage reflection coefficient seen at the input of the line

ΓL

Z0 ZL

-l 0

Zin

ΓIN

zjzj eVeVzV 00

ljlj eVeVlV 00

0

00V

VL 000 VVV

zV

zVz

0

0

lj

lj

lj

IN eeV

eVl

2

0

0 0

00 2 ljel

ljel 20

time-average Power flow along the line

zjzj eeVzV 0 zjzj ee

Z

VzI

0

0

Total power delivered to the load = Incidentpower – “Reflected” power

Return “Loss” [dB]

the input impedance seen looking toward the load

ΓL

Z0 ZL

-l0

lI

lVZin

lj

lj

ine

eZZ

2

2

01

1

ljlj eVeVlV 00

ljlj eZ

Ve

Z

VlI

0

0

0

0

Zin

lj

Llj

L

ljL

ljL

ineZZeZZ

eZZeZZZZ

00

000

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

input impedance of a length l of transmission line with characteristic impedance Z0 , loaded with an arbitrary impedance ZL

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

ΓL

Z0 ZL

-l 0

Zin

input impedance is frequency dependent through

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

ΓL

Z0 ZL

-l 0

Zin

2fv f

l

fv

ll

v

fll

ff

222

frequency dependence is periodical, imposed by the tan trigonometric function

l = k·λ/2 l = λ/4 + k·λ/2

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

ΓL

Z0 ZL

-l 0

Zin

kll

20ZZin 0tan l

ltan

L

inZ

ZZ

2

0

quarter-wave transformer

purely imaginary for any length l

+/- depending on l value

lZjZin tan0

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

purely imaginary for any length l

+/- depending on l value

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

lZjZin cot0

SWR is defined as the ratio between maximum and minimum (Voltage) Standing Wave Ratio

real number 1 ≤ VSWR < a measure of the mismatch (SWR = 1 means a matched line)

zjzj eeVzV 0

zjzj eeVzV 20 1

je

zjeVzV 20 1

12 zje maximum magnitude value for 10max VV

12 zje 10min VV

1

1

min

max

V

VVSWR

minimum magnitude value for

lZjZ

lZjZZZ

L

Lin

tan

tan

0

00

ΓL

Z0 ZL

-l 0

Zin

zz eVeVzV 00

zz eIeIzI 00

ljel 20

Impedance Matching with Impedance Transformers (Lab 1)

Feed line – input line with characteristic impedance Z0

Real load impedance RL

We desire matching the load to the fider with a second line with the length λ/4 and characteristic impedance Z1

)tan(

)tan(

1

11

ljRZ

ljZRZZ

L

Lin

lj

lj

ine

eZZ

2

2

11

1

1

1

0

0

ZR

ZR

V

V

L

LO

In the feed line (Z0) we have only progressive wave

In the quarter-wave line (Z1) we have standing waves

24

2

l

0

0

ZZ

ZZ

in

inin

L

inR

ZZ

2

1

0in LRZZ 01 L

Lin

RZZ

RZZ

0

2

1

0

2

1

1T

The Multiple-Reflection Viewpoint

The Multiple-Reflection Viewpoint

00 0

2

1 LRZZ

(only) at f0

24

2 0

0

0

l

4

0l

)tan(

)tan(

1

11

lZjZ

lZjZZZ

L

Lin

)tan( lt

not

tZjZ

tZjZZZ

L

Lin

1

11

LZZZ 01

lnot

matching quality power reflection coefficient

222 1tan1sec

cos

1sec

t

we assume that the operating frequency is near the design frequency (narrow bandwidth)

0ff 1tan1sec 22 4

0l2

cos2 0

0

L

L

ZZ

ZZ

we set a maximum value Гm for an acceptable reflection coefficient magnitude then the bandwidth of the matching transformer, θm

for TEM lines

00

0

24

12

4 f

f

f

v

v

fl

f

f

02 ff m

m

0

0

2

1

0

0

0

2

1cos

42

42

2

ZZ

ZZ

f

ff

f

f

L

L

m

mmm

When non-TEM lines (such as waveguides) are used, the propagation constant is no longer a linear function of frequency, and the wave impedance will be frequency dependent, but in practice the bandwidth of the transformer is often small enough that these complications do not substantially affect the result

We ignored also the effect of reactancesassociated with discontinuities when there is a step change in the dimensions of a transmission line (Z0 -> Z1). This can often be compensated by making a small adjustment in the length of the matching section

Bandwidth depends on the initial mismatch

increased bandwidth for smaller load mismatches

A quarter-wave matching transformer to match a 10Ω load to a 50 Ω transmission line at f0=3GHz

Determine the percent bandwidth for SWR<1.5

ADS Simulation

GHzf 88.0

51033 GHz

2933.03

88.0

0

f

f

The quarter-wave transformer can match any real load to any feed line impedance

If a greater bandwidth for the match is required we must use multiple sections of transmission lines transformers:

binomial

Chebyshev

jjj eTTeTTeTT 622

332112

42

232112

2321121

12

121

ZZ

ZZ

12

2

23

ZZ

ZZ

L

L

21

2121

21

ZZ

ZT

21

1212

21

ZZ

ZT

0

223

2321121

n

jnnnj eeTT

If the discontinuities between the impedances Z1 Z2 and Z2 ZL are small we can approximate

0

223

2321121

n

jnnnj eeTT

11

1

0

xx

xn

n

j

j

e

e2

31

231

1

je 231

We also assume that all impedances increase or decrease monotonically across the transformer

This implies that all reflection coefficients will be real and of the same sign

Previously, 1 section

01

011

ZZ

ZZ

nn

nnn

ZZ

ZZ

1

1

NL

NLN

ZZ

ZZ

1,1 Nn

je 231

jNN

jj eee 242

210

assume that the transformer can be made symmetrical

Note that this does not imply that the impedances are symmetrical

22110 ,, NNN

jNN

jj eee 242

210

442

2210

NjNjNjNjjNjNjN eeeeeee

nNNNe njN 2cos2coscos2 10

last item: even2

12/ NN

oddcos2/1 NN

Input reflection coefficient

we can choose the coefficients so we obtain a desired behavior (of the polynomial)

jNN

jj eee 242

210

xe j 2

NN xaxaxaaxf 2

210

The response is as flat as possible near the design frequency, also known as maximally flat

For N sections the first N-1 derivatives of the |Γ(θ)| functions are annuled

0;02 2

n

n

d

d

NxAxf 1

NjeA 21

NNN

jjN

j AeeeA cos2

1,1 Nn24

ll

A, θ 0 , 0 length sections, the sections disappear

Binomial expansion

Reflection coefficient:

NNN

nnNNN

NxCxCxCCxxf 101

!!

!

nnN

NC n

N

0

0

0

0 220ZZ

ZZA

ZZ

ZZA

L

LN

L

LN

jNN

jj eee 242

210

nNn CA

NjeA 21

nNn CA

n

n

nn

nnn

Z

Z

ZZ

ZZ 1

1

1 ln2

1

1

1

12ln

xx

xx

00

01 ln22222lnZ

ZC

ZZ

ZZCA

Z

Z LnN

N

L

LNnNn

n

n

0

1 ln2lnlnZ

ZCZZ Ln

NN

nn

Manual design procedure

0

02ZZ

ZZA

L

LN

Bandwidth, Γm maximum acceptable value

N

mN

mm A cos2

Nm

mA

1

1

2

1cos

N

mmm

Af

ff

f

f

1

1

0

0

0 2

1cos

42

42

2

Design a three-section binomial transformer to match a 30Ω load to a 100 Ω line at f0=3GHz, Γm=0.1

N = 3

30LZ 1000Z

0.07525ln2

12

01

0

0

Z

Z

ZZ

ZZA L

NL

LN

1!0!3

!30

3

C 3!1!2

!31

3

C 3!2!1

!32

3

C

0n

4.455100

30ln12100lnln2lnln 3

0

0301

Z

ZCZZ LN

03.861Z

1n

77.542Z

2n

552.3100

30ln3277.54lnln2lnln 3

0

2323

Z

ZCZZ LN

87.343Z

4.003100

30ln3203.86lnln2lnln 3

0

1312

Z

ZCZZ LN

74.00.07525

1.0

2

1arccos

42

2

1arccos

42

311

0

N

m

Af

f

GHzf 22.2

Similarly Lab. 1

GHzf 169.2

6105.33 GHz

The response of this multisection impedance transformer is equal-ripple in passband

optimizes (increases) bandwidth at the expense of passband ripple

We match the Γ(θ) function with an desired Chebyshev polynomial

equal-ripple

xxT 1

12 2

2 xxT

xxxT 34 3

3

188 24

4 xxxT

xTxxTxT nnn 212

111 xTx n

We can show that:

nNNNe njN 2cos2coscos2 10

jNN

jj eee 242

210

xe j 2

NN xaxaxaaxf 2

210

cosx

xnxTn arccoscos )(coshcosh)( 1 xnxTn1x 1x

nTn coscos

1x

last item:even

2

12/ NN

oddcos2/1 NN

variable change so we map:

cos

1sec

1 xm

1 xm

m

x

cos

cos

cossec mx

We search coefficients of Γ(θ) function to obtain a Chebyshevpolynomial

cosseccossec1 mmT

12cos1seccossec 2

2 mmT

cossec3cos33cosseccossec 3

3 mmmT

112cossec432cos44cosseccossec 24

4 mmmT

nNNNe njN 2cos2coscos2 10

cossec mNjN TeA

last item:even

2

12/ NN

oddcos2/1 NN

A, θ 0 , 0 length sections, the sections disappear

mN

L

L TAZZ

ZZsec0

0

0

mNL

L

TZZ

ZZA

sec

1

0

0

00

0 ln2

11sec

Z

Z

ZZ

ZZT L

mL

L

m

mN

)(coshcosh)( 1 xnxTn

m

L

L

L

m

m

ZZ

NZZ

ZZ

N 2

lncosh

1cosh

1cosh

1coshsec 01

0

01

mm

f

ff

f

f 42

2

0

0

0

Am

Design a three-section Chebyshevtransformer to match a 30Ω load to a 100 Ωline at f0=3GHz, Γm=0.1

N = 3 30LZ 1000Z

cosseccos3cos2 33

103

mjj TAee

1.0 mA

362.1

1.02

10030lncosh

3

1cosh

2

lncosh

1coshsec 101

m

Lm

ZZ

N

76.42746.0

sec

1arccos rad

m

m

mNL

L

TZZ

ZZA

sec

1

0

0

00 AZZL 1.0A

cossec3cos33cosseccos3cos2 310 mm AA

mA 30 sec2 1263.00

cos mmA secsec32 31 1747.01

3cos

simetrie: 1203 ;

0n

4.3531263.02100ln2lnln 001 ZZ

68.771Z

1n

77.542Z

2n

62.383Z

1263.00

1747.01

4.0031747.0268.77ln2lnln 112 ZZ

654.31747.0277.54ln2lnln 223 ZZ

GHzf 15.3

045.1

180

76.4242

42

2

0

0

0

mm

f

ff

f

f

Similarly Lab. 1

GHzf 096.3

51017.43 GHz

09925.0282.2 GHz

G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks,and Coupling Structures, ArtechHouse Books, Dedham, Mass. 1980

Impedance Matching

Feed line – input line with characteristic impedance Z0

Real load impedance RL

We desire matching the load to the fider with a second line with the length λ/4 and characteristic impedance Z1

)tan(

)tan(

1

11

ljRZ

ljZRZZ

L

Lin

lj

lj

ine

eZZ

2

2

11

1

1

1

0

0

ZR

ZR

V

V

L

LO

In the feed line (Z0) we have only progressive wave

In the quarter-wave line (Z1) we have standing waves

24

2

l

0

0

ZZ

ZZ

in

inin

L

inR

ZZ

2

1

0in LRZZ 01 L

Lin

RZZ

RZZ

0

2

1

0

2

1

Bandwidth depends on the initial mismatch

increased bandwidth for smaller load mismatches

ADS Simulation

GHzf 88.0

51033 GHz

2933.03

88.0

0

f

f

The quarter-wave transformer can match any real load to any feed line impedance

If a greater bandwidth for the match is required we must use multiple sections of transmission lines transformers:

binomial

Chebyshev

We also assume that all impedances increase or decrease monotonically across the transformer

This implies that all reflection coefficients will be real and of the same sign

Previously, 1 section

01

011

ZZ

ZZ

nn

nnn

ZZ

ZZ

1

1

NL

NLN

ZZ

ZZ

1,1 Nn

je 231

jNN

jj eee 242

210

Similarly Lab. 1

GHzf 169.2

6105.33 GHz

Similarly Lab. 1

GHzf 096.3

51017.43 GHz

09925.0282.2 GHz

Microwave and Optoelectronics Laboratory http://rf-opto.etti.tuiasi.ro rdamian@etti.tuiasi.ro