Lecture 3 2017/2018rf-opto.etti.tuiasi.ro/docs/files/MDCR Lecture 3.pdf · Lecture 3 2017/2018 ... 0 dBm = 1 mW 3 dBm = 2 mW 5 dBm = 3 mW 10 dBm = 10 mW 20 dBm = 100 mW-3 dBm = 0.5

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

0,0,arctan

0,arctan

anedefinit

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

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

Source matched to load ?

impedance values ? existence of

reflections ?

*iL ZZ

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 !

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

V(z,t)

I(z,t)

V(z,t)

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

tziLtziR

tzvCtzvG

CjGLjRj

022 EE

022 HH

y eEeEE

Characteristic impedance of the line

zz eVeVzV 00

zz eIeIzI 00

zz eVeVzV 00

zz eVeVLjR

LjRLjRZ

CjGLjRj

2 fv f

CLjCjGLjRj

0 Z0 is real

zjzj eVeVzV 00

zjzj eZ

voltage reflection coefficient

00 ZVV

zjzj eVeVzV 00

zjzj eZ

Z0 real

voltage reflection coefficient seen at the input of the line

zjzj eVeVzV 00

ljlj eVeVlV 00

VL 000 VVV

IN eeV

00 2 ljel

ljel 20

time-average Power flow along the line

zjzj eeVzV 0 zjzj ee

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

Return “Loss” [dB]

the input impedance seen looking toward the load

ljlj eVeVlV 00

ljlj eZ

ineZZeZZ

eZZeZZZZ

lZjZZZ

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

lZjZZZ

input impedance is frequency dependent through

lZjZZZ

frequency dependence is periodical, imposed by the tan trigonometric function

l = k·λ/2 l = λ/4 + k·λ/2

lZjZZZ

20ZZin 0tan l

quarter-wave transformer

purely imaginary for any length l

+/- depending on l value

lZjZin tan0

lZjZZZ

purely imaginary for any length l

+/- depending on l value

lZjZZZ

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

zjeVzV 20 1

12 zje maximum magnitude value for 10max VV

12 zje 10min VV

minimum magnitude value for

lZjZZZ

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

ljZRZZ

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

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

0in LRZZ 01 L

The Multiple-Reflection Viewpoint

1 LRZZ

(only) at f0

lZjZZZ

)tan( lt

tZjZZZ

LZZZ 01

matching quality power reflection coefficient

222 1tan1sec

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

0ff 1tan1sec 22 4

cos2 0

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

for TEM lines

02 ff m

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

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

232112

2321121

jnnnj eeTT

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

2321121

jnnnj eeTT

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

1,1 Nn

je 231

jj eee 242

assume that the transformer can be made symmetrical

Note that this does not imply that the impedances are symmetrical

22110 ,, NNN

jj eee 242

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)

jj eee 242

xe j 2

NN xaxaxaaxf 2

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

NxAxf 1

NjeA 21

j AeeeA cos2

1,1 Nn24

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

Binomial expansion

Reflection coefficient:

NxCxCxCCxxf 101

0 220ZZ

jj eee 242

nNn CA

NjeA 21

nNn CA

01 ln22222lnZ

1 ln2lnlnZ

ZCZZ Ln

Manual design procedure

Bandwidth, Γm maximum acceptable value

mm A cos2

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

30LZ 1000Z

0.07525ln2

C 3!1!2

C 3!2!1

4.455100

30ln12100lnln2lnln 3

ZCZZ LN

03.861Z

77.542Z

552.3100

30ln3277.54lnln2lnln 3

ZCZZ LN

87.343Z

4.003100

30ln3203.86lnln2lnln 3

ZCZZ LN

74.00.07525

1arccos

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

xxxT 34 3

188 24

4 xxxT

xTxxTxT nnn 212

111 xTx n

We can show that:

jj eee 242

xe j 2

NN xaxaxaaxf 2

xnxTn arccoscos )(coshcosh)( 1 xnxTn1x 1x

nTn coscos

last item:even

12/ NN

oddcos2/1 NN

variable change so we map:

cossec mx

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

cosseccossec1 mmT

12cos1seccossec 2

cossec3cos33cosseccossec 3

3 mmmT

112cossec432cos44cosseccossec 24

4 mmmT

cossec mNjN TeA

last item:even

12/ NN

oddcos2/1 NN

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

L TAZZ

ZZsec0

)(coshcosh)( 1 xnxTn

lncosh

1coshsec 01

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

mjj TAee

1.0 mA

10030lncosh

lncosh

1coshsec 101

76.42746.0

1arccos rad

00 AZZL 1.0A

cossec3cos33cosseccos3cos2 310 mm AA

mA 30 sec2 1263.00

cos mmA secsec32 31 1747.01

simetrie: 1203 ;

4.3531263.02100ln2lnln 001 ZZ

68.771Z

77.542Z

62.383Z

1263.00

1747.01

4.0031747.0268.77ln2lnln 112 ZZ

654.31747.0277.54ln2lnln 223 ZZ

GHzf 15.3

76.4242

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

ljZRZZ

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

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

0in LRZZ 01 L

Bandwidth depends on the initial mismatch

increased bandwidth for smaller load mismatches

ADS Simulation

GHzf 88.0

51033 GHz

2933.03

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

1,1 Nn

je 231

jj eee 242

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

Lecture 3 2017/2018rf-opto.etti.tuiasi.ro/docs/files/MDCR Lecture 3.pdf · Lecture 3 2017/2018 ... 0 dBm = 1 mW 3 dBm = 2 mW 5 dBm = 3 mW 10 dBm = 10 mW 20 dBm = 100 mW-3 dBm = 0.5

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