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EE 41139 Microwave Techniques 1

Lecture 4 Example of Signal Flow Graphs Microstrip Line Design and Matching Multisection Transformer Binomial Multisection Matching

Transformers Chebyshev Multisection Matching

Transformers


Example of Signal Flow Graphs

use signal flow graphs to find the power ratios for the mismatched three-port network shown below (Problem 5.32, Pozar)

P Port PPort

PPort11

222

33 3S

0 S12 0S12 0 S230 S23 0

=



the signal flow graph is as follows:

a1

b1

S12

S12

2b2

a2

a2

b2

S23

S23

b3

a3

3



Alternatively, we have

a1

b1

S12

S12

2b2

a2

S23

S23

b3

a3

3



to relate b2 and a1, we have the signal flow graph is as follows:

a1

b1

S12

S12

2

b2

a2

S23

S23

b3

a3

3

2

b a S

Sb a S

S2 1

12

2 3 232 1 1

122

2

2 3 2321 1

,



To relate b3 and b2,

a1

b1

S12

S12

2b2

a2

S23

S23

b3 3

2 3

2 S2322

b b S

S3 2

2 23

2 3 2321



the power ratio must be

PP

b a

a b

b

a

bain

in21

22

22

12

12

22

22

12 2

11

1

1

| | | |

| | | |

| | ( | | )

| | ( | | ),

PP

S

S S

S

21

122

22

2 3 232 12 2

2

2 3 232 2

1

1 11

| | ( | | )

| |( | |

| |)



PP

b a

a b

b

a in

31

32

32

12

12

32

32

12 2

1

1

| | | |

| | | |

| | ( | | )

| | ( | | )

PP

S S

S S

S

31

122

2 232

32

2 3 232 4 12 2

2

2 3 232 2

1

1 11

| | | | ( | | )

| | ( | |

| |)


Microstrip Line Design and Matching

to design and fabricate a 50 microstrip line

to design and fabricate a quarter-wave transformer and open-stub matching circuits for matching a 25 load to a 50 transmission line at 4 GHz

to use design curves (or computer code) for circuit design and simulations


Design of a Microstrip Line using the closed-form formulas discussed

earlier, calculate the width of a 50 microstrip line

the printed-circuit board has a dielectric constant of 2.6 and thickness of 1.59 mm

assuming the conductor thickness is small, obtain the effective dielectric constant


Design of a Microstrip Line from design curves, we found that

W=4.3980 and e= 2.1462 fabricate a microstrip transmission line

using a conducting tape the width should be close to the size W press the conducting tape to eliminate any

air gap between the substrate and the conductor


Design of a Microstrip Line for more accurate fabrication, one can use

etching techniques attach one SMA connector as shown

below:

SMAconnector

copper tape

dielectric witha ground plane

copper tape


Design of a Microstrip Line do a one-port calibration of the vector

network analyzer (VNWA) from 0.5 to 10GHz at the end of the flexible cable, assume a fixed load (50) is a broadband load in the one-port calibration

NetworkAnalyzer

calibrationplane

APC-7connector


Design of a Microstrip Line APC-7 is a sexless precision connector

which can be used up to 20 GHz to obtain an accurate amplitude and phase

of DUT(device under test), the VNWA must be calibrated at a reference point

the most commonly used OSL method utilizes three standards, Open, Short and Load (50)


Design of a Microstrip Line the front panel of VNWA has two ports which

are designated as Port 1 and Port 2 some devices have only one port and they are

called the one-port devices TV has only one input, if we want to measure

the input impedance of a TV antenna, connect it to either Port 1 or Port 2 and measure the reflection from the antenna


Design of a Microstrip Line because it is common to use Port 1 for the

one-port device measurement, we will discuss the S11 (Port 1) calibration

first we need to choose the point at which the calibration is performed

for example, if we want to perform S11 calibration at the end of a long cable, calibration standards Open, Short and Load must be connected at this point


Design of a Microstrip Line after the correct S11 one-port calibration,

Short connected at the calibration point should show the reflection coefficient of -1 (0dB and 180o phase)

the calibration point also corresponds to zero second in the time domain


Design of a Microstrip Line note that the APC-7 and the SMA connectors

are of different size and therefore, we need an APC-7-SMA adapter

NetworkAnalyzer

calibrationplane

APC-7connector

APC-7-SM Aadapter


Design of a Microstrip Line as a result, we need to shift the reference

plane to the end of the adapter two options, i.e., port extension and electrical

delay can be used port extension requires the time delay from

the original plane to the new calibration plane while electrical delay requires the round-trip time


Design of a Microstrip Line

after one-port S11 calibration has been done, attach a short to the end of the microstrip line, obtain the electrical delay to the short from the reference position



the short can be achieved by using conducting tape

SMAconnector

copper tape




remove the short and attach a SMA connector

SMAconnector

copper tape



Design of a Microstrip Line connect the 25 load to the SMA connector measure the input impedance at the load, note

that due to imperfect connections, the measured load may have a small imaginary part

we can use a Smith Chart to find out the location on the microstrip line where the input impedance becomes 25 , here let us assume it is exactly 25


Design of a Microstrip Line make a quarter-wave transformer using a

conducting tape quarter-wave transformer can be explained

by the following equation

Z Z R jZ lZ jR lin

LL

1

11

tantan



there will be no reflection is

Therefore, and

Z Z lin o , tan( ) tan( )24

Z ZRin

L 1

2

Z Z Ro L1 50 25 35 36 .



because of the presence of the SMA connector at the end of the microstrip line, it is not convenient to put the quarter-wave transformer there; we can move the 25 point to /2 from the load toward the APC-7-SMA adapter



the effective dielectric constant is 2.1462, therefore,

e/2=

3 10

4 10

12

121462

25611

9

.. mm


Design of a Microstrip Line the length of the quarter-wave transformer is e/4, however, e is different from the one for the 50 line

the characteristic impedance of the transformer is 35.25 and from the previous equations, we found W=7.2261 and e= 2.2193



the length of the quarter-wave transformer is

e mm4

3 10

4 10

14

12 2193

12 611

9

..


Design of a Microstrip Line the microstrip line and the quarter-wave

transformer are depicted below:

assuming that the total length of the transmission line is 100 mm

4.4

7.22

25.6

12.6

mm


Design of a Microstrip Line the magnitude of S11 measured at the left

SMA connector looks like:


Design of a Microstrip Line note that this result may be different from

measurement, one of the reasons is that we assume the characteristic impedance and effective dielectric constant are independent of frequency

we can make a rough estimation of the bandwidth of this quarter-wave transformer


Design of a Microstrip Line at the designed frequency fo , the reflection

coefficient is

Z ZZ Z

Z ZZ Z j t Z Z

Z Z Z jZ tZ jZ t

Z Z Zin oin o

L oL o o L

inL

Lo L2 1

11

1, ,

Z ZZ Z

Z ZZ Z j t Z Z

Z Z Z jZ tZ jZ t

Z Z Zin oin o

L oL o o L

inL

Lo L2 1

11

1, ,



Assuming a TEM line

t l l tan tan , 24 2



| | |

[( ) / ( ) ( ) / ( ) ]

| |[ / ( ) ( ) / ( ) ]

| |[ sec / ( ) ]

/

/

/

1

41

1 4 41

1 4

2 2 2 2 1 2

2 2 2 1 2

2 2 1 2

Z Z Z Z t Z Z Z Z

Z Z Z Z t Z Z Z Z

Z Z Z Z

L o L o o L L o

o L L o o L L o

o L L o



nearby the design frequency, and therefore

| | | cos | , / Z ZZ Z

L oo L2

2



this function is symmetric about the design frequency, we can define a bandwidth for a maximum value of the reflection coefficient that can be tolerated



, the lower value is while the upper value is 2

2( )

m m

1 12

2

2

| |sec

m

o LL o

Z ZZ Z

cos | |

| | | |m

m

m

o LL o

Z ZZ Z

1

22



For a TEM line,

the fractional bandwidth is given by

l fv

vf

ffp

p

o o

24 2

ff

ffomo

m 2 2 2 4



make an open stub using a conducting tape to derive the formulas for location d and

length l of the stub, consider the following equations with t=tan(d):

Z Z R jX jZ tZ j R jX t

Y G jBZo

L L oo L L

( )

( ), 1



to match the line, we need G = Yo = 1/Zo

Z R Z t X Z t R Z R Xo L o L o L o L L( ) ( ) 2 2 22 0



solving for t gives

tX R Z R X Z

R ZforR Z

t X Z forR Z

L L o L L oL o

L o

L o L o

[( ) ] /,

/ ( ),

2 2

2



the two principal solutions for d are

d t t

d t t

20

20

1

1

tan ,

( tan ),



here XL = 0,

d = 51.2(35.26/360)=5.0148mm this is too close to the SMA connector, we add

e/2 = 25.6+5.0148=30.06 mm the stub susceptance Bs must be negative of B

to cancel the imaginary part of the admittance

t RZ

Lo

0 5 0 7071. .



From the equation,

For an open stub

For a short circuit stub,

Z Z R jZ lZ jR lin

LL

1

11

tantan

l BYs

o 2

1tan ( )

l YBso

21tan ( )



if the length given by these equations is negative, /2 can be added to give a positive result

B R t Z Z t

Z R Z tl mmL o o

o L o

2

2 242 8284 10 0 12895 6 6( )( )

[ ( ) ]. , . .



attach an open-stub matching circuit to the transmission line and obtain the S11 response

mm 30.06

6.06





this result may be slightly different from measurement, the open stub has some end capacitance that is being ignored in addition to the frequency dependence of the characteristic impedance and effective dielectric constant


Multisection Transformer consider the reflection from a

segment of a transmission line discontinuity depicted below

3

2

1


Multisection Transformer If the line impedances are only

slightly different, and , Eq. (1) becomes

as

1 3

2

1 32 1 2

11e

e

j

j

( ),

1 3

2e j 1 3 0


Multisection Transformer

now let us consider a multisection transformer with N sections, each segment has a characteristic impedance slightly different from the adjacent ones, the reflection coefficient can be written as ( ) ( )

oj j

Nj Ne e e1

22

4 2 2


Multisection Transformer assuming that the segments are

symmetry so that

and so on

o N N N , ,1 1 2 2

( ) { ( ) ( ) }( ) ( ) e e e e ejNo

jN jN j N j N1

2 2

( ) { ( ) ( ) }( ) ( ) e e e e ejNo

jN jN j N j N1

2 2 ( ) { ( ) ( ) }( ) ( ) e e e e ejNo

jN jN j N j N1

2 2


Multisection Transformer it should be noted that this

does not imply , etc. for N even

N/2

( ) { cos cos( ) cos( ) }/ 2 2 2 121 2e N N N mjN

o m N

( ) { cos cos( ) cos( ) }/ 2 2 2 121 2e N N N mjN

o m N ( ) { cos cos( ) cos( ) }/ 2 2 2 121 2e N N N mjN

o m N ( ) { cos cos( ) cos( ) }/ 2 2 2 121 2e N N N mjN

o m N ( ) { ( ) ( ) }( ) ( ) e e e e ejNo

jN jN j N j N1

2 2


Multisection Transformer For N odd,

+

( ) { cos cos( ) cos( )cos }( )/

2 2 21

1 2

e N N N mjNo m

N

( ) { cos cos( ) cos( )cos }( ) /

2 2 21

1 2

e N N N mjNo m

N

( ) { cos cos( ) cos( )cos }( ) /

2 2 21

1 2

e N N N mjNo m

N


Multisection Transformer this is a Fourier cosine series

which implies that we can synthesize any desired reflection coefficient response (vs frequency) by properly choosing the with enough number of sections as the Fourier series can match any arbitrary function if enough terms are used


Binomial Multisection Matching Transformers for a given number of sections N,

the binomial matching transformer yields a flat response as much as possible near the design frequency

this is achieved by setting the first

N-1 derivatives of the reflection coefficient equal to zero at the center frequency fo


Binomial Multisection Matching Transformers

Let then the magnitude of the reflection coefficient will be

( ) ( ) A e j N1 2

| ( )| | || ||( )| | || cos | A e e e AjN j j N N N2


Binomial Multisection Matching Transformers recall that at the design

frequency fo, = /2 (quarter wavelength), the above criteria are therefore satisfied

the constant A can be obtained by letting f goes to zero at which =



the reflection coefficient will be determined by the characteristic impedance of the line and the load impedance, the matching transformer has no electrical length



or

| ( )| | | | | 0 2

N L oL o

A Z ZZ Z

A Z ZZ Z

N L oL o

2 | |



according to the binomial expansion, the reflection coefficient reads

Where

( ) ( )

A e A C ej N

nN j n

n

N1 2 2

0

C NN n nn

N

!( )! !



compare this equation with Eq. (2), we have

the characteristic impedance of each segment can be found as

and we can start from n = 0

n nNAC

nn nn n

Z ZZ Z

11



note that we assume there is only slight change in impedance among the segment and its adjacent neighbors, we can approximate n by

knowing that when x -> 1

nn nn n

nn

Z ZZ Z

ZZ

11

112

ln



the fractional bandwidth of the binomial multisection transformer is given by

note that is the maximum allowable reflection coefficient, not the reflection efficient at the junction between the mth section and (m+1)th section

ff

f ff Ao

o mo

m mN

2 2 4 2 4 12

11( ) cos | | /


Chebyshev Multisection Matching Transformers

the Chebyshev transformer optimizes bandwidth at the expense of passband ripple

the bandwidth of the Chebyshev transformer is substantially better than that of the binomial transformer if such ripple is tolerable

the Chebyshev transformer is designed by matching the coefficients of the Chebyshev polynomial



the nth order Chebyshev polynomial is a polynomial of degree n which is denoted by Tn(x), e.g.,

for |x| < 1, we have

T x T x x T x x T x x xT x xT x T x

o

n n n

( ) , ( ) , ( ) , ( ) , . . .( ) ( ) ( )

1 2 1 4 32

1 22

33

1 2

T nn (cos ) cos



the first few Chebyshev polynomial is plotted below:



it can be seen that between -1 and 1, the Chebyshev polynomials oscillate between -1 and 1, this is the equal ripple property

for |x| > 1, the region will be mapped to the frequency range outside the passband

for |x| > 1, the higher the order of the polynomial, the faster the polynomial grows



The passband is between and

, therefore we will map to

x = 1 and to x =-1

m m m

m



Consider

For | | < 1 for |x| > 1, the Chebyshev polynomial can be written as

T T nnm

n mm

( coscos

) (sec cos ) cos( cos coscos

)

1

coscos

m

T x n xn ( ) cosh( cosh ) 1



or for the first few polynomials

+1

T m m1(sec cos ) sec cos T m m2

2 1 2 1(sec cos ) sec ( cos )

T m m m33 3 3 3(sec cos ) sec (cos cos ) sec cos

T m m m44 24 4 2 3 4 2 1 1(sec cos ) sec (cos cos ) sec (cos )

T m m m44 24 4 2 3 4 2 1 1(sec cos ) sec (cos cos ) sec (cos )

T m m m33 3 3 3(sec cos ) sec (cos cos ) sec cos


Design of Chebyshev Transformer

Using the previous equations, we have

+ +

( ) { cos cos( ) cos( ) }

( ) (sec cos )

2 2 21e N N N m

Ae T

jNo m

jNN m

( ) { cos cos( ) cos( ) }

( ) (sec cos )

2 2 21e N N N m

Ae T

jNo m

jNN m

( ) { ( ) ( ) }( ) ( ) e e e e ejNo

jN jN j N j N1

2 2



once we have chosen the order of the Chebyshev polynomial, the coefficients s can be found



the constant A can be found by setting =0 corresponding to zero frequency

note that is specified for the passband which is not the length of section m

( ) (sec )0

Z ZZ Z

ATL oL o

N m

A Z ZZ Z T

L oL o N m

1(sec )



if the maximum allowable reflection coefficient magnitude is , then

Or

T Z ZZ ZN m

mL oL o

(sec )| |

1

sec cosh( cosh| |

mm

L oL oN

Z ZZ Z

1 11



once the bandwidth is known, we can determine the fractional bandwidth as

ffo

m 2 4

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zz zz zz

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mae tjno mjnn

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effective

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