View
178
Download
0
Category
Preview:
Citation preview
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
EE 41139 Microwave Techniques 2
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
=
EE 41139 Microwave Techniques 3
Example of Signal Flow Graphs
the signal flow graph is as follows:
a1
b1
S12
S12
2b2
a2
a2
b2
S23
S23
b3
a3
3
EE 41139 Microwave Techniques 4
Example of Signal Flow Graphs
Alternatively, we have
a1
b1
S12
S12
2b2
a2
S23
S23
b3
a3
3
EE 41139 Microwave Techniques 5
Example of Signal Flow Graphs
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
,
EE 41139 Microwave Techniques 6
Example of Signal Flow Graphs
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
EE 41139 Microwave Techniques 7
Example of Signal Flow Graphs
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
| | ( | | )
| |( | |
| |)
EE 41139 Microwave Techniques 8
Example of Signal Flow Graphs
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
| | | | ( | | )
| | ( | |
| |)
EE 41139 Microwave Techniques 9
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
EE 41139 Microwave Techniques 10
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
EE 41139 Microwave Techniques 11
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
EE 41139 Microwave Techniques 12
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
EE 41139 Microwave Techniques 13
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
EE 41139 Microwave Techniques 14
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)
EE 41139 Microwave Techniques 15
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
EE 41139 Microwave Techniques 16
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
EE 41139 Microwave Techniques 17
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
EE 41139 Microwave Techniques 18
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
EE 41139 Microwave Techniques 19
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
EE 41139 Microwave Techniques 20
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
EE 41139 Microwave Techniques 21
Design of a Microstrip Line
the short can be achieved by using conducting tape
SMAconnector
copper tape
dielectric witha ground plane
EE 41139 Microwave Techniques 22
Design of a Microstrip Line
remove the short and attach a SMA connector
SMAconnector
copper tape
dielectric witha ground plane
EE 41139 Microwave Techniques 23
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
EE 41139 Microwave Techniques 24
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
EE 41139 Microwave Techniques 25
Design of a Microstrip Line
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 .
EE 41139 Microwave Techniques 26
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 27
Design of a Microstrip Line
the effective dielectric constant is 2.1462, therefore,
e/2=
3 10
4 10
12
121462
25611
9
.. mm
EE 41139 Microwave Techniques 28
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
EE 41139 Microwave Techniques 29
Design of a Microstrip Line
the length of the quarter-wave transformer is
e mm4
3 10
4 10
14
12 2193
12 611
9
..
EE 41139 Microwave Techniques 30
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
EE 41139 Microwave Techniques 31
Design of a Microstrip Line the magnitude of S11 measured at the left
SMA connector looks like:
EE 41139 Microwave Techniques 32
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
EE 41139 Microwave Techniques 33
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, ,
EE 41139 Microwave Techniques 34
Design of a Microstrip Line
Assuming a TEM line
t l l tan tan , 24 2
EE 41139 Microwave Techniques 35
Design of a Microstrip Line
| | |
[( ) / ( ) ( ) / ( ) ]
| |[ / ( ) ( ) / ( ) ]
| |[ 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
EE 41139 Microwave Techniques 36
Design of a Microstrip Line
nearby the design frequency, and therefore
| | | cos | , / Z ZZ Z
L oo L2
2
EE 41139 Microwave Techniques 37
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 38
Design of a Microstrip Line
, 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
EE 41139 Microwave Techniques 39
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 40
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 41
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 42
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 43
Design of a Microstrip Line
the two principal solutions for d are
d t t
d t t
20
20
1
1
tan ,
( tan ),
EE 41139 Microwave Techniques 44
Design of a Microstrip Line
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. .
EE 41139 Microwave Techniques 45
Design of a Microstrip Line
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 ( )
EE 41139 Microwave Techniques 46
Design of a Microstrip Line
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( )( )
[ ( ) ]. , . .
EE 41139 Microwave Techniques 47
Design of a Microstrip Line
attach an open-stub matching circuit to the transmission line and obtain the S11 response
mm 30.06
6.06
EE 41139 Microwave Techniques 48
Design of a Microstrip Line
EE 41139 Microwave Techniques 49
Design of a Microstrip Line
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
EE 41139 Microwave Techniques 50
Multisection Transformer consider the reflection from a
segment of a transmission line discontinuity depicted below
3
2
1
EE 41139 Microwave Techniques 51
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
EE 41139 Microwave Techniques 52
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
EE 41139 Microwave Techniques 53
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
EE 41139 Microwave Techniques 54
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
EE 41139 Microwave Techniques 55
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
EE 41139 Microwave Techniques 56
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
EE 41139 Microwave Techniques 57
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
EE 41139 Microwave Techniques 58
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
EE 41139 Microwave Techniques 59
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 =
EE 41139 Microwave Techniques 60
Binomial Multisection Matching Transformers
the reflection coefficient will be determined by the characteristic impedance of the line and the load impedance, the matching transformer has no electrical length
EE 41139 Microwave Techniques 61
Binomial Multisection Matching Transformers
or
| ( )| | | | | 0 2
N L oL o
A Z ZZ Z
A Z ZZ Z
N L oL o
2 | |
EE 41139 Microwave Techniques 62
Binomial Multisection Matching Transformers
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
!( )! !
EE 41139 Microwave Techniques 63
Binomial Multisection Matching Transformers
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
EE 41139 Microwave Techniques 64
Binomial Multisection Matching Transformers
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
EE 41139 Microwave Techniques 65
Binomial Multisection Matching Transformers
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 | | /
EE 41139 Microwave Techniques 66
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
EE 41139 Microwave Techniques 67
Chebyshev Multisection Matching Transformers
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
EE 41139 Microwave Techniques 68
Chebyshev Multisection Matching Transformers
the first few Chebyshev polynomial is plotted below:
EE 41139 Microwave Techniques 69
Chebyshev Multisection Matching Transformers
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
EE 41139 Microwave Techniques 70
Chebyshev Multisection Matching Transformers
The passband is between and
, therefore we will map to
x = 1 and to x =-1
m m m
m
EE 41139 Microwave Techniques 71
Chebyshev Multisection Matching Transformers
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
EE 41139 Microwave Techniques 72
Chebyshev Multisection Matching Transformers
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
EE 41139 Microwave Techniques 73
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
EE 41139 Microwave Techniques 74
Design of Chebyshev Transformer
once we have chosen the order of the Chebyshev polynomial, the coefficients s can be found
EE 41139 Microwave Techniques 75
Design of Chebyshev Transformer
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 )
EE 41139 Microwave Techniques 76
Design of Chebyshev Transformer
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
EE 41139 Microwave Techniques 77
Design of Chebyshev Transformer
once the bandwidth is known, we can determine the fractional bandwidth as
ffo
m 2 4
Recommended