chebyshev multi-section matching the chebyshev transformer is optimum bandwidth to allow ripple...
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Chebyshev Multi-section MatchingThe Chebyshev transformer is optimum bandwidth to allow
ripple within the passband response, and is known as equally ripple.
Larger bandwidth than that of binomial matching.
The Chebyshev characteristics
(5.60)or (5.59) as rewritten
be can spolynomial Chebyshev
matching,for cosseclet
);()(2)(
spolynomial Chebyshev
121
m
nnn
x
xTxTxxTxT
2
N N-1
2 2
0
0
-1 0
m 0
Chebyshev response ( ) (sec cos )
1where the last term is for even , for odd .
2
1(0) (sec )
(sec )
Let to evaluate
1 1sec cosh[ cosh ( )]
N
cos
jnN m
LN m
L N m
m
Lm
L
Ae T
N N
Z ZA T A
Z Z T
A
Z Z
Z Z
-1 0
m
0
1
1 ln( / )h[ cosh ( )]
N 2
4Fraction bandwidth 2
1More accurate formula ln (5.64b)
2
L
m
nn
n
Z Z
f
f
Z
Z
Chebyshev Transformer Design
Example5.10: Design a three-section Chebyshev transformer to match a 100 load to a 50 line, with m=0.05?
Solution
2 2 33
-1 0
m
-1
33
For 3
( ) (sec cos ) (sec cos )
0.05
1 ln( / )sec cosh[ cosh ( )]
N 2
1 ln(100 / 50)cosh[ cosh ( )] 1.408 44.7
3 2 0.05
Using (5.61) and (5.60c) for (omit )
c
jn jN m m
m
Lm
m
j
N
Ae T Ae T
A
Z Z
T e
30 0
31 m 1
3 0 2 1
os3 : 2 sec 0.0698
cos : 2 3 (sec sec ) 0.1037
From symmetry ;
m
m
A
A
1 0 0
1
2 1 1
2
3 2 0
3
Using (5.64b)
0 : ln ln 2 ln50 2(0.0698) 4.051
57.5
1: ln ln 2 ln57.5 2(0.1037) 4.259
70.7
2 : ln ln 2 ln 70.7 2(0.1037) 4.466
87.0
Fraction ban
n Z Z
Z
n Z Z
Z
n Z Z
Z
0
4 44.7dwidth 2 2 ( ) 1.01
180mf
f
Using table design for N=3 and ZL/Z0=2 can find coefficient as 1.1475, 1.4142, and 1.7429. So Z1=57.37, Z2=70.71, and Z3=87.15.
Tapered Lines MatchingThe line can be continuously tapered instead of discrete multiple sections to achieve broadband matching.
Changing the type of line can obtain different passband characteristics.
Relation between characteristic impedance
and reflection coefficient
l
Z
Z
dz
de
L
z
zj
2;
)ln(2
1)(
00
2
Three type of tapered line
will be considered here
1) Exponential
2)Triangular
3) Klopfenstein
Exponential TaperThe length (L)of line should be greater than /2(l>) to minimize the mismatch at low frequency.
L
Le
ZZ
dzedz
de
Z
Z
La
eZzZ
LjL
azL
z
zj
L
az
sin
2
)ln(
)(ln2
1)(
)ln(1
)(
0
0
2
0
0
Triangular TaperThe peaks of the triangular taper are lower than the corresponding peaks of the exponential case.
First zero occurs at l=2
2
0
2/for /ln)1)/(2/4(0
2/0for /ln)/(20
]2/
2/sin[)ln(
2
1)(
)(0
2
02
L
Le
Z
Z
eZ
eZzZ
LjL
LzLZZLzLz
LzZZLz
L
L
Klopfenstein TaperFor a maximum reflection coefficient specification in the passband, the Klopfenstein taper yields the shortest matching section (optimum sense).
The impedance taper has steps at z=0 and L, and so does not smoothly join the source and load impedances.
AZ
Z
ZZ
ZZA
ALe
A
AAx
xAxA
xI
xdyyA
yAIAxAx
LzAL
zA
AZZzZ
mL
L
L
Lj
x
L
cosh );ln(
2
1cosh
)(cos)(
1cosh),( ;
2),( ;0),0(
valuesspecial the withfunction Besselmodified theis )(
1;1
)1(),(),(
0);,12
(cosh
ln2
1)(ln
0
00
00
22
0
2
1
0 2
21
200
Example5.11: Design a triangular, exponential, and Klopfenstein tapers to match a 50 load to a 100 line?
Solution
Triangular taper
2
2/for 2/1ln)1)/(2/4(
2/0for 2/1ln)/(2
]2/
2/sin)[
2
1ln(
2
1)(
100
100)( 2
2
L
L
e
ezZ
LzLLzLz
LzLz
Exponential taper
L
LL
a
eZzZ az
sin
2
)21ln()(
)2
1ln(
1
)( 0
Klopfenstein taper
cosh
)(cos)(
)346.0
(cosh)(cosh
346.0)ln(2
1
22
0
101
00
00
A
AL
A
Z
Z
ZZ
ZZ
mm
L
L
L
Bode-Fano CriterionThe criterion gives a theoretical limit on the minimum
reflection magnitude (or optimum result) for an arbitrary matching network
The criterion provide the upper limit of performance to tradeoff among reflection coefficient, bandwidth, and network complexity.
For example, if the response ( as the left hand side of next page) is needed to be synthesized, its function is given by applied the criterion of parallel RC
1ln
1ln
1ln
0
RCdωdω
mm
For a given load, broader bandwidth , higher m.
m 0 unless =o. Thus a perfect match can be achieved only at a finite number of frequencies.
As R and/or C increases, the quality of the match ( and/or m) must decrease. Thus higher-Q circuits are intrinsically harder to match than are lower-Q circuits.
Chapter 6
Microwave Resonators
RLC Series Resonant Circuit
Microwave resonators are used in a variety of applications, including filters, oscillators, frequency meters, and tuned amplifiers.
The operation of microwave resonators is very similar to that of the lumped-element resonators (such as parallel and series RLC resonant circuits) of circuit theory.
RRIP
CjLjRI
IZVIP
CjLjRZ
loss
inin
in
resistor by the dissipated Power ;2
1
resonator todeliveredPower
);1
(2
12
1
2
1
1
2
2
2*
Vout
Vin
circuitresonant a of loss theoft measuremen
;factor Quality
)1
(
21
) occur( Resonance21
)(22
)(2
capacitor thein storedenergy electric Average ;1
4
1
inductor thein storedenergy magnetic Average ;4
1
02
22
2
2
2
loss
me
lossin
em
emlossinin
emlossin
e
m
P
WWQ
LCR
I
PZ
WW
I
WWjP
I
PZ
WWjPP
CC
IW
LLIW
)2)2())(((
;2
2
)()1
1(
0 where,Let
1
21
141
22
21
41
22
resonanceAt
0020
2
0
2
20
2
2
0
02
20
2
00
0
2
2
00
RQjRLjR
LjRLC
LjRZ
CRRI
CI
P
W
R
L
RI
LI
P
WQ
in
loss
e
loss
m
CLX
R
XQ
00
1or where
;
0
2
resonator theof bandwidth fractionalpower -half The
BW
RjRRZ
QBW
BWjRQRRQ
jRZ
in
in
2
1 When
2
0