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5-1<br />
<strong>CHAPTER</strong> 5<br />
<strong>Impedance</strong> <strong>Matching</strong> <strong>and</strong> <strong>Smith</strong> <strong>Chart</strong><br />
* Pozar MW (Ch 5), “<strong>Impedance</strong> <strong>Matching</strong> <strong>and</strong> Tuning”<br />
* Pozar RF (Ch 2), “Itransmission Lines & Microwave Networks”<br />
*Ludwig, (Ch 3, Ch 8), “<strong>Matching</strong> <strong>and</strong> Biasing Networks”<br />
*Rogers, (Ch 4), “Radio Frequency Integrated Circuit Design”<br />
<strong>Matching</strong> with Lumped Elements<br />
- L Network<br />
- T & Networks<br />
- Lumped Elements for MIC : Chip R, L, C.<br />
Microstrp Single-Stub <strong>and</strong> Double-Stub Tuning<br />
Quarter-Wave Transformer<br />
* The Bode-Fano Criteria<br />
Appendix <strong>Smith</strong> <strong>Chart</strong><br />
Transmitter<br />
2011-12 H.-R. Chuang EE NCKU<br />
C<br />
ZT Z A<br />
L<br />
Z<br />
M
RF<br />
signal X<br />
RF<br />
signal X<br />
RF<br />
signal X<br />
5-2<br />
<strong>Impedance</strong> matching (or tuning) is important for the following reasons :<br />
incident<br />
(or input)<br />
reflection<br />
<br />
in<br />
Z0<br />
Zin<br />
Reflection<br />
<strong>Matching</strong><br />
Network<br />
coefficient<br />
Return<br />
( or S11) ( Zin<br />
Z0)<br />
/( Zin<br />
Z0)<br />
Loss) :<br />
2011-12 H.-R. Chuang EE NCKU<br />
(or<br />
Load<br />
minimum power loss in the feed line & maximum power delivery<br />
linearizing the frequency response of the circuit<br />
improving the S/N ratio of the system for sensitive receiver components (lownoise<br />
amplifier, etc.)<br />
reducing amplitude & phase errors in a power distribution network (such as<br />
antenna array-feed network)<br />
* Factors in the selection of matching networks<br />
- complexity -b<strong>and</strong>width requirement (such as broadb<strong>and</strong> design)<br />
- adjustability - implementation (transmission line, chip R/L/C elements ..)<br />
<br />
/4 microstrip RF Choke<br />
l<br />
Z0<br />
0.5 0.25<br />
Short-<br />
Cirucited<br />
(S.C.)<br />
* At high freq.,<br />
capacitance is like<br />
4 Short-cirucited<br />
2<br />
X sc / Z0<br />
-2<br />
-4<br />
ZL<br />
RF Choke<br />
扼流圈
5-3<br />
<strong>Matching</strong> Network Types: L-/T-/-section Networks<br />
L-section Networks (Two-component ): Lumped elements: L/C<br />
C<br />
Z S L Z L<br />
L2<br />
(a)<br />
L<br />
Z S<br />
1 Z L<br />
(e)<br />
Z S L Z L<br />
(b)<br />
L1<br />
Z S L Z<br />
2 L<br />
( f )<br />
C<br />
Z S C1 Z Z<br />
L S<br />
2 Z L<br />
5. Harmonic filtering can be done with a<br />
lowpass matching network (series L,<br />
parallel C ). This may be important, for<br />
example, for power amplifiers (PA).<br />
2011-12 H.-R. Chuang EE NCKU<br />
C2<br />
L<br />
(c)<br />
Z S<br />
Z L<br />
(g)<br />
C<br />
(d )<br />
C1<br />
C<br />
L<br />
Z S<br />
Z L<br />
(h)<br />
In any particular region on the <strong>Smith</strong> chart,<br />
several matching circuits will work <strong>and</strong> others<br />
will not.<br />
The figure shows what matching networks will<br />
work in which regions.<br />
How does one choose?<br />
There are a number of popular reasons for<br />
choosing one over another.<br />
1. Sometimes matching components can be<br />
used as dc blocks (capacitors) or to provide<br />
bias currents (inductors).<br />
2. Some circuits may result in more reasonable<br />
component values.<br />
3. Personal preference. Sometimes when all<br />
paths look equal, you just have to shoot<br />
from the hip <strong>and</strong> pick one.<br />
4. Stability. Since transistor gain is higher at<br />
lower frequencies, there may be a lowfrequency<br />
stability problem. In such a case,<br />
sometimes a highpass network (series<br />
capacitor, parallel inductor) at the input<br />
may be more stable.<br />
C
L-section Network<br />
(1) complex ZL to real Z0 matching<br />
Z0<br />
jX<br />
admittance<br />
(a)<br />
jB<br />
5-4<br />
Z L<br />
Z0<br />
jB<br />
jX<br />
admittance<br />
(b)<br />
How to determine jX & jB ? Let zL = ZL / Zo = (RL + jXL) / Zo = r + jx<br />
1. Analytic Solutions or 2. <strong>Smith</strong> <strong>Chart</strong> Solution<br />
(1) if RL Z0 ( z 1)<br />
[zL is inside the (1 + jx) circle ] => choose (a) why?<br />
1<br />
for impedance matching (to Z0) => jX <br />
Z0<br />
jB 1( RL jX L)<br />
<br />
2 2<br />
B(<br />
XRL XLZ ) R Z<br />
X<br />
0 L 0 L RL<br />
Z0<br />
RL<br />
X L Z0RL<br />
B <br />
<br />
=> <br />
2 2<br />
X(<br />
1 BXL) BZ0RL X<br />
RL<br />
X L<br />
L <br />
X<br />
( 1 B)<br />
( X LZ0<br />
RL<br />
) ( Z0<br />
B RL<br />
)<br />
(2) if RL Z0 ( z 1)<br />
[zL is outside the (1 + jx) circle ] => choose (b) why?<br />
1 1<br />
for impedance matching (to Z0) => jB <br />
<br />
RL j( X XL) Z0<br />
BZ0(<br />
X X L) Z0 RL<br />
<br />
X RL(<br />
Z0<br />
RL<br />
) X L<br />
<br />
=> <br />
(<br />
X XL) BZ0RL B ( Z R ) R Z<br />
r x<br />
r < 1<br />
r > 1<br />
0<br />
L<br />
L<br />
(1+jx) circle<br />
2011-12 H.-R. Chuang EE NCKU<br />
zL<br />
Z L / Zo<br />
r jx<br />
L<br />
( Z L Zo<br />
) /( Z L Zo<br />
) |<br />
L<br />
| <br />
0<br />
Z L
5-5<br />
EX (Pozar MW EX 5.1) (Pozar RF EX 2.5)<br />
Z0<br />
100<br />
jX<br />
jB<br />
2011-12 H.-R. Chuang EE NCKU<br />
Z L<br />
Z L RL<br />
jX L<br />
200 j100<br />
Since RL = 200 > Z0 = 100 (zL is inside the (1 + jx) circle)<br />
We choose (a) The solutions are<br />
RL 200 XL 100 Zo 100<br />
B1<br />
X1<br />
B2<br />
X2<br />
XL<br />
1<br />
B1<br />
XL<br />
1<br />
B2<br />
RL<br />
Zo<br />
RL 2 XL 2 <br />
ZoRL RL 2 XL 2<br />
Zo<br />
XL RL<br />
Zo<br />
B1RL RL<br />
Zo<br />
RL 2 XL 2 <br />
ZoRL RL 2 XL 2<br />
Zo<br />
XL RL<br />
Zo<br />
B2RL At f = 500 MHz<br />
Z0<br />
100<br />
Z0<br />
100<br />
38.8nH<br />
0.92pF<br />
Solution 1 (low pass)<br />
2.61pF<br />
46.1nH<br />
Solution 2 (high pass)<br />
Z L<br />
B1 2.899 10 3<br />
<br />
X1 122.474<br />
B2 6.899 10 3<br />
<br />
<br />
X2 122.474<br />
200 j100<br />
Z L<br />
200 j100<br />
0.<br />
33 ( &<br />
BW 0.<br />
3GHz<br />
<br />
2<br />
(high pass)<br />
<br />
<br />
0.<br />
1)<br />
Solution (1) b<strong>and</strong>width<br />
BW<br />
SWR 2<br />
0.<br />
3/<br />
0.<br />
5<br />
(low pass)<br />
60%<br />
RL <br />
9.<br />
5<br />
dB
Z0<br />
100<br />
5-6<br />
<strong>Smith</strong> <strong>Chart</strong> Representation of the <strong>Matching</strong> Process<br />
Z0<br />
100<br />
38.8nH<br />
0.92pF<br />
38.8nH<br />
0.92pF<br />
Solution 1 (low pass)<br />
Z L<br />
38.8nH<br />
200 j100<br />
Z L<br />
200 j100<br />
Z L<br />
200 j100<br />
z<br />
<br />
L<br />
L<br />
Z<br />
0.92pF<br />
2011-12 H.-R. Chuang EE NCKU<br />
L<br />
( Z<br />
/ Z<br />
L<br />
o<br />
Z<br />
o<br />
0.<br />
45<br />
<br />
2 <br />
) /( Z<br />
26.<br />
6<br />
j<br />
L<br />
o<br />
Z<br />
o<br />
)
5-7<br />
(2) Complex to complex conjugate matching (Ludwig, RF Circuit Design P401)<br />
(Conjugate <strong>Matching</strong> for maximum power transfer )<br />
ZT<br />
150<br />
j75<br />
<br />
<br />
Z<br />
A 75 j15<br />
<br />
f 2GHz<br />
*<br />
Z Z Z<br />
T<br />
M Z A A<br />
complex ZA to complex ZT<br />
conjugate matching<br />
RT<br />
( 150)<br />
RA(<br />
75)<br />
choose " "<br />
argument ( )<br />
of<br />
& vice versa<br />
2011-12 H.-R. Chuang EE NCKU<br />
Z0<br />
Transmitter<br />
jB<br />
jX<br />
admittance<br />
complex ZL to real Z0 matching<br />
( )<br />
0<br />
ZL
Transmitter<br />
ZT<br />
150 j75<br />
<br />
<br />
Z<br />
A 75 j15<br />
<br />
f 2GHz<br />
<br />
Let<br />
*<br />
Z Z Z<br />
T<br />
M Z A A<br />
complex ZT to complex ZA<br />
conjugate matching<br />
Z<br />
0<br />
75<br />
<br />
z<br />
<br />
z<br />
T<br />
A<br />
Z<br />
Z<br />
T<br />
A<br />
/ Z<br />
/ Z<br />
0<br />
0<br />
5-8<br />
<br />
<br />
( 150<br />
( 75<br />
j75)<br />
/ 75 2 j1<br />
j15)<br />
/ 75 1<br />
j0.<br />
2<br />
2011-12 H.-R. Chuang EE NCKU
A<br />
(3) General L-section matching network (complex to complex)<br />
Zs<br />
50 j25<br />
<br />
<br />
ZL<br />
25 j50<br />
<br />
f 2GHz<br />
z<br />
<br />
z<br />
<br />
z<br />
s<br />
L<br />
L<br />
Transmitter<br />
1<br />
j0.<br />
5<br />
0.<br />
5 j1<br />
* 0.<br />
5 j1<br />
zL<br />
*<br />
zL<br />
5-9<br />
complex Zs to complex ZL : conjugate matching<br />
s<br />
2011-12 H.-R. Chuang EE NCKU<br />
Zs<br />
A,<br />
B,<br />
C,<br />
D (four paths)<br />
z z<br />
B D<br />
*<br />
Zs ZL<br />
ZL<br />
C<br />
* L
5-10<br />
Ex: L-section Lumped-Elements & Microstrip <strong>Matching</strong> Networks<br />
Conjugately Matched Amplifier Design (Pozar MW EX11-3 or RF EX6-3 )<br />
Design an amplifier for maximum gain at 4.0 GHz using single-stub matching<br />
sections. Calculate <strong>and</strong> plot the input return loss & the gain from 3 to 5 GHz. The<br />
GaAs FET has the following S parameters (Z0=50 ):<br />
f (GHz) S11 S21 S12 S22<br />
30 . 080 . 89 2.86 99 003 . 56 076 . 41 4. 0 0. 72116 2.60 76 0. 0357 0. 7354 5. 0 0. 66142 2.39 54 0. 0362 0. 72 68<br />
FET S-parameters Touchstone file: Poz_11-3.s2p<br />
! poz_11-3.s2p : Pozar Ex. 11-3 transistor S parameters<br />
! Typical s-parameters at minimum attenuation setting, Ta=25°C<br />
# ghz s ma r 50<br />
3.00 0.800 -89.0 2.860 99.0 0.030 56.0 0.760 -41.0<br />
4.00 0.720 -116.0 2.600 76.0 0.030 57.0 0.730 -54.0<br />
5.00 0.660 -142.0 2.390 54.0 0.030 62.0 0.720 -68.0<br />
It cab be derived that (see chapter of RF Amplifier Design)<br />
<br />
<br />
<br />
s<br />
L<br />
0.<br />
872123<br />
0.<br />
87661<br />
&<br />
<br />
<br />
<br />
<br />
0.<br />
87<br />
61<br />
2011-12 H.-R. Chuang EE NCKU<br />
in<br />
out<br />
<br />
*<br />
S<br />
*<br />
L<br />
<br />
0.<br />
87<br />
123<br />
o<br />
o
Z<br />
<br />
Z<br />
Microstrip <strong>Matching</strong> Networks<br />
(f = 4 GHz)<br />
<br />
<br />
<br />
s<br />
L<br />
50<br />
0.<br />
872123<br />
0.<br />
87661<br />
in<br />
L<br />
4.<br />
43 j<br />
12.<br />
68 j<br />
26.<br />
97<br />
83.<br />
5<br />
&<br />
<br />
<br />
<br />
<br />
<br />
( Z<br />
0<br />
out<br />
5-11<br />
0120 . <br />
0206 . <br />
50<br />
50<br />
0206 . <br />
0206 . <br />
s in <br />
<br />
<br />
*<br />
S<br />
*<br />
L<br />
50)<br />
0.<br />
87<br />
123<br />
0.<br />
87<br />
61<br />
Lumped Elements <strong>Matching</strong> Networks<br />
50<br />
in<br />
3 1.63nH 1<br />
2.54pF<br />
0<br />
s in <br />
out<br />
<br />
50<br />
2011-12 H.-R. Chuang EE NCKU<br />
o<br />
o<br />
L<br />
2 4.19nH 4<br />
0<br />
<br />
out<br />
1.32pF<br />
L<br />
0<br />
50
* By <strong>Smith</strong>-<strong>Chart</strong> tool<br />
DP-Nr. 1(4.4 - j27.0)Ohm Q = 6.1 4.000 GHz<br />
DP-Nr. 2(4.4 + j14.1)Ohm Q = 3.2 4.000 GHz<br />
DP-Nr. 3(49.4 - j0.2)Ohm Q = 0.0 4.000 GHz<br />
DP-Nr. 1(4.4 - j27.0)Ohm Q = 6.1 4.000 GHz<br />
DP-Nr. 2(3.6 + j13.0)Ohm Q = 3.6 4.000 GHz<br />
DP-Nr. 3(50.4 + j1.4)Ohm Q = 0.0 4.000 GHz<br />
5-12<br />
rtransmission-line matching network<br />
(open-circuited stub)<br />
2011-12 H.-R. Chuang EE NCKU
DP-Nr. 1(4.4 - j27.0)Ohm Q = 6.1 4.000 GHz<br />
DP-Nr. 2(3.6 - j12.7)Ohm Q = 3.5 4.000 GHz<br />
DP-Nr. 3(48.0 - j0.0)Ohm Q = 0.0 4.000 GHz<br />
5-13<br />
Transmission-line matching network<br />
(shorted-circuited stub)<br />
2011-12 H.-R. Chuang EE NCKU
5-14<br />
Forbidden Regions for L-type <strong>Matching</strong> Networks with Z 50<br />
=> The shaded areas denote values of load impedance that cannot be matched to 50 Ω<br />
2011-12 H.-R. Chuang EE NCKU<br />
Z s<br />
0
R<br />
z<br />
L<br />
L<br />
( for Z<br />
f<br />
0<br />
5-15<br />
Design Example: Forbidden Regions for L-type <strong>Matching</strong> Networks<br />
<br />
80 <br />
<br />
1<br />
GHz<br />
Since z<br />
=> choose<br />
forbidden regions of<br />
L - network<br />
(with Z<br />
<br />
1.<br />
6<br />
0<br />
L<br />
S<br />
<br />
X<br />
j1.<br />
2<br />
= 1.6<br />
(c) or<br />
= 50)<br />
L<br />
50 )<br />
> 1<br />
60<br />
<br />
(d) from<br />
2011-12 H.-R. Chuang EE NCKU
5-16<br />
Quality factor & B<strong>and</strong>width (BW) (there are much more to be discussed!)<br />
Zs<br />
Rs<br />
jX s<br />
or<br />
YP<br />
GP<br />
jBP<br />
Qn<br />
fo<br />
f<br />
QL BW <br />
2 BW<br />
Q<br />
* T<strong>Matching</strong> Network (discussed next)<br />
o<br />
L<br />
<br />
| X s |<br />
Qn<br />
or<br />
Rs<br />
| BP<br />
|<br />
GP<br />
2011-12 H.-R. Chuang EE NCKU
5-17<br />
* T & <strong>Matching</strong> Network: The.addition of 3rd element into the two-element (L) matching<br />
network introduces an additional degree of freedom in the çi!çuit, <strong>and</strong> allows us to control<br />
the value of QL (to be discussed)by choosing an appropriate intermediate impedance. => wider<br />
(matching) b<strong>and</strong>width<br />
T <strong>Matching</strong> Network<br />
<strong>Matching</strong> Network<br />
<br />
Zin<br />
10<br />
j20<br />
<br />
<br />
Z<br />
L 60 j30<br />
<br />
f 1<br />
GHz<br />
Zin<br />
10<br />
j20<br />
<br />
<br />
ZL<br />
60 j30<br />
<br />
f 1<br />
GHz<br />
2011-12 H.-R. Chuang EE NCKU
5-18<br />
Comparison between L-, T - & - network<br />
Design a match circuit at the center frequency of 100 MHz<br />
* Prof. C.-F. Chang course note (NCCU)<br />
L<br />
<br />
T<br />
4-element ladder<br />
51<br />
0.1 H<br />
10 pF<br />
2011-12 H.-R. Chuang EE NCKU<br />
510
5-19<br />
Microstrip Line <strong>Matching</strong> Networks (Ludwig P431)<br />
<br />
In the mid-GHz <strong>and</strong> higher frequency range, the wavelength becomes<br />
sufficiently small <strong>and</strong> the distributed components are widely used. Also, the<br />
discrete R/L/C lumped elements will have more noticeable parasitic effects (see<br />
chapter 2) <strong>and</strong> let to complicating the circuit design process<br />
Distributed componenets (such as transmission line segments) can be used<br />
to mix with lumped elements<br />
From Discrete Components to Microstrip Lines<br />
<br />
Avoid using inductors (if possible) due to higher resistive loss (& higher price)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
In general, one shunt capacitor & two series transmission lines is<br />
sufficiently to transform any load to any input impedance.<br />
EX: transform load ZL to an input impedance Z in<br />
Z<br />
L 30 j10<br />
<br />
<br />
Zin<br />
60 j80<br />
<br />
f 1.<br />
5 GHz<br />
zL<br />
0.<br />
6 <br />
zin<br />
1.<br />
2 <br />
Identify input & load SWR circles<br />
j0.<br />
2 <br />
Choose A (yA= 1-j0.6) & transform zL to<br />
A by a series TL (l1)<br />
=>Transform A to B (on the input SWR circle)<br />
by a parallel C1<br />
=> Transform B to zin by a series TL (l2)<br />
zL + series-TL (l1)<br />
=> A + shunt C1<br />
=> B + series-TL (l2)<br />
=> zin<br />
j1.<br />
6 <br />
2011-12 H.-R. Chuang EE NCKU
z<br />
<br />
y<br />
<br />
z<br />
Single-Stub <strong>Matching</strong> Networks<br />
Z<br />
L 60 j45<br />
<br />
<br />
Zin<br />
75 j90<br />
<br />
Z 75 <br />
0<br />
L<br />
L<br />
in<br />
Z<br />
L<br />
/ Z<br />
0<br />
0<br />
<br />
0.<br />
8<br />
<br />
j0.<br />
6<br />
1/<br />
zL<br />
0.<br />
8 j0.<br />
6<br />
Z / Z 1<br />
j1.<br />
2 <br />
in<br />
5-20<br />
g = 0.8<br />
conductance circle<br />
zL to A (yA= 0.8 + j1.5) by adding a shunt open-circuited (O.C.)TL lSA<br />
4 adjustable parameters:<br />
l , Z l , Z )<br />
( s 0s, L 0L,<br />
Input SWR circle associated with zin<br />
has two intersected points (A & B) with<br />
g = 0.8 conductance circle<br />
yA= 0.8 + j1.05 yB= 0.8 - j1.05<br />
The corresponding susceptance for the stub : jbSA= yA- yL = (0.8 + j1.05)-( 0.8 + j0.6)=0.45<br />
O.C. point (g=0) to the point of ibSA = 0.45 is lSA = 0.067 <br />
2011-12 H.-R. Chuang EE NCKU<br />
A to zin is lLA = 0.266 <br />
g = 0<br />
(O.C.)<br />
ibSA<br />
= 0.45
2011-12 H.-R. Chuang EE NCKU<br />
5-21<br />
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<br />
<br />
<br />
<br />
<br />
s<br />
sB<br />
s<br />
sB<br />
sB<br />
sB<br />
s<br />
sB<br />
l<br />
l<br />
l<br />
l<br />
l<br />
l<br />
l<br />
l<br />
2<br />
tan<br />
2<br />
1<br />
tan<br />
2<br />
2<br />
tan<br />
2<br />
tan<br />
2<br />
//<br />
)<br />
1<br />
1<br />
:<br />
stub<br />
circuit<br />
-<br />
short<br />
:<br />
stub<br />
circuit<br />
-<br />
open<br />
(<br />
design<br />
stub<br />
Balanced
5-22<br />
Double-Stub <strong>Matching</strong> Networks<br />
Zin<br />
Z0<br />
50 <br />
<br />
Z<br />
L 50 j50<br />
<br />
l / 8 l l 3<br />
/ 8<br />
0.<br />
074<br />
0.<br />
051<br />
2011-12 H.-R. Chuang EE NCKU<br />
l<br />
1<br />
s1<br />
2<br />
<br />
l<br />
3<br />
s1
5-23<br />
Quarter-Wave Transformer 四分之一波長(傳輸線)阻抗轉換匹配<br />
( only useful for pure-resistance matching )<br />
transmission<br />
line 1<br />
V1 ( z)<br />
<br />
<br />
V1 ( z)<br />
0 X<br />
( Z 0)<br />
( Z 0)<br />
Z<br />
in<br />
quarter-wavelength<br />
transmission line 2<br />
l / 4<br />
ZL<br />
jZ0<br />
tanl<br />
( Z0<br />
) Z0<br />
Z<br />
jZ tanl<br />
l<br />
<br />
<br />
&<br />
2<br />
tan<br />
2011-12 H.-R. Chuang EE NCKU<br />
Z<br />
in<br />
Z<br />
0<br />
Z<br />
Z<br />
2<br />
0<br />
0<br />
0<br />
l <br />
<br />
4<br />
ZL<br />
jZ0<br />
( )<br />
Z0<br />
Z0<br />
Z0<br />
Z<br />
jZ ( )<br />
Z<br />
Ex: A microstrip quarter-wave trasformer that matches a 50 miscrostrip line to<br />
a 20 load at f = 4 GHz (substrate: r=2.5, thickness h = 0.75 mm)<br />
2.13[mm] 4.03[mm]<br />
50[ ]<br />
31. 62[<br />
]<br />
<br />
12.73[mm]<br />
L<br />
20[ ]<br />
* Double Quarter-Wave Transformer for wider (matching b<strong>and</strong>width)<br />
ZL<br />
L<br />
L<br />
L
5-24<br />
* (<strong>Matching</strong>) B<strong>and</strong>width (f ) of a Quarter-Wave Transformer<br />
Pozar, Mcrowave & RF Design of Wireless Systems<br />
Approximate behavior of the reflection coefficient magnitude of a quarter-wave<br />
transformer near the design frequency<br />
It can be proved that<br />
2<br />
Z Z<br />
<br />
Z<br />
f<br />
2(<br />
f <br />
( ) 0 f<br />
BW<br />
f<br />
f<br />
0<br />
1 0 L<br />
1<br />
sec<br />
2 <br />
m ZL<br />
0<br />
0<br />
m<br />
) 2 f<br />
2 <br />
f<br />
m<br />
0<br />
<br />
<br />
<br />
2<br />
4<br />
2 <br />
<br />
Increased BW for<br />
Smaller load mismatch (ZL/Z0)<br />
4<br />
2 cos<br />
<br />
2011-12 H.-R. Chuang EE NCKU<br />
m<br />
1<br />
<br />
<br />
<br />
<br />
1<br />
2<br />
Z0Z<br />
<br />
m<br />
2<br />
m ZL<br />
Z<br />
L<br />
0
5-25<br />
2011-12 H.-R. Chuang EE NCKU