CHAPTER 5 Impedance Matching and Smith Chart

5-1
CHAPTER 5
Impedance Matching and Smith Chart
* Pozar MW (Ch 5), “Impedance Matching and Tuning”
* Pozar RF (Ch 2), “Itransmission Lines & Microwave Networks”
*Ludwig, (Ch 3, Ch 8), “Matching and Biasing Networks”
*Rogers, (Ch 4), “Radio Frequency Integrated Circuit Design”
Matching with Lumped Elements
- L Network
- T & Networks
- Lumped Elements for MIC : Chip R, L, C.
Microstrp Single-Stub and Double-Stub Tuning
Quarter-Wave Transformer
* The Bode-Fano Criteria
Appendix Smith Chart
Transmitter
2011-12 H.-R. Chuang EE NCKU
C
ZT Z A
L
Z
M

RF
signal X
RF
signal X
RF
signal X
5-2
Impedance matching (or tuning) is important for the following reasons :
incident
(or input)
reflection
in
Z0
Zin
Reflection
Matching
Network
coefficient
Return
( or S11) ( Zin
Z0)
/( Zin
Z0)
Loss) :
2011-12 H.-R. Chuang EE NCKU
(or
Load
minimum power loss in the feed line & maximum power delivery
linearizing the frequency response of the circuit
improving the S/N ratio of the system for sensitive receiver components (lownoise
amplifier, etc.)
reducing amplitude & phase errors in a power distribution network (such as
antenna array-feed network)
* Factors in the selection of matching networks
- complexity -bandwidth requirement (such as broadband design)
- adjustability - implementation (transmission line, chip R/L/C elements ..)
/4 microstrip RF Choke
l
Z0
0.5 0.25
Short-
Cirucited
(S.C.)
* At high freq.,
capacitance is like
4 Short-cirucited
2
X sc / Z0
-2
-4
ZL
RF Choke
扼流圈

5-3
Matching Network Types: L-/T-/-section Networks
L-section Networks (Two-component ): Lumped elements: L/C
C
Z S L Z L
L2
(a)
L
Z S
1 Z L
(e)
Z S L Z L
(b)
L1
Z S L Z
2 L
( f )
C
Z S C1 Z Z
L S
2 Z L
5. Harmonic filtering can be done with a
lowpass matching network (series L,
parallel C ). This may be important, for
example, for power amplifiers (PA).
2011-12 H.-R. Chuang EE NCKU
C2
L
(c)
Z S
Z L
(g)
C
(d )
C1
C
L
Z S
Z L
(h)
In any particular region on the Smith chart,
several matching circuits will work and others
will not.
The figure shows what matching networks will
work in which regions.
How does one choose?
There are a number of popular reasons for
choosing one over another.
1. Sometimes matching components can be
used as dc blocks (capacitors) or to provide
bias currents (inductors).
2. Some circuits may result in more reasonable
component values.
3. Personal preference. Sometimes when all
paths look equal, you just have to shoot
from the hip and pick one.
4. Stability. Since transistor gain is higher at
lower frequencies, there may be a lowfrequency
stability problem. In such a case,
sometimes a highpass network (series
capacitor, parallel inductor) at the input
may be more stable.
C

L-section Network
(1) complex ZL to real Z0 matching
Z0
jX
admittance
(a)
jB
5-4
Z L
Z0
jB
jX
admittance
(b)
How to determine jX & jB ? Let zL = ZL / Zo = (RL + jXL) / Zo = r + jx
1. Analytic Solutions or 2. Smith Chart Solution
(1) if RL Z0 ( z 1)
[zL is inside the (1 + jx) circle ] => choose (a) why?
1
for impedance matching (to Z0) => jX
Z0
jB 1( RL jX L)
2 2
B(
XRL XLZ ) R Z
X
0 L 0 L RL
Z0
RL
X L Z0RL
B
=>
2 2
X(
1 BXL) BZ0RL X
RL
X L
L
X
( 1 B)
( X LZ0
RL
) ( Z0
B RL
)
(2) if RL Z0 ( z 1)
[zL is outside the (1 + jx) circle ] => choose (b) why?
1 1
for impedance matching (to Z0) => jB
RL j( X XL) Z0
BZ0(
X X L) Z0 RL
X RL(
Z0
RL
) X L
=>
(
X XL) BZ0RL B ( Z R ) R Z
r x
r < 1
r > 1
0
L
L
(1+jx) circle
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zL
Z L / Zo
r jx
L
( Z L Zo
) /( Z L Zo
) |
L
|
0
Z L

5-5
EX (Pozar MW EX 5.1) (Pozar RF EX 2.5)
Z0
100
jX
jB
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Z L
Z L RL
jX L
200 j100
Since RL = 200 > Z0 = 100 (zL is inside the (1 + jx) circle)
We choose (a) The solutions are
RL 200 XL 100 Zo 100
B1
X1
B2
X2
XL
1
B1
XL
1
B2
RL
Zo
RL 2 XL 2
ZoRL RL 2 XL 2
Zo
XL RL
Zo
B1RL RL
Zo
RL 2 XL 2
ZoRL RL 2 XL 2
Zo
XL RL
Zo
B2RL At f = 500 MHz
Z0
100
Z0
100
38.8nH
0.92pF
Solution 1 (low pass)
2.61pF
46.1nH
Solution 2 (high pass)
Z L
B1 2.899 10 3
X1 122.474
B2 6.899 10 3
X2 122.474
200 j100
Z L
200 j100
0.
33 ( &
BW 0.
3GHz
2
(high pass)
0.
1)
Solution (1) bandwidth
BW
SWR 2
0.
3/
0.
5
(low pass)
60%
RL
9.
5
dB

Z0
100
5-6
Smith Chart Representation of the Matching Process
Z0
100
38.8nH
0.92pF
38.8nH
0.92pF
Solution 1 (low pass)
Z L
38.8nH
200 j100
Z L
200 j100
Z L
200 j100
z
L
L
Z
0.92pF
2011-12 H.-R. Chuang EE NCKU
L
( Z
/ Z
L
o
Z
o
0.
45
2
) /( Z
26.
6
j
L
o
Z
o
)

5-7
(2) Complex to complex conjugate matching (Ludwig, RF Circuit Design P401)
(Conjugate Matching for maximum power transfer )
ZT
150
j75
Z
A 75 j15
f 2GHz
*
Z Z Z
T
M Z A A
complex ZA to complex ZT
conjugate matching
RT
( 150)
RA(
75)
choose " "
argument ( )
of
& vice versa
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Z0
Transmitter
jB
jX
admittance
complex ZL to real Z0 matching
( )
0
ZL

Transmitter
ZT
150 j75
Z
A 75 j15
f 2GHz
Let
*
Z Z Z
T
M Z A A
complex ZT to complex ZA
conjugate matching
Z
0
75
z
z
T
A
Z
Z
T
A
/ Z
/ Z
0
0
5-8
( 150
( 75
j75)
/ 75 2 j1
j15)
/ 75 1
j0.
2
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A
(3) General L-section matching network (complex to complex)
Zs
50 j25
ZL
25 j50
f 2GHz
z
z
z
s
L
L
Transmitter
1
j0.
5
0.
5 j1
* 0.
5 j1
zL
*
zL
5-9
complex Zs to complex ZL : conjugate matching
s
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Zs
A,
B,
C,
D (four paths)
z z
B D
*
Zs ZL
ZL
C
* L

5-10
Ex: L-section Lumped-Elements & Microstrip Matching Networks
Conjugately Matched Amplifier Design (Pozar MW EX11-3 or RF EX6-3 )
Design an amplifier for maximum gain at 4.0 GHz using single-stub matching
sections. Calculate and plot the input return loss & the gain from 3 to 5 GHz. The
GaAs FET has the following S parameters (Z0=50 ):
f (GHz) S11 S21 S12 S22
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
FET S-parameters Touchstone file: Poz_11-3.s2p
! poz_11-3.s2p : Pozar Ex. 11-3 transistor S parameters
! Typical s-parameters at minimum attenuation setting, Ta=25°C
# ghz s ma r 50
3.00 0.800 -89.0 2.860 99.0 0.030 56.0 0.760 -41.0
4.00 0.720 -116.0 2.600 76.0 0.030 57.0 0.730 -54.0
5.00 0.660 -142.0 2.390 54.0 0.030 62.0 0.720 -68.0
It cab be derived that (see chapter of RF Amplifier Design)
s
L
0.
872123
0.
87661
&
0.
87
61
2011-12 H.-R. Chuang EE NCKU
in
out
*
S
*
L
0.
87
123
o
o

Z
Z
Microstrip Matching Networks
(f = 4 GHz)
s
L
50
0.
872123
0.
87661
in
L
4.
43 j
12.
68 j
26.
97
83.
5
&
( Z
0
out
5-11
0120 .
0206 .
50
50
0206 .
0206 .
s in
*
S
*
L
50)
0.
87
123
0.
87
61
Lumped Elements Matching Networks
50
in
3 1.63nH 1
2.54pF
0
s in
out
50
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o
o
L
2 4.19nH 4
0
out
1.32pF
L
0
50

* By Smith-Chart tool
DP-Nr. 1(4.4 - j27.0)Ohm Q = 6.1 4.000 GHz
DP-Nr. 2(4.4 + j14.1)Ohm Q = 3.2 4.000 GHz
DP-Nr. 3(49.4 - j0.2)Ohm Q = 0.0 4.000 GHz
DP-Nr. 1(4.4 - j27.0)Ohm Q = 6.1 4.000 GHz
DP-Nr. 2(3.6 + j13.0)Ohm Q = 3.6 4.000 GHz
DP-Nr. 3(50.4 + j1.4)Ohm Q = 0.0 4.000 GHz
5-12
rtransmission-line matching network
(open-circuited stub)
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DP-Nr. 1(4.4 - j27.0)Ohm Q = 6.1 4.000 GHz
DP-Nr. 2(3.6 - j12.7)Ohm Q = 3.5 4.000 GHz
DP-Nr. 3(48.0 - j0.0)Ohm Q = 0.0 4.000 GHz
5-13
Transmission-line matching network
(shorted-circuited stub)
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5-14
Forbidden Regions for L-type Matching Networks with Z 50
=> The shaded areas denote values of load impedance that cannot be matched to 50 Ω
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Z s
0

R
z
L
L
( for Z
f
0
5-15
Design Example: Forbidden Regions for L-type Matching Networks
80
1
GHz
Since z
=> choose
forbidden regions of
L - network
(with Z
1.
6
0
L
S
X
j1.
2
= 1.6
(c) or
= 50)
L
50 )
> 1
60
(d) from
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5-16
Quality factor & Bandwidth (BW) (there are much more to be discussed!)
Zs
Rs
jX s
or
YP
GP
jBP
Qn
fo
f
QL BW
2 BW
Q
* TMatching Network (discussed next)
o
L
| X s |
Qn
or
Rs
| BP
|
GP
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5-17
* T & Matching Network: The.addition of 3rd element into the two-element (L) matching
network introduces an additional degree of freedom in the çi!çuit, and allows us to control
the value of QL (to be discussed)by choosing an appropriate intermediate impedance. => wider
(matching) bandwidth
T Matching Network
Matching Network
Zin
10
j20
Z
L 60 j30
f 1
GHz
Zin
10
j20
ZL
60 j30
f 1
GHz
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5-18
Comparison between L-, T - & - network
Design a match circuit at the center frequency of 100 MHz
* Prof. C.-F. Chang course note (NCCU)
L
T
4-element ladder
51
0.1 H
10 pF
2011-12 H.-R. Chuang EE NCKU
510

5-19
Microstrip Line Matching Networks (Ludwig P431)
In the mid-GHz and higher frequency range, the wavelength becomes
sufficiently small and the distributed components are widely used. Also, the
discrete R/L/C lumped elements will have more noticeable parasitic effects (see
chapter 2) and let to complicating the circuit design process
Distributed componenets (such as transmission line segments) can be used
to mix with lumped elements
From Discrete Components to Microstrip Lines
Avoid using inductors (if possible) due to higher resistive loss (& higher price)
In general, one shunt capacitor & two series transmission lines is
sufficiently to transform any load to any input impedance.
EX: transform load ZL to an input impedance Z in
Z
L 30 j10
Zin
60 j80
f 1.
5 GHz
zL
0.
6
zin
1.
2
Identify input & load SWR circles
j0.
2
Choose A (yA= 1-j0.6) & transform zL to
A by a series TL (l1)
=>Transform A to B (on the input SWR circle)
by a parallel C1
=> Transform B to zin by a series TL (l2)
zL + series-TL (l1)
=> A + shunt C1
=> B + series-TL (l2)
=> zin
j1.
6
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z
y
z
Single-Stub Matching Networks
Z
L 60 j45
Zin
75 j90
Z 75
0
L
L
in
Z
L
/ Z
0
0
0.
8
j0.
6
1/
zL
0.
8 j0.
6
Z / Z 1
j1.
2
in
5-20
g = 0.8
conductance circle
zL to A (yA= 0.8 + j1.5) by adding a shunt open-circuited (O.C.)TL lSA
4 adjustable parameters:
l , Z l , Z )
( s 0s, L 0L,
Input SWR circle associated with zin
has two intersected points (A & B) with
g = 0.8 conductance circle
yA= 0.8 + j1.05 yB= 0.8 - j1.05
The corresponding susceptance for the stub : jbSA= yA- yL = (0.8 + j1.05)-( 0.8 + j0.6)=0.45
O.C. point (g=0) to the point of ibSA = 0.45 is lSA = 0.067
2011-12 H.-R. Chuang EE NCKU
A to zin is lLA = 0.266
g = 0
(O.C.)
ibSA
= 0.45

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5-21
s
sB
s
sB
sB
sB
s
sB
l
l
l
l
l
l
l
l
2
tan
2
1
tan
2
2
tan
2
tan
2
//
)
1
1
:
stub
circuit
-
short
:
stub
circuit
-
open
(
design
stub
Balanced

5-22
Double-Stub Matching Networks
Zin
Z0
50
Z
L 50 j50
l / 8 l l 3
/ 8
0.
074
0.
051
2011-12 H.-R. Chuang EE NCKU
l
1
s1
2
l
3
s1

5-23
Quarter-Wave Transformer 四分之一波長(傳輸線)阻抗轉換匹配
( only useful for pure-resistance matching )
transmission
line 1
V1 ( z)
V1 ( z)
0 X
( Z 0)
( Z 0)
Z
in
quarter-wavelength
transmission line 2
l / 4
ZL
jZ0
tanl
( Z0
) Z0
Z
jZ tanl
l
&
2
tan
2011-12 H.-R. Chuang EE NCKU
Z
in
Z
0
Z
Z
2
0
0
0
l
4
ZL
jZ0
( )
Z0
Z0
Z0
Z
jZ ( )
Z
Ex: A microstrip quarter-wave trasformer that matches a 50 miscrostrip line to
a 20 load at f = 4 GHz (substrate: r=2.5, thickness h = 0.75 mm)
2.13[mm] 4.03[mm]
50[ ]
31. 62[
]
12.73[mm]
L
20[ ]
* Double Quarter-Wave Transformer for wider (matching bandwidth)
ZL
L
L
L

5-24
* (Matching) Bandwidth (f ) of a Quarter-Wave Transformer
Pozar, Mcrowave & RF Design of Wireless Systems
Approximate behavior of the reflection coefficient magnitude of a quarter-wave
transformer near the design frequency
It can be proved that
2
Z Z
Z
f
2(
f
( ) 0 f
BW
f
f
0
1 0 L
1
sec
2
m ZL
0
0
m
) 2 f
2
f
m
0
2
4
2
Increased BW for
Smaller load mismatch (ZL/Z0)
4
2 cos
2011-12 H.-R. Chuang EE NCKU
m
1
1
2
Z0Z
m
2
m ZL
Z
L
0

5-25
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