Post on 04-May-2022
Impedance MatchingRF Electronics Spring, 2020
Robert R. Krchnavek Rowan University
Objectives
• Be able to design an L-section matching network.
• Understand the importance of T and Pi matching networks.
• Understand distributed parameter matching networks.
• Be able to design biasing networks.
Matching Networks• Matching networks are often used to achieve maximum
power transfer.
• Matching networks are also used for
• minimizing noise influence.
• maximizing power handling capabilities.
• linearizing the frequency response.
• In general, matching networks simply provide impedance transformation.
L-Section Matching Networks Two-Component Networks
L2
L1
ZS
ZL
ZS
ZL
ZS
ZL
ZS
ZL
C
L
C
C C
ZS
ZL
L
L2
L1
L L
ZS
ZL
C2
C1
C2
C1
ZS
ZL
ZS
ZL
• Analytical solution
• Smith Chart solution
L-Section Matching Networks
ZT = RT + ȷXT ZA = RA + ȷXA
ZM = RA − ȷXA
ZM = ZT ∥ȷXC + ȷXL
Zin = RT − ȷXT
Zin = (ZA + ȷXL) ∥ȷXC
Zin
L-Section Matching Networks
• Analytical method
• precise but slow.
• best used with the aid of a computer.
• Graphical method (Smith Chart)
• quick.
• not precise.
• yields a visual indication of the design space.
L-Section Matching Networks Analytical Solution
ZT = RT + ȷXT = 50 + ȷ25 Ω
ZA = RA + ȷXA = 25 − ȷ50 Ω
For maximum transfer of power, ZM = ZA
L-Section Matching Networks Analytical Solution
ZM = ZT k1
|!C+ |!L = Z
A
ZM = (RT + |XT ) k1
|!C+ |!L = RA |XA = Z
A
Separate into real and imaginary parts and solve for L and C.
L-Section Matching Networks Smith Chart Solution
ZT = RT + ȷXT = 50 + ȷ25 Ω
zT =ZT
Z0
=50 + ȷ25
50= 1 + ȷ0.5
ZA = RA + ȷXA = 25 − ȷ50 Ω
zA =ZA
Z0
=25 − ȷ50
50= 0.5 − ȷ1
L-Section Matching Networks Smith Chart Solution
L-Section Matching Networks Smith Chart Solution – General Approach• Find the normalized source and load impedance.
• Plot constant R and G circles that pass through zS. Use dashed lines or a unique color.
• Plot constant R and G circles that pass through zL*. Use solid lines or a different color.
• Find the intersection points between the dashed and solid lines. These are unique solutions to the matching problem.
• Find the normalized values of reactances and susceptances by tracing a path from zS to intersection point, to zL*.
• Determine actual values of L and C.
L-Section Matching Networks Smith Chart Solution – Specific Example
zT =ZT
Z0
=50 + ȷ25
50= 1 + ȷ0.5
zA =ZA
Z0
=25 − ȷ50
50= 0.5 − ȷ1
• Find the normalized source and load impedance.
L-Section Matching Networks Smith Chart Solution – Specific Example
• Plot constant R and G circles that pass through zS. Use dashed lines or a unique color. More specifically, if the first component is a shunt component, plot a constant G circle. If it is a series component, plot a constant R circle. For this example, zS is zT.
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
L-Section Matching Networks Smith Chart Solution – Specific Example
• Plot constant R and G circles that pass through zL*. Use solid lines or a different color. More specifically, if the component near the load is a series element, plot a constant R circle. In this example, zL* is zA*.
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
L-Section Matching Networks Smith Chart Solution – Specific Example
• Find the intersection points between the dashed and solid lines. These are unique solutions to the matching problem.
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
Forbidden Regions, Frequency Response, and Q
• Not every matching condition will allow all 8 L-type matching networks.
• Although the L-type matching network is matched at one frequency, the frequency response of the different networks varies.
• The loaded Q for each matching network will also, in general, vary.
Forbidden Regions ZS = Z0 = 50 Ω
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100
SWR 1∞
12345681015203040dBS 1∞
1234571015 ATTEN. [dB]
1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 S.W. L
OSS COEFF
1 ∞
0 1 2 3 4 5 6 7 8 9 10 12 14 20 30
RTN. LOSS [dB] ∞
0.010.050.10.20.30.40.50.60.70.80.91
RFL. COEFF, P0
0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 RFL. LOSS
[dB]
∞0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 S.W. P
EAK (CONST
. P)
0 ∞
0.10.20.30.40.50.60.70.80.91
RFL. COEFF, E or I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. C
OEFF, P
1
CENTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 TRANSM
. COEFF, E
or I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.10.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.70.7
0.70.8
0.8
0.80.9
0.9
0.9
1.01.0
1.0
1.21.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.50.5
0.5
0.60.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.61.6
1.6
1.81.8
1.8
2.02.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.01.0
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
±
90-9
085
-85
80-8
0
75-7
5
70-7
0
65-6
5
60-6
0
55-55
50-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.20.2
0.21
0.210.22
0.220.23
0.230.24
0.240.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
00
AN
GLE
OF
TRAN
SMISS
ON
CO
EFFICIEN
T IN D
EGREES
AN
GLE
OF REFLEC
TION
CO
EFFC
IENT IN
DEG
REES
—>
WA
VEL
ENG
THS
TOW
ARD
GEN
ERA
TOR
—>
<— W
AVE
LEN
GTH
S TO
WA
RDLO
AD
<—
IND
UCT
IVE
REA
CTAN
CE C
OMPO
NENT (+jX/Zo), O
R CAPACITIVE SUSCEPTANCE (+jB/Yo)
EACTANCE COMPONENT (-jX/Zo),
ORIN
DUCTIVE
SUSC
EPTA
NCE
(-jB
Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
SMITH CHART FORM ZY-01-N
NAME TITLE
Microwave Circuit Design - EE523 - Fall 2000
DWG. NO.
DATE
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
R VE I TI CAPAC
T and Pi Matching Networks
• T and Pi matching networks have an additional element compared to an L matching network.
• The additional element allows more control over the Q of the network.
• Can be used to control the bandwidth of the matching network.
Microstrip Matching Networks
• At higher frequencies, distributed element networks are used to achieve matching.
• Single-stub and double-stub matching networks.
• Mid-GHz range one often uses transmission lines + discrete capacitors.
Mixed Design Matching Network
• Assume a characteristic impedance for the transmission lines of 50Ω.
• Smith Chart solution -> normalize everything to 50 Ω.
• Traversing the length of the series transmission line elements results in a rotation on the Smith Chart on a constant radius circle.
Convert a load impedance of ZL=30+j10 Ω to an input impedance of Zin=60+j80 Ω
Mixed Design Matching Network
• In general, these networks are very sensitive to precise placement of the discrete capacitor.
Convert a load impedance of ZL=30+j10 Ω to an input impedance of Zin=60+j80 Ω
Completed design:
Fully Distributed-Element Matching Networks
• Single-stub tuning. See undergraduate notes for Engineering Electromagnetics. Also see textbook. Easily done on a Smith Chart.
• When a tunable matching network is required, double-stub tuning has an advantage over single-stub tuning because only the length of the stub and not its position needs to be varied.
Summary
• Impedance matching is necessary to maximize transfer of power from one stage to the next in an RF system.
• Two-component L-section matching has up to 8 different configurations. However, not all 8 are available in any given matching problem.
• The different L-types have different frequency response.
• T or Pi matching networks have an additional degree of freedom to control bandwidth of the matching network.
• At higher frequencies, matching may include C and X-lines. At still higher frequencies, X-lines with stubs.