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2.1. Main principles2.1. Main principles (conjugate matching, maximum delivered power) (conjugate matching, maximum delivered power)
LECTURE 2. IMPEDANCE MATCHINGLECTURE 2. IMPEDANCE MATCHING
2.2. Smith chart2.2. Smith chart
2.3. Matching with lumped elements2.3. Matching with lumped elements
2.4. Matching with transmission lines2.4. Matching with transmission lines
2.6. Types of transmission lines2.6. Types of transmission lines (coaxial line, stripline,microstrip line,slotline, (coaxial line, stripline,microstrip line,slotline, coplanar waveguide) coplanar waveguide)
2.5. Determination of active device 2.5. Determination of active device impedancesimpedances
2
2.1. Main principles2.1. Main principlesImpedance matching is necessary to provide
maximum delivery of RF power to load from source
L
2
LS
L2S
L
2in
1 Re 21 1 Re
21
ZZZZV
ZVP
VS Zin = ZL
ZS
Vin
Iin
IS YS Yin =
YL
Iin
Vin
ZS = RS + jXS - source impedance
ZL = RL + jXL - load impedance
- power delivered to load
( substitution of real and imaginary parts of source and load impedances)
2LS2
LS
L2S
21
XXRRRVP
- power delivered to load as
function of circuit parameters
3
2.1. Main principles2.1. Main principles
For fixed source impedance ZS, to maximize output power
2LS2
LS
L2S
21
XXRRRVP
0 L
RP 0
L
XP
. 0
0
SLL
2SL
2L
2S
XXXXXRR
SL
LS
XXRR
or SL ZZ
S
2S
8
RVP - maximum power
delivered to load
- impedance conjugate matching conditions
SL
LS
BBGG
or SL YY
- admittance conjugate matching conditions
SL WW - immitance conjugate matching conditions (Z or Y)
4
2.2. Smith chart2.2. Smith chart
Smith chart represents relationships between load impedance Z and reflection coefficient
with real and imaginary parts of
1 1
0ZZ
000
ZXj
ZR
ZZ
ir j
ir
ir
00 1 1
jj
ZXj
ZR
2
0
02i
2
0r
ZR
ZZR
R
2
02
0 i
2r 1
XZ
XZ
Equating real and imaginary parts:
- constant-(R/Z0) circles: family of circles centered at points r = R/(R + Z0) and i = 0
with radii of Z0/(R + Z0)
- constant-(X/Z0) circles: family of circles centered at points r = 1 and i = Z0/X with
radii of Z0/XIn admittance form:
2
0
02i
2
0r
YG
YYG
G 2
02
0 i
2r 1
BY
BY
5
2.2. Smith chart2.2. Smith chart
At Z Smith chart, curve from point A to pint C indicates
impedance transformation from resistance 25 Ohm to inductive
impedance (25 +j25) Ohm
At Y Smith chart, curve from point C to point D indicates admittance transformation from inductive
admittance (20 - j20) mS to conductance 20 mS (50 Ohm)
.
1 -10.51.02.05 .0 Г r
Гx
C
D
O=
-1.00.5 1.0 2.0 5.0 Г r
Гx
-1
C
A1
ZO=
1.0
6
2.2. Smith chart2.2. Smith chart
C
D A Y Smith chart
provides transformation from point C to
point D
At combined Z-Y Smith chart:
Z Smith chart provides
transformation from point A to
point C
7
2.3. Matching with lumped elements2.3. Matching with lumped elements
X1
X2
L-transformer
R1 Z1 X1 R2 Z2
X2
Equivalence when Z1 = Z2: 21
21
12
12
12
1
211
22 XR
XRjXR
XRjXR
221 1 QRR 2
21 1 QXX
where Q Q = RR11//XX11=XX22//RR22
- quality factor equal for series and parallel circuits
Impedance parallel and series circuits
8
2.3. Matching with lumped elements2.3. Matching with lumped elements
For conjugate matching with reactance compensation :
221 1 QXX
221 1 QRR
. 1 / /
21
22
11
RRQQRXQRX
X1
X2
R2 Zin = R1
Input impedance Zin will be resistive and equal to R1 when :
where Q Q = RR11//XX11=XX22//RR22
- quality factor equal for series and parallel circuits
9
2.3. Matching with lumped elements2.3. Matching with lumped elements
Bandwidth properties
Two L-type matching circuits
22
11
2
1
22
11
/1 /
1 /
RQCQRL
RR
QRQL
RQC
L 2
R 1 C 1 R 2 R 1
C 2
R 2 L 1
a ) . b ) .
Resistance R1 connected to
parallel reactive element must be
greater than resistance R2 connected to
series reactive element
1 2/
22n
00
nQFffQ
where Fn - out-of-band suppression factor
n - harmonic number
Zin
R1
0.707R1
1 f /f0 0
2f
10
2.3. Matching with lumped elements2.3. Matching with lumped elements
- transformer
• for each L-transformer, resistances R1 and R2 are transformed to some intermediate resistance R0 with value of R0 < (R1, R2)
X1
X3
X2
X3
X2 X1
X1
X3
R0 R1 X2 R2
X3
X1
X3 R0 R1
X2
R2 X3
Connection of two L-transformers
T- transformer
• for same resistances R1 and R2, T- and -transformers have better filtering properties, but narrower bandwidth compared with single L-transformer
11
2.3. Matching with lumped elements2.3. Matching with lumped elements
-type matching circuits
111 / RQC 222 / RQC
212113 1/ QQQRL
1 1 21
1
22 Q
RR
Q 1 2
121
RR
Q
L3
R1 C1 R2 C2
• widely used as output matching circuit to provide
Class B operation with sinusoidal collector voltage
• useful for interstage matching when active device input and output capacitances can be easily incorporated inside
matching circuit
• provides significant level of harmonic suppression
• with additional series LC-filter, can be directly applied to realize Class E mode with
shunt capacitance
12
111 / QRL 222 / RQC
221223 1/ QQQRL ,
1 1 22
2
11 Q
RRQ 1
1
222
RR
Q
2.3. Matching with lumped elements2.3. Matching with lumped elements
-type matching circuits
111 / QRL 222 / RQC
212223 / 1 QQRQC
1 1 21
1
22 Q
RR
Q 1 2
121
RR
Q
L3
R1 L1 R2 C2
C3
R1 L1 R2 C2
13
2.3. Matching with lumped elements2.3. Matching with lumped elements
111 /1 QRC 222 RQL
222123 1 / QRQQC
1 1 22
1
21 Q
RR
Q 1 2
122
RR
Q
L 2
R 1 R 2 C 3
C 1
T-type matching circuits
• widely used as input, interstage and output
matching circuits in high power amplifiers
• can incorporate active device lead and bondwire
inductances within matching circuit
• provides significant level of harmonic suppression
• can be directly applied to realize Class F mode
providing high impedances at harmonics
14
111 /1 QRC 222 RQL
21
2223 / 1 QQQRL ,
1 1 21
2
12 Q
RR
Q 1 1
221
RR
Q
L2
R1 L3 R2
C1
2.3. Matching with lumped elements2.3. Matching with lumped elementsT-type matching circuits
L2
R1 R2 C3
L1
111 RQL 222 RQL
222213 1 / QRQQC
1 1 22
1
21 Q
RR
Q 1 2
122
RR
Q
15
2.3. Matching with lumped elements2.3. Matching with lumped elements
Matching design example
L3 + Lin
Rsource = R1 Rin C3
L2 Cin
Zin
C1
L1 C2 R2 R3
132-174 MHz 150 W MOSFET power amplifier:
three-section input matching
MHz 152174132c f
2.1 9.0in jZ
Q = 152/(174 - 132) = 3.6
in
3
3
2
2
1 RR
RR
RR
For Rin = 0.9 Ohm and R1 = 50 Ohm: R3 = 3.5 Ohm, R2 = 13 Ohm
Q = 1.7
Two low-pass and one high-pass L-sections
16
2.4. Matching with transmission lines2.4. Matching with transmission lines
For conjugate matching with reactance compensation when ZS = Zin* :
Transmission-line transformer
For quarter-wave transmission line with = 90° :
tan tan
L0
0 L0in jZZ
jZZZZ
Z0,
ZL Zin ZS
L20in / ZZZ
L
jj
ZZ
2exp 12exp 1
L
L
0
in
Impedance at input of loaded transmission line:
L
L
0
L
1 1
ZZ
Input impedance for loaded transmission line with electrical length of , normalized to its characteristic impedance Z0, can be found by rotating this
impedance point clockwise by 2 around Smith chart center point with radius L
SL
2S
2SL
2L
2L S
0
RRXRRXRRZ
LS LS
L S0
1
tan
RXXRRRZ
17
2.4. Matching with transmission lines2.4. Matching with transmission linesFor pure resistive source
impedance ZS= RS :
ZL ZS
Z01, /8 Z02, /4 Z03, /8
0 tan tan 1 2L
2 L
2 0
20L RXZZX
2L
2L L 0 XRZZ
L0
0L S
XZZRR
For electrical length = 45°
Any load impedance can be transformed into real source impedance using /8-transformer whose impedance is equal to magnitude of load impedance
To match any source impedance ZS and load impedance ZL, matching
circuit can be designed with two /8-transformers and one /4-transformer
L Z0,
< 90
C Z0,
< 90
tan 0ZL
Lumped and transmission line single-frequency equivalence
0 tan
ZC
18
Conjugate matching: L-type transformer
Second implicit equation : numerical or graphical solution
2.4. Matching with transmission lines2.4. Matching with transmission lines
R1 Z1 C R2 Zin
Z0,
21
21
12
12
12
1
211
inin XR
XRjXR
XRjXR
2in1 1 QRR
2in1 1 QXXReal and imaginary parts of
tan tan
20
0 20in jRZ
jZRZZ
2220
2
220in tan
tan 1
RZRZR
2220
22
20
0in tan tan
RZRZZX
1 / RQC
2202
2
222
0
2
2
0
2
1
sin/ cos
cos sin 1
ZRZR
RZ
RR
where X1 = -1/ C
Matching for any ratio of
R1/R2
19
Matching design example 470-860 MHz 150 W LDMOSFET power amplifier:
three-section input matching
Q = 635/(860 - 470) = 1.63
in
3
3
2
2
1 RR
RR
RR
For Rin = 1.7 Ohm and R1 = 50 Ohm: R3 = 5.25 Ohm, R2 = 16.2 Ohm
Q = 1.2
For Z01= Z02 = Z03 = 50 Ohm 1 = 30°, 2 = 7.5°, 3 = 2.4°
2.4. Matching with transmission lines2.4. Matching with transmission lines
MHz 635860470c f
3.1 7.1in jZ
Lin Rsource = R1
Rin C3 Zin C2 R2 R3
Z03, 3 in
C1
Z02, 2 Z01, 1
For 1 = 2 = 3 = 30° Z01 = 50 Ohm, Z02 = 15.7 Ohm, Z03 = 5.1 Ohm
20
Analytical evaluation
where Vsat is defined from load line analysis
2.5. Determination of active device impedances2.5. Determination of active device impedances
2
out
2satcc)B(
out PVVR
Output resistance in Class B :
Output capacitance : cout CC
gddsout CCC
- bipolar device
- FET device
c
coc 1 v
CC /
tVEv sin ccc cccc dt
dvvCi
2
0
2cc
c
c1c1
sin 1 cos td
ttEC
VIC
0 0.25 0.5
Cc(Vc)/Cc(Ec)
0.75
1.0
1.1
= 1/2
1/3
- junction capacitance
Large-signal collector capacitance
where cc EV /
where
21
S-parameter measurements2.5. Determination of active device impedances2.5. Determination of active device impedances
in
in0
in
inin 1
1
ZIVZ
out
out0
out
outout 1
1
ZIVZ
incin
reflin
in VV
incout
reflout
out VV
VS Vin
Iin ZS = Z0 Iout
Vout ZL = Z0 Zin Zout
reflinV refl
outV inc
inV incoutV
Z0 Z0
where
where
To define Zout, source with nominal power is placed instead of load, and load becomes source
22
Power measurements
2.5. Determination of active device impedances2.5. Determination of active device impedances
ZS ZL
Source Directional
coupler
Power meter
Impedance transformer
Impedance transformer
Power meter
Incident power
Zout Zin
• tune input impedance transformer to maximize incident power, I.e., power delivery from source to active device
• tune output impedance transformer to maximize output power delivered to load
• calculate input and output active device impedances according to
*Sin ZZ *
Lout ZZ
• measure transformer impedances seen from the active device input and output, I.e., ZS and ZL
23
2.6. Types of transmission lines2.6. Types of transmission lines
Coaxial line
- characteristic impedance
2b
2a
abZ ln
20
/ where - wave impedance of lossless line equal to intrinsic medium impedance
Main wave type for coaxial line - transverse
electromagnetic TEM wave
• widely used for hybrid high power applications: combiners, dividers, transformers
24
2.6. Types of transmission lines2.6. Types of transmission lines
Main wave type for stripline - transverse electromagnetic
TEM wave
r
W
b
t
Stripline
bWbZ
441.0 30
er0
. 35.0 for 35.0
35.0 for 0 2
e
bb
Wb
W
bb
W
bW
bW
1
10
100
Z0, Ohm
0.1 1 10
r = 2
20
W/b
10 6
4
- characteristic impedance
• provides lower characteristic impedance
25
2.6. Types of transmission lines2.6. Types of transmission lines
Microstrip line
- characteristic impedance
r
W
h
t 836.00724.0rr
0 /735.1 11120
hWW
hZ
10
100
0.1 1 10
r =2
20
Z0
W/h
4
10
6 8
26
2.6. Types of transmission lines2.6. Types of transmission lines
Coplanar waveguideSlotline
r
W
h
30
50
70
90
110
0.02
0.1 1
20
9.7
W/h
Z0, Ohm
r
W
h
W s
Characteristic impedance
• widely used for hybrid and monolithic integrated
circuits
• provide higher characteristic impedance
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