1 2.1. Main principles (conjugate matching, maximum delivered power) LECTURE 2. IMPEDANCE MATCHING 2.2. Smith chart 2.3. Matching with lumped elements

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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

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2.1. Main principles2.1. Main principles

Impedance matching is necessary to provide maximum delivery of RF power to load from source

L

2

LS

L2S

L

2in

1 Re

2

1

1 Re

2

1

ZZZ

ZV

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

2

1

XXRR

RVP

- 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

2

1

XXRR

RVP

0 L

R

P0

L

X

P

. 0

0

SLL

2SL

2L

2S

XXX

XXRR

SL

LS

XX

RRor SL ZZ

S

2S

8

R

VP - maximum power

delivered to load

- impedance conjugate matching conditions

SL

LS

BB

GGor SL YY

- admittance conjugate matching conditions

SL WW - immitance conjugate matching conditions (Z or Y)

4


Smith chart represents relationships between load impedance Z and reflection coefficient

with real and imaginary parts of

1

1

0Z

Z

000

Z

Xj

Z

R

Z

Z

ir jir

ir

00 1

1

j

j

Z

Xj

Z

R

2

0

02i

2

0r

ZR

Z

ZR

R

2

0

2

0 i

2r 1

X

Z

X

Z

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/X

In admittance form:

2

0

02i

2

0r

YG

Y

YG

G 2

0

2

0 i

2r 1

B

Y

B

Y

5


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


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

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X1

X2

L-transformer

R1 Z1 X1 R2 Z2

X2

Equivalence when Z1 = Z2: 21

21

12

12

12

1

211

22 XR

XRj

XR

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

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For conjugate matching with reactance compensation :

221 1 QXX

221 1 QRR

. 1 /

/

21

22

11

RRQ

QRX

QRX

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

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Bandwidth properties

Two L-type matching circuits

22

11

2

1

22

11

/1

/ 1

/

RQC

QRL

R

RQ

RQL

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

2/22

n

00

nQF

ffQ

where Fn - out-of-band suppression factor

n - harmonic number

Zin

R1

0.707R1

1 f /f0 0

2f

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- 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

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-type matching circuits

111 / RQC 222 / RQC

212113 1/ QQQRL

1 1 21

1

22 Q

R

RQ 1

2

121

R

RQ

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

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111 / QRL 222 / RQC

221223 1/ QQQRL ,

1 1 22

2

11 Q

R

RQ 1

1

222

R

RQ


-type matching circuits

111 / QRL 222 / RQC

212223 / 1 QQRQC

1 1 21

1

22 Q

R

RQ 1

2

121

R

RQ

L3

R1 L1 R2 C2

C3

R1 L1 R2 C2

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111 /1 QRC 222 RQL

222123 1 / QRQQC

1 1 22

1

21 Q

R

RQ 1

2

122

R

RQ

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

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111 /1 QRC 222 RQL

21

2223 / 1 QQQRL ,

1 1 21

2

12 Q

R

RQ 1

1

221

R

RQ

L2

R1 L3 R2

C1


T-type matching circuits

L2

R1 R2 C3

L1

111 RQL 222 RQL

222213 1 / QRQQC

1 1 22

1

21 Q

R

RQ 1

2

122

R

RQ

15


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 R

R

R

R

R

R

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

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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

j

j

Z

Z

2exp 1

2exp 1

L

L

0

in

Impedance at input of loaded transmission line:

L

L

0

L

1

1

Z

Z

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

RR

XRRXRRZ

LS LS

L S0

1

tan

RXXR

RRZ

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For 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

XZ

ZRR

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

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Conjugate matching: L-type transformer

Second implicit equation : numerical or graphical solution


R1 Z1 C R2 Zin

Z0,

21

21

12

12

12

1

211

inin XR

XRj

XR

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

RZ

RZZX

1 / RQC

2202

2

22

2

0

2

2

0

2

1

sin/ cos

cos sin 1

ZR

ZR

RZ

R

R

where X1 = -1/ C

Matching for any ratio of

R1/R2

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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 R

R

R

R

R

R

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°


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

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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 P

VVR

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

t

tEC

V

IC

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

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S-parameter measurements


in

in0

in

inin 1

1

ZI

VZ

out

out0

out

outout 1

1

ZI

VZ

incin

reflin

in V

V

incout

reflout

out V

V

VS Vin

Iin ZS = Z0 Iout

Vout ZL = Z0 Zin Zout

reflinV refl

outV

incinV inc

outV

Z0 Z0

where

where

To define Zout, source with nominal power is placed instead of load, and load becomes source

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Power measurements


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

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2.6. Types of transmission lines2.6. Types of transmission lines

Coaxial line

- characteristic impedance

2b

2a

a

bZ 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

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Main wave type for stripline - transverse electromagnetic

TEM wave

r

W

b

t

Stripline

bW

bZ

441.0

30

er

0

. 35.0 for 35.0

35.0 for 0 2

e

bb

W

b

W

bb

W

b

W

b

W

1

10

100

Z0, Ohm

0.1 1 10

r = 2

20

W/b

10 6

4


• provides lower characteristic impedance

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Microstrip line


r

W

h

t 836.00724.0rr

0/735.1 1

1120

hWW

hZ

10

100

0.1 1 10

r =2

20

Z0

W/h

4

10

6 8

26


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

Documents

1 2.1. Main principles (conjugate matching, maximum delivered power) LECTURE 2. IMPEDANCE MATCHING 2.2. Smith chart 2.3. Matching with lumped elements