rf circuit design - [ch2-1] resonator and impedance matching

Chapter 2-1

Resonators and Impedance Matching

with Lumped Elements

Chien-Jung Li

Department of Electronics Engineering

National Taipei University of Technology

Department of Electronic Engineering, NTUT

Resonators

• Lump Resonators

Series resonant circuit I R L

C

inZ

CV

1

inZ R j Lj C

21

2RP I R

21

4mW I L

2 2 2

2 2 2

1 1 1 1

4 4 4e C

CW V C I I

C C

At resonant frequency 0 02 f

m eW W 0

1

LC

0inZ R

Resonance occurs when the

average stored magnetic and

electric energies are equal, and

the input impedance is a purely

real impedance.

2/51


Series Resonant Circuit

• Quality Factor:

average energy stored

energy loss/secm e

R

W WQ

P

At resonance 0

2

00 0

2

12

2 41

2

m

R

I LLW

QP R

I R

2

2

00 0

20

1 12

2 4 1

1

2

e

R

IW C

QP CR

I R

X

QR

where

0

0

1 or X L

C

Q is a measure of the loss of

a resonance circuit, and lower

loss implies higher Q.

3/51


Comparisons

X LQ

R R

L R

C R

1XQ

R CR

Although the Q is defined, but the resonant frequency can’t be defined

(they are not resonators).

• For series RL and series RC Circuits

4/51


Define the Bandwidth

• For series resonant circuits, as 0

1

inZ R j Lj C

0Let 0 0

1

LC

2 2

0

22 2

2

0

1 11 1inZ R j L R j L R j L

LC

0 0 0

2

0

2R j L R jL

0

2 2QR

R j L R j

0LQR

I L R

C

inZ

CV

where

5/51


3-dB Bandwidth

0

2BW

0

0

22

in

BW QRZ R j R jBW QR

when 1

BWQ

2inZ R jR R

0

03-dB Bandwidth BWQ

1

3-dB Bandwidth in % BWQ

inZ

0

R

2R

2

• Define the bandwidth

When resonance occurs, the absolute

value of impedance is minimum in the

series resonant circuit.

6/51


Parallel Resonant Circuit

Parallel resonant circuit

1 1

inY j CR j L

2

2R

VP

R

21

4eW V C

2 2

2

2 2 2

1 1 1

4 4 4m L

V VW I L L

L L

At resonance, the angular resonant frequency 0 02 f

m eW W 0

1

LC

0

1inY

R

LI

LR C

inY

V

7/51


Parallel Resonant Circuit

• Quality Factor:

average energy stored

energy loss/secm e

R

W WQ

P

At resonance 0

B

QG

where

0

0

1 or B C

L

1G

R

Comparisons:

B RQ

G L

B

Q CRG

L

R

C

R

and

8/51


3dB Bandwidth

• For parallel resonant circuits, as 0

1 1 1

2inY j C jCR j L R

0

2BWDefine

0

0

1 12

2in

BW Q QY j jBW

R R R R

when 1

BWQ

1 1 2

inY jR R R

0

03-dB Bandwidth BWQ

1

3-dB Bandwidth in % BWQ

inY

0

1

R

2

R

2

When resonance occurs, the absolute

value of admittance is minimum

(maximum impedance) in the parallel

resonant circuit.

9/51


Loaded and Unloaded Q

LRResonant

Circuit

Unloaded Q

• For series RLC circuit

• For parallel RLC circuit

0

0

1L

L L

LQ

R R C R R

0

0

////L

L L

R RQ C R R

L

• Define the external Q eQ

0

0

1e

L L

LQ

R R Cfor series RLC

0

0

Le L

RQ R C

L

1 1 1

L eQ Q Q

for parallel RLC

for both cases

10/51


Impedance Matching

Matching

Network

in sZ R

sV

sR

LZo sZ R

0in

Goal:

• The matching network is ideally lossless, to avoid unnecessary loss of

power, and is usually designed so that the impedance seen looking

into the matching network is Zo (matched with the transmission line) or Rs (matched with the source impedance when no line connected).

• Maximum power is delivered when the load is matched to the line

(assuming the generator is matched), and power los in the feed line is

minimized.

• Impedance matching may be used to improve the signal-to-noise

ratio or amplitude/phase errors in some applications.

11/51


Eight Ell Matching Sections (L-Shape)

LZ1C

2C

LZL

C

LZ1L

2L

LZC

L

LZC

L

LZ2C

1C

LZL

C

LZ2L

1L

12/51


Lump Element Matching

• Two element (L-shape) matching

sV

sRjX

jBLY L L LY G jB

inZ

X : matching reactance

B : matching susceptance

We want to find X and B

Case (a) 1

or s s L

L

R R RG

in sZ R

0in s

in s

Z R

Z R

Goal:

13/51


L-Shape Matching – Case (a) Rs < RL

1in s

L L

Z jX RG jB jB

Real part: 1s L LR G X B B

Imaginary part: 0s L LR B B XG

sV

sRjX

Lj B B LG

lQsQ

Use the resonator concept:

Series RC or RL

Parallel RC or RL

s

s

XQ

Rl

L

L

B BQ

G

When the impedances are matched,

the imaginary part of the input

impedance looking into the matching

network is equal to zero.

14/51


L-Shape Matching – Case (a) Rs < RL

Imaginary part: 0s L LR B B XG

L

s

s L

X B BQ Q

R GlReal part: 1s L LR G X B B

2 1 1s LR G Q 1

1s L

QR G

Choose l 1

1s

s L

Q Q QR G

1

1s s

s L

X R Q RR G

and

1

1L L L L

s L

B G Q B G BR G

(>0, capacitance)

(<0, inductance)

(>0, inductance)

or 1L

s

R

R

When Q is chosen, X and B are also

decided.

15/51


L-Shape Matching – Case (b) Rs > RL

sV

sRjX

jB LZ L L LZ R jX

inY

X : matching reactance

B : matching susceptance 1

in

s

YR

1

01

in

s

in

s

YR

YR

Again, we want to find X and B

Case (b) s LR R

Goal:

16/51



1 1in

L L s

Y jBR jX jX R

Real part: s L s LBR X X R R

Imaginary part: 0L s LX X BR R

Use resonator concept:

s sQ BR l

L

L

X XQ

R

sV

sR Lj X X

jB LR

lQsQ

Parallel

RC or RL

Series RC or RL

When the impedances are matched,

the imaginary part of the input

admittance looking into the matching

network is equal to zero.

17/51



Imaginary part: l

L

s s

L

X XQ BR Q Q

R

Real part:

2

L s LQ R R R 1s

L

RQ

R

Choose l 1ss

L

RQ Q Q

R

1sL L L L

L

RX R Q X R X

R(>0, inductance)

1

1s

s s L

RQB

R R R(>0, capacitance)

(<0, capacitance)

0L s LX X BR R

s L s LBR X X R R

When Q is chosen, X and B are also

decided.

18/51


Series-to-Parallel Transformation

• Series-to-Parallel transformation

sR

sjX

pR

pjX

ss

s

XQ

R s s sZ R jX

p

p

p

RQ

X

p p

p

p p

R jXZ

R jX

ps

s p

s p

RXQ Q Q

R X

p p

s p s s

p p

R jXZ Z R jX

R jX

21s pR Q R

19/51


Matching Bandwidth (I)

Case (a) 1

or s s L

L

R R RG

sV

sRjX

jBLY L L LY G jB

inZ

in s

in s

Z R

Z R

sV

sRjX

Lj B B LG

sV

sRjX eqjB

eqR

2

1 1

1eq

L

RG Q

211 L

eq

eq

G QB

QR Q

20/51


Matching Bandwidth (II)

sV

sRjX eqjB

eqR

Impedance matching at 0

0

1in eq s

eq

Z jX R RjB

Imaginary part: 1

0eq

jXjB

1eqXB

Let 0X L 0eqB C 0

1

LC

Real part: eq sR R 2

1 1

1s

L

RG Q

1

1L s

QG R

and ,where

and

Series resonant circuit

21/51


Matching Bandwidth (III)

sV

sRjX eqjB

eq sR R

inQ

1

2L inQ Q

• Define QL and Qin for RLC resonator

21in s L

eq

XQ R Q G Q Q

R

Find 3-dB bandwidth for

Let X L eqB C

as 0 ,let 0 0

1

LC 0

1

2

1 2 22

eqin s

in s ss

eq

jXjBZ R jL

Z R jL RjX R

jB

and

where and

22/51


Matching Bandwidth (IV)

3-dB bandwidth for 2 occurs when

2 2 sL R

02 22 s sR R

L X

3-dB Bandwidth in % BW

0

22 2 1 2

11

s

L

s L

RBW

X Q Q

R G

Similarly, for case (b)

1 2 2

11L

s L

BWQ Q

R G

0

1

2

12

1

1s L

QR G

Case (a)

1

s

L

RG

1s

L

RQ

R

Case (b)

s LR R

23/51


Example – L-Shape Matching (I)

• Find element values and 3-dB fractional bandwidth of the two-element

matching network.

1 2000

1 150s L

QR G

s

L

1R < , choose case (a)

G

6.245Q

6

6

50 6.245 2 100 10

6.245 / 2000 0 2 100 10

s

L L

X R Q L

B G Q B C

1st Solution: choose

Matching

Network

100in sZ f MHz R

sV

50sR 2000LR

100 0in f MHz

Goal:

LR

LR 4.9696C pF

496.96L nH

sV

50sR

-9

12

496.96 10 ( ) 496.96 ( )

4.9696 10 ( ) 4.9696 ( )

L H nH

C F pF

6.245

24/51


Example – L-Shape Matching (II)

6

6

50 ( 6.245) 1/(2 100 10 )

( 6.245) / 2000 0 1/(2 100 10 )

s

L L

X R Q C

B G Q B L

6.245Q 2nd Solution: choose

12

9

5.097 10 ( ) 5.097 ( )

509.7 10 ( ) 509.7 ( )

C F pF

L H nH

LR

sV

50sR

509.7L nH

5.097C pF

3dB

1 2 232%

| | 6.245L

BWQ Q

9dB 15dB16%, 8%BW BWand

25/51


Three Elements Matching (High Q)

Case (a) Pi-Shape Matching

sV

sR

2jX

3jBLY L L LY G jB

inZ in sZ R

0in s

in s

Z R

Z R

1jB

sV

sR

1jX

LZ L L LZ R jX

inZ in sZ R

0in s

in s

Z R

Z R

2jB

3jX

Case (b) T-Shape Matching

26/51


Pi-Shape Matching

sV

sR

2ajX

3jBLY L L LY G jB

VR

1jB

2bjX

VR

2 2 2a bX X X

VR : virtual resistance (designed by yourself)

1

V

L

RG

V sR R

Splitting into 2 “L-shape” matching networks

sV

sR

2ajX

3jBLY

L L LY G jB

1jB

2bjX

VR

sV

sR

in sZ R 0 in VZ R 0

27/51


Pi-Shape Matching

sV

sR

2ajX

3jBLY

L L LY G jB

1jB

2bjX

VR

sV

VR

in sZ R 0 in VZ R 0

1 1s

V

RQ

R 2

11

V L

QR G

Since is small, you can get a high Q VR

1

1

2BW

Q2

2

2BW

Q

1 2min ,BW BW BW

28/51


T-Shape Matching

VR : virtual resistance (designed by yourself)

sV

sR

1jX

LZ L L LZ R jX

VR

2ajB

3jX

2bjB

VR V LR RV sR R

sV

sR

1jX

LZ

L L LZ R jX

in sZ R 0

2ajB

3jX

VR 2bjB

sV

VR

in VZ R 0


29/51


T-Shape Matching

sV

sR

1jX

in sZ R 0

2ajBVR LZ

L L LZ R jX

3jX

2bjB

sV

VR

in VZ R 0

1 1V

s

RQ

R 2 1V

L

RQ

R

Since is large, you can get a high Q VR

1

1

2BW

Q2

2

2BW

Q

1 2min ,BW BW BW

30/51


Example – Pi and T Matching Networks

• Use Pi and T matching networks to achieve a matching BW<5%.

Matching

Network

100in sZ f MHz R

sV

50sR 2000LR

100 0in f MHz

Goal:

LR

31/51


Example – Pi Matching Network (I)

sV

50sR

2ajX

3jB1jB

2bjX

1.249VR

sV

1.249VR

100 50inZ f MHz

100 0f MHz

100 1.2492inZ f MHz

100 0f MHz

VR LG

1

501 1 6.247

1.2492s

V

RQ

R 2

1 20001 1 40

1.2492V L

QR G

1

2000LG

Pi-section matching network

1 2max(| |,| |) max( 50 / 1, 2000 / 1) 2000 / 1v v vQ Q Q R R R

25% 1.249

(2000 / ) 1v

v

BW RR

32/51


Example – Pi Matching Network (II)

sV

50sR

2ajX

1jB

1.249VR

100 50inZ f MHz

100 0f MHz

VR

1 6.247Q

11 0

s

QB C

R198.85C pF

2 1 0a VX R Q L 12.42L nH

1 6.247Q

1

1

0

1

s

QB

R L

203.95C pF

2 1

0

1a VX R Q

C

12.74L nH

33/51


Example – Pi Matching Network (III)

3jB

2bjX

sV

1.249VR

100 1.2492inZ f MHz

100 0f MHz

1

2000LG

2 40Q

3 2 0L LB G Q B C 31.83C pF

2 2 0b VX R Q L 79.53L nH

2 40Q

3 2

0

1L LB G Q B

L 79.58L nH

2 2

0

1b VX R Q

C 31.85C pF

34/51


Example – Pi Matching Network (IV)

2k198.85pF

12.42nH

sV

50 79.53nH

31.83pF 2k198.85pF

12.42nH

sV 79.58nH

31.85pF

2k12.74nH

203.95pF

sV

50 79.53nH

31.83pF 2k12.74nH

203.95pF

sV

5031.85pF

79.58nH

50

35/51


Example – Pi Matching Network (V)

36/51


Example – T Matching Network (I)

sV

50

1jX

100 50inZ f MHz

100 0f MHz

2ajBVR

3jX

2bjB

sV

T-section matching network

1 2max(| |,| |) max( ( / 50) 1, ( / 2000) 1) ( / 50) 1v v vQ Q Q R R R

25% 80050

( / 50) 1v

v

BW RR

80050VR

100 80050inZ f MHz

100 0f MHz

80050VR

LR

2LR k

1

800501 1 40

50V

s

RQ

R 2

800501 1 6.247

2000V

L

RQ

R

37/51


Example – T Matching Network (II)

sV

50

1jX

100 50inZ f MHz

100 0f MHz

2ajB 80050VR

1 40Q

1 1 0sX R Q L 3.183L H

12 0a

V

QB C

R 0.795C pF

1 1

0

1sX R Q

C 0.796C pF

1

2

0

1a

V

QB

R L 3.185L H

1 40Q

38/51


Example – T Matching Network (III)

3jX

2bjB

sV

100 80050inZ f MHz

100 0f MHz

80050VR

2LR k

2 6.247Q 3 2 0L LX R Q X L 19.884L H

22 0b

V

QB C

R 0.124C pF

3 2

0

1L LX R Q X

C 0.127C pF

2

2

0

1b

V

QB

R L 20.394L H

2 6.247Q

39/51


Example – T Matching Network (IV)

2k0.795pF

3.183 H

sV

50 19.884 H

0.124pF

2k3.185 H

0.796pF

sV

50 19.884 H

0.124pF

2k0.795pF

3.183 H

sV

50 0.127pF

20.394 H

2k3.185 H

0.796pF

sV

500.127pF

20.394 H

40/51


Example – T Matching Network (V)

41/51


Cascaded L-Shape Matching (Low Q)

Case (a) 1

s V

L

R RG

sV

sR

1jX

1jBLY L L LY G jB

in sZ R

2jX

2jB

0VRVR

sV

VR

2jX

LY

L L LY G jB

2jB

sV

sR

1jX

1jB VR

in sZ R

0

in VZ R

0


42/51



sV

VR

2jX

2Lj B B LG

sV

sR

1jX

1jB VR

in sZ R

0

in VZ R

0

1 1V

s

RQ

R 2

11

L V

QG R

We want to have the maximum bandwidth (find minimum Q)

Let 1 2Q Q Q sV

L

RR

G

2 2 2

2 11

s L

BWQQ

R G

43/51



Case (b) s V LR R R

sV

sR

1jX

1jB LZ L L LZ R jX

in sZ R

2jX

2jB

VRVR 0

sV

sR

1jX

1jBLZ

L L LZ R jX

in sZ R

2jX

2jB

0

VR

sV

VR

in VZ R 0


44/51



sV

sR

1jX

1jB

in sZ R

2 Lj X X

2jB

0

VR

sV

VR

in VZ R 0

VR

1 1s

V

RQ

R 2 1V

L

RQ

R

Let 1 2Q Q Q V s LR R R

2 2 2

2 11

s L

BWQQ

R G

We want to have the maximum bandwidth (find minimum Q)

45/51


Example – Cascaded L-Shape Matching (I)

• Use Cascaded L-shape matching networks to achieve BW>60%.

Matching

Network

100in sZ f MHz R

sV

50sR 2000LR

100 0in f MHz

Goal:

LR

Choose RV to let

50 2000 316.23V s LR R R

2 261.29%

1 2000 50 11s L

BW

R G

s V LR R R

46/51


Example – Cascaded L-Shape Matching (II)

sV

50sR

1jX

1jB

100 50inZ f MHz

2jX

2jB

100 0f MHz

VR

sV

316.23VR

100 0f MHz

100 316.23inZ f MHz

LR

1

316.231 1 2.3075

50V

s

RQ

R

2

20001 1 2.3075

316.23L

V

RQ

R

1 2.3075Q

1 1 0sX R Q L 183.63L nH

11 0

V

QB C

R11.61C pF

1 2.3075Q

1 1

0

1sX R Q

C13.79C pF

1

1

0

1

V

QB

R L 218.11L nH

2 2.3075Q

2 2 0VX R Q L 1.161L H

22 0L

L

QB B C

R1.836C pF

2 2.3075Q

2 2

0

1VX R Q

C 2.181C pF

2

2

0

1L

L

QB B

R L1.379L H

316.23VR 2LR k

47/51


Example – Cascaded L-Shape Matching (III)

2k11.61pF

183.63nH

sV

50 1.161 H

1.836pF

2k218.11nH

13.79pF

sV

501.161 H

1.836pF

2k11.61pF

183.63nH

sV

502.181pF

1.379 H

2k218.11nH

13.79pF

sV

502.181pF

1.379 H

48/51


Example – Cascaded L-Shape Matching (IV)

49/51


Matching For Larger Bandwidth

L-Shape

Matching

Network

sV

50sR

LR

L-Shape

Matching

Network

L-Shape

Matching

Network

• Use multi-section L-shape matching networks to extend BW.

50/51


Summary

• Impedance matching is a procedure of transforming the complex load impedance ZL to match with the source impedance Zs or characteristics impedance Zo of the transmission line (Generally, Zs = Rs = Zo).

• The lossless lumped element matching networks that are commonly used include L-section, Pi-section, T-section, and cascaded L-section. L-section has medium but not adjustable matching bandwidth. Pi and T sections have rather small and adjustable matching bandwidth. Cascaded L-section can increase bandwidth as the number of stages increases.

51/51

rf circuit design - [ch2-1] resonator and impedance matching

Engineering

q r r c r r

r r q c r r

r q r c

iny j r r r

r v p r

bw q q y j jbw r r r

inz r jr r

q r cr