RF Circuit Design - [Ch2-2] Smith Chart

Chapter 2-2

The Smith Chart

Chien-Jung Li

Department of Electronics Engineering

National Taipei University of Technology

Department of Electronic Engineering, NTUT

The Smith Chart

• The analysis of transmission-line problems and of

matching circuits at microwave frequencies can

be cumbersome in analytical form. The smith

chart provides a very useful graphical aid to the

analysis of these problems.

• Matching circuits can be easily and quickly

designed using the normalized impedance and

admittance Smith chart (Z and Y charts).

• The Smith chart is also used to present the

frequency dependence of scattering parameters

and other amplifier characteristics.

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Development of the Smith Chart (I)

o

o

Z Zx

Z Z

• The Smith chart is the representation in the reflection coefficient plane,

called the plane, of the relation

for all values of Z, such that Re{Z}≥0. Zo is the characteristic impedance

of the transmission line or a reference impedance value.

• Defining the normalized impedance z as

o o

Z R jXz r jx

Z Z

11

1 1

r jxzU jV

z r jx

2 2

2 2

1

1

r xU

r x

2 2

2

1

xV

r xwhere and

• Reflection Coefficient

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Development of the Smith Chart (II)

r

x

U jV Γ-plane

U

V

1z j 1z

0z

1

1

z

z

1 1 1 90z j j

0 1 1 180z

1 0z

1 90

0

1

z r jxz-plane

1 1 1 90z j j

1z j

Short Load Open

1z

1

Pure Imaginary: inductive

1 90

Pure Imaginary: capacitive

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Constant Resistance Circles (I)

r

x

U jV Γ-plane

U

V

1 1z j

1 1z j 0z

0.447 63.4

0.447 63.4

z r jxz-plane

1 1z j

1 1z j

0.447 63.43

0.447 63.43

1 2z j

1 2z j

1 2z j

1 2z j

0.707 45

0.707 45

1j

2j

1j

2j

0.707 45

0.707 45

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Constant Resistance Circles (II)

r

x

z r jxz-plane

U

V

0z jx

0z r

0.5r 1r 3r

0.5z jx

1z jx 3z jx

0r 3r 1r

0.5r

6/42


Constant Reactance Loci

r

x

z r jxz-plane

U

V

0.5z j

0.5z j

1z j

3z j

0.5z j 1z j

3z j

0j

0.5j1j

3j

0.5j1j

3j

0.5 0.5z j

1 0.5z j

1.5 0.5z j

1 126.87

0.447 116.56

0.243 75.97

0.2773 33.69

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Complete the Smith Chart

Short Open Load

+jx

-jx

Inductive

Capacitive

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Reactance in the Smith Chart

Short Open Load

+jx

-jx

Inductive

Capacitive

+j0.1

+j0.2

+j0.3

+j0.4

+j0.5

+j0.6 +j1.6 +j1.7

+j1.8 +j2.0

+j3.0

+j4.0

+j5.0

+j6.0

0.4x

0.4x

0.4x

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Example – Impedance in the Smith Chart

1 1 1z j

2 0.4 0.5z j

3 3 3z j

4 0.2 0.6z j

5 0z 1z2z

3z

4z

5z

10/42


Example – Find from Impedance

19.44

1 3 3z j

1z

0.721 19.44

11/42


Example – Find Impedance from

0.447 26.56

2 1z j

26.56

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Use Smith Chart as an Admittance (Y) Chart

y g jb 1 1 1y j

2 0.4 0.5y j

3 2 1.4y j

4 0.5 0.2y j

5y 1y2y

3y4y

5y

13/42


Show Z and Y in One Chart

y g jb

U

V

U

V z r jx

1 1

1y g jb

z

1

1z

Impedance Chart (Z-Chart) Admittance Chart (Y-Chart)

jx

jx jb

jb

Short Load Open Short

Load Open

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The ZY Chart

U

V

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Adding a Series Inductor

0.8Lz j

0.3 0.3z j

0.3 0.5inz j

0.3 0.3z j

0.3 0.5inz j

0.8x

-j0.3

+j0.5

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Adding a Series Capacitor

0.8Cz j

0.3 0.3z j

0.3 1.1inz j

0.3 0.3z j

0.3 1.1inz j 0.8x

-j0.3

-j1.1

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Adding a Shunt Inductor

1.6 1.6y j

1.6 0.8iny j

2.4Ly j

1.6 1.6y j

1.6 0.8iny j

2.4y

+j1.6

-j0.8

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Adding a Shunt Capacitor

1.6 1.6y j

1.6 5iny j

3.4Cy j

1.6 1.6y j

1.6 5iny j

3.4y

+j1.6

+j5

19/42


Series/Shunt Inductor or Capacitor

Higher impedance Lower impedance

Series L

Series C

Shunt L

Shunt C

+jx

-jx

Inductive

Capacitive

Short

Open

Lower admittance Higher admittance

-jb

+jb

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Matching Networks (Two-Element L-Shape)

LZ1C

2C

LZL

C

LZ1L

2L

LZC

L

LZC

L

LZ2C

1C

LZL

C

LZ2L

1L

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Match to the Reference Impedance

• Usually the goal is to transform a particular impedance to the reference

impedance (center of the Smith chart). In practical systems, the

reference impedance .

50 refZ

1z2z

3z

4z

5z

Goal

Goal circle (r=1)

Goal circle (g=1)

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Matching from Load to the Reference Impedance (I)

10 10 LZ j

0.2 0.2Lz j

Goal

0.2j

0.4j

0.2x j

2j

0j

2y j

0.2 0.4z j

50 refZ

C

L

01@ 500 MHzinz f

0.2

0.2j

0.2j

0.5j

02 0.2 50 10f L

0

12 2 0.04

50f C

3.18 nHL

12.74 pFC

C

L 10

3.18 nH

3.18 nH

12.74 pF

23/42


Matching from Load to the Reference Impedance (II)

10 10 LZ j

0.2 0.2Lz j

Goal

0.2j

0.4j

0.6x j

2j

0j

2y j

0.2 0.4z j

L

C 0.2

0.2j

01@ 500 MHzinz f

0.6j

1

02 0.6 50 30f C

1

0

12 2 0.04

50f L

10.6 pFC

7.95 nHL

L

C

10.6 pF

7.95 nH

10

3.18 nH

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Matching from the Reference Impedance

1 L

C

8 12 mSoutY j

Goal 50

0.4 0.6outy j

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Matching from Load to an Arbitrary Impedance

LZC

L

50 20 inZ j

100 100 LZ j

Goal

100 refZ

LZC

L

0.5 0.2 inZ j

1 1 Lz j

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Impedance with Frequency Increasing

L

R

C

R

L

RC

LR

C

1inZ R j L

1

150

in

in

Zz r jx

1in aZ

1in bZ

2inZ

2in aZ

2in bZ

3inZ

3in aZ

1

1inZ R j

C

3in bZ 4inZ

4in bZ

4in aZ

27/42


Impedance with Frequency Increasing

L R

C R

L R

C

C R

L

2inZ 1inZ

4inZ 3inZ

1in aZ

1in bZ

2in aZ

2in bZ

3in aZ

3in bZ

4in aZ

4in bZ

28/42


Constant Q Contour (I)

n

X xQ

R r

1nQ

2nQ

Short Open

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Constant Q Contours (II)

Short Open

very intensive very intensive

intensive

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Matching with Particular Q Requirement (I)

• At matched condition: 2

nL

QQ

• For certain BW spec., the designed QL meets 0

1

L

f BWQ

• Design a T-shape matching networks to transform to

. The matching should meet relative bandwidth

requirement of 40%.

50 LZ

10 15 inZ j

10.4

LQ

12.5

0.4LQ

At matched condition: 2.52

nL

QQ

5nQ Thus in the design stage, the network should have a node Q:

31/42


Matching with Particular Q Requirement (II)

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Low Q Matching with 5% LC Variations

1.2 nHL 1.8 pFC

1.1nQ 1nQ 1nQ 1.06nQ

LZ1.8 pFC

1.2 nHL

24.26 11.62 LZ j 50 inZ

• Application example: Match a certain

impedance to 50-Ohm in a 1800 MHz

GSM handset front-end with node Q = 1.

1.26 nHL 1.8 pFC

5% L variation

1.2 nHL 1.89 pFC

5% C variation

1.26 nHL 1.89 pFC

5% L+C variation

50.4 0.61inZ j 51.8 0.57inZ j 50.34 1.97inZ j 51.75 2.16inZ j

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High Q Matching with 5% LC Variations

50inZ 44 5inZ j 40 8inZ j 35.4 13.5inZ j

LZ1.8 pFC

5.5 nHL

24.26 11.62 LZ j 50 inZ

8.8 nHL

5% L variation 5% C variation 5% L+C variation


impedance to 50-Ohm in a 1800 MHz

GSM handset front-end with node Q = 3.

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Small Impedance Matched to 50 Ohm (I)

4.9nQ 5.1nQ 4.9nQ 5.1nQ

LZ8.6 pFC

0.78 nHL

2 1 LZ j 50 inZ

50.2 1.26inZ j 52.8 10inZ j 48.2 9.76inZ j 45.94 0.44inZ j

0.78 nHL 8.6 pFC

0.82 nHL 8.6 pFC

5% L variation

0.78 nHL 9 pFC

5% C variation

0.82 nHL 9 pFC

5% L+C variation


small impedance to 50-Ohm in a 1800

MHz GSM handset front-end. (node Q = 4.9)

In this case, the major problem is not easy

to find a small inductor for matching.

Practically, a higher value of inductor would

be used. (see next page)

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Small Impedance Matched to 50 Ohm (II)

11.2nQ 2.2nQ 11.8nQ 1.15nQ 11.8nQ 1.62nQ

LZ3.2 pFC

1.9 nHL

2 1 LZ j 50 inZ

50.15inZ 74.3 22inZ j 119.6 41.8inZ j

To avoid a small inductor, use a higher value of L

with increasing the node Q.

5% L variation 5% L+C variation

Problems arise:

(1) Fail to meet broadband spec.

(not a case for GSM in this example)

(2) Sensitive to component variations

(3) Use parallel-connected Ls to maintain

a low-Q matching (area consuming)

How about using a series-C and shunt-L?

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Small Impedance Matched to 50 Ohm (III)

5.1nQ 5.8nQ

LZ1 pFC

1.3 nHL

2 1 LZ j

50 inZ

12 pFC

9.3 pFC

Use more components to trade the

matching bandwidth. (area consuming)

Variations affect node Q easily in

low-impedance region.

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High Impedance Matched to 50 Ohm

9.94nQ 10.44nQ

LZ0.176 pFC

44 nHL

5000 60 LZ j 50 inZ

50.08 0.32inZ j 45.47 23inZ j


small impedance to 50-Ohm in a 1800

MHz GSM handset front-end. (node Q = 9.9)

In this case, the major problem is not easy to find

a small capacitor for matching. (In ICs, it is

possible)

The components variation affects.

5% L+C variation

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Frequency Sweeping (Low Q v.s. High Q)

5.7nQ

2.2nQ

LZ0.94 pFC

9.6 nHL

160.5 44 LZ j 50 inZ

LZ3.9 pFC

6.1 nHL

160.5 44 LZ j 50 inZ

2.0 pFC

47.8 2 @1.8 GHzinZ j

44.2 10 @1.9 GHzinZ j

57.3 13 @1.7 GHzinZ j

[email protected] GHzinZ

23.6 4.7 @1.7 GHzinZ j

47.8 46 @1.9 GHzinZ j

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Frequency Sweeping Low Q with 5% LC Variations

LZ0.94 pFC

9.6 nHL

160.5 44 LZ j 50 inZ

LZ0.99 pFC

10.1 nHL

160.5 44 LZ j 50 inZ

47.8 2 @1.8 GHzinZ j

44.2 10 @1.9 GHzinZ j

57.3 13 @1.7 GHzinZ j

44.7 9 @1.8 GHzinZ j

39.4 20.7 @1.9 GHzinZ j

51.1 3 @1.7 GHzinZ j

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Frequency Sweeping High Q with 5% LC Variations

LZ3.9 pFC

6.1 nHL

160.5 44 LZ j 50 inZ

2.0 pFC LZ4 pFC

6.4 nHL

160.5 44 LZ j 50 inZ

2.1 pFC

[email protected] GHzinZ

23.6 4.7 @1.7 GHzinZ j

47.8 46 @1.9 GHzinZ j 53 44 @1.8 GHzinZ j

46.4 3 @1.7 GHzinZ j

17.6 49 @1.9 GHzinZ j

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Summary

• Although the Smith chart is seldom used nowadays for the computation of reflection coefficients. It is very useful and helpful for the engineers on the high-frequency circuit designs.

• Just remember that a higher-Q circuit corresponds to a narrower bandwidth, and a lower-Q circuit corresponds to a wider bandwidth. Thus a higher-Q circuit is more sensitive to the frequency and components variations.

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RF Circuit Design - [Ch2-2] Smith Chart

Engineering

Transcript of RF Circuit Design - [Ch2-2] Smith Chart