lecture #6 contents - guc · 2016. 3. 20. · guc (dr. hany hammad) 3/20/2016 comm (603) lecture#6...

GUC (Dr. Hany Hammad) 3/20/2016

COMM (603) Lecture#6 1

© Dr. Hany Hammad, German University in Cairo

Lecture #6 Contents

• Review of Smith Chart

• Impedance matching.

– Matching with lumped elements (L-Networks).


General Rules

jXRZ

jBGY

jxrz

jbgy

Impedance

Normalized

Resistance

Reactance

Conductance

Susceptance Normalized

Admittance




Smith Chart Construction

• It is a transformation from the impedance graph, to a reflection coefficient graph.

LZj

LL eZjXRZ

Where

)(

)(

L

L

ZmX

ZeR

Note

X

R0

to

to

- to

X

R)0,0(

),( XR

LZ

LZ

ir

j

LL je

Where

)(

)(

m

e

i

r

Note

180180

10

L

i

r)0,0(

),( ir

L



oL

oLL

ZZ

ZZ

• The relation between both domains are given by

1

1

1

1

L

L

oL

oL

oL

oLL

z

z

ZZ

ZZ

ZZ

ZZ

Instead of having separate Smith Charts for TL with different

characteristic impedances such as Zo=50, 75, 100 , etc; only one

normalized Smith chart do the job!

, Where

jxrZ

Xj

Z

R

Z

Zz

ooo

LL




Smith Chart

• The transformation

ir

L

LL j

z

z

1

1Rearrange

jxrj

jz

ir

ir

L

LL

1

1

1

1

Equating Both Sides

22

22

1

1

ir

irr

& 22

1

2

ir

ix

Rearrange

2

2

2

1

1

1

rr

rir

22

2 111

xxir

Circle Equations

22

0

2

0 ayyxx

Convert a constant r line from Z to Convert a constant x line from Z to

rradius 11

0,1 rrcenter

xradius 1

xcenter 1,1


Transforming “r”

i

r

x

1

r

rradius 11 0,1 rrcenter

1/2 2/3 (1/3,0)

0 1 (0,0)

1 1/2 (1/2,0)

0 (1,0)

2 1/3 (2/3,0)

0r

rL=0

(0,0)

1L

r Radius Center

5.0

(1/3,0)

2/3

(1/2,0)

1/2

2

(2/3,0)

1/3

.




Transforming “x”

0.5 2 (1,2)

0 (1,)

1 1 (1,1)

0 (1,0)

2 0.5 (1,0.5)

x Radius Center

xradius 1

xcenter 1,1

x

0x r

)5.0,1(

)5.0,1(

5.0

5.0

2x

2x

x

x

-0.5 2 (1,-2)

0 (1,-)

-1 1 (1,-1)

- 0 (1,0)

-2 0.5 (1,-0.5)

x Radius Center

1x

1x

1

1

)1,1(

5.0x

5.0x

)2,1(

2

1L

r

i



Constant resistance

Circles

Constant reactance

Circles




General Rules

Z

Y

o180

Impedance To Admittance

Short Circuit (Admittance

Chart)

Short Circuit (Impedance

Chart)

Open Circuit (Impedance

Chart)

Open Circuit (Admittance

Chart)


General Rules

Adding Parallel (Shunt) Components

Admittance Smith Chart

Adding Series Components

Impedance Smith Chart




General Rules

Moving on a constant Transmission line

inZ

5050 jZL

0z2z

Constant circle

500Z


General Rules

Adding Resistance in series to an impedance Conductance in parallel to an Admittance

3000R

50jjX L 50 0 Z

R

02.0jjB

50 0 Z

50jRZL

G

jBGYL

60G

Moving on a constant reactance (X) or susceptance (B) circles




General Rules

Adding Reactance in series to an impedance. Susceptance in parallel to an Admittance.

400400LX

LjX 50 0 Z

50

jB

50 0 Z

LL jXZ 50

G

Moving on a constant resistive (R) or conductance (G) circles

jBYL 02.0

02.0G


Impedance Matching

• Matching with lumped elements (L-Networks).

• The quarter-wave transformer.

• Broadband Matching

– The theory of small reflections.

– Binomial multi-section matching transformers.

– Chebyshev multi-section matching transformers.




Impedance Matching

• Impedance matching is important for the following reasons:

– Maximum power is delivered when the load is matched to the line (assuming the generator is matched), and power loss in is minimized.

– Impedance matching sensitive receiver components (antenna, low-noise amplifier, etc.) improves the signal-to-noise ratio of the system.

– Impedance matching in a power distribution network (such as an antenna array feed network) will reduce amplitude and phase errors.

Microwave Engineering, 3rd Edition by David M. Pozar Copyright © 2004 John Wiley & Sons


Impedance Matching

• Factors that may be important in the selection of a particular matching network include the following:

– Complexity:

• A simpler matching network is usually cheaper, more reliable, and less lossy than a more complex design.

– Bandwidth:

• Narrow or broadband.

– Implementation:

• According to the technology used, the matching network can be decided on.

– Adjustability:

• Required if dealing with variable load.




Impedance Matching Last Semester

• Single Stub Matching.

• Double Stub Matching.

LZoo RZ

inZ

oZ

1d

2d

oZ

LZoZ

inZ

oZ

4

3,

4,

8

orl

2d

oZ

1d

oZ


Matching with lumped elements (L-Networks)

• To be considered a "lumped element", no feature of its structure can exceed 1/10 of a wavelength at the maximum frequency of its usage.

• That being said, of the three primary lumped elements, resistors are the most well behaved as microwave elements, followed by capacitors, then inductors. It is possible to make lumped resistors and capacitors that work up to 100 GHz, but inductors usually stop being useful at X-band or lower.

oZ oZ

jX

jBLZ

jX

jBLZ

jX jB Susceptance Reactance




Matching with lumped elements (L-Networks)

Network for zL outside the 1 + jx circle

Network for zL inside the 1 + jx circle

jx1

8 possibilities for matching

jX and jB Can be capacitors or

inductors



Solution Using Smith Chart

Design an L-section matching network to match a series RC load with an impedance ZL=200-j100 , to a 100 , at a frequency of 500 MHz.

Solution

100200 jZL

12 jZ

Zz

o

LL

oL ZR





lz

1st Step 12 jzL


lz

ly180o

P

O

Q

2st Step

OP==OQ

2.04.0 jyl




3rd Step 2.04.0 jyl

ly


4th Step

ly

5.04.01 jy

5.04.02 jy

2.04.0 jyl

3.01 jjb

7.02 jjb




4th Step

ly

5.04.01 jy

2.04.0 jyl

3.01 jjb Sol. 1

2.111 jz

2.11 jjx


4th Step

ly

5.04.02 jy

2.04.0 jyl

7.02 jjb Sol. 2

2.112 jz 2.12 jjx




Example

3.01 jjb 2.11 jjx 7.02 jjb 2.12 jjx

CjZ

jbjBo

1

pF

Z

bC

o

955.0100105002

3.06

LjZjxjX o

nH

xZL o 2.38

105002

1002.16

L

j

ZjbjB

o

1

nH

b

ZL o 5.45

7.0105002

1006

C

jZjxjX o

pF

zZC

o

65.2

1002.1105002

116



Example


lecture #6 contents - guc · 2016. 3. 20. · guc (dr. hany hammad) 3/20/2016 comm (603) lecture#6...

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