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ENE 490Applied Communication

Systems

ENE 490Applied Communication

Systems

Lecture 2 circuit matching on Smith chart

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Review (1)Review (1)

High frequency operation and its applications Transmission line analysis (distributed elements)

– Use Kirchholff’s law to obtain general equations for transmission lines

– Voltage and current equations are the combination of incident and reflected waves.

0 0

0 0

0 0

( )

( )

j z j z

j z j z

V z V e V e

V VI z e e

Z Z

where Z0 is a characteristic impedance of a transmission line.

Assume the line is lossless.

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Review (2)Review (2)

Terminated lossless line– voltage reflection coefficient

– impedance along a transmission line

or

20

0

( )

j dV

d eV

00

0

tan( )

tan

L

L

Z jZ dZ d Z

Z jZ d

1 ( )( )

1 ( )

d

Z dd

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Review (3)Review (3)

- voltage standing wave ratio

source and loaded transmission line

max max

min min

( ) ( ) 1

( ) ( ) 1

L

L

V d I dVSWR

V d I d

200

0

( )

j lin

inin

Z Zd l e

Z Z

0

0

S

SS

Z Z

Z Z

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Review (4)Review (4)

power transmission of a transmission line

for lossless and a matched condition

power in decibels

*1Re2avP VI

218

Sin avs

S

VP P

Z

10log

1P W

P dBmmW

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Impedance matching network (1) Impedance matching network (1)

The need for matching network arises because amplifiers, in order to deliver maximum power to a load or to perform in a certain desired way, must be properly terminated at both the input and the output ports.

+

-VS

ZL

Inputmatching network

TransistorOutput

matching network

ZS

WZ1=50

WZ2=50

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Impedance matching network (2) Impedance matching network (2)

Effect of adding a series reactance element to an impedance or a parallel susceptance are demonstrated in the following examples.

Adding a series reactance produces a motion along a constant-resistance circle in the ZY Smith chart.

Adding a shunt susceptance produces a motion along a constant-conductance circle in the ZY Smith chart.

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Ex1 Adding a series inductor L (zL = j0.8) to an impedance z = 0.3-j0.3.

Ex1 Adding a series inductor L (zL = j0.8) to an impedance z = 0.3-j0.3.

= 0.3-j0.3

zL= j0.8

z

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Ex2 Adding a series capacitor C (zC = -j0.8) to an impedance z = 0.3-j0.3.

Ex2 Adding a series capacitor C (zC = -j0.8) to an impedance z = 0.3-j0.3.

= 0.3-j0.3

zC =-j0.8

z

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Ex3 Adding a shunt inductor L (yL = -j2.4) to an admittance y = 1.6+j1.6.

Ex3 Adding a shunt inductor L (yL = -j2.4) to an admittance y = 1.6+j1.6.

= 1.6+j1.6yyL=-j2.4

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Ex4 Adding a shunt capacitor C (yC = j3.4) to an admittance y = 1.6+j1.6.

Ex4 Adding a shunt capacitor C (yC = j3.4) to an admittance y = 1.6+j1.6.

C = 1.6+j1.6yyC=j3.4

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Examples of matching network designExamples of matching network design

Ex5 Design a matching network to transform the load Zload = 100+j100 W to

an input impedance of Zin = 50+j20 W.

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Ex6 Design the matching network that provides YL = (4-j4)x10-3 S to the transistor. Find the element values at 700 MHz.

Ex6 Design the matching network that provides YL = (4-j4)x10-3 S to the transistor. Find the element values at 700 MHz.

C

L 50W

yL=(4-j4)x10 S-3

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L matching networksL matching networks

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Forbidden regionsForbidden regions

Sometimes a specific matching network cannot be used to accomplish a given match.

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Load quality factor The developed matching networks can also be viewed as resonance circuits with f0 being a resonance frequency. These networks may be described by a loaded quality factor, QL.

The estimation of QL is simply accomplished through the use

of a so-called nodal quality factor Qn. At each node of the

L-matching networks, there is an equivalent series input impedance, denoted by RS +jXS. Hence a circuit node Qn can

be defined at each node as

0L

fQ

BW

S Pn

S P

X BQ

R G

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Circuit node Qn and loaded QL

Circuit node Qn and loaded QL

L-matching network is not a good choice for a design of high QL circuit since it is fixed by Qn.

For more complicated configurations (T-network, Pi-network), the loaded quality factor of the match network is usually estimated as simply the maximum circuit node quality factor Qn.

2n

LQ

Q

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The example of Q calculationThe example of Q calculation

At 500 MHz Qn = 2

then QL = 1.

and = 500 MHz0

L

fBW

Q

C=12.7 pF 10W

L=3.18 nH L=3.18 nH

VS

50W

Z IN = 50W

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Ex7 The low pass L network shown below wa s designed to transform a 200 W load to an in

put resistance of 200 W . Determine the loade d Q of the circuit at f = 500 MHz .

Ex7 The low pass L network shown below wa s designed to transform a 200 W load to an in

put resistance of 200 W . Determine the loade d Q of the circuit at f = 500 MHz .

ACC

=4

.77

5 p

F

R =

20

020 W

W

L = 19.09 nH

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Constant Qn contoursConstant Qn contours

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The upper and lower part of Q contours satisfy a circle equation.

The upper and lower part of Q contours satisfy a circle equation.

Since

then

which can be written as

2 2

2 2 2 2

1 1 21 (1 ) (1 )

U V

z r jx jU V U V

2 2

2

1

n

x UQ

r U V

2 22

1 1( ) 1

n n

U VQ Q

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Contour equationsContour equations

The equations for these contours can be derived from the general derivation of the Smith chart. By following the derivation, Qn contours follow this circle equation,

22 2

2

1 11i r

n nQ Q

where the plus sign is taken for positive reactance x and the minus sign for negative x.

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Qn circle parametersQn circle parameters

For x > 0, the center in the plane is at (0, -1/Qn).

For x < 0, the center in the plane is at (0, +1/Qn).

the radius of the circle can be written as

2

11 .

n

rQ

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Ex8 Design two T networks to transform the load impedance ZL = 50 W to the input impedance Zin = 10-j15 W with a Qn of 5.

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Ex9 Design a Pi network to transform the load impedance Zload = 50 W to the input impedance Zin = 150 W with a Qn of 5.

Ex9 Design a Pi network to transform the load impedance Zload = 50 W to the input impedance Zin = 150 W with a Qn of 5.