IMPEDANCE MATCHING IN HIGH FREQUENCY LINES

UNIT - III

Impedance Matching

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Maximum power is delivered when the load is matched the line and the power loss in the feed line is minimizedImpedance matching sensitive receiver components improves the signal to noise ratio of the systemImpedance matching in a power distribution network will reduce amplitude and phase errors

ComplexityBandwidthImplementation

Adjustability

Half and Quarter wave transmission lines

• The relationship of the input impedance at the input of the half-wave transmission line with its terminating impedance is got by letting L = wavelength/2in the impedance equation.

Zinput = ZL

• The relationship of the input impedance at the input of the quarter-wave transmission line with its terminating impedance is got by letting L =wavelength/4 in the impedance equation.

Zinput = (Zinput Zoutput)0.5

Series Stub

Input impedance=1/S

Voltage minimum

S

S

Xjj

dS

X

S

Sd

djjS

djSXjZ

SZ

in

in

1tan

2

tan

tan)1

1(

1

1cos

4

tan1

tan1

/1

10

0

0

10

01

01

Single Stub Tunning

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Shunt Stub

Series Stub

G=Y0=1/Z0

Single Shunt Stub Tuner Design Procedure

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1. Locate normalized load impedance and draw VSWR circle (normalized load admittance point is 180o from the normalized impedance point).

2. From the normalized load admittance point, rotate CW (toward generator) on the VSWR circle until it intersects the r = 1 circle. This rotation distance is the length d of the terminated section of t-tline. The nomalized admittance at this point is 1 + jb.

3. Beginning at the stub end (rightmost Smith chart point is the admittance of a short-circuit, leftmost Smith chart point is the admittance of an open-circuit), rotate CW (toward generator) until the point at 0 - jb is reached. This rotation distance is the stub length l.

Smith Chart

• Impedances, voltages, currents, etc. all repeat every half wavelength

• The magnitude of the reflection coefficient, the standing wave ratio (SWR) do not change, so they characterize the voltage & current patterns on the line

• If the load impedance is normalized by the characteristic impedance of the line, the voltages, currents, impedances, etc. all still have the same properties, but the results can be generalized to any line with the same normalized impedances

Smith Chart

• The Smith Chart is a clever tool for analyzing transmission lines

• The outside of the chart shows location on the line in wavelengths

• The combination of intersecting circles inside the chart allow us to locate the normalized impedance and then to find the impedance anywhere on the line

Smith ChartReal Impedance Axis

Imaginary Impedance Axis

Smith Chart Constant Imaginary Impedance Lines

Constant Real Impedance Circles

Impedance

Z=R+jX

=100+j50

Normalized

z=2+j for

Zo=50

Smith Chart

• Impedance divided by line impedance (50 Ohms)

• Z1 = 100 + j50 • Z2 = 75 -j100 • Z3 = j200 • Z4 = 150 • Z5 = infinity (an open circuit)• Z6 = 0 (a short circuit)• Z7 = 50 • Z8 = 184 -j900

• Then, normalize and plot. The points are plotted as follows:

• z1 = 2 + j• z2 = 1.5 -j2• z3 = j4• z4 = 3• z5 = infinity• z6 = 0• z7 = 1• z8 = 3.68 -j18S

Smith Chart

• Thus, the first step in analyzing a transmission line is to locate the normalized load impedance on the chart

• Next, a circle is drawn that represents the reflection coefficient or SWR. The center of the circle is the center of the chart. The circle passes through the normalized load impedance

• Any point on the line is found on this circle. Rotate clockwise to move toward the generator (away from the load)

• The distance moved on the line is indicated on the outside of the chart in wavelengths

Toward Generator

Away From Generator

Constant Reflection Coefficient Circle

Scale in Wavelengths

Full Circle is One Half Wavelength Since Everything Repeats

Single-Stub Matching

Load impedance jBYin 1

1tan

2length stub The

1

1cos

4

1

epoint wher minimum- voltage thefrom distance thebe Let

real ist coefficien reflection then thereal, is If1

1

10

10

0

S

S

S

Sd

jBY

d

Y

SY

in

L

in

Input admittance=S

Single Stub Tuning

Single-stub tuning circuits.

(a) Shunt stub. (b) Series stub.

• 2 adjustable parameters• d: from the load to the stub position.• B or X provided by the shunt or series stub.

• For the shunt-stub case, • Select d so that Y seen looking into the line at

d from the load is Y0+jB

• Then the stub susceptance is chosen as –jB.• For the series-stub case,

• Select d so that Z seen looking into the line at d from the load is Z0+jX

• Then the stub reactance is chosen as –jX.

Shunt Stubs• Single-Stub Shunt Tuning

ZL=60-j80

.

(b) The two shunt-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency

for the tuning circuits of (b).

• To derive formulas for d and l, let ZL= 1/YL= RL+ jXL.

• Now d is chosen so that G = Y0=1/Z0,

00

0

( ) tan

( ) tanL L

L L

R jX jZ dZ Z

Z j R jX d

2

2 20

20 02 2

0 0

1

(1 tan )where

( tan )

tan ( tan )( tan )

[ ( tan ) ]

L

L L

L L L

L L

Y G jBZ

R dG

R X Z d

R d Z X d X Z dB

Z R X Z d

2 2 20 0 0 0

2 20 0

00

( ) tan 2 tan ( ) 0

[( ) ] /tan , for

L L L L L

L L L LL

L

Z R Z d X Z d R Z R X

X R Z R X Zd R Z

R Z

• If RL = Z0, then tanβd = -XL/2Z0. 2 principal solutions are

• To find the required stub length, BS = -B.

for open stub

for short stub

1

0 0

1

0 0

1tan for - 0

2 2 2

1tan for - 0

2 2 2

L L

L L

X X

Z Zd

X X

Z Z

1 10

0 0

1 1tan tan

2 2Sl B B

Y Y

1 10 0 01 1tan tan

2 2S

l Y Y

B B

Series Stubs• Single Stub Series Tuning

ZL = 100+j80

(a) Smith chart for the series-stub

tuners.

(b) The two series-stub tuning

solutions. (c) Reflection coefficient

magnitudes versus frequency for the

tuning circuits of (b).

• To derive formulas for d and l, let YL= 1/ZL= GL+ jBL.

• Now d is chosen so that R = Z0=1/Y0,

00

0

( ) tan

( ) tanL L

L L

G jB jY dY Y

Y j G jB d

2

2 20

20 02 2

0 0

1

(1 tan )where

( tan )

tan ( tan )( tan )

[ ( tan ) ]

L

L L

L L L

L L

Z R jXY

G dR

G B Y d

G d Y B d B Y dX

Y G B Y d

2 2 20 0 0 0

2 20 0

00

( ) tan 2 tan ( ) 0

[( ) ] /tan , for

L L L L L

L L L LL

L

Y G Y d B Y d G Y G B

B G Y G B Yd G Y

G Y

• If GL = Y0, then tanβd = -BL/2Y0. 2 principal solutions are

• To find the required stub length, XS = -X.

for short stub

for open stub

1

0 0

1

0 0

1tan for - 0

2 2 2

1tan for - 0

2 2 2

L L

L L

B B

Y Yd

B B

Y Y

1 10

0 0

1 1tan tan

2 2Sl X X

Z Z

1 10 0 01 1tan tan

2 2S

l Z Z

X X

Analytic Solution• To the left of the first stub in Fig. 5.7b,

Y1 = GL + j(BL+B1) where YL = GL + jBL

• To the right of the 2nd stub,

• At this point, Re{Y2} = Y0

1 02 0

0 1

( ) where tan

( )L L

L L

G j B B Y tY Y t d

Y jt G jB jB

222 0 1

0 2 2

2 220 1

0 2 2 2 20

( )10

4 ( )11 1

2 (1 )

LL L

LL

Y B t B ttG G Y

t t

t Y B t B ttG Y

t Y t

• Since GL is real,

• After d has been fixed, the 1st stub susceptance can be determined as

• The 2nd stub susceptance can be found from the negative of the imaginary part of (5.18)

2 20 12 2 2

0

4 ( )0 1

(1 )Lt Y B t B t

Y t

2

00 2 2

10

sinL

YtG Y

t d

2 2 20 0

1

(1 ) L LL

Y t G Y G tB B

t

• B2 =

• The open-circuited stub length is

• The short-circuited stub length is

2 2 20 0 0(1 )L L L

L

Y Y G t G t G Y

G t

10

0

1tan

2

l B

Y

10 01tan

2

l Y

B

For a load impedance ZL=60-j80Ω, design two single-stub (short circuit) shunt tunning networks to matching this load to a 50 Ω line. Assuming that the load is matched at 2GHz and that load consists of a resistor and capacitor in series.

Single Stub Tunning

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yL=0.3+j0.4

d1=0.176-0.065=0.110λ

d2=0.325-0.065=0.260λ

y1=1+j1.47

y2=1-j1.47

l1=0.095λl1=0.405λ

Single Stub Tunning

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Results

For a load impedance ZL=25-j50Ω, design two single-stub (short circuit) shunt tunning networks to matching this load to a 50 Ω line.

Single Stub tunning

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yL=0.4+j0.8

d1=0.178-0.115=0.063λ

d2=0.325-0.065=0.260λ

y1=1+j1.67

y2=1-j1.6

l1=0.09λl1=0.41λ

Single Series Stub Tuner Design Procedure

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1. Locate normalized load impedance and draw VSWR circle

2. From the normalized load impedance point, rotate CW (toward generator) on the VSWR circle until it intersects the r = 1 circle. This rotation distance is the length d of the terminated section of t-tline. The nomalized impedance at this point is 1 + jx.

3. Beginning at the stub end (leftmost Smith chart point is the impedance of a short-circuit, rightmost Smith chart point is the impedance of an open-circuit), rotate CW (toward generator) until the point at 0 ! jx is reached. This rotation distance is the stub length l.

For a load impedance ZL=100+j80Ω, design single series open-circuit stub tunning networks to matching this load to a 50 Ω line. Assuming that the load is matched at 2GHz and that load consists of a resistor and inductor in series.

Single Stub Tunning

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zL=2+j1.6

d1=0.328-0.208=0.120λ

d2=0.5-0.208+0.172=0.463λ

z1=1-j1.33

z2=1+j1.33

l1=0.397λl1=0.103λ

Single Stub Tunning

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Single Stub Tunning

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Double Stub Matching Network

jB1jB2YL

ab

2

1

P tocircle econductancconstant along

point themoves which Bj esusceptanc a adds stubfirst The

aa plane at the

into ed transformis

LLL

LL

BjGY

YY

b a

Double-Stub Tuning• If an adjustable tuner was desired, single-tuner

would probably pose some difficulty.

Smith Chart Solution

• yL add jb1 (on the rotated 1+jb circle) rotate by d thru SWR circle (WTG) y1 add jb2 Matched

• Avoid the forbidden region.

Double Stub Tunning

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The susceptance of the first stub, b1, moves the load admittance to y1, which lies on the rotated 1+jb circle; the amount of rotation is de wavelengths toward the load. Then transforming y1 toward the generator through a length d of line to get point y2, which is on the 1+jb circle. The second stub then adds a susceptance b2.

Design a double-stub shunt tuner to match a load impedance ZL=60-j80 Ω to a 50 Ω line. The stubs are to be open-circuited stubs and are spaced λ/8 apart. Assuming that this load consists of a series resistor and capacitor and that the match frequency is 2GHz, plot the reflection coefficient magnitude versus frequency from 1 to 3GHz.

Double Stub Tunning

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yL=0.3+j0.4

b1=1.314

b1’=-0.114

y2=1-j3.38

l1=0.46λ

l2=0.204λ

Double Stub Tunning

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Double-stub tuning.

(a) Original circuit with the

load an arbitrary

distance from the first stub. (b) Equivalent-circuit with load at the first stub.

Smith chart diagram for the operation of a double-stub

tuner.

Solution to Example 5.4. (a) Smith chart for the double-

stub tuners.

ZL = 60-j80

Open stubs, d = λ/8

(b) The two double-stub tuning solutions. (c) Reflection coefficient magnitudes versus

frequency for the tuning circuits of (b).

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit Popen circuit

r=0.5

Smith Chart

YL

. cancel willstub The circle. 1G on the liemust P The

.Y is admittanceinput theb-b plane At the

d2

anglean through circle radiusconstant a along P toP from Move

3

b

32

b

bb

Bj

BjG

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit Popen circuit

r=0.5

Smith Chart

YL

Rotate the the G=1 circle through an angle -

The intersection of G=1 and the GL circle determine the point P2

P2

P3

G1=1

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit Popen circuit

r=0.5

Smith Chart

YL

The shaded range is for the load impedance whichcannot be matched when d=1/8 wavelength

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