Impedance Matching - Amanogawa.com Interactive …amanogawa.com/archive/docs/H-tutorial.pdfa...

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Transmission Lines © Amanogawa, 2006 – Digital Maestro Series 141 Impedance Matching A number of techniques can be used to eliminate reflections when the characteristic impedance of the line and the load impedance are mismatched. Impedance matching techniques can be designed to be effective for a specific frequency of operation (narrow band techniques) or for a given frequency spectrum (broadband techniques). A common method of impedance matching involves the insertion of an impedance transformer between line and load Impedance Transformer Z 0 Z R

Transcript of Impedance Matching - Amanogawa.com Interactive …amanogawa.com/archive/docs/H-tutorial.pdfa...

Page 1: Impedance Matching - Amanogawa.com Interactive …amanogawa.com/archive/docs/H-tutorial.pdfa different transmission line with appropriate characteristic impedance. A widely used approach

Transmission Lines

© Amanogawa, 2006 – Digital Maestro Series 141

Impedance Matching A number of techniques can be used to eliminate reflections when the characteristic impedance of the line and the load impedance are mismatched. Impedance matching techniques can be designed to be effective for a specific frequency of operation (narrow band techniques) or for a given frequency spectrum (broadband techniques). A common method of impedance matching involves the insertion of an impedance transformer between line and load

Impedance TransformerZ0

ZR

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Transmission Lines

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An impedance transformer may be realized by inserting a section of a different transmission line with appropriate characteristic impedance. A widely used approach realizes the transformer with a line of length 4λ . The quarter-wavelength transformer provides narrow-band impedance matching. The design goal is to obtain zero reflection coefficient exactly at the frequency of operation.

The length of the transformer is fixed at 4λ for design convenience, but is also possible to realize generalized transformer lines for which the length of the transformer is a design outcome.

ZR

λ/4

Zλ/4 Z0 Z0

f0

|Γ|

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Transmission Lines

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A broadband design may be obtained by a cascade of 4λ line sections of gradually varying characteristic impedance. It is not possible to obtain exactly zero reflection coefficient for all frequencies in the desired band. Therefore, available design approaches specify a maximum reflection coefficient (or maximum VSWR) which can be tolerated in the frequency band of operation.

f0

|Γ|

|Γ|max

ZR

λ/4

Z4 Z0 Z0 Z3 Z2 Z1

λ/4 λ/4 λ/4

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Transmission Lines

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Another broadband matching approach may use a tapered line transformer with continously varying characteristic impedance along its length. In this case, the design obtains reflection coefficients lower than a specified tolerance at frequencies exceeding a minimum value. Various taper designs are available, including linear, exponential, and raised-cosine impedance profiles. An optimal design (due to Klopfenstein) involves discontinuity of the impedance at the transformer ends.

Z0(x)

x

|Γ|

|Γ|max

fmin

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Transmission Lines

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Another narrow-band approach involves the insertion of a shunt imaginary admittance on the line. Often, the admittance is realized with a section (or stub) of transmission line and the technique is commonly known as stub matching. The end of the stub line is short-circuited or open-circuited, in order to realize an imaginary admittance. Designs are also available for two or three shunt admittances placed at specified locations on the line.

Other narrow-band examples involve the insertion of a series impedance (stub) along the line, and the insertion of a series and a shunt element in L-configuration.

The theory for several basic narrow-band matching techniques is detailed in the following. Note that the effect of loss in the transmission lines is always neglected.

ZR ZR ZR

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Transmission Lines

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Matching I: Impedance Transformers • Quarter Wavelength Transformer – A simple narrow band

impedance transformer consists of a transmission line section of length 4λ

The impedance transformer is positioned so that it is connected to a real impedance ZA. This is always possible if a location of maximum or minimum voltage standing wave pattern is selected.

ZA

dmax or dmin

ZR

λ/4

Z01 Z02 Z01

ZB

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Transmission Lines

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Consider a general load impedance with its corresponding load reflection coefficient

( )0101

; expRR R R R R

R

Z ZZ R jX jZ Z

−= + Γ = = Γ φ

+

If the transformer is inserted at a location of voltage maximum dmax

( )( )01 01

1 d 11 d 1

RA

RZ Z Z

+ Γ + Γ= =

− Γ − Γ

If it is inserted instead at a location of voltage minimum dmin

( )( )01 01

1 d 11 d 1

RA

RZ Z Z

+ Γ − Γ= =

− Γ + Γ

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Transmission Lines

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Consider now the input impedance of a line of length 4λ Since:

( )( )01 01

1 d 11 d 1

RA

RZ Z Z

+ Γ − Γ= =

− Γ + Γ

we have

( )

20 0

0tan L 0

tan( L)limtan( L)

Ain

A A

Z jZ ZZ ZjZ Z Zβ →∞

+ β= →

β +

Z0Zin

ZA

L = λ/4

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Transmission Lines

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Note that if the load is real, the voltage standing wave pattern at the load is maximum when ZR > Z01 or minimum when ZR < Z01 . The transformer can be connected directly at the load location or at a distance from the load corresponding to a multiple of 4λ .

d1

ZA=Real

n λ/4 ; n=0,1,2…

ZR=Real

λ/4

Z01 Z02 Z01

ZB

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Transmission Lines

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If the load impedance is real and the transformer is inserted at a distance from the load equal to an even multiple of 4λ , then

1; d 24 2A RZ Z n nλ λ

= = =

but if the distance from the load is an odd multiple of 4λ

201

1; d (2 1)4 2 4A

R

ZZ n nZ

λ λ λ= = + = +

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Transmission Lines

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The input impedance of the impedance transformer after inclusion in the circuit is given by

202

BA

ZZZ

=

For impedance matching we need

202

01 02 01 AA

ZZ Z Z ZZ

= ⇒ =

The characteristic impedance of the transformer is simply the geometric average between the characteristic impedance of the original line and the load seen by the transformer. Let’s now review some simple examples.

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Transmission Lines

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Real Load Impedance

202

01 02 01 50 100 70.71B RR

ZZ Z Z Z RR

= = ⇒ = = ⋅ ≈ Ω

ZA

λ/4

RR = 100 Ω Z01 = 50 Ω Z02 = ?

ZB

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Transmission Lines

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Note that an identical result is obtained by switching Z01 and RR

202

01 02 01 100 50 70.71B RR

ZZ Z Z Z RR

= = ⇒ = = ⋅ ≈ Ω

ZA

λ/4

RR = 50 Ω Z01 = 100 Ω Z02 = ?

ZB

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Transmission Lines

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Another real load case

202

01 02 01 75 300 150B RR

ZZ Z Z Z RR

= = ⇒ = = ⋅ = Ω

ZA

λ/4

RR = 300 Ω Z01 = 75 Ω Z02 = ?

ZB

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Transmission Lines

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Same impedances as before, but now the transformer is inserted at a distance 4λ from the load (voltage minimum in this case)

2 201 75 18.75

300AR

ZZR

= = = Ω

202

01 02 01 75 18.75 37.5B AA

ZZ Z Z Z ZZ

= = ⇒ = = ⋅ = Ω

λ/4

ZA

RR = 300 Ω

λ/4

Z01 = 75 Ω Z02 Z01

ZB

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Transmission Lines

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Complex Load Impedance – Transformer at voltage maximum

0

100 100 50 0.62100 100 50

1213.28

1

R

RA

R

jj

Z Z

+ −Γ = ≈

+ +

+ Γ= ≈ Ω

− Γ

02 01 50 213.28 103.27AZ Z Z= = ⋅ = Ω

dmax

ZA

ZR = 100 + j 100Ω

λ/4

Z01 = 50 Ω Z02 Z01

ZB

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Transmission Lines

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Complex Load Impedance – Transformer at voltage minimum

0

100 100 50 0.62100 100 50

111.72

1

R

RA

R

jj

Z Z

+ −Γ = ≈

+ +

− Γ= ≈ Ω

+ Γ

02 01 50 11.72 24.21AZ Z Z= = ⋅ = Ω

dmin

ZA

ZR = 100 + j 100Ω

λ/4

Z01 = 50 Ω Z02 Z01

ZB

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Transmission Lines

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• Generalized Transformer

If it is not important to realize the impedance transformer with a quarter wavelength line, one may try to select a transmission line with appropriate length and characteristic impedance, such that the input impedance is the required real value

( )02

01 0202

tan( L)tan( L)

R RA

R R

R jX jZZ Z ZZ j R jX

+ + β= =

+ + β

ZR = RR + jXR

L

Z01 Z02

ZA

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Transmission Lines

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After separation of real and imaginary parts we obtain the equations

02 01 01

022

01 02

( ) tan( L)

tan( L)

R R

R

R

Z Z R Z XZ X

Z R Z

− = β

β =−

with final solution

( )( )( )

2 201

0201

2 201 01

1 /

1 /tan L

R R R

R

R R R R

R

Z R R XZ

R Z

R Z Z R R X

− −=

− − −=

The transformer can be realized as long as the result for Z02 is real. Note that this is also a narrow band approach.

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Transmission Lines

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Matching II – Shunt Admittance We wish to insert a parallel (shunt) reactance on the transmission line to obtain impedance matching. Since the design involves a parallel circuit, it is more convenient to consider admittances: The shunt may be inserted at locations ds where the real part of the line admittance is equal to the characteristic admittance Y0

'1 0Y Y jB= +

Matching is obtained by using a shunt susceptance −jB so that 1 '

1 1 0[ ( )]sY Z d jB Y jB Y−= − = − =

ZR Z0

ds

jX

ds

−jB YR = 1/ZR Y0 = 1/Z0

Y1’Y1

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Transmission Lines

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To solve this design problem, we need to find the suitable locations ds (where the real part of the line admittance is equal to Y0) and the corresponding values of the shunt susceptance −B.

The shunt element may be also realized by inserting a segment of transmission line of appropriate length, called a stub.

In order to obtain a pure susceptance, the stub element may consist of a short-circuited or an open-circuited transmission line with input admittance –jB.

ds

−jB YR Y0

Y1’Y1

ds

−jB YR Y0

Y1’Y1

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Transmission Lines

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The line admittance at location ds can be expressed as a function of reflection coefficient

1'1 0 0 0

1 ( ) 1 ( )1 ( ) 1 ( )

s ss s

d dY Y jB Z Yd d

− + Γ − Γ= + = = − Γ + Γ

For more general results, we introduce normalization:

'1 '10

normalized admittance

normalized suscep

1 ( )1

tance1 ( )

ss

dY y jbY db

− Γ= = + = =

+ Γ

=

Then, the line reflection coefficient can be expressed in terms of b

'1'1 '1

1 ( ) 1 1 (1 )( )1 ( ) 1 1 1 2

ss

s

d y jb jby dd y jb jb

− Γ − − + −= ⇒ Γ = = =

+ Γ + + − +

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Transmission Lines

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Since we know that ( ) exp( 2 )s R sd j dΓ = Γ − β

( )

( )( )12

exp exp( 2 )2

exp 2 tan 2 exp( 22 )4

R R s

s

jbj j djb

b j b j db

j n−

−Γ = Γ θ = β

+

= π − β ++

π∓

for 0; for 0b b− > + <

The absolute value of the load reflection coefficient provides b

22

22 2

0 0

4 24 1 1

/

R RR

R R

b b bb

B b Y b Z

Γ ΓΓ = ⇒ = ⇒ = ±

+ − Γ − Γ

= ⋅ =

Added to account for periodic behavior

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Transmission Lines

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Finally, the phase of the load reflection coefficient yields ds

( )

( )

( )

1

1

1

2 2 tan 2 242 2 tan 2 2

( 2 tan 2 2 )4

R s

s s

s

d b n

d d b n

d b n

∠ Γ = θ = π + β − + π

π⇒ β = = θ π − − π

λλ

= θ ± π + − ππ

for 0; for 0b b+ > − <

The last term accounting for periodic behavior of the solution gives

24 2

n nλ λπ =

π

indicating that the solutions repeat every 2λ along the line.