Impedance Matching Z0 ZR

Transmission Lines

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 Z0 ZR Transformer

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 λ 4.

The quarter-wavelength transformer provides narrow-band impedance matching. The design goal is to obtain zero reflection coefficient exactly at the frequency of operation.

λ/4 |Γ|

Z0 Zλ/4 Z0 ZR

f0 The length of the transformer is fixed at λ 4 for design convenience, but is also possible to realize generalized transformer lines for which the length of the transformer is a design outcome.

A broadband design may be obtained by a cascade of λ 4 line sections of gradually varying characteristic impedance.

λ/4 λ/4 λ/4 λ/4 |Γ|

Z0 Z1 Z2 Z3 Z4 Z0 ZR |Γ| max

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.

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.

Z0(x) |Γ|

|Γ| max

x fmin

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.

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.

ZR Z ZR R

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.

Matching I: Impedance Transformers • Quarter Wavelength Transformer – A simple narrow band impedance transformer consists of a transmission line section of length λ 4

ZB ZA

Z Z ZR Z01 02 01

λ/4 dmax or dmin

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.

Consider a general load impedance with its corresponding load reflection coefficient

ZZR − 01 ZRjXRR=+ R;exp Γ= R =Γ R () j φ ZZR + 01

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

1d+ Γ( ) 1 + ΓR ZZ== Z A 01() 01 1d− Γ−Γ 1R

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

1d+ Γ( ) 1 − ΓR ZZ== Z A 01() 01 1d− Γ+Γ 1R

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

Zin Z0 ZA

L = λ/4

Since: 1d+ Γ( ) 1 − ΓR ZZ== Z A 01() 01 1d− Γ+Γ 1R we have 2 ZjZA +β00tan( L) Z ZZin =→lim 0 tan()β→∞ L jZA tan(β+ L) Z0 ZA

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 λ 4.

ZB ZA=Real

Z =Real Z01 Z02 Z01 R

λ/4 n λ/4 ; n=0,1,2…

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

λλ ZZ===;d2 n n AR 1 42

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

2 Z01 λ λλ ZnnA ==+=+;d(21)1 ZR 424

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

2 Z02 ZB = ZA

For impedance matching we need

2 Z02 ZZZZ01=⇒= 02 01 A ZA

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.

Real Load Impedance

ZB ZA

Z = 50 Ω Z02 = ? RR = 100 Ω 01

λ/4

2 Z02 ZZZZRBR==⇒01 02 = 01 =⋅≈50 100 70.71 Ω RR

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

ZB ZA

Z01 = 100 Ω Z02 = ? RR = 50 Ω

λ/4

2 Z02 ZZZZRBR==⇒=01 02 01 =100 ⋅≈Ω 50 70.71 RR

Another real load case

ZB ZA

Z01 = 75 Ω Z02 = ? RR = 300 Ω

λ/4

2 Z02 ZZZZRBR==⇒01 02 = 01 =⋅=Ω75 300 150 RR

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

Z Z B A

Z01 = 75 Ω Z02 Z01 RR = 300 Ω

λ/4 λ/4 22 Z01 75 ZA ===18.75 Ω RR 300

2 Z02 ZZZZZBA==⇒=01 02 01 =⋅75 18.75 =Ω 37.5 ZA

Complex Load Impedance – Transformer at voltage maximum ZB ZA

Z01 = 50 Ω Z02 Z01 ZR = 100 + j 100Ω

λ/4 dmax 100+ j 100− 50 Γ= ≈0.62 R 100++j 100 50

1 +ΓR ZZA =≈Ω0 213.28 1 −ΓR ZZZ02==⋅=Ω 01 A 50 213.28 103.27

Complex Load Impedance – Transformer at voltage minimum ZB ZA

Z01 = 50 Ω Z02 Z01 ZR = 100 + j 100Ω

λ/4 dmin 100+−j 100 50 Γ= ≈0.62 R 100++j 100 50

1 −ΓR ZZA =≈Ω0 11.72 1 +ΓR ZZZ02==⋅=Ω 01 A 50 11.72 24.21

• 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

Z01 Z02 ZR = RR + jXR

RjXjZRR+ +β02 tan( L) ZZZ01==A 02 ZjRjX02 + ()R +βR tan( L)

After separation of real and imaginary parts we obtain the equations

ZZ02() 01− RRR=β ZX 01 tan(L) ZX tan(β= L) 02 R 2 ZR01R − Z 02 with final solution 22 ZR01 RR−− R X R Z02 = 1/− RZR 01

22 ()1/−−−RZR 01() ZRR 01 RR X R tan()β L = X R

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

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:

Y1’

Z0 jX ZR Y0 = 1/Z0 −jB YR = 1/ZR

ds ds

The shunt may be inserted at locations ds where the real part of the line admittance is equal to the characteristic admittance Y0 ' YY10= + jB Matching is obtained by using a shunt susceptance −jB so that −1' YZd110=−=−=[(s )] jBYjBY

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.

Y1 Y1 Y1’ Y1’ −jB −jB Y0 YR Y0 YR

ds ds

The line admittance at location ds can be expressed as a function of reflection coefficient −1  ' 1()+Γdds 1() −Γ s YY10=+ jBZ = 0 = Y 0 1()−Γdds 1() +Γ s For more general results, we introduce normalization:

' Y1 ' 1()− Γ ds ==+=yjb1 1 =normalized admittance Yd0 1()+Γ s b = normalized susceptance

Then, the line reflection coefficient can be expressed in terms of b ' 1()−Γ ds 11(1)− yjbjb−+ − yd' =⇒Γ===() 1 1 s ' 1()+Γds 1 + y1 11 + − jb 2 + jb

Since we know that Γ()ds =ΓRs exp(2 −j βd ) − jb Γ=Γexp()jjd θ= exp( 2 β ) RR2 + jb s

b −1 =π−β+exp()∓ jbjd 2 tan() 2 exp( 2)s jn2 π 4 + b2

Added to account for −>+< for bb 0; for 0 periodic behavior

The absolute value of the load reflection coefficient provides b b 42ΓΓ2 Γ= ⇒bb2 =RR ⇒ =± R 2 4 + b221−Γ R 1−ΓR BbYbZ=⋅00 = /

Finally, the phase of the load reflection coefficient yields ds −1 ∠ΓRs =θ=∓ π22 + βdbn − tan() 2 + 2 π 4π −1 ⇒β=22tan22dd =θπ−∓ () b −π n ssλ λ dbn=θ±π+(2tan22)−1() −π s 4π

+>−< for bb 0; for 0

The last term accounting for periodic behavior of the solution gives λ λ 2nnπ= 42π

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