MATCHING TECHNIQUES

PRESENTED TO: PRESENTED BY: Er. GARIMA SAINI PRIYA KAUSHAL ROLL NO. 152615 ECE (ME) REGULAR

Matching Techniques

The operation of an antenna system over a frequency range is not completely dependent upon the frequency response of the antenna element itself but rather on the frequency characteristic of the transmission line-antenna element combination.

In practice, the characteristic impedance of the transmission line is usually real whereas that of the antenna element is complex. Also the variation of each as a function of frequency is not the same.

Thus efficient coupling-matching network must be designed which attempt to couple-match the characteristics of the two devices over the designed frequency range.

Types

Stub Matching

Quarter-Wavelength Transformer• Single Section • Multiple Sections• Binomial Design• Tschebyscheff Design

Stub Matching

Ideal matching at a given frequency can be accomplished by placing a short- or open-circuited shunt a distance s from the transmission-line-antenna element connection.

Assuming a real characteristic impedance, the length s is controlled so as to make the real part of the antenna element impedance equal to the characteristic impedance.

Stub Matching

The length l of the shunt line is varied until the susceptance of the stub is equal in magnitude but opposite in phase to the line input susceptance at the point of the transmission line–shunt element connection.

The short-circuited stub is more practical because an equivalent short can be created by a pin connection in a coaxial cable in a waveguide. This preserves the overall length of the stub line for matchings which may require longer length stubs.

A single stub with a variable length l cannot always match all antenna (load) impedances. A double-stub arrangement positioned a fixed distance s from the load, with the length of each stub variable and separated by a constant length d, will match a greater range of antenna impedances. However, a triple-stub configuration will always match all loads.

Quarter-Wavelength Transformer

Single Section

Multiple Sections

Binomial Design

Tschebyscheff Design

Single Section

Another technique that can be used to match the antenna to the transmission line is to use a λ/4 transformer. If the impedance of the antenna is real, the transformer is attached directly to the load.

However if the antenna impedance is complex, the transformer is placed a distance s0 away from the antenna. The distance s0 is chosen so that the input impedance toward the load at s0 is real and designated as Rin .

Single Section

To provide a match, the transformer characteristic impedance Z1 should be Z1 =√RinZ0, where Z0 is the characteristic impedance (real) of the input transmission line.

The transformer is usually another transmission line with the desired characteristic impedance.

Because the characteristic impedances of most off-the-shelf transmission lines are limited in range and values, the quarter-wavelength transformer technique is most suitable when used with microstrip transmission lines.

In microstrips, the characteristic impedance can be changed by simply varying the width of the center conductor.

Multiple Section

Matchings that are less sensitive to frequency variations and that provide broader bandwidths, require multiple λ/4 sections.

In fact the number and characteristic impedance of each section can be designed so that the reflection coefficient follows, within the desired frequency bandwidth, prescribed variations which are symmetrical about the center frequency.

The antenna (load) impedance will again be assumed to be real; if not, the antenna element must be connected to the transformer at a point s0 along the transmission line where the input impedance is real.

Multiple Section

The total input reflection coefficient Γin for an N-section quarter-wavelength transformer with RL > Z0 can be written approximately as

(1)

Multiple Section

ρn represents the reflection coefficient at the junction of two infinite lines with characteristic impedances Zn and Zn+1

f0 represents the designed center frequency f the operating frequency.

If RL < Z0, the ρn’s should be replaced by −ρn’s. For a real load impedance, the ρn’s and Zn’s will also be real.

Binomial Design

One technique, used to design an N-section λ/4 transformer, requires that the input reflection coefficient of (1) have maximally flat passband characteristics.

For this method, the junction reflection coefficients (ρn’s) are derived using the binomial expansion.

Doing this, we can equate (1) to

(3)

Binomial Design

For this type of design, the fractional bandwidth 3f/f0 is given by

From (1)

where

since

(4)

Binomial Design

(4) reduce using (3) to

where ρm is the maximum value of reflection coefficient which can be tolerated within

the bandwidth. The usual design procedure is to specify the

1. load impedance (RL)2. input characteristic impedance (Z0)3. number of sections (N)4. maximum tolerable reflection coefficient (ρm) [or fractional bandwidth (3f/f0)]

Binomial Design

and to find the1. characteristic impedance of each section2. fractional bandwidth [or maximum tolerable reflection coefficient (ρm)]

Tschebyscheff Design

The reflection coefficient can be made to vary within the bandwidth in an oscillatory manner and have equal-ripple characteristics. This can be accomplished by making Γin behave according to a Tschebyscheff polynomial.

For the Tschebyscheff design, the equation that corresponds to (3) is

where TN(x) is the Tschebyscheff polynomial of order N. The maximum allowable reflection coefficient occurs at the edges of the

passband where θ = θm and TN(sec θm cos θ)|θ=θm= 1. Thus

(5)

(6)

Tschebyscheff Design

Or

Using (6), we can write (5)

And its magnitude as

For this type of design, the fractional bandwidth 3f/f0 is also given by (4).To be physical, ρm is chosen to be

(6-a)

(7)

Tschebyscheff Design

Since |TN(sec θm)| > 1, then

or using (6-a) as

Thus

Or

Tschebyscheff Design

Using (7) we can write the reflection coefficient as

For a given N, replace TN(sec θm cos θ) by its polynomial series expansion and then match terms. The usual procedure for the Tschebyscheff design is the same as that of the binomial as outlined previously.

For z = sec θm cos θ, the first three polynomials reduce to

Matching Techniques

In general, the multiple sections (either binomial or Tschebyscheff) provide greater bandwidths than a single section. As the number of sections increases, the bandwidth also increases.

The advantage of the Binomial design is that the reflection coefficient values within the bandwidth monotonically decrease from both ends toward the center. Thus the values are always smaller than an acceptable and designed value.

For the Tschebyscheff design, the reflection coefficient values within the designed bandwidth are equal or smaller than an acceptable and designed value.

.

Matching Techniques

The number of times the reflection coefficient reaches the maximum ripple value within the bandwidth is determined by the number of sections.

In fact, for an even number of sections the reflection coefficient at the designed center frequency is equal to its maximum allowable value, while for an odd number of sections it is zero

For a maximum tolerable reflection coefficient, the N-section Tschebyscheff transformer provides a larger bandwidth than a corresponding N-section binomial design, or for a given bandwidth the maximum tolerable reflection coefficient is smaller for a Tschebyscheff design.