The document discusses different techniques for matching the impedance of antennas and transmission lines, including stub matching and quarter-wavelength transformers. Stub matching uses short or open circuited stubs to match the real part of the antenna impedance to the characteristic impedance of the transmission line. Quarter-wavelength transformers use transmission line sections with specific characteristic impedances to transform and match impedances. Multiple section designs provide broader bandwidth matching than single sections. Binomial and Tschebyscheff designs calculate the sections to give maximally flat or equal ripple response over the bandwidth.

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

2. 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.

3. Types
Stub Matching
Quarter-Wavelength Transformer
• Single Section
• Multiple Sections
• Binomial Design
• Tschebyscheff Design

4. 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.

5. 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.

6. Quarter-Wavelength Transformer
Single
Section
Multiple
Sections
Binomial
Design
Tschebyscheff
Design

7. 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 .

8. 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.

9. 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.

10. Multiple Section
The total input reflection coefficient Γin for an N-section
quarter-wavelength transformer with RL > Z0 can be written
approximately as
(1)

11. 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.

12. 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)

13. Binomial Design
For this type of design, the fractional bandwidth 3f/f0 is given by
From (1)
where
since
(4)

14. 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)]

15. Binomial Design
 and to find the
1. characteristic impedance of each section
2. fractional bandwidth [or maximum tolerable reflection coefficient (ρm)]

16. 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)

17. 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)

18. Tschebyscheff Design
 Since |TN(sec θm)| > 1, then
 or using (6-a) as
 Thus
Or

19. 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

20. 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.
.

21. 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.