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
Z0
Impedance Transformer
ZR
Amanogawa, 2006 Digital Maestro Series
141
Transmission Lines
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 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.
/4 Z0 Z/4 Z0
||
ZR
f0
The length of the transformer is fixed at 4 for design convenience, but is also possible to realize generalized transformer lines for which the length of the transformer is a design outcome.
Amanogawa, 2006 Digital Maestro Series 142
Transmission Lines
A broadband design may be obtained by a cascade of 4 line sections of gradually varying characteristic impedance.
/4 Z0 Z1
/4 Z2
/4 Z3
/4 Z4 Z0
||
ZR
||max
f0
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.
Amanogawa, 2006 Digital Maestro Series 143
Transmission Lines
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)
||
||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.
Amanogawa, 2006 Digital Maestro Series 144
Transmission Lines
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
ZR
ZR
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.
Amanogawa, 2006 Digital Maestro Series 145
Transmission Lines
Matching I: Impedance Transformers Quarter Wavelength Transformer A simple narrow band
impedance transformer consists of a transmission line section of length 4
ZB
ZA
Z01
Z02
Z01
ZR
/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.
Amanogawa, 2006 Digital Maestro Series 146
Transmission Lines
Consider a general load impedance with its corresponding load reflection coefficient
ZR = RR + jX R ;
ZR Z01 R = = R exp ( j ) ZR + Z01
If the transformer is inserted at a location of voltage maximum dmax
1 + (d) 1 + R = Z01 ZA = Z01 1 (d) 1 R
If it is inserted instead at a location of voltage minimum dmin
1 + (d) 1 R = Z01 ZA = Z01 1 (d) 1 + R
Amanogawa, 2006 Digital Maestro Series 147
Transmission Lines
Consider now the input impedance of a line of length 4
Zin
Z0
L = /4
ZA
Since:
1 + (d) 1 R = Z01 ZA = Z01 1 (d) 1 + R
lim ZA + jZ0 tan( L) Z0 jZA tan( L) + Z0
2 Z0 ZA
we have
Zin =
tan( L )
Amanogawa, 2006 Digital Maestro Series
148
Transmission Lines
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 4 .
ZB
ZA=Real
Z01
Z02
Z01
d1
ZR=Real
/4
n /4 ; n=0,1,2
Amanogawa, 2006 Digital Maestro Series
149
Transmission Lines
If the load impedance is real and the transformer is inserted at a distance from the load equal to an even multiple of 4 , then
ZA = ZR ;
d1 = 2 n = n 4 2
but if the distance from the load is an odd multiple of 4
2 Z01 ZA = ZR
d1 = (2 n + 1) = n + 4 2 4
Amanogawa, 2006 Digital Maestro Series
150
Transmission Lines
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 Z01 = ZA
Z02 = Z01 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. Lets now review some simple examples.
Amanogawa, 2006 Digital Maestro Series
151
Transmission Lines
Real Load Impedance
ZB
ZA
Z01 = 50
Z02 = ?
RR = 100
/4
2 Z02 ZB = = Z01 Z02 = Z01 RR = 50 100 70.71 RR
Amanogawa, 2006 Digital Maestro Series
152
Transmission Lines
Note that an identical result is obtained by switching Z01 and RR
ZB
ZA
Z01 = 100
Z02 = ?
RR = 50
/4
2 Z02 ZB = = Z01 Z02 = Z01 RR = 100 50 70.71 RR
Amanogawa, 2006 Digital Maestro Series
153
Transmission Lines
Another real load case
ZB
ZA
Z01 = 75
Z02 = ?
RR = 300
/4
2 Z02 ZB = = Z01 Z02 = Z01 RR = 75 300 = 150 RR
Amanogawa, 2006 Digital Maestro Series
154
Transmission Lines
Same impedances as before, but now the transformer is inserted at a distance 4 from the load (voltage minimum in this case)
ZB
ZA
Z01 = 75
Z02
Z01
RR = 300
2 Z01 752 ZA = = = 18.75 RR 300
/4
/4
2 Z02 ZB = = Z01 Z02 = Z01 ZA = 75 18.75 = 37.5 ZA
Amanogawa, 2006 Digital Maestro Series 155
Transmission Lines
Complex Load Impedance Transformer at voltage maximum
ZB
ZA
Z01 = 50
Z02
Z01
ZR = 100 + j 100
/4
dmax
100 + j100 50 R = 0.62 100 + j100 + 50 1 + R 213.28 ZA = Z0 1 R Z02 = Z01 ZA = 50 213.28 = 103.27
Amanogawa, 2006 Digital Maestro Series 156
Transmission Lines
Complex Load Impedance Transformer at voltage minimum
ZB
ZA
Z01 = 50
Z02
Z01
ZR = 100 + j 100
/4
dmin
100 + j100 50 R = 0.62 100 + j100 + 50 1 R ZA = Z0 11.72 1 + R Z02 = Z01 ZA = 50 11.72 = 24.21
Amanogawa, 2006 Digital Maestro Series 157
Transmission Lines
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
ZA
Z01
Z02
ZR = RR + jXR
RR + jX R + jZ02 tan( L) Z01 = ZA = Z02 Z02 + j ( RR + jX R ) tan( L)
Amanogawa, 2006 Digital Maestro Series 158
Transmission Lines
After separation of real and imaginary parts we obtain the equations
Z 02 ( Z 01 RR ) = Z 01 X R tan( L) tan( L) =
with final solution
2 Z 01RR Z 02
Z 02 X R
Z 02 =
2 2 XR Z 01RR RR
1 RR / Z 01
tan ( L ) =
2 2 R Z Z R R X 1 / ( 01 ) 01 R R R R
XR
Z02 is real.
The transformer can be realized as long as the result for Note that this is also a narrow band approach.
Amanogawa, 2006 Digital Maestro Series
159
Transmission Lines
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 Y1 Z0 jX ds ZR Y0 = 1/Z0 jB ds YR = 1/ZR
The shunt may be inserted at locations ds where the real part of the line admittance is equal to the characteristic admittance Y0
Y1' = Y0 + jB
Matching is obtained by using a shunt susceptance jB so that
Y1 = [ Z (d s )]1 jB = Y1' jB = Y0
Amanogawa, 2006 Digital Maestro Series 160
Transmission Lines
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 Y0 jB ds YR Y0 jB ds Y1 Y1 YR
Amanogawa, 2006 Digital Maestro Series
161
Transmission Lines
The line admittance at location ds can be expressed as a function of reflection coefficient
Y1' = Y0 + jB = Z 0
1 + (d s ) 1 (d s )
1 (d s ) = Y0 1 + (d s )
For more general results, we introduce normalization:
1 (d s ) Y1' ' = y1 = 1 + jb = = normalized admittance Y0 1 + (d s ) b = normalized susceptance
Then, the line reflection coefficient can be expressed in terms of b
1 (d s ) 1 ' y1 = (d s ) = 1 + (d s ) 1+
Amanogawa, 2006 Digital Maestro Series
' 1 (1 + jb) y1 jb = = ' y1 1 + 1 jb 2 + jb
162
Transmission Lines
Since we know that
(d s ) = R exp( j 2 d s ) jb R = R exp ( j ) = exp( j 2 d s ) 2 + jb
b 4 + b2
exp ( j 2 tan 1 ( b 2 ) ) exp( j 2 d s + j 2n ) + for b < 0
2 2
Added to account for periodic behavior
for b > 0;
The absolute value of the load reflection coefficient provides b
R =
b 4 + b2
b2 =
4 R
1 R
b=
2 R
2
1 R B = b Y0 = b / Z 0
163
Amanogawa, 2006 Digital Maestro Series
Transmission Lines
Finally, the phase of the load reflection coefficient yields ds
R = = 2 + 2 d s tan 1 ( b 2 ) + 2 n 4 2 ds = d s = 2 tan 1 ( b 2 ) 2 n ds = ( 2 + tan 1 ( b 2 ) 2 n ) 4 + for b > 0; for b < 0
The last term accounting for periodic behavior of the solution gives
2n = n 4 2
indicating that the solutions repeat every 2 along the line.
Amanogawa, 2006 Digital Maestro Series 164
