Transmission Lines
Impedance Matching
A number of techniques can be used to eliminate reflections when
line characteristic impedance and 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).
One method of impedance matching involves the insertion of an
impedance transformer between line and load
Z0
Impedance
Transformer
ZR
In the following, we neglect effects of loss in the lines.
©Amanogawa, 2000 – Digital Maestro Series
164
Transmission Lines
A simple narrow band impedance transformer consists of a
transmission line section of length /4
ZA
ZB
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, 2000 – Digital Maestro Series
165
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
ZA Z01
Z01
1 d
1 R
If it is inserted instead at a location of voltage minimum dmin
1 d
1 R
ZA Z01
Z01
1 d
1 R
©Amanogawa, 2000 – Digital Maestro Series
166
Transmission Lines
Consider now the input impedance of a line of length /4
Zin
Z0
ZA
L = /4
Since:
1 d
1 R
ZA Z01
Z01
1 d
1 R
we have
Zin lim
tan L ©Amanogawa, 2000 – Digital Maestro Series
ZA jZ0 tan( L)
Z0
jZA tan( L) Z0
Z02
ZA
167
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 .
ZA=Real
ZB
Z01
Z02
Z01
ZR=Real
d1
/4
n /4 ; n=0,1,2…
©Amanogawa, 2000 – Digital Maestro Series
168
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
©Amanogawa, 2000 – Digital Maestro Series
;
d1 (2 n 1)
n
4
2 4
169
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.
Let’s now review some simple examples.
©Amanogawa, 2000 – Digital Maestro Series
170
Transmission Lines
Real Load Impedance
ZA
ZB
Z01 = 50 Z02 = ?
RR = 100 /4
2
Z02
ZB Z01 Z02 Z01 RR 50 100 70.71 RR
©Amanogawa, 2000 – Digital Maestro Series
171
Transmission Lines
Note that an identical result is obtained by switching Z01 and RR
ZA
ZB
Z01 = 100 Z02 = ?
RR = 50 /4
2
Z02
ZB Z01 Z02 Z01 RR 100 50 70.71 RR
©Amanogawa, 2000 – Digital Maestro Series
172
Transmission Lines
Another real load case
ZA
ZB
Z01 = 75 Z02 = ?
RR = 300 /4
2
Z02
ZB Z01 Z02 Z01 RR 75 300 150 RR
©Amanogawa, 2000 – Digital Maestro Series
173
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
Z01 = 75 2
752
Z01
ZA 18.75 RR 300
ZA
Z02
/4
Z01
RR = 300 /4
2
Z02
ZB Z01 Z02 Z01 ZA 75 18.75 37.5 ZA
©Amanogawa, 2000 – Digital Maestro Series
174
Transmission Lines
Complex Load Impedance – Transformer at voltage maximum
ZA
ZB
Z01 = 50 Z02
Z01
/4
dmax
ZR = 100 + j 100
100 j100 50
R 0.62
100 j100 50
1 R
ZA Z0
213.28
1 R
Z02 Z01 ZA 50 213.28 103.27 ©Amanogawa, 2000 – Digital Maestro Series
175
Transmission Lines
Complex Load Impedance – Transformer at voltage minimum
ZA
ZB
Z01 = 50 Z02
Z01
/4
dmin
ZR = 100 + j 100
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, 2000 – Digital Maestro Series
176
Transmission Lines
If it is not important to realize the impedance transformer with a
quarter wavelength line, we can 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
L
RR jX R jZ02 tan( L)
Z01 ZA Z02
Z02 j RR jX R tan( L)
©Amanogawa, 2000 – Digital Maestro Series
177
Transmission Lines
After separation of real and imaginary parts we obtain the equations
Z02 ( Z01 RR ) Z01 X R tan L tan L Z02 X R
2
Z01 RR Z02
with final solution
2
2
Z01 RR RR
XR
Z02 1 RR / Z01
tan L 1 RR / Z01 Z01 RR RR2 X R2 XR
The transformer can be realized as long as the result for Z02 is real.
Note that this is also a narrow band approach.
©Amanogawa, 2000 – Digital Maestro Series
178