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1. TRANSMISSION LINES AND RADIOFREQUENCY CIRCUITS
Smith chart and impedance matching• Smith Chart• Impedance matching
o lumped elementso Single-stub matchingo Quarter-wave impedance transformer
Transmission line design• Balanced and unbalanced lines• Homogeneous and non-homogeneous lines• Coupled lines• Line design
Application notes• Coaxial cables• Connectors
1Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
[COLLIN] R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York)[POZAR] D.M. Pozar, Microwave Engineering, Addison-Wesley Publishing Company, 2nd Edition, 1993 (Reading, Massachusetts)
• : attenuation constant [m-1]• : phase constant [rad·m-1]• Cd : distributed capacitance per unit length [F/m]• CNS : Communications, Navigation, Surveillance• 0 : electric permittivity of vacuum [8.85·10-12 F/m]• f0 : frequency [Hz]• : propagation constant [m-1]• G : distributed conductance per unit length [S/m]• i(z,t) : current in time domain [V]• I0
+ : current amplitude of progressive wave at z=0 [A]• l : transmission line length [m]• Ld : distributed inductance per unit length [H/m]• : wavelength [m]• 0 : magnetic permeability of vacuum [4·10-7 H/m]• : angular frequency [rad/s]• R : distributed resistance per unit length [/m]• RF : radiofrequency• T : period [s]
2
GLOSSARY
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
• RS : surface resistivity [/square]• S : skin depth [m]• : conductivity [Sm]• d : dielectric conductivity [Sm]• tan : loss tangent [adim]• RL : return loss [dB]• : (voltage) reflection coefficient [adim]• G : (voltage) generator reflection coefficient [adim]• IN : (voltage) reflection coefficient at input port [adim]• L : (voltage) load reflection coefficient [adim]• v(z,t) : voltage in time domain [V]• VG : voltage at generator [V]• V0
+ : voltage amplitude of progressive wave at z=0 [V]• vp : phase velocity [m/s]• VSWR : Voltage Standing Wave Ratio [adim]• ZG : generator impedance []• ZIN : impedance at the input port of the transmission line []• ZL : load impedance []• Z0 : transmission line characteristic impedance []
3
GLOSSARY
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Smith Chart
• A graphical tool very helpful when dealing with impedance transformation and matching network design is the Smith Chart (invented by Phillip H. Smith in 1939 while working for Bell Telephone Laboratories). Because working with (almost) infinite values for resistances and reactances is usual in
Microwaves, a graphical plot of these impedances is not practical in a rectangular coordinate system. Nevertheless, operating with the reflection coefficient associated to a given normalized passive impedance (impedance Z with possitive resistance normalized to the characteristic impedance of the line Z0) leads to a graphical representation of the loads inside a circle of unity radius.
• The transformation between impedances and reflection coefficients leads to a Z-chart:
• The transformation between admitances and reflection coefficients leads to a Y-chart.
11
11 z
zz
0ZZz being:
11
11 y
yy
00
YZYYy being:
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
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SMITH CHART AND IMPEDANCE MATCHING Smith Chart
• Lines of constant reactance (susceptance) map into circumferences
Z-Chart
Y-Chart
xjrz
bjgy
0x
0x
0b
0b
impedances corresponding to
active loads 11
zz
11z
yy
11
11y
f g h i
a
b
c
d e
r
x
g
b
f g h i
abcde
abcde
f g h i
a b
c
d
e
fg
hi
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
Im
Im
Re
Re
6
SMITH CHART AND IMPEDANCE MATCHING Smith Chart
[Images from: http://upload.wikimedia.org/wikipedia/commons/d/df/Smith_chart_explanation.svg]
• Z-Smith chart: main loads and their correspondence with the reflection coefficient.
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Smith Chart
transmission line length in wavelengths
(periodicity /2)
phase of reflection coefficient in degrees
(periodicity 180)
shift towards LOAD
shift towards GENERATOR
• Z-Smith chart.
ljLIN e 2
angle
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Smith Chart
50 5050
0ZjZ
Example: Impedance on Smith Chart. Plot the load ZL=50+j50 on the Smith chart (reference impedance 50 ). Indicate the value of the reflection coefficient, the return loss, and the VSWR.
jz 1
63.5º
Graphically from the Smith Chart:
dB 7RLdB 8.5VSWR 2.6 VSWR
63.5º)angle( 46.0
Numerically from equations:
dB 99.6log20RL
º4.63 45.011 je
zz
dB 36.862.211
VSWR
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Smith Chart
Example: Loaded coaxial. A 2 cm-length coaxial operating at 3 GHz is loaded with ZL=50+j50 . Use the Smith chart to find the input impedance seen from the coaxial transmission line. Consider the dielectric constant of the line is 2.56.
Graphically from the Smith Chart:
32.0·103
56.2·103 02.0 8
9
c
flfv
ll r
p
jz 1Normalized load impedance (point A):Transmission line electrical length:
0.32 from load to generator
A
B
Normalized input impedance (point B): 10.041.0 jz Unnormalized input impedance: 0.55.20 jZin
Numerically from equations: º20.115360 ll
j
ljZZljZZZZ
L
LIN 9.43.19
tantan
0
00
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching• Though the Smith chart is useful to avoid tedious computations involving complex
numbers, nowadays this is not a problem thanks to the widespread use of calculators and computers.
• The main advantage of using Smith chart is that it allows drawing conclusions without requiring complex calculations.
• Smith chart is very useful when designing matching networks. Several alternatives are possible when trying to match a load (or a generator) to the reference impedance of a transmission line. Some of them are: Impedance matching with lumped elements. Single-stub matching. Double-stub matching. Triple-stub tuner. Quarter-wave impedance transformer.
• In a practical design of a matching network the technology to be used and the frequency bandwidth of the solution should be considered.
• Conjugate matching can also be treated by using the Smith Chart.
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
out of the scope of this course
11
SMITH CHART AND IMPEDANCE MATCHING Impedance matching
• When designing matching networks use to be helpful employing CAD tools.
• An unlicensed tool (with limited functionalities) called Smith (designed by Prof. Fritz Dellsperger and Michael Baud from Bern University) is an example.
[Image from: http://www.fritz.dellsperger.net/]
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching
• Impedance matching using reactive elements is desirable due to the absence of losses.
• The matching solution depends on the technology to be used for its implementation.
A
BL
moving CW or CCW a load along a constant-R circle is equivalent to add
a L or a C in series, respectively
-1/(C)
DZ-Smith chart
moving CCW or CW a load along a constant-G circle is equivalent to add a
L or a C in shunt, respectively
A
D
B -1/(L)
C
Y-Smith chart
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
• Impedance matching using two lumped elements. ZL inside the circle can be matched using shunt-series elements.xj1
CL
0Z LZCL0Z LZ
YL inside the circle can be matched using series-shunt elements.bj1
C
L0Z LZ
CL0Z LZ
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
1C0Z LZ
2C1C
0Z LZ2C
1C0Z LZL C
L0Z LZ
ZL outside the circle and YL outside the circle can be matched using series-shunt and shunt-series elements.
xj1 bj1
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
• Impedance matching using three lumped elements networks (only some possible solutions are shown).
2CL0Z LZ1C
3C0Z LZ1C
2C
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
• Impedance matching using four lumped elements networks (only some possible solutions are shown).
• When designing matching networks, remember that shorter paths in the Smith chart provide a wider operational bandwidth.
2C
2L0Z LZ
1C
1L
4C0Z LZ2C
3C 1C
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
• Lumped elements (should be smaller than /10) used to design the matching networks can be:
Capacitors: chip capacitors, MIM capacitors, interdigital capacitors, open-circuited stubs.
Inductors: chip inductors, loop inductors, spiral inductors, short-circuited stubs.
Resistor: chip resistors, planar resistors.
• Lumped elements have parasitics in the microwave range.• The standard units used when describing the size of a lumped element is the mil:
1 mil=0.001 in=25 m=1/40 mm
[Imag
es fr
om:
http
://w
ww
.ad-
mte
ch.c
om/p
rodu
cts/
thin
_fil
m/in
dex.
htm
l]
[Imag
es fr
om:
http
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ww
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com
/eng
lish/
prod
ucts
/ca
paci
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/tant
al/k
_chi
p.ht
mll]
[Images from [POZAR]]
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
Example: Matching a dipole. Consider a dipole with input impedance 82+j45 and operating at 2.45 GHz. Consider all the possibilities of matching the dipole to the line using a two-lumped elements network when fed with a 50 transmission line. Solve the problem analytically and check the results using the application Smith.exe.
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: lumped elements
Example: Matching a monopole. Consider a monopole with input impedance 30+j20 operating at 2.45 GHz. Consider all the possibilities of matching the monopole to the line using a two-lumped elements network when fed with a 50 transmission line. Solve the problem using the application Smith.exe. Make a frequency sweep from 2 to 3 GHz and decide which solution has a better bandwidth (assume that the monopole impedance does not change in the proposed frequency bandwidth).
worst RL is around 20 dB
worst RL is around 14 dB
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: single-stub matching
• Single-stub matching. A stub is a short-circuited or open-circuited section of a transmission line. There are two alternatives: single-shunt-stub and single-series-stub. There are two parameters to adjust: distance l from load to stub and the shunt impedance.
ZL
l
Z0Z0 Z0
ZL
l
Z0Z0
Z0
ZL
l
Z0Z0
Z0
ZL
l
Z0Z0 Z0
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching: /4 transformer
A resistive load RL can be matched to a transmission line with reference impedance Z0 by means of a /4 section of a transmission load having a reference impedance of Z0’: 0
'0 ZRZ L
• Quarter-wave impedance transformer.
If the load is not purely resistive a series or paraler reactive element (lumped element or transmission line section) should be added to make it purely resistive before including the quarter-wave transformer.
ZL
l
Z0
/4
Z0’
ZL=RL +jXL
/4
Z0’ -jXL
YL=GL +jBL
/4
Z0’ -jBL
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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SMITH CHART AND IMPEDANCE MATCHING Impedance matching
Example: Matching a monopole with a microstrip single-stub network. Consider a monopole with input impedance 30+j20 operating at 2.45 GHz. Consider the possibility of matching the monopole using a single-stub network made with microstrip technology (avoid using vias) and a Rogers Duroid 4003C substrate (r=3.55). Solve the problem using the application Smith.exe.
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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High impedance matching. Which of the following options are possible to match an impedance of 188 to a 50 transmission line?
23
Take your time…
Z0=50 188 Matching Network
Z0’=97
/4
C
L CL
LL
a) b) c) d)
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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TRANSMISSION LINE DESIGN Balanced and unbalanced lines
• A balanced line is a transmission line having two conductors with the same voltage magnitude but a phase shift of 180º with respect to ground. Impedance of both conductors is equal with respect to ground.
• The balanced and unbalanced character of transmission lines has to be accounted for proper connection to circuits or devices.
• An example of a balanced line is a twin-lead line, whereas an unbalanced line is a coaxial cable.
• The transition from a balanced to an unbalanced structure, or viceversa, requires a BALUN transformer.
• BALUNS are used to connect balanced to unbalanced lines or structures. They are required irrespective of transmission line technology.
• Baluns are usually narrowband devices. It is difficult to design wide bandwidth baluns.
V/2 -V/2V
0 Volts
UNbalancedBALanced
0 Volts
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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TRANSMISSION LINE DESIGN Homogeneous and non-homogeneous lines
• Homogeneous dielectric media are uniform in all points and its physical properties are unchanged. A transmission line in a homogeneous medium has a propagation velocity that
depends only on material properties (dielectric permittivity r and magnetic permeability r). The principal wave existing in these kind of transmission lines is a TEM
(transversal electromagnetic) wave.
• Non-homogeneous media contain multiple materials with different dilectric constants. Wave propagation velocity in non-homogeneous transmission lines depends
on material properties and structure dimensions An effective r,eff dielectric constant is often used to represent an average
dielectric constant. These line do not propagate pure TEM modes.
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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TRANSMISSION LINE DESIGN Homogeneous and non-homogeneous lines
• Cross sections of some common transmission lines:
microstrip
embedded microstrip
coupled microstrip
coplanar
coplanar strip
non-
hom
ogen
eous
centered stripline dual stripline
coaxial shielded two-conductor
two-conductor
circular WGrectangular WG
hom
ogen
eous
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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TRANSMISSION LINE DESIGN Coupled lines
• Coupled lines are balanced-type lines.
• These structures are analyzed by means of an odd and an even excitation mode that superpose. In the even mode the currents in the strip conductors are equal in amplitude
and flow in the same direction. In the odd mode the currents in the strip conductors are equal in amplitude
and flow in opposite directions. Each strip conductor is characterized by its
characteristic impedance (relative to ground) and its propagation constant. Both parameters are different for the excited
modes.
eZ0
oZ0
e0
o0 odd mode parameters
even mode parameters
[Imag
e fro
m: [
PO
ZAR
]]
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
EETAC-UPC
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TRANSMISSION LINE DESIGN Line design
• Homogeneous lines propagate purely TEM modes and have analytical equations helping to match the characteristic impedance and propagation constants of the lines to given specific requirement.
• Non-homogeneous lines do not propagate purely TEM modes (but under certain circumstances they propagate quasi-TEM modes). The characteristic parameters of the line have to be numerically computed or other approximate techniques have to be used.
http://web.awrcorp.com/Usa/Products/Optional-Products/TX-Line/
• Nowadays, CAD software helps the designer... (e.g. TX-Line Calculator of AWR, LineCalc part of ADS of Agilent Technologies).
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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TRANSMISSION LINE DESIGN Line design
• TX-Line Calculator of AWR is a license free software for transmission line design.
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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APPLICATION NOTES Coaxial cables
• The coaxial cable was invented by Oliver Heaviside at the end of the 19th century for transmitting telegraphic signals.
• The first propagating mode in a coaxial cable is TEM. It has a cut-off frequency of zero.• Next mode is TE11. Its cut-off frequency depends on the mean radius between the
conductors and the material filling this gap.• The characteristic impedance of the cable also depends on the geometry of the cable
section and the materials.
[Imag
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http
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idso
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u/st
uhom
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lein
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tml]
electric field distribution inside a coaxial
TEM mode TE11
rrinout
TEc rr
f
22
111
dDZ
r
r ln21
0
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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APPLICATION NOTES Coaxial cables
• Characteristic impedance, frequency bandwidth, attenuation, wave speed of propagation, and maximum power handling capability should be carefully considered when adquiring a cable for a given application.
[Table from SSi Cable Corp.]Aeronautical Communications
C. Collado, J.M. González-ArbesúEETAC-UPC
TNC
32
APPLICATION NOTES Connectors
• Some 50 impedance connectors typically used in RF and microwave equipment. Others exist.
Warning! RP connectors(RP: reverse polarity)
11 (18) GHz 11 GHz 18 GHz2 GHz
BNC N
18 (26.5) GHz
SMA
APC-7
6 GHz
MCX
6 GHz
MMCX
40 (45) GHz
K BNC: Bayonet Neill-ConcelmanMCX: Micro CoaXialMMCX: Micro-Miniature CoaXialN: Neill connectorTNC: Threaded Neill-ConcelmanAPC-7: Amphenol Precission Connector
with 7 mm diameterSMA: SubMiniature version AAPC-3.5: Amphenol Precission
Connector with 3.5 mm diameterSMK: SubMiniature version K (also
called 2.92 mm)
26.5 (34) GHz
3.5
Aeronautical CommunicationsC. Collado, J.M. González-Arbesú
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