• : 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]
• 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 []
• 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.
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.
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:
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.
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.
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.
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).
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.
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.
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.
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.
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.
• 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
• 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.
• 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
• 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.
