1. Chapter I Transmission Lines and Microwave Networks

2. Outline Distributed Circuits and Transmission-Line Effects
Transmission-Line Parameters and the Smith Chart
Microstrip Lines
Microwave Network Analysis

3. 1. Distributed Circuits and Transmission-Line EffectsZL Rs l
m, e, s sc
Equivalent circuit
At DC or very low frequencies, the equivalent circuit can be simplified as ZL Rs
R/2 R
Balanced type
Unbalanced type
At medium high frequencies, the equivalent circuit becomes Rs R L ZL
Unbalanced type G C

4. ZL Rs R/2 L/2 G C ZL Rs R L G/2 C/2 ZL Rs R/4 L/4 G C ZL Rs G/2 C/2
Unbalanced T-type ZL Rs R L
G/2
C/2
Unbalanced p-type ZL Rs
R/4
L/4 G C
Balanced T-type ZL Rs
G/2
C/2
R/2
L/2
Balanced p-type

5. R: Conductor resistance (Series resistance)
where
R: Conductor resistance (Series resistance)
I2R/2: Time-average power dissipated due to conductor loss
L: Self inductance (Series inductance)
I2L/4: Time-average magnetic energy stored in the transmission line
C: Self capacitance (Shunt capacitance)
V2C/4: Time-average electric energy stored in the transmission line
G: Dielectric Conductance (Leakage conductance, shunt conductance)
V2G/2: Time-average power dissipated due to dielectric loss
At very low frequencies:
(s represents dielectric conductivity)
Thus, L,C,G can be ignored at very low frequencies. But at medium high frequencies, parasitic effects due to L,C,G have to be considered.

6. Solutions of Static L,C,G parameters
PDE:
BCs:
z = l C S
z = 0

7. Distributed Parameters
For distributed parameters of TEM transmission lines

8. Example: Coaxial Line PDE: b BCs: a Due to symmetry, PDE becomes ODE:
BCs become

9. General solutions for electric potential at z=0
Substitute BCs into general solutions to find the coefficients C1 and C2
Final solution
Electric and magnetic fields at z= 0

10. Current along the inner conductor at z=0
Find distributed parameters L,C,G
Check the following relations between LC and C/G

11. Conductor resistance per unitlength
Loss tangent of dielectric
Material
e=ere0
tandc
FR4
er= 4.5
0.014
Ceramic
er= 9.9
0.0001
Teflon
er= 2.2
0.0003
GaAs
er= 12.9
0.002
Silcon
er= 11.9
0.015
Conductor resistance per unitlength t C1 C2

12. Skin effect: At high frequencies, currents tend to concentrate on surface of the conductor within a skin depth d d
Effective conductor thickness t
d(f)
tec
fec f

13. Example: Two-wire lineD a
Example: Parallel-plate line d w

14. Distributed equivalent circuit
At RF and microwave frequencies, a general two-conductor uniform line divided into many sections can be used to describe the transmission-line behavior. ZL Rs
l = Nz z
N sections
Unbalanced-type distributed equivalent circuit Rs
L z
R z
G z
C z ZL
I(z) +
V(z) -

15. Example: Simulation of transmission-line effects
10 cm
50W
I(z) +
V(z) -
1 V
D=1 cm
air
50W
PL= power dissipated
by load
2a=0.1 cm
Calculate the L,C,G,R distributed parameters and Z:

16. H1 Voltage standing wave
One can use TOUCHSTONE (ADS) to simulate the frequency-domain responses of this distributed circuit. Also, the SPICE can be used to simulate the time-domain responses.
Voltage standing wave
N=1000

17. H1
Current standing wave
N=1000

18. H1

19. H1

20. H1
Convergence of number of stages

21. Summary
Voltage and current on the transmission exhibit the form of standing wave.
The input impedance deviates from the load impedance. This is due to the parasitic effects of the wires.
The dissipated power of the load deviates from its maximum value. This is called as the impedance mismatch.
All the above effects belong to the transmission line effects, which become more obvious as electrical length (    l ) increases.

22. 2. Transmission-Line Parameters and the Smith Chart
Transmission-Line Equations Rs
L z
R z
G z
C z ZL
I(z) +
V(z) -
I(z+ z)
V(z+ z)
According to KVL:
According to KCL:
After rearrangement:

23. As Z0,
Solve for either V(z) or I(z) by eliminating either I(z) or V(z):
where

24. Low-Loss Transmission Lines
General solutions (traveling-wave solutions) yield:
where
Low-Loss Transmission Lines
Low-loss conditions: R<<w L, G<< w C,

25. Therefore,
where ac is attenuation due to conductor loss
ad is attenuation due to dielectric loss
Parameters related to the attenuation constant:
Parameters related to the phase constant:
Up (Phase velocity)  w 0, t (Time delay)  w 0, U p = l / t

26. a b
For low-loss coaxial lines,
For low-loss two-wire lines, D a
For low-loss parallel-plate lines, d w

27. Why 50 ohm for coaxial-line systems?b
The aspect of loss:
Material factor Geometrical factor Material factor
The geometrical factor related to the loss is
For fixed material parameters, the total loss increases as g(a,b) increases.

28. H2
Assuming that the outer dimension b is fixed, g(a,b) has a minimum at b/a= The corresponding Z0 is

29. The aspect of breakdown:
The maximum electric field inside the dielectric region of the coaxial cable is at inner conductor surface (r=a). Assume that Ed represents the dielectric strength at breakdown. The breakdown voltage (Vb) can be expressed as
The maximum power capacity is defined as the time-average propagating power at breakdown, which is expressed as
The maximum power capacity is proportional to the geometrical factor h(a,b). Assuming that the outer dimension b is fixed, h(a,b) has a maximum at b/a= The corresponding Z0 is

30. H2
Why 50 ohm 
For the air-filled coaxial lines, the characteristic impedance compromises between 77 W and 30 W to 50 W.

31. G(z) G Terminated Transmission Lines Standing-wave solutions:
Applying the first boundary condition at z=0:
Therefore,

32. where G represents the voltage reflection coefficient
where G represents the voltage reflection coefficient. In general, G is a function of z, which can be expressed as
Assuming lossless transmission lines, thus, g = j b,
Incident power Reflected power

33. Zin Zin(z) The magnitude of the standing-wave voltage is expressed as
The standing-wave ratio is defined as the ratio of Vmax to Vmin, which can be expressed as
Zin(z)
Zin
The input impedance can be expressed as

34. Gg Zin Generator and Load Mismatches
In general, the input impedance is a function of z, which is expressed as
The general relation between G(z) and Zin(z) is
Generator and Load Mismatches
Applying the second boundary condition at z=-l :
Zin Gg

35. After substituting Zin that has been expressed in terms of Z0, ZL, and b l, the expression can be rewritten as
Once V0+ has been determined, V(z) & I(z) can be determined subsequently.
Consider the generator and load mismatches for a lossless transmission line, the power delivered to the load becomes
Substitute V(z=- l ) that was expressed previously and let Zg = Rg + j Xg and Zin = Rin +j Xin. Then

36. To maximize Pl , we differentiate with respect to the real and imaginary parts of Zin. Thus,
It is concluded that the maximum Pl occurs at conjugate matching condition. That is
The Smith Chart
P.H. Smith devised the Smith chart as a graphical tool for transmission-line calculations in 1939.

37. Smith chart is used to convert reflection coefficient into normalized input impedance, and vice versa.
G(z’), zin(z’)
toward load
toward generator
In this day of scientific calculators and power CAD software, graphical solutions have no place in modern engineering practice. However, the Smith chart is still important because it provides an extremely useful way of visualizing transmission-line phenomenon, and so is also important for pedagogical reasons.

38. Derivation of the Smith Chart
Display G(z’):
Display zin(z’):
Rearrangement gives the loci with respect to r and x on the Smith chart

39. Summary
Under low-loss condition, the transmission-line parameters can be expressed as
The following parameters are related to the phase constant:
In the early days, the Smith chart is used to convert the reflection coefficient into normalized input impedance, and vice versa. Today, it is primarily used in visualizing the transmission-line phenomenon and display of data for pedagogical reasons.

40. 3. Microstrip Lines Quasi-Static Approach
Microstrip line is the most popular type of planar transmission lines, primarily because it can be fabricated by photolithographic processes and is easily integrated with other passive and active RF devices.
Quasi-Static Approach
For low-loss microstrip lines,
For air-filled microstrip lines,
We can derive

41. er er w h w h Procedure for calculating the distributed capacitance:
Effective dielectric constant er w h
For very wide lines, w / h >> 1 er w h
For very narrow lines, w / h << 1

42. Planar Waveguide Model
We can express eeff as
where filling factor q represents the ratio of the EM fields inside the substrate region, and its value is between ½ and 1. Another approximate formula for q is
(provided by K.C. Gupta, et. al.)
Planar Waveguide Model

43. Curve-Fitting Formulae
However, the dielectric loss should occur in the substrate region only, not the whole region. Therefore, ad should be modified as
Curve-Fitting Formulae
(provided by I.J. Bahl, et. al.)
Analysis procedure: Give w / h to find eeff and Z0.

44. Dispersion in Microstrip
Synthesis procedure: Give Z0 to find w / h.
Dispersion in Microstrip
As frequency goes higher, EM fields tend to distribute in the substrate region in a higher ratio. This causes eeff to increase with frequency.

45. Getsinger’s frequency-dependent expression:
In above expression, G is an empirical parameter and fp represents transverse resonant frequency.
Example: Design a 50-W microstrip line on a mm thick Alumina substrate (er=9.9). Calculate the wavelength of the line at 1 and 10 GHz. Assume that G = Z0 in Getsinger’s expression.

47. Summary
At low frequencies, microstrip lines possess the characteristics of low loss and dispersion and are good for MIC (Microwave Integrated Circuit) applications.
For the majority of microstrip lines suitable for MICs, the statically derived results including curve-fitting formulae are quite accurate where the frequency is below a few GHz.
At higher frequencies, up to the limits for the useful operation of microstrip, these static results combined with frequency-dependent functions can be used to characterize the dispersion in microstrip.

48. 4. Microwave Network Analysis
Impedance, Admittance and Scattering Matrices
At reference planes zk=tk, k=1,2, …,N, the impedance matrix is defined as
Abbreviation
Matrix element

49. Similarly, the admittance matrix is defined as
Abbreviation
Matrix element
Conversion between [Z] and [Y]

50. The scattering matrix is defined as
Abbreviation
Matrix element
Conversion between [S] and [Z] can be derived as follows.

51. At reference planes zk=tk, k=1,2, …,N, the voltage and current on the transmission lines can be expressed as

52. The Transmission (ABCD) matrix of a Two-Port Network
Matrix element

53. The ABCD matrix is good for analysis of cascaded microwave networks.
Equivalent ABCD matrix
Conversion between [ABCD] and [Z]
In derivation of B using Z parameters, please note that the directions of I2 in ABCD and Z parameters are in opposite directions.

54. ABCD parameters of some useful two-port networks
In a similar derivation

55. Conversion between two-port network parameters

56. Reciprocal Networks
A microwave network is reciprocal when it has no active devices, ferrites, or plasmas inside the network.
Reciprocity theorem
source a S V V S
source b
For ABCD matrix of a two-port reciprocal network

57. Lossless Networks
A microwave network is lossless when it has no resistors or active devices inside the network. Besides, the network has no losses due to conductor skin effects, dielectric leakage, radiation, and surface waves, etc.
Time-average power dissipated by the network is zero. That is
All the port currents are independent. Besides, the network is assumed to be reciprocal. For all m, n
(Imaginary matrix)

58. For the ABCD matrix of a lossless and reciprocal network, A and D will be real. B and C will be imaginary.
The scattering matrices of lossless networks
real
imaginary
real
(Conservation of power)
incident power
reflected power
(Unitary matrix)

59. Equivalent Circuits for Reciprocal Two-Port Networks

60. Summary
For reciprocal networks, [Z],[Y] and [S] are symmetric. ABCD matrix has determinant equal to 1.
For reciprocal and lossless networks, [Z] and [Y] are imaginary. ABCD matrices have elements A and D that are real. The other two elements, B and C, are imaginary.
For lossless networks, [S] is unitary.
Bibliography
D.M. Pozar, Microwave Engineering, 4th Ed., Ch. 2,3,4
T.C. Edwards, Foundations of Interconnect and Microstrip Design, 3rd Ed., Ch. 4,5