1. Transmission Line Theory
Chapter 2
Transmission Line Theory

2. Transmission-Line (TL) Theory
TL theory bridges the gap between field analysis and basic circuit theory. ZL Rs l
m, e, s sc
Lumped-element equivalent circuit
At DC or very low frequencies, the equivalent circuit can be simplified as Rs R ZL
At medium and high frequencies, the equivalent circuit becomes Rs R L G C ZL

3. 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 Rs
L Z
R Z
G Z
C Z ZL
L,C,R,G are called distributed parameters.

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

5. Solutions of L,C,G parameters
PDE: (Laplace’s Equation)
BCs:
Z=l C S
Z=0

6. L,C,G Distributed parameters can be found as
For distributed parameters of TEM transmission lines

7. Example2.1: Find the TL parameters of coaxial Line?
Solution( another solution can refer to p.54 of the text book) a b
PDE:
BCs:
Due to symmetry,
PDE becomes ODE:
BCs become

8. 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

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

10. 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

11. Skin effect: At high frequencies, currents tend to concentrate on surface of the conductor within a skin depth d or penetration depth
(Defined as amplitude of fields decay to 1/e) d
Effective conductor thickness t
d(f)
tec
fec f

12. Example2.2: Two-wire lineD a
Example2.3: Parallel-plate line d w

13. Wave Equations of Transmission Line
Telegrapher equations
General solutions (traveling-wave solutions) of transmission-line equations yield:
where
ps. The parameters  and Z0 are called transmission line parameters.

14. Lossless Transmission Lines
(Lossless conditions: R= G=0)
Low-Loss Transmission Lines
(Low-loss conditions: R<<w L, G<< w C,)
Therefore,
where ac is attenuation due to conductor loss
ad is attenuation due to dielectric loss

15. Example2.4: a b For low-loss coaxial lines,
For low-loss two-wire lines, D a
For low-loss parallel-plate lines, d w

16. Distortionless Lines
Lossy line has a linear phase factor as a function of frequency.
Relation :
Verification :
Advantage : Distortionless line transmitted signal without dispersion.
Dispersion : If the phase velocity is different for different frequencies,
then the individual frequency components will not maintain their original
phase relationships as they propagate down the TL,
and signal distortion will occur.

17. Why 50 characteristic impedance for coaxial lines?b
From distributed parameters
From transmission-line parameters
Attenuation constant due to conductor loss
Attenuation constant due to dielectric loss
Assuming that the outer dimension b is fixed, c has a minimum when b/a= The value comes from

18. Considering the breakdown voltage Vb
Then the maximum power capacity Pmax
Assuming that the outer dimension b is fixed, Pmax has a minimum when b/a= The value comes from
Therefore, use 50 to compromise between 77 and 30.
(Also reference textbook p.130 “point of interest” and problem 2.28 and 3.28.)

20. The Terminated Lossless TL
Zin
Match conditions
Match Mismatch
(Total Reflection)
Reflection coefficient
Return loss (RL,dB) 
Standing wave ratio 
Transmission line impedance equation

21. Terminated in short circuit
Terminated in open circuit

22. Quarter-Wave Transformer
A useful and practical circuit for impedance matching.
Defined as TL with length equals to ℓ=/4(+ n/2).
Perfect matching occurs at one frequency (odd multiple) but mismatch will occur at other frequencies.
Impedance matching is limited to real load impedances (complex load impedance can be transferred to real one, by transformation through an appropriate length of line.)
Substituting ℓ=(2/)(/4)= /2
into equation Zin can find
In order for =0, one must have Zin = Z0, then
Insertion Loss (IL)

23. Formulations for TL

24. Generator and Load Mismatch
Load matched to line
Generator matched to line
Zin=Zg ; l 0 ; SWR>1
Zl=Z0 ; l=0; Zin=Z0 ; SWR=1
Conjugate matching
Zin=Zg* ; g  0 ; l 0 ; 0 ; SWR>1
Multiple reflections may add in phase to deliver more power P to the load.

25. The Smith Chart The Smith chart is a plot of constant-r circles
constant-x circles

27. Compressed Smith Chart
It is applied for both active and passive networks. While a standard Smith chart is used in passive networks where Re(Z)0.
Example2.5: A load impedance of 50+j100 terminates a lossless /4 line (Z0=50). Find the input impedance, the load reflection coefficient, and VSWR?
Solve 1
Solution from equations

28. Solve 2
Solution from Smith chart

30. Example2.6: A lossless 75 line is terminated by an impedance of 150+j150. Find (a) L , (b) VSWR, (c) Zin at a distance of 0.375 from the load, (d) the shortest length of the line for which impedance is purely resistive, and (e0 the value of this resistance?
Solve

32. Transverse Electromagnetic (TEM) Wave (Chapter 3)
Characterized as
TEM wave have a uniquely defined voltage, current, and characteristic impedance.
TEM wave exists in TL consisted of two or more conductors.
single-conductor
closed waveguide
two-conductor TL

33. Transverse Electric (TE) and Transverse Magnetic (TM) Wave
TE Wave :
kcis cutoff wavenumber.
Wave propagation needs:
 is real
k>kc
f>fc (cutoff frequency)
Cutoff or evanescent mode
f<fc  is imaginary
TM Wave :
TE and TM waves have not a uniquely defined voltage, current, and characteristic impedance.
TE and TM waves often exist in single-conductor structure.
Equation must be solved subject to the boundary conditions of specific geometry

34. Parallel Plate Waveguide

35. Parallel Plate Waveguide