1. Lecture 6

2. 2.3 Coaxial line
Advantage of coaxial design: little electromagnetic leakage outside the shield and a good choice for carrying weak signals not tolerating interference from the environment or for higher electrical signals not being allowed to radiate or couple into adjacent structures or circuits.
Common application: video and CATV distribution, RF and microwave transmission, and computer and instrumentation data connections。

3. TEM mode in coaxial cables
The transverse fields satisfy Laplace equations, i.e.
and
Equivalent to the static case, electric field , here the electric potential also satisfies Laplace equation
(in Cartesian coordinate )
or (in cylindrical coordinate)
Boundary conditions

4. 2.3 Rectangular waveguide
Closed waveguide, propagate Transverse electric (TE) and/or transverse magnetic (TM) modes.

5. Maxwell’s equations (no sources)
expansion

6. to solve EZ、HZ Helmholtz equation TE TEmn mode Boundary conditions
TM: HZ= 0, to solve Ez
to solve EZ、HZ
TE: Ez= 0, to solve Hz
Helmholtz equation TE
TEmn mode
Boundary conditions
Separate variable
Amn = C1C2
General solution
Separated PDE

7. xx x x x x xxx l’ TEmn mode Fields TE10 mode (the fundamental mode)
Propagation constant:
Cutoff frequency:
Wave impedance:
Phase velocity:
TE10 mode (the fundamental mode) x
x x x xx
xxx
Propagation direction l’

8. TM11 mode (the lowest mode)
TMmn mode
fields
Propagation constant:
Cutoff frequency:
Wave impedance:
Phase velocity:
TM11 mode (the lowest mode)

9. Mode patterns-- Rectangular waveguide

10. 2.5 Circular waveguide

11. Mode patterns _ Circular waveguide

12. 2.6 Surface waves on a grounded dielectric slab
a field that decays exponentially away from the dielectric surface
most of the field contained in or near the dielectric
more tightly bound to the dielectric at higher frequencies
phase velocity: Vdielectric < Vsurface < Vvacuum
Geometries:

13. 2.7 Stripline Stripline as a sort of “flattened out” coaxial line.
Stripline is usually constructed by etching the center conductor to a grounded substrate of thickness of b/2, and then covering with another grounded substrate of the same thickness.

14. Propagation constant :
with the phase velocity of a TEM mode given by
Characteristic impedance for a transmission:
Laplace's equation can be solved by conformal mapping to find the capacitance. The resulting solution, however, involves complicated special functions, so for practical computations simple formulas have been developed by curve fitting to the exact solution. The resulting formula for characteristic impedance is
with

15. Inverse design Attenuation loss
When designing stripline circuits, one usually needs to find the strip width, given the characteristic impedance and permittivity. The inverse formulas could be derived as
Attenuation loss
(1) The loss due to dielectric filler.
(2) The attenuation due to conductor loss (can be found by the perturbation method or Wheeler's incremental inductance rule).
with
t thickness of strip

16. 2.8 Microstrip
For most practical application, the dielectric substrate is electrically very thin and so the field are quasi-TEM.
Phase velocity:
Propagation constant:

18. Effective dielectric constant:
Formula for characteristic impedance: (numerical fitting )
Inverse waveguide design with known Z0 and r:
Home work.

19. Attenuation loss
Considering microstrip as a quasi-TEM line, the attenuation due to dielectric loss can be determined as
Filling factor:
which accounts for the fact that the fields around the microstrip line are partly in air (lossless) and partly in the dielectric.
The attenuation due to conductor loss is given approximately
For most microstrip substrates, conductor loss is much more significant than dielectric loss; exceptions may occur with some semiconductor substrates, however.

20. 2.9 Wave velocities and dispersion
So far we have encountered two types of velocities:
The speed of light in a medium
The phase velocity (vp = /)
Dispersion: the phase velocity is a frequency dependent function.
The “faster” wave leads in phase relative to the “slow” waves.
Group velocity: the speed of signal propagation (if the bandwidth of the signal is relatively small, or if the dispersion is not too sever)

21. Consider a narrow-band signal f(t) and its Fourier transform:
A linear transmission system (TL or WG) with transfer function Z(w)
Transfer
Z()
F()
Fo()
Input
Output
If and (ie., a linear function of ),
f(t) is a replica of f(t) except for an amplitude factor and time shit. A lossless TEM line (=/v) is disperionless and leads to no signal distortion.

22. Consider a narrow-band signal s(t) representing an amplitude modulated carrier wave of frequency 0:
Assume that the highest frequency component of f (t) is m, where m << 0 .The Fourier transform is
The output signal spectrum:
m << 0

23. The output signal in time domain:
For a narrowband F(),  can be linearized by using a Taylor series expansion about 0:
From the above, the expression for so(t):
(a time-shifted replica of the original envelope s(t).)
The velocity of this envelope (the group velocity), vg:

24. 2.10 Summary of transmission lines and waveguids
Comparison of transmission lines and waveguides

25. 2.10 Summary of transmission lines and waveguids
Other types of lines and guides
Ridge waveguide
Dielectric waveguide
(TE or TM mode, mm wave to optical frequency, with active device)
(lower the cutoff frequency, increase bandwidth ad better impedance characteristics)
Coplanar waveguide
Covered microstrip
Slot line
(Quasi-TEM mode, useful for active circuits)
(quasi-TEM mode, rank behind microstrip and stripline)
(electric shielding or physical shielding)

26. Homework
1. An attenuator can be made using a section of waveguide operating below cutoff, as shown below. If a =2.286 cm and the operating frequency is 12 GHz, determine the required length of the below cutoff section of waveguide to achieve an attenuation of 100 dB between the input and output guides. Ignore the effect of reflections at the step discontinuities.
2 Design a microstrip transmission line for a 100  characteristic impedance. The substrate thickness is cm, with r = What is the guide wavelength on this transmission line if the frequency is 4.0 GHz?