2 2.3 Coaxial lineAdvantage 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.andEquivalent 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.
6 to solve EZ、HZ Helmholtz equation TE TEmn mode Boundary conditions TM: HZ= 0, to solve Ezto solve EZ、HZTE: Ez= 0, to solve HzHelmholtz equationTETEmn modeBoundary conditionsSeparate variableAmn = C1C2General solutionSeparated 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)xx x xxxxxxPropagation directionl’
8 TM11 mode (the lowest mode) TMmn modefieldsPropagation constant:Cutoff frequency:Wave impedance:Phase velocity:TM11 mode (the lowest mode)
12 2.6 Surface waves on a grounded dielectric slab a field that decays exponentially away from the dielectric surfacemost of the field contained in or near the dielectricmore tightly bound to the dielectric at higher frequenciesphase velocity: Vdielectric < Vsurface < VvacuumGeometries:
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 byCharacteristic 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 iswith
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 asAttenuation 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).witht thickness of strip
16 2.8 MicrostripFor 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 lossConsidering microstrip as a quasi-TEM line, the attenuation due to dielectric loss can be determined asFilling 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 approximatelyFor 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 mediumThe 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)TransferZ()F()Fo()InputOutputIf 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 isThe 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 guidesRidge waveguideDielectric waveguide(TE or TM mode, mm wave to optical frequency, with active device)(lower the cutoff frequency, increase bandwidth ad better impedance characteristics)Coplanar waveguideCovered microstripSlot line(Quasi-TEM mode, useful for active circuits)(quasi-TEM mode, rank behind microstrip and stripline)(electric shielding or physical shielding)
26 Homework1. 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?