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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。
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
2.3 Rectangular waveguide Closed waveguide, propagate Transverse electric (TE) and/or transverse magnetic (TM) modes.
Maxwell’s equations (no sources) expansion
TM: HZ= 0, to solve Ez to solve EZ、HZ TE: Ez= 0, to solve Hz TE TEmn mode Boundary conditions Helmholtz equation Separate variable Separated PDE General solution Amn = C1C2
Propagation constant: Cutoff frequency: Fields Wave impedance: Phase velocity: x xx Propagation direction x x x xx x x xx xxx xx x l’ TEmn mode TE10 mode (the fundamental mode)
Propagation constant: Cutoff frequency: Wave impedance: Phase velocity: fields TMmn mode TM11 mode (the lowest mode)
2.6 Surface waves on a grounded dielectric slab • Surface waves • 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:
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.
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
Inverse design • 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
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:
Effective dielectric constant: Formula for characteristic impedance: (numerical fitting ) Inverse waveguide design with known Z0 and r:
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.
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)
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() Output Input F() Fo() • 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.
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
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:
2.10 Summary of transmission lines and waveguids Comparison of transmission lines and waveguides
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)
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 0.158 cm, with r = 2.20. What is the guide wavelength on this transmission line if the frequency is 4.0 GHz?
