Chapter2_Notes

# V0 ez ej tz notes based on fundamentals

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Unformatted text preview: s −z traveling. • What is the phase (propagation) velocity? up = f λ = ω β (58) • Important realization: waves traveling in opposite directions on transmission lines form standing waves ! Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 31 1.5. The lossless microstrip line • The microstrip line is a type of transmission line for RF and microwave circuits. • Microwave circuits are found in many applications including cellular communications, wireless networking, satellite communications and radar. • These transmission lines are easy to fabricate on a circuit board consisting of just a thin copper strip printed on a dielectric substrate that is over a ground plane. • It is similar to a parallel plate waveguide that supports TEM modes but since it has limited dimensions is only approximately TEM or quasi-TEM. • There are two geometric parameters- the width of the strip w and the height (thickness) of the dielectric layer h. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 32 • The thickness of the strip is neglected because it is generally much smaller than w. • We assume the substrate is a perfect dielectric σ = 0. • We assume the strip and ground plane are perfect conductors σ ≈ ∞. • These approximations and assumptions simplify the analysis quite a bit but do not introduce signiﬁcant error. • The three parameters that will determine the characteristics of the transmission line are w, h and . • With these assumptions the phase speed of the wave is given by, c up = √ (59) r with r the relative permittivity and c the speed of light in free space. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 33 • Even though the electric ﬁeld is mostly in the dielectric substrate, some is in the surrounding air. This mixture of where the electric ﬁeld is can be accounted for by using an eﬀective permittivity eﬀ which leads to, up = √ c (60) eﬀ • Getting the exact eﬀective permittivity gives a complicated expression but we can get a good approximation by curve ﬁtting to this, eﬀ where s = w h = r +1 + 2 r −1 2 1+ 10 s −xy (61) and, x = 0.56 r − 0.9 r +3 0.05 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (62) Electromagnetics I: Transmission lines y = 1+0.02 ln 34 s4 + 3.7 × 10−4 s2 + 0.05 ln 1 + 1.7 × 10−4 s3 s4 + 0.43 (63) • The characteristic impedance is given by, 60 6 + (2π − 6)e−t Z0 = √ + ln s eﬀ with t= 30.67 s 1+ 4 s2 (64) 0.75 (65) • The ﬁgure shows the relationship between Z0 and s for various dielectric materials. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 35 Figure 9: Plots of Z0 as a function of s for various dielectric m...
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## This note was uploaded on 09/25/2013 for the course ECE 331 taught by Professor Martinsiderious during the Fall '12 term at Portland State.

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