Chapter2_Notes

L c notes based on fundamentals of applied

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Unformatted text preview: ω LC LC • Using the relationship, up = √1 µ (83) , √ β = ω µ (rad/m) Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (84) Electromagnetics I: Transmission lines 42 • Typical materials for transmission lines will have permeability µ = µ0 = 4π × 10−7 H/m, while permittivity is given via relative permittivity r = / 0 . Free space permittivity 0 = 8.854 × 10−12 F/m. • Some simple manipulations leads to c up = √ ⇒λ= r up c1 λ0 = √ =√ f f r r (85) One more property of transmission lines: dispersive or not? If the phase velocity is independent of frequency ⇒ medium is nondispersive. Lossless TEM lines are of this type. Why do we care? It gets to the signal integrity issues and how faithfully is the shape of the signal preserved (fig. 2). Summary of several cases of transmission lines is given in table 2.2. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 43 • Voltage reflection coefficient After all this, we still don’t know the voltage! Actually, to solve eqs. 87 we need a full circuit, as presented in fig. 10. ˜ V (z ) = V0+ e−γz + V0− eγz ˜ I (z ) = I + e−γz + I − eγz 0 0 (86) (87) Remember that for lossless lines γ = jβ . Set up coordinate system so that z = 0 at the load end and z = −l at the generator end. • We know that at the load ˜˜ ZL = VL /IL (88) This has nothing to do with the waves! • Now we make a connection with the wave picture: V− V+ ˜ ˜ ˜ ˜ VL = V (z = 0) = V0+ + V0− , IL = I (z = 0) = 0 − 0 (89) Z0 Z0 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines Zg ~ Vg + ~ Ii 44 Transmission line + + ~ Vi Z0 ~ VL ~ IL ZL - - Generator Load z = -l z=0 Figure 2-9 Figure 10: Transmission line connected to a general load and generator. • Plug this into eq. 88 ⇒ ZL = V0+ + V0− V0+ − V0− Z0 ⇒ V0− = ZL − Z0 ZL + Z0 V0+ (90) • Very interesting result: the ratio of incident and reflected wave at the load depends only on the load impedance ZL and characteristic impedance Z0 ! Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 45 • This we call voltage reflection coefficient Γ= ZL /Z0 − 1 V0− ZL − Z0 = = ZL + Z0 ZL /Z0 + 1 V0+ (91) • Similarly, for current waves − I0 V0− + = − + = −Γ I0 V0 (92) • Note that Γ is a complex number! Z0 may be real (for lossless lines), but ZL is generally complex. ⇒ Γ = |Γ|ej Θr (93) • For passive loads |Γ| ≤ 1 • The load is considered matched if ZL = Z0 ⇒ no reflection at the load ⇒ V0− = 0 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines Transmission line 46 A RL 50 W CL Z0 = 100 W 10 pF A' Figure 2-10 Figure 11: Transmission line for example 2-3. • Open...
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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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