Chapter2_Notes

Chapter2_Notes - 1 Contents 1 Transmission lines 1.1...

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1 Contents 1 Transmission lines 3 1.1 Transmission Lines: General Considerations . . . . . . 3 1.1.1 Wavelength and transmission lines . . . . . . . 4 1.1.2 Propagation modes . . . . . . . . . . . . . . . . 8 1.2 Lumped element model . . . . . . . . . . . . . . . . . 11 1.3 Transmission line equations . . . . . . . . . . . . . . . 17 1.4 Wave propagation on a transmission line . . . . . . . . 22 1.5 The lossless microstrip line . . . . . . . . . . . . . . . 31 1.6 Lossless transmission line . . . . . . . . . . . . . . . . 40 1.6.1 Voltage refection coefficient . . . . . . . . . . . 43 1.6.2 Standing waves . . . . . . . . . . . . . . . . . . 47 1.7 Input Impedance . . . . . . . . . . . . . . . . . . . . . 55 1.8 Special Cases . . . . . . . . . . . . . . . . . . . . . . . 63 1.8.1 Short-circuited line . . . . . . . . . . . . . . . . 63 1.8.2 Open-circuited line . . . . . . . . . . . . . . . . 70 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU.
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2 1.8.3 Application of short-circuit and open-circuit mea- surements . . . . . . . . . . . . . . . . . . . . . 72 1.8.4 One-half wavelength lines . . . . . . . . . . . . 73 1.8.5 Quarter wave transformer . . . . . . . . . . . . 74 1.8.6 Matched line . . . . . . . . . . . . . . . . . . . 76 1.9 Power Fow . . . . . . . . . . . . . . . . . . . . . . . . . 77 1.10 Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . 83 1.10.1 Input impedance . . . . . . . . . . . . . . . . . 96 1.10.2 Impedance to Admittance . . . . . . . . . . . . 104 1.11 Impedance matching . . . . . . . . . . . . . . . . . . . 107 1.12 Transients on Transmission lines . . . . . . . . . . . . 114 1.12.1 Transient response . . . . . . . . . . . . . . . . 115 1.12.2 Bounce diagrams . . . . . . . . . . . . . . . . . 122 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU.
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Electromagnetics I: Transmission lines 3 1. Transmission lines 1.1. Transmission Lines: General Considerations What is a transmission line? For us it will be a pair of wires (or a waveguide) used to guide electromagnetic signals, e.g. telephone wires, coaxial cables, optical ±bers etc. Schematically, Fig. 1 presents a transmission line as a two port circuit with input source and output load connected. Various loads are possible. Sending-end port A A' B B' Transmission line Load circuit Generator circuit Receiving-end port + - V g R g R L Figure 2-1 Figure 1: Transmission line as a black box. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU.
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Electromagnetics I: Transmission lines 4 C R i V g V AA' l A A' B B ' + + - - V BB' + - Transmission line Figure 2-2 Figure 2: Transmission line in a circuit. Wavelength and transmission lines Why didn’t we worry about transmission lines when studying basic circuits? Check out Fig. 2 : simple AC generator connected to R-C load via a transmission line. Can we just take out the transmission line? It depends. .. The generator sends a cosinusoidal signal V AA 0 = V g ( t ) = V 0 cos ωt . Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU.
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Electromagnetics I: Transmission lines 5 Assume that the signal (current or voltage) travels with speed of light in vacuum c = 3 × 10 8 m / s How much is the signal delayed going from AA 0 to BB 0 ? If there are no ohmic losses V 0 ( t ) = V AA 0 ( t - l/c ) = V 0 cos[ ω ( t - l/c )](V) (1) Take a wire length of l = 5 cm . Set the time to t = 0 s. For f = 1 kHz, this case gives V 0 = 0 . 999 ...V 0 , i.e. V 0 and V AA 0 are indistinguishable. In the second case take l = 20km. This gives V 0 = 0 . 91 V 0 , i.e. quite a difference between V 0 and V AA 0 . Where is this coming from? The term ωl/c . That can be re- expressed using u p = c (here) so c = (m / s), so that ωl c = 2 πfl c = 2 π l λ radians (2) Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU.
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Electromagnetics I: Transmission lines 6 If l/λ is small transmission line effects negligible If l/λ & 0 . 01
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Chapter2_Notes - 1 Contents 1 Transmission lines 1.1...

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