Download presentation
Presentation is loading. Please wait.
Published byMilton Warren Modified over 9 years ago
1
Prof. D. R. Wilton Notes 6 Transmission Lines (Frequency Domain) ECE 3317 [Chapter 6]
2
Frequency Domain Why is the frequency domain important? Most communication systems use a sinusoidal carrier signal (that may be modulated). A solution in the frequency-domain allows us to solve for an arbitrary time- varying signal on a lossy line (by using the Fourier transform method). Arbitrary signals can be expressed as a superposition of phasors of different frequencies:
3
Frequency Domain Amplitude modulation: Spectrum of a single square pulse Video or audio spectrum
4
Telegrapher’s Equations RzRz LzLz GzGz CzCz z zz Transmission-line model for a small length of transmission line: C = capacitance/length [ F/m ] L = inductance/length [ H/m ] R = resistance/length [ /m ] G = conductance/length [ S/m ]
5
Telegrapher’s Equations (cont.)
6
Hence to convert to the phasor domain: Must hold for all t !
7
Telegrapher’s Equations (cont.) = series impedance/length = parallel admittance/length Define Frequency domain telegrapher’s equations
8
Telegrapher’s Equations (cont.) To eliminate I, differentiate the first equation and substitute from the second:
9
Frequency Domain (cont.) Solution: Then Define Note: We have an exact solution, even for a lossy line!
10
Propagation Constant Convention: we choose the (complex) square root to be the principal branch: Choosing the principal branch means that For a lossless line, we consider this as the limit of a lossy line, in the limit as the loss tends to zero: (lossless case) (lossy case) is called the propagation constant, with units of [1/m]
11
Physical interpretation of waves: (This will be shown shortly.) = propagation constant [ m -1 ] = attenuation constant [nepers/m] = phase constant [radians/m] (forward traveling wave) (backward traveling wave) Forward traveling wave: Note: For a lossless line = 0, Propagation Constant (cont.) Denote:
12
Current Use the first Telegrapher equation: Next, use so
13
Current (cont.) Solving for the current I, we have
14
Current (cont.) Define the (complex) characteristic impedance Z 0 : Then we have
15
Special case of a lossless line: Hence we realize that Compare with Propagation Constant for Lossless Line
16
Forward Wave Forward traveling wave: Denote Hence we have In the time domain we have: Then Note that
17
Snapshot of Waveform The distance is the distance (in meters) over which the waveform repeats itself. Forward Wave (cont.) = wavelength The distance 1/ α is the distance (in meters) over which the (envelope) amplitude drops by a factor of e
18
Wavelength The oscillation repeats i.e., Hence: (ignoring the amplitude decay) when:
19
Attenuation Constant The attenuation constant controls how fast the wave decays.
20
Snapshot of Waveform Spatial vs. Time-Varying Signals The distance is the distance (in meters) over which the (envelope) amplitude drops by a factor of e Oscilloscope signal trace The distance is the time (in seconds) over which the (envelope) amplitude drops by a factor of e
21
Phase Velocity The wave forward-traveling wave is moving in the positive z direction. Consider a lossless transmission line for simplicity ( = 0 ): z [m] v = velocity t = t 1 t = t 2 crest of wave:
22
The phase velocity v p is the velocity of a point on the wave, such as the crest. Set We thus have Take the derivative with respect to time: Hence Note: this result holds also for a lossy line.) Phase Velocity (cont.) Hence note that
23
Let’s calculate the phase velocity for a lossless line: Also, we know that Hence (lossless line) Phase Velocity (cont.)
24
Backward Traveling Wave Let’s now consider the backward-traveling wave (lossless, for simplicity): z [m] v = velocity t = t 1 t = t 2
25
Attenuation in dB/m Attenuation in dB: Hence we have: Recall:
26
Attenuation (cont’d) Attenuation in dB: Forward traveling wave measured at two points on line:
27
Example: Coaxial Cable Example (coaxial cable) a b z (skin depth) a = 0.5 [mm] b = 3.2 [mm] r = 2.2 tan d = 0.001 m = 5.8 10 7 [S/m] f = 500 [MHz] (UHF) Dielectric conductivity is specified in terms of the loss tangent: copper conductors (nonmagnetic: = 0 )
28
a b z Example: Coaxial Cable (cont.)
29
a b z R = 2.147 [ /m] L = 3.713 E-07 [H/m] G = 2.071 E-04 [S/m] C = 6.593 E-11 [S/m] = 0.022 +j (15.543) [1/m] = 0.022 [nepers/m] = 15.543 [rad/m] attenuation = 8.686 x 0.022 = 0.191 [dB/m] = 0.404 [m] Example: Coaxial Cable (cont.)
30
Propagation Wavenumber Alternative notations: (propagation constant) (propagation wavenumber)
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.