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 RzRz LzLz GzGz CzCz z zz 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)