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Transmission Line Equations

Transmission Line Equations. Alan Murray. Agenda. We now have a voltage/current/impedance “lumped” circuit mode Work toward expressions for V( x,t ) and I( x,t ) Similar to E ( x,t ) and H ( x,t ) in the plane wave

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Transmission Line Equations

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  1. Transmission Line Equations Alan Murray

  2. Agenda We now have a voltage/current/impedance “lumped” circuit mode Work toward expressions for V(x,t) and I(x,t) Similar to E(x,t) and H(x,t) in the plane wave Look at V(x,t) and I(x,t) in two adjacent “lumps” of the lumped-circuit model Results/understanding examinable Derivations NOT examinable

  3. Current Variation Discard ΔVΔx RΔx GΔx Ix Ix+Δx LΔx Vx+Δx Vx CΔx ΔIC ΔIG Δx

  4. Voltage Variation RΔx GΔx Ix Ix+Δx LΔx Vx+Δx Vx CΔx ΔIC ΔIG Δx

  5. Δx → 0 These "Telegrapher's Equations"describe the variation of I and V (ie how they depend upon one another) as a function of distance along the line (x) and time (t). Remember these?

  6. Variation of V and Iwith space and time? • As with Maxwell’s equations • Put these together … • Eliminate V and get a wave equation for I • Eliminate I and get a wave equation for V • This manipulation (slides 7-9) isNOT EXAMINABLE

  7. I(x,t)? Differentiate. d/dt & d/dx d/dt d/dx Rearrange …

  8. V(x,t)? Differentiate. d/dt & d/dx Rearrange …

  9. Put them together … Rememberthese? Just as we did with Maxwell 3 and 4

  10. Start with a lossless cable J …. Set G=0, R=0, … lossless cable (cf non-conductor for plane waves)

  11. Form of solution? (cfPlanewave) In fact, any solution of the form V(x,t) = F(ω±x) fits. “-” → left-right “+” → right-left “-” → left-right “+” → right-left “-” → left-right “+” → right-left For example It is not surprising that F can be any function since we can propagate sine waves, square waves, triangular waves and waves of arbitrary shape down e.g. a coaxial cable.

  12. Speed of Transmission? For a lossless line, G=0, R=0

  13. Real Signals … Fourier Real signals have many and various shapes All periodic signals can be represented as a Fourier series

  14. Dispersion – this derivation not examinable – set R>0, G=0! X X X Solve forαand β

  15. Dispersion – this derivation not examinable – set R>0, G=0!

  16. Dispersion – this derivation not examinable – set R>0, G=0! So – waves of different frequencies travel at different velocities Increased frequency → lower velocity Now back to examinable material!

  17. Dispersion - examinable again! Fourier components travel at different velocities Increased frequency → lower velocity What does this do to (eg) a square wave?

  18. What does this do to a pulse? For a lossless line v (= ±1/√LC) will be independent of frequency if L and C are independent of frequency. In this case, any waveform F(t + x/v) will travel along the lossless line without distortion and the line is said to have zero dispersion If the velocity, v, depends on frequency the signal will distort as it travels down the line

  19. Example 2.1 Wave velocity on a transmission line. • Derive an expression for the velocity, v, of a wave on a lossless coaxial transmission line in terms of the medium’s • Relative and absolute permittivity • εr and ε0 • Relative and absolute permeability • μr and μ0 • (i) for an air-spaced line • (ii) for a line filled with a non-magnetic, dielectric medium.

  20. Example 2.1 X X X X

  21. Example 2.1 • In an air-spaced line • εr=1, μr=1 • In a dielectric-spaced line … • say εr=3

  22. Example 2.2 - Effect of a medium on the wavelength of an EM wave x=0 x=d E↑ E↑ 3GHz EM radiation is incident on a sheet of polystyrene ( εr = 2.7) with a hole. How thick should the sheet be in order that the wave passing through the sheet and that through the hole are in phase when they emerge from the sheet? from Kraus, 4th ed.

  23. Example 2.2 - Effect of a medium on the wavelength of an EM wave X X X • In air – • In polystyrene – • The phase difference after passing through the dielectric and the hole is • This must be an whole number of 2π’s • ie • For n=1 (thinnest sheet)

  24. Example 2.2 - Effect of a medium on the wavelength of an EM wave Plug in the numbers εr

  25. 2.3 Properties of a wave on a transmission line • A wave of frequency 4.5 MHz and phase constant, =0.123 radm-1 propagates down a lossless transmission line of length 500m and spaced by a non-magnetic material. Find: • (i) the wavelength, λ, of the wave • (ii) the velocity of the wave • (iii) the phase difference between the voltages at the two ends of the line • (iv) the time required for a reference point on the wave to travel down the line • (v) εrof the dielectric in the line

  26. 2.3 Properties of a wave on a transmission line • f=4.5 MHz • =0.123 radm-1 • length 500m • (i) the wavelength, λ, of the wave • (ii) the velocity of the wave • (iii) the phase difference between the voltages at the two ends of the line • (iv) the time required for a reference point on the wave to travel down the line • (v) εrof the dielectric in the line

  27. Summary • Equations governing waves on a transmission line • Relate ω, λ, speed • Waves are sums of sinusoids • Loss-free line → no dispersion • Lossy line → different waves at different speeds • Dispersion • Distortion • Also attenuation

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