1Note 2 Transmission Lines (Time Domain) ECE 3317Prof. D. R. WiltonNote Transmission Lines (Time Domain)
2Note about Notes 2 Disclaimer: Transmission lines is the subject of Chapter 6 in the book. However, the subject of wave propagation on transmission line in the time domain is not treated very thoroughly there, appearing only in the latter half of section 6.5.Therefore, the material of this Note is roughly independent of the book.Approach:Transmission line theory can be developed starting from either circuit theory or from Maxwell’s equations directly. We’ll use the former approach because it is simpler, though it doesn’t reveal the approximations or limitations of the approach.
3Transmission LinesA transmission line is a two-conductor system that is used to transmit a signal from one point to another point.Two common examples:abzcoaxial cabletwin lineA transmission line is normally used in the balanced mode, meaning equal and opposite currents (and charges) on the two conductors.
4Transmission Lines (cont.) Here’s what they look like in real-life.coaxial cabletwin line
5Transmission Lines (cont.) Another common example (for printed circuit boards):wrhmicrostrip line
7Transmission Lines (cont.) Some practical notes:Coaxial cable is a perfectly shielded system (no interference).Twin line is not a shielded system – more susceptible to noise and interference.Twin line may be improved by using a form known as “twisted pair.”l-l+-EEcoaxtwin line
8Transmission Lines (cont.) Transmission line theory must be used instead of circuit theory for any two-conductor system if the travel time across the line, TL, (at the speed of light in the medium) is a significant fraction of a signal’s period T or rise time for periodic or pulse signals, respectively.Load
9Transmission Lines (cont.) symbol:z4 parametersNote: We use this schematic to represent a general transmission line, no matter what the actual shape of the conductors.
10Transmission Lines (cont.) Four fundamental parameters that characterize any transmission line:zThese are “per unit length” parameters.4 parametersC = capacitance/length [F/m]L = inductance/length [H/m]R = resistance/length [/m]G = conductance/length [S/m]capacitance between the two wiresinductance due to stored magnetic energyresistance due to the conductorsconductance due to the filling material between the wires
12Coaxial Cable Example (coaxial cable) a zExample (coaxial cable)d = conductivity of dielectric [S/m].m = conductivity of metal [S/m].(skin depth of metal)
13Coaxial Cable (cont.)Overview of derivation: capacitance per unit lengthl-l+-E
14Coaxial Cable (cont.) Js Overview of derivation: inductance per unit lengthyExJs
15Coaxial Cable (cont.)Overview of derivation: conductance per unit lengthRC Analogy:
16Coaxial Cable (cont.) Relation between L and C: Speed of light in dielectric medium:This is true for ALL two-conductortransmission lines.Hence:
17Telegrapher’s Equations Apply KVL and KCL laws to a small slice of line:+V (z,t)-RDzLDzGDzCDzI (z,t)V (z+Dz,t)I (z+Dz,t)zz+Dz
18Telegrapher’s Equations (cont.) HenceNow let Dz 0:“Telegrapher’s Equations (TE)”
19Telegrapher’s Equations (cont.) Take the derivative of the first TE with respect to z.Substitute in from the second TE.To combine these, take the derivative of the first one with respect to z:
20Telegrapher’s Equations (cont.) Hence, we have:There is no exact solution to this differential equation, except for the lossless case.The same equation also holds for i.
21Telegrapher’s Equations (cont.) Lossless case:Note: The current satisfies the same differential equation:The same equation also holds for i.
22Solution to Telegrapher's Equations Hence we haveSolution:This is called the D’Alembert solution to the Telegrapher's Equations (the solution is in the form of traveling waves).The same equation also holds for i.
23Traveling Waves Proof of solution: General solution: It is seen that the differential equation is satisfied by the general solution.
24Traveling Waves Example: z z0 z t = 0 t = t1 > 0 t = t2 > t1 z0 + cd t1z0z0 + cd t2V(z,t)
25Traveling Waves Example: z z0 t = t1 > 0 t = t2 > t1 t = 0 V(z,t)t = 0zz0 - cd t2z0 - cd t1z0
26Traveling Waves (cont.) Loss causes an attenuation in the signal level, and it also causes distortion (the pulse changes shape and usually becomes broader).t = 0V(z,t)t = t1 > 0t = t2 > t1zz0z0 + cd t1z0 + cd t2(These effects can be studied numerically.)
27Current (first TE) lossless Our goal is to now solve for the current on the line.Assume the following forms:The derivatives are:
28Current (cont.) This becomes Equating terms with the same space and time variation, we haveHence we haveConstants represent independentDC voltages or currents on the line.Assuming no initial line voltage orCurrent we conclude Constants =0
29Current (cont.) Define characteristic impedance Z0: Observation about term:Define characteristic impedance Z0:The units of Z0 are Ohms.Thenor
30Current (cont.) General solution: For a forward wave, the current waveform is the same as the voltage, but reduced in amplitude by a factor of Z0.For a backward traveling wave, there is a minus sign as well.Note that without this minus sign, the ratio of voltage to current would be constant rather than varying from point-to-point and over time along the line as is generally the case!
31Current (cont.) Picture for a forward-traveling wave: z +-
32Current (cont.)Physical interpretation of minus sign for the backward-traveling wave:backward-traveling wavez+-The minus sign arises from the reference direction for the current.
33Coaxial Cable Example: Find the characteristic impedance of a coax. a z
34Coaxial Cable (cont.)abz(intrinsic impedance of free space)
38Microstrip Line (cont.) t = strip thicknessMore accurate CAD formulas:Note: the effective relative permittivity accounts for the fact that some of the field exists outside the substrate, in the air region. The effective width w' accounts for the strip thickness.
39Some CommentsTransmission-line theory is valid at any frequency, and for any type of waveform (assuming an ideal transmission line).Transmission-line theory is perfectly consistent with Maxwell's equations (although we work with voltage and current, rather than electric and magnetic fields).Circuit theory does not view two wires as a "transmission line": it cannot predict effects such as signal propagation, distortion, etc.