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Prof. D. R. Wilton Note 2 Transmission Lines (Time Domain) ECE 3317

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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. Note about Notes 2 Approach: Transmission line theory can be developed starting from either circuit theory or from Maxwells equations directly. Well use the former approach because it is simpler, though it doesnt reveal the approximations or limitations of the approach.

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A transmission line is a two-conductor system that is used to transmit a signal from one point to another point. Transmission Lines Two common examples: coaxial cable twin line a b z A transmission line is normally used in the balanced mode, meaning equal and opposite currents (and charges) on the two conductors.

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Transmission Lines (cont.) coaxial cable Heres what they look like in real-life. twin line

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Another common example (for printed circuit boards): microstrip line w h r Transmission Lines (cont.)

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microstrip line Transmission Lines (cont.)

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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 E coax twin line E

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Transmission Lines (cont.) Transmission line theory must be used instead of circuit theory for any two-conductor system if the travel time across the line, T L, (at the speed of light in the medium) is a significant fraction of a signals period T or rise time for periodic or pulse signals, respectively. Load

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4 parameters symbol: z Note: We use this schematic to represent a general transmission line, no matter what the actual shape of the conductors. Transmission Lines (cont.)

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4 parameters C = capacitance/length [ F/m ] L = inductance/length [ H/m ] R = resistance/length [ /m ] G = conductance/length [ S/m ] Four fundamental parameters that characterize any transmission line: z These are per unit length parameters. capacitance between the two wires inductance due to stored magnetic energy resistance due to the conductors conductance due to the filling material between the wires Transmission Lines (cont.)

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Circuit Model Circuit Model: R z L z G z C z z z z z

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Coaxial Cable Example (coaxial cable) a b z (skin depth of metal) d = conductivity of dielectric [S/m]. m = conductivity of metal [S/m].

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Coaxial Cable (cont.) Overview of derivation: capacitance per unit length l - l E

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Coaxial Cable (cont.) Overview of derivation: inductance per unit length y E x JsJs

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Coaxial Cable (cont.) Overview of derivation: conductance per unit length RC Analogy:

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Coaxial Cable (cont.) Relation between L and C : Speed of light in dielectric medium: Hence: This is true for ALL two-conductor transmission lines.

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Telegraphers Equations Apply KVL and KCL laws to a small slice of line: + V (z,t) - R z L z G z C z I (z,t) + V (z+ z,t) - I (z + z,t) z z z+ z

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Hence Now let z 0: Telegraphers Equations (TE) Telegraphers Equations (cont.)

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To combine these, take the derivative of the first one with respect to z : Take the derivative of the first TE with respect to z. Substitute in from the second TE. Telegraphers Equations (cont.)

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The same equation also holds for i. Hence, we have: There is no exact solution to this differential equation, except for the lossless case. Telegraphers Equations (cont.)

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The same equation also holds for i. Lossless case: Note: The current satisfies the same differential equation: Telegraphers Equations (cont.)

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The same equation also holds for i. Hence we have Solution: Solution to Telegrapher's Equations This is called the DAlembert solution to the Telegrapher's Equations (the solution is in the form of traveling waves).

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Traveling Waves Proof of solution: It is seen that the differential equation is satisfied by the general solution. General solution:

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Example: z z0z0 z t = 0 t = t 1 > 0 t = t 2 > t 1 z 0 + c d t 1 z0z0 z 0 + c d t 2 V(z,t)V(z,t) Traveling Waves

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Example: z z0z0 z t = 0 t = t 1 > 0 t = t 2 > t 1 z 0 - c d t 1 z0z0 z 0 - c d t 2 V(z,t)V(z,t) Traveling Waves

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Loss causes an attenuation in the signal level, and it also causes distortion (the pulse changes shape and usually becomes broader). z t = 0 t = t 1 > 0 t = t 2 > t 1 z 0 + c d t 1 z0z0 z 0 + c d t 2 V(z,t)V(z,t) Traveling Waves (cont.) (These effects can be studied numerically.)

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Current lossless (first TE) Our goal is to now solve for the current on the line. Assume the following forms: The derivatives are:

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Current (cont.) This becomes Equating terms with the same space and time variation, we have Hence we have Constants represent independent DC voltages or currents on the line. Assuming no initial line voltage or Current we conclude Constants =0

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Then Define characteristic impedance Z 0 : The units of Z 0 are Ohms. Current (cont.) Observation about term: or

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General solution: For a forward wave, the current waveform is the same as the voltage, but reduced in amplitude by a factor of Z 0. 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! Current (cont.)

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z Picture for a forward-traveling wave: + - forward-traveling wave

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z Physical interpretation of minus sign for the backward-traveling wave: + - backward-traveling wave The minus sign arises from the reference direction for the current. Current (cont.)

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Coaxial Cable Example: Find the characteristic impedance of a coax. a b z

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Coaxial Cable (cont.) a b z (intrinsic impedance of free space)

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Twin Line d a = radius of wires

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Twin Line (cont.) coaxial cable twin line These are the common values used for TV [ ] transformer 300 [ ] twin line 75 [ ] coax

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w h r Microstrip Line parallel-plate formulas:

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Microstrip Line (cont.) More 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. t = strip thickness

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Some Comments Transmission-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.

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