Prof. Ji Chen Notes 6 Transmission Lines (Time Domain) ECE Spring 2014

Note: Transmission lines is the subject of Chapter 6 in the Shen & Kong book. However, the subject of wave propagation in the time domain is not treated very thoroughly there. Chapter 10 of the Hayt book has a more thorough discussion. Note about Notes 6 2

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 lead a b z A transmission line is normally used in the balanced mode, meaning equal and opposite currents (and charges) on the two conductors. 3

Transmission Lines (cont.) Coaxial cable Here’s what they look like in real-life. Twin lead 4

Transmission Lines (cont.) 5 CAT 5 cable (twisted pair)

Transmission 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” (e.g., CAT 5 cable). This results in less interference. ll -l-l E Coax Twin lead E 6

Another common example (for printed circuit boards): Microstrip line w h rr Transmission Lines (cont.) 7

Transmission lines are commonly met on printed-circuit boards. A microwave integrated circuit (MIC) Microstrip line 8 Transmission Lines (cont.)

Microstrip line Transmission Lines (cont.) 9

Microstrip h w rr rr w Stripline h Transmission lines commonly met on printed-circuit boards Coplanar strips h rr w w Coplanar waveguide (CPW) h rr w 10 Transmission Lines (cont.)

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

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.) 12

Circuit Model Circuit Model: zz 13 z zz + - RzRz LzLz GzGz CzCz z

Coaxial Cable Example (coaxial cable) a b z (skin depth of metal)  d = conductivity of dielectric [ S/m ]  m = conductivity of metal [ S/m ] 14

Coaxial Cable (cont.) Overview of derivation: capacitance per unit length 15 ll -l-l E a b

Coaxial Cable (cont.) Overview of derivation: inductance per unit length 16 y E x JsJs         b a

Coaxial Cable (cont.) Overview of derivation: conductance per unit length RC Analogy: 17 a b

Coaxial Cable (cont.) Relation Between L and C : Speed of light in dielectric medium: Hence: This is true for ALL transmission lines. 18 ( A proof will be seen later.)

Telegrapher’s Equations Apply KVL and KCL laws to a small slice of line: + V (z,t) - RzRz LzLz GzGz CzCz I (z,t) + V (z+  z,t) - I (z +  z,t) z z z+  z 19

Hence Now let  z  0: “Telegrapher’s Equations (TEs)” Telegrapher’s Equations (cont.) 20

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. Telegrapher’s Equations (cont.) 21

The same equation also holds for i. Hence, we have: There is no exact solution to this differential equation, except for the lossless case. Telegrapher’s Equations (cont.) 22

The same equation also holds for i. Lossless case: Note: The current satisfies the same differential equation. Telegrapher’s Equations (cont.) 23 “wave equation”

The same equation also holds for i. Hence we have Solution: where f and g are arbitrary functions. Solution to Telegrapher's Equations This is called the D’Alembert solution to the wave equation (the solution is in the form of traveling waves). 24

Traveling Waves Proof of solution It is seen that the differential equation is satisfied by the general solution. General solution: 25

Example (square pulse): 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) 26 z f (z) z0z0 t = 0 Traveling Waves (cont.) “snapshots of the wave”

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) 27 z g (z) z0z0 t = 0 Traveling Waves (cont.) Example (square pulse): “snapshots of the wave”

Loss causes an attenuation in the signal level, and it also causes distortion (the pulse changes shape and usually gets 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.) 28

Current Lossless (first Telegrapher’s equation) Our goal is to now solve for the current on the line. Assume the following forms: The derivatives are: 29

Current (cont.) This becomes Equating like terms, we have Hence we have 30 Note: There may be a constant of integration, but this would correspond to a DC current, which is ignored here.

Then Define the characteristic impedance Z 0 of the line: The units of Z 0 are Ohms. Current (cont.) Observation about term: 31

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. Current (cont.) 32

Current (cont.) z Picture for a forward-traveling wave: + - forward-traveling wave 33

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.) 34

Coaxial Cable Example: Find the characteristic impedance of a coax. a b z 35

Coaxial Cable (cont.) (intrinsic impedance of free space) 36 a b z

Twin Lead d a = radius of wires 37

Twin Line (cont.) Coaxial cable Twin line These are the common values used for TV [  ] transformer 300 [  ] twin line 75 [  ] coax 38

w h rr Microstrip Line Parallel-plate formulas: 39

Microstrip Line (cont.) More accurate CAD formulas: Note: The effective relative permittivity accounts for the fact that some of the fields are outside of the substrate, in the air region. The effective width w' accounts for the strip thickness. t = strip thickness 40

Some Comments  Transmission-line theory is valid at any frequency, and for any type of waveform (assuming an ideal straight length of 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 single propagation, reflection, distortion, etc. 41 One thing that transmission-line theory ignores is the effects of discontinuities (e.g. bends). These may cause reflections and possibly also radiation at high frequencies, depending on the type of line.