1. CSE 464 Transmission Line Equations (Dally/Poulton Notation) January 2009 Based on Notes by Fred U. Rosenberger David M. Zar Washington University in St. Louis dzar@wustl.edu

2. T-Line Equations 2 - David M. Zar - 11/18/2015 Transmission Line Equations Superposition (if linear)

3. T-Line Equations 3 - David M. Zar - 11/18/2015 General Equations For Parallel Plates:

4. T-Line Equations 4 - David M. Zar - 11/18/2015 Transmission Line Equations Z0Z0 ZsZs ZTZT V0V0 +-+- +-+- VsVs L 0 +-+- VlVl (Z S -Z 0 )/(Z S + Z 0 )= K rS (Z T -Z 0 )/(Z T + Z 0 )= K rT

5. T-Line Equations 5 - David M. Zar - 11/18/2015 General Equations (V T = 0) Use Superposition for Multiple Sources 1) 2) 2) into 1), 4) 3) 5) At arbitrary x, From 2) & 3), 2 l since we have to go “there and back” to see it at the start of the line

6. T-Line Equations 6 - David M. Zar - 11/18/2015 Equations Continued Repeat 3) into 4), 3) into 5), At x = 0, Now can find V 0, V l at any time. At x = l,

7. T-Line Equations 7 - David M. Zar - 11/18/2015 Example Driving T-Line Apply to problem, Z0Z0 ZsZs ZTZT This is the time to go from l  0  l for the last (N) time.

8. T-Line Equations 8 - David M. Zar - 11/18/2015 Steady State Value and Error Term, Example With algebra, Attenuation in steady state Error0 as N For We get

9. T-Line Equations 9 - David M. Zar - 11/18/2015 Example Continued n For 2 foot line, this could be on the order of 85ns (or even 3.55 ns for a 1 inch line!) n What to do? »Wait a long time »Real wire will have some resistance (Z s ≠0) » CS solution : use 1900 Ohm transmission Line »Reduce resistance of receiver to match t-line –Increased power –May be a problem with driver resistance (e.g. 10 to 50 Ohms) »Increase resistance of driver to match t-line –100 Ohms in series with driver gives 2.5% error after only l/v, or after 2ns for one foot line (this is the minimum possible)

10. T-Line Equations 10 - David M. Zar - 11/18/2015 Transmission Line Equations for Wire r r2r2 r1r1 s Wire over ground plane (warning, not accurate for r close to s) Between 2 Wires: 1/2 capacitance of wire over ground plane

11. T-Line Equations 11 - David M. Zar - 11/18/2015 PC Board Equations w s Parallel Plates (ignoring fringing)

12. T-Line Equations 12 - David M. Zar - 11/18/2015 PC Board Equations w t hdielectric air w t h2=h1 b Stripline Microstrip h1 (Careful here, approximations grossly wrong for some values: large w/h)