6.976 High Speed Communication Circuits and Systems Lecture 2 Transmission Lines

Transcription

1 6.976 High Speed Communication Circuits and Sstems Lecture 2 Transmission Lines Michael Perrott Massachusetts Institute of Technolog Copright 2003 b Michael H. Perrott

2 Mawell s Equations General form: Assumptions for free space and transmission line propagation - No charge buildup ρ = 0 - No free current J = 0 Note: we ll onl need Equations 1 and 2

3 Assumptions Orientation and direction - E field is in -direction and traveling in z-direction - H field is in -direction and traveling in z-direction - In freespace: z For transmission line (TEM mode) direction of travel H E b direction of travel H E a z

4 Solution Fields change onl in time and in z-direction - Assume comple eponential solution

5 Solution Fields change onl in time and in z-direction - Assume comple eponential solution Implications:

6 Solution Fields change onl in time and in z-direction - Assume comple eponential solution Implications: But, what is the value of k?

7 Evaluate Curl Operations in Mawell s Formula Definition

8 Evaluate Curl Operations in Mawell s Formula Definition Given the previous assumptions

9 Now Put All the Pieces Together Solve Mawell s Equation (1)

10 Now Put All the Pieces Together Solve Mawell s Equations (1) and (2)

11 Now Put All the Pieces Together Solve Mawell s Equations (1) and (2)

12 Connecting to the Real World Current solution is comple But the following comple solution is also valid And adding them together is also a valid solution that is now real-valued

13 Calculating Propagation Speed The resulting cosine wave is a function of time AND position E (z,t) direction of travel t z z Consider riding one part of the wave Velocit calculation

14 Freespace Values Constants Impedance Propagation speed Wavelength of 30 GHz signal

15 Voltage and Current Definitions: b H E a z I w H t E b a

16 Parallel Plate Waveguide E-field and H-field are influenced b plates b H E a z

17 Current and H-Field Assume that (AC) current is flowing I b H E a z I

18 Current and H-Field Current flowing down waveguide influences H-field I b H E a z I H

19 Current and H-Field Flu from one plate interacts with flu from the other plate I b H E a z I

20 Current and H-Field Approimate H-Field to be uniform and restricted to lie between the plates I b H E a z I b a

21 Voltage and E-Field Approimate E-field to be uniform and restricted to lie between the plates J b H E a z J b V E a

22 Back to Mawell s Equations From previous analsis These can be equivalentl written as Where

23 Wave Equation for Transmission Line (TEM) Ke formulas Substitute (2) into (1) Characteristic impedance (use Equation (1))

24 Connecting to the Real World Current solution is comple But the following solution is also valid And adding them together is also a valid solution

25 Calculating Propagation Speed The resulting cosine wave is a function of time AND position direction of travel t E (z,t) z z Consider riding one part of the wave Velocit calculation

26 Integrated Circuit Values Constants Impedance (geometr dependant) Propagation speed (geometr independent) Wavelength of 30 GHz signal in silicon dioide

27 LC Network Analog of Transmission Line (TEM) LC network analog L L L L Z in C C C Calculate input impedance

28 How are Lumped LC and Transmission Lines Different? In transmission line, L and C values are infinitel small - It is alwas true that L L L L Z in C C C For lumped LC, L and C have finite values - Finite frequenc range for

29 Loss Transmission Lines Practical transmission lines have losses in their conductor and dielectric material - We model such loss b including resistors in the LC model R L R L R L R L Z in 1/G C 1/G C 1/G C The presence of such losses has two effects on signals traveling through the line - Attenuation - Dispersion (i.e., bandwidth degradation) See Chapter 5 of Thomas Lee s book for analsis