08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 1 Transmission Lines Transmission line effects must be considered when length is comparable to ¼ wavelength We will ignore the energy loss on transmission lines Concentrate on time-domain description rather than frequency domain egeg + - zgzg zlzl i i e xd l

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 2 Transmission Lines C:Capacitance / Unit Length [F/m] L:Inductance / Unit Length [H/m] R:Conductor Resistance / Unit Length [Ω/m] G:Insulation Conductance /Unit Length [ /m] Note G ≠ 1/R ! Ω

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 3 Transmission Lines By KVL: In the limit as :

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 4 Transmission Lines By KCL: In the limit as :

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 5 Lossless Case Take partial derivative w.r.t. x in (1) and partial derivative w.r.t. t in (2), then substitute. Recognize as wave equations

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 6 Lossless Case Show that the solution is in the form.

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 7 Interpretation is a wave traveling to right with velocity is a wave traveling to left with velocity Solving for i, we obtain is characteristic of line

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 8 Interpretation Note: Each traveling wave direction, e and i are related by z 0.

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 9 Reflections Look at terminations with real impedances  frequency independent Wave of voltage and current traveling to right At termination zlzl z0z0 e - + elel + - ilil

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 10 Reflections Hence, there must be reflected waves e - and i - such that In terms of voltage Reflection coefficient

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 11 Reflections Special cases: –z l = z 0  k=0 Matched, no reflections. Line looks infinite. –z l = 0, short circuit  k = -z 0 /z 0 = -1 –z l =, open circuit  k = 1 8

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 12 Multiple Reflections R=3z 0 z0z0 t=0 x=0 x=l E Hitting load, a wave of E/2 is produced E t=.3T E t=1.3T 3E/2 E t=2.3T 3E/2 Arriving wave of E/2 is reflected toward load T2T3T4T5T Voltage at load 1.5E.75E 1.125E.9375E 6T 7T E

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 13 Time-Space (Bounce) Diagram Mark reflection coefficients Write initial voltages Write wave amplitudes Update –Wave amplitudes when reflected –Voltages as waves cross x k 0 =-1k l =.5 e=E e=0 e=E E E/2 -E/2 -E/4 E/4 E/8 T 2T 3T 4T 5T e=3E/2 e=3E/4 e=9E/8 Time

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 14 Example 3z 0 E z0z0 Magnitude of the first wave: x k 0 =.5k l =1 e=E/4 e=0 e=5E/8 e=13E/16 E/4 E/8 E/16 T 2T 3T 4T 5T e=E/2 e=3E/4 e=7E/8 Time z0z0 t=0 k 0 =.5 E R=3z 0 l k l =1 E t At x=0 T2T3T 4T 5T

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 15 Reflections in Digital Lines Consider one source and one load Option 1: Do not terminate either end. Ringing will stop eventually. –Pro:Simple, no additional power loss –Con:Limited speed

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 16 Reflections in Digital Lines Option 2: Matched termination at the end –Pro:No reflections –Con:Excessive power consumption For z 0 =150Ω, power consumption 135mW Reduce by duty factor (0.5 for regular lines, 0.05 for floppy drives) Multiply by number of lines z0z0 Open-collector driver

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 17 Reflections in Digital Lines Option 3: Matched termination at the source end –Pro:Can be run at the same speed as load termination If receiver has very high input impedance, full voltage appears at the receiver No power dissipated at constant voltage level –Con:Special, high input impedance line receivers required (not suited for standard TTL) Look at multiple terminations z0z0

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 18 Reflection in Digital Lines Problem for multiple receivers –Assume the ideal case, where taps are infinitely short and have infinite impedances –Even for this case, intermediate taps do not get full signal immediately z0z0 E x k 0 =0k l =1 e=.5E e=0 E/2 T 2T e=E E t At x=.5L.5TT1.5T 2T

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 19 Reflection and Transmission at Junctions e 1 +, i 1 + e 1 -, i 1 - e 2 +, i 2 + e 3 +, i 3 +

08 - Winter 2005 ECE ECE 766 Computer Interfacing and Protocols 20 Reflection and Transmission at Junctions 3/2 L 2/3 L L z0z0 t=0 T 2T 3T 4T k=0 k=1 E/2 -E/6 2E/9 E/3 2E/9 E/3 -E/9 Many multiple reflections, eventually come to rest