Transmission Lines Transmission line effects must be considered

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 + eg - i zg x Winter 2005 ECE zl e i l ECE 766 Computer Interfacing and Protocols d 08 - 1

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

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

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

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 Winter 2005 ECE 766 Computer Interfacing and Protocols 08 - 5

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

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 Winter 2005 ECE 766 Computer Interfacing and Protocols 08 - 7

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

Reflections z 0 + e - + il el zl - • Look at terminations with real impedances frequency independent • Wave of voltage and current traveling to right • At termination Winter 2005 ECE 766 Computer Interfacing and Protocols 08 - 9

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

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

Multiple Reflections t=0 R=3 z 0 E x=0 x=l t=. 3 T E t=1. 3 T E 3 E/2 t=2. 3 T Voltage at load Hitting load, a wave of E/2 is produced Arriving wave of E/2 is reflected toward load 1. 5 E 1. 125 E E T Winter 2005 ECE . 9375 E 2 T 3 T 4 T ECE 766 Computer Interfacing and Protocols 5 T 6 T 7 T 08 - 12

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

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

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 Winter 2005 ECE 766 Computer Interfacing and Protocols 08 - 15

Reflections in Digital Lines • Option 2: Matched termination at the end z 0 Open-collector driver – Pro: No reflections – Con: Excessive power consumption For z 0=150Ω, power consumption 135 m. W Reduce by duty factor (0. 5 for regular lines, 0. 05 for floppy drives) Multiply by number of lines Winter 2005 ECE 766 Computer Interfacing and Protocols 08 - 16

Reflections in Digital Lines • Option 3: Matched termination at the source end z 0 – Pro: – Con: Winter 2005 ECE 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 Special, high input impedance line receivers required (not suited for standard TTL) Look at multiple terminations ECE 766 Computer Interfacing and Protocols 08 - 17

Reflection in Digital Lines • Problem for multiple receivers z 0 E – 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 k 0=0 T e=. 5 E x kl=1 E/2 2 T Winter 2005 ECE e=0 E At x=. 5 L e=E. 5 T ECE 766 Computer Interfacing and Protocols T 1. 5 T 2 T 08 - 18 t

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

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