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ECE 124a/256c Transmission Lines as Interconnect

ECE 124a/256c Transmission Lines as Interconnect. Forrest Brewer Displays from Bakoglu, Addison-Wesley. Interconnection. Circuit rise/fall times approach or exceed speed of light delay through interconnect Can no longer model wires as C or RC circuits Wire Inductance plays a substantial part

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ECE 124a/256c Transmission Lines as Interconnect

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  1. ECE 124a/256cTransmission Lines as Interconnect Forrest Brewer Displays from Bakoglu, Addison-Wesley

  2. Interconnection • Circuit rise/fall times approach or exceed speed of light delay through interconnect • Can no longer model wires as C or RC circuits • Wire Inductance plays a substantial part • Speed of light: 1ft/nS = 300um/pS

  3. Fundamentals • L dI/dt = V is significant to other effects • Inductance: limit on the rate of change of the current • E.g. Larger driver will not cause larger current to flow initially R L G C

  4. v V +I -I dx Lossless Tranmission R=G=0 • Step of V volts propagating with velocity v • Initially no current flows – after step passes, current of +I • After Step Voltage V exists between the wires

  5. Lossless Tranmission R=G=0 • Maxwell’s equation: • B is field, dS is the normal vector to the surface • F is the flux • For closed surface Flux and current are proportional

  6. Lossless Tranmission R=G=0 • For Transmission Line I and F are defined per unit length • At front of wave: • Faraday’s Law: V = dF/dt = IL dx/dt = ILv • Voltage in line is across a capacitance Q=CV • I must be: • Combining, we get: • Also:

  7. Units • The previous derivation assumed e = m = 1 • In MKS units: • c is the speed of light = 29.972cm/nS • For typical IC materials mr = 1 • So:

  8. Typical Lines • We can characterize a lossless line by its Capacitance per unit length and its dielectric constant.

  9. Circuit Models

  10. Circuit Models II • Driver End • TM modeled by resistor of value Z • Input voltage is function of driver and line impedance: • Inside Line • Drive modeled by Step of 2Vi with source resistance Z • Remaining TM as above (resistor) • Load End • Drive modeled by Step of 2Vi with source resistance Z • Voltage on load of impedance ZL:

  11. Discontinuity in the line (Impedance) • Abrupt interface of 2 TM-lines • Incident wave: Vi = Ii Z1 Reflected Wave: Vr = IrZ1 • Transmitted Wave: Vt = ItZ2 • Conservation of charge: Ii = Ir + It • Voltages across interface: Vi + Vr = Vt • We have:

  12. Reflection/Transmission Coefficients • The Coefficient of Reflection G = Vr/Vi • Vi incident from Z1 into Z2 has a reflection amplitude Gvi • Similarly, the Transmitted Amplitude = 1+G

  13. Inductive and Capacitive Discontinuities

  14. Typical Package Pins

  15. Discontinuity Amplitude • The Amplitude of discontinuity • Strength of discontinuity • Rise/Fall time of Impinging Wave • To first order Magnitude is: • Inductive: Capacitive:

  16. Critical Length (TM-analysis?) • TM-line effects significant if: tr < 2.5 tf • Flight time tf = d/v

  17. Un-terminated Line Rs = 10Z

  18. Un-terminated Line Rs = Z

  19. Un-terminated Line Rs = 0.1Z

  20. Unterminated Line (finite rise time) • Rise Time Never Zero • For • Rs>Zo, tr>tf • Exponential Rise • Rs<Z0, • “Ringing” • Settling time can be much longer than tf.

  21. Line Termination (None)

  22. A B V VA V VB tF Line Termination (End) Z G = 0 R = Z V

  23. Line Termination: (Source)

  24. Piece-Wise Modeling • Create a circuit model for short section of line • Length < rise-time/3 at local propagation velocity • E.G. 50mm for 25pS on chip, 150nm wide, 350nm tall • Assume sea of dielectric and perfect ground plane (this time) • C = 2.4pF/cm = 240fF/mm = 12fF/50mm • L = 3.9/c2C = 1.81nH/cm = 0.181nH/mm = 9.1pH/50mm • R = r L/(W H) = 0.005cm*2.67mWcm/(0.000015cm*0.000035cm) = 25W/50mm 25W 9.1p 12f 200mm:

  25. Lossy Transmission • Attenuation of Signal • Resistive Loss, Skin-Effect Loss, Dielectric Loss • For uniform line with constant R, L, C, G per length:

  26. Conductor Loss (Resistance)

  27. Conductor Loss (step input) • Initial step declines exponentially as Rl /2Z • Closely approximates RC dominated line when Rl >> 2Z • Beyond this point, line is diffusive • For large resistance, we cannot ignore the backward distrbitued reflection

  28. Conductor Loss (Skin Effect) • An ideal conductor would exclude any electric or magnetic field change – most have finite resistance • The depth of the field penetration is mediated by the frequency of the wave – at higher frequencies, less of the conductor is available for conducting the current • For resistivity r (Wcm), frequency f (Hz) the depth is: • A conductor thickness t > 2d will not have significantly lower loss • For Al at 1GHz skin depth is 2.8mm

  29. Skin Effect in stripline (circuit board) • Resistive attenuation: • At high frequencies:

  30. Dielectric Loss • Material Loss tangent: • Attenuation:

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