ECE 124a256c Transmission Lines as Interconnect

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ECE 124a/256cTransmission Lines as Interconnect

• Forrest Brewer
• Displays from Bakoglu, Addison-Wesley

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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

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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
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Lossless Tranmission RG0

• 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

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Lossless Tranmission RG0

• Maxwells equation
• B is field, dS is the normal vector to the
  surface
• F is the flux
• For closed surface Flux and current are
  proportional

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Lossless Tranmission RG0

• For Transmission Line I and F are defined per
  unit length
• At front of wave
• Faradays Law V dF/dt IL dx/dt ILv
• Voltage in line is across a capacitance QCV
• I must be
• Combining, we get
• Also

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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

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Typical Lines

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

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Circuit Models
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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

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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

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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 1G

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Inductive and Capacitive Discontinuities
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Typical Package Pins
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Discontinuity Amplitude

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

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Critical Length (TM-analysis?)

• TM-line effects significant if tr lt 2.5 tf
• Flight time tf d/v

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Un-terminated Line Rs 10Z
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Un-terminated Line Rs Z
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Un-terminated Line Rs 0.1Z
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Unterminated Line (finite rise time)

• Rise Time Never Zero
• For
• RsgtZo, trgttf
• Exponential Rise
• RsltZ0,
• Ringing
• Settling time can be much longer than tf.

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Line Termination (None)
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Line Termination (End)
Z
R Z
G 0
V
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Line Termination (Source)
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Piece-Wise Modeling

• Create a circuit model for short section of line
• Length lt 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)
• 0.005cm2.67mWcm/(0.000015cm0.000035cm)
  25W/50mm

25W
9.1p
12f
200mm
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Lossy Transmission

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

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Conductor Loss (Resistance)
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Conductor Loss (step input)

• Initial step declines exponentially as Rl /2Z
• Closely approximates RC dominated line when Rl gtgt
  2Z
• Beyond this point, line is diffusive
• For large resistance, we cannot ignore the
  backward distrbitued reflection

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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 gt 2d will not have
  significantly lower loss
• For Al at 1GHz skin depth is 2.8mm

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Skin Effect in stripline (circuit board)

• Resistive attenuation
• At high frequencies

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Dielectric Loss

• Material Loss tangent
• Attenuation