This document discusses different electrical wire models used to analyze circuit behavior. It begins by introducing lumped models that simplify distributed parasitic elements into single circuit components. A common lumped model is the RC model, which approximates a wire's distributed resistance and capacitance. For long wires, a distributed RC model more accurately captures the wire's continuous resistance and capacitance per unit length. The document concludes by comparing lumped and distributed RC wire models.

1. Electrical Wire Model
Dr. Varun Kumar
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2. Outlines
1 Electrical Wire Models
2 The Lumped Model
3 The Lumped RC Model
4 The Distributed rc Line
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3. Electrical Wire Models
⇒ Parasitic elements (resistance, capacitance, inductance) have an
impact on the electrical behavior of the circuit.
⇒ It influence the delay, power dissipation, and circuit reliability.
⇒ To study these effects requires
⇒ Electrical models that estimate the real behavior of the wire as a
function of its parameters.
The Ideal Wire
⇒ A voltage change at one end of the wire propagates immediately to
its other ends.
⇒ The same voltage is present at every segment of the wire at the every
point in time.
⇒ Ideal wire concept was the early phase of design process.
⇒ Wire parasitics play a vital role for more complex models.
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4. The Lumped Model
⇒ The circuit parasitics of a wire are distributed along its length and are
not lumped into a single position.
⇒ For better circuit analysis, it is useful to lump the different fractions
into a single circuit element.
⇒ Advantage of this approach
The effects of parasitic can be described by an ordinary differential
equation.
⇒ Requirement for analysis
The description of a distributed element requires partial differential
equations.
⇒ Dominant component selection
When wire length is small and larger cross sectional area, the resistive
component does not dominate.
When switching frequencies are in the low to medium range, the
capacitive component of the wire dominate.
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5. Continued–
⇒ Distributed parameter to single one
Lump the distributed capacitance into a single capacitor, because
capacitor is dominating.
⇒ Quality of this capacitive lumped model
Simple
Effective
Model of choice for the analysis of most interconnect wires in digital
integrated circuits.
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6. The Lumped RC model
⇒ The equipotential assumption in the lumped-capacitor model is no
longer adequate.
⇒ A resistive-capacitive model has to be adopted.
⇒ Approach for analysis (Short wire)
Lumps the total wire resistance of each wire segment into one single R
Combines the global capacitance into a single capacitor C
This simple model, called the lumped RC model
Note: Inaccurate for long interconnect wires.
⇒ Challenges and approach for analysis (Long wire)
The distributed rc-model is complex and no closed form solutions exist.
The distributed rc-line can be adequately modeled by a simple RC
network.
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7. RC Tree
⇒ The network has a single input node.
⇒ All the capacitors are between a node and the ground
⇒ The network does not contain any resistive loops (which makes it a
tree)
Elmore delay:
τDi
= R1C1 + R1C2 + (R1 + R3)C3 + (R1 + R3)C4 + (R1 + R3 + Ri )Ci
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8. The Distributed rc Line
⇒ The lumped RC model is a pessimistic model for a resistive-capacitive
wire.
⇒ A distributed rc model is more appropriate.
Figure: Distributed model
L → Total length of the wire.
r → Resistance per unit length.
c → Capacitance per unit length.
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9. Continued–
Figure: Schematic symbol for distributed RC line
c∆L
∂Vi
∂t
=
(Vi+1 − Vi ) − (Vi − Vi−1)
r∆L
⇒ No closed-form solution exists for this equation.
⇒ These equations are difficult to use for ordinary circuit analysis.
⇒ Distributed rc line can be approximated by a lumped RC ladder network.
Dr. Varun Kumar (IIIT Surat) IIIT Surat-Lecture-3 9 / 9