Distributed rc Model

Distributed rc Model

1. Distributed rc Model
Dr. Varun Kumar
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 1 / 9

2. Outlines
1
Lumped RC Model vs Distributed rc Model
2
RC Tree vs RC Ladder
3
Eect of Length of Wire on Delay and Time Constant
4
Distributed rc Line
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 2 / 9

3. Lumped RC Model vs Distributed rc Model
)The equipotential assumption in the lumped-capacitor model is no
longer adequate.
)Aresistive-capacitivemodel 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.
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 3 / 9

4. RC Tree vs RC Ladder
Elmore delay:
Di
=R1C1+R1C2+ (R1+R3)C3+ (R1+R3)C4+ (R1+R3+Ri)Ci
DN
=
N
X
i=1
Ci
N
X
j=1
Ri=R1C1+C2(R1+R2) +:::+CN(R1+R2+::+RN)
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 4 / 9

5. Eect of Length of Wire on Delay and Time Constant
)=RC!(Time constant)!Simple RC network.
)!Time required to charge 0 to 63.2% of the value on applied DC
voltage and discharge to 36.8% from its initial value.
Eect of length of wire:
)Let RC ladder model is taken andR1=R2=:::RN=R
)The wire with a total length is L.
)It is partitioned into N parts.
)Resistance and capacitance per unit length, i.er=
R
N
andc=
C
N
.
)Overall delay
DN
=

L
N

2
(rc+ 2rc+:::+Nrc) =
N+ 1
2N
L
2
rc
Note:The delay of in distributed rc-line is half of the delay in lumped RC
model.
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 5 / 9

6. Distributed rc Line
)Lumped RC model:
)A
Figure:
L!Total length of the wire.
r!Resistance per unit length.
c!Capacitance per unit length.
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 6 / 9

7. Continued{
Figure:
cL
@Vi
@t
=
(Vi+1Vi)(ViVi1)
rL
The correct behavior of the distributed rc line is obtained by reducing L!0.
Diusion equation:
rc
@V
@t
=
@
2
V
@x
2
)V!Voltage at a particular point in the wire
)x!Distance between reference point and the signal source.
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 7 / 9

8. Diusion Equation Interpretation
rc
@V
@t
=
@
2
V
@x
2
Dr. Varun Kumar(IIIT Surat) IIIT Surat-Lecture-4 8 / 9

9. Comparison between Lumped RC
)No closed-form solution exists for this equation.
)These equations are dicult 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-4 9 / 9