IEEE Solid-States Circuits Magazine - Summer 2019 - 7

C IRCU IT INTU ITIONS

Ali Sheikholeslami

The Electrical Length of a Wire

W

Welcome to the 21st article in the
"Circuit Intuitions" column series.
As the title suggests, each article
provides insights and intuitions into
circuit design and analysis. These
articles are aimed at undergraduate
students but may serve the interests
of other readers as well. If you read
this article, I would appreciate your
comments and feedback as well as
your requests and suggestions for
future articles in this series. Please
e-mail me your comments: ali@ece
.utoronto.ca.
We use wires to connect various
electronic building blocks, for example, to connect microchips, such as
microprocessors and memories, on a
printed circuit board. These wires are
expected to transfer a signal from one
point on the board to another, without changing the signal. However,
there is always a delay and attenuation associated with this transfer.
A delay occurs because the signal
cannot travel at an infinite speed; in
fact, it cannot travel faster than the
speed of light. This is true even if
one uses a superconducting material
for the wire. The attenuation occurs
because the wire (which may be
made of copper) has a nonzero resistance and parasitic capacitances and
inductances, and these will attenuate
the signal (in addition to delaying it)
at high frequencies.
As a result, a piece of wire may
alter the signal at high frequencies
with a transfer function that is not
unity, and so it may not be considDigital Object Identifier 10.1109/MSSC.2019.2922885
Date of publication: 27 August 2019

form, the Tx sends binary data to the
channel, and the Rx reliably detects
the data at the other end. As an exam+
+
vRx (t) Rx
Tx vTx (t)
ple, consider a binary signal at 40 Gb/s
-
-
being sent over a 1-cm piece of wire.
At this speed, one bit period is 25 picoseconds (ps) in duration; that is, the
FIGURE 1: A simplified block diagram of a
period during which the Tx maintains
high-speed wireline transceiver.
a constant voltage corresponding to a
ered a short circuit. In this article,
digital zero or one is only 25 ps. Howwe look at the circumstances under
ever, the time of flight, which is the
which a wire can and cannot be
time it takes for this signal to propadeemed a short circuit. We conclude
gate through 1 cm of wire to arrive
intuitively that we need to design
at the Rx, is about 50 ps (this is close
circuits to compensate for the sigto the time it takes for light to travel
nal attenuation of the wire.
the same distance).
Figure 1 shows a simplified block
Figure 2(a) presents a case where
diagram of what is known as a wirewe send a digital one at t = 0 ps, followed by a digital zero at t = 25 ps.
line transceiver [transmitter (Tx)
Because the time of flight is 50 ps,
and receiver (Rx)]. In its simplest
this sequence arrives at
the Rx 50 ps later, as seen
in Figure 2(b). The voltv Tx (t )
age along the 1-cm wire
1V
at exactly 50 ps, v ^x h,
where x is the distance
t
from the Tx, looks like
0
25 ps
50 ps
the waveform pictured
(a)
in Figure 2(c). Note that,
v Rx (t )
while the waveforms in
1V
Figure  2(a) and (b) relate
to voltages as functions
t
0
25 ps
50 ps
75 ps
100 ps
of time (at the ends of the
(b)
wire), the waveform in
v (x )
Figure 2(c) is a voltage as
1V
a function of "distance"
from the Tx. In other
x (Distance From Tx)
words, Figure 2(c) gives a
0
5 mm
1 cm
snapshot of the voltage
(c)
on the wire (at one instant of time, t = 50 ps) .
FIGURE 2: The voltage waveform at (a) the Tx and (b) the
This figure clearly shows
Rx. (c) The voltage as a function of distance from the Tx at
t = 50 ps.
that the voltage along
Channel

IEEE SOLID-STATE CIRCUITS MAGAZINE

SU M M E R 2 0 19

7
IEEE Solid-States Circuits Magazine - Summer 2019

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