Digital Signal Processing Reference
In-Depth Information
have an impedance of 75
per meter. This is different from the resistance, where
the transmission line length does matter. For example, a trace might have a resis-
tance of 0.1
Ω
/inch.
The delay (
tpd
) has units of time (typically picoseconds or nanoseconds per
unit of length. A 10-inch-long trace having a delay of 180 ps per inch has a total
delay of 1.8 ns.
In Section 6.8 and the Problems we explore why the impedance does not de-
pend on the lines length but the time delay does.
Equations (6.1) and (6.2) are only true when the effect of the resistance (
R
) is
very small compared to the effect of the inductance (
L
), and when the capacitor
losses (
G
) are very small compared to the effects of the capacitance (
C
). In fact,
these equations calculate the lossless impedance and lossless delay time because the
losses in the conductor (represented by
R
) and in the capacitor (represented by
G
)
are assumed to be small enough to drop out of the general equations for impedance
and delay. This is a good assumption for circuit board transmission lines carrying
signals with rise times of about 10 nS or faster [7].
For instance, the effect of the inductance is some 25 times higher than the ef-
fect of the series resistor for a pulse having a 1-ns rise time traveling down a 50
Ω
Ω
,
5-mil (127
m)-wide half-ounce stripline trace on FR4. In contrast, the effect of the
inductance is only about eight times larger than the resistance when the rise time
increases to 10 ns.
μ
6.4.1 Units and Electrical Length
The units of length associated with the values for
L
and
C
are important when using
these equations. For instance, field solvers usually give these values per unit length
(sometimes referred to as PUL), such as picofarads per centimeter or nanohenry per
inch. Total values for the entire length of the line are also sometimes given.
As an example, a 50
transmission line might have a PUL capacitance of 3 pF
per inch and an inductance of 6.5 nH/inch. A 10-inch-long line would have a total
capacitance and inductance equal to the PUL value times the total length: 30 pF
and 65 nH.
The electrical length of a transmission line having PUL values of 3.5 pF and 6.4
nH is 150 ps. It increases to 1.5 ns when the PUL values are 10 times as large. We
will see the advantage of looking at transmission lines in this way later on in this
chapter and in Chapters 11 and 12 when we discuss reflections and terminations.
Ω
6.5
How Does a Signal Travel Down the Line?
Because we took results from circuit theory rather than field theory, we have not
seen that energy travels as waves along transmission lines. We can develop an intui-
tive understanding of this by thinking about the transmission line circuit shown in
Figure 6.4 in the following way [8].
The circuit shows the resistance, inductance, and capacitance of a microstrip
over a return path such as a ground plane. For simplicity, the capacitor losses (the
G
element in Figure 6.3) are not included.
