# Reflections and transmission lines

So what makes transmission lines so special? Consider a long, straight wire or trace with its return wire or trace nearby. The wire has some inductance along its length. There is also some capacitive coupling between the wire and its return (see Figure 10-2). Figure 10-2 shows what we call a lumped model of the wire pair. It is called a lumped model because we show the capacitors and inductors as individual, lumped components. In reality, the inductance and capacitance are spread continuously along the wires. We don’t know how to show that in a drawing, so we approximate it with a lumped approximation. On the other hand, this is fine because we are going to think of this wire as being infinitely long. Therefore, the part shown in Figure 10-2 is just an infinitely small part of the total length.

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*Figure 10-2: A transmission line is made up of an infinitely long network of capacitors and inductors.*

Now, if these wires are infinitely long, there will be no reflection. At least if there is a reflection, it will take an infinitely long time for it to return, so we can assume it doesn’t exist.

It takes at
least one other thing to avoid reflections. The wires must be absolutely
uniform. The little Ls and Cs must be identical everywhere along the
wire. If the wires are not uniform, then we can consider them to be
multiple sets of wires, each with different characteristics. Individual
sets of wires will not be infinitely long, and therefore there will be
reflections from each individual set. Therefore, the way to avoid
reflections is to use an infinitely long, absolutely uniform wire or
trace pair. We give such a wire or trace pair the special name *transmission line*.

It
can be shown mathematically that if we look into the front of this wire
pair there is an input impedance we can calculate. We give it the
symbol Zo, and call it the *intrinsic impedance* of the line. If we
could calculate the “lumped” values of inductance (L) and capacitance
(C ), the impedance would be calculated as shown in Equation 10-1:

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*Equation 10-1*

Now
here is a clever twist. If you look into the transmission line at the
front, it looks like it has an impedance of Zo. Let’s take our
infinitely long transmission line and break it in two parts. If we look
into the second part, it also looks like an infinitely long transmission
line with an input impedance of Zo. What if we simply replaced the
second part of the line with an impedance equal to Zo (see Figure 10-3)?
From the front of the first line, it still looks exactly like an
infinitely long line. It turns out it behaves that way, too. *A
transmission line of finite length, terminated in its characteristic
impedance, Zo, looks like an infinitely long transmission line.*
Therefore, even though it has a finite length, it still will have no
reflections. In this case it is not the infinite length that makes
reflections irrelevant. It is the fact that the energy traveling down
the line is exactly absorbed (dissipated) in the termination. There is
no energy left to reflect back. The net effect is the same thing. There
are no reflections to worry about.

* *

*Figure 10-3: If we terminate a finite-length transmission line in its characteristic impedance, it looks infinitely long.*

These are the two actions we need to take to control reflections on PCBs:

1. We need to make our traces look like transmission lines.

2. We need to terminate them in their characteristic impedances, Zo.

Of course, we can also make our traces short enough that reflections don’t interfere with the receiver’s ability to hear the signal in the first place, or we can slow down the rise time enough that the receiver can hear through the echoes.

There are certain types of transmission lines that are commonly used around us everyday. The coaxial cable leading to our cable TV is a 75-Ω transmission line. If you use 10Base2 coaxial cable for networking, that is a 50-Ω transmission line. The 300-Ω “twin lead” cable from your TV “rabbit ear” antenna to your TV is a transmission line. And it is no accident that those high-power electrical lines from the power generating plants to our cities are called transmission lines, strung along transmission line towers. Even power system engineers need to worry about reflections if the lines are long enough (about 480 miles or so).

The rest of this chapter is concerned with how to make our traces look like transmission lines, and what the implications are if we don’t do that.

Author

Robotics Developer 6/18/2013 8:42:21 PM