Chapter 23: Transmission Lines
Chapter Learning Objectives: After completing this chapter the student will be able to:

Explain the definition of a transmission line and determine whether a particular
signal being transmitted over a particular medium requires consideration as a
transmission line.

Calculate the inductance and capacitance per unit length for coaxial cables,
microstrip lines, and twin lead conductors.

Draw an equivalent circuit representation of a transmission line.

Use the transmission line equations to model the electrical behavior of a
transmission line.
You can watch the video associated
with this chapter at the following link:
Historical Perspective: Samuel Morse (1791-1872) was an
American inventor who developed the concept of a singlewire telegraph, helped to create the relays that boost the
signal (allowing it to travel more than a few hundred feet) and
developed Morse Code for telegraph communication.
Photo credit: https://commons.wikimedia.org/wiki/File:Samuel_Morse_1840.jpg, [Public domain], via Wikimedia
Commons.
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23.1
Introduction to Transmission Lines
We are going to turn our attention away from propagation of electromagnetic waves in free space
and/or dielectric materials for a few chapters, and we will instead focus on better understanding
transmission of electrical signals over physical cables, called transmission lines.
Not every signal nor every wire needs the special attention we are giving to transmission lines.
For example, the electric wires inside the walls of a building do not require this special attention,
because we can assume that the signals are changing so slowly that the voltage is essentially
constant along the entire length of the wire. However, if the signal is changing rapidly enough,
then the voltage at different points along the wire will vary. This will be the case if the signal is
changing so rapidly that the wavelength of the signal is comparable to the length of the cable
being considered. In particular, we will assume that we must treat a wire as a transmission line if
the length of the wire is at least 10% of the wavelength of the highest frequency being
transmitted.
(Equation 23.1)
Example 23.1: Determine whether each of the following scenarios would qualify as a
transmission line. Assume the propagation velocity is 3x108 m/s.
a. A 60Hz signal being transmitted over a wire that is 10km long.
b. A 100KHz signal being transmitted over a wire that is 10km long.
c. A 1MHz signal being transmitted over a wire that is 100m long.
d. A 10GHz signal being transmitted over a wire that is 1mm long.
23.2
Equivalent Electrical Circuits
Recall that all physical devices will exhibit parasitic, or stray, capacitance and inductance. We
calculated the stray capacitance of a coaxial cable in Example 9.4, and we found the stray
inductance of a coaxial cable in Example 15.3:
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(Equation 23.2)
(Equation 23.3)
Here, a is the radius of the inner conductor, b is the radius of the outer conductor, and z is the
length of the wire.
It is this stray capacitance and inductance that prevent a signal from propagating along a wire
with infinite velocity. (Since we know that it is impossible for any signal in any medium to
move faster than the speed of light in a vacuum, we also now know that it is impossible to
completely eliminate stray capacitance and inductance.)
Of course, this stray capacitance and inductance do not appear all in one lump at the end of the
wire. It is distributed along the length of the wire, and our model for a transmission line
distributes these quantities in that way. We will add a hat over the C and L to represent the
quantity per unit length:
(Equation 23.4)
(Equation 23.5)
As shown in Figure 23.1, the stray capacitance and inductance per unit length are then
distributed along the length of the transmission line to form a distributed model:
V(z)
I(z)
z-z
z
z+z
Figure 23.1. Equivalent Circuit for a Lossless Transmission Line
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Example 23.2: Draw a distributed model for a one-meter piece of coaxial cable with an
inner radius 2 cm and an outer radius 7 cm. The relative dielectric constant of the dielectric
between the conductors is 5. Use four capacitors and four inductors in the model, as shown in
Figure 23.1.
23.3
The Lossless Transmission Line Equations
We can apply Kirchhoff’s Laws to the equivalent circuit shown in Figure 23.1, and this will yield
equations describing the behavior of the transmission line. First, we will apply Kirchhoff’s
Current Law at the point labeled V(z):
(Equation 23.6)
Rearranging this equation, we obtain:
(Equation 23.7)
Taking the limit of the left side as z approaches zero gives the partial derivative of I with
respect to z:
(Equation 23.8)
Combining Equations 23.7 and 23.8, we find:
(Equation 23.9)
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Next, we will apply Kirchhoff’s Voltage Law around the loop to the left of the same node:
(Equation 23.10)
Rearranging this equation gives:
(Equation 23.11)
Taking the limit of the left side as z goes to zero gives the partial derivative of V with respect to
z:
(Equation 23.12)
Combining Equations 23.11 and 23.12 yields:
(Equation 23.13)
Equations 23.9 and 23.13 taken as a pair are known as the Telegrapher’s Equations or the
Transmission Line Equations. They will help us to fully characterize the behavior of lossless
transmission lines and how signals propagate along them. (Note that the original circuit diagram
in Figure 23.1 does not include any resistance, so these equations assume that the transmission
line has no resistance. We will improve this model in a future chapter, but it is an excellent first
step.)
Now, let’s take the derivative of both sides of Equation 23.9 with respect to time:
(Equation 23.14)
Next, we’ll take the derivative of both sides of Equation 23.13 with respect to space and divide
both sides by inductance per unit length:
(Equation 23.15)
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The left side of Equation 23.14 is equal to the right side of Equation 23.15, so we can set the
other two sides equal to each other:
(Equation 23.16)
A bit of algebra to rearrange this equation gives:
(Equation 23.16)
This is the wave equation, following the same form as the one we derived way back in Equation
18.42. This means that electrical signals along transmission lines will flow as waves, just as
electromagnetic fields flow as waves in free space or dielectrics.
A similar procedure can be used to derive the wave equation for current:
(Equation 23.17)
Both of these equations take the same form, meaning that voltage and current will both flow
along the transmission line in the same way and with the same velocity. In fact, the voltage and
current reinforce each other and allow each other to propagate, in precisely the same what that
the electric field and the magnetic field allow each other to propagate in free space or dielectrics.
Without both, neither can propagate.
The velocity of propagation for lossless transmission lines is:
(Equation 23.18)
Example 23.3: What is the propagation velocity for a wave passing along the coaxial cable
of Example 23.2?
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23.4
Transmission Line Parameters
As we have just seen, it will be essential to be able to calculate the inductance per unit length and
capacitance per unit length for a transmission line in order to analyze it. We have already
derived these quantities for a coaxial cable. Figure 23.2 shows the geometry of a coaxial cable,
and the capacitance and inductance per unit length can be easily found from Equations 23.2 and
23.3.
a
z
b
Figure 23.2. Geometry of a Coaxial Cable
(Equation 23.19)
(Equation 23.20)
A second commonly used type of transmission line is a microstrip line, which is frequently used
in integrated circuits and printed circuit boards. The geometry and dimensions of a microstrip
line are shown in Figure 23.3.
w
d
z
Figure 23.3. Geometry of a Microstrip Line
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Relatively simple applications of Gauss’s Law and Ampere’s Law can be used to derive the
capacitance and inductance per unit length of a microstrip line. The results are:
(Equation 23.21)
(Equation 23.22)
The third and final type of transmission line we will consider here is a twin lead, which is simply
two wires running parallel to each other, as shown in Figure 23.4.
D
a
Figure 23.4. Geometry of a Twin Lead
It is important to note that a is the diameter of each lead, not the radius. A derivation of the
capacitance and inductance per unit length for the twin lead transmission line is quite
challenging, and it is beyond the scope of our work here. But it has been done, and we can apply
the results:
(Equation 23.23)
(Equation 23.24)
The inverse hyperbolic cosine can be calculated by the following relationship:
(Equation 23.25)
But your calculator can almost certainly perform this calculation for you.
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Example 23.4: A microstrip line has a width of 1mm with 2mm between the conductors.
The material between the conductors has a dielectric constant of 4. What are the
capacitance and inductance per unit meter for this line?
Example 23.5: What is the speed of propagation along the line in Example 23.4?
Example 23.6: A twin lead transmission line has two conductors 1cm apart, and each
conductor has a diameter of 1mm. They are separated by free space. What are the capacitance
and inductance per unit meter for this line?
Example 23.7: What is the speed of propagation along the line in Example 23.6? How
long would it take for a signal on this line to travel 3000 miles?
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23.5
Summary

A transmission line is any physical cable with a length more than 10% of the shortest
wavelength being transmitted:

Transmission lines exhibit stray capacitance and inductance. We will specify these quantities
on a per-unit-length basis.

Lossless transmission lines can be modeled by distributed capacitance and inductance:
V(z)
I(z)
z-z
z
z+z

Analysis of this model results in the transmission line equations:

Further analysis demonstrates that transmission lines follow the wave equation with a
propagation velocity calculated as follows:
Coaxial Cable
Microstrip Line
Capacitance
Inductance
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Twin Lead