PHYSICS COURSE NAME
LAB 13
SIGNAL TRANSMISSION THROUGH A COAXIAL CABLE
Lab format: This lab is performed via an internet connection with the Remote Web-based Science
Laboratory (RWSL).
Relationship to theory: This lab corresponds to the theory of transmission lines or signal propagation
through an inductive medium.
OBJECTIVES
To study signal transmission through a coaxial cable
To study and measure the signal travel speed through a coaxial cable
To study reflections at the termination of a transmission line
EQUIPMENT LIST



Coaxial cable spool (30 m or longer)
Oscilloscope - ELVIS II
Function generator – ELVIS II
INTRODUCTION
Figure 1. Inside a coaxial cable.
[Credit: Tkgd2007, Wikimedia Commons]
A coaxial cable is a two-conductor cable constructed of a central wire and a cylindrical “shield” as shown
in Figure 1. Known as “coax” for short, the word coaxial refers to the geometry in which the axis of the
wire and the axis of symmetry of the cylinder are one and the same. A common sight in the laboratory
as well as in the home, coaxial cables are often the cable of choice for connecting instruments,
computer networking, or sending RF signals such as for television.
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LAB 13
We know from our study of E & M in first year physics that the electric field inside a conducting
cylinder depends only on the charges inside it, and also that the electric field outside the cylinder due to
charges inside is zero. This means that the current carried by the coaxial cable is shielded from outside
fields and that the outside world is shielded from electrical currents inside. Not only does this supress a
signal carrying cable from emitting RF noise into its environment, one can lay many coaxial cables beside
each other, say along a cable tray, and not be affected by crosstalk of signals among separate cables.
There are many practical benefits to this shielding property but for the purposes of this lab, as
scientists, we will place our focus on studying the properties of this cable, rather than its practical, and
“proper” use and applications. One property is that the signal travels through this cable rather slowly.
The other has to do with signal reflections that can occur at the ends of the cable, which are the only
places where the shielding is absent so that inside signals can interact with the outside environment.
Wave Travel Speed in a Coaxial Cable
The geometry of the conducting elements of a coaxial cable is well described by concentric cylinders.
From E & M textbooks, we recall the well-known result that a cylindrical capacitor with inner radius a
and outer radius b has capacitance per unit length given by
C
2
ln b a 
F/m
Equation 1
and a corresponding inductance per unit length of
L

ln b a 
2
H/m
Equation 2
where  and  are the dielectric constant and permeability of the medium between the conductors. We
find the resistances of the conductors are negligible in comparison such that C and L dominate the
overall impedance.
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LAB 13
Figure 2. LC model of a coaxial cable
Consider a model of a cable consisting of an infinite line of capacitors and inductors, as in Figure
2. We focus our attention on a section of this cable of length dx between points x and x+dx, with
capacitance Cdx and inductance Ldx. Using standard circuit theory and a second order partial
differential equation, we can derive (see Lab 13 Appendix A) a wave speed through this system as
v 1
LC
Equation 3
Equation 3 directly counters misconceptions on electrical signal travel speed such as (a) that it occurs at
c = 3.0 x 108 m/s or even that (b) it is instantaneous. The signal transmission speed is finite and is a
direct result of the electrical properties L and C of the cable.
Using Equation 1 and Equation 2, we can rewrite Equation 3 as
v



1
 
 2 

ln b a 

 2
 ln b a  
1

1
 0  r  0 
1
1
 0 0
 r
Equation 4
The terms in the second square root, dielectric constant relative permeability are each equal to unity
when the medium is a vacuum and the remaining terms give the speed of light in vacuum,
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c  1  0  0 . Thus we can simplify the equation one further step as
vc
1
 r
Equation 5
Again, if the medium is a vacuum, r = 1 and v = c. We can also see that for media where
and/or r are large (or L and C are large), v can be quite slow in comparison to c.
Characteristic Impedance
Coaxial, and other, cables are generally referred to by their characteristic impedances, as described
here. We assume that the cable is long enough that we need not concern ourselves with what happens
to the signal at the far end (we probe this further in the next section). When a square wave pulse of
height V0 is applied, a step up in voltage travels a distance of dx = v dt down the cable during time dt.
Another way to look at the situation is that a length dx of the cable is elevated to a potential V0.
Equation 1 gives us the capacitance per unit length, so the capacitance of this section of cable is Cdx.
The charge on this “capacitor” is given by
dQ  (Cdx)V0  CV0 dx  CV0 vdt .
The current starting to flow down this near end of the cable is the ratio of dQ to the time interval dt.
I
dQ CV0 vdt

 CV0 v
dt
dt
Cancelling dt on the top and the bottom. Using Ohm’s Law, the apparent resistance or the impedance
of the cable is
R0 
V0
V
1
.
 0 
I
CV0 v Cv
We have already determined v  1
R0 
LC in Equation 3 and making this substitution gives us
L
C
Equation 6
Further substitutions for C and L from Equation 1 and Equation 2 lead to the following construction.
R0 

ln b a 
ln b a  
2

2
2

ln b a 
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The term under the square root can be separated into their relative and free space values, so that
R0 

ln b a   r  0 ln b a   r

2

2

0
0
ln b a   r
337
2

Equation 7
where 337   0  0 is sometimes called the “impedance of free space.” Values of cable impedance
can range usually from few tens to few hundreds of ohms. Some of the more exotic and contrived
cables with deliberately high inductance can have impedance of a few thousand ohms. Signals travel
very slowly in such cables.
Boundary Conditions and Reflections
If the cable is not infinitely long, the signal, say a square pulse, will eventually reach the end of the cable.
What happens there depends on the boundary condition; three special cases are examined.
a) Matched Impedance
R = R0
If the end of the cable is terminated with a load resistor R = R0, the signal approaching it
effectively does not see an end, in that no special event happens there as it will continue to
see the same impedance. The signal is simply absorbed by the load resistor and there is no
reflection.
b) Short Circuit
R=0
The two conductors of a cable can simply be connected to each other (short circuit) at the
end of the run. Because the resistance between the two conductors is zero, the boundary
condition that is forced is V = 0 between the two conductors. When the pulse of height V0
arrives at the end, a new pulse of opppsite polarity –V0 is needed. This pulse travels back up
the cable as a reflection. (See Figure 3.)
c) Open Circuit
R=∞
The two conductors of a cable can also be left open. With an open circuit, no current can
flow, making I = 0 the boundary condition. The physical consequence is that there needs to
be a flow of charge of the same polarity as the original pulse but travelling in the opposite
direction. This is the positive reflection. (See Figure 4.)
Of course, termination impedances are not restricted to these three special cases and cables
can, in principle, be terminated with any arbitrary impedance. A configuration worthy of further
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attention is “terminating” a cable with another cable, i.e. connecting two cables together. If the
impedances of the two cables are matched to each other, signal transmission is optimised. Mismatched
cables result in reflections which are generally, though not always, undesired.
Figure 3. Reflection at a short circuit.
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Figure 4. Reflection at an open circuit
WARNINGS

There are no special safety warnings associated with this lab.

If connecting your own instruments rather than RWSL, keep in mind that many instruments are
connected to each other through a common ground. Unintentionally creating a ground loop is
a common rookie mistake.
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PROCEDURE
Figure 5 shows the circuit to be examined. Positions 1, 2 and 3 of the switch allows us to select R = 0,
R = ∞, and R = Ro, respectively.
Figure 5. Coaxial cable with three different terminations
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Part I: Measurement of Signal Speed
 For this portion of the study, we first eliminate reflections in the cable using a matched load by
selecting switch position 3. (See sample traces on Figure 6.)
 Measuring the signal speed is a very straight forward exercise. Measure T by comparing
oscilloscope channels 0 and 1. Compute v using the provided length of the cable.
 Compare your result with predictions.
Figure 6: Sample oscilloscope traces when reflections are suppressed using a matched load. Blue and red
traces are input and output, respectively.
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Part II: Boundary Conditions and Reflections
 We can examine the effects of a short circuit termination (R = 0) by turning the switch to
position 1 in Figure 5.
 Channel 1 is a short circuit, so there will be no signal to see.
 What can you expect to see on channel 0, in addition to the original pulse? When? Compare
your predictions with observations.



We can examine the effects of an open circuit termination (R = ∞) by turning the switch to
position 2 in Figure 5.
Channel 1 should see a pulse +2 V0 compared to the original pulse V0. In order for the cable end
to be truly an open circuit, there must be no current flow through the channel 1 input of the
oscilloscope, meaning that the input impedance of the oscilloscope must be really high.
Discuss.
What can you expect to see on channel 0, in addition to the original pulse? When? Compare
your predictions with observations.
ANALYSIS AND/OR QUESTIONS

In Part II, in order for us to observe the reflections at the originating end of the cable, what
conditions must exist on this end? Specifically, what must be the output impedance of the
function generator and the input impedance of the oscilloscope channel 0?

We have studied the transmission line from the point of view of scientific curiosity. From the
point of view of an RF engineer or a technician, what would be some practical implications on
what one might call the “proper” installation of coax cables? For example, what are some
considerations when running coax cables around a building to connect equipment? What
happens when a cable needs to be extended?
REFERENCES
Horowitz, P., and Hill, Winfield., 1989, The Art of Electronics, 2nd Ed., Cambridge University Press,
Cambridge (section 13.09).
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LAB 13 APPENDIX A - DERIVATION OF THE WAVE SPEED
For voltage and current expressed as functions of both position and time, V(x,t) and I(x,t), we can write
V ( x  dx)  V ( x)  dV
I ( x  dx)  I ( x)  dI
Equation 8
Equation 9
From the definition of capacitance
Q  CV
and current can be determined by taking the time rate of change
I
dQ
dV
C
dt
dt
Over a distance dx, the change in current can be written as
I
V
dx  (Cdx)
x
t
Finally, eliminating dx from both sides and taking the partial derivative with respect to t gives us
2I
 2V
 C 2
tx
t
Equation 10
Similarly, from the definition of inductance
V  L
dI
dt
and over a distance dx, the change in voltage can be written as
V
I
dx  ( Ldx)
x
t
By eliminating dx from both sides and taking the partial derivative with respect to x gives us
 2V
2I


L
x 2
xt
Equation 11
Substituting Equation 10 into Equation 11 yields
 2V
 2V


LC
x 2
t 2
or
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 2V 1  2V

x 2 v 2 t 2
Equation 12
where
v 1
LC
Equation 3
Those who have studied partial differential equations will recognise Equation 12 as the standard form of
the wave equation with propagation speed v and no attenuation. A solution to this PDE has form
V(x±vt).
Developed for Remote Delivery by T. Sato under the Remote Science Labs for Second Year
Physics Project (2012 – 2013) funded by BCcampus.
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