Speed of Electromagnetic
Signal Along a Coaxial Cable
Se-yuen Mak,
Chinese University of Hong Kong, Shatin, N.T. Hong Kong
O
ne common paradox students find perplexing in learning about electric current is the
apparent contradiction between the tiny
drift speed of free electrons in a conductor, say about
1 m/h, and the response of a current “in no time”
when the circuit is switched on or off. These phenomena can be understood in terms of the speed of
the electrical signal, which travels at or near the speed
of light. As soon as the circuit is closed, apart from inductive delay, an electric field is set up almost simultaneously throughout the circuit. It is the electric field
that causes electrons to start drifting at all points in
the circuit. This paper describes an experiment for
measuring the speed of an electromagnetic signal in a
coaxial cable.
An experiment to estimate the speed of an electrical
pulse in a cable has been explained by T. Duncan.1 In
this paper we describe a setup using more updated
measuring instruments. The setup and procedure of
our method are given in some detail but the theoretical framework is kept to a minimum. The validity of
our method is based on phenomenological reasoning
and self-consistence.
Experimental Setup
Our sample is a 15-m long coaxial cable with characteristic resistance2 75 . A function generator3
with a frequency range of 0 to10 MHz is used to provide the signal, and a student-grade oscilloscope4 with
a sweep frequency of 20 MHz is used for time measurement. A 10-to-200- noninductive rheostat is
connected across the output end of the cable (Fig. 1).
The rheostat provides a damping to the outgoing signal and prevents the formation of a standing wave
along the cable. In general, the length of the cable re46
Dual Trace CRO
20 MHz
Trigger Hor. Mag.
Channel 1 Trace
x10
Time
Channel 2 Trace
Base
0.2 µs
Signal
Generator
V1
Gain Y1 shift
0-10MHz
Channel 1
V2
Gain Y2 shift
Channel 2
0-200 Ω
Rheostat
15 m of
Antenna or
AV coaxial
cable
10 Ω
Fig. 1. Measurement of speed of EM wave along a coaxial cable. (The ground cable is replaced by a symbol and
the traces are drawn separately for clarity.)
quired is inversely proportional to the fastest sweeping
frequency of the oscilloscope. For a 20-MHz oscilloscope, a length of 15 to 30 m, or the length of a typical physics laboratory, is desirable.
Method
A square wave at 2 V ( 600 kHz) is applied to
Channel 1 and the input end of the cable. The output
signal is applied to Channel 2. The rheostat is adjust-
DOI: 10.1119/1.1533966
THE PHYSICS TEACHER ◆ Vol. 41, January 2003
Fig. 2. Display of oscilloscope screen when r is adjusted
to the characteristic resistance of the cable. (Time base
is set at 0.2 s cm-1. The input trace is slightly larger
than the output trace.)
ed until the input trace and the output trace look the
same but differ only in phase (Fig. 2). For the sake of
easy comparison of waveform and amplitude between
the traces, the same gain and zero-level should be used
in both channels. The time taken by an EM wave to
travel along a single cored coaxial cable is the difference in zero crossings, T, between the traces (Fig. 3).
Theory
The voltage along a coaxial line assumes a general
form:
v(x,t) =
V+(x
– ct) +
V -(x
+ ct),
where V+ represents a wave going in the + x direction (away from the source) and V- a wave in the –x
direction (toward the source, reflected from the far
end).2 Empirically, the functional values of v(0,t)
and v(l,t) are represented respectively by the input
and output trace displayed on an oscilloscope screen.
If the transmission line were infinitely long, there
would be no reflected wave and
v(x,t) = V+(x – ct).
Although an infinitely long transmission line does
not really exist, it can be replaced by connecting a
resistive load, r0 (the characteristic resistance of the
line), across the far end of a cable of finite length.
THE PHYSICS TEACHER ◆ Vol. 41, January 2003
Fig. 3. Display of oscilloscope screen when 10 horizontal magnification is applied.
Most coaxial cables available in the market have r0 in
the range 50 to 200 . By changing the resistance
of the rheostat and observing the corresponding
changes in waveforms and separation of the traces
v(0,t) and v(l,t), one can show that the reflected wave
indeed disappears when r = r0. Since the difference
in zero crossings can be found without knowing the
value of r0, the depth of treatment on characteristic
resistance can be adjusted or even skipped at teachers’ own discretion in this measurement exercise. As
long as the waveform and amplitude of the traces are
identical irrespective of frequency changes, both
traces must be represented only by V+ with the same
argument (x–ct = constant).
Why Square Wave?
For three reasons, a square wave produces better results than a sine wave or a pulse. First, variations and
47
Table I. Summary of results.
Experimental Parameters
Value
Length of coaxial cable / l
(15.0
Dielectric constant () of material in the insulating layer (Estimated)
~2.3
Maximum time base sensitivity of a 20-MHz oscilloscope
20 ns/cm
Characteristic resistance and
~75
range of rheostat recommended
10 to 200
Difference in zero crossings / T (Result based on Figs. 3a and 3b)
(63
0.1) m
0.3
10) ns
An error of 0.4 ns is produced by the finite trace width, 0.3 ns each by
distortion of oscilloscope and incorrect judgment of similarity
Speed of electric pulse
Measured value / c = l /T
(2.44 0.39) ms-1
Calculated value
(1.99 0.13) ms-1
distortions in the shape of waveforms are more easily
detected. Second, the difference in zero crossings can
be measured with higher accuracy using the “vertical
part” of these traces. Third, a fairly wide frequency
range, say from 50 kHz to 6 MHz, can be used. Below 50 kHz, i.e., when the sweeping frequency of the
oscilloscope is much faster than the signal frequency,
the traces may become too dim for clear observation.
D. Rheostat
If a noninductive rheostat with range 0 to 100 is
not available, a more common 500 rotary carbon
film B-type (with r ) rheostat can be used as a
substitute. In the latter case, less than 15% of its full
range is used. Adjustment requires practice and
some patience.
Experimental Precautions
With a suitably long coaxial cable, the uncertainty
due to length measurement is less than 1%. Using an
oscilloscope with maximum time base sensitivity 0.2
s cm-1 and 10 horizontal magnification, the random error generated in time measurement is about
5% [from Fig. 3, trace width/trace separation (0.2
cm/3.2 cm) 100% 6%]. Errors also arrive from
A. Choice of frequency
In theory, the result of our experiment should be
independent of frequency, so the best frequency is
the highest frequency with the least distortion in the
input square wave. The “highest frequency” is used
because it gives rise to the brightest trace.
B. Oscilloscope probes
Since oscilloscope probes are by themselves coaxial
cables, identical probes5 should be used in both
channels so that the time lag generated by these leads
cancel one another.
C. Oscilloscope adjustment
In the experiment described here, a student-grade
oscilloscope was used. With such an instrument,
one should increase the time base sensitivity stepwise
in order to continually keep the traces on screen.
The horizontal magnification 10 is applied in the
last step.
48
Result and Uncertainty Analysis
(a) distortion produced by the CRO,
(b) nonidentical probes, and
(c) defects in a real and finite transmission line,
such as flux leakage at both ends.
These errors render the traces v(0,t) and v(l,t) not
exactly identical. By comparing our results here with
those obtained from a research-grade oscilloscope ,6
we found that the distortion in the oscilloscope may
introduce a percentage error of about 5% in the
result. Incorrect judgment of similarity of the input
and output trace will introduce an error of approxiTHE PHYSICS TEACHER ◆ Vol. 41, January 2003
mately the same magnitude. Results of our experiment are summarized in Table I.
Acknowledgment
The author is indebted to Kenneth Young of the
Physics Department, CUHK, for enlightening discussions in this investigation.
References
1. T. Duncan, Advanced Physics (John Murray, London,
1994), Vol. I, Appendix 2.
2. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman
Lectures on Physics (Addison-Wesley, Reading, MA,
1964), Vol. II, Chaps. 22–24.
3. Radio Frequency Function Generator (Topward Model: 8140, Taiwan) [frequency range 0 to 10 MHz].
4. E.g. Kenwood 20-MHz Oscilloscope Model: CS-4125
(Japan) [max. time base sensitivity = 20 ns cm-1 when
“10 horizontal magnification” is used].
5. A pair of IWATSU Probes SS-082R is used in our
experiment.
6. E.g. IWATSU 250-MHz Oscilloscope Model: SS-7825
(Japan) [max. time base sensitivity = 1 ns cm-1 when
“10 horizontal magnification” is used].
Se-yuen Mak is a professor of physics in the Department
of Curriculum and Instruction, Faculty of Education,
Chinese University of Hong Kong. His research interests
include teaching methods in physics and junior science at
the high school level. His research findings can be found
in The Physics Teacher, Journal of Science Education and
Technology (U.S.) and Physics Education (UK). The
Chinese University of Hong Kong, Shatin, N.T. Hong
Kong; symak@cuhk.edu.hk
THE PHYSICS TEACHER ◆ Vol. 41, January 2003
49