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Transcript
Brown University
Physics Department
Physics 0060
Lab C-260
ELECTROMANETIC PULSE PROPAGATION IN A
COAXIAL CABLE
The mechanical waves on a stretched string are easily generated and observed but not
easily studied in quantitative detail. The propagating waves in an electrical system are
however easily generated and easily studied with standard instrumentation.
The purpose of this lab is to measure the speed of light using the properties of
electromagnetic waves and pulses in a coaxial cable.
THEORY:
If we simultaneously have two signals(electric circuit is demonstrated in Fig.2): one is
input to the oscilloscope directly, the other is reflected from a long cable before input to
the oscilloscope, there will be a phase difference between the two signals because the
wave inside the cable travels a longer distance. The speed of light is determined by
measuring the propagation velocity in the cable.
The velocity is given by
=
v 2* L / ∆t
Where L is the length of the cable and ∆t is the time interval of two signals. Note that
the factor 2 implies the reflected signal by the cable. Compare the velocity of the signal v
with its theoretical value given below:
v = c/ ε
Where c is the speed of light in vacuum and ε is the dielectric constant of the plastic
insulator. You will experimentally determine the dielectric constant from a measurement
of the capacitance of the coax cable (See Appendix for further details).From the above
equation we can get an electrical measurement of the speed of light.
A coaxial cable has a characteristic impedance that relates the voltage to the current in
the cable (in our experiment, it’s 50 Ohms). A cable terminated with this impedance will
behave equivalently to a pure resistance equal to the characteristic impedance. So if we
connect an variable termination resistor (VTR) at the end of the cable and adjust the
impedance( in this case the impedance is the same as resistance), the ratio of the
amplitudes of reflected and incident signal is
Vreflected R − Z 0
=
Vincident R + Z 0
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Brown University
Physics Department
Physics 0060
Lab C-260
Z 0 is the characteristic impedance.
A detailed derivation of the electromagnetic theory is shown in the appendix.
Experimental Procedure
1. Measure the radii of the inner copper conductor and inner plastic insulator of the short
sample cable. It has its insulation stripped back and conductors exposed.
Fig. 1
2.
Determine the dielectric constant of the insulation separating the inner and outer
conductors by measuring the capacitance of the coiled cable. Use the capacitance meter
and solve for the dielectric constant in CGS units (1 Farad = 9 × 1011cm) using equation 9
of the Appendix.
3.
Measure velocity of wave in the cable:
(a) Be familiar with the TDS2012B oscilloscope and connect it to PC. Use
the software Tektronix OpenChoice Desktop to acquire data on PC. (See details
on our website: --Lab Manuals--Introduction to the Oscilloscope)
(b) Use the SRS DS335 3.1MHz synthesized function generator (please
check for availability) to generate a square wave function. Connect the filter
circuit directly to the function generator. Set the function generator to ≈ 100
kHz, ≈ 10 V square waves. (To set the frequency, press [FREQ] and then enter
“100.00000” and press [Vpp/kHz]. To set the amplitude, press [AMPL] and
then enter “10.0” and press [Vpp/kHz].)
The filter circuit acts to
“differentiate” the square wave, giving short pulses when the amplitude
changes. Use a short cable to connect the output of this pulse generator to one
side of a Tee connected to the scope input. Connect the coiled cable you used
in Part 2 to the other side of the Tee.
Important: Do not connect the end of the filter with the 50 Ω resistor to the
function generator.
(c) Connect SYNC OUT of the function generator to the oscilloscope
external trigger input and set the oscilloscope to trigger on this external trigger
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Brown University
Physics Department
Physics 0060
Lab C-260
signal to offer a synchronous reference. Note that you have to select the SYNC
ON option on the panel of function generator by pressing [SHIFT] and then [.]
Fig. 2
(d) Observe the two pulses on the oscilloscope and output the data on PC.
You are supposed to see waveforms as is shown below:
Fig.3 Measurement of the time interval between incident and reflected pulses.
4.
Measure the length of the cable and calculate the velocity of propagation in the cable.
5.
Calculate the constant c using the velocity of propagation and the dielectric constant
for the cable. Compare your result with the known speed of light. (c = 2.998 × 1010
cm/sec)
6.
If you have extra time, you may want to investigate some of the other options.
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Brown University
Physics Department
Physics 0060
Lab C-260
Adjust the scope so that you see both the original pulse and a reflected signal
from the end of the cable. Connect the variable termination resistor (VTR) to the
far end of the cable. Connect an ohmmeter to the other BNC jack on the VTR.
(Use the switch on the VTR to toggle between the ohmmeter and the cable.)
Notice how the reflected signal changes as you change the termination resistance.
Plot the reflected signal amplitude versus termination resistance; be sure to make
several measurements in the region where the reflected amplitude is rapidly
changing. Use this plot to determine the characteristic impedance of the cable.
Compare your results with the predicted impedance for your cable. (1 Ohm =
1.113 × 10-12 cm-1 sec in CGS units)
Connect to Ω Meter
Switch
Fine Adjust
Coarse Adjust
Connect to far
end of cable
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Variable Terminator
Fig. 4
4
Brown University
Physics Department
Physics 0060
Lab C-260
APPENDIX
Theory of Electromagnetic Wave Propagation in a Coaxial Cable
While the effects of signal propagation can usually be neglected for low frequency
circuits, propagation effects become very important when the signal changes appreciably
during the time it takes the signal to propagate in the circuit. Electromagnetic wave
propagation is governed by Maxwell’s equations, and the theory behind electromagnetic
wave propagation in a coaxial cable is developed in the next section.
The electric and magnetic fields inside the cable must obey Maxwell’s equations:
 4πρ
∇⋅E =
,
(1)

∇ ⋅ B = 0,
(2)
ε

 4π  ε ∂E
∇× B =
J+
,
c
c ∂t


1 ∂B
∇× E = −
,
c ∂t
(3)
(4)
where the dielectric constant ε is needed to take care of the polarization of the material in
response to an applied field (see Purcell, Chapter 10, for example). The dominant
“mode” for electromagnetic waves in a coaxial cable is one where the electric and
magnetic fields are transverse to the axis of the cable. Given this assumption, Eqs. 1 and
3 require that
E r ( z, t ) =
2λ ( z , t )
,
εr
(5)
Bφ ( z , t ) =
2 I ( z, t )
,
cr
(6)
where λ (z,t) is the charge density on the inner conductor and I(z,t) is the current carried
by the inner conductor.
The potential difference between the inner and outer conductors, V, is found by
integrating Eq. 5
V ( z, t ) =
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2λ ( z , t )
ε
ln(b / a),
(7)
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Brown University
Physics Department
Physics 0060
Lab C-260
where a and b are the radii of the inner and outer conductors, respectively. For a fixed
potential difference between conductors with the far end of the cable unconnected, the
cable acts like a capacitor with a capacitance of
C=
εL
2 ln b / a
(8)
,
where L is the length of the cable. Thus, by measuring the capacitance of the cable we
can determine the dielectric constant ε
ε=
2C ln b / a
.
L
(9)
From Eqs. 3 and 4 we have the requirement
−
∂Bφ
=−
∂z
ε ∂E r
c ∂t
,
∂E r
1 ∂Bφ
.
=−
c ∂t
∂z
(10)
(11)
By differentiating Eq. 10 with respect to t, rearranging the order of differentiation, and
substituting Eq. 11 yields a wave equation field:
∂ 2 Er
ε ∂ E
= 2 2 2r .
2
∂z
c ∂t
(12)
The solutions of Eq. 12 are travelling waves that propagate in either the +z or –z
directions with a velocity given by
v=
c
ε
.
(13)
The magnetic field obeys a similar wave equation and is related to the electric field as
follows:
Bφ = ε E r .
(14)
This relation between the electric and magnetic fields requires that there be a fixed ratio
between the voltage across the inner and outer conductors and the current on the
conductors
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Brown University
Physics Department
Physics 0060
Lab C-260
Z=
V 2 ln b / a
,
=
I
εc
(15)
where Z is the “characteristic impedance” of the cable. If the end of the cable is
“terminated” by connecting a resistor whose value is equal to the characteristic
impedance, this relation between the voltage and current is maintained. Such a cable
looks like a pure resistance, independent of the length of the cable. A cable terminated
by a resistance R ≠ Z will generate a reflected wave so that the ratio between voltage and
current is maintained at the characteristic impedance. The reflected wave will propagate
back down the cable with the same propagation velocity as the incident wave. At any
point in the cable, the voltage and current are determined by superimposing the incident
and reflected waves. For an incident wave of amplitude VI , the amplitude of the
reflected wave VR can be shown to be
VR =
R−Z
VI .
R+Z
(16)
Such reflections will occur wherever there is a change in impedance seen by signals
propagating down the cable.
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