Download AC Circuits

PHYSC 3322
Experiment 2.5
26 June, 2017
Coaxial Transmission Line
Purpose
Electromagnetic waves are the means by which much of the information that forms a
part of daily life is transmitted. This information may be in the form of a telephone call,
radio and television broadcasts, digital signals inside a computer, etc. In this
experiment, you will examine the propagation of transient and steady-state signals in a
transmission line.
Background
Figure 1 shows a voltage source, V , connected to a load resistor, R , by a coaxial cable.
If the source is a DC source, a current, I , flows down the center conductor, through R ,
and back to the source via the outer conductor. Elementary electromagnetic theory
states that there are corresponding E and H fields (as shown in the figure) inside the
cable, and that there are no electromagnetic fields outside of the cable. For lowfrequency AC sources, the description is essentially the same as for DC. However, at
higher frequencies, where the wavelength,  , is comparable to the length, L , of the
transmission line, it is often more useful to describe the problem in terms of an
electromagnetic wave traveling on the line.
Figure 1. A coaxial transmission line.
For the moment, imagine that the source is connected to a coaxial line of infinite length.
The electromagnetic wave travels away from the source at a phase velocity, v p , which
is determined by the characteristics of the line; generally, the phase velocity will be less
than the speed of light in a vacuum. The propagation of energy in the line is given by
the Poynting vector, S  E  H . E and H are related through Maxwell's equations.
Their magnitudes are related by E  Z0 H , where Z 0 is the characteristic impedance of
the line. This can also be written as V  Z0 I . For a coaxial line, Z 0 is given by
Z0 
1
2
0
ln b / a ,
k 0
(1)
where  0  1.26  10 6 H m -1 is the permeability of free space,  0  8.85  10 12 F m -1 is
the permittivity of free space, k is the dielectric constant of the insulator located
between the inner and outer conductors, b is the inside diameter of the outer
conductor, and a is the outside diameter of the inner conductor. Numerically, this is
Z0  60 
ln b / a
k 1/2
.
(2)
1
PHYSC 3322
Experiment 2.5
26 June, 2017
When dealing with a line of finite length, the boundary conditions at the end of the line
become important. In this experiment, only lines terminated in pure resistances (i.e., no
reactive components) will be considered. If the load resistance is equal to Z 0 , then no
signal is reflected and all of the electromagnetic energy is transferred to the load;
consequently, the magnitudes of V and I will be the same at any point along the line.
If instead the line is terminated in an infinite resistance (open circuit), then I  0  0 and
the signal will be reflected. Figure 2 shows the resulting V and I patterns along the
line. If the line is terminated in a zero resistance (short circuit), then V  0  0 and the
signal will once again be reflected. The V and I patterns for this case are shown in
Figure 3.
Figure 4 shows the V patterns for other load resistance values, in which case only part
of the signal is reflected. If R  Z0 , then the voltage standing wave ratio is given by
VSWR  Vmax /Vmin  Z0 / R .
(3)
If instead R  Z0 , then the VSWR is given by
VSWR  Vmax /Vmin  R / Z0 .
(4)
The VSWR can also be expressed in terms of the reflection coefficient, r , defined as the
square root of the reflected power divided by the transmitted power:
VSWR  1  r  / 1  r  ;
(5)
r  Prefl / Pxmit 
(6)
1/2
.
Figure 2. Voltage and current standing
wave patterns for an open-circuit
terminated line (x=0 is at the open
end).
Procedure
Figure 3. Voltage and current standing
wave patterns for a short-circuit
terminated line (x=0 is at the short
end).
Transient waves: In this part of the experiment, you will observe the propagation of
short impulses along a transmission line. Using the BNC tee, connect the pulse
generator and channel 1 of the oscilloscope to the L end of the transmission line. The 0
end of the line should be left open-circuit. Observe the transmitted pulse along with
any reflections on the oscilloscope. Print the waveform using the print function of your
oscilloscope. Measure the time delay between the original pulse and the first reflection,
and record this time. Make sure that the pulse width is smaller than the delay time
(using the smallest pulse width on the generator).
2
PHYSC 3322
Experiment 2.5
26 June, 2017
Attach the short-circuit termination to the 0 end of the transmission line, and repeat the
above procedure. Repeat the measurements using the 4.7  , 47  and 470
terminations.
Figure 4. Voltage standing wave
patterns for various load resistances
(neglecting phase).
Standing waves: In this part of the experiment, you will observe the standing wave
patterns produced by interference between transmitted and reflected sine waves.
Remove the pulse generator and replace it with the sine-wave oscillator. Attach channel
2 of the oscilloscope to the L/2 point on the line. With no load resistance attached to the
0 end of the line, vary the sine wave frequency and locate the lowest frequency null at
the L/2 point. Record this frequency. Increase the frequency and locate at least two
more nulls at higher frequencies.
Return to the first null frequency and measure the voltages (using the oscilloscope) at
all five tap points on the transmission line.
Attach the short-circuit termination to the 0 end of the line and repeat the voltage
measurements at the five tap points. Repeat this procedure using the three resistive
terminations.
Questions
The transmission line is 81.5 m long. How long does it take an impulse to travel two
lengths of the line? What is the propagation velocity of an electromagnetic wave in the
line? What is the propagation factor (propagation velocity/speed of light in a vacuum)?
Measure the a and b diameters of the RG-58/U cable used in the transmission line
(assume that b is equal to the outside diameter of the inner insulation). The impedance
of RG-58/U cable is 50  . What is the dielectric constant of the insulation?
3
PHYSC 3322
Experiment 2.5
26 June, 2017
According to electromagnetic theory, the square root of the dielectric constant should
be equal to the inverse of the propagation factor. How close is it?
Discuss and explain the impulse waveforms you observed as you changed the
termination resistance. Why the waveform changes its sign when R  Z 0 ?
In the standing wave portion of the experiment, what are the wavelengths of the waves
at the three null frequencies?
Sketch the probable current and voltage waveforms for each value of termination
resistance. Calculate the VSWR for each case.
4