FILE (SILVER) D:\TEACHING\L5EMT&RFID10\PART9(OTHERNOTESTEXT)
\OUTLINELECTURE10.DOC
OUTLINE OF LECTURE 10: 12:00 Mon 23 Aug 2010
SMITH CHARTS AND SLOTTED LINES
Objective
To assist students in understanding the relationships between Smith charts, slotted lines and
voltage standing wave ratio concepts.
This will be achieved by providing a demonstration of how to use a slotted line to determine
an unknown impedance.
The demonstration will involve some work on the part of the class.
Voltage standing wave ratio
The concept of voltage standing wave ratio was introduced in the Part 3 Distributed Circuits
and Radiation Notes in Section 2.72 of Chapter 2 on page 43.
There it is pointed out that forward and backward waves on a line will produce an
interference pattern in the magnitude of the total voltage as seen by an observer who moves
along the line. The pattern is seen to repeat itself each half wavelength.
A definition of voltage standing wave ratio (VSWR) S is given in equation 2.39 and
illustrated by Figure 2.5. It should be realised that S is a positive constant describing this
interference pattern; it does not vary as we move along the line, and is always 1 or greater.
Exercise for students If a transmission line of characteristic impedance 50 ohm, is terminad
in a real impedance of 100 ohm, calculate the voltage reflection factor at the load end, and
calculate the VSWR.
Equation 2.40 gives the relations that allow the magnitude of voltage reflection factor to be
calculated from the WSWR, and vice versa. Note that we cannot get the phase of the voltage
reflection factor from the VSWR.
Equations 2.41 and 2.42 give the relations between the maximum and minimum impedances
(the ratio of total voltage phasor to total current phasor) seen as we move along the line to
various points in the interference pattern.
Other features of the interference pattern are the positions along the line at which the
interference minima and maxima of total voltage are seen. The positions for maxima and
minima are one quarter of a transmission line wave length apart.
A feature of importance is the actual position at which a minimum of the standing wave
pattern is achieved. It is at this point that we can state that the impedance (which varies from
point to point on the line) seen on the line (looking toward the load) is both real and has the
value Z0 /S. At this point the voltage reflection factor v is also real and negative.
So we see from this section of the notes that if we have an instrument that allows us to
determine the VSWR of the interference pattern and the position of minim of the interference
pattern, we can work out the load impedance as transformed to that reference plane, and
hence the load impedance transformed to any other reference plane.
The slotted line
Such an instrument is called a slotted line. An illustration of a coaxial slotted line is provided
in the Figure below, together with some equipment with which it is often used.
2
It is fair to state that the slotted line is not now a commonly used instrument. Its discussion
here is for the reasons that (i) it provides an introduction to important concepts in
transmission lines, (ii) measurements using it can very accurate, and (iii) measurements using
it depend on simple principles in which the errors can be identified.
In the diagram can be seen the slotted line at front centre, a signal source at the left connected
to the input end of the line, and a voltage stranding wave indicating meter at the right.
Relevant additional components are a sliding (i.e. adjustable) short circuit, and a range of
terminations that can be placed at the load (right hand) end of the line. The photograph also
shows some things that will not be discussed.
Smith Charts
The Smith chart was introduced in Part 3 the Distributed Circuits and Radiation notes in
Section 3.2 of Chapter 3 on page 49. There the important statement is made that the Smith
Chart is a plot of voltage reflection factor in the complex plane with some extra information
added. Soon it becomes clear that the extra information added consists of contours of the real
and imaginary parts of the normalised impedance on the line.
As the exposition unfolds, it is made clear that the Smith Cart could equally well be regarded
as a plot of current reflection factor on the line, with extra information representing contours
of real and imaginary parts of the normalised admittance on the line.
Questions for students
If the line is terminated in a short circuit, what is the nature of the VSWR you expect, and
where would be the minimum, if any, of the standing wave pattern?
If the line is terminated in an open circuit, what is the nature of the VSWR you expect, and
where would be the minimum, if any, of the standing wave pattern?
If the line is terminated in an inductor, whose reactance at the frequency of test is equal to the
characteristic impedance of the line, what is the nature of the VSWR you expect, and where
would be the minimum, if any, of the standing wave pattern?
Where do the currents flow inside the line? Are they likely to meet and enter the slot?
Sampling the line voltage
3
The slotted line contains a probe that enters the line through its longitudinal slot, and samples
the electric field in the line. The assumption is made that the electric field is proportional to
the voltage between the inner and outer conductor of the line.
Incorporation of a detector
The sampling probe that senses the electric field is connected to a low threshold
semiconductor diode to convert the sensed field to a small voltage representative of the
envelope of the radio frequency voltage on the line.
Square law assumption
The signal reaching the diode is so small that it may be assumed it is operating in the square
law region, i.e., to diode output voltage is proportional to the square of the radio frequency
input voltage that it senses.
The scales of the voltage standing wave indicating instrument are adjusted to take this
assumption in to account.
Questions for students again
If the voltage standing ratio is 2, what is the ratio of the maximum to minimum signal
reaching the voltage standing ratio indicating meter?
Some relevant facts and formulae
See Chapter 2 page 41 of the part 3 Distributed Circuits and Radiation notes for the following
facts that are given in Equations 2.27 and 2.28.
Normalised impedance z  r  jx 
Voltage reflection factor Γ v 
VSWR S 
Z
Z0
z 1
z 1
1 | Γ v |
S 1
or | Γ v |
1 | Γ v |
S 1
At the position of maximum of the standing wave pattern, the reflection factor is real and
positive.
At the position of minimum of the standing wave pattern, the reflection factor is real and
negative.
Examining the scales on the voltage standing wave ratio indicating meter shows that the
VSWR scale S and the VSWR(dB) scale are related by
VSWR(dB) = -10log10-|S|2 = -20log10|S|
so we see that at a VSWR of 2 we read -6dB on the VSWR(dB) scale. The inverse of this
equation is
| S | 10
 VSWR(dB) 


20


This equation is needed when we have changed the gain of VSWR meter by the stepped gain
control which changes the gain in 10dB steps.
PHC. 22 Aug 2010.