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AMATEUR RADIO
BY GARRY CRATT, VK2YBX
The Smith Chart – what it is &
how you use it
Possibly one of the most useful graphic tools
available today to the RF engineer is the Smith
Chart. This chart, named after its inventor, Mr
Phillip Smith, an engineer at Bell Laboratories
during the 1930s, first appeared in “Electronics”
magazine in the USA in January 1939.
problems, the precise reason for its
creation. In fact, the Smith Chart is
really a special type of graph, having
curved coordinate lines, instead of
the rectangular lines encountered on
standard graph paper. Quite complex
mathematical reasons exist behind the
construction of the chart but these do
not need to be understood by the user.
It is easy to be put off by the Smith
Chart. At first glance, it looks like a
nightmare, with all those apparently
spiralling curves, but after you’ve read
this article you should be quite at ease
with it. It really is a very useful chart
Different curves
for the amateur radio operator.
Essentially, the Smith Chart is used
to graphically repre
sent the reflection characteristics and impedance
of an RF circuit. It is ideally suited
for the solution of transmission line
To make the Smith Chart easy to
understand, we’ll show its different
curves separately and then bring them
all together as a simplified composite
chart. Fig.1 is the first set of curves
0
REACTANCE
AXIS
0.2
+0.2
0
RESISTANCE
AXIS
0.2
0.
5
0.5
PRIME
CENTRE
.5
+0
RESISTANCE
CIRCLES
1
1
+1
2
h
Fig.1: the first components of a Smith Chart are the
resistance circles. The values are “normalised” to a
value of 1.00, the prime centre of this plot.
+5
50
+2
2
5
5
h
Fig.2: the second component of the Smith Chart is
the reactance plot. Again the reactance values are
normalised.
June 1993 53
tance lines plotted on the one graph.
Note that this is a greatly simplified
Smith Chart, as those normally pub
lished have the circles at much closer
intervals which is why they look so
complicated at first glance.
Now let’s see how you might plot
a particular impedance on the chart.
Consider an example where we have
an impedance consisting of 50Ω resistive and 100Ω inductive reactance
(50 + j100), and a prime centre value
of 50Ω. This particular impedance can
be plotted at the intersection of 1.0
on the resistive scale and 2.0 on the
positive reactance circle.
AMATEUR RADIO – CTD
0 + j0 (SHORT CIRCUIT)
0.2
+0.2
0
0.2
0.
5
.5
+0
0.5
1
1
+1
SWR circles
2
0.5 + j1
+2
2
5
1 + j2
50
+5
5
1  j2
OPEN CIRCUIT
Fig.3: this simplified Smith Chart shows the resistance and
reactance circles plotted together. Note that the resistance
axis coincides with the zero reactance line.
which are resistance circles.
Each resistance circle is assigned
a particular value, shown where the
circle cuts the vertical resistance axis.
That value remains the same for all
points along that circle. In fact, the
values range from zero at the top of
the axis to infinity at the bottom and
represent a ratio with respect to the
centre point of the chart which is mark
ed “1”. By assigning the centre point or
“prime centre” of the chart a particular
value, each circle represents a value of
resistance scaled in accordance with
the ratio for that circle.
For example, if you allocate a value
of 100Ω to the centre point, any point
lying on the 0.5 circle has a value of
100 x 0.5 = 50Ω. Similarly, any point
on the 2.0 circle has a value of 200Ω.
This also means that the resistance
value of any point on the chart can be
calculated by multiplying the ratio
of the particular line with the value
assigned to the prime centre. The
value you would normally assign to
the centre point is the same as the
value of the characteristic impedance
of the line being matched, typically
54 Silicon Chip
50Ω. In fact, special printed charts are
available having a prime centre of 50Ω.
All resistance and reac
tance values
can then be plotted directly, without
having to “normalise” impedances.
Reactance circles
Fig.2 shows curves which are
reactance circles. Note that these
circles originate from the left and
right hand sides of the vertical zero
reactance line. The circumference of
the circle is the reactance axis. Just as
each resistance circle was assigned a
particular value, so are the reactance
lines. Any point along a reactance
circle has the same value and these
values can be multiplied (or normalised). Points located to the right of
the zero reactance axis are positive
(inductive) and values to the left are
negative (capacitive).
Note also that the vertical zero reactance line on Fig.2 coincides with
the vertical resistance axis on Fig.1.
That makes sense because any “pure”
resistance will have zero reactance.
Fig.3 is the composite Smith Chart,
with the resistance circles and reac
Now we come to the nub of the
matter, as far as most amateur radio
operators will be concerned. A useful addition to the Smith chart is the
standing wave circle. A series of these
can be drawn on the chart using a
draughting compass, centred on the
prime centre. The point at which a
circle for a given SWR crosses the
resistance axis is the value of SWR.
So the circle representing an SWR of
2:1 has its centre at the prime centre
and the radius crossing 2.0 on the
resistance axis.
Fig.4 shows a simplified Smith
Chart with SWR circles added.
If we wish to match a 50Ω transmission line, having a length of 2¼ wavelengths to a terminating impedance
of 25Ω resistive and 25Ω inductive
reactance (25 + j25), the following
procedure should be used. First, we
“normalise” the terminating impedance by dividing both components by
50. This equates to 0.5 +j0.5. We then
plot this impedance at the intersection
of the 0.5Ω resistance line and the 0.5Ω
reactance circle.
We know the reactance is positive
(inductive), so it must be located on the
right hand side of the resistance axis.
By drawing a circle, whose centre is at
the prime centre and whose radius is
the distance from the prime centre to
the impedance point, we have plotted
(0.5 + j0.5).
By noting where the circle intersects
the resistance axis, it can be seen that
a voltage ratio of 2.6:1 exists at that
point.
Wavelength scale
A comprehensive Smith Chart, as
distinct from the simplified examples
used here, also bears a wavelength
5.0 SWR
CIRCLE
0.2
+0.2
0
2.0 SWR
CIRCLE
0.2
0.
5
.5
+0
0.5
1
1
+1
2
GW
QUALITY
SCOPES
100MHz
+2
2
50
PLUS
FREE
DMM
+5
5
5
Fig.4: plotting SWR circles on a Smith Chart is a
useful step in the process of matching a transmitter
to an antenna, while avoiding the need for tedious
mathematical calculations.
scale around the perimeter of the chart. The scale is
marked in fractions of a wavelength of a transmission line.
One scale runs anticlockwise, starting at the “generator”
end, which is normally the input end, and running towards
the load. Another scale runs in the opposite direction from
load to generator. The complete circumference equals one
half wavelength.
Using our matching example above, we could further
progress towards a solution by drawing a line from the
prime centre, through the plotted 0.5 + j0.5 point, and to
the wavelength scale. As our plotted impedance point
is looking from the load end of the network, we use the
“towards generator” scale to read 0.088 wavelength at the
point of intersection.
We know that our 50ohm cable has a length of 2.25
wavelengths, and as the complete scale on the chart
represents a half wavelength and any impedance reflections will be repeated every half wavelength, we
need only use 0.25 as our transmission line length for
this calculation.
By adding 0.25 to the 0.088 indicated on the wavelength scale, we can locate the resultant 0.338 on the
wavelength scale and draw a line from that point to the
prime centre. The point where this line intersects our
2.6:1 SWR circle is the line input impedance, in this
example 1.0  j1.0. To find the line impedance, we simply multiply by 50, and this gives 50Ω resistive and 50Ω
capacitive. This is the impedance that the transmitter
must match.
Line loss & multielement matching
The Smith Chart can be used to calculate line loss and
also to facilitate the design of multielement matching
networks.
A comprehensive guide to the use of Smith Charts can
be found in the Sams publication “RF Circuit Design”
by Chris Bowick. Good background material can also be
SC
found in the ARRL Antenna Handbook.
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June 1993 55
