Smith
Chart – plots with Matlab
1.
Aid for RF problems
2.
Calculate Reflection Coefficient, VSWR and Return loss
3.
Video: alternative code to draw the chart
1.
Useful for visualization of radio frequency
and transmission line problems
The Smith chart
was
created by Phillip H. Smith in 1939. It’s a graphical tool designed for
electrical engineers specializing in radio frequency
(RF) to solve problems related to transmission
lines and matching circuits.
Use of the Smith chart has grown over the years and it’s still used
today, not only as a problem solving tool, but as a graphical display
to show how RF parameters behave at one or more frequencies.
The Smith chart can be
used to represent many parameters
including impedances, admittances, reflection and transmission
coefficients,
scattering parameters...
Normalized scaling allows
the Smith chart to be used for
problems involving any characteristic or system impedance which is
represented
by the center point of the chart. The most commonly used normalization
impedance is 50 ohms. Once an answer is obtained through the graphical
method,
it is easy to convert between the normalized impedance and the
corresponding
unnormalized value by multiplying by the characteristic impedance.
Reflection
coefficients can be read directly from the chart.
Use of the Smith chart
and the meaning of the results
obtained requires a good understanding of AC circuits and transmission
line
theory, both of which are needed by RF engineers.
In the chart, there are
horizontal circles that represent
constant resistances; vertical circles represent constant reactances.
The
complex impedance (or admittance )is normalized using a reference,
which
usually is the characteristic impedance Z_{0}.
The chart is part of
current CAD tools and modern
measurement equipments. It’s a graphic of the reflection coefficient in
polar
coordinates.
Let’s create a chart with
Matlab.
function
draw_smith_chart
%
Draw outer circle
t =
linspace(0, 2*pi, 100);
x =
cos(t);
y =
sin(t);
plot(x,
y, 'linewidth', 3); axis equal;
%
Place title and remove ticks from axes
title('
Smith Chart ')
set(gca,'xticklabel',{[]});
set(gca,'yticklabel',{[]});
hold on
%
Draw circles along horizontal axis
k = [.25
.5 .75];
for i = 1 :
length(k)
x(i,:)
= k(i) + (1  k(i)) * cos(t);
y(i,:)
= (1  k(i)) * sin(t);
plot(x(i,:),
y(i,:), 'k')
end
%
Draw partial circles along vertical axis
kt
=
[2.5 pi 3.79 4.22];
k = [.5
1 2 4];
for i = 1 :
length(kt)
t
= linspace(kt(i), 1.5*pi, 50);
a(i,:)
= 1 + k(i) * cos(t);
b(i,:)
= k(i) + k(i) * sin(t);
plot(a(i,:),
b(i,:),'k:',
a(i,:), b(i,:),'k:' )
end
2.
Calculate the Reflection Coefficient, VSWR and Return Loss
Now, let’s prepare some
formulas to calculate typical values
used in RF theory. To plot a reflection coefficient we need a
normalized
impedance with magnitude and angle. It’s a good idea to deliver the
angle in
degrees to make it easier to visualize, and also let’s calculate the
Voltage
Standing Wave Ratio (VSWR) and the return loss.
function [m,
thd, SWR, rloss] = smith_ch_calc(Z0, Zl)
%
Draw appropriate chart
draw_smith_chart
%
Normalize given impedance
zl =
Zl/Z0;
%
Calculate reflection, magnitude and angle
g = (zl
 1)/(zl + 1);
m =
abs(g);
th = angle(g);
%
Plot appropriate point
polar(th,
m, 'r*')
%
Change radians to degrees
thd = th
* 180/pi;
%
Calculate VSWR and return loss.
%
We can add epsilon to magnitude, to avoid div by 0 or log(0)
SWR = (1
+ m)/(1  m + eps);
rloss =
20 * log10(m + eps);
Now, let’s try our functions:
clear,
clc, close all, format compact
[m1, d1,
VSWR1, Rloss1] = smith_ch_calc(50, 50)
[m2, d2,
VSWR2, Rloss2] = smith_ch_calc(50, 100 + 50j)
[m3, d3,
VSWR3, Rloss3] = smith_ch_calc(50, 30  j*47)
and the
results are:
m1 =
0
d1 =
0
VSWR1 = 1.0000
Rloss1 = 313.0712
m2 =
0.4472
d2 = 26.5651
VSWR2 = 2.6180
Rloss2 = 6.9897
m3
=
0.5505
d3
=
82.6171
VSWR3
=
3.4494
Rloss3 =
5.1848
3. Video: alternative code to draw
the chart
This
video shows an alternative code to draw the Smith Chart. You can pause
the video to take appropriate notes...
From
'Smith Chart' to home
From
'Smith
Chart' to '2D Graphics Menu'
