<s>
The	O
Smith	B-Application
chart	I-Application
,	O
invented	O
by	O
Phillip	O
H	O
.	O
Smith	O
(	O
1905	O
–	O
1987	O
)	O
and	O
independently	O
by	O
Mizuhashi	O
Tosaku	O
,	O
is	O
a	O
graphical	O
calculator	O
or	O
nomogram	B-Application
designed	O
for	O
electrical	O
and	O
electronics	O
engineers	O
specializing	O
in	O
radio	O
frequency	O
(	O
RF	O
)	O
engineering	O
to	O
assist	O
in	O
solving	O
problems	O
with	O
transmission	O
lines	O
and	O
matching	O
circuits	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
can	O
be	O
used	O
to	O
simultaneously	O
display	O
multiple	O
parameters	O
including	O
impedances	O
,	O
admittances	O
,	O
reflection	O
coefficients	O
,	O
scattering	O
parameters	O
,	O
noise	O
figure	O
circles	O
,	O
constant	O
gain	O
contours	O
and	O
regions	O
for	O
unconditional	O
stability	O
,	O
including	O
mechanical	O
vibrations	O
analysis	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
is	O
most	O
frequently	O
used	O
at	O
or	O
within	O
the	O
unity	O
radius	O
region	O
.	O
</s>
<s>
While	O
the	O
use	O
of	O
paper	O
Smith	B-Application
charts	I-Application
for	O
solving	O
the	O
complex	O
mathematics	O
involved	O
in	O
matching	O
problems	O
has	O
been	O
largely	O
replaced	O
by	O
software	O
based	O
methods	O
,	O
the	O
Smith	B-Application
chart	I-Application
is	O
still	O
a	O
very	O
useful	O
method	O
of	O
showing	O
how	O
RF	O
parameters	O
behave	O
at	O
one	O
or	O
more	O
frequencies	O
,	O
an	O
alternative	O
to	O
using	O
tabular	B-Application
information	O
.	O
</s>
<s>
Thus	O
most	O
RF	O
circuit	O
analysis	O
software	O
includes	O
a	O
Smith	B-Application
chart	I-Application
option	O
for	O
the	O
display	O
of	O
results	O
and	O
all	O
but	O
the	O
simplest	O
impedance	O
measuring	O
instruments	O
can	O
plot	O
measured	O
results	O
on	O
a	O
Smith	B-Application
chart	I-Application
display	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
is	O
a	O
mathematical	B-Algorithm
transformation	I-Algorithm
of	O
the	O
two-dimensional	O
Cartesian	O
complex	O
plane	O
.	O
</s>
<s>
The	O
transformation	B-Algorithm
,	O
for	O
an	O
impedance	O
Smith	B-Application
chart	I-Application
,	O
is	O
:	O
</s>
<s>
The	O
impedance	O
Smith	B-Application
chart	I-Application
is	O
then	O
an	O
Argand	O
plot	O
of	O
impedances	O
thus	O
transformed	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
is	O
plotted	O
on	O
the	O
complex	O
reflection	O
coefficient	O
plane	O
in	O
two	O
dimensions	O
and	O
may	O
be	O
scaled	O
in	O
normalised	O
impedance	O
(	O
the	O
most	O
common	O
)	O
,	O
normalised	O
admittance	O
or	O
both	O
,	O
using	O
different	O
colours	O
to	O
distinguish	O
between	O
them	O
.	O
</s>
<s>
These	O
are	O
often	O
known	O
as	O
the	O
Z	O
,	O
Y	O
and	O
YZ	O
Smith	B-Application
charts	I-Application
respectively	O
.	O
</s>
<s>
Normalised	O
scaling	O
allows	O
the	O
Smith	B-Application
chart	I-Application
to	O
be	O
used	O
for	O
problems	O
involving	O
any	O
characteristic	O
or	O
system	O
impedance	O
which	O
is	O
represented	O
by	O
the	O
center	O
point	O
of	O
the	O
chart	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
has	O
a	O
scale	O
around	O
its	O
circumference	O
or	O
periphery	O
which	O
is	O
graduated	O
in	O
wavelengths	O
and	O
degrees	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
may	O
also	O
be	O
used	O
for	O
lumped-element	O
matching	O
and	O
analysis	O
problems	O
.	O
</s>
<s>
Use	O
of	O
the	O
Smith	B-Application
chart	I-Application
and	O
the	O
interpretation	O
of	O
the	O
results	O
obtained	O
using	O
it	O
requires	O
a	O
good	O
understanding	O
of	O
AC	O
circuit	O
theory	O
and	O
transmission-line	O
theory	O
,	O
both	O
of	O
which	O
are	O
prerequisites	O
for	O
RF	O
engineers	O
.	O
</s>
<s>
As	O
impedances	O
and	O
admittances	O
change	O
with	O
frequency	O
,	O
problems	O
using	O
the	O
Smith	B-Application
chart	I-Application
can	O
only	O
be	O
solved	O
manually	O
using	O
one	O
frequency	O
at	O
a	O
time	O
,	O
the	O
result	O
being	O
represented	O
by	O
a	O
point	O
.	O
</s>
<s>
This	O
is	O
often	O
adequate	O
for	O
narrow	O
band	O
applications	O
(	O
typically	O
up	O
to	O
about	O
5%	O
to	O
10%	O
bandwidth	B-Algorithm
)	O
but	O
for	O
wider	O
bandwidths	B-Algorithm
it	O
is	O
usually	O
necessary	O
to	O
apply	O
Smith	B-Application
chart	I-Application
techniques	O
at	O
more	O
than	O
one	O
frequency	O
across	O
the	O
operating	O
frequency	O
band	O
.	O
</s>
<s>
Provided	O
the	O
frequencies	O
are	O
sufficiently	O
close	O
,	O
the	O
resulting	O
Smith	B-Application
chart	I-Application
points	O
may	O
be	O
joined	O
by	O
straight	O
lines	O
to	O
create	O
a	O
locus	O
.	O
</s>
<s>
A	O
locus	O
of	O
points	O
on	O
a	O
Smith	B-Application
chart	I-Application
covering	O
a	O
range	O
of	O
frequencies	O
can	O
be	O
used	O
to	O
visually	O
represent	O
:	O
</s>
<s>
The	O
accuracy	O
of	O
the	O
Smith	B-Application
chart	I-Application
is	O
reduced	O
for	O
problems	O
involving	O
a	O
large	O
locus	O
of	O
impedances	O
or	O
admittances	O
,	O
although	O
the	O
scaling	O
can	O
be	O
magnified	O
for	O
individual	O
areas	O
to	O
accommodate	O
these	O
.	O
</s>
<s>
The	O
SI	O
unit	O
of	O
impedance	O
is	O
the	O
ohm	O
with	O
the	O
symbol	O
of	O
the	O
upper	O
case	O
Greek	O
letter	O
omega	B-General_Concept
( Ω	O
)	O
and	O
the	O
SI	O
unit	O
for	O
admittance	O
is	O
the	O
siemens	O
with	O
the	O
symbol	O
of	O
an	O
upper	O
case	O
letter	O
S	O
.	O
Normalised	O
impedance	O
and	O
normalised	O
admittance	O
are	O
dimensionless	O
.	O
</s>
<s>
Actual	O
impedances	O
and	O
admittances	O
must	O
be	O
normalised	O
before	O
using	O
them	O
on	O
a	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
is	O
used	O
with	O
one	O
frequency	O
(	O
)	O
at	O
a	O
time	O
,	O
and	O
only	O
for	O
one	O
moment	O
(	O
)	O
at	O
a	O
time	O
,	O
so	O
the	O
temporal	O
part	O
of	O
the	O
phase	O
(	O
)	O
is	O
fixed	O
.	O
</s>
<s>
If	O
the	O
line	O
is	O
lossy	O
(	O
is	O
non-zero	O
)	O
this	O
is	O
represented	O
on	O
the	O
Smith	B-Application
chart	I-Application
by	O
a	O
spiral	O
path	O
.	O
</s>
<s>
In	O
most	O
Smith	B-Application
chart	I-Application
problems	O
however	O
,	O
losses	O
can	O
be	O
assumed	O
negligible	O
(	O
)	O
and	O
the	O
task	O
of	O
solving	O
them	O
is	O
greatly	O
simplified	O
.	O
</s>
<s>
The	O
outer	O
circumferential	O
scale	O
of	O
the	O
Smith	B-Application
chart	I-Application
represents	O
the	O
distance	O
from	O
the	O
generator	O
to	O
the	O
load	O
scaled	O
in	O
wavelengths	O
and	O
is	O
therefore	O
scaled	O
from	O
zero	O
to	O
0.50	O
.	O
</s>
<s>
These	O
are	O
the	O
equations	O
which	O
are	O
used	O
to	O
construct	O
the	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
Mathematically	O
speaking	O
and	O
are	O
related	O
via	O
a	O
Möbius	O
transformation	B-Algorithm
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
is	O
actually	O
constructed	O
on	O
such	O
a	O
polar	O
diagram	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
scaling	O
is	O
designed	O
in	O
such	O
a	O
way	O
that	O
reflection	O
coefficient	O
can	O
be	O
converted	O
to	O
normalised	O
impedance	O
or	O
vice	O
versa	O
.	O
</s>
<s>
Using	O
the	O
Smith	B-Application
chart	I-Application
,	O
the	O
normalised	O
impedance	O
may	O
be	O
obtained	O
with	O
appreciable	O
accuracy	O
by	O
plotting	O
the	O
point	O
representing	O
the	O
reflection	O
coefficient	O
treating	O
the	O
Smith	B-Application
chart	I-Application
as	O
a	O
polar	O
diagram	O
and	O
then	O
reading	O
its	O
value	O
directly	O
using	O
the	O
characteristic	O
Smith	B-Application
chart	I-Application
scaling	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
graphical	O
equivalent	O
of	O
using	O
the	O
transmission-line	O
equation	O
is	O
to	O
normalise	O
to	O
plot	O
the	O
resulting	O
point	O
on	O
a	O
Smith	B-Application
chart	I-Application
and	O
to	O
draw	O
a	O
circle	O
through	O
that	O
point	O
centred	O
at	O
the	O
Smith	B-Application
chart	I-Application
centre	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
uses	O
the	O
same	O
convention	O
,	O
noting	O
that	O
,	O
in	O
the	O
normalised	O
impedance	O
plane	O
,	O
the	O
positive	O
-axis	O
extends	O
from	O
the	O
center	O
of	O
the	O
Smith	B-Application
chart	I-Application
at	O
to	O
the	O
point	O
The	O
region	O
above	O
the	O
x-axis	O
represents	O
inductive	O
impedances	O
(	O
positive	O
imaginary	O
parts	O
)	O
and	O
the	O
region	O
below	O
the	O
-axis	O
represents	O
capacitive	O
impedances	O
(	O
negative	O
imaginary	O
parts	O
)	O
.	O
</s>
<s>
If	O
the	O
termination	O
is	O
perfectly	O
matched	O
,	O
the	O
reflection	O
coefficient	O
will	O
be	O
zero	O
,	O
represented	O
effectively	O
by	O
a	O
circle	O
of	O
zero	O
radius	O
or	O
in	O
fact	O
a	O
point	O
at	O
the	O
centre	O
of	O
the	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
If	O
the	O
termination	O
was	O
a	O
perfect	O
open	O
circuit	O
or	O
short	B-Application
circuit	I-Application
the	O
magnitude	O
of	O
the	O
reflection	O
coefficient	O
would	O
be	O
unity	O
,	O
all	O
power	O
would	O
be	O
reflected	O
and	O
the	O
point	O
would	O
lie	O
at	O
some	O
point	O
on	O
the	O
unity	O
circumference	O
circle	O
.	O
</s>
<s>
The	O
normalised	O
impedance	O
Smith	B-Application
chart	I-Application
is	O
composed	O
of	O
two	O
families	O
of	O
circles	O
:	O
circles	O
of	O
constant	O
normalised	O
resistance	O
and	O
circles	O
of	O
constant	O
normalised	O
reactance	O
.	O
</s>
<s>
In	O
the	O
complex	O
reflection	O
coefficient	O
plane	O
the	O
Smith	B-Application
chart	I-Application
occupies	O
a	O
circle	O
of	O
unity	O
radius	O
centred	O
at	O
the	O
origin	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
is	O
constructed	O
in	O
a	O
similar	O
way	O
to	O
the	O
Smith	B-Application
chart	I-Application
case	O
but	O
by	O
expressing	O
values	O
of	O
voltage	O
reflection	O
coefficient	O
in	O
terms	O
of	O
normalised	O
admittance	O
instead	O
of	O
normalised	O
impedance	O
.	O
</s>
<s>
The	O
Smith	B-Application
chart	I-Application
appears	O
like	O
the	O
normalised	O
impedance	O
,	O
type	O
but	O
with	O
the	O
graphic	O
nested	O
circles	O
rotated	O
through	O
180°	O
,	O
but	O
the	O
numeric	O
scale	O
remaining	O
in	O
its	O
same	O
position	O
(	O
not	O
rotated	O
)	O
as	O
the	O
chart	O
.	O
</s>
<s>
Again	O
,	O
if	O
the	O
termination	O
is	O
perfectly	O
matched	O
the	O
reflection	O
coefficient	O
will	O
be	O
zero	O
,	O
represented	O
by	O
a	O
'	O
circle	O
 '	O
of	O
zero	O
radius	O
or	O
in	O
fact	O
a	O
point	O
at	O
the	O
centre	O
of	O
the	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
If	O
the	O
termination	O
was	O
a	O
perfect	O
open	O
or	O
short	B-Application
circuit	I-Application
the	O
magnitude	O
of	O
the	O
voltage	O
reflection	O
coefficient	O
would	O
be	O
unity	O
,	O
all	O
power	O
would	O
be	O
reflected	O
and	O
the	O
point	O
would	O
lie	O
at	O
some	O
point	O
on	O
the	O
unity	O
circumference	O
circle	O
of	O
the	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
A	O
point	O
with	O
a	O
reflection	O
coefficient	O
magnitude	O
0.63	O
and	O
angle	O
60°	O
represented	O
in	O
polar	O
form	O
as	O
,	O
is	O
shown	O
as	O
point	O
P1	O
on	O
the	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
To	O
plot	O
this	O
,	O
one	O
may	O
use	O
the	O
circumferential	O
(	O
reflection	O
coefficient	O
)	O
angle	O
scale	O
to	O
find	O
the	O
graduation	O
and	O
a	O
ruler	O
to	O
draw	O
a	O
line	O
passing	O
through	O
this	O
and	O
the	O
centre	O
of	O
the	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
The	O
length	O
of	O
the	O
line	O
would	O
then	O
be	O
scaled	O
to	O
P1	O
assuming	O
the	O
Smith	B-Application
chart	I-Application
radius	O
to	O
be	O
unity	O
.	O
</s>
<s>
The	O
following	O
table	O
gives	O
some	O
similar	O
examples	O
of	O
points	O
which	O
are	O
plotted	O
on	O
the	O
Z	O
Smith	B-Application
chart	I-Application
.	O
</s>
<s>
The	O
conversion	O
may	O
be	O
read	O
directly	O
from	O
the	O
Smith	B-Application
chart	I-Application
or	O
by	O
substitution	O
into	O
the	O
equation	O
.	O
</s>
<s>
Solving	O
a	O
typical	O
matching	O
problem	O
will	O
often	O
require	O
several	O
changes	O
between	O
both	O
types	O
of	O
Smith	B-Application
chart	I-Application
,	O
using	O
normalised	O
impedance	O
for	O
series	O
elements	O
and	O
normalised	O
admittances	O
for	O
parallel	O
elements	O
.	O
</s>
<s>
For	O
these	O
a	O
dual	O
(	O
normalised	O
)	O
impedance	O
and	O
admittance	O
Smith	B-Application
chart	I-Application
may	O
be	O
used	O
.	O
</s>
<s>
To	O
graphically	O
change	O
this	O
to	O
the	O
equivalent	O
normalised	O
admittance	O
point	O
,	O
say	O
Q1	O
,	O
a	O
line	O
is	O
drawn	O
with	O
a	O
ruler	O
from	O
P1	O
through	O
the	O
Smith	B-Application
chart	I-Application
centre	O
to	O
Q1	O
,	O
an	O
equal	O
radius	O
in	O
the	O
opposite	O
direction	O
.	O
</s>
<s>
Reading	O
the	O
value	O
from	O
the	O
Smith	B-Application
chart	I-Application
for	O
Q1	O
,	O
remembering	O
that	O
the	O
scaling	O
is	O
now	O
in	O
normalised	O
admittance	O
,	O
gives	O
.	O
</s>
<s>
Once	O
a	O
transformation	B-Algorithm
from	O
impedance	O
to	O
admittance	O
has	O
been	O
performed	O
,	O
the	O
scaling	O
changes	O
to	O
normalised	O
admittance	O
until	O
a	O
later	O
transformation	B-Algorithm
back	O
to	O
normalised	O
impedance	O
is	O
performed	O
.	O
</s>
<s>
Again	O
,	O
these	O
may	O
be	O
obtained	O
either	O
by	O
calculation	O
or	O
using	O
a	O
Smith	B-Application
chart	I-Application
as	O
shown	O
,	O
converting	O
between	O
the	O
normalised	O
impedance	O
and	O
normalised	O
admittances	O
planes	O
.	O
</s>
<s>
The	O
choice	O
of	O
whether	O
to	O
use	O
the	O
Z	O
Smith	B-Application
chart	I-Application
or	O
the	O
Y	O
Smith	B-Application
chart	I-Application
for	O
any	O
particular	O
calculation	O
depends	O
on	O
which	O
is	O
more	O
convenient	O
.	O
</s>
<s>
For	O
distributed	O
components	O
the	O
effects	O
on	O
reflection	O
coefficient	O
and	O
impedance	O
of	O
moving	O
along	O
the	O
transmission	O
line	O
must	O
be	O
allowed	O
for	O
using	O
the	O
outer	O
circumferential	O
scale	O
of	O
the	O
Smith	B-Application
chart	I-Application
which	O
is	O
calibrated	O
in	O
wavelengths	O
.	O
</s>
<s>
This	O
is	O
plotted	O
on	O
the	O
Z	O
Smith	B-Application
chart	I-Application
at	O
point	O
P20	O
.	O
</s>
<s>
As	O
the	O
transmission	O
line	O
is	O
loss	O
free	O
,	O
a	O
circle	O
centred	O
at	O
the	O
centre	O
of	O
the	O
Smith	B-Application
chart	I-Application
is	O
drawn	O
through	O
the	O
point	O
P20	O
to	O
represent	O
the	O
path	O
of	O
the	O
constant	O
magnitude	O
reflection	O
coefficient	O
due	O
to	O
the	O
termination	O
.	O
</s>
<s>
An	O
alternative	O
shunt	O
match	O
could	O
be	O
calculated	O
after	O
performing	O
a	O
Smith	B-Application
chart	I-Application
transformation	B-Algorithm
from	O
normalised	O
impedance	O
to	O
normalised	O
admittance	O
.	O
</s>
<s>
A	O
generalization	O
of	O
the	O
Smith	B-Application
chart	I-Application
to	O
a	O
three	O
dimensional	O
sphere	O
,	O
based	O
on	O
the	O
extended	O
complex	O
plane	O
(	O
Riemann	O
sphere	O
)	O
and	O
inversive	O
geometry	O
,	O
was	O
proposed	O
by	O
Muller	O
,	O
et	O
al	O
in	O
2011	O
.	O
</s>
<s>
The	O
3D	O
Smith	B-Application
chart	I-Application
has	O
been	O
further	O
extended	O
outside	O
of	O
the	O
spherical	O
surface	O
,	O
for	O
plotting	O
various	O
scalar	O
parameters	O
,	O
such	O
as	O
group	O
delay	O
,	O
quality	O
factors	O
,	O
or	O
frequency	O
orientation	O
.	O
</s>
<s>
The	O
visual	O
frequency	O
orientation	O
(	O
clockwise	O
vs.	O
counter-clockwise	O
)	O
enables	O
one	O
to	O
differentiate	O
between	O
a	O
negative	O
/	O
capacitance	O
and	O
positive	O
/	O
inductive	O
whose	O
reflection	O
coefficients	O
are	O
the	O
same	O
when	O
plotted	O
on	O
a	O
2D	O
Smith	B-Application
chart	I-Application
,	O
but	O
whose	O
orientations	O
diverge	O
as	O
frequency	O
increases	O
.	O
</s>
