Smith chart
basics
Updated January 15,
2012
Click
here to go to our main page on Sparameters
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here to go to our page on VSWR
Click
here to go to a page on plotting Smith Charts with Excel
New for January 2012! Here's a new page on a threedimensional Smith Chart!
We've got our own Smith chart
tutorial here, thanks to a fan from Florida, Mike Weinstein, who
really knows this subject, and is a fine writer too. If anyone else
wants to be a technical contributor on their favorite microwave
subject, please contact
us.
If you want to download a Smith
chart in pdf or gif format, we have several different ones in our
download area!
The Smith chart was developed
by Philip Smith at Bell Telephone's Radio Research Lab during the
1930s. Be sure to check out our entry on Philip
Smith in our Microwave Hall of Fame! Phil's wife still operates
Analog Instruments, the company that sells the chart. Their snail
mail address is:
Analog Instruments Company,
P.O. Box 950,
New Providence, NJ 07974,
(908) 4644214.
A clickable index to our growing
Smith chart page:
What's a Smith
Chart?
Impedance,
admittance
Which
way is up and where's that short circuit?
"Yes
sir!" and please don't flip me!
Normalization
How much
is that stub worth?
Singlestub
matching
Some Smith
chart links
What's a
Smith chart?
What is a Smith chart? It's really
just a plot of complex reflection overlaid with an impedance
and/or admittance grid referenced to a 1ohm characteristic
impedance. That's it! Transmission coefficient, which equals unity
plus reflection coefficient, may also be plotted (see below). You
can find books and articles describing how a Smith chart is a graphical
representation of the transmission line equations and the mathematical
reasons for the circles and arcs, but these things don't really
matter when you need to get the job done. What matters is knowing
the basics and how to use them, like always.
The Smith chart contains almost
all possible impedances, real or imaginary, within one circle. All
imaginary impedances from  infinity to + infinity are represented,
but only positive real impedances appear on the "classic"
Smith chart. Yes, it is possible to go outside the Smith chart "unity"
circle, but only with an active device because this implies negative
resistance.
One thing you give up when plotting
reflection coefficients on a Smith chart is a direct reading of
a frequency axis. Typically, plots that are done over any frequency
band have markers calling out specific frequencies.
Why use a Smith chart?
It's got all those funny circles and arcs, and good ol' rectangular
plots are much better for displaying things like VSWR, transmission
loss, and phase, right? Perhaps sometimes a rectangular plot is
better, but a Smith chart is the RF engineer's best friend! It's
easy to master, and it adds an air of "analog coolness"
to presentations, which will impress your friends, if not your dates!
A master in the art of Smithcharting can look at a thoroughly messed
up VSWR of a component or network, and synthesize two or three simple
networks that will impedancematch the circuit in his head!
Impedance
and admittance
A quick refresher on the basic
quantities that have units of ohms or its reciprocal, Siemens (sometimes
called by its former name, mhos), is helpful since many of them
will be referenced below. We all think of resistance (R) as the
most fundamental of these quantities, a measure of the opposition
to current flow that causes a potential drop, or voltage, according
to Ohms Law: V=I*R. By extension, impedance (Z) is the steady state
AC term for the combined effect of both resistance and reactance
(X), where Z=R+jX. (X=jwL for an inductor, and X=1/jwC for a capacitor,
where w is the radian frequency or 2*pi*f.) Generally, Z is a complex
quantity having a real part (resistance) and an imaginary part (reactance).
We often think in terms of impedance
and its constituent quantities of resistance and reactance. These
three terms represent "opposition" quantities and are
a natural fit for seriesconnected circuits where impedances add
together. However, many circuits have elements connected in parallel
or "shunt" that are a natural fit for the "acceptance"
quantity of admittance (Y) and its constituent quantities of conductance
(G) and susceptance (B), where Y=G+jB. (B=jwC for a capacitor, and
B=1/jwL for an inductor.) Admittances add together for shuntconnected
circuits. Remember that Y=1/Z=1/(R+jX), so that G=1/R only if X=0,
and B=1/X only if R=0.
When working with a seriesconnected
circuit or inserting elements in series with an existing circuit
or transmission line, the resistance and reactance components are
easily manipulated on the "impedance" Smith chart.
Similarly, when working with a parallelconnected circuit or inserting
elements in parallel with an existing circuit or transmission line,
the conductance and susceptance components are easily manipulated
on the "admittance" Smith chart. The "immittance"
Smith chart simply has both the impedance and admittance
grids on the same chart, which is useful for cascading seriesconnected
with parallelconnected circuits.
Which
way is up and where's that short circuit?
The most common orientation of
the Smith chart places the resistance axis horizontally with the
short circuit (SC) location at the far left. There's a good reason
for this: the voltage of the reflected wave at a short circuit must
cancel the voltage of the incident wave so that zero potential exists
across the short circuit. In other words, the voltage reflection
coefficient must be 1 or a magnitude of 1 at an angle of 180 degrees.
Since angles are measured from the positive real axis and the real
axis is horizontal, the short circuit location and horizontal orientation
make sense. ("Voltage" is underlined above because the
current reflection coefficient of a short circuit being +1 would
place the short circuit location at the right end, but let's not
go there.)
For an open circuit (OC), the
reflected voltage is equal to and in phase with the incident voltage
(reflection coefficient of +1) so that the open circuit location
is on the right. In general, the reflection coefficient has a magnitude
other than unity and is complex. For reasons we won't bore you with
here, anywhere above the real axis is inductive (L) and anywhere
below is capacitive (C).
"Yes
sir!" and please don't flip me!
Can't remember which way to rotate
the locus when moving along the transmission line? Well, it's
clockwise toward the generator because generals make you go like
clockwork. Also keep in mind that moving "x" degrees
along the line moves a point on the locus "2x" degrees
on the chart because the reflected wave must transverse the roundtrip
distance moved (remember, it's the reflection coefficient).
Alternately, you could remember that the impedance repeats itself
every half wavelength along a uniform transmission line, so you
must move one time around the chart to wind up at the same impedance.
Of course, a physical line length has variable electrical length
over a frequency band, so a fixed impedance will spread out to a
locus when viewed through a connected transmission line. This is
why it is always easier to obtain a wideband match when you're close
to the device or discontinuity.
Many older RF engineers advocate
reflecting through the origin to "convert" from impedance
to admittance and vice versa. That's why you see the same axis labeled
"INDUCTIVE REACTANCE OR CAPACITIVE SUSCEPTANCE" on the
original Smith chart, for example. This can be confusing, you've
got to do the flip, you need to remember what the grid currently
represents, and SC, OC, L & C are moving targets! Why not just
keep the reflection coefficient where it belongs and use the appropriate
grid? We have computers, color printers, and immittance charts these
days. (If you still like to do things manually and either can't
deal with all those lines on an immittance chart or are color blind,
use a transparency overlay and a blank piece of paper.)
Normalization
Moving along a uniform transmission
line doesn't change the magnitude of the reflection coefficient
or its radial distance plotted on the Smith chart. But what about
when the impedance of the line changes, for example, when a quarterwavelength
transformer is used? Reflection coefficient (Gamma) is, by definition,
normalized to the characteristic impedance (Z_{0}) of the
transmission line:
Gamma = (Z_{L}Z_{0})
/ (Z_{L}+Z_{0})
where Z_{L} is the load
impedance or the impedance at the reference plane. Note that Gamma
is generally complex. Likewise, the impedance (admittance) values
indicated on the grid lines are normalized to the characteristic
impedance (admittance) of the transmission line to which the reflection
coefficient is normalized.
When Z_{0} changes just
past the junction between two different transmission lines, so does
the reflection coefficient. Determining the new impedance (admittance)
is simple: multiply by the characteristic impedance (admittance)
of the current line (this yields the unnormalized value), then
divide by the characteristic impedance (admittance) of the new line
to obtain the new renormalized value.
The new Gamma may be calculated
with the formula above or graphically determined by drawing a line
from the origin to the new renormalized value. This example ignores
the effect of the step discontinuity encountered in physical (nonideal)
transmission lines, which typically introduces some shunt capacitance.
How
much is that stub worth?
Transmission line stubs are essential
for impedance matching, introducing small amounts of phase delay
(in pairs to cancel
reflections), biasing, etc. Are you sometimes unsure that a
shortcircuited stub that's less than a quarter wavelength is inductive,
or whether a wide, low impedance stub in shunt with the main line
has low or high Q? A smith chart can tell you these things and give
you hard numbers in a jiffy.
For example, a shortcircuited
stub is just a short circuit seen through a length of transmission
line. Place your pencil at the SC point on the chart and move clockwise
toward the generator (at the other end of the stub) on the rim by
an amount less than a quarter wavelength (180 degrees on the chart).
This is in the inductive region; moving more than 180 degrees makes
the stub input look capacitive. At exactly onequarter wavelength,
the impedance is infinite, an open circuit. You can do the same
for an opencircuited stub by starting at the OC point on the chart.
The real power of the Smith chart
comes into play for analysis over a frequency band. Suppose you
want to know the susceptance variation of a 50ohm shortcircuited
stub over a 3:1 band. This stub could be placed in shunt with the
main line at the proper point to doubletune a seriesresonant locus,
for instance. (We'll cover doubletuning, a very powerful technique,
in a future update.) Shown in the admittance chart below is a shortcircuited
stub that's oneeight wavelength long at the low end and thus is
threeeighths wavelengths long at the high end of the 3:1 frequency
band. The normalized susceptance varies from 1.0 siemens
(inductive) at f_{low} to zero (open circuit) at midband
to +1.0 siemens (capacitive) at f_{high}. Therefore, the
unnormalized susceptance varies between ±1.0*Y_{0} siemens,
where Y_{0} (=1/Z_{0}) is the characteristic admittance
of the stub. When the characteristic admittance (Y_{0})
of the stub is the same as the main line, the normalized susceptance
of the stub may be added to the normalized admittance of the load
at each frequency to yield the normalized admittance of the parallel
combination. When Y_{0} of the stub differs from that of
the main line, renormalize the stub's susceptance by Y_{0}
of the main line before adding.
Generally, the desired susceptance
variation is other than ±0.02 siemens (±1.0*Y_{0}), which
a 50ohm stub would provide in this example. Suppose a 50ohm main
line locus needs a normalized susceptance variation of only
±0.4 siemens instead of ±1.0 siemens. Achieve this simply by making
the characteristic admittance of the stub equal to 0.4 times that
of the main line or Y_{0}=0.4*0.02=0.008 siemens. The stub
is now a 125ohm line (50/0.4) and its susceptance varies less over
the band, so it has lower Q. Note that the unnormalized values are
rarely needed, normalized values may be renormalized by the ratio
of the characteristic impedances involved.
Next, consider a stub for changing
the transmission phase of a mainline signal. We know that an opencircuited
stub less than a quarter wavelength long retards the phase (adds
phase delay), and this is readily seen on the Smith chart: Moving
clockwise from the OC position, an opencircuited stub has a transmission
coefficient (1 + Gamma) with a negative phase angle. Similarly,
a shortcircuited stub less than a quarter wavelength long will
advance the phase. The following figure illustrates the phase delay
of 50ohm and 25ohm opencircuited stubs in shunt with a 50ohm
main line. Note that the result is mismatched, which is why stubs
should be added in pairs to cancel
reflections. Also note that the amount of phase delay increases
as the characteristic impedance of the stub decreases (a larger
Y_{0} produces a larger unnormalized susceptance), which
makes sense since a wider stub looks like a larger capacitor.
Singlestub
matching
The ability to obtain a reasonable
match over a frequency band depends upon the magnitude of the mismatch,
the desired bandwidth, and the complexity of matching circuit. But
at any one frequency any impedance mismatch can be perfectly matched
to the characteristic impedance of the transmission line, as long
as it's not on the rim of the chart (perfect reflection, Gamma
= 1). And this always can be done with one stub that's less than
a quarterwavelength long. The technique is simple: move along
the transmission line to rotate the mismatch to the unity resistance
(conductance) circle and insert the appropriate type and length
of stub in series (shunt) with the main line to move along this
circle to the origin. If the far end of the stub is either a
short or open circuit (or generally, any pure reactance), its input
end is also a pure reactance (susceptance) so that it doesn't affect
the resistance (conductance) component of the mainline impedance
(admittance).
Since it's usually easier to
add a stub in parallel with a transmission line, the example shown
below uses an admittance chart because, at the attachment point,
the resulting admittance is the sum of the stub's input susceptance
and the main line admittance. First, the mismatched point is rotated
around the origin until it reaches the unity conductance circle.
Then, the characteristic impedance and length of the stub is chosen
such that its input susceptance is equal and opposite to the main
line susceptance indicated on the unity conductance circle. The
example shows two cases: move toward the generator 39 degrees of
line and add a shortcircuited stub that provides 0.8 siemens normalized
inductive susceptance, or move toward the generator 107 degrees
of line and add an opencircuited stub that provides 0.8 siemens
normalized capacitive susceptance.
There are an infinite number
of possible solutions because, at one frequency, a stub of any characteristic
impedance can provide the necessary normalized susceptance simply
by adjusting its length. The differences show up when looking over
a frequency band. For example, the stub's length may be increased
by an integer multiple of halfwavelengths at a particular frequency
and its input susceptance at this frequency will not change. But
over a frequency band, the susceptance will vary considerably more
than if the extra length had not been added.
Some Smith
chart links
Here are some links on Smith
charts for anyone that wants additional info or needs a Smith
chart clock!
http://www.sssmag.com/smith.html
