The Smith chart, also known as
a polar impedance plot, was invented by Philip Smith in 1939 to plot the
characteristics of microwave components such as reflection coefficient,
impedance, and admittance. With the help of a Smith chart, complex mathematical
equations can be simplified.
The Smith Chart is the
graphical representation of a complex mathematical equation. It is the circular
plot of the characteristics of microwave components. The Smith Chart is the
most used tool for microwave engineers to visualize complex-valued quantities
and calculate the mapping between them. It consists of two sets of circles for
plotting various parameters of mismatched transmission lines. One is the set of
complete circles whose centers lie on the straight line and the other one is
the set of two arc circles which lie on the either sides of the straight line.
The figure to the right shows
the Smith Chart. The horizontal axis represents the normalized resistance and
the normalized line reactance is shown on the outer edge of the circles. The
complete circle of the Smith Chart represents a half wavelength along the
straight line.
The important applications of
a Smith Chart are as follows:
1. Admittance calculations on any
transmission line, on any load.
2. Impedance calculations on any
transmission line, on any load.
3. Calculation of the length of a short
circuited piece of transmission line to provide a required capacitive or
inductive reactance.
4. Impedance matching.
Before plotting on a Smith
Chart we need to study a few terms such as transmission line, characteristic
impedance, standing wave, etc.
Transmission lines are the
circuits that deliver power from a transmitter to an antenna and from an
antenna to a receiver. These are the impedance matching circuits. While
transferring the RF waves on a finite transmission line, for maximum power
transfer from source to load (i.e. transmitter to antenna or antenna to
receiver), the impedance of source must be equal to the impedance of load. This
is known as impedance matching. If there is impedance mismatching, part of the
energy in the incident wave is reflected back giving rise to a standing wave.
The ratio of the voltage of reflected wave and the voltage of incident wave is
known as the reflection coefficient. It is denoted by Γ.
It is expressed as Γ=
Vref/ Vinc.
The amount of reflected wave
depends upon the mismatching of the source and load impedances. Therefore, from
the fig., gamma can be expressed as,
Γ = (ZL - Zo) / ( ZL + Zo
) where, ZL is load impedance and Zo is the characteristic impedance of
transmission line, a constant.
ZL= RL + jXL.
Normalized impedance is used
for plotting on Smith chart. This is because the behavior of the transmission
line depends on load impedance as well as characteristic impedance.
Normalized impedance, Z = ZL/
Zo = r+ jx, where r = R / Z0 and x = X / Z0.
Therefore, Reflection
coefficient can expressed as
Γ = (Z-1) / (Z+1) or
Γ = (ZL-1) / (ZL+1) as Zo
takes constant values such as 50Ω, 100Ω, etc.
Γ = [(r +jx) - 1] / [(r +
jx) + 1]
Γ = [(r - 1) + jx] / [(r
+ 1) + jx] for r lies between 0 and ∞ , x lies between -∞ and
+∞.
The above equation can also be
written as
Γ = u + jv = Rejθ ,
where R is the radius and θ is the angle of incidence.
Z = r + jx
From the last two equation, we
can see that there is one-to-one correspondence and for every Z we can obtain a
unique Γ. Following fig. shows mapping of Z to Γ- plane.
lΓl ≤ 1. Therefore,
the possible values of Γ will remain within the unit circle.
Z = (1 + Γ) / (1 - Γ)
r + jx = [1 + (u + jv)] / [1 -
(u + jv)]
Separating the real and
imaginary part, we get
r = (1 - u2 - v2) / [(1 - u)2
+ v2] and
x = 2v / [(1- u)2 + v2]
Furher simplifying the above
two equations, we can re-write them in standard circle equation form as
[u- r/(r+1)]2 + (v - 0)2 =
[1/(r+1)]2 with center at [r/(r+1), 0] and radius = 1/(r+1) and
(u - 1)2 + (v - 1/x)2 = (1/x)2
with center at (1, 1/x) and radius = 1/x.
The circles with centers at
[r/(r+1), 0] and radii = 1/(r+1) are known as constant resistance circles as
shown in following fig.The circles with centers at (1, 1/x) and radii = 1/x are
known as constant reactance circles as shown in following fig.
The Smith Chart is drawn by
superimposing these two types of circles.