Mathematical Foundations of the Smith Chart
The Smith Chart is based on a mathematical transformation between the complex reflection coefficient and normalized impedance. Understanding these relationships explains why the Smith Chart consists of constant resistance circles, constant reactance arcs, and rotational properties used in transmission line analysis.
Mathematical Justification of the Smith Chart Grid
The Smith Chart represents a conformal mapping of the complex reflection coefficient to normalized impedance.
We begin with the reflection coefficient:
\[
\Gamma = \Gamma_r + j\Gamma_i
\]
and the normalized impedance:
\[
z = r + jx
\]
The fundamental relationship is:
\[
\Gamma = \frac{z - 1}{z + 1}
\]
Rearranging for normalized impedance:
\[
z = \frac{1 + \Gamma}{1 - \Gamma}
\]
Substituting:
\[
z = \frac{1 + (\Gamma_r + j\Gamma_i)}
{1 - (\Gamma_r + j\Gamma_i)}
\]
Separating the real and imaginary components gives:
\[
r =
\frac{1 - \Gamma_r^2 - \Gamma_i^2}
{(1-\Gamma_r)^2+\Gamma_i^2}
\]
\[
x =
\frac{2\Gamma_i}
{(1-\Gamma_r)^2+\Gamma_i^2}
\]
These equations can be rearranged into the standard circle form:
\[
(x-h)^2+(y-k)^2=R^2
\]
Constant Resistance Circles
The constant resistance circles are given by:
\[
\left(
\Gamma_r-\frac{r}{1+r}
\right)^2
+
\Gamma_i^2
=
\left(
\frac{1}{1+r}
\right)^2
\]
The circle center is:
\[
\left(
\frac{r}{1+r},
0
\right)
\]
The radius is:
\[
R_r=
\frac{1}{1+r}
\]
These circles form the resistance grid of the Smith Chart.
Constant Reactance Arcs
The constant reactance arcs are given by:
\[
(\Gamma_r-1)^2
+
\left(
\Gamma_i-\frac{1}{x}
\right)^2
=
\left(
\frac{1}{x}
\right)^2
\]
The arc center is:
\[
\left(
1,
\frac{1}{x}
\right)
\]
The radius is:
\[
R_x=
\frac{1}{|x|}
\]
These arcs form the reactance grid of the Smith Chart.
Justification of 180° Phase Inversion for Admittance
For shunt matching applications, impedance must be converted into admittance.
We know:
\[
y=\frac{1}{z}
\]
Substituting the reflection coefficient relationship:
\[
y=
\frac{1-\Gamma}
{1+\Gamma}
\]
This may be rewritten as:
\[
y=
\frac{1+(-\Gamma)}
{1-(-\Gamma)}
\]
Comparing with:
\[
z=
\frac{1+\Gamma}
{1-\Gamma}
\]
shows that converting from impedance to admittance simply replaces:
\[
\Gamma
\rightarrow
-\Gamma
\]
The negative reflection coefficient becomes:
\[
-\Gamma
=
|\Gamma|
e^{j(\phi+\pi)}
\]
Therefore:
\[
\Delta\phi=\pi=180^\circ
\]
A phase shift of 180° corresponds to moving to the diametrically opposite point on the same SWR circle.
This is why admittance is obtained by rotating 180° through the center of the Smith Chart.
Justification for Moving Clockwise Toward the Generator
When moving along a transmission line toward the generator by a distance \(d\), the reflection coefficient changes according to:
\[
\Gamma(d)
=
\Gamma_0
e^{-j2\beta d}
\]
or
\[
\Gamma(d)
=
|\Gamma_0|
e^{j(\phi_0-2\beta d)}
\]
Constant Radius Property
The magnitude remains unchanged:
\[
|\Gamma(d)|
=
|\Gamma_0|
\]
Therefore the point moves along a circle of constant radius.
This circle is the SWR circle.
Phase Change Property
The phase angle changes by:
\[
-2\beta d
\]
As distance increases, the phase decreases.
A decreasing phase angle corresponds to clockwise rotation on the complex plane.
Therefore moving toward the generator always corresponds to clockwise movement around the Smith Chart.
Periodicity of the Smith Chart
A complete revolution occurs when:
\[
2\beta d
=
2\pi
\]
Substituting:
\[
\beta
=
\frac{2\pi}{\lambda}
\]
gives:
\[
2
\left(
\frac{2\pi}{\lambda}
\right)
d
=
2\pi
\]
Solving for distance:
\[
d
=
\frac{\lambda}{2}
\]
Therefore:
\[
360^\circ
\text{ rotation}
=
\frac{\lambda}{2}
\]
This explains why the wavelength scales on the Smith Chart repeat every:
\[
0.5\lambda
\]
The Smith Chart is derived directly from the mathematical relationship between normalized impedance and reflection coefficient.
Constant resistance circles and constant reactance arcs originate from transforming impedance equations into standard circle equations.
Admittance conversion corresponds to replacing:
\[
\Gamma
\rightarrow
-\Gamma
\]
which creates a 180° rotation on the SWR circle.
Movement toward the generator follows:
\[
\Gamma(d)
=
\Gamma_0e^{-j2\beta d}
\]
resulting in clockwise rotation while maintaining a constant SWR circle radius.
A complete revolution corresponds to:
\[
\frac{\lambda}{2}
\]
which establishes the periodic nature of all Smith Chart calculations.