Smith Chart Fundamentals and Key Equations

Smith Chart Fundamentals and Key Equations

The Smith Chart is a circular graphical tool widely used in RF and microwave engineering to represent complex reflection coefficients, normalized impedances, and normalized admittances. It provides a convenient graphical method for solving transmission line and impedance matching problems without lengthy calculations.

Complex Reflection Coefficient

The reflection coefficient relates the reflected wave to the incident wave on a transmission line.

\[
\Gamma = |\Gamma|e^{j\phi} = \frac{Z_L - Z_0}{Z_L + Z_0} = \frac{z_N - 1}{z_N + 1}
\]

At the center of the Smith Chart:

\[
z_N = 1 + j0
\]

\[
\Gamma = 0
\]

This represents a perfect impedance match with no reflected power.

At the outer rim of the Smith Chart:

\[
|\Gamma| = 1
\]

This represents total reflection and corresponds to purely reactive loads.

Voltage Standing Wave Ratio (VSWR)

The Voltage Standing Wave Ratio is directly related to the magnitude of the reflection coefficient.

\[
\text{SWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}
\]

The radius of the SWR circle is equal to:

\[
|\Gamma|
\]

The SWR value is read at the point where the SWR circle intersects the positive real axis.

General Step-by-Step Smith Chart Analysis Procedure

## Step 1: Calculate the Normalized Load Impedance

Convert the load impedance into a normalized value by dividing it by the characteristic impedance.

\[
z_N = \frac{Z_L}{Z_0}
\]

\[
Z_L = R_L + jX_L
\]

\[
z_N = r_N + jx_N
\]

This normalized impedance is plotted directly on the Smith Chart.

## Step 2: Plot the Normalized Impedance

Locate the intersection of the constant resistance circle and constant reactance arc.

\[
r = r_N
\]

\[
x = jx_N
\]

For inductive reactance:

\[
+jx_N
\]

The point lies in the upper half of the chart.

For capacitive reactance:

\[
-jx_N
\]

The point lies in the lower half of the chart.

## Step 3: Draw the SWR Circle and Measure Reflection Coefficient

Using the Smith Chart center as the pivot point, draw a circle passing through the normalized impedance point.

The magnitude of the reflection coefficient is determined from the radius of the SWR circle.

\[
|\Gamma|
\]

The SWR is then obtained from:

\[
\text{SWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}
\]

The numerical value is read where the SWR circle intersects the positive real axis.

## Step 4: Find the Normalized Admittance

The normalized admittance is found by rotating through:

\[
180^\circ
\]

around the center of the chart.

The resulting admittance is:

\[
y_N = g_N + jb_N
\]

\[
y_N = \frac{1}{z_N}
\]

This point lies directly opposite the impedance point on the same SWR circle.

## Step 5: Determine Wavelength Scale Readings

The Smith Chart contains wavelength scales around its outer perimeter.

The primary scales are:

\[
\text{WTG}
\]

and

\[
\text{WTL}
\]

WTG represents movement toward the generator, while WTL represents movement toward the load.

Clockwise movement corresponds to motion toward the generator.

Counter-clockwise movement corresponds to motion toward the load.

## Step 6: Determine Input Impedance at Distance d

To determine the input impedance at a distance from the load, rotate clockwise along the SWR circle.

The target wavelength reading is obtained from:

\[
\text{WTG}_{\text{target}}
=
\text{WTG}_{\text{load}}
+
\frac{d}{\lambda}
\pmod{0.5\lambda}
\]

The corresponding point on the SWR circle gives the normalized input impedance.

\[
z_{in}(d)
\]

This impedance can then be denormalized to obtain the actual impedance value.

Smith Chart Summary Parameter Lookup Table                                                                                                                                                                                                                                                                                                                                                                                                                                                                      

Parameter	Symbol	Smith Chart Location / Calculation
Normalized Impedance	\(z_N\)	Direct point intersection of \(r\) and \(x\) arcs
Normalized Admittance	\(y_N\)	Directly opposite \(z_N\) on SWR circle (\(180^\circ\) point)
Voltage Standing Wave Ratio	\(\text{SWR}\)	Intersection of SWR circle with right horizontal axis
Reflection Angle	\(\arg(\Gamma)\)	Angle measured from the positive real axis
Voltage Maxima	\(V_{\max}\)	Intersection of SWR circle with right real axis (\(r > 1\))
Voltage Minima	\(V_{\min}\)	Intersection of SWR circle with left real axis (\(r < 1\))
Rotation Direction	Clockwise	Movement toward generator
Periodicity	\(0.5\lambda\)	One complete chart revolution corresponds to \(\lambda/2\)

The Smith Chart provides a graphical relationship between impedance, admittance, reflection coefficient, wavelength, and standing wave ratio. By understanding the normalized impedance point, SWR circle, admittance conversion, and wavelength scales, engineers can quickly analyze transmission lines and design impedance matching networks without extensive mathematical computation.