Part 2: Resistance and Reactance Circles

This is the second of the three articles devoted to the topic of a Smith Chart. The previous article, [1], introduced the concept of normalized load impedance and concluded with two equations describing the resistance and reactance circles. This article explains the creation of the resistance and reactance circles which are the basis of the graphical operations on the Smith Chart, like the one shown in Figure 1 [2].

The next two sections discuss the resistance and reactance circles and are based on the material presented in [3].

Relevant Background

Figure 2 shows a typical model of a lossless transmission line.

With this transmission line we associate the load reflection coefficient, , given by

  (1.1)

This load reflection coefficient can be expressed in terms of the normalized load impedance by dividing the numerator and denominator by the characteristic impedance of the line, Z_C.

  (1.2)

or

  (1.3)

where

  (1.4)

is the normalized load impedance, r_L is the normalized load resistance and x_L is the normalized load reactance.

From Eq. (1.3) we can express the normalized load impedance in terms of the load reflection coefficient as

  (1.5)

or, in terms of the real and imaginary parts as

  (1.6)

Equation (1.6) leads to the equation describing the resistance circle

  (1.7)

and another equation describing the reactance circle, [1],

  (1.8)

Resistance Circles

The resistance circle, described by Eq. (1.7) has a radius 

  (2.1)

and is centered at

  (2.2)

Let us calculate the radii and centers of the resistance circles for typical values of the normalized resistance r_L, [3]; this is shown in Table 1. 

Figure 3 shows the plots of these circles on the complex Γ plane.

Observations: All circles pass through the point (Γ_r, Γ_i) = (1,0). The largest circle is for r_L = 0, (which is the unit circle corresponding to ). All circles lie within the bounds of Γ = 1 unit circle. 

Let us identify some of these circles on the actual Smith Chart; this is shown in Figure 4.

The values of the normalized resistances corresponding to these circles are shown on the horizontal axis (and other places) of the Smith Chart as shown in Figure 5.

Reactance Circles

The reactance circle is described by 

  (3.1)

with the radius of

  (3.2)

centered at

  (3.3)

Let us calculate the radii and centers of the reactance circles for typical values of the normalized reactance x_L, [3].Note that unlike the normalized resistance (which is always non-negative), the normalized reactance can be positive (inductive load) or negative (capacitive load). Thus, Eq. (3.1) represents two families of circles, as shown in Table 2 and Figure 6.

Of interest to us is the part of a given reactance circle that falls within the bounds of 

Γ = 1 unit circle. 

Observations: The centers of all the reactance circles lie on the vertical Γ_r = 1 line. All reactance circles also pass through the (Γ_r, Γ_i) = (1,0) point (just like the r_L circles). 

Let us identify some of these partial circles on the actual Smith Chart; this is shown in Figure 7.

The values of the normalized reactances corresponding to these partial circles are shown on the perimeter circle (and other places) of the Smith Chart, as shown in Figure 8.

When we superimpose the resistance and reactance circles onto each other, we obtain Smith Chart shown in Figure 1.

The intersection of any r-circle with any x-circle corresponds to a normalized load impedance, as shown in Figure 9.

The next article will discuss the use of the Smith Chart in determining the input impedance to the transmission line at a given distance from the source or the load. 

References

1. Adamczyk, B., “Smith Chart and Input Impedance to Transmission Line – Part 1: Basic Concepts,” In Compliance Magazine, April 2023. 
2. https://upload.wikimedia.org/wikipedia/commons/7/7a/Smith_chart_gen.svg
3. Fawwaz Ulaby and Umberto Ravaioli, “Fundamentals of Applied Electromagnetics,” Pearson Education Limited, 7^th Ed., 2015.