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Smith Chart and Input Impedance to Transmission Line

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].

Figure 1: Basic Smith Chart

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.

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Figure 2: 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, ZC.

  (1.2)

or

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  (1.3)

where

  (1.4)

is the normalized load impedance, rL is the normalized load resistance and xL 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 rL, [3]; this is shown in Table 1. 

Table 1: Radii and Centers of r-Circles

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

Figure 3: Typical r-circles

Observations: All circles pass through the point (ΓrΓi) = (1,0). The largest circle is for rL = 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.

Figure 4: Selected r-circles on the Smith Chart

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.

Figure 5: Normalized resistance values on the Smith Chart

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 xL, [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.

Table 2: Radii and Centers of x-Circles

 

Figure 6: Typical x-circles

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 Γr = 1 line. All reactance circles also pass through the (Γr, Γi) = (1,0) point (just like the rL circles). 

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

Figure 7: Parts of the selected x-circles on the Smith Chart

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.

Figure 8: Normalized reactance values on the Smith Chart

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.

Figure 9: Normalized load impedance on the Smith Chart

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, 7th Ed., 2015.

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