**3. Construction of the Smith chart**

The Smith chart is constructed based on the voltage reflection coefficient and can be considered as parameterized plot, on polar coordinates, of the generalized voltage reflection coefficient , *<sup>j</sup> e* , within a circle of unit radius 1 .

It is well known from transmission line theory, that the voltage reflection coefficient at the load is given by:

$$
\rho\_L = \frac{Z\_L - Z\_0}{Z\_L + Z\_0} \tag{1}
$$

where *ZL* is the load impedance and *Z*0 is the characteristics impedance of the line. According to the transmission line theory, *Z*0 is a real value but in general *ZL* is a complex value. Equation 1 can be written as:

$$\rho \lrcorner \rho \lrcorner\_{\perp} = \|\rho\_{\perp}\| \angle \phi = \rho\_r + \mathrm{j}\rho\_i \tag{2}$$

where *<sup>r</sup>* and *<sup>i</sup>* are respectively the real and imaginary parts of the reflection coefficient. Instead of having separate Smith chart for transmission lines with different characteristics impedances, it is preferable to have just one that can be used for any line. This is achieved using a normalized chart in which all impedances are normalized to the characteristic impedance *Z*0 of the particular line under consideration. For example, for the load impedance *ZL* , the normalized impedance *Lz* is given by,

$$z\_L = \frac{Z\_L}{Z\_0} = r + j\infty \tag{3}$$

where *r* and *x* are respectively the real and imaginary parts of the normalized impedance. Substituting equations 2 and 3 into equation 1 gives,

$$j\,\rho\_r + j\,\rho\_i = \frac{z\_L - 1}{z\_L + 1} \tag{4}$$

or

$$\mathbf{r} + j\mathbf{x} = \frac{\left(\mathbf{1} + \rho\_r\right) + j\rho\_i}{\left(\mathbf{1} - \rho\_r\right) - j\rho\_i} \tag{5}$$

Equating real and imaginary parts, we obtain

$$r = \frac{1 - \rho\_r^2 - \rho\_i^2}{(1 - \rho\_r)^2 + \rho\_i^2} \tag{6}$$

Using the Smith Chart in an E-Learning Approach 103

values of the normalized reactance *x*. Notice that while *r* is always positive, *x* can be positive (inductive impedance) or negative (capacitive impedance). From this figure we see that there are symmetry about the horizontal central axis of the chart. Only the portion of the

any passive loaded line is one. The circles given by equations 8 and 9 are orthogonal circles

If the *r* circles and the *x* circles are superimposed, the result is the Smith chart shown in the

On a Smith chart there are some important points, lines and contours that should be mentioned. In Figure 3 some of these important features are indicated. The outer circle is the locus of the pure reactive impedances, that is, those with zero resistance. The horizontal axis is the locus of the real impedances. The left radius is the locus of the resistances less than *Z*<sup>0</sup> for which the reflection coefficient has a phase of 180º. The left extreme of this radius is the zero resistance and zero reactance point, that is, the short circuit point (SC). The right radius is the locus of the resistances greater than *Z*0 for which the reflection coefficient has a phase of 0º. The right extreme of this radius is the infinite resistance and infinite reactance point,

For a lossless transmission line terminated in a load with a reflection coefficient

appearing along the line, normalized to the characteristic impedance *Z*0 of the line. These impedances can be obtained moving along the line either toward the load (counter

*<sup>L</sup>* (known as the circle or S circle), is the locus of all impedances

for

*<sup>L</sup>* , the

circles inside the central circle of radius one is shown, since the maximum value of

that make a conform mapping chart.

Fig. 2. Basic Smith chart.

**3.1 Important features on a Smith chart**

that is, the open circuit point (OC).

circle with radius

Figure 2.

$$\alpha = \frac{2\rho\_i}{(1-\rho\_r)^2 + \rho\_i^2} \tag{7}$$

rearranging the terms in equation 6 leads to:

$$\left(\rho\_r - \frac{r}{1+r}\right)^2 + \rho\_i^2 = \left(\frac{1}{1+r}\right)^2\tag{8}$$

rearranging the terms in equation 7 leads to:

$$(\mathfrak{p}\_r - 1)^2 + \left(\mathfrak{p}\_i - \frac{1}{\mathfrak{x}}\right)^2 = \left(\frac{1}{\mathfrak{x}}\right)^2\tag{9}$$

Each of equations 8 and 9 is similar to the circle equation. Equation 8 is a r-circle (resistance circle) with center at 0 <sup>1</sup> , *r <sup>r</sup>* and radius equal to 1 *<sup>r</sup>* <sup>1</sup> . Several of these circles for various values of normalized resistance *r*, are plotted in Figure 1a). From the Figure 1a), we see that all circles pass the point (1,0).

Fig. 1. Basic Smith chart. a)- normalized resistance circles. b)- normalized reactance curves.

Similarly, equation 9 is an *x* circle (reactance circle) with center at *x* <sup>1</sup> <sup>1</sup>, and radius equal to *x* <sup>1</sup> . Several of these circles are plotted in Figure 1b), this time for positive and negative

 <sup>2</sup> <sup>2</sup> 22

 <sup>2</sup> <sup>2</sup> 1 2

*ir*

*ir*

2

*r r*

<sup>1</sup>

<sup>2</sup> <sup>11</sup> <sup>1</sup>

 

Each of equations 8 and 9 is similar to the circle equation. Equation 8 is a r-circle (resistance

values of normalized resistance *r*, are plotted in Figure 1a). From the Figure 1a), we see that

a) b)

Fig. 1. Basic Smith chart. a)- normalized resistance circles. b)- normalized reactance curves.

<sup>1</sup> . Several of these circles are plotted in Figure 1b), this time for positive and negative

Similarly, equation 9 is an *x* circle (reactance circle) with center at

1 1

22

(6)

(7)

*<sup>r</sup> <sup>i</sup>* (8)

*xx <sup>r</sup> <sup>i</sup>* (9)

<sup>1</sup> . Several of these circles for various

 *x*

<sup>1</sup> <sup>1</sup>, and radius equal to

2

 

 

1 1

*<sup>i</sup> x*

2

*<sup>r</sup>* and radius equal to 1 *<sup>r</sup>*

*r*

 

 0 <sup>1</sup> , *r*

*ir r* 

Equating real and imaginary parts, we obtain

rearranging the terms in equation 6 leads to:

rearranging the terms in equation 7 leads to:

circle) with center at

all circles pass the point (1,0).

*x*

values of the normalized reactance *x*. Notice that while *r* is always positive, *x* can be positive (inductive impedance) or negative (capacitive impedance). From this figure we see that there are symmetry about the horizontal central axis of the chart. Only the portion of the circles inside the central circle of radius one is shown, since the maximum value of for any passive loaded line is one. The circles given by equations 8 and 9 are orthogonal circles that make a conform mapping chart.

If the *r* circles and the *x* circles are superimposed, the result is the Smith chart shown in the Figure 2.

Fig. 2. Basic Smith chart.
