EE-611

Properties of Smith Chart

Prof. A.M.Ferendeci

• 1. The radius of the Smith Chart is 1.0 unit whatever the size is.

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• 2. For impedances with positive real parts ( Z=R+jX, that is R >= 0.0) (all passive impedance elements), corresponding impedances lie inside the Smith chart. They all have a reflection coefficient magnitude 0.0 <= |G| <= 1.0.

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• 3. If there is a negative real part (negative resistance), the corresponding impedance lies outside the Smith chart. Note all impedances with negative real parts ( Z= -R+jX, that is R<0.0) have a reflection coefficient magnitude r_L = |G_L| >1.0.
  

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• 4. For a given load, locate the corresponding normalized impedance on the Smith chart. Z₁ = R₁ + jX₁ or Z₂ = R₂ - jX₂ in the figure.
  

  Note: Upper part of the Smith chart is positive reactance, lower part is negative reactance.

  
  Read the magnitude of the reflection coefficient r₁ or r₂ The angle from the horizontal axis toward the load is the angle of the load reflection coefficient, q. This can be read from the angular divisions of the Smith chart.

  Note: The distance from the center of the Smith chart to the R=0.0 circle is one unit irrespective of the size of the Smith chart. (To find the magnitude of the reflection coefficient, measure it's length with a ruler and scale it with respect to the radius of the Smith chart).

  Note: The angle for the negative reactance can be read as a negative angle, i.e., q₂=-112^o, or q₂=360^o-112^o=248^o.

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• 5. If the reflection coefficient of any load is given, locate the angle q measured form the horizontal axis. From center of the Smith chart, draw a line whose length is equal to r_L at this angle and read the corresponding normalized impedance from the Smith chart.

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• 6. If you want to find the corresponding admittance for a given impedance, extend the reflection coefficient line from the center of the Smith chart in the opposite direction (180^o). Measure the same length of r_L and read the admittance Y_L = 1/Z_L.

  Note: You can not read reflection coefficient at this point. For the IMPEDANCE Chart, the reflection coefficient is always determined from the impedance.

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• 7. If a load is connected to a transmission line of length d, to find the input impedance remember that the reflection coefficient on the line is given by

  Note: the magnitude of the reflection coefficient is still determined by the load impedance. The presence of the line effects only the phase angle. And the angle changes by q-2bd). 2bd is subtracted from the angle q.

  To find the input impedance and the reflection coefficient at the input of the line, draw a circle with a radius equal to r_L centered at the origin of the Smith chart. Move through an angle -2bd . This means that you move in the clockwise direction an angle equal to -2bd beginning from the location of q (toward the generator on the Smith chart) At the intersection point of the r_L circle and the -2bl line, read the input impedance. The angle read with respect to the horizontal axis is the phase angle of the reflection coefficient at the input of the line.
  
  Note: In addition to the angular division of the Smith, there is also an angular scale which is labelled as wavelengths toward the generator (or toward load). This is more general than calculating the angle -2bd in degrees or radians. This is so general that it applies to any arbitrary transmission line.

  If the length of the line is equal to (l/2)l,, then -2bl.= -2p. This is consistent with the result that the impedance repeats itself with every half wavelength.