Smith Chart

                                                                                                                                                            Iulian Rosu, YO3DAC / VA3IUL, http://www.qsl.net/va3iul

    Named after its inventor, Phillip H. Smith (Bell Laboratories), the Smith Chart was originally described in Electronics for January 1939. (Interview with Philip H. Smith)

 Rs + jXs = RL - jXL

 

 

     For better understanding let give an example. Suppose we have an impedance consisting of 50 ohms resistance and 100 ohms inductive reactance (Z = 50 +j100). If we assign a value of 100 ohm to prime center, we normalize the above impedance by dividing each component of the impedance by 100 (Normalization must be used, in order to facilitate the plotting of larger impedances. Each impedance to be plotted is divided by a convenient number that will place the new normalized impedance near the center of the chart where increased accuracy in plotting is obtained). The normalized impedance is then 50/100 + j(100/100) =
0.5 + j1.0. This impedance is plotted on the Smith Chart at the intersection of 0.5 resistance circle and the +1.0 reactance circle.
    Instead of assigning 100 ohms to prime center, we assign a value of 50 ohms. With this assignment, the 50 + j100 ohm is plotted at the intersection of the
50/50 = 1.0 resistance circle, and the 100/50 = 2.0 positive the same impedance value, 50 + j100 ohms. This example shows that the same impedance may be plotted at different points on the chart, depending upon the value assigned to prime center. But two plotted points cannot represent the same impedance in the same time.
Prime center is a point of special significance. It is customary when solving problems to assign the Zo value of the line to this point on the chart, 50 ohms for a 50 ohms line, for example. The center point of the chart now represents 50 + j0 ohms, a pure resistance equal to the characteristic impedance of the line, that it represents a perfect match, with no reflected power and with a 1.0 to 1 SWR.
    When plotting impedances two cases can be. These are short circuits and open circuits. A true short circuit has zero resistance and zero reactance, or 0 + j0. This impedance is plotted at the left of the chart, at the intersection of the resistance and reactance axes. An open circuit has infinite resistance, and therefore is plotted at the right of the chart, at the intersection of resistance and reactance axes.

 

             
     

    In other words the zero ohms circles (r = 0) is the largest one and the infinite resistor circle is reduced to one point at (1,0). There should be no negative resistance. If one (or more) should occur, we will be faced with possibility of oscillatory conditions.
     It can be seen that all of the circles of one family will intersect all of the circles of the other family. Knowing the impedance, in form of: r + jx, the corresponding reflection coefficient can be determined. It is only necessary to find the intersection point of the two circles, corresponding to the values r and x.
    The reverse operation is also possible. Knowing the reflection coefficient, find the two circles intersecting at that point and read the corresponding values r and x on the circles. The procedure for this is as follows:
 
    - Determine the impedance as a spot on the Smith Chart
    - Find the reflection coefficient
Γ (Gamma) for the impedance.
The reflection coefficient is defined as the ratio between the reflected voltage wave and the incident voltage wave:

Gamma = Vref / Vinc
    - Having the characteristic impedance and
Γ (Gamma), find the impedance
Any point on the Smith Chart represents a series combination of resistance and reactance of  Z = R + jX. 

Thus , to locate the impedance Z = 1 +j1, you would find R = 1 constant resistance circle and follow until it crossed the X = 1 constant reactance circle. The junction of these two circles would then represent the needed impedance value)
    - Convert the impedance to admittance.
The equivalent admittance of a plotted impedance value lies diametrically opposite the impedance point on the chart. In other words, an impedance plot and its corresponding admittance plot will lie on a straight line then passes through prime center, and each point will be the same distance from prime center (on the same SWR circle)
    - Find the equivelent impedance.
    - Find the components values for the wanted reflection coefficient
 
    The third family of circles, which are not printed on the main chart, but are added during the process of solving problems, are SWR circles. Each circle represents a value of SWR, with every point on a given circle representing the same SWR. The SWR for a given circle may be determined directly from the chart coordinate system, by reading the resistance axis to the right prime center.

 

     

The Smith Chart has the following futures:

  • It represents all passive impedances on a grid of constant r and x circles.

  • It contains the corresponding reflection coefficients in polar co-ordinates; the angle being read on the peripheral scale and the magnitude being calculated using:

  • The upper half of the diagram represents positive reactance values (inductive elements).

  • The lower half of the diagram represents negative reactance values (capacitive elements).

  • Impedance or Admittance values read from the chart are normalized values.

  • Moving away from the load (i.e. toward the generator) corresponds to moving in a clockwise direction.

  •  A complete revolution around the chart is made moving a distance L = λ/2 along the transmission line.

  • The same chart can be used for reading admittance.

  • The center of the chart corresponds to the impedance-matched condition since Γ = 0.

  • A circle centered at the origin is a constant Γ circle.

  • Moving along the lossless transmission line is equivalent to moving along Γ circle.

  • For impedance Z reading, the point (Γr = 1 and Γi = 0) corresponds to an open circuit.

  • For admittance Y reading, the same point corresponds to a short circuit.

  • The impedance distance λ/4 from ZL is equal to YL.

  • The SWR can be found by reading R at the intersection of the constant Γ circle with real axis.

  • The Smith Chart can be used to find: ΓL from ZL and vice-versa, Zin from ZL and vice-versa, Z from Y and vice-versa, the SWR, and the impedance matching.

  • The horizontal radius to the left of the centre corresponds to voltage minimum and current maximum (Vmin, Imax).

  • The horizontal radius to the right of the centre corresponds to the standing wave ratio (SWR), the voltage maximum and the current minimum (Vmax, Imin).

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References:
    1. RF Circuit Design - C. Bowick

    2. RF and Microwave Wireless Systems - K. Chang

    3. Microwave Communications Engineering - Glover, Pennock, Shepherd

    4. RF Design Magazine, 1988-2000
    5. Microwave Journal, 1998-2000
    6. Applied Microwave Magazine, 1995-2000
    7. ARRL Handbook, 1990-2000