The Smith Chart
The Smith chart began its existence as a very useful graphical calculator for the analysis and design of TLs. It was developed by Phillip H. Smith in the 1930s. The Smith chart remains a useful tool today to visualize the results of TL analysis, oftentimes combined with computer analysis and visualization as an aid in design. The development of the Smith chart is based on the normalized TL impedance defined as
where Z(z)= V(z)/ I (z) is the total TL impedance at z and
is the generalized reflection coefficient at z. The real and imaginary parts of the generalized reflection coefficient will be defined as Substituting this definition into (1) gives
Now, we will define z (z) = r + jx and separate (3) into its real and imaginary parts
Equating the real and imaginary parts of this last equation gives
Rearranging both of these leads us to the final two equations
And
We will use (4) and (5) to construct the Smith chart. Definition: The Smith chart is a plot of normalized TL resistance and reactance functions drawn in the complex, generalized reflection coefficient [T(z)] plane. To understand this, first notice that in the T_{r} - T_{i} plane:
1. Equation (4) has only r as a parameter and (5) has only x as a parameter.
2. Both (4) and (5) are families of circles. Consequently, we can plot (4) and (5) in the T_{r}- T_{i} plane while keeping either r or x constant, as appropriate.
Plot (4) in the T_{r} - T_{i} plane:
Plot these curves in the T_{r} - T_{i}plane:
Plot (5) in the T_{r} - T_{i} plane:
Plot these curves in the T_{r} - T_{i}plane:
Combining both of these curves (or “mappings”), as shown on the next page, gives what is called the Smith chart. As quoted from the text (p. 65): “The real utility of the Smith chart, however, lies in the fact that it can be used to convert from reflection coefficients to normalized impedances (or admittances), and vice versa, using the impedance (or admittance) circles printed on the chart.” Additionally, it is very easy to compute the generalized reflection coefficient and normalized impedance anywhere on a homogeneous section of TL.
Notice that the T_{r} and T_{i} axes are missing from the “combined” plot. This is also the case for the Smith chart.
Important Features of the Smith Chart
From this result, we can show that if r £ 0 then T(z) ³ 1. This condition is met for passive networks (i.e., no amplifiers) and lossless TLs (real Z_{0} ). Consequently, the standard Smith chart only shows the inside of the unit circle in the T_{r} - T_{i} plane. That is, T(z) ³ 1 which is bounded by the r = 0 circle described by T_{r2} + T_{i2} = 1.
2. If z(z) is purely real (i.e., x = 0), then since
we deduce that T_{r}= 0 (except possibly at T_{r} =1 ). Consequently, purely real z (z) values are mapped to T(z)values on the T_{r} =x e [T(z)] axis.
3. If z(z) is purely imaginary (i.e., r = 0) then from (4) T_{r2}+ T_{i2} =2 which is the unit circle in the T_{r}- T_{i} plane. Consequently, purely imaginary z (z) values are mapped to T(z) values on the unit circle in the T_{r}- T_{i} plane.
Example N6.1: Using the Smith chart, determine the voltage reflection coefficient at the load and the TL input impedance.
VSWR and the Smith Chart
It was shown in the previous lecture that the voltage magnitude anywhere on the TL can be written as As derived in the text (Section 2.3)
|V(z)|_{max} = |V_{0}^{ +}|| 1 + T_{L} (z)|
and
|V(z)|_{min} = |V_{0}^{ +}|| 1 - T_{L} (z)|
So, when positioned along the TL at a maximum voltage magnitude:
Using the definition of VSWR from the last lecture
then from (7) at a voltage magnitude maximum on the TL
Z (z) = Z_{0} .VSWR
Z(z) = Z_{0} .VSWR
Because of this last result, we can read the VSWR of a TL directly from the Smith chart. Similarly, we can show that at a minimum voltage magnitude
In the previous example, we can read VSWR=2 directly from the Smith chart by drawing the constant VSWR circle. This is the circle traced by T(z) as z varies. However, notice that depending on where we “stop” this rotation of T(z) versus z, we obtain different z (z) values. This happens because T(z) is not traversing circles of constant r and/or x as z varies.
Smith Admittance Chart
The Smith chart can be used as an admittance chart as well as an impedance chart. To see this, recall that we derived the mapping upon which the Smith chart is based [ z (z) « T(z)] from the normalized TL Impedance
From this, we can express the normalized TL admittance as
We can repeat the construction of the Smith chart with y = g + jb and T = T_{r} + jT_{i} , as we did originally for the impedance chart. Substituting these quantities into (11) we find
A Smith admittance chart can be constructed based on these two equations for circles in the complex T(z) plane:
This Smith admittance chart looks very similar to the Smith impedance chart. In fact, if we rotated one 180º we obtain the other. This is actually an easily proved result. Consider the definition of the negative generalized reflection coefficient from
That is
If we now substitute (14) into (11) we find that
But what is z + l/ 4 It’s a half rotation around the Smith chart.
Discussion
From (13) we can deduce that:
1. If z(z) is known, then y(z) is the point on the constant VSWR circle that is diametrically opposite the z(z) point on the Smith chart. (In this context, remember that a QWT is an impedance inverter device.)
2. The Smith chart can be used either as an impedance chart or as an admittance chart. Rather than keeping these two types of charts around, we can use one for either impedance or admittance calculations. The following example should help you understand this.
3. One subtlety with these mixed Smith charts is that generalized reflection coefficients are only correctly represented on impedance charts when plotting normalized impedances and on admittance charts when plotting normalized admittances. You’ll read negative generalized reflection coefficients otherwise (for admittances on impedance charts and impedances on admittance charts).