Sunday, March 29, 2015

A Brief Tutorial on Smith Charts

Recently I've become interested in Antenna Tuners and I want to post some notes on them to this blog.  I thought that I'd start off these posts with a quick Smith Chart review/tutorial, given their helpfulness in understanding and analyzing impedance-matching.  So here we go...


A Brief Tutorial on Smith Charts

The foundation of the Smith Chart is a calculated quantity known as the "Reflection Coefficient."  The Reflection Coefficient (usually symbolized with Γ, the Greek "Gamma"), is a measure of impedance mismatch between a load impedance (Zload) and a desired target impedance (the target impedance is usually the Characteristic Impedance, Zo, of a transmission line, but it could also be the Source Impedance of a signal generator, such as a transmitter).  For this discussion I will use 50 ohms as our target impedance.

The Reflection Coefficient can be calculated with this formula:

Γ = (Zload - Zo) / (Zload + Zo)

Note that Zload is a complex quantity with both real and imaginary parts:

Zload = R + jX

Therefore, Γ is also a complex quantity.  Let's look at an example.  Consider the following load impedance:

 Zload = 50 + j200

Its impedance mismatch (as represented with its Reflection Coefficient) from our target impedance of Zo = 50 ohms calculates to be:

Γ = 0.8 + j0.4

And we can plot its position (using Cartesian coordinates) on the complex plane: 

(click on image to enlarge)

In addition to expressing Γ's position on the complex-plane using Cartesian coordinates, we can also express its position in terms of Polar-coordinates using rho (symbol: ρ), representing the magnitude of Γ, and theta (symbol: θ), the angle it subtends from the positive x-axis.

Again using our example of the Reflection Coefficient for Zload = 50 + j200 (i.e. Γ = 0.8 + j0.4), this Γ, in Polar coordinates (rho, theta), would be:

(0.894, 26.6°)

where ρ = 0.894 and θ = 26.6 degrees:

(click on image to enlarge)

Let's plot some more Reflection Coefficients on our Complex plane.  I'm going to chose eight more values for Zload, all with a resistance (i.e. "real") value of 50 ohms, and various reactance (i.e. imaginary) values, and then use Excel to calculate Γ:

(click on image to enlarge)

Here's how these eight Reflection Coefficients look when plotted on the Complex plane:

(click on image to enlarge)

If we remove the annotation from the image above, our points look like this:

(click on image to enlarge)

Looks like they form a crude circle, doesn't it?

If we calculate Γ for all possible Zloads of 50 ohms resistive and anything reactive, that is, Zload = 50 + jX, where X ranges from -∞ to + ∞, and we add these values to our plot of Γ above, it would become:


It's a circle!  This is a circle of "constant resistance".  That is, it is the locus of all possible Γ values calculated from the equation Zload = 50 + jX, where R, the resistance of Zload, is fixed at 50 ohms.

Now, if we were to calculate Γ for all possible values of Zload = R + jX (i.e. both R and X range independently from -∞ to + ∞) and then plotted the results on the complex plane, they would plot into a region bounded by (and including) a circle of radius 1.0, centered at 0,0:

 (click on image to enlarge)

Let's add a few more "constant resistance" plots of  Γ to our plot of Γ for Zload = 50 + jX.  I'll chose Zload = 150 + jX and Zload = 16.67 + jX, where again, X ranges from -∞ to + ∞:

(click on image to enlarge)


We have three circles, each circle representing Γ for a Zload with "constant" Resistance and all possibilities of Reactance.

Lets look at what happens to our plot of  Γ if we add additional series Reactance (either inductive or capacitive) to an existing value of Zload.  I'll again use the example of Zload = 50 + j200.

First, let's put  Γ for Zload = 50 + j200 on our "Resistance" circle:

(click on image to enlarge)

What happens if we add series reactance?  We are creating a new Zload:

Zload = 50 + j200 + jX, where X is the reactance we're adding.

When we add jX to our Zload, we are going to move Γ along our "resistance" circle.  The movement will be clockwise if we add inductance and counter-clockwise if we add capacitance.  If we add just enough series capacitance to our "j200" term, we will cancel it out and be left with Zload = 50 + j0, a perfect match.  But if we add too much capacitance, we will continue to move counter-clockwise away from that point of perfect match.  To illustrate these points:

(click on image to enlarge)




How does the value of the reactance we're adding correlate to the amount Γ moves along the circle?   We are adding a series reactance to Zload.  What happens if I add a series-element that is a short circuit (zero ohms)?  Nothing happens -- Γ doesn't move (that is, the impedance at that point doesn't change).  So if I add a series element that has a very small reactance, that is, it looks almost a short circuit (such as a large capacitor or a small inductor), there should be little movement along the circle.

In contrast, the more the series element looks like an open-circuit (that is, the higher its reactance, e.g. small capacitor or large inductor), the greater will be Γ's movement around the circle. 

(Also note that in the plot above, Γ at its limit (when jX is either -j∞ or +j∞) is at coordinate 1,0, on the circumference of the unit-circle.)

But suppose we add reactance in parallel with Zload, rather than in series as was done above?  How do we visually represent this?

First, recognize that any representation of Zload impedance in series form (i.e. R + jX, where R is resistance and X is reactance) can be transformed into an equivalent parallel form in which:

Yload = G + jB

Where Y is Admittance, G is Conductance, and B is Susceptance.

Yload is simply 1/Zload.  Therefore, G and B can be easily calculated by inverting the equation for Zload (Zload = R + jX) which results in these two equations:

G = R/(R2 + X2), and

B = -X/(R2 + X2)

 Let's calculate the equivalent Yload admittances for our example of nine impedances (all with R = 50 ohms) and add them to the table that's earlier in this post:

(click on image to enlarge)

 The first thing you should note:  although R = 50 for all nine values, G is not constant.  But take a look at G when Zload = 50 + j0 (our ideal match).  It's equal to 0.02 Siemens.

So...what happens to Γ if, say,  we hold Yload's Conductance (G) constant but vary Yload's Susceptance (B)?

As an example, what happens if we fix G at 0.02 Siemens?  Let's calculate Γ for nine different Yloads, each with its G term set to 0.02 Siemens:

(click on image to enlarge)

Plotted on the complex plane, the coordinates for these 9 Γ are:

(click on image to enlarge)

Now, again with G fixed at 0.02 Siemens, let's plot Γ for Yload = 0.02 + jB, where B ranges from -∞ to + ∞:

(click on image to enlarge)

This is a circle of "constant admittance" (in this case, G = 0.02).  Note that it looks exactly like the circle of "constant resistance" that we drew above for Zload = 50 + jX, except this new circle is shifted to the left-hand side of the complex-plane.

Adding either inductance or capacitance in parallel with Zload moves us along this circle.  Adding inductance will move us counter-clockwise and adding capacitance moves us clockwise.

(click on image to enlarge)

How does the value of the reactance we're adding correlate to the amount Γ moves along the circle?   We are adding a parallel reactance across Zload.  What happens if I add a parallel-element across Zload that is an open circuit (infinite ohms)?  Nothing happens -- Γ doesn't move (that is, the impedance at that point doesn't change).  So if I add a parallel element that has a very large reactance, that is, it looks almost an open circuit (such as a small capacitor or a large inductor), there should be little movement along the circle.

In contrast, the more the parallel element looks like a short-circuit (that is, the lower its reactance, e.g. large capacitor or small inductor), the greater will be Γ's movement around the circle. 

(Also note that in the plot above, Γ at its limit (when jB is either -j∞ or +j∞) is at coordinate -1,0, on the circumference of the unit-circle.)

For the sake of symmetry (because we did it above for circles of "constant-resistance"), let's add two more "constant-conductance" circles to the one above:
(click on image to enlarge)

And let's add our previous Γ plots of circles of constant-resistance:

(click on image to enlarge)

In addition to Γ plots of circles of constant-resistance or circles of constant-conductance, I could also plot Γwith Zload's reactance term (X) held constant and varying R from -∞ to + ∞, or I could plot Γwith Yload's susceptance term (B) held constant and varying G from -∞ to + ∞).  The resulting curves would not be circles, though.  Let's look at them:

Here are plots of Γin the complex-plane with  R held constant (circles) and with X held constant (not circles):

(click on image to enlarge)

This is a Smith Chart. (I hope the circles look familiar!).  With the addition of the "constant-reactance" curves, I can easily plot Γ for a Zload = R + jX.  (This is made even easier with Smith-charting programs such as Smith V3.10, which I will  use throughout this blog post).

For example, let's plot Γ for an impedance of Zload = 25 + j25:

(click on image to enlarge)

I can plot the same point in terms of admittance:
(click on image to enlarge)

And here are the admittance and impedance charts, combined:

(click on image to enlarge)

The point represents a unique Γ, but the load associated with it can be represented either as an impedance (Z=R+jX) or as an admittance (Y=G+jB).  The two forms are equivalent.

One more quick topic:  circles of constant SWR.

SWR can be calculated from Γ:

SWR = (1 + |Γ|) / (1 - |Γ|)

That is, SWR depends on the magnitude of Γ.  But the magnitude of Γ is also rho, ρ (see discussion at the beginning of this post).  And so we could also write the equation in terms of ρ:

SWR = (1 + ρ) / (1 - ρ)
 
(Note:  some authors use ρ in lieu of Γ to represent the Reflection Coefficient.  I believe this is a mistake.  I prefer to represent the Reflection Coefficient with Γ and its magnitude with  ρ).

The SWR for any  ρ (that is, |Γ|) can be plotted as an overlay on the Smith Chart.  For any given ρ, the set of impedance (or admittances) that result in this ρ will lie on a circle centered at (0,0) and with radius ρ.  Here are some examples:

(click on image to enlarge)


 
 That's it for the tutorial.  To recap, the important points are these:
First, a load's Reflection Coefficient (Γ) can be plotted on the complex-plane.  This plotted point will have a magnitude (also known as ρ), when referenced to the center of the plane (0,0), that is between (and inclusive of) 0 and 1.
Second, the Smith Chart itself is simply an overlay over this plot of the Reflection Coefficient, and this overlay identifies the impedance (or admittance) of the load associated with Γ at any point.
Third, if we add capacitance or inductance in series with our load we can visualize this as moving Γ along Smith Chart circles of constant-resistance (the direction will depend on the sign of the reactance: inductive or capacitive).  And in an analogous fashion, if we add capacitance or inductance in parallel with our load, we will move Γ along circles of constant-conductance.
Fourth, when we add capacitance or inductance to our circuit, this movement of  Γ (and thus the change in impedance or admittance) is only along these circles.  It is not along the other lines representing constant-reactance or constant susceptance.
Fifth, circles of constant SWR can be plotted on the Smith Chart.  These circles are centered at (0,0) -- the chart's Zo, and their radius is ρ, the magnitude of Γ.

Resources:

You can download the Smith-chart software here:  Smith V3.10


Related Posts:

Antenna Tuners and Impedance Matching with L Networks


Caveats: 

I could have easily made a mistake anywhere in this blog post.  If something looks wrong, is unclear, or doesn't make sense, please feel free to contact me!

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