Smith chart of λ/4 Transmission Line

I came across the following slide on power combiner circuit and I have questions about it.

1. Why traverse across the upper half circle ? In other words, why is Zin located at the right side of Zout ?

2. Why when translating an impedance towards the generator we move clockwise around the chart ?

3. Why "ZT-line is the geometric mean of Zout and Zin." ?

4. Why overlay the Smith chart scaled for ZT on top of the Z0 Smith chart ?

Deeper understanding about Smith chart may come from analysis of formulas behind it. Complex impedance transformation may be calculated using formulas (for line, capacitance, inductance) without using Smith chart. I try to answer your questions in more detail later.

promach said:

1. Why traverse across the upper half circle ? In other words, why is Zin located at the right side of Zout ?
2. Why when translating an impedance towards the generator we move clockwise around the chart ?

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From Smith chart it looks like Zout ≈ 2 Ohm, Zin ≈ 100 Ohm. Zin may be located at any side depending on actual values.
Traverse direction comes from actual impedance change. Impedance change may be calculated without using Smith chart (which is only a graphical aid). For example, you may calculate impedance change by adding 10 degree Zt=sqrt(Zout*Zin) line pieces and then draw points on Smith chart.

promach said:

3. Why "ZT-line is the geometric mean of Zout and Zin." ?

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It turns out that matching occurs when Zt=sqrt(Zout*Zin), not when Zt=Zout/Zin. Quarterwave matching formula Zt=sqrt(Zout*Zin) certainly may be derived without using Smith chart. Again, Smith chart is only a graphical aid. I do not see how using "geometric mean" term may be useful here, maybe for emphasizing that it is not an arithmetic mean.

promach said:

4. Why overlay the Smith chart scaled for ZT on top of the Z0 Smith chart ?

Click to expand...

I think it is not scaled Smith chart. This circle shows possible impedance values for Zt line of length 0 to λ/2. For 0 to λ/4 impedance moves from Zout to Zin. For λ/4 to λ/2 impedance moves back from Zin to Zout. You may stop at some arbitrary line length and end up with a complex impedance too.

My thoughts on "moving towards the generator" / "moving towards the load impedance" rules. Rules are:
1) Moving towards the generator - clockwise rotation on a Smith chart
2) Moving towards the load -counter-clockwise rotation on a Smith chart

I do not like these "toward generator" rules.
My explanation: on a Smith chart reflection coefficient Г phase decreases in clockwise direction.
Let's start from some reflection coefficient Г for this impedance. Real physical place on a PCB board, for example output of transistor.
When we physically add λ/4 line line to output of transistor, we will delay reflection signal two times by 90° (signal enters newly added λ/4 line, travels 90°, reflects from transistor s22, travels 90° degree in opposite direction through the same λ/4 line). So adding line with some characteristic impedance Z0 with θ phase length will delay reflected signal by 2*θ degrees (on a smith chart with center at Z0). So when adding physical length θ, we go clockwise (delaying output by 2*θ). When we removing physical length by cutting θ length of line, signal will come earlier be 2*θ, we go counter-clockwise.

Regarding scaling you referred to: It may be just using non 50 Ohm characteristic impedance for a Smith chart. If you use Z0=Zt=sqrt(Zout*Zin), then Zt will be in the center of Smith Chart, so maybe it useful somehow for using Smith chart without any formulas.

In third message i've made a mistake:

It turns out that matching occurs when Zt=sqrt(Zout*Zin), not when Zt=Zout/Zin.

Click to expand...

must be:

It turns out that matching occurs when Zt=sqrt(Zout*Zin), not when Zt=(Zout+Zin)/2.

Click to expand...

@Georgy.Moshkin

on a Smith chart reflection coefficient Г phase decreases in clockwise direction.

Click to expand...

Why is that so ?

When we physically add λ/4 line line to output of transistor, we will delay reflection signal two times by 90° (signal enters newly added λ/4 line, travels 90°, reflects from transistor s22, travels 90° degree in opposite direction through the same λ/4 line). So adding line with some characteristic impedance Z0 with θ phase length will delay reflected signal by 2*θ degrees (on a smith chart with center at Z0).

Click to expand...

Why 90° ? Why reflects from transistor s22 ?

promach said:

Why is that so ?

Click to expand...

Smith chart uses polar coordinate system with angle increase in counter-clockwise direction.

promach said:

Why 90° ? Why reflects from transistor s22 ?

Click to expand...

It's just an example. You may start from any reflection coefficient Г.

@Georgy.Moshkin

I do not understand why reflects from transistor s22 ? There is only one transistor in the circuit, right ?

S22 is s-parameter. What exactly do you want to achieve? studying or maybe designing some device? I think you may start from some tutorials on transistor matching, there are many materials online on this topic, you may Google for it.

I think I got what you explained on s22.

I think it is not scaled Smith chart. This circle shows possible impedance values for Zt line of length 0 to λ/2. For 0 to λ/4 impedance moves from Zout to Zin. For λ/4 to λ/2 impedance moves back from Zin to Zout. You may stop at some arbitrary line length and end up with a complex impedance too.

Click to expand...

But why use two smith charts together ?

Like Georgy.Moshkin I don't see two Smiths charts.

@FvM

Visually, we can see two smith charts with black and red colours.

Could you "see" them now ?

black and red are constant resistance and constant conductance circles

@Georgy.Moshkin @FvM

Look carefully. Black and red lines in the background belong to two separate smith charts.

As stated by Georgy.Moshkin, the brown circles are constant conductance curves. But they are normalized for Z0.

You can also draw circles that represent the impedance transformation by a transmission line. Transformation by Z0 transmission lines are circles around the origin. The red half circle shows the transformation by a λ/4 line with Zt<Z0. It's a circle within the Z0 Smith chart, not a smith chart scaled for Zt.

Instead of guessing about a single page in lecture 31, you better review the T-Line transformer chapter in lecture 28 which explains everything in detail.

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As done in the lecture, you can use SimSmith to simulate Smith charts. **broken link removed**

Instead of guessing about a single page in lecture 31, you better review the T-Line transformer chapter in lecture 28 which explains everything in detail.

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Wait, you must be visiting some other school websites because I CANNOT find any lecture 31 or 28 in any of my posts.

The picture in post #1 without the handwritten supplement is copied from lecture module 31, page 14. For the verbose explanation, see module 28 Smith Chart on this page https://rickettslab.org/radio-system-design/lectures/lecture/

Anyway, did you understand the rest of my post?

You can also draw circles that represent the impedance transformation by a transmission line. Transformation by Z0 transmission lines are circles around the origin.

Click to expand...

I understand the red circle, but why "circles around the origin" ?

The red half circle shows the transformation by a λ/4 line with Zt<Z0. It's a circle within the Z0 Smith chart, not a smith chart scaled for Zt.

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I do not understand this.

Reference impedance (Smith chart center) is 50 Ohm in all plots below

line=50, load=100


line=25, load= 50


line=200, load= 50


and a case where all 3 impedances are different: line=25, load=200, reference=50

promach said:

I came across the following slide on power combiner circuit and I have questions about it.

1. Why traverse across the upper half circle ? In other words, why is Zin located at the right side of Zout ?

2. Why when translating an impedance towards the generator we move clockwise around the chart ?

3. Why "ZT-line is the geometric mean of Zout and Zin." ?

4. Why overlay the Smith chart scaled for ZT on top of the Z0 Smith chart ?

Click to expand...

The theoretical answer to your questions is: The smith chart is a bilinear transform from the impedance to the reflection coefficient domains. Such transforms follow certain rules.

It is an obscure branch of mathematics called "COMPLEX VARIABLES". Back in the day, the rules of complex transforms were used to solve thermal transfer, and other electromagnetic field problems, without the use of computers. Basically some problems are hard to solve in one domain, but trivial to solve in the new domain once the transform is made.

Take a good course in Complex Variables and it will make more sense.

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here is one book that appears to be in the public domain on Complex Variables, and has a section on conformal mapping

http://www.baileyworldofmath.org/uploads/Schaums-outline-complex-variables.pdf

looking thru amazon, there are a bunch of other books on the topic