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

  1. Why traverse across the upper half circle ?

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

Smith chart of λ/4 Transmission Line

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    \$\begingroup\$ have you read the math (the derivations) behind the Smith Chart? \$\endgroup\$ – analogsystemsrf Mar 18 at 14:32
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Why traverse across the upper half circle ?

When translating an impedance towards the generator we move clockwise around the chart.

When translating an impedance towards the load, we move counter-clockwise around the chart.

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

This is a design decision that was made to get the desired result. If you choose a different \$Z_T\$, you won't get the equivalent load (\$Z_T\$ line plus actual load) to match your input impedance.

Notice how they overlaid the Smith chart scaled for \$Z_T\$ on top of the \$Z_0\$ Smith chart. You need to be able to rotate the load around from the 0-ohm point to the \$\infty\$-ohm point on the \$Z_T\$ Smith chart and have that end up being the correct place on the \$Z_0\$ Smith chart. That only happens if you choose \$Z_T=\sqrt{Z_{out} Z_{in}}\$

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  • \$\begingroup\$ Why overlay the Smith chart scaled for ZT on top of the Z0 Smith chart ? \$\endgroup\$ – kevin998x Mar 19 at 1:08
  • \$\begingroup\$ why is Zin located at the right side of Zout ? \$\endgroup\$ – kevin Mar 19 at 4:55
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Why traverse across the upper half circle ?

Traverse is across the upper half of a circle is because of adding physical line with length θ=90° delays reflection from Г(Zout) by 2*θ=180° (90° before reflection and 90° after reflection). On a Smith chart phase delay corresponds to clockwise rotation around some center depending on Zline(phase decreases clockwise). Cutting physical line length will result in counter-clockwise direction. Rotation go through upper or lower half circle depending on particular values of Zout and quarterwave transformer line impedance Zt.

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

Maybe for emphasizing that it is not an arithmetic mean (Zout+Zin)/2.

Formula for Zt is derived without using Smith chart. Smith chart is a graphical aid.

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

Maybe it is more convenient to use Z0=Zt Smith chart for obtaining impedance transformation graphically without formulas (Smith chart center at Zt, not at 50 Ohm)

For deeper understanding of Smith chart I recommend to calculate impedance change for different line lengths using formulas, then calculate corresponding reflection coefficient Г (vector) for each case and draw it on a Smith chart.

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  • \$\begingroup\$ "it is more convenient to use Z0=Zt Smith chart for obtaining impedance transformation graphically without formulas" <-- could you explain more on this reason of putting two smith charts together on top of each other ? \$\endgroup\$ – kevin998x Mar 20 at 15:53
  • \$\begingroup\$ What do you exactly mean by "Smith chart center at Zt, not at 50 Ohm" ? \$\endgroup\$ – kevin998x Mar 20 at 16:01
  • \$\begingroup\$ @kevin998x , you may draw Smith chart for any impedance. Usually Z0=50 Ohm is chosen. But you may use any other Z0 in this formula: Г=(Z-Z0)/(Z+Z0). If Z0=Zt, then Zt will be in the center of Smith chart, and quarter-wave transforming will go through circle around this center. \$\endgroup\$ – Georgy Moshkin Mar 21 at 2:24
  • \$\begingroup\$ "use Z0=Zt Smith chart for obtaining impedance transformation graphically without formulas" <-- what do you exactly mean by impedance transformation ? And how is normalizing Z0 with respect to Zt related to the use of two smith chart overlaid together ? \$\endgroup\$ – kevin998x Mar 22 at 12:57

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