# 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 ?

2. Why "ZT-line is the geometric mean of Zout and Zin." ? • have you read the math (the derivations) behind the Smith Chart? – analogsystemsrf Mar 18 '19 at 14:32

## 2 Answers

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}}\$$

• Why overlay the Smith chart scaled for ZT on top of the Z0 Smith chart ? – kevin998x Mar 19 '19 at 1:08
• why is Zin located at the right side of Zout ? – kevin Mar 19 '19 at 4:55

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.

• "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 ? – kevin998x Mar 20 '19 at 15:53
• What do you exactly mean by "Smith chart center at Zt, not at 50 Ohm" ? – kevin998x Mar 20 '19 at 16:01
• @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. – Georgy Moshkin Mar 21 '19 at 2:24
• "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 ? – kevin998x Mar 22 '19 at 12:57