Smith chart of λ/4 Transmission Line

Question

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." ?

Answer 1

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}}\$

Answer 2

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(Z_out) 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 Z_out and quarterwave transformer line impedance Z_t.

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=Z_t Smith chart for obtaining impedance transformation graphically without formulas (Smith chart center at Z_t, 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.