My Circle is Like a Smith Chart

Smith chart is used be mysterious to me. My daughter has an interesting book named my heart is like a zoo. This makes me come up with the story – my circle is like a Smith chart.

Figure 1. My circle is like a Smith chart

I can find lots of reference on Smith chart online, among which one of the most useful one is explained by Prof. F. Delssperger and he also develops a handy software to help us with matching and etc. After reading his slides, I kind of understand that Smith chart is trying to combine the impedance in Z plane with the reflection coefficient in Polar diagram. However, I am still confused about the exact size and position of the circles until I happen to find this derivation.

I extract the main flow of derivations here. For details, please click the above link.

First, the reflection coefficient is a complex number.

\Gamma=|\Gamma|\angle \theta=\Gamma_R+j\Gamma_I

Second, this coefficient corresponds directly to a specific impedance as seen at the point it is measured. It can be calculated based on a load impedance ZL (using a reference impedance Z0). And the load impedance is further normalized to the reference impedance zL=ZL/Z0.

\Gamma=\frac{Z_L-Z_0}{Z_L+Z_0}=\frac{z_L-1}{z_L+1}

Third, the normalized load impedance, which is also a complex number, can be expressed by the reflection coefficient.

z_L=z_R+jz_I=\frac{1+\Gamma}{1-\Gamma}=\frac{1+\Gamma_R+j\Gamma_I}{1-\Gamma_R-j\Gamma_I}

Fourth, after rationalizing, the normalized load resistance zR and load reactance zI can be expressed by the following two circle equations.

z_R=\frac{1-\Gamma^2_R-\Gamma^2_I}{(1-\Gamma_R)^2+\Gamma^2_I}

z_I=\frac{2\Gamma_I}{(1-\Gamma_R)^2+\Gamma^2_I}

Finally, thanks to the author’s derivation, the equations of the two circle can be rewritten in a familiar format.

For normalized load resistance:

(\Gamma_R - \frac{z_R}{1+z_R})^2+\Gamma^2_I=(\frac{1}{1+z_R})^2

\boxed{\textrm{Circle with center } (\frac{z_R}{1+z_R},0) \textrm{ , radius }\frac{1}{1+z_R}}

For normalized load reactance:

(\Gamma_R-1)^2 +(\Gamma_I-\frac{1}{z_I})^2=(\frac{1}{z_I})^2

\boxed{\textrm{Circle with center } (1, \frac{1}{z_I}) \textrm{ , radius }\frac{1}{z_I}}

Now we can draw some circles on Smith chart.

Figure 2. Some circles on Smith chart

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