for the coaxial cable is ¼ wavelength, everything in transmission line theory will be simplified.

the input impedance will be

Z_{in} = Z_0 \frac{1-\Gamma}{1+\Gamma}

where Γ is the characteristic impedance:

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

and the reflection wave and input wave ratio is :

\rho = \left| \frac{V_-}{V_+} \right| = |\Gamma|

these are 3 important equations on smith chart.

we can use normalized impedance, which is defined as

z_{in} = Z_{in}/Z_0

z_L = Z_L / Z_0

than  2 of the  3 equations will be normalized to :

z_{in}= \frac{1-\Gamma}{1+\Gamma}

$latex \Gamma = \frac{ z_L – 1 }{z_L + 1} $

by setting z_{in} = r + i x , we have :

\Gamma = \frac{ 1- r^2 - x^2 }{(1+r)^2+x^2} - i \frac{2x}{(1+r)^2+x^2}

by some algebra, we have :

z_{in} = 1/ z_L

which is the result for ¼ wavelength cable. by this,we have:

z_L = \frac{r}{r^2+x^2}- i \frac{x}{r^2+x^2}

after many equations, for impedance matching, we have following equivalent statements.

  • impedance matching
  • z_L = 1
  • \Gamma = 0
  • z_{in} = 1
  • \rho = 0 , no reflected wave

OK. the math is over. The Smith Chart is the Cartesian coordinate for Γ, real axis on horizontal, imagine axis on vertical. and the circles, are the transformed coordinate of the input impedance. the transformation is called Möbiüs  transform.

a good impedance matching can be found at the origin of the chart, where \Gamma = 0 + 0 i , and if we read the input impedance coordinate, it is z_{in} = 1 + 0 i , which mean, impedance matched.

However, there is never so ideal in real world. the input impedance always has some imaginary part, or real part not equal to 1. so, what if, both of them are different by 0.2 or 10 Ohm?

 

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