Editing Smith chart (section)

====Circles of constant normalised resistance and constant normalised reactance====
The normalised impedance Smith chart is composed of two families of circles: circles of constant normalised resistance and circles of constant normalised reactance. In the complex reflection coefficient plane the Smith chart occupies a circle of unity radius centred at the origin. In cartesian coordinates therefore the circle would pass through the points (+1,0) and (&minus;1,0) on the {{mvar|x}}-axis and the points (0,+1) and (0,&minus;1) on the {{mvar|y}}-axis.

Since both <math>\,\Gamma\,</math> and <math>\,z_\mathsf{T}\,</math> are complex numbers, in general they may be written as:
:<math>z_\mathsf{T} = a + j b \,</math>
:<math>~ \Gamma ~= c + j d \,</math>

with ''a'', ''b'', ''c'' and ''d'' real numbers.

Substituting these into the equation relating normalised impedance and complex reflection coefficient:
:<math>\Gamma = \frac{z_\mathsf{T} - 1}{\, z_\mathsf{T} + 1 \,} = \frac{(a - 1) + j\,b}{\, (a + 1) + j\,b \,} \,</math>
gives the following result:
:<math>\Gamma = c + j d = \left[\frac{a^2 + b^2 - 1}{\,(a + 1)^2 + b^2\,}\right] + j \left[\frac{2b}{\,(a + 1)^2 + b^2\,}\right]  = \left[ 1 + \frac{ -2(a + 1) }{\,(a + 1)^2 + b^2\,}\right] + j \left[\frac{+2b}{\,(a + 1)^2 + b^2\,}\right] \,.</math>
This is the equation which describes how the complex reflection coefficient changes with the normalised impedance and may be used to construct both families of circles.<ref name="Davidson_1989"/>