# Stability Equation

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#### shermaine

##### New Member
Hi,

Given conditionaly stability and s parameter of a equation, how do we know it's laying inside the circle or outside the circle?
Like |rL| = | S11 + [S21 S12 rL / 1 - S22 rL] | = 1 and |rout| = | S22 + [S21 S12 rs / 1 - S11 rs] | = 1 ?

#### Papabravo

##### Well-Known Member
Any complex quantity can be expressed in Cartesian coordinates like a + jb where a is the real part and b is the imaginary part. That same quantity can be expressed in polar coordinates (r, θ) where the radial component r is equal to:√a²+b², and θ= arctan (b/a).

If the magnitude of r > 1 you are outside the unit circle, and if the magnitude of r < 1 you are inside and if r = 1 you are on the unit circle. For circles of other sizes located at places other than the origin you can apply constants of translation or scaling but the idea is the same as for the unit circle.

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