The Nyquist criterion determines the closed-loop stability of a feedback system from its open-loop transfer
function 
.
For unity negative feedback, the closed-loop poles are the
zeros of the characteristic function 
.
Let 
be a positively oriented Nyquist contour, let
be the number of poles
of 
inside 
,
and let 
be the number of zeros of 
there. The argument principle
states that the winding number of 
about 0 is 
. Translating the curve by 
gives the equivalent statement that the Nyquist
plot 
winds 
times about the critical point 
. This point is critical because 
is equivalent to 
, so a preimage of 
is a closed-loop pole.
To use the criterion, first count the open-loop right-half-plane poles to obtain 
,
then draw the Nyquist plot, and finally count its
winding about 
.
The closed-loop system is stable exactly when the resulting
value gives 
,
so with the stated orientation the required winding
number is 
.
Reversing the contour orientation or the sign
convention for winding reverses this sign. If 
has a pole on the imaginary
axis, the Nyquist contour is deformed by a
small semicircular detour around that pole, preventing any
pole from lying on the contour
itself.
