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The concept of the s-plane in control systems engineering, including the definitions of zeros and poles, system characteristic polynomial and equation, and pole-zero patterns. It also covers the transient response and inverse laplace transform, as well as graphical determination of residues and root locus gain.
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We write
C(s) = G(s)R(s)
where C, G and R are each ratios of polynomials in s, i.e. G(s) =
num G . den G
Consider the following definitions:
Note that the roots of the C.E. are
Since, the polynomials have real coefficients, the poles and zeros are
We plot the poles and zeros in the s(σ + jω) plane.
Example:
1 Assume R(s) = s
. Then the pole-zero pattern of C(s) = R(s)G(s) is the
superposition of the patterns of R(s) and G(s):
K(s + 2) C(s) = s(s + 4)
(a) Typical factor in PFE is
(a is positive and b is negative)
We can write
where b − (−a) is
So in the s-plane:
(b) The general expression for K 1 in the example above is
(c) Using the actual values, we have:
and for K 2 :
(d) So as before,
− 4 t c(t) = K( + e ) 2 2
