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3.2.5Loca on of the Poles in the s-plane
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3.2.5Loca on of the Poles in the s-plane
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1. A real pole as shown below
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in the le half of the s-plane defines an exponen ally decaying component
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the origin correspond to slowly decaying components.
in
the homogeneous phase. The rate of decay is determined by the pole loca on poles; poles far from the origin in the left-half plane correspond to components that decay rapidly, while poles near
2. A pole at the origin
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Modality for 2nd
defines a component that is constant in amplitude and defined by the ini al condi ons.
3. A real pole in the right-half plane
corresponds to an exponen ally increasing component in the homogeneous response ; thus defining the system to be unstable.
4. A complex conjugate pole pair σ±jω in the le -half of the s-plane
combine to generate a response component that is a decaying sinusoid of the form Ae−σtsin (ωt+φ) where A and φa re determined by the ini al condi ons. The rate of decay is specified by σ; the frequency
of oscilla on is determined by ω.
5. An imaginary pole pair, that is a pole pair lying on the imaginary axis,±jω generates an oscillatory component with a constant amplitude determined by the ini al condi ons.
6. A complex pole pair in the right half plane
generates an exponen ally increasing component.
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