BeginnerTopic 19 of 101Signals, Control & Comms
Poles and zeros
Roots of the denominator and numerator of a transfer function.
In depth
Poles (roots of denominator) and zeros (roots of numerator) of H(s) determine system behaviour: pole positions give natural frequencies and damping; left half-plane means stable; zeros shape magnitude and can cancel poles. Pole-zero plots allow quick frequency-response sketches.
Key points
- Dominant poles
- Complex conjugate pairs
- Minimum phase systems
ExamplePoles at -1±j3 give an oscillating decay with ω=3 rad/s.
What is Poles and zeros?
Roots of the denominator and numerator of a transfer function.
Why is it important to understand poles and zeros?
Poles and zeros summarise a transfer function in two sets of numbers, which is how engineers compare designs at a glance.
What is a dominant pole?
The pole closest to the imaginary axis, which usually controls the slowest part of the response and therefore the settling time.
Can I cancel a pole with a zero?
In theory yes, but a small error in the model means the cancellation is imperfect, so it is risky for unstable poles.
How is Poles and zeros used in practice?
Typical applications include plotting poles and zeros of a filter on the complex plane, identifying the dominant pole that controls settling time and finding the unstable pole that makes a control loop diverge.
What should I learn before Poles and zeros?
Study "Laplace transform in systems" first (topic 18 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Poles and zeros?
Dominant poles. Complex conjugate pairs. Minimum phase systems.
Can you give an example of Poles and zeros?
Poles at -1±j3 give an oscillating decay with ω=3 rad/s.
What comes after Poles and zeros?
The next topic is "Block diagrams", which builds directly on this one.
How difficult is Poles and zeros?
It is classed as Beginner in Signals, Control & Comms (topic 19 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.