Transfer function's poles and Auxiliary Equation

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In summary, a transfer function's pole is a point in the complex plane where the denominator of the function becomes zero, representing the system's natural frequency and affecting its stability and response. Poles are related to the auxiliary equation and their solutions give the values of the poles in the complex plane. They indicate the stability and response of the system, with poles in the left half representing stability and poles in the right half representing instability. Changing the poles can alter the system's response and stability, with closer poles to the origin making the system more stable but less responsive, and farther poles making it less stable but more responsive. A transfer function can have multiple poles, each representing a different natural frequency and determining the function's order.
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unseensoul
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Why does the homogeneous of a second order differential equation/system (i.e. a series RLC circuit) is identical to the transfer function (i.e. H(jw)) denominator in its standard form? Therefore the poles of the transfer function are also the solutions for the auxiliary equation...

I cannot see any link between these two things, but they seem to be interrelated somehow. Is there any proof for this?
 
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The Laplace transform transforms a differential equation into an algebraic equation. We can work it out in more detail if you wish.
 

1. What is a transfer function's pole?

A transfer function's pole is a point in the complex plane where the denominator of the transfer function becomes equal to zero. It represents the natural frequency of the system and can affect the stability and response of the system.

2. How are poles related to the auxiliary equation?

The poles of a transfer function are the roots of the auxiliary equation, which is obtained by setting the denominator of the transfer function equal to zero. The solutions to the auxiliary equation give the values of the poles in the complex plane.

3. What is the significance of poles in a transfer function?

Poles in a transfer function can indicate the stability and response of a system. If all the poles are in the left half of the complex plane, the system is stable. If any poles are in the right half of the complex plane, the system is unstable. The distance of the poles from the origin can also affect the system's response.

4. How does changing the poles affect the transfer function?

Changing the poles of a transfer function can change the system's response and stability. Moving poles closer to the origin can make the system more stable and less responsive, while moving poles farther from the origin can make the system less stable and more responsive.

5. Can a transfer function have multiple poles?

Yes, a transfer function can have multiple poles, each representing a different natural frequency of the system. The number of poles also determines the order of the transfer function.

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