gives a matrix of roots of the denominators in the TransferFunctionModel tfm.


only gives the roots inside the region reg on the complex plane.


  • A rectangular region reg can be given as {{remin,remax},{immin,immax}}.
  • The default region is the entire complex plane.


open allclose all

Basic Examples  (3)

The poles of a notch filter:

The poles for specific values of damping ratio and natural frequency:

The poles of a MIMO system are the poles of the elements of the transfer function:

Time-delay systems may have an infinite number of poles:

But they always have a finite number of poles in a bounded region:

Scope  (5)

The poles of a fourth-order Butterworth filter:

They lie on the unit circle:

The poles of a tenth-order Bessel filter:

The poles of a discrete-time second-order system:

The system is stable because the poles lie within the unit circle:

The poles of a time-delay system in a square around the origin:

The poles trace out paths on the complex plane:

The poles of the elements of a MIMO system:

Applications  (2)

Use TransferFunctionPoles to determine if a system is asymptotically stable:

Asymptotic stability of various systems:

Find the poles of the time-delay system near the origin:

Create a delay-free approximation using the poles:

Compare step responses of the time-delay system and the delay-free approximation:

Properties & Relations  (4)

For a SISO system, the poles of the transfer function are the eigenvalues of its state matrix:

The poles determine the natural response of a system:

The exponentials in the response are the real parts of the poles:

The root locus plot gives the closed-loop poles as any parameter is varied:

The root locus plot as k is varied:

A stable third-order system with one symbolic pole:

It can be better approximated by a second-order system if the third pole is further to the left:

Possible Issues  (1)

TransferFunctionPoles may not find solutions for time-delay systems:

Specify a region:

Plot the poles:

Wolfram Research (2010), TransferFunctionPoles, Wolfram Language function, (updated 2012).


Wolfram Research (2010), TransferFunctionPoles, Wolfram Language function, (updated 2012).


Wolfram Language. 2010. "TransferFunctionPoles." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012.


Wolfram Language. (2010). TransferFunctionPoles. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2021_transferfunctionpoles, author="Wolfram Research", title="{TransferFunctionPoles}", year="2012", howpublished="\url{}", note=[Accessed: 24-May-2022 ]}


@online{reference.wolfram_2021_transferfunctionpoles, organization={Wolfram Research}, title={TransferFunctionPoles}, year={2012}, url={}, note=[Accessed: 24-May-2022 ]}