A professor is writing the general form of a Transfer Function as:

$$H(s)=k\cdot s^{l}\cdot \frac{\prod\limits_{i=1}^{m}\left(1+\frac{s}{z_{i}}\right)}{\prod\limits_{i=1}^{n}\left(1+\frac{s}{p_{i}}\right)}$$

but in every book and in every source that I have found is never mentioned that way. The poles and zeros are never in fractions in the numerator and denominator. Am I missing something? Has anyone any source to explain the logic behind this specific form of Transfer Function?


1 Answer 1


There is nothing to be surprised at. A transfer function can be shown in one of the two following notations.

  1. $$ H(s) = K\dfrac{(s-z_1)(s-z_2)\dots(s-z_n)}{(s-p_1)(s-p_2)\dots(s-p_m)} $$

  2. $$ H(s) = \dfrac{b_ns^n + b_{n-1}s^{n-1}+ \dots + b_0s^0}{a_ms^m + a_{m-1}s^{m-1}+ \dots + a_0s^0} $$

The first one is good for seeing the locations of poles and zeros at first glance, and for performing partial fraction expansion. The second one is good for easily extracting the differential equation between input and output, also for transforming into state space form. Each notation has their own advantages and handicaps under every topic.

  • \$\begingroup\$ Thank you for your answer, but I know those two notations. The one thing that I don't understand is how the poles came to appear in the denominator of the fraction appeared in the denominator. How can someone reach at this form of the Transfer Function? \$\endgroup\$
    – Adam
    Oct 29, 2014 at 22:56
  • \$\begingroup\$ As far as I understood, you are asking how to write down a polynomial in form of product of fractions. It is not very easy to do by hand. I remember finding very complex formulas for up to 5th order polynomials in Wikipedia (example). It may be possible to solve some 3rd order polynomials by paper and pencil, but we usually use computer software (e.g.; Matlab) for these kind of difficult calculations. \$\endgroup\$ Oct 29, 2014 at 23:04
  • \$\begingroup\$ Yes of course I don't want you to do it as I know it is painstaking. I only find strange this form of Transfer Function and I would like to see if there is actually any source mentioning it and explaining it and yes maybe proving how can someone reach to this form. Also thank you for those links! \$\endgroup\$
    – Adam
    Oct 29, 2014 at 23:10
  • \$\begingroup\$ @Adam, each frequency-dependent function (containing capacitive and inductive componenets) can be written as a ratio of polynominals. And each polynominal can be set to zero and solved for the corresponding (mostly) complex parameters (in our case: s). Thus, we can always write the numerator and the denominator as a product of expressions like (s-z). Because all zeros of the denomionator are poles of the whole function we replace these terms by the symbol "p". That´s all. \$\endgroup\$
    – LvW
    Oct 30, 2014 at 8:07

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