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When writing the transfer function for a system, the Characteristic equation in the denominator which carries information about the poles of the system is equated with the standard second order transfer function denominator to find out the Resonant Frequency and Quality factor of the system.

Control to Output Transfer Function

The author( User: [Verbal Kint][1]) does the same when trying to find out the quality factor of Buck converter Control to Output transfer function(Page 117) and Buck converter Output impedance(Page 95) in the attached link.

But in the Page 126 , when deriving the transfer function of the Buck converter Input impedance, the author equates the numerator which carries the information about zeros in the system to find out the Quality factor and Resonant frequency of Buck converter Input impedance.

Buck converter Input Impedance

Am a beginner when it comes to writing the transfer function and control system in general.

Can someone explain, why it is equated differently for different kinds of Transfer function?

Thank you.

user15174
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  • Hello, there are no differences between the two approaches. Whether the 2nd-order polynomial lies in the denominator or in the numerator, you can always write it in a canonical form, highlighting a \$Q\$ and an \$\omega_0\$. When both the numerator and the numerator host a 2nd-order polynomial, I rewrite them with \$Q_N\$ and \$\omega_{0N}\$ for the numerator and \$Q_D\$ and \$\omega_{0D}\$ for the denominator. Let me know if it makes sense to you. – Verbal Kint Aug 30 '23 at 13:45
  • Also keep in mind that if \$\frac{V_{_\text{OUT}}}{V_{_\text{IN}}}=\frac{\left[\left(\frac{s}{\omega_{_{N_0}}}\right)^2+\frac1{Q}\left(\frac{s}{\omega_{_{N_0}}}\right)+1\right]}{\left[\left(\frac{s}{\omega_{_{D_0}}}\right)^2+\frac1{Q}\left(\frac{s}{\omega_{_{D_0}}}\right)+1\right]}\$ then it follows that \$V_{_\text{OUT}}\cdot\left[\left(\frac{s}{\omega_{_{D_0}}}\right)^2+\frac1{Q}\left(\frac{s}{\omega_{_{D_0}}}\right)+1\right]=V_{_\text{IN}}\cdot\left[\left(\frac{s}{\omega_{_{N_0}}}\right)^2+\frac1{Q}\left(\frac{s}{\omega_{_{N_0}}}\right)+1\right]\$. – periblepsis Aug 30 '23 at 14:18
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    It's also possible to convert the entire system so that only the denominator matters, converting the rest into gain constants. See [this EESE answer from jonk](https://electronics.stackexchange.com/a/632840/330261) for details. – periblepsis Aug 30 '23 at 14:19
  • Am able to understand what is Q factor in the context of filters. Could you explain what is Q factor in the context of those three transfer functions?(Control to output, Output impedance and Input impedance). Can a transfer function contain Q factor in both numerator and denominator? Sorry if the question sounds stupid. Am trying to learn these concepts.Also kindly share any references/links to understand these theoretical concepts better. Thank you. – user15174 Aug 30 '23 at 17:20
  • @user15174, have a look at the [answer](https://electronics.stackexchange.com/questions/675851/analysis-of-an-rcl-filter-utilizing-a-potentiometer-as-r/676144#676144) I provided on SE. There is 2nd-order polynomials in the numerator and the denominator. In this example, the \$Q\$ are different but \$\omega_0\$ is common to both expressions. In the control-to-output transfer function of the voltage-mode buck, you have a \$LC\$ filter showing a resonance and a peaking depending on the losses in the circuit: this is what the 2nd-order polynomial describes. – Verbal Kint Aug 30 '23 at 18:26

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