Modeling first-order dynamics of a switching converter

Question

I have created a simple, first-order model of a switching regulator in steady state (it's basically a transformer with a loss element). Its only parameters are desired output voltage \$V_o\$ and efficiency \$\eta\$:

^{simulate this circuit – Schematic created using CircuitLab}

$$ I=-\frac{V_o}{V(\rm{in})} \cdot I(\rm{Vsns}) \\ V = V_o \\ R_{\rm loss} = V(\rm{in})^2 \frac{1}{I(\rm{Vsns})} \frac{1}{V_o} \frac{\eta}{1-\eta} $$

This works well for both Buck and Boost. However, this is in steady state, or assuming an infinite loop bandwidth with infinite loop gain.

What is the simplest way to add the behavior of a feedback loop to this model, parameterized by loop bandwidth (unity cross over frequency of \$T(s)\$; \$T(s)\$ can be assumed single pole rolloff to first order)?

Answer 1

The PWM-switch model has been introduced in 1986 by Vatché Vorpérian (JPL) and Larry Meares (Intusoft) - see V. Vorpérian, Simplified Analysis of PWM Converters using Model of the PWM Switch, parts I and II, IEEE Transactions 1990 - and object of numerous publications since then. It is a three-terminal device and you may want to look at it if you have interest in the modeling of switching converters. The circuit was originally described for continuous conduction mode (CCM) operation and later for discontinuous conduction mode (DCM). I have derived an auto-toggling version in 2005 working for voltage- or current-mode control. There is also a current-mode version predicting subharmonic instabilities. The device is shown below in a compensated buck converter (excerpt from my book):

The PWM-switch model lends itself amazingly well to analyzing many switching converters operated in different modes. To that respect, it is simpler to handle than the state-space averaging (SSA) technique which requires heavy matrices manipulations. It is a nonlinear model which, once linearized, lets you determine the transfer functions of your choice. If you want to dig further, you can have a look at my APEC 2013 seminar on the subject