## Small-Signal Analysis

Compute a linear approximation of power electronics model

Small-signal analysis approximates the behavior of a nonlinear power electronics system, such as a switched-mode power supply, with a linear time-invariant (LTI) model that is valid around an operating point of interest. Small-signal analysis is an enabling step to apply classic control theory to power electronics systems, which requires an LTI representation such as a transfer function or a state-space model of the system.

For well known, simple topologies such as a boost or a buck converter, you can derive their equivalent LTI systems analytically. However, for nonstandard converter topologies and for converters integrated in complex power-electronics-based systems analytical derivation becomes very time-consuming and error-prone.

An industry accepted approach to do small-signal analysis is to build a simulation model of a power electronics system, and then use frequency response estimation. Frequency response estimation starts with superimposing a small perturbation signal of defined amplitude and frequency content to the input of the power electronics system around the operating point and measuring the system response to this perturbation. You then use perturbation signal and measured output signal to compute the frequency response or a transfer function that represents the system dynamics in the vicinity of the operating point.

You can inject different types of input signals into a model to compute frequency response:

- Sinestream, a series of sinusoidal perturbations applied one after another.
- Chirp, a swept-frequency signal that excites the system at a range of frequencies, such that the input frequency changes instantaneously.
- Random input signals.
- Step input signal.

Once you have computed the frequency response or a transfer function of the system, you can design a compensator and evaluate it against the linear model. By repeating small-signal analysis for different operating conditions (for example, different desired output voltage levels or different duty cycle ratios), you can develop a gain-scheduled controller to operate the power electronics system across the desired operating range.

Using Simulink, you can:

- Build accurate simulation models of switched-mode power supplies, AC motors, and other loads in the distribution systems.
- Conduct small-signal analysis of power electronics model using a choice of several perturbation input signals.
- Design and tune a compensator for the obtained linear model using techniques such as automated PID tuning or interactive loop shaping with root locus and Bode diagrams.
- Design a gain-scheduled compensator to control power electronics system across the range of operating condition.
- Verify and test controller design by simulating it against a nonlinear model of power electronics system.
- Automatically generate ANSI, ISO, or processor-optimized C code and HDL for rapid prototyping and production implementation of the controller.

#### Estimating the Frequency Response of a Power Electronic Model

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