Overview

A small-signal is applied to a system to investigate the small-signal behavior around its equilibrium. Small-signal analysis is often carried out in the frequency-domain because the resulting data can be succinctly displayed in various graphical forms, such as semi-log Bode plots and direct or inverse polar plots. Various characteristics, such as impedances, transfer functions, and stability information, can be determined from these graphical plots.

The application of small-signal frequency-domain analysis to switching piecewise-linear systems presents tremendous challenges. A Laplace transform or Laplace-transformed equivalent-circuit analysis, which is normally applied to a non-switching system to extract the small-signal frequency-domain characteristics, cannot be easily applied to a switching piecewise-linear system due to the inherent switching actions. SIMPLIS-FX is a small-signal frequency-domain analyzer specifically designed for the analysis of switching piecewise-linear systems. The analysis is based on the time-domain simulation of the switching piecewise-linear systems without having to resort to any circuit averaging or derivation of equivalent non-switching models. Instead of removing the switching actions to derive the small-signal frequency-domain characteristics, SIMPLIS-FX includes the switching action in its calculation of the small-signal frequency-domain characteristics.

While the computational aspects of SIMPLIS-FX may be daunting and tedious, the mathematical basis from which it is formulated is very simple. First, SIMPLIS-POP is used to compute the periodic operating point/trajectory of a switching piecewise-linear system. This operating point/trajectory represents the large-signal equilibrium of the system. SIMPLIS-FX can then be applied to study the small-signal behavior of the system around the large-signal equilibrium. The small-signal frequency-domain analysis is actually a sequence of analyses at discrete analysis frequencies. At each analysis frequency, the procedure of the analysis can be summarized as follows:

  1. Apply small-signal stimuli in the form of voltage/current sources to the system under study. These small-signal stimuli are called the small-signal AC sources and their waveforms are time-domain sinusoidal. At each analysis frequency, the frequencies of all small-signal AC sources are set to the same value, the analysis frequency, and their amplitudes are set to infinitesimally small values. Since the frequency of all small-signal AC sources are set to the analysis frequency, the analysis frequency is also frequently called the excitation frequency.
  2. The equilibrium of the system under perturbation from the small-signal is computed next. This new equilibrium, although at an infinitesimally small distance away from the large-signal equilibrium computed by the periodic operating-point (POP) analysis, is definitely not the same as the large-signal equilibrium. While the large-signal equilibrium is periodic with a frequency equal to the periodic operating frequency, or the switching frequency, as determined by the POP analysis, this new equilibrium is periodic with a frequency that is equal to the highest common factor between the analysis frequency and the periodic operating frequency.
  3. Fourier analysis is then applied to the new equilibrium to extract the small-signal response of the system at the analysis frequency.
Since SIMPLIS-FX is based on time domain simulations, it can handle a switching piecewise-linear system with any structure (topology), any mode of operation, and any control scheme, and under fixed or variable switching frequency as long as the following two conditions are satisfied:
  1. The system can be simulated in the time domain via SIMPLIS-TX, and
  2. SIMPLIS-POP is able to successfully compute the periodic operating point/trajectory of the system.
For example, multiple-switch multiple-output, multiple feedback loop converters are easily handled by SIMPLIS-FX because the analysis is constructed for general switching piecewise-linear systems without any assumption or restriction placed on the number of switches, outputs, or feedback loops.

Some switching power supplies are designed to have very high DC gain to improve the line/load regulation of the output voltage(s). The measurement of the loop gain of these systems cannot be carried out with the loop opened because the high DC gain of the system may drive the opened-loop system to operate at a vastly different operating equilibrium from the original closed-loop equilibrium. Thus, the ability to evaluate the loop-gain of a switching power supply with the feedback loop closed is very useful. The effect of the parasitic elements on the frequency response is usually minimal. If it is suspected that a few parasitic elements are significantly affecting the frequency response of the system, SIMPLIS-FX can be relied upon to verify such a hypothesis. SIMPLIS-FX can accurately predict the impact that the parasitic elements might have on the frequency response because SIMPLIS-FX does not assume the system variables to be slowly varying within one switching period and does not remove the switching actions during the process of deriving the small-signal response.

The algorithm behind SIMPLIS-FX is rigorously derived, making it accurate and robust. For example, the analysis frequency is not limited to less than half of the switching frequency. Undeniably, aliasing is going to be present when a switching system is excited at a frequency over half of the switching frequency. In such a situation, SIMPLIS-FX can accurately compute the response of the system at the excited frequency, whether it is below or above the switching frequency. Theoretically, the small-signal frequency-analysis algorithm behind SIMPLIS-FX is accurate to within 0.5 dB and 1 degree at each analysis frequency from DC to infinity. Practically, the accuracy of the analysis and the highest analysis frequency that can be applied and still maintain a prescribed accuracy depend on how accurately the physical components are modeled in the switching piecewise-linear system. If there is a noticeable discrepancy between the measured frequency response in the laboratory and the data generated by SIMPLIS-FX, you can trust SIMPLIS-FX and concentrate on

  1. determining that the simulated system represents, with reasonable accuracy, the system measured in the laboratory,
  2. improving the device models of any components that you believe have not been adequately modeled, and
  3. checking the laboratory measurement setup to make sure that the measurements are valid, since laboratory measurements of switching systems with small-signal excitation are inherently noisy and noise can easily lead to measurement errors.
In summary, the features of SIMPLIS-FX are as follows:
  1. It is based on time-domain simulation via SIMPLIS-TX.
  2. It relies on SIMPLIS-POP to compute the large-signal periodic operating equilibrium of the system.
  3. It is general and versatile:

    it handles any pulse-width modulated (PWM) circuit topologies such as boost, buck, buck-boost, Cuk, SEPIC, half-bridge, full-bridge, etc.,

    it handles any resonant circuit topologies such as series resonant, parallel resonant, quasi resonant, phase-shifted resonant, etc.,

    it handles any mode of operation such as continuous-mmf (CMM) mode, discontinuous-mmf (DMM) mode, etc.,

    it handles any control scheme such as voltage-mode control, peak-current-mode control, average-current-mode current, charge control, etc.,

    it handles both fixed-frequency as well as variable-frequency systems, and

    it easily handles multiple-switch multiple-output converters

    it can evaluate the loop-gain of a system while the loop is closed

    it can evaluate the effect of the parasitic elements on the frequency response,

    it is accurate for analysis frequencies above the switching frequency, and

    it is accurate to within 0.5 dB and 1 degree
◄ Example of Applying the POP Analysis Tool   Statements Relating AC Analysis ▶