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# Small-Signal Frequency Response Theory for Ideal Dc-to-Dc Converter Systems

## Citation

Lau, Billy Ying Bui (1987) Small-Signal Frequency Response Theory for Ideal Dc-to-Dc Converter Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/mqpe-vz16. https://resolver.caltech.edu/CaltechETD:etd-03012008-134552

## Abstract

The frequency response problem of switching dc-to-dc converter systems is the problem of computing the small-signal frequency response of the system with respect to its inputs. It arises in the study of the small-signal behavior and in the design of a feedback controller for the dc-to-dc converter system. There are two approaches in tackling the problem: the numerical approach and the analytical approach. This thesis is limited to the analytical approach. There are previous efforts in developing approximate analytical methods for solving the problem; however, these methods are unsatisfactory in one way or another because they are applicable only to few special cases, and valid only in limited range of frequency — less than half the switching frequency in many cases.

The Small-Signal Frequency Response Theory presented in this thesis is developed to overcome the problems encountered in the application of the approximate analytical methods. Instead of finding an approximate model for a dc-to-dc converter system and postulating that the response of the model is the same as that of the converter system, as in the approximate analytical methods, the new theory computes the frequency response of the perturbed output with respect to perturbations at the control-inputs by the direct application of Fourier Analysis. Hence, the theory is exact in the small-signal limit. Unlike the approximate analytical methods, the results given by the theory are valid at all frequencies provided that the system model used in the calculation of frequency response is valid at all frequencies. In short, the Small-Signal Frequency Response Theory is a mathematical theory for the linearization of an ideal dc-to-dc converter system in the vicinity of its periodic steady state solution.

In the derivation of the results of the Small-Signal Frequency Response Theory, two steps are taken: First, find a difference equation that describes the small-signal motion of the system in the vicinity of the given steady state solution. Second, find the equivalent hold that relates the samples of the perturbed state of the system, given by the difference equation, to the analog output signal. The z-transform of the difference equation with z = esT. is used to relate the spectrum of the sampled perturbed control-input to the spectrum of the sampled perturbed output. The frequency response of the converter system given by the theory resembles the frequency response of a classical single-rate sampled-data system.

The prediction given by the theory and the experimental results for three converter circuits are compared. These three converter circuit have the same basic circuit topology, but different control strategies. The control strategies in these three examples are: constant-switching-frequency PWM, constant-switching-frequency programmed, and bang-bang controlled. It is found that the theory consistently gives good predictions, even up to many times of the switching frequency, while, in many cases, the approximate analytical methods break down.

The theory has the best of both the time domain approach and the frequency domain approach for the analysis of switching dc-to-dc converter systems. It has the exactness of the time domain approach, which uses a difference equation to describe the system, and the measurability of the of frequency domain approach. The exactness and the uniformity of the theory, which has not been achieved before, results in significant impact in the fields of computer-aided design and modelling and analysis in power electronics.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Electrical Engineering
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Electrical Engineering
Thesis Availability:Public (worldwide access)
• Middlebrook, Robert David
Thesis Committee:
• Middlebrook, Robert David (chair)
• Caughey, Thomas Kirk
• Martel, Hardy Cross
• Rutledge, David B.
• Vaidyanathan, P. P.
Defense Date:11 September 1986
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
Garret AiResearchUNSPECIFIED
IBMUNSPECIFIED
GTEUNSPECIFIED
EmersonUNSPECIFIED
Record Number:CaltechETD:etd-03012008-134552
Persistent URL:https://resolver.caltech.edu/CaltechETD:etd-03012008-134552
DOI:10.7907/mqpe-vz16
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:828
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:13 Mar 2008  Preview