A method for modelling switching converters in the discontinuous conucction mode is developed, whose starting point is the unified state-space representation, and whose end results is a complete linear circuit model which correctly represents all essential features, namly, the input, output, and transfer properties (static dc as well as dynamic ac small signal). While the method is generally applicable to any switching converter operating in the discontinuous conduction mode, it is extensively illustrated for the three common power stages (buck, boost, and buck-boost). The results for these converters are then easily tabulated owing to the fixed equivalent circuit topology of their canonical circuit model. The outlined method lends itself easily to investigation of the discontinuous conduction mode in more complex structures (cascade connection of buck and boost converters, for example), in which more thean one inductor current may become discontinuous. As opposed to other modelling techniques, the new method considers the discontinuous conduction mode as a special case of the continuous conduction mode.

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A general unified approach to modelling switching DC-tO-DC converters in discontinuous conduction mode

A general sampled-data representation of the dynamics of arbitrary power electronic circuits is proposed to unify existing approaches. It leads, via compact and powerful notation, to disciplined modeling and straightforward derivation of small-signal models that describe perturbations about a nominal cyclic steady state. Its usefulness is further illustrated by considering the representation and analysis of a class of symmetries in circuit operation. The results of the application of this methodology to modeling the small-signal dynamics of a series resonant converter are described. The results correlate well with simulation results obtained on the Massachusetts Institute of Technology's Parity Simulator. What is of greater significance is the fact that the small-signal model is obtained in a completely routine way, starting from a general formulation and working down to the actual circuit; this contrasts with the circuit-specific analyses that are more typical of the power electronics literature. The automatability of this procedure is also discussed, and it is pointed out that the key ingredients for automatic generation of dynamic models from a circuit specification are now available.

A General Approach to Sampled-Data Modeling for Power Electronic Circuits

The letter presents an exact small-signal discrete-time model for digitally controlled pulsewidth modulated (PWM) dc-dc converters operating in constant frequency continuous conduction mode (CCM) with a single effective A/D sampling instant per switching period. The model, which is based on well-known approaches to discrete-time modeling and the standard Z-transform, takes into account sampling, modulator effects and delays in the control loop, and is well suited for direct digital design of digital compensators. The letter presents general results valid for any CCM converter with leading or trailing edge PWM. Specific examples, including approximate closed-form expressions for control-to-output transfer functions are given for buck and boost converters. The model is verified in simulation using an independent system identification approach.

Small-Signal Discrete-Time Modeling of Digitally Controlled PWM Converters

The state-space averaging method, associated with power converters, is discussed, Emphasis is on the motivation and objectives of modeling by equivalent circuits based on physical interpretation of state-space averaging. The discussion is limited to pulse-width-modulated DC-to-DC converters. >

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Small-signal modeling of pulse-width modulated switched-mode power converters

State-space average-value modeling of pulsewidth modulation converters in continuous and discontinuous modes has received significant attention in the literature, and various models have been developed. This paper presents a new approach for generating the state-space average-value model. In the proposed methodology, the so-called duty-ratio constraint and the correction term are extracted numerically using the detailed simulation and are expressed as nonlinear functions of the duty cycle and average-value of the fast state variable. The parasitic effects of circuit elements are readily included. The resulting average-value model is compared to a hardware prototype, a detailed simulation, and several previously published models. The proposed model is shown to be very accurate in predicting the large-signal time-domain transients as well as the small-signal frequency-domain characteristics.

Numerical state-space average-value modeling of PWM DC-DC converters operating in DCM and CCM

A generalized discrete time-domain modeling and analysis technique is presented for all types of switching regulators using any type of duty-cycle controller and operating in both continuous and discontinuous inductor currents. State-space techniques are employed to derive an equivalent nonlinear discrete time model that describes the converter exactly. The system is linearized about its equilibrium state to obtain a linear discrete time model for small signal performance evaluations, such as stability, audiosusceptibility, and transient response. The analysis makes extensive use of the digital computer as an analytical tool. It is universal, exact, and easy to use.

Generalized Computer-Aided Discrete Time-Domain Modeling and Analysis of DC-DC Converters

A generalized discrete time domain modeling and andlysis technique is presented for all types of switching regulaors using any type of duty-cycle controller, and operat1nq 1n both continuous and discontinuous inductor current. State space techniques are employed to derive an equivalent nonlinear discrete time model that describes the converter exactly. The system is linearized about its equilibrium state to obtain a linear discrete time model for small signal performance evaluations, such as stability, audiosusceptibility and transient response. The analysis makes extensive use of the digital computer as an analytical tool. It is universal, exact and easy to use.

Generalized computer-aided discrete time domain modeling and analysis of DC-DC converters

Using state variable representation a nonlinear, discrete-time system is derived that models the converter exactly. This system is linearized about its steady state solution, and converter stability, transient response and audio susceptibility are studied. The steady state solution of the converter is stable if and only if all the roots of the linearized system are absolutely less than unity. Excellent agreement with laboratory test data has been observed.

Time domain modelling and stability analysis of an integral pulse frequency modulated DC to DC power converter

State space techniques are employed to derive accurate models for buck, boost, and buck/boost converter power stages operating with and without zero-inductor-current dwell time. A generalized procedure is developed which treats the continuous-inductor-current mode without the dwell time as a special case of the discontinuous-current mode, when the dwell time vanishes. An abrupt change of system behavior including a reduction of the system order when the dwell time appears is shown both analytically experimentally.

Modeling of switching regulator power stages with & without zero-inductor-current dwell time

Utilizing the demonstrated capability of nonlinear programming algorithms, a practical design optimization approach for power converters is established to conceive a design to meet all power-circuit performance requirements and concurrently optimize a defined quantity such as weight or losses. In addition to facilitate a cost-effective design, the computer-aided approach provides a means to readily assess (1) the weight-efficiency tradeoff, (2) impacts of converter requirements and component characteristics on a given design, and (3) optimum power system configurations.

James E. Triner

Papers

Generalized Computer-Aided Discrete Time-Domain Modeling and Analysis of DC-DC Converters

Generalized computer-aided discrete time domain modeling and analysis of DC-DC converters

Time domain modelling and stability analysis of an integral pulse frequency modulated DC to DC power converter

Modeling of switching regulator power stages with & without zero-inductor-current dwell time

Power converter design optimization