Pii: S0026-2692(00)00024-0

Recent advances in semiconductor technology have greatly increased the performance and range of application of switched mode circuits. Periodic switching can give rise to acoustic noise [Y.-S. Lai, Random switching techniques for inverter control, Electronics Letters 33 (9) (1977) 747–749] or undesirable electromagnetic radiation. These problems can be reduced through the use of random switching policies [S.Y.R. Hui, S. Sathiakumar, K.-K. Sung, Novel random pwm schemes with weighted switching decision, IEEE Transactions on Power Electronics 12(0885-8993) (1997) 945–951], but it is not always clear how this could be done without affecting other performance measures, such as RMS ripple or stability. We use the buck/boost regulator as an example for analysis and determine some simple techniques for choosing appropriate component values. The circuit is simulated and it is shown that strict adherence to the formal limits of stability, suggested by control theory, does not always guarantee a satisfactory output. We demonstrate that if switching is performed quickly enough then a state-space averaged model may be used for the buck/boost controller. This model is stable within wide bounds. It is possible to use some of this freedom to optimise EMC performance through the use of a control law which is random within certain limits. In the popular mind, the idea of “randomness” seems to be completely opposed to the idea of “control.” We show that not necessarily the case. Some randomness can beneficial, from the point of view of minimising the maximum power spectral density of the noise waveforms in the output current. This can be done without compromising the stability of the system. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Switched mode; Control; Random; Stochastic resonance 1. The buck/boost regulator as an example for analysis In this paper we use a state variable approach to systems, which is very general. Rather than discussing all systems purely in the abstract, we illustrate the important points using an example system. We have selected the buck/ boost regulator for the following reasons: • It is very commonly used. Applications are found with the data sheets of many of the commercially available integrated circuits, such as the LM78S40. • Simple analysis can readily be found in the literature [1,2]. • Since this regulator can “boost” voltages, it has interesting stability properties. It can appear to be unstable if an inappropriate control rule is used. • This regulator is composed from linear elements and can be readily formulated and analysed in state space. The buck/boost regulator is basically a switched inductor circuit. The topology is in Fig. 1. The regulator has two modes. In mode 1, called the “on” time, S1 is closed and S2 is open. In mode 2, called the “off” time, S1 is open and S2 is closed. We can denote the “on” time by DT1 and the “off” time by DT2. The use of the symbol “D” implies that the switching times are small compared with all of the time constants in the regulator. This switched-mode system only has two modes and only a very simple control rule is needed, or even possible. The system can be viewed as a finite state machine and the control law can be represented using a state transition diagram, shown in Fig. 2. The “modes” of operation of the buck/boost regulator are shown as “states” of the state transition diagram. In practice, S1 is often a bipolar transistor and S2 is a diode [1,2]. The buck/boost circuit is an inverting regulator. The average DC values of Vs and Vc are opposite in sign. 2. Normal operating conditions and a simple approach to design The natural state variables to use for this type of circuit are the capacitor voltages and the inductor currents. Microelectronics Journal 31 (2000) 515–522 Microelectronics Journal 0026-2692/00/$ see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0026-2692(00)00024-0 www.elsevier.com/locate/mejo * Corresponding author. E-mail address: [email protected] (A. Allison). Together they characterise the total stored energy of the system. These variables are also preserved across switching boundaries. If DT1 and DT2 are “small” then simulations show that the responses have a small triangular wave, or “ripple”, superimposed on top of them. The rise and fall times of this superimposed wave are tied to the switching times, DT1 and DT2. If we consider the quiescent or DC case (after all transients have been attenuated), then we expect to get regular triangular waveforms, as shown in Fig. 3. We can develop a piecewise linear model. The symbol Vc is used here to denote the median value of the capacitor voltage and DVc denotes the ripple voltage across the capacitor. The capacitor voltage is also equal to the output voltage, delivered to the load, Rl. Similarly, the symbol Il is used to denote the median value of the inductor current and DIl denotes the ripple current through the inductor. During the “on” time, the inductor current is also equal to the input source current. During the “off” time, the input source current is zero. The simple application of nodal and mesh equations to the system, in both modes, leads to the following formulation: during the “on” time: IlRs 1 L DIl DT1 ˆ Vs …1† C …2DVc† DT1 ˆ Vc Rl …2† and during the “off” time: Il 2 C DVc DT2 1 Vc Rl ˆ 0 …3† 2L DIl DT2 ˆ Vc: …4† We can eliminate terms involving DVc and DIl and write the equations in matrix form: 21 RlC 2…1 2 d† C 1…1 2 d† L 2dRs L 6664 7775 Vc Il " #