Transactions on Engineering Sciences vol 11, 1996 WIT Press, ISSN

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1 The design and modelling of resonant switched mode power supply (SMPS) using Simulink and Matlab B.Baha,»D.C.Hamill* "Department ofelectrical and Electronic Engineering, University of Brighton, Brighton, Sussex, UK *Department ofelectronic and Electrical Engineering, University of Brighton, Brighton, Sussex, UK Abstract A method of modelling DC-DC converters has been developed using SIMULINK. For flyback quasi-resonant converter (QRC), the equations for the state variables and the active and passive switches have been derived and a corresponding SIMULINK model has been constructed. Sub-models for the different parts of the controller have also been constructed, and the complete system has been simulated. This modelling technique requires less memory and CPU time than SPICE. 1 Introduction In many industrial applications, there is an increasing demand to reduce the size and weight of power supply circuits and to improve their efficiency. With advances in semiconductor devices (such as BIT, MOSFET, GTO, IGBT and MCT) and the arrival of new DC-DC converter topologies, it is now possible to increase the switching frequency of resonant converters, quasi-resonant converters (QRCs) and multi-resonant converters to the MHz range. By increasing the switching frequency, the size of inductors and capacitors in resonant switched mode power supplies (SMPS) will be reduced considerably and consequently the size and weight of these power supplies will also be reduced. The analysis, design and practical implementation of such circuits without computer simulation is extremely laborious, time-consuming and expensive. Thus computer simulation plays an important role in the design and analysis of such circuits. Traditionally various types of SPICE have been used to predict the operation of such circuits. Although SPICE can analyse converter circuits, it is less well suited to the design and analysis of controllers, which SIMULINK can handle with ease. In addition, to establish the transient response of a DC-DC converter, it is necessary to carry out long term simulation. It is well known that SPICE is very slow and is not practical for this purpose. In this paper, step-by-step modelling of a QRC and its control circuit has been carried out and the performance evaluated from the SIMULINK simulation. The converter and the controller are first represented by a set of mathematical equations; the derived equations are fed to SIMULINK; and the complete system is then simulated. This technique is expected to facilitate the design, practical implementation and testing of all types of DC-DC converters. 2 What is Simulink? SIMULINK is a computer program developed by Math Works [1] for the analysis of dynamic systems. There are two important steps in running SIMULINK: model definition and model analysis. In the definition phase a graphical editor is used to set-up hierarchical block diagram models of control systems. Blocks can be selected form the SIMULINK library and blocks can be nested within other blocks. Then the blocks can be wired together to establish the model

2 36 Software for Electrical Engineering of a system. In the analysis phase, the defined model can be simulated from the SIMULINK menu. SIMULINK can use any function of MATLAB or its related toolboxes. These toolboxes include control systems, non-linear control, robust control, optimization, system identification, neural networks, fuzzy logic, quantitative feedback theory, partial differential equations, signal processing, symbolic mathematics, and many more. MATLAB and its toolboxes are well known and widely used by control engineers. The simulation process in SIMULINK consists of numerical integration of ordinary differential equations. The available algorithms include Euler's method, Runge-Kutta third and fourth order methods, Gear's predictor-corrector method for stiff systems and Adams predictorcorrector methods. For more detailed information, the user guide of SIMULINK should be consulted [1]. SIMULINK and MATLAB are well known and widely used by control engineers and it is believed that its application can be extended to power electronics, in particular to the dynamic modelling of DC-DC converters. In the next section, SIMULINK is employed to model a resonant switched mode circuit. Sub-models for the converter and control circuit are derived, and the converter is tested in an open-loop and closed-loop system. 3 The model of Flyback QRC To illustrate the application of SIMULINK to DC-DC converters, the zero-current-switching (ZCS)flybackQRC as shown in figure (1) was used. This was derived from the PWM flyback converter by replacing the active switch by a resonant switch [2]. Figure (1), non-isolated flyback QRC Modelling is carried out in two steps: first, the state variables (ilo» vgo, il, converter are modelled, then the active and passive switches. of the 3.1 Modelling the state variables of the flyback QRC In a QRC circuit four state variables can be identified. Of these four state variables two (i vgo), associated with the resonant tank are fast changing, while the other two (i associated with the output filter, are slowly changing. The following equations represent the state variables: =J^ J VLO dt + ilo(o)

3 Software for Electrical Engineering 37 = I 'Co dt (2) =L J VL dt +il(0)- -(3) vc=c J ic dt + vc(0) (4) To determine the state of the switch and the diode, the following variables are defined: _ f 1 when S is on or Ds is conducting I 0 when S and Ds are not conducting -{ 1 when D is conducting 0 when D ts not conducting The variables A and P stand for active and passive switches respectively, and are binary variables, i.e. they are restricted to the set { 1, 0). An equation should be derived for each state variable, the active switch and passive switches throughout a complete switching cycle. From the steady state analysis, the circuit operation can be divided into four distinct stages analysed in [3]. A brief summary of each stage follows. Only the relevant equations for the purpose of simulation are presented. Finally, a single equation is derived for each state variable and for each switch. - Linear stage [A = 1, P = 1] This stage begins when S turned on and D is still conducting. The equations can be derived from figure (2): vlo = ico= 'C = - - VL (5) +io) (6) +io) (7) 'Lo- IL *" I -Co' >\ FL c " " c" I Re* "l Figure (2), equivalent circuit of linear stage - Resonant stage [A = 1, P = 0] Either S or Ds are conducts, but D is off. This stage is governed by equation (5) and the following equations: I ic = -io (9)

4 38 Software for Electrical Engineering - Recovery stage [A = 0, P = 0] In this stage none of S, Ds and D conduct, and the following equations can be written: VLO = (10) ilo = (11) ico = -il (12) together with equation (9) for the output capacitor. - Free-Wheeling stage [A = 0, P = 1] S and Ds are still off but the diode D is now forward biased. The equations during stage are (10), (11) and ico= Transactions on Engineering Sciences vol 11, 1996 WIT Press, ISSN Having considered the four stages, the equations can be combined into a single one for each state variable. For the resonant inductor current, it is necessary to consider the condition of the active switch and the integrator used equation (1). The voltage across the resonant inductor depends on the condition of the switch. When the switch is on (A = 1) the voltage across the resonant inductor is given by equation (5), and when the switch is off (A = 0) the voltage is zero. This function is represented by the Switch block in figure (3). To ensure that the resonant inductor current is zero when the switch is off, a reset integrator block has been included in figure (3). The voltage across the resonant capacitor is described by equation (2). The current in the resonant capacitor depends on the diode condition. When the diode is conducting (P = 1 ), the current is given by equations (6) and (12), and when the diode is not conducting (P = 0), it is given by equation (8). Hence, the current during a switching cycle can be represented by equation (15). Co ico = [OLo -il + io) CQ + C ' ^ " ^o - il)- P (15) where P =0 when P = 1 and vice versa. Similarly, the SIMULINK model for the filter capacitor current can be written as follows: ic = - [(%Lo -il + io) QT+~C ^ " <*Lo - il). P ( 1 6) In figure (3), the ESR of the filter capacitor has also been included in the model. 3.2 Modelling the active and passive switches The full-wave switch (S in parallel with Ds) can be represented by a switching variable A' given by A = A OR ( ilo < 0) AND A (17)

5 Software for Electrical Engineering 39 If a half-wave switch (S in series with Ds) were used, equation (17) would become: A = A AND (ilo > 0) (18) The logical equations (17) is used here. Similarly the diode D is described by P'= [P AND A AND (iq > 0)] OR ( A AND P) where if) is the diode current. (19) In figure (3), memory blocks have been inserted in the diode model to break algebraic loops. It is necessary to break algebraic loops in order to prevent SIMULINK from solving them iteratively according to SIMULINK Release notes [4]. (The memory blocks performs a sample and hold function). To test and simulate the model, the QRC Input voltage: Vdc Output voltage: 20V dc Output power: 12-50W Maximum frequency: 100kHz Ripple: 1% was designed to the following specification: The design procedure proposed in [3] was used to obtain the values of the converter components. The converter can then be simulated open-loop under steady-state and transient conditions. Here, the emphasis is on the transient analysis of the converter and its controller which will be discussed in the next section. Figure (3): The SIMULINK model of the flyback ZCS-QRC

6 40 Software for Electrical Engineering 4 Simulink Modelling of the Control Circuit Figure (4) shows an isolated version of flyback ZCS-QRC with its control circuit which includes a current sensing network, a compensation network, an integrator, a comparator and a timer circuit. The control circuit has been designed on the basis of the small-signal model of the flyback QRC proposed in [5]. ^ ] - J-c r " T I HResr Vout Rfl Rf2 S = BUK , Ds = D =BYR , Vin = 40-60V, Vout = 20V, LQ = 5.8uH, Co = 109nF, L = 88p.H, C = 1200^F, R = 8Q- 33O, Rfi= 17.3kft, Rf? = 2.7kO, RI = 18kQ, R] = 300Q, R3 = 27kQ, R4 = 5.6KQ, R$ = lookn, Cjc= l^f, C\ = loopf, C? = 85nF, REXT = loko, CEXT = loopf, IC1 =LF356, IC2 = LM319, IC3 = 54LS123 Figure (4), schematic of the control circuit for flyback To model the control circuit, each part was considered separately then combined and simulated. To simulate the network which senses the inductor current, the voltage across the filter inductor N2 (vl3) divided by the turns ratio n = ^z is integrated. The model of this network shown in figure (5) can obtained derived using equation (20): where RI and Q resistor. (20) are the components of the integrator circuit and Rf is the finite gain limiting The transfer function of the compensation network is of the following form: QRC

7 Software for Electrical Engineering 41 Vr = - HT 1+sTz (21) For the purposes of SIMULINK, equation (21) can be divided into two parts, where the first part represents the integrator for the inductor current and the second part represents the compensation network. The complete model for the flyback QRC and its control circuit is shown in figure (5). This model has been used to obtain the transient response and other simulation results which will be discussed in the next section. s 1^1 Vin2 RLoad ^ -^ ^ H Flybjick IL_ U [^>-»jdu/dt 1/n Derwatwe Oairi2 Integrator 1/RC 7" i ^.+_ J ^ Timer r\ ^.. _._. -V um2 Comparator ^~lj[ 3.3) Referenocl num(«) dent) Tranafer Fen Kg «J nn Surn Reference Figure (5): The SIMULINK model of control system

8 42 Software for Electrical Engineering 5 Simulation Results The model shown in figure (5) was simulated for both steady-state and transient conditions. Long term simulation was carried out to show the transient response of the system. The response of the system to a change in the load from 20Q to 10O is shown in figure (6). The response to a step change in the input voltage from 20V to 30V with a fixed load of 8D is shown in figure (7). The output voltage response of the system is well damped in each case, the maximum overshoot is 200m V and the settling time is 3ms. The model has also been tested in the steady-state condition. The error voltage, the inverted AC part of the filter inductor current, the reference signal (which are the two inputs to the comparator) and the output of the comparator which is the input to timer circuit have been simulated; the results are shown in figure (8). These results indicate that the control circuit is working as predicted by the frequency domain and time analysis presented in [5]. Several attempts were made to carry out long term simulation on the same computer using PSPICE, but because of memory limitations, the computer was not able to finish the simulation. How ever, it was obvious that the simulation was proceeding very slowly. output voltage Figure (6): The response of the system due to load change (20 fc to 10Q)

9 Software for Electrical Engineering 43 output *c!tage Figure (7): The response of the system due to the input voltage change (20V to 30V) control signal 2 3 output comparator < 10"' 0.5 fl -if n i 1 j H PI I Figure (8): The inverted AC part of the filter inductor current, the reference signal, comparator output which is input to the timer

10 44 Software for Electrical Engineering 6 Conclusion A new modelling technique has been developed for conventional and resonant power supplies. First, mathematical equations for the state variables and the switches are used to construct a SIMULINK dynamic model of the switched mode circuit. Next, models are derived for the compensation network and other parts of controller. Finally, long term simulations can be carried out to obtain the transient response of the system. The advantages of using SIMULINK are: 1- SIMULINK has access to sophisticated routines embedded in various MATLAB toolboxes. 2- The simulation is faster than SPICE and requires less memory. However, the disadvantages are: 1- In deriving the equations, human error may be made invalidating the simulation results. 2- Because the switching devices are idealised, the detailed waveforms of the real devices will differ from the simulations. Despite of these disadvantages, there is a place for this technique in the dynamic analysis of converters and in the design of suitable controllers, where SPICE performs unsatisfactorily. This modelling technique could easily be extended to other power electronics applications, such as motor drives. References 1- The Math Works Inc., "SIMULINK, Dynamic System Simulation Software," Liu, K.H. and Lee, F.C., "Resonant Switches - Topologies and Characteristics," IEEE Power Electronics Specialist Conference (PESC) Record, 1985, pp Baha, B., "Analysis of quasi-resonant converters using the state-plane method,"international Journal on Circuit Theory and Applications, Vol.21, no.6, November- December, pp The Math Works Inc., "SIMULINK 1.3 Release Notes," Baha, B., "The small-signal modelling and design of trough current control of QRCs," Internal report, University of Brighton, January 1996.