Development of a power electronics converter dynamics toolbox for MATLAB

Providing the required load’s power using Switch Mode Power Supplies (SMPS), leads to smaller and more efficient converters. Correct operation of a SMPS needs some form of control. Applying the well-known controller design techniques like root locus and Bode needs a dynamical model for system under control. Power electronics converters are dynamical variable structure systems. Extraction of small signal dynamical equations by hand is cumbersome, time consuming and error prone. Development a software to automate the small signal model extraction process of power electronics converters is the aim of this paper. Parameter’s change and uncertainty’s effects on overall system performance can be studied easily using this software. Multi graph property of developed software, allow drawing results of different simulations on the same graph. This makes comparison possible. Available commercial softwares cannot calculate the algebraic transfer functions while developed software can do this job. Contact correspondence author to receive the software.


Introduction
*Power electronics is the science and art of control and efficient conversion of electric power using solid state semiconductor devices. Power range can vary from miliwatts up to several megawatts. Switch Mode Power Supply (SMPS) is among the most common technologies in use. SMPS's working principle is based on storing energy temporarily in passive elements like inductors and capacitors and releasing it to output at different voltage level. SMPS are composed of two parts, i.e., power and control circuitry.
Power circuitry, is responsible for providing required and suitable form of power for load. Control circuitry, provides control signals required for power circuitry's switches, i.e. signals which make the switches on and off. When a switch is made on (off), a new current path is added to (removed from) circuit. In fact, SMPS is an example of variable structure systems. Fig. 1 shows a buck converter and its equivalent circuit for switch is closed and opened.
In order to obtain a dynamical model of variable structure systems, averaging techniques which takes average of sub circuit's state space model has been suggested (Middlebrook and Cuk, 1976). Obtaining a dynamical model for plant is the first step toward designing a controller in model base controller design techniques like Bode diagram and root locus. In this paper a MATLAB ® base toolbox is designed to extract converter's dynamical model. Available commercial softwares give only converter's frequency response diagram, while developed software can give either frequency response diagram or the algebraic transfer function. Effect of uncertainties in system parameters can be studied easily with the aid of this toolbox. Developed toolbox can analyze buck, boost, buck-boost, Cuk, SEPIC, fly back, forward and full bridge topologies.

Previous works
Modeling is the process of formulating a mathematical description of the system. Generally speaking, a dynamical system can be described in two forms, i.e., input-output equation, and state space equation.
Input-output equation gives no information about the internal details of system. State space equation, gives more information about internal dynamics of system of system. Various techniques can be found in literature to obtain a linear continuous time invariant (LTI) model of a DC-DC converter. The most well-known methods are: Current injected approach, circuit averaging and state space averaging (Middlebrook and Cuk, 1976;Sokal et al., 1991;Mohan et al., 2007;Vorperian, 1990). Averaging and small signal linearization is key steps of these methods.
State space averaging described in Middlebrook and Cuk (1976) is appropriate to describe converters that work in CCM while is less suitable for converters work in DCM. The current injected method (Sokal et al., 1991;Mohan et al., 2007) can do the job of modeling in either CCM or DCM. Circuit averaging gained a lot of attention recently due to its generality (Hren and Slibar, 2005).
To avoid time and financial losses converter's performance must be ensured before realization. This job can be done with the aid of simulation softwares. Some of the most important softwares are: PSpice ® , NI Multisim ® , PSIM ® , SABER ® and MATLAB ® /Simulink ® . PSpice and NI Multisim are general purpose circuit simulation's programs. Either analogue or digital simulations can be done. Although PSIM can be used to simulate any electronic circuits, it is designed specifically for use in power electronics and motor drive simulations. SABER is a nonlinear dynamic system simulation program. Dynamical equation of power electronics converters is nonlinear in nature. So, after extraction of circuit's dynamical equations SABER can be used to solve these equations. Simulink and power system toolbox of MATLAB provide an environment to analyze power electronics circuits by drawing the schematic (Asadi and Abut, 2016). MATLAB/Simulink can be used to extract small signal Bode plot of converters. It gives no information about system's poles and zero locations.

CCM, DCM, and state space averaging (SSA)
SMPS is nothing more than passive elements, semiconductor switches and a controller. In steady state, converters voltages and currents are periodic. An inductor current that remains positive throughout the switching period is known as continuous current. Conversely, in discontinuous current, inductor current's return to zero during each switching period. Table 1, compares two modes of operation: Where is state vector, i.e. capacitors' voltage and inductors' current, is control input and is output voltage of converter. Applying State Space Averaging (SSA) to Eqs. 1 and 2 leads to: (Erickson and Maksimovic, 2001): where tilde ( ,̃,̃) shows small signal variables and D shows steady state duty ratio. Table A in appendix, shows state space of Eqs. 1 and 2 for different type of DC-DC converters. Equations in Table B and Eqs. 3 and 4 make the working principles of developed software.   Obtaining the dynamical model of converter in presence of these non-idealities is quite cumbersome for pencil-and-paper analysis. Assume a SEPIC converter with the following parameters: Vin = 9V, rinternal = 0.9Ω, L1 = L2 = 90μH, rL1 = rL2 = 10mΩ, C1 = C2 = 80μF, rC1 = rC2 = 15mΩ, VDiode_on = 0.7, rDiode_on = 0.05 Ω, rMOSFET = 40 mΩ, RLoad = 3Ω After entering component's values and selecting the desired transfer function (for example ( ) = 2( ) ( ) ), following results are obtained in Fig. 6. transfer function from control input to output voltage is shown in Fig. 7.  Table 2 shows some of the obtained results for basic converters:

Conclusion
Computers play an important role in designing and analyzing modern power electronics converters. In this paper software has developed to extract small signal dynamical model of famous DC-DC topologies. Developed software can give the algebraic transfer function of converter which makes it different from available commercial softwares. Developed software can be used to design the control loop of converters.