Power electronic systems are electrical or electro-mechanical systems that contain one or more power semi-conductor switches which are switched between two states. The switches are switched between a conducting state (ON state) during which the device can conduct current with small voltage drop across it and a non-conducting state (OFF state) during which the device can block large voltage with negligible current flow through it. The device will go through a transient state during which there will be significant current flow through it with significant voltage across it whenever it is switched between the two states. Thus, the device will dissipate power during ON state depending on the voltage drop across it and current flow through it and it will dissipate a certain amount of energy every time it is switched between the two states. Thus the total power losses in the device will depend on the ON state v-i characteristics of the device, the exact nature of switching process and the number of times the device is switched between the two states in every second - i.e, the switching frequency.

The switches in a Power electronic system may be switched at a fixed switching frequency as in the case of a Sinusoidal PWM Inverter or a Voltage-mode controlled PWM Buck converter. Or, they may be switched at a variable switching frequency as in the case of a current-regulated inverter employing hysteresis current control.

The switches may be externally controlled ones or internally controlled ones. A MOSFET is an externally controlled switch - it can be switched on by applying a positive gate-source voltage of suitable magnitude and it can be switched off by applying zero or negative gate-source voltage. A diode is an internally controlled switch. It goes ON when the voltage across it reaches cut-in value as the result of variation of some input in the system or because of switching performed on some other externally-controlled switch. It goes OFF when the current flowing through it becomes zero and tries to become negative as a result of input variation or switching of other devices.

The semi-conductor device used to implement a switch in a Power electronic system will have a static input-output-control description that is usually a non-linear relationship. In addition, the devices contain dynamic elements like various junction capacitances and lead inductances that play their role whenever the voltage/current in the device is time-varying. Thus, a Power electronic system containing switches is essentially a non-linear circuit.

However, one can often assume that the switch static v-i characteristic is reasonably linear once the switch is fully in the ON state or OFF state and that the variations of dynamic elements in the equivalent circuit of the switch are small once the switch is fully in the ON state or OFF state. For instance, the transition capacitance of a reverse biased diode is a non-linear function of the reverse voltage across it. However, it may be treated as a constant if the reverse voltage appearing across a diode during its OFF state in a Power electronic system is reasonably constant - an example would be a SPWM inverter running from a battery source. Thus, the Power electronic system changes from one linear circuit structure to another linear circuit structure whenever any switch in that Power electronic system changes its state from ON to OFF or OFF to ON. However, the Power electronic system changes from one linear circuit structure to another linear circuit structure only by assuming a thoroughly non-linear behaviour during the time interval taken to accomplish the switching.

Power electronic systems can be simulated by circuit level simulators like PSPICE, SABER etc. They may also be simulated by a system level simulator like MATLAB/SIMULINK. The objective of simulation rules the choice of simulation tool. Quite often, a mix of various simulation tools may be needed during design evaluation stage of a complex Power electronic system. For instance, a designer may use extensive PSPICE simulation for assessing the transient voltage and current waveforms across various switching devices in the proposed Power electronic system and for estimating the power losses (conduction loss and switching loss) in various devices. She will use detailed device models for the actual devices that she plans to use in the system (usually provided by the manufacturer of the device) for this simulation. She will also carry out reasonably detailed design of magnetic components that she needs in the Power electronic system so that she can arrive at the values of series resistance and shunt resistance of the inductors to be used in simulation. She will estimate the ESR of capacitors for inclusion in simulation. Further, she has to have a reasonably detailed idea of how the various components in the proposed system are going to be laid out in the final hardware so that she can estimate the values of wiring inducatnces at various points in the system to be included in simulation. The voltage transients across switching devices is critically dependent on parasitic inductances in the system and snubber design can not be attempted at all without a reasonably accurate estimate of parasitic inductance values.

The same Power electronic system will subsequently be simulated in MATLAB/SIMULINK for designing the control loops. At that stage the designer may use very simple models for switches ignoring all the details of device switching. She may even replace the switched Power electronic system by its state space averaged model or by some other averaged model for designing the control loops with the help of SIMULINK.

It is not that details of switching waveforms can not be obtained from SIMULINK. It is not that control loops can not be simulated in PSPICE. In fact, an expert can simulate everything about a Power electronic system in PSPICE. She can do that in MATLAB/SIMULINK too. For that matter, she may do that better with C++ programming language. The choice simulation tool for simulating a Power electronic system is secondary to a thorough understanding of principles of operation of the Power electronic system and clarity as to the objectives of simulation.

Four broad categories of objectives can be thought of in this context.

Fig. 1 below shows a DC-DC Buck Converter with two stray wiring inductances, series resistance of buck inductor, series resistance of capacitor and device capacitances identified. The load is a resistor. The device capacitances and wiring inductances have profound influence on the switching behaviour of the converter. The parasitic resistances in inductor and capacitor affect efficiency of the converter and the damping during transient conditions. However, all these effects may be ignored in a first level model of the converter to be used to explain the principle of operation of Buck converter to a learner.

Fig.1 A DC-DC Buck Converter with stray inductances and device capacitances shown.

The circuit to be used to explain the basic principle of Buck Converter is shown in Fig.2. The switch in this circuit is ideal and the diode in the circuit is ideal.

Fig. 2 Ideal Buck Converter

The waveforms of various voltages and currents in this circuit can be derived analytically and the relation between the steady-state average output voltage and the input voltage may be derived easily by applying circuit theory relations. A simulation tool is quite unnecessary at this level even for instructional purposes. In fact, using simulation on this circuit in order to understand how this circuit works will be wrong pedagogy.

The switch current in this circuit will be piece-wise linear. The switched voltage across D will be rectangular. The inductor current and capacitor current will be piece-wise linear. Output voltage will contain a DC component and ripple component.

The second level model of Buck Converter for instructional purposes will include the series resistance of inductor and series resistance of capacitor. The switch will be modeled by a small resistor in ON condition and diode will be modeled by a battery of value equal to cut-in voltage in series with a small resistance representing the forward resistance of the diode. Now the inductor current waveform and switch current waveform will contain exponential waveform segments. The simple relation between input voltage and output voltage under continuous conduction mode (V_o = dV_in) will be affected by the resistor values. A simulation study with various values of these second order components will help the learner to know Buck Converter more closely. She can gain insight into the effect of ON-resistance of S, forward resistance of diode, parasitic resistances etc., on the regulation of output at various duty-ratio values. Simulation study can be used to prepare Vo vs V_in curves with duty ratio d and load resistance R_load as the parameters. Such curves will help her to estimate the range of duty-ratio variation needed to maintain a fixed output against a varying input and varying load. The range of duty-ratio variation required will be a direct indication of gain in the system.

The simulation of second level model can be done in PSPICE (or in any SPICE based circuit simulator) easily. It can be done in MATLAB by using MATLAB commands alone. It can be done in SIMULINK by modeling the differential equations of the system by employing standard SIMULINK blocks. And it can be done in SIMULINK along with Power System Block Set (PSB).

In the case of SPICE-based simulation, the schematics editor of simulation program will be used to draw the circuit. The Simulation Software in the simulator prepares the network equations of the circuit using Nodal Analysis or Modified Nodal Analysis. These equations will be non-linear differential equations in general. The Equation Solver in Simulation Software solves these equations numerically using a variable time-step integration algorithm. The post-processing module of the simulator allows visualisation and analysis of simulation results.

Simulation in MATLAB command window can be done by deriving the differential equations governing the circuit and writing MATLAB code to solve the equations numerically. The numerical results can be post-processed using MATLAB graphics and analysis functions. Obviously, simulating a circuit in MATLAB requires equation preparation and code preparation.

Simulation using SIMULINK (without PSB) requires preparation of differential equations of the circuit and preparation of a SIMULINK diagram using standard SIMULINK blocks like Summer, Product, Gain, Integrator etc. The SIMULINK diagram will represent the equations of the circuit straightaway. The diagram will in no way resemble the original circuit diagram. In fact, MATLAB/SIMULINK is a system level simulator and does not know that it is simulating a circuit in this case. SIMULINK does not know volts and amps. There are no units associated with variables in SIMULINK. It converts a simulation diagram into state-space description and solves the state-space equations by using one of the many numerical algorithms available for solving a set of non-linear coupled first-order ordinary differential equations. Many choices of fixed time-step or variable time-step Solvers are available in SIMULINK.

Power System Block set in SIMULINK allows the user to draw a schematics of the circuit in SIMULINK window. The task of preparing the network differential equations and setting up the state-space description is now done by the simulator. The user can draw the circuit in SIMULINK much the same way she does it in a SPICE-based circuit simulator. SIMULINK along with PSB code will parse the circuit diagram and prepare state-space description prior to simulation. PSB models the non-linear elements (the switches in a Power electronic system) in a special way. The linear portion of the circuit is translated into a linear time-invariant state-space block. Currents are inputs to this state-space description and voltages are the outputs. The switches are modeled as controlled current sources feeding their output currents as inputs to the state-space description of linear portion of the circuit. The output currents of switch-models are controlled by voltage outputs from the state-space block. Thus, PSB models switches (and other non-linear elements) as feedback elements across a LTI system described by state-space equations (with current source input and voltage output), feeding back a value of current decided by the voltages in the circuit. Due to this reason PSB does not allow the user to (i) put two switches in series unless RC snubber is employed across at least one of them (ii) put a switch and inductance in series unless the switch has a RC snubber connected across it.

Second level model of a Power electronic system as explained here can be simulated for instructional purposes in MATLAB/SIMULINK in any of the three ways suggested above. PSB makes it easy by reducing the preparation work and by retaining the circuit structure intact in the simulation diagram. However, it does not render any more information than the MATLAB simulation or SIMULINK (without PSB) simulation since the switches are represented by ideal+ON resistance model in all the three cases.

The third level model of Buck Converter will aim at exposing the details of switching transients in S and D. The wiring inductances, device capacitances etc., have to be included in this model. The aim of the simulation will be to obtain the detailed waveforms of MOSFET voltage and current and Diode voltage and current. A study of these waveforms under different loading and input voltage conditions and under abnormal conditions is needed to estimate the voltage and current stresses in the devices, switching loss in the devices, conduction loss in the devices, EMI level of the system and snubber requirements. The switching devices can no longer be modeled as ideal + resistance model.

Detailed device models for PSPICE employing macro modeling which involves preparation of a subcircuit (generally non-linear) using standard linear and non-linear PSPICE components to emulate the behaviour of the device under steady and dynamic conditions is available from various manufacturers. And many such models come along with PSPICE simulation packages. These device models approximate the actual behaviour of device to a close degree even under switching conditions.

Thus, detailed analysis of switching transients and device loss calculations can be carried out in SPICE-based circuit simulators. However, there are no such manufacturer-supplied device models in PSB in SIMULINK. Thus PSPICE will know that IRF840 is a plastic-package 8A, 500V n-channel Power MOSFET from IR and will have its model in the library. But SIMULINK/PSB does not know IRF840. It knows only some generic MOSFET which is modeled as a small resistance (which we have to specify) in series with a inductance (which we have to specify and which can not be set to 0) when its gate is given any signal >0. And when the gate is given 0 it is a simple open circuit. A diode is also modeled similarly in PSB. Thus, a diode from PSB does not conduct in reverse direction at all. But a real diode in a Power electronic system conducts heavily in the reverse direction under reverse recovery transient. In fact reverse recovery transient (of diodes and thyristors) is the most important transient in a Power electronic system which results in over-voltages across devices, EMI and extra switching losses. A Power electronic system modeled in SIMULINK/PSB will be unaware of reverse recovery of diodes.

The power devices in PSB library are essentially instantaneously switching devices. This is why PSB insists on a non-zero inductance in series with the device in the model. The device switches on instantaneously as soon as the gate is >0, but the device current is limited in its rate of change by the inductance in series with it. This kind of a model may give the shape of switching waveforms correctly provided we know how much of inductance to be assigned to the device. This inductance is to be calculated from the actual switching speed of the device. But that is not known usually. Essentially, the simulation software is asking for a piece of data that is at a level even higher than what information the software can produce after simulation!

However, devices can be switched so fast that the approximation of instantaneous switching is not inappropriate if there is RC snubber or RCD snubber connected across the device. Thus, PSB modeling of Power electronic system can give satisfactory waveform results (at least for instructional purposes) provided RC snubbers are put across all devices. This is why PSB switch models treat RC snubbers as integral part of the switch. The RC snubber parameters appear in parameter dialog box of the switch. However, better simulation result will be obtained if (i) make R value of RC snubber in the switch model = 0 and C value = device capacitance of the actual device (ii) make the inductance in series with switch = the package inductance of actual device (iii) connect extra inductance in series with switch representing wiring inductance (iv) connect a RC snubber across the switch using Series RLC Branch of PSB and use R and C such that switching transients come under control.

The fourth level model of Buck Converter is for studying its control characteristics and for designing and evaluating a closed loop control system for it. Design of closed loop control requires the knowledge of open loop gain and phase characteristics. There are two ways to obtain these.

The relevant characteristics in the case of Buck Converter is the relation between average value of output voltage and the duty ratio signal for the switch. In the first method, we add a sine wave source of fixed amplitude to the operating point duty ratio value and simulate the circuit till the circuit reaches a periodic steady-state. Once steady-state is achieved we obtain the sinusoidal component in the output voltage by Fourier analysis. Repeat simulations with different values of frequency for the added sinusoidal duty ratio and collect data. Plot the frequency response data off-line and thereby obtain the bode plot of open loop gain. All kinds of losses in the circuit go towards damping the natural modes of the circuit and towards reducing loop gain. Hence, all loss generating components should be included in the circuit. Therefore, switching losses should be included and that implies that PSPICE simulation is needed. However, switching losses may be ignored in a conservative control design and simulation using SIMULINK or SIMULINK/PSB will be satisfactory for the purpose of loop gain evaluation around an operating point.

However, the basic issue in simulation of Power electronic systems emerges at this point. A Power electronic system is a system that usually undergoes periodic switching at high frequency. For instance, let us assume that the Buck Converter under consideration is being switched at 100kHz. The natural frequency of the second order filter circuit formed by the buck inductor and output capacitor will usually be around 1/20 of switching frequency or less. The damping factor associated with the poles contributed by the filter will usually be small. Thus it takes 20 to 40 cycles of about 5kHz oscillations for the initial transient to die down to near zero levels. There are 800 device switchings in 40 cycles of 5kHz. The frequency of sinusoidal duty ratio disturbance will vary from near zero to about 10kHz in this case. Assume that this frequency is 10Hz. Thus at least (800+10000) switching cycles will have to be simulated to get at least one steady-state sinusoidal cycle at output. Thus, the problem is that a huge number of switchings will have to be simulated in order to obtain control gain characteristics in a Power electronic system in this cycle-by-cycle simulation strategy.

State variables (or circuit variables) undergo rapid changes during switching of a device. PSPICE and SIMULINK (with variable time-step Solver) will detect this and reduce the time-step in order to improve the accuracy of simulation. The result is that simulation becomes extremely slow and may suffer from convergence issues. Mixing devices with widely different voltage and current levels will accentuate this problem. For instance, let the switches and power circuit of a Power electronic system be working at 100's of volts and 100's of amps and let that system contain low power electronics that carry mAs of current. Simulation speed suffers heavily - especially in PSPICE. Similarly, if the same Power electronic system involves widely different time constants in various parts the simulation speed suffers heavily in SIMULINK.

We may want to simulate the performance of the closed loop control system that we design based on open loop gain simulations later. There too the same problem emerges. The closed loop control bandwidth is usually much lower than the switching frequency. This implies that a large number of switchings will have to be simulated to let the control loop reach its steady-state.

Thus, the first method - cycle-by-cycle simulation - is not suited for the purpose of control system studies on Power electronic systems.

After all, the response we wanted was the average value of output voltage averaged over a cycle of switching when the duty ratio is varying sinusoidally. Why don't we develop an Averaged Model for Buck Converter which suppresses the switching frequency domain behaviour and describes the behaviour of average value of various voltages and currents? Such a model will not suffer from the difficulty explained above. At the best it will be a linear time-invariant continuous model and at the worst it will be non-linear continuous model. It can be simulated in MATLAB or MATLAB/SIMULINK or SIMULINK/PSB in either case. If it is a linear model, then, the full power of MATLAB/SIMULINK will be availbale for control design. If it is a non-linear model, it may be linearised around an operating point and linear control system design tools like LTI Viewer and SISO Tool in MATLAB can be used on the linearised model.

State Space Averaging is a technique to derive this kind of an averaged model for Power electronic systems. The technique will be described later in the examples section. This technique usually leads to a non-linear time-invariant continuous system that is described by a state-space equation of the standard form. It is also possible to arrive at an equivalent circuit schematic that will have this state-space description in the case of many Power electronic systems. In that case, the model can be drawn in a SIMULINK diagram using PSB. If only state-space description is available, it may be implemented in SIMULINK by equation simulation or by S-Functions.

Testing out a proposed new concept in the design of power stage of a Power electronic system under various normal and abnormal operating conditions can be the major objective of simulation. An example would be a new modulation scheme for a multi-level inverter. The new proposed modulation scheme has to be tested out for its correctness under all circumstances. Further, it has to be evaluated and compared with other existing modulation schemes for similar inverters on its harmonic performance, DC voltage utilisation, switching frequency etc. SIMULINK simulation using PSB can be the best tool for this kind of work. The devices may be modeled by ideal switch model for this objective.

A proposed control strategy for a Power electronic system can be evaluated and compared with other know control strategies by SIMULINK simulation with or without PSB. The Power electronic system will be represented by an averaged model for this purpose. In fact, SIMULINK comes to its full power when used in this kind of problems. It is somewhat difficult to simulate control strategies in PSPICE. Circuit level simulation is natural to SPICE-based simulators. System level simulation is what MATLAB/SIMULINK best at.

However, SIMULINK/PSB will not help us to evaluate a proposed Power electronic system topology which involves getting into the details of switching. For instance, consider a new active clamp circuit aimed at clamping the over-voltage appearing across the switch in a fly-back DC-DC converter and returning the energy trapped in leakage inductance of fly-back transformer to the supply. Obviously, the switch models in PSB simply will not do for this purpose. This problem can be simulated only in SPICE-based circuit simulators.

In general, if a new proposed concept or control strategy in a Power electronic system involves performance claims only at the level of averaged variables, it can be validated by SIMULINK/PSB simulation. If the proposed concept involves claims at device level (lower switching loss, lower device stress, higher efficiency, energy recovery from parasitic elements, improvements in switching locus, reduction in size of snubber elements, reduction in snubber loss etc.), it requires a SPICE-based simulator with detailed device models to test out the claims before a hardware evaluation.

SIMULINK/PSB can not be used to correctly estimate the switching transients and losses in devices. However, this does not mean that SIMULINK can not provide any help in power circuit design.

The Power electronic system can be simulated in SIMULINK/PSB using ideal switch models to obtain the waveforms of current that will flow through a device and the voltage that will appear across it under a steady-state switching cycle at a given operating condition. These waveforms may be post-processed to obtain average value, rms value, peak value, harmonic content etc. These values are needed to fix the rating of components.But the product of voltage that appears across a device and the current through a device does not give the power loss in the device correctly in this simulation. This is so due to an idealised model that has nothing much to do with the actual device. The current flows and voltage variables across various elements are not affected much by this model. This is so since the devices are not expected to drop significant voltages across them when they are in the ON state. However, the power loss in the devices have to be calculated more carefully.

It is possible to prepare a look-up table containing conduction loss and switching energy loss of a given device as a function of voltage and current that is switched and temperature assuming a particular kind of gate drive. This can in turn be done in a SPICE-based simulator by simulating a simple circuit that will subject the chosen device to switching at same voltage and current level and at same frequency. Once this data table is available, MATLAB post-processing code may be prepared to read the look-up table and calculate the energy lost in a device during one switching cycle by reading the conduction loss at various current levels from the look-up table and switching energy at relevant voltage and current from the same table.

SIMULINK excels in this objective and SPICE-based simulators more or less fail.

However, the power converter has to be modeled by some averaging technique. Averaged Models replace the switched Power electronic system by a non-linear continuous time-invariant system which describes the evolution of averaged variables (averaging done over switching cycle) after suppressing the switching ripple. The non-linear continuous model may be linearised around a given operating point by employing the Linearisation Module in Control Design Toolbox in SIMULINK. The linearised model (a state-space description) may then be exported to MATLAB workspace from Control Design Toolbox. The linear model available in workspace may then be viewed in LTI Viewer and closed loop design (i.e., compensator design) can then be carried out in SISO Tool invoked from command window. The designed system may then be evaluated at other operating points by simulation of the non-linear model (after making it a closed loop system) with the designed compensator.

The most difficult part of this is the preparation of the averaged model. Analytical derivation of state-space averaged model is possible in the case of simple Power electronic systems. However, some help from SIMULINK itself may be required to prepare the averaged model before attempting control loop design in the case of complex Power electronic systems.

Continuous conduction mode is assumed. The circuit flips between two structures when the switch is operated periodically with a period of T and duty ratio of d. The two structures are shown in Fig. 3. (a) shows the circuit when S is ON and (b) shows the circuit when S is OFF. V_d is the cut-in voltage of diode. Rs is the ON resistance of switch, R_d is the ON resistance of the diode, R_L is the series resistance of the inductor, R_C is the series resistance of the capacitor and R is the load resistor.

Fig. 3 (a) Buck converter with switch ON (b) Buck converter with switch OFF

The switch and diode are taken to be ideal in this simulation. Hence R_S = R_d = 0 and V_d = 0 are assumed. With these assumptions, the system equations are

(1)

These equations are realised in SIMULINK as below.

Fig. 4 Simulink diagram to implement Eq. 1

However, an important point was missed in this simulation diagram. The simulation diagram assumes continuous conduction mode of operation. But the system can get into discontinuous conduction mode during transient conditions or for operation with higher values of load resistance. Simulation of this system will show that the inductor current takes on negative values during large transient condition. Thus the Buck Converter represented by this simulation diagram is one that uses a switch instead of a diode with that switch kept ON whenever the buck switch is kept OFF. But that is not the system that we wanted to simulate! The next example shows how to correct this mistake. The Buck Converter system in this example had L=20m H , R_L = 0.08W , R_C=0.015W , C = 2.2mF, V_g = 12V, d=0.45, V_{o »} 5V and R=1W .

Fig. 5 Simulink Diagram of Buck Converter Subsystem with discontinuous conduction mode accounted

The relay block switches to zero output when its input goes < 0.01. The d input through summing block will ensure that relay input is > 0.01 throughout the switch ON period. During the switch OFF condition (d input = 0) if the inductor current goes below 0.01 the relay outputs 0 and the integrator input becomes 0. Therefore the integrator output does not change anymore and relay output continues at 0. The next time d goes high, the relay output switches to 1 and integrator is allowed to integrate its input.

Fig. 6 Top level Simulink Diagram for Buck Converter Model using PSB Blocks

The subsystem level diagram of "Buck Converter" block using PSB blocks is shown in Fig. 7. MOSFET from PSB has a series inductance parameter that can not be set at 0. It is set at 100nH to simulate the effect of wiring inductance. MOSFET in PSB comes with a built-in snubber and this is set at 5Ohms in series with 10nF. The ON resistance of MOSFET is set as 0.05Ohms and the ON resistance of integral diode is also set at 0.05Ohms. The diode cut-in voltage is taken as 0.5V. Similar values are used in the case of the Diode element too. Power devices from PSB will have measurement ports that can be enabled from parameter dialog box. These measurements will come out in the form of a vector of voltage and current values converted into Simulink signals. A current measurement units that converts the current signal in the PSB model into unit-less Simulink signal has to be used to access the currents flowing in various paths. Similar Voltage Measurement PSB blocks are needed for converting PSB block signals into Simulink signals.

Fig. 7 Simulink/PSB Simulation diagram for a Buck Converter

The simulation results using PSB model with wiring inductances modeled will show typical overshoot and ringing in the voltage appearing across devices when they switch off.

The time taken by the switches in a Power electronic system to switch from ON to OFF state and OFF to ON state is generally very small compared to the durations for which they remain in ON or OFF condition.(Otherwise the Power electronic system will be very inefficient in power conversion). The switching process is highly non-linear; but it lasts for very small time. Therefore, switchings may be assumed transient-free and instantaneous as a first approximation when a model for control study purposes is attempted. Of course, with instantaneous switchings, there will be no switching loss. Thus the model will miss out some damping. But the error due to this is usually tolerable.

With the assumption of instantaneous switching, the Power electronic system can be thought to flip from one linear structure to another periodically. In the case of Buck Converter under continuous conduction mode the system spends dT seconds as the circuit in (a) in Fig. 2 and spends (1-d)T seconds as the circuit in (b) in Fig. 2. The circuit equations of both these circuits can be cast in LTI state-space form choosing inductor current and capacitor voltage as the state variables.

Let

be the state-space description of first circuit where

and let

be the similar description for second circuit. Then the total change in the state vector over one switching period T may be approximated as
.
Then the average rate of change of X will be
.
This can be written as

where is the vector of averaged state variables with averaging done over a cycle and is the cycle average input vector. ; the output equation for cycle averaged output may similarly be derived. Thus the two equations

will be the state space representation for the dynamics of averaged variables in a Power electronic system that switches between two linear structures.

The results of derivation of the four averaged matrices for a Buck Converter are given below.

These equations can be obtained from the following circuit and hence the following circuit can be simulated in SIMULINK/PSB to implement the state-space averaged model of a Buck Converter.

Fig. 8 State-space Averaged Model of a Buck Converter

Fig. 9 shows the SIMULINK/PSB diagram for implementing this circuit. V_d in the above circuit stands for cut-in voltage of diode and is referred to as V_gamma in Fig. 9.

Fig. 9 State-space Averaged Model of a Buck Converter in SIMULINK/PSB

This model can be linearised around an operating point with d =0.5 using the Control Design Toolbox accessed from SIMULINK.