Feedback controller design for a boost converter ... - IEEE Xplore

www.ietdl.org Published in IET Power Electronics Received on 16th September 2011 Revised on 7th August 2013 Accepted on 17th August 2013 doi: 10.1049/iet-pel.2013.0266

ISSN 1755-4535

Feedback controller design for a boost converter through evolutionary algorithms Kinattingal Sundareswaran, Vadakke Devi, Selvakumar Sankar, Panugothu Srininivasa Rao Nayak, Sankar Peddapati Electrical and Electronics Engineering Department, National Institute of Technology, Tiruchirappalli-620015, Tamil Nadu, India E-mail: [email protected]

Abstract: This study explains a systematic design procedure for the output voltage regulation of a boost-type DC–DC converter employing evolutionary algorithms. The feedback controller design for output voltage regulation is formulated as an optimisation problem and the controller constants are identified via evolutionary search. The design procedure employing genetic algorithm, differential evolution and artificial immune system is lucidly described. Computer simulation results supported by experimental evidence clearly demonstrate that the controllers estimated through evolutionary algorithms are capable of delivering enhanced output voltage regulation under different types of load and supply disturbances.

1

Introduction

The three basic, single switch–diode–inductor, non-isolated, DC–DC converters namely buck, boost and buck–boost converters have been presented and analysed extensively in the literature and texts [1]. The boost converter increases a given input DC voltage and is employed in many applications such as laptop computers, personal communication devices [2], photovoltaic applications [3], power factor correction methods [4], hybrid photovoltaic (PV)/wind power system applications [5], piezoelectric energy harvesting systems [6], vehicular systems [7], fuel cell applications [8] etc. To name specific applications, which require DC bus voltage control include, a DC distribution system or distributed generation modules aggregated by cascade boost converters [9]. In such systems, the DC bus voltage should be maintained at the set value so as to avoid abnormal performance and instability phenomena. In the case of parallel PV string applications, when a string is shaded or malfunctions, the DC voltage derived from this string rapidly drops to a low value and can eventually cause the whole system to shut down. Hence, it becomes obvious that in many cases, a suitable control scheme is required to regulate the DC bus voltage of the boost converter in a manner that ensures stability and convergence to the desired equilibrium. When boost converter is employed in open-loop mode, it exhibits poor voltage regulation and unsatisfactory dynamic response, and hence, this converter is generally provided with closed-loop control for output voltage regulation. The feedback controllers are designed either through conventional methods [10–15] or through non-linear controller design procedures [9, 16–27]. Conventional methods of controller designs make use of averaged IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

transfer function models linearised at a typical operating point. The computation of proportional integral derivative (PID) controller elements for a boost-type DC–DC converter is generally carried out using averaged transfer function model. This model is approximate and valid only for a typical operating point. The other solution is to go for non-linear control methods. However, implementation of non-linear controller is complex and costly. The non-linear methods reported in the literature are predictive control [16], sliding mode control [17], adaptive control [18], passivity-based control [19], H ∞ control [20], robust control [21–23], fuzzy control [24–26] and gain schedule [27]. Optimisation methods were also employed for feedback controller design [28–31]. This paper suggests an optimisation model for feedback controller design of a boost-type DC–DC converter with the objective of enhancing transient behaviour of output voltage profile. The exact state-space model of boost converter is employed for computing output voltage dynamics, whereas all earlier published works made use of averaged transfer model. Thus the attributes of exact model of boost converter together with evolutionary algorithms lead to an optimal PID controller, which is realised using cheaper analogue components. The methods include genetic algorithm (GA) [32–37], differential evolution (DE) [38– 40] and artificial immune systems (AISs) [41]. Towards this goal, the controller design is redrafted as an optimisation task and each of the method is used to tune the PID constants for optimum output voltage dynamics at a typical operating point with a step change in reference voltage. The novelty of paper lies in integrating exact ON- and OFF-state model of boost converter with evolutionary algorithms. This process is new and bilinear property of boost converter is absorbed in the process; further, we employ elitism in GA, 903

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www.ietdl.org which guarantees faster convergence. A suitable objective function is tailored and constraints on controller parameters are set using transfer function model of the converter. The procedure for controller design employing each method is lucidly explained and the results are compared qualitatively and quantitatively. It is shown that the attributes of biologically inspired algorithms together with the exact model of the boost converter yield excellent dynamic response at different operating points of the DC–DC converter.

During the OFF-state of the power electronic switch, the state-space equation is given below ⎡

−rL + R//rc ⎢ x˙ 1 L =⎢ ⎣ R x˙ 2 C(R + rc )

⎤ ⎡ ⎤ 1 −R ⎥ ⎢ ⎥ x L L(R + rc ) ⎥ 1 ⎥Vin +⎢ ⎦ ⎣ ⎦ −1 x2 0 C(R + rc ) (3)

The output voltage in this mode is given by

2

Problem formulation

The experimental setup of the closed-loop boost converter system employed in this work is shown in Fig. 1. In the feedback control system, a fraction, β of actual output voltage vo, and its reference value bv∗0 are first compared using comparator and the error, e(t) so obtained is processed by the PID controller. The output voltage of the PID controller, u(t) is an analogue signal and must be converted into gating pulse to metal oxide semiconductor field effect transistor (MOSFET) switch with adjustable duty cycle. This task is performed by the modulator which compares the PID controller output voltage with a ramp signal, so that the output of the modulator is a gating pulse with its duty cycle varying in accordance with PID controller output voltage. Let x1 = iL and x2 = vc be the state variables where iL is the inductor current and vc is the capacitor voltage. During the ON-state of the MOSFET, the state-space model of the boost converter [10, 11] is given below

x˙ 1 x˙ 2

⎡ −r

L

0

⎢ L =⎣ 0

⎤

⎥ x1 −1 ⎦ x 2 C(R + rc )

⎡1⎤ ⎢ ⎥ + ⎣ L ⎦Vin

(1)

0

The output voltage in this mode is given by the following equation v0 (t) = 0

R (R + rc )

x1 x2

(2)

v0 (t) = rc //R

R (R + rc )

x1 x2

(4)

When the boost converter works in a closed-loop mode with a PID controller, let the reference voltage change by a small value, DVo∗ such that the error signal, e(t) is

e(t) = b DVo∗ − Dv0 (t)

(5)

This signal is processed by the PID controller and the output of the PID controller is de(t) u(t) = Kp e(t) + Ki e(t) dt + Kd dt

(6)

In the above, Kp, Ki and Kd are controller constants and realised using R1, R2, C1 and C2. Equations (2) and (4) describe the output voltage of boost converter in the ON-state and OFF-state of the MOSFET switch. As indicated in (5) and (6), the transient variation of output voltage, v0(t) depends on the values of Kp, Ki and Kd such that best dynamic response is achieved at a typical operating point with optimum values of these constants. Consider that a step change of small amplitude DVo∗ is fed to the closed-loop system of boost converter shown in Fig. 1. Let Δv0(t) be the time variation of output voltage for this step input. The goal here is to identify the suitable values of Kp, Ki and Kd such that the sum-squared error between DVo∗ and Dv0 (t) is minimised to the least value to obtain excellent dynamic response. This is formulated as an optimisation problem and is given below minimise F(f) =

ts

(e(t))2

(7)

t=0

subject to,

f(lower) ≤ f ≤ f(upper)

Here, φ = {Kp, Ki and Kd} is the controller structure, ts is the time to settle the output voltage transients and φ(lower) and φ(upper) are lower and upper bounds of controller constants and are obtained using transfer function model.

3 Application of evolutionary algorithms towards controller design

Fig. 1 Experimental circuit of boost-type DC–DC converter in closed-loop mode 904 & The Institution of Engineering and Technology 2014

This paper employs GA, DE and AISs as tools for controller design. The steps of solution to the problem at hand using these optimisation methods are described in the following sections. IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

www.ietdl.org 3.1

GA-based controller design

Step1: Create a population of initial solution of parameters φ = {Kp, Ki and Kd} This step primarily requires the population size. Each variable in the problem is called as a gene and in the present problem, there are three genes. A chromosome consists of the genes and thus each chromosome represents a solution to the problem. The population consists of a set of chromosomes. It is well articulated in literature that a population size of 10–30 is an ideal one and hence population size is selected as 10 in this work. Step 2: Evaluation of fitness function The degree of ‘goodness’ of a solution is qualified by assigning a value to it. This is done by defining a proper fitness function to the problem. The fitness function is given below fitness function =

1 1 + F(f)

(8)

For each chromosome structure, (i.e. Kp, Ki and Kd), the closed-loop output voltage dynamics is now computed using (1)–(6) and corresponding fitness function is evaluated. Thus, each chromosome is made to associate with fitness function value. Step 3: Generation of offspring Offspring is a new chromosome obtained through the steps of selection, crossover and mutation. After fitness of each chromosome is computed, parent solutions are selected for reproduction. It emulates the survival of the fittest mechanism in nature. The Roulette wheel selection is the most common and easy-to-implement selection mechanism. A virtual wheel is implemented for this selection process. Each chromosome is assigned a sector in this virtual wheel and the area of the sector is proportional to their fitness value. Thus, the chromosome with largest fitness value will occupy largest area, whereas the chromosome with a lower value takes the slot of a smaller sector. Let there be five chromosomes labelled as A, B, C, D and E and their fitness value increases in the order of D, B, A, E and C. Then, Fig. 2a shows a typical allocation of five sectors of chromosomes in the Roulette wheel. In Roulette wheel selection, an angle is generated randomly and the chromosome corresponding to this angle

is selected. Fig. 2b shows a randomly generated angle of 4π/3 rad. In this case, chromosome C is selected. The chromosomes thus selected are called parent population and are subjected to undergo crossover and mutation to produce offspring for the next generation. Conventional method adopted in GA is Roulette wheel selection and in this work, this selection method is modified by combining it with elitism. Using elitism, a definite number of best solutions are retained and are re-used in the next generation without undergoing the steps of mutation and crossover. Following the selection of parent population, crossover and mutation are performed to generate offspring population. In crossover, randomly selected sub-sections of two individual chromosomes are swapped to produce the offspring. In this work, multipoint crossover is adopted for increased efficiency since three variables are embedded in one chromosome. Mutation is another genetic operation by which a bit within a chromosome may toggle to the opposite binary. Fig. 2 illustrates crossover and mutation. The crossover and mutation are performed based on the probability of crossover and mutation. Step 4: Replace the current population with the new population. Step 5: Terminate the programme if termination reached; else go to step 2. 3.2

Differential evolution-based controller design

The following section describes the development of DE algorithm for PID controller design. Step 1: Create a population of initial solution of parameters (Kp, Ki and Kd) The first step involves generating an initial population of size N. If g denotes generation number, the parent population, Xg can be written as X g = x1g , x2g

···

xig

···

xNg

(9)

where i = 1, 2, 3 … N. This is shown in Fig. 3a. Here, x1g x2g · · · xig · · · xN g are called vectors and each vector represents a possible solution to the problem at hand. Then in the present case, there are three

Fig. 2 Steps of GA a Typical allocation of sectors for chromosomes b Roulette wheel selection c Crossover d Mutation IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

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Fig. 3 Steps of DE a Initial population b Mutation c Crossover d Selection

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IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

www.ietdl.org parameters namely Kp , Ki

and

Kd

xig = Kpig ,

and

Kiig ,

Kdig (10)

This is represented in Fig. 3a. Step 2: Generate mutant population This step generates another population named as mutant population which also has the same population size ‘N’. Mutant population is generated from the parent population based on a probability, PDE. For a fixed value of PDE (0 ≤ PDE ≤ 1), a random number Pi is generated and a mutant population is guaranteed by the following way if

Pi , PDE

Mig = xr1g + Q∗ xr2g − xr3g

else

Mig = xr1g + K ∗ xr2g + xr3g − 2∗ xr1g

Mij, g , Xij, g ,

if CDEi ≤ CDEr otherwise

or

Xi, g+1 =

Tig , if F(Tig ) ≤ X (Xig ) Xij, g , otherwise

(14)

Step 5: Stop the computation if termination criterion is reached; else go to step 2.

(11) 3.3

(12)

where K = 0.5 × (Q + 1) and Q is known as scale factor. Here, the indices r1, r2 and r3 are so chosen that for a target vector xig, i ≠ r1 ≠ r2 ≠ r3. This is shown in Fig. 3b. Step 3: Crossover operation In this step, new population labelled as trial population Tg is generated by suitably combining parent population and mutant population. This process is based on crossover probability CDE (0 ≤ CDE ≤ 1). For each component in each vector of Xig and Mig, a random Pci is generated and crossover operation is carried out as per the following equation. Tij, g =

Step 4: Selection In this step, each vector, (i.e. Kp, Ki and Kd) in both initial population and trial population is substituted in (1)–(6) and the corresponding objective function is computed. For ith vector in the initial population and trial population in the generation, let F(Tig) and F(Xig) be the objective function value. A new population is now formed for the next generation as given below

j = jrand

(13)

In the above i = 1, 2, … N, j = 1, 2, … n and index jrand is a randomly chosen integer within the range [1, n], where n is the dimension of the variable and is 3 in the present work. This process is shown in Fig. 3c. The use of jrand ensures that trial vector Tig is different from both Xig and Mig at least by one parameter.

AIS-based design

Various steps in AIS-based scheme are given below: Step 1: Antigen identification In AIS, antigen is identified as the objective function to be minimised. Step2: Create a population of initial antibodies In this algorithm, possible solutions are represented by binary-valued strings of finite length. The string consists of three sectors which represents Kp, Ki and Kd parameters of the PID controller, and in this work each one spans for 8 bits so that one antigen is 24 bits long. This is illustrated in Fig. 4a. The strings are accepted as antibodies of the artificial immune system. A collection of such antibodies forms a population of prospective solution to the given problem. The algorithm goes to solution by the evolution of these antibodies. Step 3: Evaluation of objective function In this step, each antibody (i.e. Kp, Ki and Kd) is substituted in (1)–(6) and the respective objective function F(φ) of each antibody is computed. Step 4: Affinity assessment This step involves determination of the affinity of each antibody towards the antigen or in other words the capability of the antibody to destroy the antigen. It is desirable to have antibodies with maximum affinity towards the antigen. The present case is a minimisation task and hence the affinity is defined as follows

Fig. 4 AIS-based scheme a Antibody structure b Mutation principle IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

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www.ietdl.org affinity,

A=

1 1 + F(f)

(15)

After affinity measurement, the antibodies are arranged in decreasing order of their affinity value. Step 5: Proliferation and affinity maturation The first best 50% of the antibodies are selected for proliferation. Proliferation is the process of making multiple copies of binary structure of each antibody. The number of clones for each antibody is given by the following expression n (g × N b ) Nc = ni i=1

(16)

where Nc is the total number of clones, Nb is the number of antibodies, ni is the ith antibody in the re-arranged population of antibodies and γ is the multiplying factor. Step 6: Somatic hypermutation Somatic hypermutation is the process of flipping bits in the binary structure of each cloned antibody. The rate of mutation for each antibody is given by the following expression

s = e−(d×A)

Table 1 Parameters of GA population size crossover probability, CGA mutation probability, PGA ending criterion

10 0.7 0.05 100 iterations

Table 2 Parameters of DE population size scale factor, Q crossover probability, CDE mutation probability, PDE ending criterion

10 0.8 0.5 0.5 100 iterations

Table 3 Parameters of AIS population size percentage of population selected for cloning multiplication factor, γ decay control factor, δ ending criterion

10 50 1 10 000 100 iterations

(17)

where σ is the rate of hypermutation and δ is the control factor of decay. Based on these rules and axioms, the algorithm generates the hypermutated antibodies. Fig. 4b represents the initial cloned antibody and the maturated antibody produced after hypermutation. Step 7: Reselection

The newly created mutated clones are exposed to the antigen and their affinity is determined. The combined population of parent antibody and cloned antibody is now arranged as a column in the decreasing order of affinity. The best top 50% of this column together with randomly generated 50% antibodies constitute the new population. This step ensures diversification in the antibody population and helps in exploring greater regions for global minima.

Fig. 5 Convergence graphs a Convergence graph with modifications in GA b Convergence graphs of all methods 908 & The Institution of Engineering and Technology 2014


www.ietdl.org Table 4 Controller constants and time-domain specifications Method

GA DE AIS

Kp

Ki

Kd

Rise time, ms

Settling time, ms

Steady-state error, V

Peak overshoot, V

Averaged objective function F(φ)

3.7835 1.3722 2.9382

33.0293 23.6918 49.0615

0.04651 0.0532 0.04163

24 30 19

32 50 25

0 0 0

6 0 0

81.21 110.7 72.52

Step 8: Replace the current population with the new population. Step 9: Terminate the program if termination criterion is reached; else go to step 2.

4

Results and discussion

A dedicated MATLAB program is developed for each method for the controller design. The values of parameters employed in each algorithm is given in Tables 1–3 and are obtained by the general guidelines available in the literature. The evolutionary algorithms are now sufficiently matured such that these algorithms are no more complex and can be easily programmed [42–45]. The range of parameters involved in these algorithms is well known now. For example, the mutation constant can be assumed to be any value between 0.1 and 0.01 which guarantees confirmed convergence. Similar logic applies to all parameters in all the evolutionary algorithms. While DE and AIS methods are used as such available in the literature, certain modifications were carried out in conventional GA. The modifications are the use of elitism and multipoint crossover. Fig. 5a shows variation of objective function with and without these modifications. The convergence graphs in Fig. 5a clearly illustrate that use of elitism and multipoint crossover leads to faster convergence and convergence to lower value of objective function value. Since numerical value of objective function is a direct index of quality of solution, it can be seen that elitism and multipoint crossover improve faster convergence and quality of solution. A total of 20 trial runs are conducted with GA, AIS and DE methods and the population size and initial population in each trial run are kept common for fairness of comparison. The averaged curve of 20 trial runs of each method is plotted in Fig. 5b. The convergence graphs in this figure clearly indicate that all the optimisation algorithms considered in this work are successful in obtaining near-global optima, since the objective function values are low and are nearer to each other. The controller constants obtained with each method is tabulated in Table 4. A closer examination of controller constants obtained through the optimisation methods reveals that the values of the same controller type varies largely; this shows that solution to a feedback controller design for a boost converter possesses multiple sets. In order to verify the stability of the entire system with gains selected through evolutionary algorithms, the Bode plots are now obtained and are shown in Fig. 6. These plots show that the system is stable with newly obtained controller gains. In order to evaluate the controller performance, simulation and experiments are carried out on a prototype boost converters. The specifications of the boost converter are given in Appendix. The boost converter dynamics with a step input are now simulated at a typical operating point with each PID controller and the output voltage transients IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

Fig. 6 Bode plot of closed-loop boost converter obtained through a GA b DE c AIS

are shown in Fig. 7. In Fig. 7a, the output voltage dynamics are shown when reference value, v∗o changes from 16 to 24 V. It is seen that the feedback controllers designed through GA, AIS and DE offer excellent output voltage dynamics. Although the feedback controller is designed for optimum dynamic response, their capabilities in optimising the output voltage dynamics during two types of disturbances are now studied. The two disturbances are a step change in source voltage from 13 to 15 V and then 15 to 13 V and a step change in load current from 0.1 to 1 A and vice versa. The transient responses of the boost converter under these perturbations with controller designed through each method are now computed and are included in Fig. 7. From this figure, it can be seen that evolutionary algorithms produce very good dynamic response for different types of disturbances. 909


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Fig. 7 Computed dynamic response of boost converter with a Step change in reference output voltage V0∗ from 16 to 24 V (upper trace: output voltage and lower trace: step input. Scale Y-axis: upper trace: 20 V/div; lower trace: 20 V/div; and X-axis: 200 ms/div) b Change in input voltage Vin from 13 to 15 V with 24 V as reference (upper trace: output voltage and lower trace: variation in source voltage. Scale Y-axis upper trace: 10 V/div; lower trace: 10 V/div; and X-axis: 200 ms/div) c Step change of load current from 0.1 to 1 A with 24 V as reference (upper trace: output voltage and lower trace: change in load current. Scale Y-axis: upper trace: 20 V/div; lower trace: 2 A/div; and X-axis: 200 ms/div)

In order to validate the simulation findings, a prototype boost converter was fabricated in the laboratory and output voltage transients for various disturbances are recorded individually with each optimisation method tuned controller under identical conditions of simulation study. The measured results are shown in Fig. 8. It can be seen that there is a good agreement between computed and measured results, validating in the proposed study. In order to make a quantitative assessment, the time-domain specification of the step response of output voltage dynamics with a step change of reference voltage from 16 to 24 V of the boost converter are now evaluated from measured results and are included in Table 4. The values of time-domain specifications indicate that respective values obtained with evolutionary algorithms vary a little from each other and each controller structure gives near critically damped response. It may be noted that in the 910 & The Institution of Engineering and Technology 2014

experimentation the source voltage is disturbed because of high-frequency switching of converter. Even with such an input voltage, the output voltage is observed to remain at the set value. Owing to limited laboratory facilities, the simulation and experimentation were carried out on a low-power boost-type DC–DC converter. It is interesting to evaluate the application of the proposed optimisation algorithms on a high-power boost converter. Towards this goal, extensive simulation results were carried out on 1.5 kW boost converter taken from [9]. The specifications of this boost converter are given in Table 5. Dedicated computer programs for the above-mentioned boost converter were developed in MATLAB for the estimation of feedback controller gains based on GA, DE and AIS. The programs were executed and each method converged satisfactorily and the controller coefficients obtained through the methods are given in Table 6. IET Power Electron., 2014, Vol. 7, Iss. 4, pp. 903–913 doi: 10.1049/iet-pel.2013.0266

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Fig. 8 Measured dynamic responses of boost converter a Step change in reference output voltage V0∗ from 16 to 24 V (upper trace: output voltage and lower trace: step input. Scale Y-axis: upper trace: 20 V/div; lower trace: 20 V/div; and X-axis: 200 ms/div) b Change in input voltage Vin from 13 to 15 V with 24 V as reference (upper trace: output voltage and lower trace: variation in source voltage. Scale Y-axis upper trace: 10 V/div; lower trace: 10 V/div; and X-axis: 200 ms/div) c Step change of load current from 0.1 to 1 A with 24 V as reference (upper trace: output voltage and lower trace: change in load current for a load change. Scale Y-axis: upper trace: 20 V/div; lower trace: 2 A/div; and X-axis: 200 ms/div)

Table 5 Parameters of boost converter with higher rating boost converter power rating, Pn output voltage, Vo inductance, L capacitance, C switching frequency, fs load resistance, RL

1.5 kW 500 V 4 mH 470 µF 40 kHz 180 Ω

With these parameters, the dynamic response of the boost converter is computed for the same disturbances as explained earlier. The simulation results are presented in Fig. 9 and for brevity, these results are not elaborated. This figure suggests that the proposed algorithm performs successfully in identifying optimal PID constants irrespective of the power level of DC–DC converter. For

better understanding, the time-domain specifications of output voltage dynamics of 1.5 kW boost converter for a steep rise in reference voltage from 400 to 500 V is included in Table 6.

5

Conclusion

This paper has explained the application of a few evolutionary algorithms for the feedback controller design of boost-type DC–DC converter. The steps of each algorithm towards optimal controller estimation are systematically and vividly explained. The computed results together with experimented data are projected and analysed. It is shown that while each optimisation algorithm varies in its origin of biological structure, each method successfully identifies near-optimal solution and improves output voltage response.

Table 6 Controller constants and time-domain specifications of higher rating boost converter Method GA DE AIS

Kp

Ki

Kd

Rise time, ms

Settling time, ms

Steady-state error, V

Peak overshoot, V

9.01 12.16 24.33

23.99 29.06 22.74

0.020 0.010 0.045

21 18 17

28 20 20

0 0 0

0 0 0


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Fig. 9 Computed dynamic response of boost converter with a Step change in reference output voltage V0∗ from 400 to 500 V (upper trace: output voltage and lower trace: Step input. Scale Y-axis: upper trace: 100 V/div; lower trace: 100 V/div; and X-axis: 0.2 s/div) b Change in input voltage Vin from 150 to 160 V with 500 V as reference (upper trace: output voltage and lower trace: variation in source voltage. Scale Y-axis upper trace: 100 V/div; lower trace: 100 V/div; and X-axis: 0.2 s/div) c Step change of load current from 2.8 to 5.6 A with 500 V as reference (upper trace: output voltage and lower trace: change in load current for a load change. Scale Y-axis: upper trace: 100 V/div; lower trace: 2 A/div; and X-axis: 0.2 s/div)

6

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7 Appendix: specifications of boost converter Input DC voltage, Vin = 15 V. Output DC voltage, Vo = 24 V. Output power, W = 24 W. Switching frequency = 40 kHz. MOSFET IRF840. Inductance, L = 2.17 mH. Equivalent resistance of inductance, rL = 0.5 Ω. Capacitance, C = 4700 μF. Equivalent resistance of capacitance, rC = 0.2 Ω. Load resistance, R = 100 Ω. Fraction, β = 0.1.

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