Quadratic Boost Converter for Thermo-Electric Energy Harvesting

Transcription

1 1 Quadratic Boost Converter for Thermo-Electric Energy Harvesting N.G.A.M. Brás, MSc Student, IST, J.F.A Silva, IST Abstract In this thesis, a study is carried out to design a quadratic boost convertor DC-DC and its application when connected with a thermoelectric generator to feed a battery of an electric car. The energy wasted by the car electric motor as heat will be used to recharge the battery. The thesis describes the topology of the thermoelectric generator that will be used as a way to produce energy and the design and sizing of a quadratic boost converter considering losses. The quadratic boost converter is used to get a high conversion ratio between the thermoelectric output voltage generator and the voltage needed to recharge the battery. Two control systems for the quadratic boost converter are also designed. In the first one, the maximum power point tracking algorithm MPPT to optimize the power from the thermoelectric generator is used. In the second system, to avoid overvoltages a dc output voltage controller is used. Simulations in MATLAB/Simulink are shown and discussed to verify the operation of the converter and to evaluate the stability of the system. Index Terms DC-DC Converters, MPPT, Quadratic Boost Converter, Thermoelectric Generators, Voltage Controller W I. INTRODUCTION ITH an expectant market for new forms of energy production, there is a strong demand and integration of alternative sources. With the rapid evolution of technological advances, it becomes possible to reuse part of the energy that was previously wasted, as the energy dissipated as heat. Thus, thermoelectric phenomena which for years had been used only as a means of heating or cooling, being limited to a small market in recent years have begun to reveal itself as a possible source for the reuse of wasted energy by other processes. The interest in the new thermoelectric energy generation applications then grew, and its varied applications and new vertical markets benefited from new devices very broad, ranging from monitoring in industrial environments to thermal applications in vehicles. The market for energy produced thermoelectrically will reach over 980 million dollars by 04 [1]. Focusing on the transport sector, there is now a consensus that the next 0 years, hybrid and pure electric cars will be the fastest growing sector of the automotive industry. Alongside this growth then comes a need for greater energy efficiency []. In this work, a study is carried out to design a quadratic boost convertor DC-DC to be able to extract about 1000W through thermoelectric modules, the heat produced in the electric motor, and thus enable recharging the car battery. The necessary steps are the following: 1) Study the operation of the thermoelectric generator; ) Study the operation and design the quadratic boost converter; 3) Implement and simulate the MPPT algorithm to control the quadratic boost converter, causing the system to simultaneously control the converter and the operation point of maximum power of thermoelectric generator; 4) Study, design and simulate the control system of the quadratic boost converter output voltage, from current control; 5) Implement a monitoring system composed of the MPPT and the output voltage controller simultaneously. II. THERMOELECTRIC GENERATOR (TEG) A thermoelectric generator (TEG) is an array of thermoelectric modules connected in series and/or parallel mode according to the load needs. They consists in a solid-state devices, using thermocouples that convert heat directly into electricity. The thermocouple is placed between two thermal isolators electrically connected and sandwiched between two ceramic plates, to allow a low thermal conductivity and high mechanical strength. It consists of semiconductor junctions p and n that are connected in a single junction with a conductive metal. The p-type semiconductor has a positive Seebeck coefficient α p and the n-type has a negative coefficient α n, therefore the overall Seebeck coefficient of p-n junction is positive α pn = α p α n [3]. The Figure 1 represents the basic structure of a thermoelectric generator module based on thermocouples. The p-n junction generates an electrical voltage which is proportional to the temperature difference between the hot and the cold sides of the generator and the output voltage multiplied N. Brás is with the Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal ( nuno.bras@ist.utl.pt). J. F. Silva is with the Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal ( fernando.alves@ist.utl.pt).

2 by the number of joints (number of links in series of several p- n junctions (n)) given by: The higher the ZT m factor, designated Figure of Merit, the greater the M, which gives efficiencies closer to the efficiency of Carnot cycle (T hot T cold ) T hot. The Figure of Merit is given by: ZT m = α ρ K t T (8) Figure 1 - Thermoelectric generator module [4]. V out = n (α p α n ) T (1) Where α is called the Seebeck coefficient, ρ the electrical conductivity, K the thermal conductivity and T the temperature. The thermoelectric modules can be associated to a converter like presented in Figure : A thermoelectric generator internal resistance (R i ) is also considered for the power available at the output and is given by: R i = R n + R p + 4 R c + R j () Where the internal resistance is obtained by the sum of the resistances of the thermoelectric elements (R n and R p ), the resistance of the metal contact junctions (R j ) and contact resistances (R c ). As relevant to a thermoelectric generator factors are considered the electric power delivered to the load and the efficiency, which is the ratio of output power to the rate at which heat is removed from the source. The thermal electromotive force (e.m.f) is given by ((α p α n )(T hot T cold )) that gives the current I which may be expressed as: I = (α p α n )(T hot T cold ) R n + R p + R L (3) It follows from this that the power delivered to the load is: P out = I R L = { (α p α n )(T hot T cold ) } R R n + R p + R L L There is also the flow of heat conduction between the branches. Then, the total rate of heat flow [W] is given by: q hot = (α p α n )I T hot + (K p + K n )(T hot T cold ) (5) (4) Figure - Thermoelectric modules associated to a converter. The thermoelectric generator here used is the combination of 4 thermoelectric modules TE-G (Tellurex) in series and 10 in parallel. The Table 1 presents the characteristics of a single module provided by the manufacturer, which are the operating values for T = 100. Table 1 TE-G characteristics provided by manufacturer Peak power P out 4,6 W Rated Current I máx,05 A Rated Voltage V máx,5 V Internal Resistance R i 1,1 Ω Short-Circuit Current I cc 4,09 A Open Circuit Voltage V ca 4,5 V III. QUADRATIC BOOST CONVERTER To raise the value of output voltage of the thermoelectric modules, a quadratic boost converter was chosen because it allows high values of the input/output relation. For example, it makes possible to have a thermoelectric generator with low voltage, in the order of 54V and this voltage may be raised to 400V in the load. The efficiency can be calculated as: η = (T hot T cold )(M 1) T hot (M + T cold T hot ) Where M is the ratio between the load resistance and the resistance of the generator and can be given by: M = (1 + ZT m ) (7) (6) Figure 3 Quadratic boost converter with thermoelectric generator (TEG). In this analysis we consider the converter in the continuous conduction mode and semiconductor as ideal switches. The diodes for a positive terminal voltage to its anode and cathode

3 3 behave as short circuit and there is a zero voltage drop and a positive current. In the case of being negative, diodes work as an open circuit with no current flow. In this converter there are two points of operation, one when the MOSFET S is on, since it is driven on, and the other where it is OFF. It is considered that the MOSFET S is brought into conduction during t on and is blocked for the rest of the period T, with a cycle factor δ = t on T, and a frequency f PWM = 1 T. During the fraction of the period δt or (0 < t < t on ) it is assumed that the MOSFET S is in conduction and (1 δ)t or (t on < t < T) it is blocked. Adding these two intervals we get one period. Assigning γ as a variable that defines these two intervals we get: i D = { 0 γ = 1 i L γ = 0 i D3 = { i L1 γ = 1 0 γ = 0 i MOS = { i L1 + i L γ = 1 0 γ = 0 (18) (19) (0) The relationship between V 0 Vout,TEG, as function of δ can be calculated knowing that, in steady state, the average value of voltages at the terminals of the coils L 1 and L are zero. γ = { 1, (0 < t < t on) 0, (t on < t < T) (9) V L1av = V C1 = 1 V out,teg 1 δ (1) When the MOSFET S is ON, γ = 1, the diodes D 1 and D are OFF and the diode D 3 is ON. As the diode D is blocked, the load represented by a resistance R is only in parallel with the capacitor C 0 not receiving power from the rest of the circuit. Thus the capacitor will discharge into the resistance. Due to the diode D 1 also blocked, the current will go through the MOSFET S and the current il 1 will circulate through the diode D 3. The capacitor C 1 will transfer energy to the coil L. When MOSFET S is OFF, γ = 0, the diodes D 1 and D are ON, causing D 3 to stay blocked. The current il 1 will pass through D 1 which will be ON ensuring continuity of the magnetic energy in the coil L 1. With the MOSFET S OFF, the capacitors will be able to charge [5]. The voltages and currents of the circuit have the following variations: V out,teg = U V req (10) V L1 = { V out,teg γ = 1 V out,teg V C1 γ = 0 V L = { V C1 γ = 1 V C1 V 0 γ = 0 V AK1 = { V C1 γ = 1 0 γ = 0 V AK = { V 0 γ = 1 0 γ = 0 V AK3 = { V C1 γ = 1 V C1 V 0 γ = 0 V MOS = { 0 γ = 1 V 0 γ = 0 i D1 = { 0 γ = 1 i L1 γ = 0 (11) (1) (13) (14) (15) (16) (17) V Lav = V O V C1 = 1 1 δ Therefore the converter input/output relation given by: V out,teg = V 0 (1 δ) () V 0 1 (3) = V out,teg (1 δ) Defining δ in function of the input and output voltages and considering the efficiency of the converter: δ = 1 η V out,teg V o (4) We can reach the expression of the output current of the thermoelectric generator by calculating the duty cycle δ by the method of power conservation: I L1 = I 0 (1 δ) = I U η (P U R eq I Uef ) = V 0 I 0 η (V U I U R eq I 0 (1 δ) ) = V 0 I 0 I 0 η (V U (1 δ) R eq (1 δ) 4) = V 0 I 0 I 0 (5) η V U (1 δ) η R eq I 0 = V 0 (1 δ) (6) Taking the value of δ from the previous equation we obtain the value of I U using the equation (5) and then proceed to the following calculations: V Req = R eq I U,rms (7)

4 4 V TEG,out = U V Req (8) A. Converter Design Despite the voltages and currents presented in the converter being almost continuous, they exhibit ripple due to reactive elements in the circuit. In accordance with allowed current and voltages ripples, for the inductive elements we have the coils L 1 and L and can be calculated by: L 1 = δv out,teg i L1 f PWM = δ(1 δ) V 0 i L1 f PWM (9) L = δv out,teg i L f PWM (1 δ) = δ(1 δ) V 0 i L f PWM (30) For the capacitive elements we have de capacitors C o and C 1 and can be calculated by: C 0 = C 1 = δv out,teg (1 δ) V 0 R 0 f PWM (31) δv out,teg (1 δ) V C1 R 0 f PWM (3) Where f PWM is the commutation frequency. To assure the operation in continuous conduction mode we get: L 1 δ(1 δ) 4 R 0 f PWM (33) L δ(1 δ) R 0 f PWM (34) B. Converter Model Including Losses In the quadratic converter there are power losses which have to be taken into account. The efficiency is no longer equal to 100%. To reduce ON state losses, the diodes were replaced by MOSFETS. The equivalent circuit is present in Figure 4 when the MOSFET is ON a) and OFF b). b) Figure 4 Quadratic boost converter equivalent circuit including dissipative elements when MOSFET is ON a) and when MOSFET is OFF b). The efficiency is given as: η = P 0 P i = P i P perdas P i (35) The power losses in the coils are given by P rl1 in coil L 1 and P rl in coil L, representing losses by Joule effect, in their internal resistances, r L1 and r L. Besides the losses in the coils we have the ON state and the switching losses in the semiconductors P CMOS and P S, respectively. P perdas = P rl1 + P rl + P CMOS + P S The power losses in r L1 and r L can be given by: (36) P rl1 = r L1 I L1,rms (37) P rl = r L I L,rms (38) When the MOSFETs are conducting, they can be represented by their equivalent resistance and exhibit Joule losses which are given by: P CMOS = R DSon I MOS,rms (39) P CMOS1 = R DSon1 I MOS1,rms (40) P CMOS = R DSon I MOS,rms (41) P CMOS3 = R DSon3 I MOS3,rms (4) The switching losses result from changes in the status of semiconductors (OFF-ON and ON-OFF) and results in overlap of both the current and the voltage of the semiconductor during the state transition. Figures 5 and 6 represent the semiconductor switching voltages and currents. a)

5 5 It is intended to have 1000W and 400V in output power and voltage of the converter. The circuit was simulated taking in account the theoretical calculated efficiency of 91.64%. Figure 7 and 8 show the output power and voltage of the converter, respectively. Figure 5 Switching time OFF-ON of the MOSFETS [6]. Figure 6 - Switching time ON-OFF of the MOSFETS [6]. Figure 7 Output power in steady state. The time t ON is represented in Figure 5 and is appointed by the time of establishing the conditions of the MOSFETs ON state, being the sum of the time of current rise and fall time of the voltage and is given by: t ON = t r (i MOS ) + t f (v MOS ) (43) The time t OFF is represented in Figure 6 corresponds to the cut-off time of the MOSFETS, being the sum of the rise time of the voltage and the fall time of the current and is given by: t OFF = t r (v MOS ) + t f (i MOS ) (44) The power losses associated to the switching of each MOSFET is given by: P S = t ON + t OFF (v T MOS i MOS ) + 1 T C OSSv MOS (45) Wherein C OSS is the output capacitance of the MOSFET and T the period of operation. C. Voltage gain The new relation of transformation that enters with the losses in the coils is: V 0 V out,teg = 1 (1 δ) + r L1 R 0 1 (1 δ) + r L R 0 (46) It is noted that the value of the transfer ratio is limited by the loss resistor of the coils. Figure 8 Output voltage in steady state. In the simulation we reached an efficiency of 91.37%, which is a quiet acceptable value compared to the value obtained theoretically. The difference between the theoretical value and the simulated value is mainly because of factors not considered theoretically. IV. MPPT To the main objective, the thermoelectric modules must deliver to the load the maximum available power at any moment. The MPPT is an algorithm that allows to reach that point of operation. This operating point is characterized by a current and an output voltage of the TEG. The MPPT will provide reference values of voltage and current to a switched converter and the converter will control the voltage and current in order to put the TEG running at its maximum power point. A. MPPT Algorithm The IV characteristic of the thermoelectric is presented in Figure 9. D. Application example

6 6 dp di < 0 I > I MPP I (lower I) MOSFET off (γ = 0) In the case where the power time derivative is zero, the generator is running at maximum power, so no state changes are done in MOSFET (γ = γ (t-1)): dp di = 0 I = I MPP I (remain I) Figure 9 - IV curve showing the relationship between the output current and output voltage of the thermoelectric generator [7]. To obtain the maximum power is needed to calculate the time derivative of the expression of the thermoelectric power generator, P = VI, and equal it to zero: MOSFET remain (γ = γ(t 1)) In Figure 10 the flowchart of the MPPT operation is shown. dp di = 0 d dv (VI) = 0 V = I di di (47) As the changes in output current and voltage of the thermoelectric generator are small in each period, the following approach is considered: dp dv V = V + I V + I di di I v(t) v(t 1) v(t) + i(t) i(t) i(t 1) (48) Where v(t 1) and i(t 1) are the voltages and currents sampled at the previous instant. It can be concluded from Figure 9, that the derivative of the power is positive when the power is less than the maximum power and negative in the opposite case. When the derivative is equal to zero we are operating at maximum power point. It is intended to control the current i L1 through the switching of MOSFET to find the maximum power point. When the derivative of the power is positive, the MPPT will have to raise the current i L1, i.e., will have to increase the duty cycle of the MOSFET so that the derivative approaches zero and thus take the maximum power of the generator [8], [9], [10]. For that to happen the MOSFET will have to be driven ON (γ = 1): Figure 10 - Flowchart of the method used for the MPPT. B. Simulation To make simulations a block diagram in Simulink/MATLAB was created. In this block diagram, the power time derivative is calculated in order to the current and according to the derivative sign, an action is performed to drive the MOSFET, as seen previously. Figure 11 shows the illustrative block diagram of the MPPT created in Simulink. dp di > 0 I < I MPP I (raise I) MOSFET on (γ = 1) When the derivative of the power is negative, the MPPT will have to lower i L1. In this case, the opposite will happen when the derivative was positive, i.e., the MOSFET will be driven to the OFF state (γ = 0): Figure 11 Diagram MPPT Figure 1 shows the operating principle of the MPPT. The values shown are the power derivative and the control signal of the converter MOSFET, which was originated through the MPPT. As observed, when the power derivative is positive the

7 7 converter control signal is one, causing an increase in the input current. When the derivative is negative, the control signal goes to zero, which will decrease the input current. Figure 15 MPPT currents. Figure 1 Command of the MPPT The maximum power extracted by the generator will be calculated by: P TEG,out = V TEG,out I U (49) For the values of the generator we have a output power of 1104W as it can be observed in Figure 13. Figure 13 TEG output power with MPPT. The voltages and currents for the nominal load are shown in the Figures 14 and 15, respectively. Figure 14 MPPT Voltages. As it can be seen it is possible to implement the MPPT and control the quadratic boost converter, using a single control system. The system is always running at maximum power point. With the MPPT working, if the load is lighter than the rated, the output voltage will be higher than desired. To change this situation an output DC voltage control will be used to retain the 400V output voltage. V. OUTPUT VOLTAGE CONTROL Switched converters usually need to be controlled in closed loop in order to supply the desired voltages and currents [11], [1]. In the converter under study there is a degree of freedom, which determines the ON state or OFF state of MOSFETS characterized by γ, and only the input current of the converter is controlled for any load conditions. As the dynamics of the output voltage is much slower than the input current, the output voltage control is done by slowly varying the reference input current. The voltage control loop is designed considering a linear time-invariant equivalent (LTI) of the current controlled converter. A. Current Control The system used to control the current il1 is a non-linear current control, since it is easy to implement and takes advantage of the fact that the discrete system is a switched converter with reduced levels () in the control quantity. This control determines the error between the reference value and the current il1 trying to enforce it to zero. As the switched system is in a finite frequency in the instantaneous values the error e il1 cannot always be zero and current il1 will present an associated ripple. This ripple is identified as an error, which can vary in the range presented in (50). The current il1 determined according to (51) and its respective error by (5). L 1 di L1 dt = V TEG,out (1 γ)v C1 di L1 dt = V TEG,out (1 γ)v C1 L 1 (50)

8 8 e il1 = i L1ref i L1, ε il1 < e il1 < ε il1 (51) expression of the converter output voltage (57) and the block diagram shown in Figure 17: ε il1 = i L1 (5) The convergence of the system to the reference value is guaranteed using stability condition [8], [9], [10]. v 0 = V TEG,out V 0 i L1 i 0 sc 0 = Gi L1 i 0 sc 0 (57) e il1 de il1 dt < 0 (53) With the use of a comparator with hysteresis width ε, γ can be controlled. This will compare the current i L1 with the reference current so deciding which device to put on driving: e il1 e il1 > +ε i L1ref > i L1 i L1 di L1 dt > 0 γ = 1 (54) < +ε i L1ref < i L1 i L1 di L1 dt < 0 γ = 0 B. Voltage Control Analyzing the output circuit of this converter, shown in Figure 16, it is possible to writer the current through the capacitor C o (45). Figure 17 Block diagram of the voltage controller. In closed loop control, the information of how the controlled output is evolving is used to determine the control signal to be applied to the process at a specific instant. To make the system more precise and cause it to react to external disturbances, the output signal is compared with the reference signal and the error between them is used to determine the control signal. In closed loop it is obtained: v 0 (s) = (1 + st z ) G v 0ref (s) (1 + st d) (st p ) i 0 (s) s 3 T p T d C 0 + s T p C 0 + (1 + st z ) G (58) The T z and T p parameters are calculated by equations (60) and (61), respectively. The T d value is initially assumed. ξ = (59) T z = C 0 R 0 (60) T p = 4 ξ a G R 0 T d (61) a = 1 To find the constant T d, an equivalent RLC circuit (Figure 18) expressions are used: (6) Figure 16 Representative circuit of the converter output. i C0 = C 0 dv 0 dt = i MOS I 0 (55) Determining i mos in order to i L1 and applying the Laplace transform, it is obtained: sc 0 v 0 = V TEG,out V 0 i L1 i 0 (56) By manipulation of the previous expressions we arrive at the Z = L C ξ = Z L R = C R ω = 1 LC t = π ωd = (63) (64) π ω 1 ξ = π LC (65) 1 ξ

9 9 VI. MPPT AND OUTPUT VOLTAGE CONTROLLER Figure 18 Equivalent RLC circuit. The calculations give T d = 3,3 ms. C. Steady-state Simulation In this simulation the load is considered as the nominal value and the Figure 19 shows the correct operation of the output voltage controller enforcing the output voltage to the theoretical value of 400 V. After completion of the study of the isolated converter, we proceed to the study and simulation of the whole system. The system works as follows: To v 0 < 400 V v 0 the MPPT controller is operating, with the voltage controller off. To v 0 > 400 V + v 0 the voltage controller is operating, with MPPT controller being off. In a first approach, the system analyzed the output voltage using a comparator connected to a switch, which made the controller switch depending on the output voltage presented and the desired load on the converter output. In the simulation, a problem arose when the system alternated the MPPT controller for the voltage controller and vice versa, introducing unwanted overvoltage in output voltage. This problem arose due to the integrators of the two controllers are running constantly, even when not required, which leads to values already present in the transition from one controller to another which causes unwanted overvoltage arise in output voltage. As a solution for this problem we opted for the use of a common integrator to two controllers, which caused the system to avoid overvoltage. For faster response to change the controller, it was decided to also decrease the gain of the MPPT to slow the rise of current and thus prevent the voltage rise far above the value of the desired output voltage until the voltage controller operates in system. Figure 1 shows the block diagram of the controller. D. Dynamic Simulation Figure 19 Steady state output voltage. To simulate the converter in dynamic regime, the load resistance was varied at 10 ms of the simulation in order to verify the correct operation of the voltage controller (Figure 0). To vary the load resistance, the value of R o was given to the resistance in the beginning and set up in parallel an equal resistance which is connected to the circuit at the time 10 ms. Figure 1 Block Diagram of the controller. To simulate and test the new system used a load of R o, so that there is transitions of controllers presented in Figure. Figure 0 Dynamic state output voltage. As can be seen the output voltage converges to the reference value of 400V. Figure - Simulation of the output voltage controller with MPPT and voltage under load Ro. With rated load R o, we obtained the following graphs present in Figures 3 and 4.

10 10 REFERENCES Figure 3 - Simulation of the output voltage controller with MPPT and voltage at rated load. Figure 4 Simulation of the output power with MPPT controller and voltage at rated load. It can be seen that the results are as expected. As the system is working with rated load, there is 400 V voltage and 1000 W output power. VII. CONCLUSION In the study of the thermoelectric system was designed a generator using 40 thermoelectric modules, coupling 4 in series and 10 in parallel to give an output voltage of 54V and a maximum current of 0,5A, resulting in a maximum power of 1104W. An analysis of a quadratic converter losses was made with the coils and semiconductor devices. After simulating the converter the efficiency was %, with a difference of 0.7 percentage points to the theoretically calculated value. A maximum power point tracking MPPT algorithm was implemented to optimize the power harvested from the thermoelectric generator and the converter. An output power of W has been achieved, slightly above 1000 W and average voltage of V being the desired value of 400 V. An output voltage control system was implemented, which has been tested both in a steady state and in dynamic operation. In the steady state, simulation values equaled the desired ones, and the dynamic operation showed fast response times in the order of 0 ms to voltage stabilization. Finally the converter was tested with two controllers working together, achieving the desired power of 1000 W and an average voltage of 400 V to the converter output. [1] Dr. Harry Zervos, "Energy harvesting for automotive applications," IDTechEx, 011. [] Dr. Peter Harrop and Raghu Das, "Hybrid And Pure Electric Cars ," IDTechEx, 010. [3] H. Julian Goldsmith, Introduction to thermoelectricity, Springer, 010. [4] "Direct Conversion of Heat Energy to Electrical Energy," [Online]. Available: [Accessed ]. [5] Dinis Honrado and Ricardo Santos, "Conversor Elevador Quadrático para Aproveitamento de Energia Renovável," Instituto Superior Técnico, 006. [6] José Fernando Silva, "Eletrónica Industrial", Fundação Calouste Gulbenkian, [7] Ryan Mayfield, "EC&M electrical constrution & Maintenance," 8 Nov 01. [Online]. Available: [Accessed 6 Set 014]. [8] V. Fernão Pires, S. Pinto, J. Barros José Fernando Silva, Advanced control methods for power electronics systems, Mathematics and computers in simulation 63 (3), [9] José Fernando Silva, PWM audio power amplifiers: sigma delta versus sliding mode control, Internacional Conference on Electronics, Circuits and Systems, 1998, Volume 1, pp [10] José Fernando Silva, Sliding mode control of voltage sourced boost-type reversible rectifiers, Proceedings of the IEEE International Symposium on Industrial Electronics, 1997, ISIE'97, Volume, pp [11] Katsuhiko Ogata, Modern Control Engineering, Prentice-Hall, [1] José Fernando Silva, "Eletrónica Industrial: semicondutores e conversores de potência", Fundação Calouste Gulbenkian, 013.