Non-linear average electrothermal models of buck and boost converters for SPICE

Microelectronics Reliability 49 (2009) 431–437

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Non-linear average electrothermal models of buck and boost converters for SPICE Krzysztof Górecki * Gdynia Maritime University, Department of Marine Electronics, Morska 83, 81-225 Gdynia, Poland

a r t i c l e

i n f o

Article history: Received 26 August 2008 Received in revised form 5 December 2008 Available online 20 March 2009

a b s t r a c t The new method of formulating non-linear average electrothermal models of dc–dc converters is presented in the paper. These models take into account both: non-linearity of diode and transistor characteristics and electrical inertia of these elements. The form of the models for buck and boost converters is presented and their correctness is verified by comparing the characteristics of the considered converters obtained with the proposed model and the classic transient analysis. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Buck and boost converters are commonly used in switchedmode power supplies of the output power up to a few hundred watts. These converters contain: semiconductor devices (unipolar and bipolar transistors and diodes) and passive elements RLC. While designing and analysing electronic systems, computer software is generally used. The most popular among this software is SPICE [1,2]. In the literature one can distinguish two groups of methods analysing switched-mode systems, to which dc–dc converters belong: the method of the transient analysis [3–10] and the dc analysis with average models of dc–dc converters [11–20]. In order to calculate the characteristics of converters in the steady state, the method of average models is often used. In this method the average model of the analysed converter must be formulated. In such a kind of model the values of terminal voltages and currents are equal to the average values of the time runs of terminal voltages and currents existing in the real converter. This method ensures short duration time of calculations, even a few orders of magnitude shorter than in the case when the transient analysis is used [21]. An important phenomenon occurring in semiconductor devices contained in dc–dc converters is selfheating [2,21–24]. As a result of this phenomenon the internal temperatures of semiconductor elements are higher (sometimes considerably) than the ambient temperature [25–32]. Due to an increase in the internal temperature the characteristics of these elements and the systems with these elements change significantly. The characteristics measured with the phenomenon of selfheating taken into account are called non-isothermal characteristics and the models taking this phenomenon into account – electrothermal models [6,26,30–33]. The paper [23] proposes the average electrothermal model of the diode-transistor switch designed for the electrothermal analy* Tel.: +48 58 6901448; fax: +48 58 6217353. E-mail address: [email protected] 0026-2714/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2009.01.009

sis of dc–dc converters in SPICE. Unfortunately, this model takes into account neither non-linearity of the semiconductor devices characteristics nor their electrical and thermal inertia. Yet, as it was shown in the works [34,35]: as a result of the negligence of the electrical inertia in semiconductor devices, the use of the average model in the analysis of dc–dc converters (within the frequency of the controlling signal exceeding 200 kHz) causes great inaccuracy of the calculations. On the other hand, it was shown in the work [36] that non-linearity of semiconductor devices characteristics influences significantly the characteristics of buck and boost converters in the steady state, especially for the low value of the magnitude of the signal controlling transistor included in these converters. The present paper proposes the method of formulating non-linear average electrothermal models of dc–dc converters dedicated for SPICE. On the example of buck and boost converters, the form of such models is shown and the verification results of their correctness are presented. 2. Formulating non-linear average electrothermal models of dc–dc converters In order to formulate a non-linear average model of the analysed converter one performs seven steps, where the first four steps are similar to the classic model of formulating average models described in the work [15]. In the new method one makes use of: voltage drops between the terminals of semiconductor devices instead of their on-state resistances. Moreover, non-linearity of semiconductor devices, their electrical inertia and selfheating are taken into account. The realization of the method involves: 1. Making three equivalent circuits of the examined converter, where one circuit corresponds to the switching-on state of the transistor. The second constitutes the switching-off state of the transistor and switching-on state of the diode and finally, the third constitutes the switching-off state of the transistor

432

2.

3.

4.

5.

6.

7.

K. Górecki / Microelectronics Reliability 49 (2009) 431–437

and the switching-off state of the diode. In the switching-on state the transistor is represented by the voltage source VON and in the switching-off state – by the open. The diode is represented in the reverse range by the open, and while conducting – by the voltage source VD. As it was demonstrated in the works [37], neglecting the reverse currents of semiconductor devices practically does not influence the terminals characteristics of dc–dc converters and makes the analysis of these elements far simpler. Formulating equations describing voltage on the inductor and the capacitor current for the formulated in point 1 networks with the assumption that variable components of voltages and currents resulting from switching on and off semiconductor devices are negligibly small in relation to their constant components. Formulating and comparing to zero expressions describing the mean value of the voltage on the inductor with the use of the principle of flux continuity in the coil and expressions describing the mean value of the capacitor current making use of the principle of charge protection in the capacitor. Formulating the circuit model of the converter based on the equations obtained in point 3 describing behavior of the converter in the steady state taking into account losses in the semi-conductor elements and the choking coil. Making the circuit consisting of a series connection of the switching transistor, the diode, the controlled current source, the current of which is equal to the mean value of the inductor current in the steady state, two controlled voltage sources modeling changes in the output voltage of the transistor and in the voltage of the diode conduction caused by a change in the internal temperature of these elements. This circuit contains also the voltage source of the voltage equal to the value of the high state voltage controlling the switching transistor connected by the resistor with the controlling electrode of this transistor. Voltages on the output terminals of the transistor and the diode are equal to the values of voltages VON and VD, occurring in the description of the controlled sources contained in the main circuit of the converter, which was formulated in point 4. Formulating in the circuit form the static thermal models of the transistor and the diode, describing the dependence of their internal temperatures on the power dissipated in them. According to the considerations in the work [38], it can be a static thermal model because of the high values of thermal time constants in the model of transient thermal impedance of semiconductor elements in comparison with the period of the signal controlling the transistor. Then, the value of the element internal temperature in the steady state constitutes the sum of the ambient temperature and the product of thermal resistance of this element and the mean value of the power dissipated in it. Formulating formulas describing time tp, in which the inductor current flows in the discontinuous conducting mode of the converter operation and formulas describing the mean values of the diode current and the main transistor current.

The model of the examined converter dc–dc, formulated in the above way enables calculating its characteristics by the dc analysis in SPICE.

3. Non-linear average electrothermal models of dc–dc converters The presented method of formulating non-linear average electrothermal models of dc–dc converters is used to simulate characteristics of the boost (Fig. 1a) and the buck (Fig. 1b) converters, in which the switching element is the MOS power transistor. The fur-

a

L D RG

Vout

R0

T C

Vin

b

Vster

L

T

RG Vin

C

R0

Vout

D Vster

Fig. 1. Diagrams of the considered dc–dc converters: (a) boost and (b) buck.

ther part of this chapter presents the method of constructing circuits that appear in the equivalent circuit of the considered converters and justifies the form of the formulas describing efficiencies of particular controlled sources contained in these circuits. Assuming that the times of switching on and off the semiconductor devices are negligibly short in relation to the period of the controlling signal generated by the voltage source Vster, it is possible to distinguish in each period of this signal three phases of the converter operation. In the first one the current of the inductor L flows through the transistor T operating within linear range [39,40]. In the second one – the inductor current flows through the forward-biased diode D. In the third phase, which appears only in the discontinuous operation mode, the inductor current does not flow. Due to short switching on and off times of semiconductor devices in relation to the period of the switching signal, the analysis takes into account, in a simplified way, the shape of the time characteristics of currents and voltages of semiconductor devices while they are switched on and off. When using the presented method, non-linear average electrothermal models of buck and boost converters are formulated; the network representations of these models are presented in Figs. 2 and 3, respectively. In these figures it is possible to distinguish: the main circuit of the converter resulting from the realization of points 1–4 of the algorithm (Figs. 2a and 3a), the auxiliary circuit to measure a drop in the voltage VD on the forward biased diode and the voltage VON on the switched-on transistor resulting from the algorithm (Figs. 2b and 3b), the circuit thermal models of the transistor and the diode, formulated in point 6 of the algorithm and the circuit to measure time tp, in which the inductor current flows after switching off the transistor formulated in point 7 of the algorithm (Figs. 2c and 3c). The boost converter operates in the switching mode only when the voltage VON on the switched on transistor is lower than the input voltage Vin. If this condition is not satisfied, the diode D conducts the current in the continuous way and the output voltage is equal to the difference between the input voltage and the voltage drop on the forward biased diode. Then, the circuit from Fig. 3d should substitute the circuit from Fig. 3a. In Figs. 2 and 3, the symbol w designates the duty factor of the transistor current, which is different from the duty factor of the controlling signal d, Ts is the period of controlling signal, whereas VH – the value in the high state of the voltage controlling the transistor, RL is series resistance of the inductor L, through which the

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K. Górecki / Microelectronics Reliability 49 (2009) 431–437

a

b RL ⋅

ETD

D

w ⋅ TS + t p TS

VD

IL VON⋅w

VT

ETT

VD⋅tp/TS

VO Vout⋅

Vin⋅d

G1

VON

w ⋅ TS + t p

RG

T

TS

ERo

VH

c

ΔTD

ΔTT

p D ⋅ RthD

tp

pT ⋅ RthT

Vout

tp

Eout

η

ESPRAW

Fig. 2. Network representation of the non-linear average electrothermal model of the buck converter.

a

b RL1

ETD

D

Iin ERL

VT

VT1

VD

w ⋅ TS + t p Vin ⋅ TS

ET1

ETT

VON⋅w VD⋅tp /TS VO ERo

Vout 0 ⋅

tp

GT1

VON

T2

T1

E20

TS

RG VH

VD1

c

ΔTD

ΔTT

tp

Vout

d

Pout Pin

RL

Iin1

D EX1

p D ⋅ RthD

pT ⋅ RthT

tp

Eout

EPout E Pin

V11

GXT1

R0

Vout1

Fig. 3. Network representation of the non-linear average electrothermal model of the boost converter.

current of the mean value IL flows. The controlled voltage sources ETT, ET1 and ETD present voltage changes on the transistor and the diode, caused by an increase of their internal temperatures over the ambient temperature DTT and DTD, respectively. The mean values of the power dissipated in the diode and the transistor are equal to pD and pT, respectively. The current IP is the mean value of the diode and transistor currents during their conduction. It can be easily noticed that when the considered converters operate in the continuous conduction mode (CCM) (then tp = (1  w)TS) and when we neglect resistance RL of the inductor, the circuits from Figs. 2a and 3a will be reduced to the form presented in the work [41]. While formulating the systems from Figs. 2b and 3b (point 5 of the method), it is taken into account that the mean value of the diode and the transistor currents during the time of their conduction is the same and equal to IT, because the instantaneous value of these currents changes linearly during the conduction of the mentioned elements and assumes the identical maximal and minimal values. The current IT is forced by the controlled current sources GT1 and G1 (appearing in Figs. 2b and 3b), the efficiencies of which are given by the formulas

TS G1 ¼ IL  w  T S þ tp ( if tp ¼ ð1  wÞ  T S Iin GT1 ¼ V in V ON w  2  ðw  T S þ tp Þ if tp < ð1  wÞ  T S L

ð1Þ ð2Þ

In the formula (2) the first dependence refers to the boost converter operating in the CCM, whereas the other dependence – of the same converter operating in the discontinuous conduction mode (DCM). The values of the voltage sources modeling the temperature voltage changes in the conduction of the diode ETD or EX1 and the transistor ETT or ET1 are given by the formulas [36]

ET1 ¼ ETT ¼ RON0  IT  aTT  DT T

ð3Þ

ETD ¼ aUD  DT D þ RD  IT  aTD  DT D V out1 EX1 ¼ aUD  DT D þ RD   aTD  DT D R0

ð4Þ ð5Þ

where R0N0 designates the on-state resistance of the transistor MOS in the ambient temperature, aTT is the temperature coefficient of relative voltage changes VON, aUD – the temperature coefficient of voltage changes in the forward biased diode, RD – series resistance of the diode, aTD – the temperature coefficient of relative changes in the diode series resistance. The transistors and the diodes appearing in Figs. 2b, 3b and 3d are described with the help of the models of these elements builtin in the program SPICE, whereas the values of the models parameters of the considered elements correspond to the elements used in the analysed converter. The voltage sources of zero value VT and VT1 are designed to monitor the mean value of the semiconductor elements currents. The resistance RG corresponds to the resistance connected to the gate of the switching transistor in the real converter.

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K. Górecki / Microelectronics Reliability 49 (2009) 431–437

In the real converter, as a result of non-zero capacities of the transistor, the time in which the current of the drain flows is different from the time, in which the controlling voltage is higher than the threshold voltage of the transistor. Therefore, the pulse duty factor of the drain current is equal to

w¼d

tON t OFF þ TS TS

ð6Þ

where d is the duty factor of the voltage controlling the transistor gate, tON and tOFF designate the times of switching-on and switching-off the transistor. These times are calculated by assuming that they depend only on the input capacity of the transistor Cin, resistance RG in the gate circuit and the voltage drop on this resistor. They can be described with the formulas

C in  RG  V TO V H  V TO ¼ C in  RG

t ON ¼

ð7Þ

t OFF

ð8Þ

where VTO designates the transistor threshold voltage, the input capacity Cin constitutes the sum of capacities occurring between the gate and the source and between the gate and the transistor drain and is described by the following approximated dependence obtained on the basis of Dang’s model [42] with the assumption that the voltage of the drain-transistor source is considerably smaller than the difference of the voltages VH  VTO. This assumption is typically satisfied in the on-state of the transistor

C in ¼ Cgso  ww þ Cgdo  ww þ ww  ll 

eSiO2  e0 tox

ð9Þ

where Cgso, Cgdo, tox are parameters of Dang’s model, ww is width and ll – length of the channel of the MOS transistor, eSiO2 designates relative dielectric permittivity of silicon dioxide, whereas e0 – dielectric permittivity of free space. The voltage source V11 has the value equal to the input voltage of the boost converter. The output current of the controlled current source GXT1 is given by the formula

GXT1 ¼ w  iT1

ð10Þ

where iT1 is the current flowing through the voltage source VT1. The voltage source E20 has the capacity equal to the voltage on the source GXT1. Time tP (point 7 of the method) is measured by comparing the mean load current and the mean current of the inductor. The following dependence describing time tp is obtained for the buck converter [36]

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! V out  T S  L t p ¼ 0:5  w  T S þ w2  T 2S þ 8  R0  ðV out  V D Þ

V out  L  R0  ðV out  V D Þ  w

trr ¼ TT  lnð1 þ IF =IR Þ

ð13Þ

where TT is the time of the carrier flight through the diode basis, IF – the forward current of the diode and IR – the maximal reverse current of the diode while switching-off. The current switching-off the diode flows also through the transistor causing, especially in the buck converter, the occurrence of the essential element of the power dissipated in this element. Assuming the piecewise-linear time characteristics of the drain current and the drain – source voltage of the transistor power losses connected with switching-on and switching-off the transistor are taken into account in the model of heat generation. Eventually, in the buck converter the mean value of the power pT dissipated in the transistor is given by the formula

  V out þ V D tON þ t OFF  tp  pT ¼ w  V ON  IT þ IT þ 2L TS     V ON V in þ V D TT V D þ V ON þ   ln 1 þ þ TS 3 6 V in  V ON 

ðV in  V D Þ  ðV in þ V ON Þ  IT V ON þ V D

ð14Þ

and the power pD dissipated in the diode – by the formula

pD ¼ V D  IT 

   2 t p TT V D þ V ON V in þ V ON  þ  ln 1 þ  IT TS TS V in  V ON V ON þ V D

 RD  ð1 þ aRD  DT D Þ

ð15Þ

In turn, for the boost converter the powers pT and pD are respectively given by the formula

8     > w  V ON  IT þ IT þ V out2LþV D  w  T S  tONTþtS OFF  V 3ON þ V out6V D > > >   > < D ÞðV out V ON Þ ON þV out  ðV outVV  ln 1 þ V D þV  IT þ TT V out V D TS ON þV D þV out pT ¼ > > > if V out0 > V out1 > > : w  E20  IT1 if V out0 < V out1 ð16Þ and

pD ¼

8    2 > tp V D þV ON þV out V ON TT >  IT  V ONV out > < V D  IT  T S þ T S  ln 1 þ V out V ON þV D þV out RD  ð1 þ aRD  DT D Þ if V out0 > V out1 > > > : V D1  V out1 if V out0 < V out1 R0

ð11Þ

ð17Þ

ð12Þ

The voltage of the controlled voltage source Eout is equal to the output voltage of the converter Vout. In the case of the boost converter this voltage is given by the formula

and for the boost converter [36]

tp ¼ 2 

simplification is assumed that in the switching-off time the current and voltage characteristics have the rectangular shape and the time of switching-off trr is given by the formula [43]

The formulas (11) and (12) are right only for operation in the DCM. If the value of time tp obtained from these formulas is higher than (1  w)TS, it means that the converter operates in the CCM and then, tp = (1  w)TS. As the power dissipated in semiconductor devices in the offstate is in the considered systems negligible [15], the mean value of the power dissipated in these devices occurring in their thermal models constitutes the sum of the powers dissipated in the onstate of these devices and the power dissipated in the time of their switching on and off. The first of the mentioned constituents is equal to the product of the mean current of the element during its conduction measured in the circuit from Figs. 2b, 3b or 3d and the voltage on this element and the product of the element conduction time and the period of the controlling signal. While describing the power connected with switching-off the diode a

V out ¼



V out0

if V out0 > V out1

V out1

if V out0 < V out1

ð18Þ

where the voltage Vout0 is obtained in the system from Fig. 3a, the voltage Vout1 in the system from Fig. 3d. In turn, the watt–hour efficiency of the considered converters is expressed by the formula [36]

g¼

V 2out w  R0  V in  IL

ð19Þ

for the buck converter and by the formula

g¼

8 < :

V 2out R0 V in Iin

if V out0 > V out1

V 2out R0 V in Iin1

if V out0 < V out1

ð20Þ

for the boost converter. The currents Iin and Iin1 are marked in Fig. 3.

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K. Górecki / Microelectronics Reliability 49 (2009) 431–437

results obtained with the proposed calculation method using nonlinear average models, points – the results obtained from the classic electrothermal transient analysis [44], dashed lines – the results of electrothermal analyses obtained with the linear average model of the diode-transistor switch presented in [23]. In this section, the results of calculations obtained with the use of the method proposed in [44] are treated as the exemplary results, because during the elaboration of this method nothing simplifications are taken into account and the correctness of this method is proved experimentally in [44]. The model from [23], which is later called the linear average model, does not take into account the influence of the voltage VH on the converter characteristics. Therefore, in Figs. 4–7 there appears only one line corresponding to this model. The results obtained from the transient analyses are assumed as the standards, because for this analysis the fewest simplifications are made. Fig. 4 presents the calculated dependences of the output voltage of the buck converter (Fig. 4a) and the internal temperature increase in the transistor DTT over the ambient temperature (Fig. 4b) on the duty factor d for load resistance R0 = 3 X. As seen, the results obtained with the use of the transient analysis and the method of non-linear average models are in good agreement. The characteristics Vout(d), which corresponds to different values of the voltage VH, differ among each other even by 30%. These differences are most visible for the duty factor d < 0.6. An increase in the voltage VH causes an increase of the module of the output voltage of the buck converter. The results obtained with the linear average model are convergent with the results obtained with the non-linear average model at the high value of the voltage VH. The differences between the values of the voltage Vout calculated with the linear average model and the results of the transient analysis exceed even 30%, whereas

4. Calculations results In order to verify the proposed non-linear average electrothermal models the characteristics of buck and boost converters in the steady state are calculated with the use of the classic electrothermal transient analysis described in the work [36,44], the dc analysis with the use of the proposed in this paper non-linear electrothermal models of the considered converters and the authors’ own linear average electrothermal model of the diode-transistor switch described in [23]. In the analyses performed with SPICE, the following values of passive elements are assumed: for the buck converter: Vin = 20 V, L = 100 lH, RL = 0.1 X, C = 470 lF and for the boost converter: Vin = 12 V, L = 50 lH, RL = 1 mX, C = 470 lF. In the transient analyses and in the new method the following parameters of the model of the transistor IRF 840 are made use of: Level = 3, j = 0.2, tox = 100 nm, Uo = 600 cm2 V1s1, U = 0.6 V, RS = 6.382 mX, Kp = 20.85 lA/V2, W = 0.68 m, L = 2 lm, Vto = 3.879 V, Rd = 0.6703 X, TT = 710 ns, Rds = 2.222 MX, Cbd = Mj = 0.5, Fc = 0.5, Cgso = 1.625 nF/m, 1.415 nF, Pb = 0.8 V, Cgdo = 133.4 pF/m, Rg = 0.6038 X, Is = 56 pA and the diode BY229: Is = 53.4 pA, N = 1.185, RS = 0.12 X, trs1 = 3103 K1, Ikf = 3.5 mA, Cjo = 325 pF, M = 0.3333, Vj = 0.75 V, Fc = 0.5, Isr = 100 pA, Nr = 2, TT = 145 ns. The values of thermal resistance of the diode and the transistor are assumed as equal to 20 K/W, which accounts for placing these elements on the small radiator. The switching period is assumed as equal to TS = 10 ls. In order to illustrate the influence of non-linearity of semiconductor devices on the characteristics of the considered converters, Figs. 4–7 present the analyses results of the considered converters for two different values of the high level of the voltage VH controlling the switching transistor gate, equal to 5 V and 15 V, respectively. In these figures solid lines designate the calculations

a

1 -1

VH = 5 V

-2

transient analysis

-3 -4

ΔTT [K]

Vout [V]

b

nonlinear average model linear average model

0

VH = 15 V

-5

200 160

nonlinear average model linear average model

140

transient analysis

180

Vin = -20 V

120

VH = 15 V

100 80

-6

BUCK

60

-7

R0 = 3 Ω Vin = -20 V

40

-8

BUCK R0 = 3 Ω

VH = 5 V

20

-9

0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

d

0.6

0.8

1

d

Fig. 4. Calculated dependences of the output voltage of the buck converter (a) and the internal temperature increase of the transistor over the ambient temperature (b) on the pulse duty factor.

a

0

BUCK d = 0.5 Vin = -20 V

-2

VH = 5 V

-4 -8 -10 -12

VH = 15 V nonlinear average model linear average model

-14 -16 -18

80

1

60

VH = 5 V

40 20

transient analysis

-20

VH = 15 V

100

η [%]

Vout [V]

-6

b 120

transient analysis

0

10

100

R0 [Ω]

1000

nonlinear average model linear average model

1

10

BUCK d = 0.5 Vin = -20 V

100

R0 [Ω]

Fig. 5. Calculated dependences of the output voltage (a) and watt–hour efficiency (b) of the buck converter on load resistance.

1000

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K. Górecki / Microelectronics Reliability 49 (2009) 431–437

a

17

BOOST R0 = 10 Ω Vin = 12 V

VH = 15 V

16 15

VH = 5 V

12 11 9

transient analysis

8 0

0.2

Vin = 12 V

60

VH = 15 V

50 40 20

nonlinear average model linear average model

10

transient analysis

30

nonlinear average model linear average model

10

VH = 5 V

70

14 13

BOOST R0 = 10 Ω

80

η [%]

Vout [V]

b 100 90

18

0

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

d

d

Fig. 6. Calculated dependences of the output voltage (a) and watt–hour efficiency (b) of the boost converter on the pulse duty factor.

50 45

Vout [V]

40

BOOST d = 0.5

nonlinear average model linear average model

Vin= 12 V

transient analysis

b

35

ΔTT [K]

a

VH = 15 V

30 25

200 160

BOOST d = 0.5

140

Vin= 12 V

180

120

VH = 15 V

100 80

VH = 5 V

60

20

nonlinear average model linear average model transient analysis

40

15

20

VH = 5 V

10 1

10

0

100

1000

1

R0 [Ω]

10

100

1000

R0 [Ω]

Fig. 7. Calculated dependences of the output voltage of the boost converter (a) and the internal temperature increase over the ambient temperature (b) on load resistance.

the differences between the calculation results obtained with the non-linear average model and the results of the transient analysis do not exceed 3%. In turn, as seen in Fig. 4b, good agreement is achieved between the calculations made with the non-linear average model and the transient analysis. On the other hand, the linear average model causes that we obtain the values of DTT smaller even by 30% than with the use of the transient analysis. Big values of internal temperature increases appear for small load resistances. Fig. 5 presents the dependence of the output voltage and the watt–hour efficiency of the buck converter on load resistance at the duty factor of the controlling pulses d = 0.5. At load resistance R0 > 50 X the converter operates in the DCM, which results in a considerable increase of the output voltage module and an increase of the watt–hour efficiency. The influence of the voltage VH on the dependences Vout(R0) and g(R0) is most often seen within the changes of resistance R0 from 3 to 50 X. An increase in the value of VH causes an increase in the voltage Vout module even by 16% and an increase in the converter watt–hour efficiency even by 4%. The differences between the calculated values of the output voltage obtained from the transient analysis and with the use of average models come up to 3% for the non-linear average model and up to 20% for the linear average model. On the other hand, in the case of the watt–hour efficiency these deviations are 5% and 7%, respectively. Fig. 6 shows the calculated dependences of the output voltage and the watt–hour efficiency of the boost converter on the duty factor for load resistance equal to R0 = 10 X. With the help of average models one obtains higher values of the voltage Vout and the watt–hour efficiency; yet, the differences are the biggest for d  0.45 and are up to 5% for the non-linear average model and up to 12% for the linear average model. It is worth noticing that for d > 0.6, as a result of a big voltage drop on the switched-on tran-

sistor switch, the converter stops raising the voltage. Within this range the output voltage is equal to the difference between the input voltage and voltages drops on the diode and resistance of the inductor. The converter watt–hour efficiency decreases from nearly 90% to merely 18%. Fig. 7 presents the dependence of the output voltage of the boost converter (Fig. 7a) and the internal temperature increase in the transistor over the ambient temperature (Fig. 7b) at the duty pulse factor d = 0.5. At load resistance R0 > 80 X the converter operates in the DCM, which leads to a considerable increase in the output voltage. The results of the analyses obtained with the non-linear average models are practically identical with the results of the transient analysis, whereas the results obtained with the linear average model are different from them even by a dozen or so percent. For the resistance R0 < 6 X, switching on and off does not occur in the system and the output voltage is smaller than the input voltage and is equal to about 11 V. In turn, as seen in Fig. 7b, good agreement is achieved between the calculation results obtained with the new method and the transient analysis. On the other hand, the use of the linear average model causes the values of the internal temperature of the transistor to be lower even by 20 K. Big values of the internal temperature increases are observed for small load resistances and high values of the duty factor. 5. Conclusions The presented results confirm the correctness of the formulated non-linear average electrothermal models of buck and boost converters operating both in the CCM and in the DCM. The advantage of these models is a possibility of assessing the internal temperatures of semiconductor devices, measuring the

K. Górecki / Microelectronics Reliability 49 (2009) 431–437

output voltage and the watt–hour efficiency of the converter with selfheating present and while taking into account non-linearity and inertia of semiconductor devices. In comparison with the transient analysis, the calculations time is considerably shorter, which for the non-linear average electrothermal model does not exceed 1 s, whereas for the electrothermal transient analysis it does not exceed a few hours. The advantage of the new model is especially visible within high values of the resistance R0. The non-linear average electrothermal model ensures obtaining such calculation results, which are much closer to the results of the electrothermal transient analysis than the method based on the average linear electrothermal models. Moreover, it requires only a few times longer calculation time. The presented non-linear average electrothermal models of dc– dc converters are universal, which means they can be made use of for the operation of the considered converters both in the continuous conduction mode and the discontinuous conduction mode and for any kind of switching semiconductor devices. Acknowledgments This work is supported by the Polish Ministry of Science and Higher Education in 2007–2008, as a Research Project No. N515 064 32/4331. References [1] Maksimovic D, Stankovic AM, Thottuvelil VJ, Verghese GC. Modeling and simulation of power electronic converters. P IEEE 2001;89(6):898–912. [2] Mohan N, Robbins WP, Undeland TM, Nilssen R, Mo O. Simulation of power electronic and motion control systems – An overview. P IEEE 1994;82: 1287–302. [3] Chung HS-H, Ioinovici A. Fast computer-aided simulation of switching power regulators based on progressive analysis of the switches’ state. IEEE T Power Electr 1994;9(2):206–12. [4] Hsiao CJ, Ridley RB, Naitoh H, Lee FC. Circuit-oriented discrete-time modeling and simulation of swiching converters. In: IEEE power electronics specialists conference pesc, 1987. p. 167-76. [5] Kelkar SS, Lee FCY. A fast time domain digital simulation technique for power converters: Application to a buck converter with feed forward compensation. IEEE T Power Electr 1986;PE-1:21–31. [6] Lu K, Halloran P, Brazil TJ. Simple method to simulate diode selfheating using SPICE. Electron Lett 1992;28(17):1667–9. [7] Pietrenko W, Janke W, Kazimierczuk MK. Application of semianalytical recursive convolution algorithms for large-signal time-domain simulation of switch-mode power converters. IEEE T Circuits-I 2001;48(10):1246–52. [8] Verghese GC, Elbuluk ME, Kassakian JG. A general approach to sampled-data modeling for power electronics circuits. IEEE T Power Electr 1986;PE-1:76–89. [9] Wong BKH, Chung H. Time-domain simulation of power electronics circuits using state variable quadratic extrapolations. IEEE T Circuits-I 1999;46(6): 751–6. [10] Wong RC, Owen HA, Wilson TG. An efficient algorithm for the time-domain simulation of regulated energy-storage dc-to-dc converters. IEEE T Power Electr 1987;2:154–68. [11] Basso ChP. Switch-mode power supply SPICE cookbook. New York: McGrawHill; 2001. [12] Ben-Yaakov S. Average simulation of PWM converters by direct implementation of behavioral relationships. In: Eighth annual applied power electronics conference and exposition - APEC ’93, San Diego; 1993. p. 510–16. [13] Ben-Yaakov S, Gaaton Z. Generic SPICE compatible model of current feedback in switch mode convertors. Electron Lett 1992;28(14):1356–8. [14] Borkowski A. Zasilanie urza˛dzen´ elektronicznych. Warszawa: WKiŁ; 1990 [in Polish].

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