AN3400 Application note

Transcription

1 Application note Analysis an simulation of a BJT complementary pair in a self-oscillating CFL solution ntrouction The steay-state oscillation of a novel zero-voltages switching (ZS) clampe-voltage (C) self-oscillating resonant riving system for compact fluorescent lamps (CFL), using a complementary pair of bipolar transistors on the half brige converter section, is analyze an simulate. One or more auxiliary winings are ae on the ballast inuctor in series with the lamp in orer to generate the perioic signal to supply the bases of the two complementary evices connecte to each other in a common emitter half brige topology. n fact, an LC network filters, with a resonant effect, the voltage generate by the seconary wining of the loa transformer, proucing a novel, perioic switching signal to accurately control the bases of the transistors. The two bipolars are supplie by a unitary control signal so it is not possible to turn on both evices at once because of their opposing base-emitter junction threshols. Self-oscillating operation is ivie into eight stages accoring to the variation over a perio of the riving voltage signal across the filter capacitor at the output of the transformer seconary winings. Stage-wise circuit analysis shows as the resonant filter action limits the lamp current an ominates the switching frequency of the ballast in steay-state working conition. The half brige of the power active evices generates a rectangular voltage waveform that rives an opportunely tune output circuit compose of a parallel loae RLC output circuit of which the R is the steay-state LAMP resistance. For the inverter self-oscillating conition, the switching frequency is etermine by all of the circuit elements relate to the oscillation frequency, such as the resonant tank, gas-ischarge lamp, riving circuit, an switching evices. Depening on the circuit esign, its oscillation frequency is typically aroun 35 to 48 khz an can be eventually increase by shortening the storage time of the bipolars. Cost benefits are achieve by the propose self-oscillating solution that allows to rive the CFL lamp eliminating the saturable core auxiliary transformer, place in the more traitional stanar solution, without sacrificing the performance or reucing the expecte life time of the lamps. n this paper, general structure an self-oscillating principle are iscusse an verifie by laboratory experiment, while analytical results are valiate by mathematical simulation using the Matlab tool. November 20 Doc D Rev /78

2 Contents AN3400 Contents Circuit escription Startup phase Bipolar transistor operating moes Principle of self-oscillating operation Steay-state stage-wise circuit analysis Driving network moeling NPN conuction phase moeling Moeling of re-circulating phase preliminary to the NPN conuction time phase NPN storage time phase moeling Moeling of the ea time after the NPN storage time phase Moeling of the re-circulating phase preliminary to the PNP conuction time phase PNP conuction phase moeling PNP storage time phase moeling Moeling of the ea time after the PNP storage time phase Experimental results Applicative parameters Simulation results with the Matlab tool Simulative parameters /78 Doc D Rev

3 Contents 7 Conclusions References Revision history Doc D Rev 3/78

4 List of figures AN3400 List of figures Figure. Complementary pair circuital topology Figure 2. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN base current (bnpn) an PNP base current (bpnp) Figure 3. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN collector current (pn) an PNP collector current (np) Figure 4. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN collector-emitter voltage (CEnpn) an PNP emitter-collector voltage (Enp) Figure 5. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN base-emitter voltage (BEnpn) an PNP emitter-base voltage(ebpnp) Figure 6. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN base series capacitor voltage (pn) an PNP base series capacitor voltage (np) Figure 7. First stage: first re-circulating phase before the NPN conuction time Figure 8. First stage: secon re-circulating phase before the NPN conuction time Figure 9. Secon stage: NPN conuction phase with filter capacitor charge Figure 0. Secon stage: NPN conuction phase with filter capacitor ischarge Figure. Thir stage: NPN storage phase Figure 2. Fourth stage: ea time after NPN conuction phase Figure 3. Fifth stage: first re-circulating phase before the PNP conuction time Figure 4. Fifth stage: first re-circulating phase before PNP conuction time with PNP evice working in reverse active region Figure 5. Fifth stage: secon re-circulating phase before the PNP conuction time Figure 6. Sixth stage: PNP conuction phase with filter capacitor charge Figure 7. Sixth stage: PNP conuction phase with filter capacitor ischarge Figure 8. Seventh stage: PNP storage phase Figure 9. Eighth stage: ea time phase after PNP conuction Figure 20. Driving network moeling for the NPN bipolar conuction phase simulation Figure 2. Driving network moeling for the simulation of the first re-circulating phase preliminary to the NPN bipolar conuction time phase Figure 22. Driving network moeling for the simulation of the secon re-circulating phase preliminary to the NPN bipolar conuction time phase Figure 23. Driving network moeling for the NPN bipolar storage time phase simulation Figure 24. ariability of the extractive current negative peak Bnpn-off versus variability of the integer n with storage time fixe Figure 25. Driving network moeling for the simulation of the ea time phase after the NPN bipolar storage time phase Figure 26. Driving network moeling for the simulation of the first re-circulating phase preliminary to the PNP bipolar conuction time phase Figure 27. Driving network moeling for the simulation of the secon re-circulating phase preliminary to the PNP bipolar conuction time phase Figure 28. Driving network moeling for the PNP bipolar conuction time phase simulation Figure 29. Driving network moeling for the PNP bipolar storage time phase simulation Figure 30. Driving network moeling for the simulation of the ea time after the PNP storage time phase Figure 3. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current (B), collector current ( C ), base-emitter voltage ( BE ) an collector-emitter voltage ( CE ) signals acquire Figure 32. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current ( B ), collector current ( C ), voltage on base series capacitor C n ( CN ) an 4/78 Doc D Rev

5 List of figures voltage on base series capacitor C p ( CP ) signals acquire Figure 33. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current ( B ), collector current ( C ), voltage on filter capacitor C2 (C) an voltage on riving resistance R2 (RS) signals acquire Figure 34. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current (B), collector current (C), voltage on base series capacitor C3 (CN) an voltage on breakown resistance R4 (RBN) signals acquire Figure 35. STX83003 (NPN) bipolar transistor uring turn-on particular in steay-state operation with 230 input voltage: base current ( B ), collector current ( C ) an collector-emitter voltage ( CE ) signals acquire Figure 36. STX83003 (NPN) bipolar transistor uring turn-off particular in steay-state operation with 230 input voltage: base current ( B ), collector current ( C ) an collector-emitter voltage ( CE ) signals acquire Figure 37. STX93003 (PNP) bipolar transistor uring turn-on particular in steay-state operation with 230 input voltage: base current ( B2 ), collector current ( C2 ) an emitter-collector voltage ( CE2 ) signals acquire Figure 38. STX93003 (PNP) bipolar transistor uring turn-off particular in steay-state operation with 230 input voltage: base current ( B2 ), collector current ( C2 ) an emitter-collector voltage ( CE2 ) signals acquire Figure 39. Current source signal s moeling for the simulation on the re-circulating phase before the NPN conuction phase Figure 40. Re-circulating phase preliminary to the NPN conuction time phase Figure 4. NPN conuction time phase Figure 42. NPN storage time phase Figure 43. Dea time phase after the NPN storage time phase Figure 44. Re-circulating phase preliminary to the PNP conuction time phase Figure 45. PNP conuction time phase Figure 46. PNP storage time phase Figure 47. Dea time phase after the PNP storage time phase Figure 48. Driving network simulation results: filter capacitor voltage signal ( s ), NPN base series capacitor voltage ( ) an PNP base series capacitor voltage ( ) signals simulate. 73 Figure 49. Driving network simulation results: NPN base current ( bn ) an PNP base current ( bp ) signals simulate Figure 50. Driving network simulation results: NPN base series capacitor current ( ) an PNP base series capacitor current ( ) signals simulate Doc D Rev 5/78

6 Circuit escription AN3400 Circuit escription The circuit comprises a fluorescent lamp connecte to a ballast inuctance accoring to a voltage fe topology with a single voltage resonant riving system supplying the bases of two complementary bipolar transistors connecte to each other in a common emitter half brige topology. Figure epicts the complementary pair CFL boar electrical schematic. Figure. Complementary pair circuital topology D D2 L R2 R3 C3 Q D5 C7 Rfuse C TB C2 R4 TA C6 LAMP D3 D4 C4 Q2 D6 R5 C5 C8 AM09789v Base on the previous figure, the current source that flows through the loa transformer TA, is fe back through a seconary wining of the current transformer itself an, upon filtering operation of the LC network (L-C2), is converte into a base current suitable for riving the bipolars in saturation state. Therefore, this source provies the excitation to the bases of the transistors, so that the converter oscillation is perpetuate by its own regenerative feeback means. Few aitional turns (aroun -3 turns) on the ballast transformer can be ae as seconary winings since the resonant effect assures that the voltage across the filter capacitor C2 results higher than that impose by the transformer seconary (see References 2). Capacitors C3-C4 in series to the base, together with the magnetizing inuctance of the transformer an the resonant effect of the LC filter, provie the require phase shift to maintain the oscillation conition. Soft-switching conition of the evices is achieve using the external snubber capacitor C5 inserte between the emitter an collector of the PNP bipolar Q2 to control precisely the collector-emitter voltage rise time uration an reach negligible switching losses uring the turn-off switching for both bipolars. The L-C2 network inserte at the output of the transformer seconary acts as a sinusoial voltage generator across the C2 capacitor itself to supply the R-C network (R2-C3-C4) connecte in series to the base of the transistors. All the system incluing the LC filter, RC riving network an RLC series loa network impeance reflecte from the primary sie back to the seconary one of the ballast transformer, is resonant at the half brige converter section working frequency while the single LC stage has the function of filtering the funamental component of the voltage signal across the transformer primary opportunely lowere by the transformation ratio. n fact, the LC network shoul ieally be esigne in orer to have a cut-off frequency inclue between the first an the secon harmonic component of the signal coming from the seconary sie to filter in orer to attenuate all the signal frequencies from the secon one. Therefore, the half brige section 6/78 Doc D Rev

7 Circuit escription working frequency an, consequently, the lamp power are influence by the sizing of both L an C2 components. Capacitor C2 has also the function of putting the base current in phase with the collector current. Resistance R2 of the series R-C network has the function of regulating the current level to provie to the bases of both transistors acting on the power lamp. n the propose riving solution, the two capacitors C3 an C4 are charge with opposite polarity to each other an assuming the same polarity of the base-emitter voltage of the relevant bipolar transistor. n the regular functionality, when a capacitor goes through a charging phase, before or uring the conuction time of the relevant bipolar, so the opposite capacitor is ischarging contemporarily with a long constant time ue to the series with the breakown resistance R4. So, this resistance R4, inserte between the bases of the two transistors, creating an alternative path for the ischarge of each capacitor, also avois that the oscillation is blocke uring startup phase when one of the two capacitors is fully charge. Regulating the ischarge time of each capacitor, this resistance allows to fine tune the working frequency an consequently the power supplie to the lamp. Two capacitors C3 an C4 are not totally ischarge uring a forcing voltage source perio but maintain always a resiual charge passing from the ischarging to the charging transient. Therefore, current across the breakown resistance R4 flows always in the same irection going from the PNP to the NPN base because the two capacitors maintain always the same polarity uring a perio of the voltage signal across the C2 filter capacitor. A breakown resistance with a too high value can increase the working frequency but also can bring the reverse biase base-emitter junction of the inactive evice in breakown conition. An vice versa, a too low value of the breakown resistance coul cause re-conuction peaks in the PNP transistor at the high temperature when the emitter-collector voltage reaches the maximum value. n particular only the PNP evice can show the phenomena above mentione because it has a lower base-emitter breakown voltage an it has a higher low current h fe than the NPN one. Setting the resistance R4 in orer to guarantee a BEoff value in the range -4 to -7, it is possible to avoi breakown phenomena for the base-emitter junctions of the bipolars an, at the same time, anomalous re-conuction effects ue to unexpecte isturbances. Doc D Rev 7/78

8 Startup phase AN Startup phase The two bipolar transistors shoul have an appropriately high gain value in orer to guarantee the correct ignition of the lamp uring the startup phase. Compact fluorescent lamps usually require about 600 as peak voltage to strike the arc. Once the arc is establishe about 00 are enough to sustain it while, electrically, the resistance of the lamp falls from about one mega ohm own to a few hunre ohm (see References ). Furthermore, the value of the loa inuctance L of the ballast must be chosen so that it oes not saturate at the operative current of the lamp, even at high temperatures. n fact the lamp is ignite by generating an overvoltage across the capacitor C6 in parallel to the tube, through the circuit forme by the series of the ballast transformer primary inuctance an C6-C7-C8 capacitors. At startup the lamp is an open circuit an the C6 capacitor imposes the resonant frequency of the circuit as C6 is much smaller than the C7-C8 ones. The impose overvoltage is high enough to ionize almost instantaneously the gas in the lamp. Once the lamp is lighte, the capacitor C6 is short-circuite by the lamp itself an the natural frequency is etermine mainly by the capacitors C7-C8 charge/ischarge through the DC-AC converter. The startup network is represente by the only resistor R3, connecte between the collector an the base of the high sie transistor, since the capacitors in series with the bases act as a high impeance elements uring the first instant of the startup. 8/78 Doc D Rev

9 Bipolar transistor operating moes 3 Bipolar transistor operating moes Bipolars switching in the resonant circuit have three ifferent operating moes in normal working conitions: re-circulating, conuction an transition phases. During the re-circulating phase, the transistor isn't yet in conuction state an passes from a first inactive state in which its B-C junction ioe is activate to allow the re-circulation of loa inuctor emagnetization current LP share with the external ioe in anti-parallel to the evice itself, to a secon inactive state in which the external ioe is turne off but both of the two B-E an B-C junctions are forwar biase in orer to complete the resiual emagnetization of the Lp inuctor until the forcing re-circulating current LP, that imposes the negative collector current, reaches the zero value. n conuction moe, the two B-E an B-C junctions of the transistor continue to be forwar biase an the bipolar transistor conucts a positive magnetization current for the inuctive loa. The turn-off mechanism occurs in three stages: storage time, or time spent to extract the storage in excess an recombine the charge store on the base, current fall time, in which the collector current passes from 90% to 0% of its maximum value, an rise time of the collector-emitter voltage. The transition moe occurs uring the voltage rise time an represents a sort of ea time phase in which both bipolars are substantially inactive since only the B-C an B-E junction capacitances of both evices are intereste in being charge/ischarge in reverse bias. This phase anticipates the re-circulating phase preliminary to the conuction phase of the complementary evice. During the turn-off process, the ynamic operation point of the transistor moves through three ifferent operating regions on the current-voltage C - CE characteristic: har saturation region, quasi saturation region an active region. Har saturation region: at the first instance of turn-off switching time, the base-collector junction is forwar biase an the base region an collector rift region are both in high-level injection conition. An excess carrier istribution fills the collector rift region an the storage time is just the time require to remove/extract this charge store in excess. Also the contribution of charge ecay ue to recombination, epening on the minority-carrier lifetime, influences this phase. During the storage time process the base-emitter voltage oes not change immeiately from its forwar bias value BESAT, ue to the excess minority carriers store in the base region, an also the collector-emitter voltage remains constant. The extent of storage time is epenent not only on the amount of excess charges remaining in the collector rift region but also on the external riving. Excess minority carriers are remove from the base region an base-collector junction at a constant rate etermine by the negative base rive voltage, as well as the base rive resistance, an proportional to the reverse base current Boff slope. So the excess charge at the base-collector junction begins to reuce ue to charge removal to make up for the reverse base current an the collector current continues to increase. Quasi saturation region: after the storage time, the forwar biase base-collector junction is out of high-level injection an the remaining charges store in the base become insufficient to support the transistor in the har saturation region. Therefore, at this point the transistor enters quasi saturation region in which the epletion region begins to expan while a voltage appears across it an the collector-emitter voltage starts rising with a small slope. After having remove the charges in the rift region holing the bipolar in the quasi saturation region, the transistor enters the active region. Active region: crossing the active region, the charges store in the base region are insufficient to support the full negative base current so the base-emitter voltage starts falling negatively an the negative base current starts reucing. Corresponingly, the store base charges can no longer support the full loa current through the collector so that also the collector current ecays exponentially resulting in current fall time require Doc D Rev 9/78

10 Bipolar transistor operating moes AN3400 to remove remaining store charge in base. Both of the slopes ictate by base current an collector current simultaneously ecay with time ue to recombination. Collector-emitter voltage increases rapily towars the rectifie voltage upper rail CC until it is exceee in orer to turn-on the anti-parallel ioe of the complementary evice at the en of the rise time interval. The increasing voltage is supporte across the collector rift region while the expaning epletion region at the base-collector junction sweeps out the excess charge until it totally sustains the bus voltage. To reuce the stress on the evices, an hence the switching losses, the half brige converter switches uner zero voltage (ZS) conitions, in which turn-on losses are absent an a favorable turn-off trajectory of power transistors can be ensure by the use of a snubber capacitor. n fact, contrary to the har switching ynamic, in which the voltage rise was a function of the amount of charge in the rift region after the storage phase, with the soft-switching ynamic the rate of voltage rise uring turn-off is controlle by the snubber capacitor connecte across the evice. While the voltage continues to raise, the collector current ecays to zero an the loa current flows into the snubber capacitor until the evice voltage reaches the bus voltage. Therefore, even though turn-off oes not occur at exactly zero volts, the stresses on the evice are much reuce compare to har-switching moe. When the collector current reaches the zero value, the bipolar turn-off process is complete an the loa current starts to freewheel through the anti-parallel ioe of the complementary evice. 0/78 Doc D Rev

11 Principle of self-oscillating operation 4 Principle of self-oscillating operation The current through the transformer primary TA an the resonant network at the half brige output consists of two portions of waveforms, one when the loa inuctor is clampe to the upper rail at the en in common with the emitter noe of the bipolars by the NPN bipolar Q an its anti parallel ioe D5, an the other one when it is clampe by the PNP bipolar Q2 an its anti parallel ioe D6 to the lower rail. The intervals, when the snubber capacitor C5 is charging/ischarging while the emitter common noe ramps between the rails, are generally short compare to the conuction perios. To achieve the ZS working conition, the bipolar transistor is turne on at the en of the re-circulating phase of the resiual emagnetization current from the loa transformer. Figure 2, 3, 4, 5 an 6, escribe the main conceptual steay-state signals relate to the theoretical working principle of the propose riving system. These figures highlight the ifferent operating phases of the two complementary bipolar transistors uring the steay-state normal working conitions: Re-circulating time phases with a time length of t A an t C respectively for the NPN an PNP transistors Conuction time phases inicate with NPN-B ontime an PNP-B ontime Storage time phases inicate with NPN-B storagetime an PNP-B storagetime Dea time phases with the time length of t B an t D respectively for the NPN an PNP transistors. As previously escribe, the ea time phase refers to the transient occurring uring the voltage rise time an in which both of the bipolars are substantially inactive since only the B-E an B-C junction capacitances on both evices are intereste in being charge/ischarge in reverse bias. Moreover, from this point on an for the whole steay-state analysis subsequently explaine, it is assume to name, for the sake of simplicity, as storage time the interval incluing both the effective storage time an fall time of the evice. Figure 2. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN base current ( bnpn ) an PNP base current ( bpnp ) AM09793v Doc D Rev /78

12 Principle of self-oscillating operation AN3400 Figure 3. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN collector current ( pn ) an PNP collector current ( np ) AM09794v Figure 4. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN collector-emitter voltage ( CEnpn ) an PNP emitter-collector voltage ( Enp ) AM09795v 2/78 Doc D Rev

13 Principle of self-oscillating operation Figure 5. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN base-emitter voltage ( BEnpn ) an PNP emitter-base voltage( EBpnp ) AM09796v Figure 6. Steay-state waveforms: filter capacitor voltage signal ((t)), NPN base series capacitor voltage ( pn ) an PNP base series capacitor voltage ( np ) AM09797v n Figure 2 the filter capacitor voltage (t) an the base currents bnpn an bpnp of the two bipolars uring a perio of the voltage signal filtere are shown. Two ifferent signal threshols fix respectively the turn-on an turn-off instants of the evices uring their working operation perio. n particular, in the previous figure it is evient that, starting from the riving source voltage, the turn-on of the NPN transistor is possible only when the (t) is higher than the pnon-char + BEsatnpn value an similarly the turn-on of the PNP transistor is possible only when the (t) is lower than the - npon-char - EBsatpnp value. On the contrary, the turn-off of the NPN transistor is possible only when the (t) is lower than the pnon-ischar + BEsatnpn value an, similarly, the turn-off of the PNP transistor is possible only when the (t) is higher than the - npon-ischar - EBsatpnp value. Figure 3 shows the filter capacitor voltage (t) an the collector currents pn an np of the two bipolars uring a perio of the voltage signal filtere. t can be seen as the collector currents pass through the zero value at the en of the re-circulating phases in which the whole resiual external emagnetization current, impose by the loa inuctor, has Doc D Rev 3/78

14 Principle of self-oscillating operation AN3400 elapse. nstants of turn-on switching are at t 2 (or t 4 ) an t 8 times respectively for the NPN an PNP evices. Figure 4 shows the filter capacitor voltage (t) an the collector-emitter voltage CEnpn signal an emitter-collector voltage signal Enp respectively of the NPN an PNP bipolars uring a perio of the voltage signal filtere. n this image the waveforms are not reporte in scale since the voltage signals CEnpn an Enp assume far higher levels compare to the (t) signal ones. Figure 5 shows the filter capacitor voltage (t) an base-emitter voltage signal BEnpn an emitter-base voltage signal EBpnp respectively of the NPN an PNP bipolars uring a perio of the voltage signal filtere. Figure 6 shows the filter capacitor voltage (t) an the voltage signals across the capacitors pn an np in series, respectively, to the bases of the NPN an PNP bipolars uring a perio of the voltage signal filtere. t is possible to see as the ifferent signal threshols impose the initial instants of the charging an ischarging phases for the capacitors in series to the bases. Observing the waveforms attache, which escribe the resonant riving functionality, it is clearly possible to istinguish the ifferent working operation phases of the two evices uring a switching perio. n particular, when (t)> pnon-char + BEsatnpn with ( () t ) > 0, NPN conuction time begins an when (t) ecreasing positively reaches the pnon-ischar + BEsatnpn value (that is (t)< pnon-ischar + BEsatnpn with ( () t ) < 0 ), the NPN storage time phase starts. After the fall time perio, NPN bipolar can no longer sustain its collector current so the voltage of the mipoint in common with the PNP emitter changes. At this time, the short transition moe ('ea time' after the NPN conuction time has elapse) begins (for 0<(t)< pnoff-char + BEsatnpn with ( () t ) < 0 ), where both bipolars are off an the snubber capacitance carries the loa resonant inuctor current while the B-C junction capacitances of the evices are respectively charge (NPN) an ischarge (PNP). Similarly, when (t)<- npon-char - EBsatpnp with ( () t ) < 0, PNP conuction time begins an when ecreasing (t), in moule, negatively reaches the - npon-ischar - EBsatpnp value (that is (t)>- npon-ischar - EBsatpnp with ( () t ) > 0 ), the PNP storage time phase starts. Short transition moe after the PNP conuction time ('ea time' after the PNP conuction time has elapse) begins when - npoff-char - EBsatpnp <(t)<0 with ( () t ) > 0 in which the B-C junction capacitances of the PNP an NPN evices are respectively charge an ischarge. Re-circulating phases elapse respectively for 0<(t)< pnon-char + BEsatnpn an ( () t ) > 0 with the NPN evice turne on at t 2 (or t 4 ) instant an for - npon-char - EBsatpnp <(t)<0 an ( () t ) < 0 with the PNP evice turne on at t 8 instant. 4/78 Doc D Rev

15 Steay-state stage-wise circuit analysis 5 Steay-state stage-wise circuit analysis This section provies a comprehensive escription of the resonant filter working operation using a stage-wise circuit analysis ivie into eight phases in accorance with the variation over a perio of the voltage signal across the filter capacitor (t) at the output of the transformer seconary winings. The ynamic behavior of the resonant riving system can be easily escribe by applying, in the time omain, Kirchhoff's voltage law on the loops containing the filter capacitor C2 an the R-C network (R2-R4-C3-C4) connecte in series to the bases of the transistors an Kirchhoff's current law on the noes with the branches of the base series capacitors connecte to it. The mathematical moel evelope is useful for the successive simulation in Matlab environment. Stage : t 0 <t<t 2 an t 2 <t<t 4 : (re-circulating phase before the NPN conuction time: 0<(t)< pnon-char + BEsatnpn with ( () t ) > 0 ) The re-circulating phase is referre to all the emagnetization transient of the loa inuctor Lp elapsing until its current LP reaches the zero value at the en of this stage. This phase, in which both Q-Q2 switches are inactive, can be subivie into a further two sub-phases istinguishe by the following: First re-circulating phase before the NPN conuction time (t 0 <t<t an t 2 <t<t 3 ): Both Q-Q2 switches are turne off an the anti-parallel ioe D of the NPN transistor conucts, sharing with the forwar biase B-C NPN bipolar junction, the resiual emagnetization current LP from the loa inuctor Lp. oltage across the filter capacitor C increases an charges the capacitor C n in series to the not yet forwar biase NPN base-emitter junction, therefore assuming the same polarity of the relevant BEnpn-on, whereas the capacitor C p begins to ischarge through the breakown resistance R b. n this re-circulating stage, currents an voltages on the riving network are represente by the information in Figure 7. Figure 7. First stage: first re-circulating phase before the NPN conuction time NPN base-collector ioe conuction DD L Rs Rs Rs BB BB BEn D Q D Ls c C Rb D Rb LP Lp First re-circulating phase before NPN conuction Rb BEp Q2 D2 Cs Cs GND AM09798v Doc D Rev 5/78

16 Steay-state stage-wise circuit analysis AN3400 Accoring to the agreement in Figure 7, the following equations are vali in the riving section uring the first re-circulating stage for t 0 <t<t an t 2 <t<t 3 : System of equations () t = C () t BEn () t = B() t D () t < BEn on where B() t > B on an D () t > D on () t t ora + () t C n t0 () t t ora () t C t0 = where ( t ) p or a = 0 = where ( t ) () t = C ( () t ) n t = t = t + t ( ) or a = 0 () () t () () () = LP () t D() t RS () t () t + () t = () t = () t () t with () t = () t 0 B = Rb BEp BEn BEp EBp > in which B an B are, respectively, the current signal an voltage signal of the NPN bipolar forwar conucting B-C junction ioe. During this phase, in general, BEn (t) voltage is so that 0< BEn (t)< BEn-on for the NPN bipolar B-E junction ioe, but if it were BEn (t)<0 the evice woul work in reverse active region as it is B (t)>0. Neglecting the reverse recovery time of the anti-parallel ioe D, the subsequent phase is given by the following secon re-circulating phase before the NPN conuction time. Secon re-circulating phase before the NPN conuction time (t <t<t 2 an t 3 <t<t 4 ): n this phase the NPN bipolar transistor acts as two uncouple/inepenent ioes. n fact, the external ioe D is turne off an the NPN evice continues to be inactive but, contrary to the previous first re-circulating phase, both of its two B-E an B-C junctions are forwar biase in orer to complete the resiual emagnetization of the Lp inuctor until the forcing re-circulating current LP, that imposes the negative (extractive) collector current, reaches the zero value. oltage across the filter capacitor C continues to increase an charge the capacitor in series to the forwar biase NPN base-emitter junction whereas the capacitor continues to ischarge through the breakown resistance Rb. When the inuctive resonant current reverses, after t 2 instant, the successive phase starts an the NPN evice Q conucts carrying a positive (coming in) collector current pn. n this secon re-circulating stage, currents an voltages on the riving network are represente by the information in Figure 8. 6/78 Doc D Rev

17 Steay-state stage-wise circuit analysis Figure 8. First stage: secon re-circulating phase before the NPN conuction time NPN base-collector ioe conuction Ls L c Rs Rs C Rs Rb Bnpn BEn Rb B DD pn D Q Enpn LP Lp Rb NPN base-emitter ioe conuction Secon re-circulating phase before NPN conuction BEp Q2 D2 Cs Cs GND AM09799v Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the secon re-circulating stage for t <t<t 2 an t 3 <t<t 4 : System of equations 2 () t = () t C BEn() t = BEn on B () t B on = an E() t = CEn() t < D on t2 () t orb + () t C t = where ( t ) t2 () t orb () t C t p n orb = = where ( t ) t + () t = C ( () t ) n ( ) orb = () t = () t () t = () () t = pn() t + Enpn() t = LP() t + Enpn () t RS() t () t + () t = () t = () t () t with () t = () t 0 Bnpn = Rb BEp BEn BEp EBp > Stage 2: t 2 <t<t 4 : (NPN conuction time: (t)> pnon-char + BEsatnpn uring filter capacitor C charging phase for t 2 <t<t 3 with ( () t ) > 0 an (t)> pnon-ischar + BEsatnpn uring filter capacitor C ischarging phase for t 3 <t<t 4 with ( () t ) < 0 ). From the t 2 instant (t) voltage is higher than the pnon-char + BEsatnpn value an increases up to reach its maximum value at t 3. NPN transistor Q is turne on uner zero voltage switching (ZS) an the current in the resonant loa inuctor increases. Capacitor C n continues to be charge up by the source (t), whereas capacitor C p ischarges through the breakown resistance R b inserte between the bases of the Doc D Rev 7/78

18 Steay-state stage-wise circuit analysis AN3400 two transistors. After t 3 instant voltage across the filter capacitor C ecreases from its maximum value while the capacitor C n continues to charge until it reaches its maximum value an the capacitor C p to ischarge maintaining a share of conuction current on the base Bn of the NPN bipolar Q. During this stage, the base current Bn is the superposition of two components, one generate by the charging transient an the other one generate by the C p ischarging transient through the breakown resistance R b. n this way, it is possible to see the voltage signal, as a voltage source when the NPN evice Q is conucting. n this stage, currents an voltages on the riving network are represente by the information in Figure 9 an 0. Figure 9. Secon stage: NPN conuction phase with filter capacitor charge DD L Rs Rs Rs pn CEn Bnpn Q D BEn Ls c C Rb Enpn Rb LP Lp Filter capacitor charge NPN conuction phase Rb BEp D2 Q2 Cs Cs GND AM09800v Figure 0. Secon stage: NPN conuction phase with filter capacitor ischarge DD L Rs Rs Rs pn CEn Bnpn Q D BEn Ls c C Rb Enpn Rb LP Lp Filter capacitor ischarge NPN conuction phase Rb BEp D2 Q2 Cs Cs GND AM0980v 8/78 Doc D Rev

19 Steay-state stage-wise circuit analysis Accoring to the agreement of the previous images, the following equations are vali in the riving section uring the NPN conuction stage for t 2 <t<t 4 : System of equations 3 () t = () t C BEn () t = BEn sat CEn () t = CEn sat t 4 () t = oc + () t C t 2 n t4 () t = oc () t C t 2 n = C p t t = t + () t = C ( () t ) p ( ) where ( ) where ( ) oc = pnon char = t 2 oc = t2 () t () on () () () t = Enpn () t LP () t RS () t where LP () t = pn () t () t + () t = Rb() t = BEp () t BEn () t with BEp () t = EBp() t > 0 () t = () t + () t () t Bnpn = C RS + BEn where BEn-sat is the voltage of the forwar conucting B-E junction ioe of the NPN bipolar an Bnpn-on is the base current for the NPN bipolar working in saturation state. Stage 3: t 4 <t<t 5 : (NPN storage time: pnoff-char + BEsatnpn <(t)< pnon-ischar + BEsatnpn with ( () t ) < 0 ) The ecreasing voltage (t) across the filter capacitor C equals the pnon-ischar + BEsatnpn value at t 4 instant. At this point the voltage on the capacitor C n starts to ecline but the store charges of the NPN bipolar base keep the NPN conucting with a negative base current (extraction base current Bnpn-off ) while the C p capacitor maintains its ischarging phase. After the store excess charges have isappeare, the operating point of the NPN enters its active region an CEnpn voltage begins to increase while the collector current pn rops. Therefore, the collector current pn fall time phase is inclue in this state until the NPN bipolar Q is graually soft switche on turn-off. n this stage, currents an voltages on the riving network are represente by the information in Figure. Doc D Rev 9/78

20 Steay-state stage-wise circuit analysis AN3400 Figure. Thir stage: NPN storage phase DD L Rs Rs Rs pn CEn Bnpn Q D BEn Ls c C Rb Enpn Rb LP Lp Rb NPN storage phase BEp D2 Q2 Cs Cs GND AM09802v Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the NPN storage time for t 4 <t<t 5 : System of equations 4 () t = () t C BEn () t = BEn sat CEn () t CEn sat t5 () t os () t C t4 = where ( t ) t5 () t os () t C t4 p n os = 4 = where ( t ) () t = C ( () t ) n = t = t ( ) () t () t os = 4 off() () () t = LP () t Enpn() t RS() t where LP () t = pn () t () t + () t = Rb() t = BEp () t BEn () t with BEp () t = EBp () t > 0 () t = () t + () t () t Bnpn = C RS + BEn where Bnpn-off is the extracting base current uring the storage time of the NPN bipolar. Stage 4: t 5 <t<t 6 : (ea time after the NPN conuction phase: 0<(t)< pnoff-char + BEsatnpn with ( () t ) < 0 ) At t 5 instant, the loa inuctor current LP continues to ecay an the resonant current is commutate from the NPN bipolar Q to the snubber capacitor C s that ischarges. As a result, the voltage on the common emitter noe of the two switches ecreases linearly from the upper rail to the lower rail until the emitter voltage of the PNP transistor Q2 begins to go negative with respect to the lower rail (groun reference) at t 6 an the anti-parallel ioe D2 of the PNP transistor starts to conuct beginning the next fifth stage. At the en of this ea time stage, the voltage on the capacitor C p is reuce 20/78 Doc D Rev

21 Steay-state stage-wise circuit analysis to the minimum value whereas the voltage on the capacitor C n keeps nearly constant. The final instant t 6, from which the capacitor current changes polarity, marks the change of the (t) signal polarity an one-half of its perio. n this stage, currents an voltages on the riving network are represente by the information in Figure 2, in which the charging/ischarging effects of the reverse biase B-E junction capacitances, respectively, for the NPN an PNP evices, are neglecte an so it is possible to approximate t t an t t. Bnpn () CCBn () Bpnp () CB() Figure 2. Fourth stage: ea time after NPN conuction phase NPN base-collector junction capacitance charge Rs L Rs Rs CCBn CCBn EBn DD CEn Q D Ls c C Rb Rb Rb LP Lp Cs PNP collector-base junction capacitance ischarge CB BEp D2 Q2 Cs Cs Dea time phase after NPN conuction CB GND AM09803v Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the ea time after the NPN conuction phase for t 5 <t<t 6 : System of equations 5 () t = () t C CEn() t > CEn sat () t = EBn () t = ( DD CCBn () t CS() t ) > 0 BEn ( t) 0 () t = () t = () t () t > 0 () t 0 BEn < EBp BEp CB CS EBp < () t t6 o () t C n t5 () t t6 o () t C t5 RS CS = where ( t ) p o = 5 = where ( t ) () t = C ( () t ) n t + ( ) () t = () t () t = CCBn() CB () t = LP() t CS() t () t = Cs ( CS() t ) () t + () t = Rb() t = BEp () t EBn () t () t = () t + () t () t + C RS EBn o = 5 Doc D Rev 2/78

22 Steay-state stage-wise circuit analysis AN3400 where CCBn is the charging current of the NPN bipolar B-C junction capacitance an CB is the ischarging current of the PNP bipolar C-B junction capacitance. Stage 5: t 6 <t<t 8 : (re-circulating phase before the PNP conuction time: - npon-char - EBsatpnp < (t)<0 with ( () t ) < 0 ) As one for the re-circulating phase before the NPN conuction time (t 0 <t<t 2 ), this phase, in which both Q-Q2 switches are inactive, can be subivie into a further two sub-phases istinguishe by the following: First re-circulating phase before the PNP conuction time (t 6 <t<t 7 ): both Q-Q2 switches are turne off an the loa inuctor resiual magnetization current LP circulates freely through the PNP transistor anti-parallel ioe D2 an the irectly biase C-B junction of the PNP bipolar Q2. oltage across the filter capacitor C increases in the opposite polarity than the NPN turn-on phase an charges the capacitor C p in series to the not yet forwar biase base of the PNP, assuming the same polarity of the relevant EBonpnp, while the capacitor C n begins to ecrease its charge through the breakown resistance R b. n this re-circulating stage, currents an voltages on the riving network are represente by the following Figure 3. Figure 3. Fifth stage: first re-circulating phase before the PNP conuction time DD L Rs Rs Rs EBn D Q Ls c C Rb Rb Rb LP D2 Lp PNP collector-base ioe conuction First re-circulating phase before PNP conuction CBp CBp EBp GND D2 Cs Q2 D2 Cs AM09804v Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the first re-circulating stage for t 6 <t<t 7 : 22/78 Doc D Rev

23 Steay-state stage-wise circuit analysis System of equations 6 () t = C () t () t CBp () t D2() t < EBp on = where CBp () t > CBp onan D 2() t > D2 on t7 () t = or 2a () t C where or 2a = ( t6 ) t6 EBp n t7 () t or 2a + () t C t6 = where ( t ) () t = C ( () t ) n t + p ( ) () t = C () t p or2a = 6 () t = () () t = LP () t D2 () t RS() t () t + () t = () t = () t () t with () t = () t 0 CBp = Rb EBn EBp EBn BEn > where CBp an CBp are, respectively, the current an voltage signal of the forwar biase C-B junction ioe of the PNP bipolar. During this phase, in general, EBp (t) is so that 0< EBp (t)< EBp-on for the PNP bipolar E-B junction ioe but if it were EBp (t)<0 the evice woul work in reverse active region since it is CBp (t)>0. Therefore, in this last case, currents an voltages on the riving network are epicte by the information in Figure 4. Figure 4. Fifth stage: first re-circulating phase before PNP conuction time with PNP evice working in reverse active region DD L Rs Rs Rs EBn D Q Ls c C Rb Rb LP Lp PNP evice in reverse active region First re-circulating phase before PNP conuction Rb Bpnp Bpnp BEp CBpnp CBpnp D2 Epnp D2 Cs Q2 np D2 Cs GND AM09805v Doc D Rev 23/78

24 Steay-state stage-wise circuit analysis AN3400 Then, the relations below between the current an voltage signals woul be vali: Equation () t = () t () t = () t + () t = () t () t () t () t Bpnp np Epnp LP D2 Epnp = RS () t + () t = () t = () t () t Rb EBn + Neglecting the reverse recovery time of the anti-parallel ioe D2, the subsequent phase is given by the following reporte secon re-circulating phase before the PNP conuction time. Secon re-circulating phase before the PNP conuction time (t 7 <t<t 8 ): n this phase the PNP bipolar transistor acts as two uncouple/inepenent ioes. n fact, the external ioe D2 is turne off an the PNP evice continues to be inactive but, contrary to the previous first re-circulating phase, both of its two E-B an C-B junctions are forwar biase in orer to complete the resiual emagnetization of the L p inuctor until the forcing re-circulating current LP, that imposes the negative (coming in) collector current, reaches the zero value at the en of this stage. oltage across the filter capacitor C continues to increase an charge the capacitor C p in series to the forwar biase PNP emitter-base junction whereas the capacitor C n continues to ischarge through the breakown resistance R b. When the inuctive resonant current reverses, after t 8 instant, it is transferre naturally as a positive (extractive) collector current np to the PNP bipolar Q2 an the successive PNP conuction phase begins. n this stage, currents an voltages on the riving network are epicte by the information in Figure 5. BEp with () t = () t 0 an () t = () t 0 EBn BEn > BEp EBp > Figure 5. Fifth stage: secon re-circulating phase before the PNP conuction time DD L Rs Rs Rs EBn D Q Ls c C Rb Rb LP Lp PNP emitter-base ioe conuction Rb EBp Epnp PNP collector-base ioe conuction Secon re-circulating phase before PNP conuction Bpnp CBp CBp GND D2 Q2 np Cs Cs AM09806v 24/78 Doc D Rev

25 Steay-state stage-wise circuit analysis Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the secon re-circulating stage for t 7 <t<t 8 : System of equations 7 () t = () t C EBp() t = EBp on CBp () t CBp on = an CEp() t = E () t < D2 on t8 () t or2b () t C t7 = where ( t ) t8 () t or2b + () t C t7 n p or 2b = 7 = where ( t ) () t = C ( () t ) n t + ( ) () t = C () t p or2b = 7 () t = () () t = np () t + Epnp () t = LP() t + Enpn () t RS() t () t + () t = () t = () t () t with () t = () t 0 Bpnp = Rb EBn EBp EBn BEn > Stage 6: t 8 <t<t 0 : (PNP conuction time: (t)<- npon-char - EBsatpnp uring filter capacitor C charging phase for t 8 <t<t 9 with ( () t ) < 0 an (t)<- npon-ischar - EBsatpnp uring filter capacitor C ischarging phase for t 9 <t<t 0 with ( () t ) > 0 ). From the t 8 instant (t) voltage is lower than the - npon-char - EBsatpnp value an ecreases reaching its minimum value at t 9. PNP transistor Q2 turns on uner zero voltage switching an the current in the resonant loa inuctor increases in moule. Capacitor C p continues to be charge up by the source (t), whereas capacitor C n ischarges through the breakown resistance R b. At t 9 instant the voltage on filter capacitor C reaches its maximum inverse (negative) value. After this time (t) ecreases negatively in moule whereas the capacitor C p continues to charge until it reaches its maximum value an the capacitor C n to ischarge maintaining a share of conuction current on the base Bp of the PNP bipolar Q2. During this stage the base current Bp of the PNP bipolar transistor is the sum of the currents an in both capacitors. n this way, it is possible to consier the voltage signal, as a voltage source when the PNP evice Q2 is conucting. n this stage, currents an voltages on the riving network are represente by the information in Figure 6 an 7. Doc D Rev 25/78

26 Steay-state stage-wise circuit analysis AN3400 Figure 6. Sixth stage: PNP conuction phase with filter capacitor charge DD L Rs Rs Rs EBn D Q Ls c C Rb Rb Rb LP Epnp Lp Filter capacitor charge PNP conuction phase Bpnp EBp E D2 Q2 np Cs Cs GND AM09807v Figure 7. Sixth stage: PNP conuction phase with filter capacitor ischarge DD L Rs Rs Rs EBn D Q Ls c C Rb Rb Rb LP Epnp Lp Filter capacitor ischarge PNP conuction phase Bpnp EBp E D2 Q2 np Cs Cs GND AM09808v 26/78 Doc D Rev

27 Steay-state stage-wise circuit analysis Accoring to the agreement of the previous images, the following equations are vali in the riving section uring the conuction stage of the NPN for t 8 <t<t 0 : System of equations 8 () t = () t C EBp() t = EBp sat E() t = E sat t0 () t oc2 () t C t8 = where ( t ) t0 () t oc2 + () t C t8 p n oc 2 = 8 = where = ( t ) () t = C ( () t ) n = t t = t + ( ) oc 2 = npchar 8 () t () on () () () t = Epnp() t LP () t RS() t where LP () t = np() t () t + () t = Rb() t = EBn () t EBp () t with EBn () t = BEn() t > 0 () t = () t + () t () t Bpnp = C RS + EBp where EBp-on is the voltage of the forwar conucting E-B junction ioe of the PNP bipolar an Bpnp-on is the base current with PNP bipolar working in saturation state. Stage 7: t 0 <t<t : (PNP storage time: - npon-ischar - EBsatpnp <(t)<- npoff-char - EBsatpnp with ( () t ) > 0 ) oltage (t) across the filter capacitor C ecreases, in moule, negatively up to equal the - npon-ischar - EBsatpnp value at t 0 instant. At this point the capacitor voltage C p starts to ecline an the excess charges store in the PNP evice Q2 base keeps it conucting with a negative base current (coming-in base current Bpnp-off ) while the C n capacitor maintains its ischarging phase. After the store excess charges have isappeare the operating point of the PNP enters its active region an the Enp voltage begins to increase while the collector current np falls own. Therefore, the collector current np fall time phase is inclue in this state until the PNP bipolar Q2 is graually soft-switche on turn-off. n this stage, currents an voltages on the riving network are represente by the information in Figure 8. Doc D Rev 27/78

28 Steay-state stage-wise circuit analysis AN3400 Figure 8. Seventh stage: PNP storage phase DD L Rs Rs Rs EBn D Q Ls c C Rb Rb Rb LP Epnp Lp PNP storage phase Bpnp EBp E D2 Q2 np Cs Cs GND AM09809v Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the PNP storage time for t 0 <t<t : System of equations 9 () t = () t C EBp() t = EBp sat E() t E sat t () t os2 () t C t0 = where ( t ) t () t os2 () t C t0 p n os 2 = 0 = where ( t ) n = t = t () t = C ( () t ) ( ) os 2 = 0 () t () t off() () () t = LP () t Epnp () t RS() t where LP () t = np () t () t + () t = Rb() t = EBn () t EBp () t with EBn () t = BEn() t > 0 () t = () t + () t () t Bpnp = C RS + EBp where Bpnp-off is the coming-in current uring the storage time of the PNP bipolar. Stage 8: t <t<t 2 : (ea time after the PNP conuction phase: - npoff-char - EBsatpnp <(t)<0 with ( () t ) > 0 ) At t instant, the current LP in the loa inuctor has starte to ecay an the resonant current is commutate from the PNP bipolar Q2 to the snubber capacitor C s that charges. As a result, the voltage on the common emitter noe of the two switches increases linearly 28/78 Doc D Rev

29 Steay-state stage-wise circuit analysis from the lower rail to the upper rail until the emitter voltage of the NPN transistor Q begins to go positive with respect to the higher rail at t 2 an the anti-parallel ioe D of the NPN transistor Q begins to conuct, commencing again a new cycle starting from the first stage. At the en of this stage, the voltage on the capacitor is reuce to the minimum value whereas the voltage on the capacitor C p keeps nearly constant. The en of this 'ea time' (instant t 2 ), when the capacitor current changes polarity, marks the change of the (t) signal polarity an its complete evolution perio. n this stage, currents an voltages on the riving network are represente by the information in Figure 9 in which the charging/ischarging effects of the reverse biase B-E junction capacitances, respectively for the PNP an NPN evices, are neglecte an so is it possible to approximate t t an t t. Bnpn () CCBn () Bpnp () CB() Figure 9. Eighth stage: ea time phase after PNP conuction NPN base-collector junction capacitance ischarge Rs L Rs Rs CCBn CCBn EBn DD CEn Q D Ls c C Rb Rb Rb LP Lp Cs PNP collector-base junction capacitance charge CB BEp D2 Q2 Cs Cs Dea time phase after PNP conuction CB GND AM0980v Doc D Rev 29/78

30 Steay-state stage-wise circuit analysis AN3400 Accoring to the agreement of the previous image, the following equations are vali in the riving section uring the ea time after the PNP conuction phase for t <t<t 2 : System of equations 0 () t = () t C E () t > E sat BEn () t = EBn () t = ( DD CCBn () t CS () t ) > 0 BEn ( t) < 0 () t = () t = () t () t > 0 () t < 0 EBp BEp CB t2 () t = () t o 2 C t n t2 () t = o 2 () t C t () t = C ( () t ) n p t + ( ) () t = C p () t () t = CCBn () CB () t = LP () t CS () t () t = Cs ( () t ) CS () t + () t = Rb () t = BEp() t EBn() t () t = () t + () t () t RS CS + C RS BEp CS where CCBn is the ischarging current of the NPN bipolar B-C junction capacitance an CB is the charging current of the PNP bipolar C-B junction capacitance. The en of this stage represents the completion of one full conversion cycle an then the process repeats returning to the original operational stage in the complementary irection using the anti-parallel ioe D before the NPN bipolar transistor Q conuction phase. After the operating moes of the bipolar transistors on the switching converter section have been ientifie, the corresponing Matlab moels of the resonant riving network are create as explaine in the following. EBp where ( ) o 2 = t where ( ) o 2 = t 30/78 Doc D Rev

31 Driving network moeling 6 Driving network moeling n orer to simulate, in Matlab environment, the self-oscillating resonant network riving the bases of the complementary pair of bipolars in the eight ifferent steay-state working operation phases, a possible metho may be to combine, with each other, the circuital equations escribing the working conitions of the evices in each stage so as to obtain only one (ifferential) equation in the specific state variable or. For example, the following circuital scheme can be taken as a reference to give a moel of the NPN bipolar uring the conuction time in which the filter capacitor voltage is represente by an inepenent voltage generator s. Figure 20. Driving network moeling for the NPN bipolar conuction phase simulation DD s Rs Rs Rs Rb pn Bnpn Q BEn Enpn Rb LP s Rs Rs Rs Rb Bnpn-on Rb BEn-on Lp Rb Rb AM098v By combining, with each other, the circuital equations of the previous System of equations 3, which escribes the working conitions of the NPN bipolar uring the conuction time phase, it is possible to obtain the following secon orer linear ifferential equation with constant coefficients on the state variable vali on the (t 2,t 4 ) time interval: Equation 2 2 m 2 ( () t ) + h ( () t ) + k ( t) = + τ ( () t ) τ ( () t ) S BEn B S B BEn Doc D Rev 3/78

32 Driving network moeling AN3400 in which: System of equations S b 2 m = R R C with C = C n = ( ) h = C 2R S + R b k = τ = C B R b = BEn BEn on S () t = () t C filter Two ifferent conitions at the initial instant t 2, respectively, one for the voltage signal an the other for its function erivative, are neee to impose in orer to obtain one, an only one, solution of the Equation 2, therefore solving the following Cauchy problem: System of equations 2 m 2 2 ( () t ) + h ( () t ) + k ( t ) = + τ ( () t ) τ ( () t ) ( t 2 ) = γ ( () t ) = γ 2 t=t2 S BEn B S B BEn As a consequence of the LC filtering action on the voltage signal at the output of the seconary winings, s signal can be expresse, with a goo approximation, as a sinusoial voltage generator having the below reporte mathematical expression vali in the entire time interval (t 0, t 2 ): Equation 3 S () t = A sen( 2πft) with t 0 < t< t 2 Then, imposing BEn = BEsat =constant for the NPN evice base-emitter junction uring its conuction phase (that is ( () t ) 0 ), it is possible to change the System of equations 2 into the form with ω=2πf: BEn = 32/78 Doc D Rev

33 Driving network moeling System of equations h m k m A m m A ω BEn B ( () t ) + ( () t ) + () t = sen( ωt) + cos( ωt) ( t ) 2 = γ ( () t ) = γ2 t= t2 τ m Therefore, the secon orer linear ifferential equation with constant coefficients on the state variable takes the form of the physics law that rules the motion of a umpe harmonic oscillator riven by an externally applie forcing signal given by the superimposition of two inepenent sinusoial generators S2 τ = B A m Equation ω in which it is fixe: System of equations 4 S sen cos( ωt). After having re-written this equation in the form: = A m m ( ωt) 2 BEn B ( () t ) + 2ξω ( () t ) + ω ( t) = sen( ωt) + cos( ωt) 0 0 A m τ m an A ω 2 k ω 0 = m h h 2ξω 0 = ξ = = m 2ω m 2 0 h m k it is possible to obtain the general solution as the sum of a transient solution (associate homogeneous equation solution or unforce equation) that epens on initial conitions, an a steay-state solution (particular solution) that is inepenent by the initial conitions A but epens only on the amplitues of the riving signals (, an BEn ), riving m m frequency (ω), unampe angular frequency (ω 0 ), an the amping ratio (ξ). Steay-state solution is proportional to the riving forces with an inuce phase change of φ as follows: Equation 5 steaystate A m A ω m B () t = G( jω) sen( ωt + φ) + cos( ωt + φ) where G( jω ) = an ( ) 2ωω0ξ φ = G jω = arctg are respectively ( 2ω ) ( ) ωξ + ω0 ω ω ω0 moule an phase of the frequency response function G jω. t ( ) ω BEn 2 0 m Doc D Rev 33/78

34 Driving network moeling AN3400 Resolving the polynomial characteristic of Equation 4, the following can be obtaine: Equation 6 λ , ( ξω ) + 2ξω λ + ω = 0 λ = ξω ± ω 0 0 with Δ = istinct real roots an then the particular solution is on the forms: Equation R + R 0 0 > m 4 RS RB C S B ( ξω ) ω = ( h 4mk) = 0 transient λ t λ 2 t () t = c e + c e implying that λ an λ 2 are The general solution is the sum of steay-state solution Equation 5 an the transient solution Equation 7 reporte as follows: () () () t = steaystate t + transient t vali on the (t 2,t 4 ) time interval in which BEn =constant for hypothesis. n orer to free the signal calculation by the imposition of its erivative value at the initial instant t 2 on the Cauchy problem resolution referre to in the System of equations 2, the metho use to simulate in Matlab environment the riving network, in each working phase for the bipolar evices, is base on resolving a system of two first orer linear ifferential equations on the two state variables an combine with each other as explaine in the following. By simulating with the Matlab tool, the ODE45 solver algorithm has been use for the resolution of the initial value problem concerning the system of orinary ifferential equations relate to each operation phase of the opportunely moele bipolars uring the steay-state working conition. The ODE45 function uses the syntax expresse by the comman: [t,y]= oe45 (oefun, tspan, y0, options) nput arguments to the solver are liste as: oefun Function, expresse with a Matlab function-file, that evaluates the right sie of the ifferential equations. tspan ector specifying the interval of integration (t 0,t f ). The solver imposes the initial conitions at t 0 =t span () an integrates from t 0 to t f =t span (en). y0 ector of initial conitions. options Structure of optional parameters that change the efault integration properties. Output arguments for the solver are the following: t Column vector of time points. Y Solution array. Each row in y correspons to the solution at a time returne in the corresponing row of t. The solver integrates the system of ifferential equations y'=f(t,y) from time t 0 to t f with initial conitions y0. Function f=oefun(t,y), for a scalar t an a column vector y, must return a column vector f corresponing to f(t,y). Each row in the solution array Y correspons to a time returne in column vector T. The ODE45 solves with efault integration parameters 2 34/78 Doc D Rev

35 Driving network moeling replace by property values specifie in 'options', an argument create with the 'oeset' function having the following syntax: options = oeset ('RelTol',value,'AbsTol',value2) in which scalar relative error tolerance RelTol is set equal to e-5 (e-3 by efault) an the vector of absolute error tolerance AbsTol is set equal to e-8 (all components are e-6 by efault). Doc D Rev 35/78

36 NPN conuction phase moeling AN NPN conuction phase moeling n orer to aequately simulate the NPN conuction time phase through the Matlab tool, the function solution of the System of equations 2 can be equivalently obtaine, together with function, resolving the following system of two first orer linear ifferential equations on the two state variables an vali on the [t 2, t 4 ] time interval with the initial hypothesis of BEn =constant an C=C n =C p : System of equations 5 ( () t ) npncon = npncon npncon C + RS R b C Rb ( npncon () t ) = npncon npncon C R ( t2+ ) npnric ( t ) ( t ) ( t ) npncon = 2 npncon 2+ = npnric 2 b C R b S + C R S C R BEn S As seen, only two conitions, respectively, for the an voltages at the initial instant t 2, are neee to be impose for the resolution of the System of equations 5. These two initial conitions ( t ) = ( t ) an, are fixe npnric 2 npncon 2+ npnric ( t2 ) = npncon ( t2+ ) in orer to guarantee the continuity for the an waveforms passing from the re-circulating phase to the NPN bipolar conuction time phase, being npnric ( t2 ) an npnric ( t2 ) the values for an functions obtaine at the re-circulating phase final instant t 2. After that, the two an functions have been obtaine by resolving the previous system, the NPN bipolar base current Bnpn-on is irectly given by the following formula: Equation 8 in which Bnpn on () t = () t + () t = C () t npncon ( () t ) 0 npncon < npncon () t npncon ( ) C ( () t ) is a ischarging voltage on the C p capacitor (that is ). After having etermine, through the ODE45 simulator, in a rather large time interval (one half of the total working perio), the an voltage functions that resolve the System of equations 5, then a research of the maximum value for the function was impose in orer to eterminate the final instant of the NPN conuction time t 4. This NPN conuction time phase is precee by the re-circulating phase which follows. npncon npncon 36/78 Doc D Rev

37 Moeling of re-circulating phase preliminary to the NPN conuction time phase 8 Moeling of re-circulating phase preliminary to the NPN conuction time phase The re-circulating phase before the NPN bipolar conuction time is moele taking into consieration the following circuital schemes in which the LP signal represents the emagnetization current of the inuctor Lp an B is the NPN evice base-collector ioe junction forwar biase current on the first re-circulating phase: Figure 2. Driving network moeling for the simulation of the first re-circulating phase preliminary to the NPN bipolar conuction time phase DD DD Rs Rs Rs B B BEn D Q D Rs Rs Rs B B D D s Rb D s Rb BEn D Rb LP Rb LP Lp Rb Rb AM0982v Doc D Rev 37/78

38 Moeling of re-circulating phase preliminary to the NPN conuction time phase AN3400 Figure 22. Rs Rs Driving network moeling for the simulation of the secon re-circulating phase preliminary to the NPN bipolar conuction time phase Rs Bn B BEn DD C Q Rs Rs Rs Bn B BEn DD C s2 Rb E s2 Rb E Rb LP Rb LP Lp Rb Rb Equations of the two System of equations an System of equations 2, escribing the working conitions of the NPN bipolar uring the re-circulating phases before the NPN bipolar conuction time, can be combine with each other in orer to obtain the following system of two first orer linear ifferential equations vali on all the [t 0,t 2 ] time interval with C=C n =C p : System of equations 6 ( () t ) S npnric = npnric npnric + C R C R C b ( npnric () t ) = npnric npnric ( t0 ) ( t ) C R npnric = γ 0 (initial value fixe) npnric 0 = ϑ0 (initial value fixe) b b C R b AM0983v where: System of equations 7 S = = = = + = for t0<t<t; S S2 RS RS LP LP D E B = = + = + = for t<t<t2; Bn During the simulation phase, solutions for the an voltage functions have been calculate with the ODE45 simulator in a time interval as large as the length of the s forcing 38/78 Doc D Rev

39 Moeling of re-circulating phase preliminary to the NPN conuction time phase signal opportunely moele in orer to take into consieration the effect of emagnetization current for the inuctor L p. Also in this case, for the NPN bipolar base current Bnpn, the formula below is vali uring the whole re-circulating phase preliminary to the NPN conuction time phase: Equation 9 Bnpn () t = () t + () t = C () t npnric npnric ( ) C ( () t ) npnric npnric in which npnric () t is a ischarging voltage on the C p capacitor (that is ( () t ) 0 npnric < ). Doc D Rev 39/78

40 NPN storage time phase moeling AN NPN storage time phase moeling During the NPN storage time phase the base-emitter voltage oes not change from its forwar bias value ( BEn on = BEn sat ) ue to the excess minority carriers store in the base region. A negative base current starts removing this excess carrier at a rate etermine by the base riving network. NPN bipolar working conition uring this phase is moele by the following circuital scheme in which R base-npn is an opportunely size variable resistor that takes into consieration the effect of the base resistance moulation (increase) ue to the removal of the excess carriers: Figure 23. Driving network moeling for the NPN bipolar storage time phase simulation DD Rs Rs Rs pn Bnpn Q Rs Rs Rs Rbase Bnpn-off Rbase-npn s Rb BEn Enpn s Rb BEn-on Rb LP Rb Lp Rb Rb AM09965v By combining, with each other, the equations of the System of equations 4, escribing the working conitions of the NPN bipolar uring the NPN bipolar storage time phase, the following system of two first orer linear ifferential equations vali on all the [t 4,t 5 ] time interval with C=C n =C p : 40/78 Doc D Rev

41 NPN storage time phase moeling System of equations 8 + C R ( ()) b t + + npnsto S = npnsto npnsto C Rb RS + RBase npn C Rb BEn ( RS + RBase npn ) C ( RS + RBase npn ) ( npnsto () t ) = npnsto npnsto C R b ( t4+ ) = npncon ( t ) ( t ) = ( t ) npnsto 4 npnsto 4+ npncon 4 C R b in which BEn =constant, an for R Base-npn (t) an increasing variability law, of the following type, is impose: Equation 0 R Base npn (t) = α t n with 0 t Δ npn where coefficients α an n (this last integer number) are etermine so that R Base-npn (t) has a fixe an constant mean value R Base-npn-mean uring the NPN storage time perio of length equal to Δ npn secons: Equation R = Δ Δ Δ( n + ) ( n + ) Maintaining constant the T Storage-npn =t 5 t 4 time, with an increase of the function R Base-npn(t) orer n, so also the base resistance moulation effect approximation changes with a proportional increase of the negative peak of Bnpn-off current increases, as the following shows: R = Δnpn Base npn mean Base npn mean RBase npn () t α 0 npn npn R (by imposing Δ = T = t t ) = npn Storage npn 5 4 Base npn npn mean ( n + ) ( t t ) n 5 4 = Figure 24. ariability of the extractive current negative peak Bnpn-off versus variability of the integer n with storage time fixe AM0984v Doc D Rev 4/78

42 NPN storage time phase moeling AN3400 Therefore, it is important to fix correctly the two α an n parameters in orer to opportunely moel the transistor base resistance moulation effect uring the storage time phase simulation. The following mathematical formula has been consiere to make a connection among the storage time uration an the base currents eveloping uring the conuction an storage time phases of the evice (see References 3): Equation 2 n which: storage npn A on npn with A on-npn area subtene to the Bnpn-on current curve Bnpn on average = T uring the T on conuction perio; T on npn = σ npn ln + Bnpn on average Bnpn off average A storage npn with A storage-npn the moule of the area Bnpn off average = T (positive) subtene to the Bnpn-off current curve uring the T storage-npn storage time perio; σ npn is a time constant irectly connecte to the lifetime of the minority carriers (electrons) in the transistor base. Constant σ npn can be experimentally calculate, with a goo approximation, supposing two triangular areas subtene respectively to the Bnpn-on an Bnpn-off current curves an, in this case, Equation 2 assumes the following simplifie form: Equation 3 T storage npn being Bnpn-on-max an storage npn = σ npn ln + Bnpn off max Bnpn on max Bnpn off max σ T = ln + storage npn Bnpn on max Bnpn off max the maximum values of the conuction an extraction (in moule) currents. Consequently, the σ npn value can be mathematically an, in an approximate manner, etermine if the values of the T storage-npn, Bnpn-on-max an Bnpn-off-max assume (measure) by the evices uring the applicative steay-state working conitions are well-known. Substituting the efinitions of the Bnpn-on-average, Bnp-off-avarage an σ npn parameter to Equation 2, the following equation is obtaine: npn Equation 4 T storage npn = σ npn ln + T on npn T storage npn ε npn 42/78 Doc D Rev

43 NPN storage time phase moeling in which the following is impose: Equation 5 ε npn = A storage npn A on npn with ε npn parameter epening on the recombination phenomenon of a share of the amount of charges in base. Once the two parameters σ npn an T on-npn are fixe an well-known (the latter parameter obtaine from the simulation of the previous conuction perio) Equation 4 is use to efine the function of two inepenent variables: Equation 6 f(t storage npn, ε npn ) = T storage npn σ ln + T on npn n orer to etermine the length of the time interval to set for the NPN storage time simulation with the ODE45 simulator, the following proceure has been applie through the Matlab tool: m ifferent ε(i) npn values, increasing, equally-space out an inclue between a minimum ε min-npn an a maximum ε max-npn value, are fixe in base to the recombination characteristics of the minority carriers in base of the bipolar examine, that is: npn T storage npn ε npn Equation 7 ε (i) npn for i = m with ε min npn ε( i) npn εmax npn an also with: Equation 8 ε( j) ε(j + for j = (m ) npn ) npn Equation 9 m zeros of the efine function f(t storage-npn, ε npn ) are evaluate with the Matlab tool for each ε(i) npn fixe: Equation 20 T storage-npn final value is obtaine performing an arithmetical average operation among all the T(i) storage-npn so calculate: Equation 2 ε ( k) ε for k = 2 (m ) npn ε(k ) npn = ε(k + ) npn (k) npn f(t(i) npn, ε(i) npn) 0 for i = m storage = m Tstorage npn = mean ( T() storage npn,..., T(m) storage npn ) = T(i) storage npn m i= Doc D Rev 43/78

44 NPN storage time phase moeling AN3400 Setting the values ε min-npn, ε max-npn with the two parameters σ npn an T on-npn well-known, a realistic evaluation of the effective NPN evice storage time can be obtaine applying the proceure of calculation previously explaine. After having foun, with the ODE45 simulator, in the calculate time interval T storage-npn, the solutions of the System of equations 8 for the an voltage functions, then the NPN bipolar base current Bnpn-off is given by the following formula: Equation 22 Bnpn off () t = () t () t = C () t npnsto npnsto ( ) + C ( () t ) in which both -npnsto (t) an -npnsto (t) are ischarging voltages respectively on the ( ) 0 C n an C p capacitors (that is () t an ). npnsto < npnsto () t < npnsto ( ) 0 npnsto 44/78 Doc D Rev

45 Moeling of the ea time after the NPN storage time phase 0 Moeling of the ea time after the NPN storage time phase During the phase concerning the ea time after the NPN storage time both complementary pairs of bipolars are inactive an their B-C an B-E junctions are reverse biase. So, in orer to moel this phase, the following circuital scheme in which the B-C an B-E junctions are represente by the respective junction capacitances charging or ischarging accoring to the working conition, respectively, of the NPN an PNP evice is consiere. Figure 25. Driving network moeling for the simulation of the ea time phase after the NPN bipolar storage time phase AM0985v Rs Rs Rs CCBn CCBn Bnpn CEBn DD Q Rs Rs Rs DD CCBn CCBn CEBn Bnpn CEBn s Rb CEBn s Rb Rb LP Rb LP Lp Rb CBEp Q2 Bpnp CB CB CBEp GND Cs Cs Cs Rb CBEp Bpnp CBEp CB Cs CB GND Cs Cs Neglecting the currents CEBn an CBEp of the reverse biase B-E junction capacitances of the two bipolars, the following conitions are obtaine: Equation 23 CEBn, CBEp 0 Bnpn CCBn, Bpnp CB an CS LP where: CCBn = CCBnpn ( CCBn() t ) on the charge phase of the (variable) NPN base-collector junction capacitance C CBnpn ; ( ) CB = CBnp CB() t on the ischarge phase of the (variable) PNP collector-base junction capacitance C Bnp. Then, imposing the previous assumptions, combining, with each other, the equations of the System of equations 5, escribing the working conitions of the NPN bipolar uring this Doc D Rev 45/78

46 Moeling of the ea time after the NPN storage time phase AN3400 phase, the following system of two first orer linear ifferential equations is vali on all the [t 5,t 6 ] time interval with C=C n =C p : System of equations 9 ( () t ) npnea ( () t ) npnea npnea npnea C RS R b t = npnea npnea C + R R C R S b b = npnea npnea C Rb ( t 5+ ) = npnsto( 5 ) ( t ) ( t ) 5+ = npnsto 5 + C C CB CCBn S + C RS S C R + C R S CEBn S C R CEBn S in which BEn = CEBn <0 an EBp = CBEp <0 is an opportunely moele signal complying with the following function: Equation 24 CEBn = C CCBn CBnpn C CS S t + ψ in which ψ is a constant epening on the initial value of the CEBn voltage at t=t 5-. The length of this ea time interval fixe for the simulation, in which the an voltage functions are etermine through the ODE45 simulator, is impose to be given by the following formula in which T npnsto =t 5 t 4, T npncon =t 4 t 2 an T npnric =t 2 t 0 : Equation 25 T npnea = t 6 t 5 = 2f T npnsto npncon This last working operation phase escribe completes the resonant riving network moeling with reference to the variation over a one-half perio of the voltage signal (s) across the filter capacitor an in particular concerning the time intervals, in sequence, shortly before (re-circulating phase), uring (conuction phase) an afterwars (ea time) the NPN bipolar evice functional stage. n the secon one-half perio the voltage signal (s) has an opposite polarity compare to the first one an an analytical proceure similar to the NPN evice moeling technique is applie for the PNP bipolar evice intereste in being simulate uring its sequential working conition phases, as subsequently shown. T T npnric 46/78 Doc D Rev

47 Moeling of the re-circulating phase preliminary to the PNP conuction time phase Moeling of the re-circulating phase preliminary to the PNP conuction time phase The re-circulating phase before the PNP bipolar conuction time is moele taking into consieration the following circuital scheme in which LP signal represents the emagnetization current of the inuctor Lp an CBp is the PNP evice collector-base ioe junction forwar biase current on the first re-circulating phase: Figure 26. Driving network moeling for the simulation of the first re-circulating phase preliminary to the PNP bipolar conuction time phase Rs Rs Rs Rs Rs Rs s Rb Rb s Rb Rb LP Lp LP Rb CBp CBp EBp D2 D2 Q2 D2 Rb CBp EBp CBp D2 D2 D2 GND GND AM0986v Doc D Rev 47/78

48 Moeling of the re-circulating phase preliminary to the PNP conuction time phase AN3400 Figure 27. Driving network moeling for the simulation of the secon re-circulating phase preliminary to the PNP bipolar conuction time phase Rs Rs Rs Rs Rs Rs s2 Rb Rb s2 Rb Rb LP LP Rb EBp E2 Lp Rb EBp E2 Bp Q2 Bp CBp CBp GND C2 GND C2 Equations in the System of equations 6 an System of equations 7, escribing the working conitions of the PNP bipolar uring the re-circulating phase before the PNP bipolar conuction time, can be combine, with each other, in orer to obtain the following system of two first orer linear ifferential equations vali on all the [t 6,t 8 ] time interval with C=C n =C p : System of equations 20 AM0987v ( pnpric () t ) = pnpric pnpric ( () t ) C R b C R S pnpric = pnpric pnpric + C R C R C ( t6+ ) = npnea ( t ) ( t ) = ( t ) pnpric 6 pnpric 6+ npnea 6 b b b 48/78 Doc D Rev

49 Moeling of the re-circulating phase preliminary to the PNP conuction time phase where: System of equations 2 S = = = = + = for t6<t<t7; S S2 RS RS LP LP D2 E2 CBp = = + = + = for t7<t<t8; Bp During the simulation phase, solutions for the an voltage functions have been calculate with the ODE45 simulator in a time interval as large as the length of the forcing signal s opportunely moele in orer to take into consieration the effect of emagnetization current for the inuctor L p. For the PNP bipolar base current Bpnp, the following formula reporte is vali uring the whole re-circulating phase preliminary to the PNP conuction time: Equation 26 Bpnp () t = () t + () t = C () t pnpric pnpric ( ) + C ( () t ) in which -pnpric (t) is a ischarging voltage on the C n capacitor (that is pnpric () t < ). pnpric pnpric ( ) 0 Doc D Rev 49/78

50 PNP conuction phase moeling AN PNP conuction phase moeling The PNP conuction time phase is moele taking into consieration the following circuital scheme in which the EBp voltage is constant: Figure 28. Driving network moeling for the PNP bipolar conuction time phase simulation Rs Rs Rs Rs Rs Rs s Rb Rb s Rb Rb LP Lp Rb Bpnp EBp Q2 Epnp np Rb Bpnp-on EBp-on AM0988v By combining, with each other, the circuital equations of the previous System of equations 8, escribing the working conitions of the PNP bipolar uring the conuction time phase, the following system of two first orer linear ifferential equations is obtaine vali on all the [t 8,t 0 ] time interval in which the signal S () t = A sen( 2πft π) has the polarity shown in the previous figure an impose C=C n =C p : System of equations 22 ( pnpcon () t ) = pnpcon pnpcon ( () t ) pnpcon C R b C R = pnpcon pnpcon C Rb C + RS R b ( t 8+ ) pnpric ( t ) ( t ) ( t ) pnpcon = 8 pnpcon 8+ = pnpric 8 b S + C R S C R EBp S 50/78 Doc D Rev

51 PNP conuction phase moeling After that the two an functions have been obtaine by resolving the previous system, the PNP bipolar base current Bpnp-on is irectly given by the following formula: Equation 27 Bpnp on () t = () t + () t = C () t pnpcon in which -pnpcon(t) is a ischarging voltage on the C n capacitor ( ) 0 (that is pnpcon () t < ). pnpcon ( ) + C ( () t ) After having etermine, through the ODE45 simulator, in a rather large time interval (one half of the total working perio), the an voltage functions that resolve the System of equations 22, a research of the maximum value for the function was impose in orer to eterminate the final instant of the PNP conuction time t 0. pnpcon pnpcon Doc D Rev 5/78

52 PNP storage time phase moeling AN PNP storage time phase moeling During the PNP storage time phase the emitter-base voltage oes not change from its forwar bias value ( EBp-on = EBp-sat ) ue to the excess minority carriers store in the base region. A negative base current starts removing this excess carrier at a rate etermine by the base riving network. PNP bipolar working conition uring this phase is moele by the following circuital scheme in which R base-pnp is an opportunely size variable resistor that takes into consieration the effect of the base resistance moulation (increase) ue to the removal of the excess carriers: Figure 29. Driving network moeling for the PNP bipolar storage time phase simulation Rs Rs Rs Rs Rs Rs s Rb Rb s Rb Rb LP Lp Rb Bpnp EBp Q2 Epnp np Rb EBp-on Rbase-pnp Bpnp-off Rbase AM0989v By combining, with each other, the equations of the System of equations 9, escribing the working conitions of the PNP bipolar uring the PNP bipolar storage time phase, the following system of two first orer linear ifferential equations vali on all the [t 0,t ] time interval in which the signal S () t = A sen( 2πft π) has the polarity shown in the previous figure an impose C=C n =C p : System of equations 23 ( pnpsto () t ) = pnpsto pnpsto C R b C R R ( ()) b t C pnpsto b = pnpsto pnpsto C Rb C Rb RS + RBase pnp EBp Base pnp S + Base pnp S ( R + R ) C ( R R ) S ( t0+ ) = pnpcon ( t ) ( t ) = ( t ) pnpsto 0 pnpsto 0+ pnpcon 0 52/78 Doc D Rev

53 PNP storage time phase moeling in which EBp is a constant voltage an R Base-pnp (t) is impose to have the same increasing variability law as the R Base-npn (t) variable resistance: Equation 28 R Base pnp (t) = α 2 t p with 0 t Δ pnp in which α 2 an p (this last integer number) coefficients are etermine so that R Base-pnp (t) has a fixe an constant mean value R Base-pnp-mean uring the PNP storage time perio of length equal to Δ pnp secons: Equation 29 R Then, as alreay etermine for the NPN storage time calculation, once the two parameters σ pnp (or time constant irectly connecte to the lifetime of the minority carriers (holes) in the transistor base) an T on-pnp (obtaine from the simulation of the previous PNP bipolar conuction perio) are fixe an well-known, a function of two inepenent variables is efine for the calculation of the PNP storage time as follows: Equation 30 = Δ R = Δ Δ ( p+ ) Δpnp Base pnp mean Base pnp mean RBase pnp(t) α2 0 pnp pnp R (by imposing Δ = T = t t ) = f(t storage pnp, ε pnp pnp ) = T Storage pnp storage pnp ( p + ) in which ε pnp parameter, epening on the recombination phenomenon of a share of the amount of charges in base, is given by: σ pnp 0 ln + T Base on pnp pnp pnp mean ( p + ) ( t t ) p T 0 ε storage pnp pnp = Equation 3 ε pnp A = A storage pnp on pnp Therefore, varying the parameter ε pnp from a minimum to a maximum value, impose in base to the recombination characteristics of the minority carriers in base of the bipolar examine, the value T storage-pnp, that etermines the storage time final instant t, can be obtaine fining the zeroes of the functions f(t storage-pnp, ε pnp ) for each fixe ε pnp value an effectuating a mean operation among the T storage-pnp values so obtaine. After having foun, with the ODE45 simulator, in the calculate time interval T storage-pnp, the solutions of the System of equations 23 for the an voltage functions, then the NPN bipolar base current Bpnp-off is given by the following formula: Equation 32 Bpnp off () t = () t () t = C () t pnpsto pnpsto ( ) + C ( () t ) pnpsto pnpsto Doc D Rev 53/78

54 PNP storage time phase moeling AN3400 in which both -pnpsto (t) an -pnpsto (t) are ischarging voltages respectively on the C n ( ) 0 ( ) 0 an C p capacitors (that is () t an ). pnpsto < pnpsto () t < 54/78 Doc D Rev

55 Moeling of the ea time after the PNP storage time phase 4 Moeling of the ea time after the PNP storage time phase During the phase concerning the ea time after the PNP storage time both complementary pairs of bipolars are alreay inactive an their B-C an B-E junctions are reverse biase. So, in orer to moel this phase, the following circuital scheme in which the B-C an B-E junctions are represente by the respective junction capacitances charging or ischarging accoring to working conition respectively of the PNP an NPN evice, is consiere. Figure 30. Driving network moeling for the simulation of the ea time after the PNP storage time phase DD s Rs Rs Rs CCBn CCBn Bnpn CEBn Rb CEBn DD Q s Rs Rs Rs CCBn Bnpn Rb CCBn CEBn CEBn Rb LP Rb LP Lp Rb CBEp Q2 Bpnp CB CB CBEp GND Cs Cs Cs Rb CBEp Bpnp CBEp CB Cs CB GND Cs Cs AM09820v Neglecting the currents CEBn an CBEp of the reverse biase B-E junction capacitances of the two bipolars, the following conitions are obtaine: Equation 33 CEBn, CBEp 0 Bnpn CCBn, Bpnp CB an CS LP where: CCBn = CCBnpn ( CCBn() t ) on the ischarge phase of the (variable) NPN base-collector junction capacitance C CBnpn ; ( ) CB = CBC pnp CB () t on the charge phase of the (variable) PNP collector-base junction capacitance C Bnp. Doc D Rev 55/78

56 Moeling of the ea time after the PNP storage time phase AN3400 Then, after having impose the previously liste assumptions, combining, with each other, the equations of the System of equations 0, escribing the working conitions of the PNP bipolar uring this phase, the following system of two first orer linear ifferential equations has valiity on all the [t,t 2 ] time interval in which the signal S () t = A sen( 2πft π) has the polarity shown on the previous figure an impose C=C n =C p : System of equations 24 ( () t ) pnpea ( () t ) pnpea pnpea pnpea C RS + R b = pnpea + pnpea C Rb C RS R b = pnpea pnpea C Rb ( t + ) = pnpsto( t ) ( t ) ( t ) + = pnpsto C + C CB CCBn S C R + C S S RS CBEp C R S + C R CBEp S in which BEn = CEBn <0 an EBp = CBEp <0 is an opportunely moele signal complying with the following function: Equation 34 CBEp = C CB Bnp in which Ψ 2 is a constant epening on the initial value of the CBEp voltage at t=t -. The length of this ea time interval fixe for the simulation, in which the an voltage functions are etermine through the ODE45 simulator, is impose to be given by the following formula in which T pnpsto =t -t 0, T pnpcon =t 0 -t 8 an Tpnpric=t 8 -t 6 : C CS S t + ψ 2 Equation 35 T pnpea = t 2 t = T f pnpsto T pnpcon T pnpric T npnea T npnsto T npncon T npnric This last working operation phase escribe completes the resonant riving network moeling with reference to the variation over the whole perio of the voltage signal (s) across the filter capacitor an covering the time intervals, relating to the re-circulating, conuction an ea time phases for both NPN an PNP bipolar evices in their functional stages. The accuracy of the moeling technique propose is verifie by comparing the simulation results obtaine using the Matlab tool with experimental results of the riving network circuital implementation response in steay-state working conition for the bipolar evices. 56/78 Doc D Rev

57 Experimental results 5 Experimental results Two commercially available power bipolar junction transistors, that fin application in electronic lamp ballasts, were chosen for the experimental acquisition on the boar teste. n particular, a complementary pair of ST power bipolar transistors, the NPN evice STX83003 an PNP evice STX93003 (in TO-92 package), respectively, were employe on the common emitter half brige section to supply a 5 W CFL boar in orer to verify their compatibility on the resonant riving circuit solution. alues of the components use on the half brige voltage fe topology optimize boar setting an the main atasheet electrical specifications of the above mentione evices are the following. 5. Applicative parameters Setting of the riving network components (see Figure ): D 4 C,D2,D3,D = N4007 ; D5,D6 = BA59 = 4.7μF / 400 ; C 2 = 47nF / 00 ; = C 00nF; C 5 =.2nF C 3 4 = C 6 = 4.7nF / 630 ; = C 00nF / 250 C7 8 = R fuse = 8. 2Ω ; R 2 = 8. 2Ω ; R 3 = 330kΩ ; R 4 = 330Ω ; = 470kΩ L = 00μH T A = 2.3mH ; N T B = 3 turns. R 5 Power bipolar transistors specifications: Q High voltage fast-switching NPN power bipolar transistor STX83003 in TO-92 package with c= A, B ceo =400 an B ces =700 ; CESAT C =350 ma, B =50 ma an h FE = 25 C = 350 ma an CE =5. Q2 High voltage fast-switching PNP power bipolar transistor STX93003 in TO-92 package with C =- A, B ceo =-400 an B ces = -500 ; CESAT =-500 C = -350 ma, B =-50 ma an h FE = 25 C = -350 ma an CE =-5. An evaluation of switching performances of the evices uring the normal working operation was mae in open boar (at 25 C ambient temperature) conitions with 230 input voltage values an 50 Hz frequency. n particular, the following images epict the functionality of the high sie NPN bipolar transistor in steay-state operation. Doc D Rev 57/78

58 Experimental results AN3400 Figure 3. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current ( B ), collector current ( C ), base-emitter voltage ( BE ) an collector-emitter voltage ( CE ) signals acquire Figure 32. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current ( B ), collector current ( C ), voltage on base series capacitor C n ( CN ) an voltage on base series capacitor C p ( CP ) signals acquire 58/78 Doc D Rev

59 Experimental results Figure 33. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current ( B ), collector current ( C ), voltage on filter capacitor C2 ( C ) an voltage on riving resistance R2 ( RS ) signals acquire Figure 34. STX83003 (NPN) bipolar transistor uring steay-state operation with 230 input voltage: base current ( B ), collector current ( C ), voltage on base series capacitor C3 ( CN ) an voltage on breakown resistance R4 ( RBN ) signals acquire As previously seen, experimental waveforms acquire on the 5 W CFL optimize setting boar show a close similarity with the theoretical waveforms iscusse uring the escription of the self-oscillating operation principle (see Figure 2, 3, 4, 5 an 6). mages relate to the steay-state situation (Figure an 34) show basically a regular behavior of half brige section NPN bipolar transistor uring the open boar normal working operation at 230 ac without highlighting any critical electrical conition that coul cause bipolar case overheating phenomena. n the following images, switching particulars uring turn-on an turn-off transients respectively for both NPN an PNP bipolar transistors are reporte in open boar steay-state working operation with main equal to 230. Doc D Rev 59/78

60 Experimental results AN3400 Figure 35. STX83003 (NPN) bipolar transistor uring turn-on particular in steay-state operation with 230 input voltage: base current ( B ), collector current ( C ) an collector-emitter voltage ( CE ) signals acquire Figure 36. STX83003 (NPN) bipolar transistor uring turn-off particular in steay-state operation with 230 input voltage: base current ( B ), collector current ( C ) an collector-emitter voltage ( CE ) signals acquire 60/78 Doc D Rev

61 Experimental results Figure 37. STX93003 (PNP) bipolar transistor uring turn-on particular in steay-state operation with 230 input voltage: base current ( B2 ), collector current ( C2 ) an emitter-collector voltage ( CE2 ) signals acquire Figure 38. STX93003 (PNP) bipolar transistor uring turn-off particular in steay-state operation with 230 input voltage: base current ( B2 ), collector current ( C2 ) an emitter-collector voltage ( CE2 ) signals acquire As can be seen when observing the previous waveforms, ST transistors fit this application very well in terms of switching characteristics an power issipation. n fact, no thermal problems ue to a ba riving or behavior of the evices were foun for the two bipolars uner test in normal working conitions. Switching uring the turn-off times is goo as the cross-over points c-ce are acceptably low (within aroun 25 ma-25 values at 230 ac) an, consequently, the amounts of energy issipate by the bipolars (proportional to the areas subtene to the c-ce signals) are quite limite. Also measurements performe in close boar conitions point to a satisfactory functionality for the two complementary ST bipolars with case temperatures reaching values of aroun 64 C-68 C an working frequencies of aroun 46 khz while the absorbe input power (Pin) is fixe at aroun 5 W at 230 ac main voltage. Doc D Rev 6/78