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AND8123/D Power Factor Correction Stages Operating Critical Conduction Mode
This paper proposes detailed mathematical analysis operation critical conduction mode Power factor Corrector (PFC), with goal easing stage dimensioning. After some words specification brief presentation main critical conduction schemes, this application note gives equations necessary computing magnitude currents voltages that critical choice power components. INTRODUCTION IEC1000-3-2 specification, usually named Power Factor Correction (PFC) standard, been issued with goal minimizing Total Harmonic Distortion (THD) current that drawn from mains. practice, legislation requests current nearly sinusoidal phase with line voltage. Active solutions most effective means meet legislation. pre-regulator inserted between input bridge bulk capacitor. This intermediate stage designed output constant voltage while drawing sinusoidal current from line. practice, step-up boost) configuration adopted, this type converter easy implement. just notice that this topology requires output higher than input voltage. That output regulation level generally around universal mains conditions.
Diode Bridge Stage
Basics Critical Conduction Mode Critical conduction mode border line conduction mode) operation most popular solution power applications. Characterized variable frequency control scheme which inductor current ramps twice desired average value, ramps down zero, then immediately ramps positive again (refer Figures this control method following advantages: Simple Control Scheme: application requires external components. Ease Stabilization: boost keeps first order converter there need ramp compensation. Zero Current Turn major benefit critical conduction mode MOSFET turn when diode current reaches zero. Therefore MOSFET switch lossless soft there need diode. other hand, critical conduction mode some disadvantages: Large peak currents that result high dl/dt currents conducted throughout stage. Large switching frequency variations detailed paper.
Power Supply
Line Controller
Bulk Capacitor
LOAD
Figure Power Factor Corrected Power Converter
boost pre-converters typically require coil, diode Power Switch. This stage also needs Power Factor Correction controller that circuit specially designed drive pre-regulators. Semiconductor developed three controllers (MC33262, MC33368 MC33260) that operate critical mode NCP1650 continuous mode applications.
generally devotes critical conduction mode power factor control circuits below
Semiconductor Components Industries, LLC, 2003
September, 2003 Rev.
Publication Order Number: AND8123/D
AND8123/D
Diode Bridge
Diode Bridge Icoil
Icoil
Vout
power switch
power switch
power switch being about zero, input voltage applied across coil. coil current linearly increases with (Vin slope.
coil current flows through diode. coil voltage (Vout -Vin coil current linearly decays with (Vout -Vin slope.
Coil Current
Vin/L Icoil_pk
(Vout-Vin)/L
Critical Conduction Mode: Next current cycle starts soon core reset.
Figure Switching Sequences Stage
critical discontinuous mode, boost converter presents phases (refer Figure on-time during which power switch inductor current grows linearly according slope (Vin/L) where instantaneous input voltage inductor value. time during which power switch off. inductor current decreases linearly according slope (Vout-Vin)/L where Vout output voltage. This sequence terminates when current equals zero. Consequently, triangular current flows through coil. stage adjusts amplitude these triangles that average, coil current (rectified) sinusoid (refer Figure filter (helped input capacitor generally placed across diodes bridge output), performs filtering function. more popular scheme control triangles magnitude shape current, forces inductor peak current follow sinusoidal envelope. Figure diagrammatically portrays operation mode that could summarized follows: diode bridge output being slightly filtered, input voltage (Vin) rectified sinusoid. controller receives portion Vin. voltage this terminal shaping information necessary build current envelope. error amplifier evaluates power need response error senses between actual wished
levels output voltage. error amplifier bandwidth that error amplifier output reacts very slowly considered constant within line period. controller multiplies shaping information error amplifier output voltage. resulting product desired envelope that wished, sinusoidal, phase with line whose amplitude depends amount power delivered. controller monitors power switch current. When this current exceeds envelope level, latch reset turn power switch. Some circuitry detects core reset latch initialize MOSFET conduction phase soon coil current reached zero. Consequently, when power switch current ramps from zero envelope level. that moment, power switch turns current ramps down zero (refer Figures simplicity drawing, Figure only shows "current triangles". Actually, their frequency very high compared line one. input filtering capacitor filter averages "triangles" coil current, give:
Icoil Icoil_pk
(eq.
where <Icoil>T average current triangle (period Icoil_pk peak current this triangle.
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Icoil_pk forced follow sinusoidal envelop (k*Vin), where constant modulated error amplifier, <Icoil> also sinusoidal
Icoil sin(wt)
result, this scheme makes line current sinusoidal.
Stage Bulk Capacitor
Line
Input Filtering Capacitor
Current Sensing Resistor
Latch Zero Current Detection Output Buffer Current Envelope Error Amplifier Vref
Current Sense Comparator
Multiplier
Figure Switching Sequences Stage
controller monitors input output voltages using this information multiplier, builds sinusoidal envelope. When sensed current exceeds envelope level, Current Sense Comparator resets latch power switch turns off. Once core reset, dedicated block sets latch MOSFET conduction time starts.
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Peak
Icoil_pk
Inductor Current (Icoil)
Average (<Icoil>T)
Tac/2
(Tac line period)
MOSFET DRIVE Figure Coil Current
During power switch conduction time, current ramps from zero envelope level. that moment, power switch turns current ramps down zero. simplicity drawing, only "current triangles" shown. Actually, their frequency very high compared line one.
note that simple calculation would show that on-time constant over sinusoid:
Vac2
that switching frequency modulation brought off-time that equals:
toff sin(wt) sin(wt) (Vout sin(wt)) Vout sin(wt)
(eq.
That MC33260 developed Semiconductor does incorporate multiplier inputting portion rectified line shape coil current. Instead, this part
forces constant on-time achieve simplest manner, power factor correction.
Main Equations
Switching Frequency
power switch time (toff). During this second
phase, coil current flows through output diode feeds output capacitor load. diode voltage being considered null when voltage across coil becomes negative equal (Vin-Vout). coil current decreases then linearly with slope ((Vout-Vin)/L) from (Icoil_pk) zero, follows:
Icoil(t) Icoil_pk Vout
(eq.
power switch conduction time (ton). During this
already stated, coil current consists phases: time, input voltage applies across coil current increases linearly through coil with (Vin/L) slope:
Icoil(t)
(eq.
This phase ends when conduction time (ton) complete that when coil current reached peak value (Icoil_pk). Thus:
Icoil_pk
(eq.
This phase ends when Icoil reaches zero, then off-time given following equation:
toff Icoil_pk Vout
(eq.
conduction time then given
Icoil_pk
(eq.
total current cycle (and then switching period, toff. Thus:
toff Icoil_pk Vout (eq. (Vout Vin)
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shown next paragraph (equation 15), coil peak current expressed function input power line voltage follows:
Icoil_pk sin(wt) where
f(200W)
line angular frequency. Replacing Icoil_pk this expression equation leads
sin(wt) Vout sin(wt) (Vout Vin)
(eq.
This equation simplifies:
Vout Vac2 (Vout Vin)
(eq.
switching frequency inverse switching period. Consequently:
sin(wt) Vac2 Vout
(eq.
Figure Switching Frequency Input Power Sinusoid top)
This plot sketches switching frequency variations versus input power normalized form where f(200 switching frequency multiplied when power practice, stage propagation delays clamp switching frequency that could theoretically exceed several megaHertz very light load conditions. MC33260 minimum off-time limits load frequency around kHz.
This equation shows that switching frequency consists
term
Vac2 that only varies versus sin(wt) Vout
working point (load line voltage).
modulation factor
that
makes switching frequency vary within line sinusoid. following figure illustrates switching frequency variations versus line amplitude, power within sinusoid.
2.50
(wt)
2.00 f(90)
1.50
Figure Switching Frequency Over Line Sinusoid
This plot gives switching variations over line sinusoid Vout normalized form where taken equal line zero crossing. switching frequency approximately divided sinusoid.
Vac,
1.00
0.50
Figure Switching Frequency Over Line Voltage Sinusoid top)
figure represents switching frequency variations versus line voltage, normalized form where f(90) plot drawn Vout shows large variations (200% shape curve tends flatten Vout higher. However, minimum switching frequency always obtained line extremes (VacLL VacHL where VacLL VacHL respectively, lowest highest levels).
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Provided that line current results from averaging coil current, deduct following equation:
(wt)
lin(t) Icoil
Icoil_pk
(eq.
where <Icoil>T average considered coil current triangle over switching period Icoil_pk corresponding peak. Thus, peak value coil current triangles follows sinusoidal envelope equals:
Icoil_pk sin(wt)
(eq.
Figure Switching Frequency Over Line Sinusoid
This plot shows same characteristic Similarly what observed Figure versus Vac), higher difference between output input voltages, flatter switching frequency shape.
Since stage forces power factor close well known relationship linking average input power line current voltage lac) precedent equation leads
Icoil_pk sin(wt)
(eq.
Finally, switching frequency dramatically varies within line versus power. This probably major inconvenience critical conduction mode operation. This behavior often makes tougher filtering. also increase risk generating interference that disturb systems powered stage (for instance, produce some visible noise screen monitor). addition, variations frequency high values reach kHz) practically prevent effective tools damp reduce noise like snubbing networks that would generate high losses. also note that frequency increases when power diminishes when input voltage increases. light load conditions, switching period become (500 kHz). propagation delays within control circuitry power switch reaction times more negligible, what generally distorts current shape. power factor then degraded. switching frequency variation major limitation system that should reserved application where load does vary drastically. Coil Peak Currents
Coil Peak Current
coil current peak maximum sinusoid where This maximum value, (Icoil_pk)H, then:
(Icoil_pk)H
(eq.
From this equation, easily deduct that peak coil current maximum when required power maximum line minimum voltage:
Icoil_max VacLL
(eq.
where <Pin>max maximum input power application VacLL lowest level line voltage.
Coil Current
stage makes line current sinusoidal phase with line voltage, write:
lin(t) sin(wt)
(eq.
where Iin(t) instantaneous line current value.
value current magnitude that squared, gives dissipation produced this current within resistor. must then compute coil current First calculating "rms current" within switching period such that once squared, would give power dissipated resistor during considered switching period. Then switching period being small compared input voltage cycle, regarding obtained expression instantaneous square coil current averaging over rectified sinusoid cycle, have squared coil current. This method will used this section. above explained, current flowing through coil (IM(t) Icoil_pk ton) during MOSFET on-time, when 0<t<ton.
(ID(t) Icoil_pk- (Vout-Vin) Icoil_pk ton) during diode conduction time, that
when ton<t<T.
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Therefore, value coil current triangle over corresponding switching period given following equation:
(Icoil)rms Icoil_pk Icoil_pk
(eq.
Solving integrals, becomes:
(eq.
(Icoil)rms
Icoil_pk2 ton3 ton) Icoil_pk ton2
Icoil_pk
Icoil_pk
precedent simplifies follows:
(Icoil)rms Icoil_pk2 ton) Icoil_pk3) Icoil_pk
(eq.
Rearrangement terms leads
(eq.
(Icoil)rms Icoil_pk
Calculating term under root square sign, following expression obtained:
(Icoil)rms Icoil_pk
(eq.
gives resistive losses this given Vin. have current over rectified line period, must integrate <(Icoil)rms>T square would have proceeded deduct average resistive losses from dissipation over switching period. However, must forget extract root square result obtain value. consequence, coil current
(eq.
Replacing coil peak current expression function average input power line voltage (equation 15), write following equation:
(Icoil)rms sin(wt)
(eq.
(Icoil)rms
(Icoil)rms
This equation gives equivalent current coil over switching period, that given Vin. already stated, multiplying square coil resistance,
where 2*p/w line period Europe, 16.66 USA). stage being rectified line voltage, operates twice line frequency. That why, integrates over half line period (Tac/2).
Substitution equation (23) into precedent equation leads
(Icoil)rms sin(wt)
(eq.
This equation shows that coil current value
sin(wt), that
Therefore:
Icoil(rms)
(eq.
value sinusoidal current whose magnitude
value such sinusoidal
current well known (the amplitude divided
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Switching Losses
switching losses difficult determine with accuracy. They depend MOSFET type particular gate charge, controller driver capability obviously switching frequency that varies dramatically critical conduction mode operation. However, make rough estimation assumes following:
output voltage considered constant.
output voltage ripple being generally less than nominal voltage, this assumption seems reasonable. switching times tFR, defined Figure considered constant over sinusoid.
Dissipated Power: (IMOSFET Vdrain)
IMOSFET Vdrain
Figure Turn Waveforms
Figure represents turn sequence. observe three phases: During approximately second half gate voltage Miller plateau, drain-source voltage increases linearly till reaches output voltage. During short time that part diode forward recovery time, MOSFET faces both maximum voltage current. gate voltage drops (from Miller plateau) below gate threshold drain current ramps down zero. "dt" Figure represents total time three phases, "tFR'' second phase duration. Therefore, write:
(eq.
Vout Icoil_pk dt-tFR Vout Icoil_pk
where: switching times portrayed Figure switching period. Equation gives expression linking coil peak current switching period considered current cycle (triangle): Substitution equation into equation (27) leads
(Vout Vin) tFR)
(eq.
Icoil_pk Vout Vout
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This equation shows that switching losses over switching period depend instantaneous input voltage, difference between instantaneous output input voltages, switching time coil value. Let's calculate average losses (<psw>) integrating over half line period:
(eq.
(Vout Vin) tFR)
Rearranging terms, obtains:
Vout Vin2
(eq.
Vout being considered constant, easily solve this equation remembers that input voltage average value that
(Vac2 Vin2 dt). Applying this, becomes:
(eq.
being always specified, instead, take with half Miller plateau gate charge (Q2/2). Knowing drive capability circuit, deduct turn time Q3/Idrive (Q2/2)]/Idrive). first approach, taken equal diode forward recovery time.
DRAIN-TO-SOURCE VOLTAGE (VOLTS)
simpler manner:
tFR) Vac2
(eq.
Vout
VGS, GATE-TO-SOURCE VOLTAGE (VOLTS)
Vout Vac2
coil inductance plays important role: losses inversely proportional this value. simply because switching frequency also inversely proportional This equation also shows that switching losses independent power level. could have easily predict this result simply noting that switching frequency increased when power diminished. Equation (32) also shows that lower ratio (Vout/Vac), smaller MOSFET switching losses. That because "Follower Boost" mode that reduces difference between output input voltages, lowers switching frequency. other words, this technique enables smaller coil same switching frequency range same switching losses. instance, MC33260 features "Follower Boost" operation where pre-converter output voltage stabilizes level that varies linearly versus line amplitude. This technique aims reducing between output input voltages optimize boost efficiency minimize cost stage extract tFR? best measure them. approximate time necessary extract gate charge MOSFET (refer Figure 10).
TOTAL GATE CHARGE (nC) 25°C
Figure Typical Total Gate Charge Specification MOSFET
must note that calculation does take into account: energy consumed controller drive MOSFET (Qcc*Vcc*f), where MOSFET gate charge necessary charge gate voltage Vcc, driver supply voltage switching frequency. energy dissipated because parasitic capacitors stage. Each turn produces abrupt voltage change across parasitic capacitors MOSFET drain-source, diode coil. This results some extra dissipation across MOSFET (1/2*Cparasitic*DV2*f), where Cparasitic
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considered parasitic capacitor voltage change across
Power MOSFET Conduction Losses
Refer MC33260 data sheet more details http://www.onsemi.com/.
However, equation (32) should give sufficient first approach approximation most applications where listed sources losses play minor role. Nevertheless, losses produced parasitic capacitors become significant light load conditions where switching frequency gets high. always, bench validation key.
portrayed Figure coil current formed high frequency triangles. input capacitor together with input filter integrates coil current ripple that resulting line current sinusoidal. During on-time, current rises linearly through power switch follows:
Icoil(t)
(eq.
where input voltage (Vin sin(wt) coil inductance time.
During rest switching period, power switch off. conduction losses resulting from power dissipated Icoil during on-time, calculate power during switching period follows:
Icoil(t)2
(eq.
where MOSFET on-time drain source resistor, on-time. Solving integral, equation (34) simplifies follows:
(eq.
Either noting that off-time (toff) expressed
function (refer equation substituting this equation into Toff), considering that critical conduction mode being border continuous conduction mode (CCM), expression giving duty-cycle boost converter applies. Both methods lead same following result:
Vout Icoil_pk2 Vout
(eq.
calculate duty cycle ton/T)
coil current reaches peak value on-time, Icoil_pk precedent equation rewritten follows:
Icoil_pk2
(eq.
recognize traditional equation permitting calculate MOSFET conduction losses boost flyback Ipk2 where peak current MOSFET duty cycle).
Substitution equation (37) into equation (36) leads
(eq.
note that coil peak current (Icoil_pk) that follows sinusoidal envelop, written follows:
Icoil_pk sin(wt) (refer equation 15).
Replacing Icoil_pk their sinusoidal expression, respectively sin(wt) sin(wt)
equation (38) becomes:
sin(wt) sin(wt) Vout
(eq.
That more compact form:
2(wt) 3(wt) Vout
(eq.
Equation (40) gives conduction losses given voltage. This equation must integrated over rectified line sinusoid obtain average losses:
2(wt) 3(wt) Vout
(eq.
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average value sin2(wt) well known (0.5), calculation <sin3(wt)> requires trigonometry remembers:
sin(a) cos(b)
sin(a sin(a sin(wt) sin(3wt)
(eq.
Combining precedent formulas, obtain:
3(wt)
2(a)
cos(2a)
Substitution equation into equation (41) leads:
(eq.
sin(wt)2 sin(wt) Vout sin(3wt) Vout
Solving integral, becomes:
Vout Vout
(eq.
Equation (44) simplifies follows:
Vout
(eq.
This formula shows that higher ratio (Vac/Vout), smaller MOSFET conduction losses. That "Follower Boost" mode that reduces difference between output input voltages, enables reduce MOSFET size. instance, MC33260 features "Follower Boost" operation where pre-converter output voltage stabilizes level that varies linearly versus line amplitude. This technique aims reducing between output input voltages optimize boost efficiency minimize cost stage2. way, deduct from this equation current ((IM)rms) flowing through power switch knowing that (IM)2rms
(IM)rms Vout
(eq.
MC33260 monitors whole coil current monitoring voltage across resistor inserted between ground diodes bridge (negative sensing refer Figure 15). circuit utilizes current information both overcurrent protection core reset detection (also named zero current detection). This technique brings major benefits: need auxiliary winding detect core reset. simple coil sufficient stage. MC33260 detects in-rush currents that flow start-up during some overload conditions prevents power switch from turning that stressful condition. stage significantly safer. Some increase power dissipated current sense resistor counter part since whole current sensed while circuits like MC33262 only monitor power switch current.
Dissipation Current Sense Resistor MC33262 Like Circuits
Dissipation within Current Sense Resistor
controllers monitor power switch current either perform shaping function simply prevent from being excessive. That resistor traditionally placed between MOSFET source ground sense power switch current.
Since same current flows through current sense resistor power switch, calculation rather easy. must just square value power switch current (IM)rms calculated previous section multiply result current sense resistance.
Refer MC33260 data sheet more details http://www.onsemi.com/.
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Doing this, obtains:
Vout
(eq.
where <pRs>262 power dissipated current sense resistor
Dissipation Current Sense Resistor MC33260 Like Circuits
this case, current sense resistor derives whole coil current. Consequently, product square coil current gives dissipation current sense resistor:
(Icoil(rms)
(eq.
where Icoil(rms) coil current that expressed equation (26), equals: Icoil(rms)
current switching period level then integrate obtained result over line sinusoid. portrayed Figure coil discharges during time. More specifically, current decays linearly through diode from peak value (Icoil_pk) down zero that reached off-time. Taking beginning off-time time origin, then write:
Icoil(t) Icoil_pk toff-t toff
(eq.
Consequently:
(eq. Comparison Losses Amount Cases Let's calculate ratios:
Similarly calculation done compute coil current, calculate "diode current over switching period":
Id(rms)2T toff Icoil_pk toff-t toff
(eq.
obtains:
(eq.
Solving integral, obtains expression "rms diode current over switching period":
Id(rms)T toff Icoil_pk
(eq.
Vout
considers that (8/3 approximately equals 0.85, precedent equation simplifies:
0.85 (eq. Vout
Substitution equation (15) that expresses Icoil_pk, into precedent equation leads
Id(rms)T toff sin(wt) (eq.
where line amplitude.
Average Current through Diode
addition, easily show that toff linked following equation:
sin(wt) toff Vout Vout
(eq.
diode average current easily computed notes that load output capacitor currents:
Iload ICout
(eq.
Consequently, equation (58) changed into:
Id(rms)T Vout sin(wt)
(eq.
Then, average:
(eq.
Iload ICout Iload ICout
equilibrium, average current output capacitor must (otherwise capacitor voltage will infinite). Thus:
Iload Pout Vout
(eq.
This equation gives equivalent current diode over switching period, that given Vin. already stated Coil Peak Currents section, square this expression must integrated over rectified sinusoid period obtain square diode current. Therefore:
Id(rms)2
(eq.
diode current more difficult calculate. Similarly computation coil current instance, necessary first compute squared
3(wt) Vout
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Similarly Power MOSFET Conduction Losses section, integration (sin3 (wt)) requires some preliminary trigonometric manipulations:
3(wt) sin(wt) 2(wt) sin(wt) sin(wt) cos(2wt) (sin(-wt) sin(4wt) Then 3(wt) sin(wt) sin(3wt) cos(2wt) sin(wt) sin(wt) cos(2wt)
Consequently, equation (61) change into:
Id(rms)2 sin(wt) sin(3wt) Pin2 Vout
(eq.
solve integral write:
(cos(w0) cos(wTac cos(3wTac cos(3w0) Id(rms)2 Vout
(eq.
2p), have:
cos(p)-1 cos(p) Pin2 Id(rms)2 Vout
(eq.
simplify equation replacing cosine elements their value:
(eq. Id(rms)2 Vout
Thus:
Ic(rms)2 I1(rms)2 I2(rms)2
(eq.
square diode current simplifies follows:
Id(rms)2 Vout
(eq. (eq. Power Switch Load Stage Vout
Finally, diode current given
Id(rms) Vout
Output Capacitor Current
shown Figure capacitor current results from difference between diode current (I1) current absorbed load (I2):
Ic(t) I1(t) I2(t)
(eq.
Figure Output Capacitor Current
Thus, capacitor current over rectified line period, value difference between during this period. consequence:
Ic(rms)2 I2)2
(eq.
Rearranging (I1-I2)2 leads
Ic(rms)2
(eq.
[I12 I2)]
knows first term (I1(rms)2). This diode current calculated previous section. second third terms dependent load. cannot compute them without knowing characteristic this load. Anyway, second term (I2(rms)2) generally easy calculate once load known. Typically, this current absorbed downstream converter. other hand, third term more difficult determine depends relative occurrence currents. stage load (generally switching mode power supply) synchronized, this term even seems impossible predict. simply note that this term tends decrease capacitor current consequently, deduct that:
Ic(rms) I1(rms)2 I2(rms)2
(eq.
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Substitution equation (67) that gives diode current into precedent equation leads
Ic(rms) I2(rms)2 (eq. Vout load resistive, Vout/R where load resistance equation (71) changes into: Ic(rms)2 1(rms)2 Vout Thus, capacitor squared current Vout Ic(rms)2 Id(rms)2 Vout Vout Vout Pout Ic(rms)2 Vout Vout Pout Vout2/R, precedent equation simplifies follows: Vout Vout
(eq.
where I2(rms) load current.
Vout
(eq.
(eq.
Ic(rms)
(eq.
find more friendly expression literature:
Ic(rms) where load current. This equation
approximate formula that does take into account switching frequency ripple diode current. Only frequency current that generates frequency ripple bulk capacitor (refer next section) considered (this expression easily found using equation (88) computing Ibulk Cbulk dVout Equation (77) takes into account both high frequency ripples.
Output Voltage Ripple
output voltage bulk capacitor voltage) exhibits ripples. first traditional Switch Mode Power Supplies. This ripple results from output current pulses switching frequency pace. bulk capacitors exhibit parasitic series resistor (ESR refer Figure 12), they cannot fully filter this pulsed energy source. More specifically: During on-time, MOSFET conducts energy provided output. bulk capacitor feeds load with current needs. current together with resistor bulk capacitor form negative voltage -(ESR*I2), where instantaneous load current, During off-time, diode derives coil current towards output current across becomes ESR*(Id-I2), where instantaneous diode current.
This explanation assumes that energy that stage perfectly matches energy drawn load over each switching period that consider that capacitive part bulk constant voltage that only creates some ripple. fact, there additional frequency ripple which inherent Power Factor Correction. input current voltage being sinusoidal, power stage squared sinusoid shape. other hand, load generally draws constant power. consequence, pre-converter delivers amount power that matches load demand average only. output capacitor compensates lack (excess) input power supplying (storing) part energy necessary instantaneous matching. Figures sketch this behavior.
Stage Load
Driver
Bulk Capacitor
Figure Output Capacitor
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Vout V/div)
Load Power (100
h*Pin W/div)
(100 V/div)
Figure Output Voltage Ripple
dashed black line represents power that absorbed load. stage delivers power that squared sinusoid shape. long this power lower than load demand, bulk capacitor compensates supplying part energy stores. Consequently output voltage decreases. When power pre-converter exceeds load consumption, bulk capacitor recharges. peak power twice load demand.
Vout V/div)
(200 mA/div)
(100 V/div)
Figure Output Voltage Ripple
output voltage equals average value when input voltage minimum maximum. output voltage lower than average value during rising phase input voltage higher during input voltage decay. Similarly input power voltage, frequency capacitor current (represented case resistive load) twice line one.
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this calculation, does consider switching ripple that generally small compared frequency ripple. addition, switching ripple depends load current shape that cannot predicted general manner. already discussed, average coil current over switching period
sin(wt)
(eq.
instantaneous input power (averaged over switching period) product input voltage Iin. Consequently:
2(wt)
(eq.
average over switching period, bulk capacitor receives charge current Vout) where stage efficiency, supplies averaged load current Vout. Applying famous "capacitor formula" becomes:
Cbulk dVout Vout
(eq.
Substitution equation (79) into equation (80) leads
dVout 2(wt) Vout Vout Cbulk
(eq.
Rearranging terms this equation, obtain:
(eq. Vout dVout 2(wt) Cbulk d(Vout2) Vout dVout that Noting that cos(2wt) 2(wt), deduct square
sin(2wt) Vout Vout Cbulk Vout
(eq.
Thus:
(eq.
output voltage from precedent equation:
Vout2 Vout sin(2wt) (eq. Cbulk
Vout dVout Vout
sin(2wt) Cbulk Vout
Where dVout instantaneous output voltage ripple. Equation (85) rearranged follows:
(eq.
where <Vout> average output voltage. Dividing terms precedent equations square average output voltage, becomes:
dVout Vout
sin(2wt) Cbulk Vout
simplify this equation considering that output voltage ripple small compared average output voltage (fortunately, generally true). This leads that term words, that
sin(2wt) Cbulk Vout sin(2wt) Cbulk Vout
nearly zero other
small compared Thus, write that:
sin(2wt) sin(2wt) [1*1* Cbulk Vout Cbulk Vout
(eq.
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Substitution equation (86) into equation (87), leads simplified ripple expression that generally find literature:
sin(2wt) (eq. Cbulk Vout maximum ripple obtained when minimum when (sin(2wt) Thus, peak-to-peak dVout Conclusion
ripple that difference these values
(dVout)pk-pk (eq. Cbulk Vout
And:
Vout Vout (dVout)pk-pk sin(2wt) (eq.
Compared traditional switch mode power supplies, faces additional difficulty when trying predict currents voltages within stage: sinusoid modulation. This particularly true critical conduction mode where switching ripple cannot neglected. proposed this paper, overcome this difficulty First calculating their value within switching period, Then switching period being considered very small compared line cycle, integrating result over sinusoid period. proposed theoretical analysis helps predict stress faced main elements stages: coil, MOSFET, diode bulk capacitor, with goal easing selection power components therefore, implementation. Nevertheless, always, cannot replace bench work reliability tests necessary ensure application proper operation.
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Peak Coil Current: Icoil_pk sin(wt) Maximum Peak Current: Icoil_max VacLL Coil Current: Icoil(rms) Switching Frequency: sin(wt) Vac2 Vout
Switching Losses: tFR) Vac2 Vout
Conduction Losses: Vout
Average Diode Current: Iload Pout Vout Diode Current: Id(rms) Vout
Vout
Iload
Line
CONTROLLER
LOAD
Capacitor Frequency Ripple: MC33260 like Current Sense Resistor Dissipation:
(dVout)pk-pk Cbulk Vout
Capacitor Current:
Ic(rms) Iload(rms) Vout
MC33262 like Current Sense Resistor Dissipation:
Vout
load resistive:
Ic(rms) Vout Vout
Vac: line voltage VacLL: lowest level line angular frequency <Pin>: Average input power <Pin>max: Maximum level
Vout: Output voltage Pout: Output power Iload: Load current Iload(rms): load current Efficiency
Ron: MOSFET resistance tFR: Switching times (see Switching Losses section Figure Cbulk Bulk capacitor value Current sense resistance Coil inductance
Figure Summary
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Notes
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