Small-Signal Analysis of DCM Flyback Converter in Frequency-Foldback Mode of eration Laszlo Huber and Milan M. Jovanović Delta Products Cororation P.. Box 73 5 Davis Drive Research Triangle Park, NC 779, USA Abstract - The redicted closed-loo bandwidth of the DCM flyback converter in frequency-foldback mode (FFM of oeration obtained by the conventional averaged small-signal model that models the voltage-controlled oscillator (C with a constant gain does not change with the load. However, this rediction is not in agreement with the closed-loo measurements. In this aer, the discreancy between the conventional averaged small-signal model and measurements is systematically investigated. It is found that the source of this discreancy is the nonlinearity of the C that generates sideband frequency comonents and, therefore, cannot be modeled as a constant-gain transfer function. By including the effect of the sideband comonents in the closed loo, a multi-frequency averaged smallsignal model of the DCM flyback converter in the FFM which accurately redicts the loo bandwidth behavior with the load is roosed. I. INTRDUCTIN The flyback converter is a oular toology for low-ower alications such as, for examle, notebook adaters/chargers, because of its simlicity and cost effectiveness. A tyical oeration rofile of the flyback adaters/chargers is shown in Fig.. At heavy loads, they oerate at the boundary of the discontinuous conduction mode (DCM and continuous conduction mode (CCM, also called quasi-resonant mode (QRM, with valley itching (SW of the flyback itch. As shown in Fig., in QRM the itching frequency increases as the load decreases. To revent high-frequency oeration at lighter loads, the quasi-resonant oeration changes to DCM oeration with valley itching, where the itching frequency is aroximately constant and equal to its maximum value. To meet the challenging high-efficiency requirements in the entire load range [], [], at light loads the oeration changes from DCM with constant itching frequency and valley itching to DCM with variable itching frequency, where the itching frequency linearly decreases as the load decreases. This mode of oeration is known as frequency-foldback mode (FFM. The FFM can be imlemented with or without valley itching. IN R Cl C Cl D Cl N P SW R CS CS N S Q S R D f SW PWM C I r Co C CS,REF = const. CL L BIAS CC R CC FB C FB R D Z FB R EA v EA R,REF Fig. Simlified circuit diagram of DCM flyback converter in FFM ( - erturbation voltage source, CL - closed loo, L - oen loo The simlified circuit diagram of the DCM flyback converter oerating in the FFM is resented in Fig.. The itching frequency f is controlled by the feedback voltage FB using a voltage-controlled oscillator (C. The circuit diagram of the C is shown in Fig. 3(a; whereas, the transfer function of the C is shown in Fig. 3(b. As the load decreases, the feedback voltage decreases and, consequently, the itching frequency decreases. The main itch turns on by the C and turns off when the current-sense (CS voltage CS reaches reference voltage level CS,REF. If the eak value of the sensed current ulses is constant, the itching frequency linearly varies with the load current. It should be noted that voltage source shown in Fig. is the erturbation voltage source, which was only used in simulations aimed to study the oen- and closed-loo behavior of the converter. It does not exist in the hardware, i.e., in a real circuit node CL is directly connected to the C inut. v FB - FBTh CC I ch ~ v FB - FBTh f f,max C f SW f SWmax SW v Ram v C K C C Ram RamPeak FFM DCM QRM w/ SW w/ SW Δ FB v - FB FBTh Load Fig. Tyical oeration rofile of flyback adaters/chargers (a Fig. 3 C (a circuit diagram, and (b transfer function (b 978--4673-4355-8/3/$3. 3 IEEE 746
f = 5.85 khz (a PM= 5.3 o 3 - - -3-4 8 6 4 - -4 f = 5.5 khz LG[dB] PM= 57. o LG[deg] 4 k k 4k k k (a f = 4 khz PM= 55.7 o 3 - - -3-4 LG[dB] f = 3.95 khz (b Fig. 4 Closed-loo gain measurements obtained on an 85-W (.5-/4.5-A flyback adater/charger in DCM oerating in FFM at IN=5 and (a % load, (b 5% load For roer design of the feedback circuit, an accurate smallsignal model of the flyback converter with variable itching frequency in both DCM/CCM boundary and DCM oerating in FFM is necessary. The conventional averaged small-signal model of the variable itching frequency flyback converter [3] rovides results which are in good agreement with the exerimental results as long as the itching frequency is well above the closed-loo bandwidth. However, as the itching frequency of DCM flyback converter oerating in the FFM decreases with the load and aroaches the bandwidth frequency, measurements show that the bandwidth frequency also starts decreasing. Closed-loo gain measurements obtained on an 85- W (.5-/4.5-A flyback adater/charger in DCM oerating in FFM are resented in Fig. 4. As shown in Fig. 4, at % load (where f 8.4 khz, the closed-loo bandwidth is f = 5.85 khz, whereas, at 5% load (where f 9 khz, the closed-loo bandwidth is reduced to 4 khz. Similar results can be obtained by simulations. For examle, closed-loo gain simulations obtained in SIMPLIS are shown in Fig. 5. As can be seen, the simulation results are in close agreement with the exerimental results. However, according to the conventional averaged smallsignal model of the DCM flyback oerating in the FFM, the closed-loo bandwidth does not change when the itching frequency decreases with the load. In this aer, the discreancy between the averaged small-signal model and measurements is investigated. It is found that the source of this discreancy is the nonlinearity of the voltage 8 6 4 PM= 37. o LG[deg] - -4 4 k k 4k k k (b Fig. 5 Closed-loo gain simulations in SIMPLIS obtained on an 85-W (.5- /4.5-A flyback adater/charger in DCM oerating in FFM at IN=5 and (a % load, (b 5% load controlled oscillator. In fact, the C generates sideband frequency comonents [4]-[7] similar to the conventional ulse width modulator (PWM [4], [5], [8]-[]. By including the effect of the sideband comonents in the closed loo, a multifrequency averaged small-signal model of the DCM flyback converter in the FFM is roosed. II. AERAGED SMALL-SIGNAL MDEL F DCM FLYBACK CNERTER IN FFM F PERATIN Key waveforms of the DCM flyback converter oerating in FFM are shown in Fig. 6, where L M and N denote the magnetizing inductance and the turns ratio N P /N S of the flyback transformer. The average diode current over a itching eriod T = /f, which is equal to the load current I, can be determined as LM CS, REF idav = id = f (, T = f f. ( R CS 747
SW v CS N FF IN L M R CS CS,REF = const. t g FB vfb r Co C v i D N L M t CS,REF N Fig. 8 Simlified conventional averaged small-signal model of DCM flyback converter in FFM I Fig. 6 T N T SW T RST R CS Key waveform of DCM flyback converter in FFM As shown in (, if CS,REF is constant, the itching frequency is roortional to the load current. Then, the small-signal variation of the diode current is obtained as where, and r iˆ Dav k k fˆ vˆ r =, ( LM CS, REF RCS =, (3 LM CS, REF I f = = R CS RL =. (4 Assuming a linear C transfer function as that shown in Fig. 3(b, the relationshi between small-signal frequency change fˆ and small-signal change of feedback voltage vˆ is given by FB fˆ =. (5 KC vˆ FB Based on (-(5, the conventional averaged small-signal model of the DCM flyback in the FFM is obtained as shown in Fig. 7, which can be simlified as shown in Fig. 8, where gfb =. (6 KC k t The asymtotic Bode lots of the control-to-outut transfer function vˆ + jω / ωz Gvc( jω = = Kvc vˆ FB + jω / ω. (7 where, R K L vc = gfb, (8 ω z =, rco C (9 and ω = ( r + R / C R C, ( Co L are resented in Fig. 9. As it can be seen in Fig. 9, only the low-frequency art of the control-to-outut transfer function changes with the load. Since the closed-loo bandwidth is tyically located inside the frequency range highlighted in green in Fig. 9 and since the transfer function of the feedback circuit is indeendent of the load, it can be concluded that the closed-loo bandwidth does not change with the load. = G vc (jω K vc g FB G vc (jω ω ~ C o L - db/dec = f ( ω z = r Co C o log(ω G g vc HF = r FB Co v FB K C i D r = r Co v - 45-45 /dec + 45 /dec log(ω k SW f SW C - 9 Fig. 7 Conventional averaged small-signal model of DCM flyback converter in FFM Fig. 9 Asymtotic Bode lots of control-to-outut transfer function of DCM flyback converter in FFM 748
- [m] -.34.34.33.3 8 6 4.76.7.68.64.64 99.4 99.5 99.6 99.7 99.8 99.9 Time [ms] (a Cin [] RAMP [] [].33.3 8 6 4.76.7.68 99.4 99.5 99.6 99.7 99.8 99.9 (b Fig. Simulation waveforms of DCM flyback in FFM at IN = 5, at % load, with sinusoidal erturbation frequency oen loo and (b closed loo f = f /4 = 4.6 khz in (a.5 6 Cin.5 4 [m].5.5.5 RAMP.5 [].5.5.5 5 5 5 [] 5 5 5 5 5 5 3 35 4 5 5 5 3 35 4 f f f f f f f f f -f f f +f f -f f SW SW SW SW SW SW SW SW SW SW SW f +f SW Freq [khz] (a (b Fig. Simulation sectra of DCM flyback in FFM at IN = 5, at % load, with sinusoidal erturbation frequency loo and (b closed loo f = f /4 = 4.6 khz in (a oen III. EFFECT F SIDEBAND FREQUENCY CMPNENTS Closed-loo measurements of the DCM flyback in FFM are in disagreement with the conventional averaged small-signal model. In fact, as the itching frequency decreases with decreasing load, the closed-loo bandwidth also decreases, as shown in Fig. 4. To understand the discreancy between the conventional averaged small-signal model and measurements, the DCM flyback in FFM is simulated in SIMPLIS. An 85-W (.5-/4.5-A dc/dc flyback converter was used as the examle circuit. In Fig., v = m cos( πf t + θ. ( is the erturbation voltage source. Both, oen-loo (L and closed-loo (CL simulations were erformed. In Figs. and, relevant simulation waveforms and corresonding sectra, obtained at % load (where the 749
( f itching frequency is f = 8.4 khz with a sinusoidal erturbation frequency f = f /4 = 4.6 khz, illustrate the differences between oerations in oen loo and closed loo. In oen-loo, as shown in Fig. (a, at the inut of the C, there are only two comonents: the dc comonent, which determines the steady-state itching frequency, and the erturbation frequency comonent. The sectrum of the C ram voltage consists of a dc comonent, the itching frequency comonent and its harmonics, the erturbation frequency comonent and, in addition, sideband frequency comonents f ± f, f ± f, etc. This means that the C is a nonlinear circuit, similar to the conventional ulse width modulator (PWM [4], [5], [8]-[]. All the frequency comonents at the outut of the C aear in the sectrum of the outut voltage. In closed loo, as shown in Fig. (b, at the inut of the C, besides the dc comonent and the erturbation frequency comonent, there are all the other frequency comonents resent in the outut voltage. Consequently, the sectrum of the C ram voltage and, furthermore, the sectrum of the outut voltage in closed loo differ from the corresonding sectra in oen loo. This difference is the result of the nonlinearity of the C and can be exlained by considering the lowest sideband comonent f - f and its effect on the closed loo as illustrated in Figs. and 3. If the erturbation frequency is small such that f << f, then f - f is close to f and above the closed-loo bandwidth f. Therefore, sideband comonent f - f will be attenuated before fed back through the feedback circuit to the C inut and its effect on the loo gain is negligible. However, if f is close to f such that f - f < f, then f - f will be amlified before fed back to the C inut and its effect on the loo gain is not negligible. It is shown in Section I that the loo gain is decreased around the itching frequency, which results in a decrease of the closed-loo bandwidth. ( f C C C ( f ( f - f PWER STAGE o ( f o( f - f Fig. Frequency domain reresentation of oen-loo DCM flyback in FFM with sideband comonents C FB ( f - f ( C f C ( f - f ( C f f = f - ( f - f C ( f - f FB( f FEEDBACK CIRCUIT PWER STAGE o ( f ( o f - f Fig. 3 Effect of sideband comonents in closed-loo DCM flyback in FFM v (f LG ( av f C A = K C f B= K C - f f C = K C - f D= K C v FB - ( f - f v FB (f PWER STAGE f ( f v o (f G = G vf ( f f ( f - f f ( f f = f + ( f - f f ( f - f H = H v ( f - f H = H v ( f FEEDBACK CIRCUIT G = G vf LG av ( f - f ( f - f v o ( f - f Fig. 4 Multi-frequency averaged small-signal model of DCM flyback converter in FFM I. MULTI-FREQUENCY AERAGED SMALL-SIGNAL MDEL F DCM FLYBACK CNERTER IN FFM F PERATIN The sectrum of the frequency modulated ulses at the outut of the C was derived in [4] and [6] as vc( t = AFBdcKC + AmKC cos(πf t + θ + k + mkc kf nf A Jn k= n= f k kmkc cos π( kf + nf t + nθ sin θ f, ( where, A is the area of the narrow ulses and J n is the Bessel function of the first kind of order n. In accordance with Figs. and 3, considering only the lowest sideband comonent f - f, ( can be simlified as vc( t = AFBdcKC + AmKC cos(πf t + θ + mkc AJ ( f f.(3 f mkc cos π( f f t θ sinθ f In small-signal analysis, it can be assumed that the erturbation amlitude m is very small and, therefore, (3 can be further simlified as vc( t = AFBdcKC + AmKC cos(πf t + θ mkc A ( f f cos f [ π( f f t θ ] where the Bessel function J - is aroximated as, (4 x J ( x when x <<. (5 75
It should be noted in (4 that the sideband frequency comonent f - f at the C outut has inverted sign of initial hase angle θ. In order to avoid more comlex mathematical exressions, it is convenient to consider the negative sideband frequency comonent f - f instead of the ositive sideband frequency comonent f - f []. Then, (4 can be rewritten as vc( t = AFBdcKC + AmKC cos(πf t + θ mkc A ( f f cos f [ π( f f t + θ ], (6 Combining (6 and the block diagram in Fig. 3, the multifrequency averaged small-signal (MFASS model of the DCM flyback in FFM can be obtained as shown in Fig. 4, where Gvf ( f = vˆ / fˆ and Hv( f = vˆ FB / vˆ are the transfer functions of the ower stage and the feedback circuit, resectively. The closed-loo gain at the erturbation frequency is obtained as AG H + ( BC AD G ( HG H LG f =. (7 DG H Since, BC = AD =, (8 K C the closed-loo gain (7 is determined as AG H LG ( f =. (9 DG H Finally, the closed-loo gain (9 can be rewritten as KC Gvf ( f Hv( f LGav( f LG( f = = KC Gvf ( f f Hvf ( f f LGav( f f ( where LG av is the closed-loo gain of the conventional averaged small-signal model. Comarison of the closed-loo gains of the DCM flyback in FFM obtained with the conventional averaged small-signal model and with the MFASS model is resented in Fig. 5. The closed-loo gain obtained with the MFASS model is in a good agreement with the corresonding closed-loo gain obtained with measurements. As can be seen in Figs. 4 and 5, the closed-loo bandwidth of the DCM flyback in FFM decreases by aroximately.9 khz when the load decreases from % to 5%.. SUMMARY The conventional averaged small-signal model of the DCM flyback converter in the FFM of oeration shows that the closed-loo bandwidth does not change with the load. However, this is not in agreement with the closed-loo measurements. In this aer, the discreancy between the conventional averaged small-signal model and measurements is investigated. It is found that the source of this discreancy is the 4 3 - Δf =.98kHz - LG av (f -3 LG(f % load -4 LG(f 5% load -5 k k f [Hz] 8 o 35 o 9 o f = 4.8kHz f = 6.6kHz 45 o LG av (f o LG(f % load PM = 33.5 o LG(f 5% load PM = 59. o -45 o k k f [Hz] Fig. 5 Closed-loo gain of DCM flyback in FFM [85-W (.5-/4.5-A] obtained with conventional averaged small-signal model (blue line; and with multi-frequency averaged small-signal model at % load, where f SW = 8.4 khz, (red line, and 5% load, where f SW = 9 khz, (green line, at IN=5 voltage controlled oscillator. In the conventional averaged small-signal model, the C is reresented with a linear relationshi f = K C v Cin. However, the C is a nonlinear circuit and it generates sideband frequency comonents similar to the conventional ulse width modulator (PWM [4], [5], [8]-[]. The effect of the nonlinearity of the C on the closed-loo is exlained by considering the case when f is not significantly greater than the closed-loo bandwidth f and by considering the lowest sideband comonent f - f. If the erturbation frequency f is small such that f << f, then f - f is close to f and above f. Therefore, sideband comonent f - f will be attenuated before fed back through the feedback circuit to the C inut and its effect on the loo gain is negligible. However, if f is close to f such that f - f < f, then f - f will be amlified before fed back to the C inut and its effect on the loo gain is not negligible. Including the effect of the sideband comonents in the closed loo, a multi-frequency averaged small-signal (MFASS model of the DCM flyback in FFM is roosed. The closed-loo gain obtained with the MFASS model is in a good agreement with the corresonding closed-loo gain obtained with measurements. 75
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