Abstract: Has established the Buck electric circuit’s under uninterrupted current pattern small signal mathematical model, and has analyzed the voltage pattern and under the electric current mode control ring circuit design question according to the stable principle.
Key word: Switching power supply; Small signal model; Voltage mode control; Electric current mode control
Introduction
Designs when has the good tendency and the static property switching power supply, controls ring circuit’s design is a very important part. But ring circuit’s design and main circuit’s analysis situs and the parameter have the enormous relations. In order to carry on the stability analyses, it is necessary to establish the switching power supply complete small signal mathematical model. Under the frequency range model, the baud chart has provided one simple convenience project analysis method, may use for to carry on the loop gain the computation and the stability analyses. Because the switching power supply is a misalignment controlled member essentially, therefore, can only establish it approximately with the analysis means modelling in the stable state time small signal perturbation model, but explains the wide range perturbation with this model (e.g. startup procedure and load fierce change process) not completely accurate. Is good in the switching power supply general work in the stable state, the practice indicated that rests on the small signal perturbation pattern layout control circuit, after coordinating the soft starting circuit, the current limiting electric circuit, the clamp and other auxiliary bodies, can definitely cause switching power supply’s performance to satisfy the request. The switching power supply uses the Buck electric circuit generally, the work is deciding the frequency PWM control mode, this article carries on the analysis take this as the foundation. Uses other analysis situses the switching power supply analysis method to be similar.
1 Buck electric circuit inductance electric current continuously time small signal model
Figure 1 is the model Buck electric circuit, to simplify the analysis, hypothesis power switching valve S and D1 are the perfect switch, filter inductance L are the ideal inductance (resistance are 0), electric circuit work in uninterrupted current pattern (CCM) under. Re is the filter electric capacity C equivalent series resistance, Ro is the load resistance. In various state variable’s direction definition like chart 1 shows.
When S breakover, has to the inductance row equation of state
L(dil/dt)=Uin-Uo (1)
When S separation, D1 after flow breakover, the equation of state becomes
L(dil/dt)=-Uo (2)
When the dutyfactor is D, in a switching period process, the type (1) and the type (2) respectively has continued DTs and (1-D) the Ts time (Ts is switching period), therefore, one cyclical internal inductance’s average equation of state is
L(dil/dt)=D(Uin-Uo) (1-D)(-Uo)=DUin-Uo (3)
When stable state, =0, then DUin=Uo. This indicated when stable state the output voltage is a constant, its size and dutyfactor D and input voltage Uin is proportional.
Because the electric circuit various state variable always revolves the stable state value fluctuation, therefore, by type (3)
L [d (il il')/dt] = (D d) (Uin Uin’)- (Uo Uo’) (4)
Type (4) by type (3) the stable state value adds the small signal undulating quantity formation. The superscript is the wave symbol quantity is the undulating quantity, d is the D undulating quantity. The type (4) reduced the type (3) and has left out two undulating quantity product items
L (dil’/dt) =DUin’ dUin-Uo’ (5)
By Figure 1, also has
iL=C(duc/dt) Uo/R0 (6)
Uo=Uc ReC(duc/dt) (7)
Type (6) and type (7), no matter which condition the electric circuit work does establish. (6) and the type (7) may result in by the type
iL ReC(dil/dt)=1/Ro(Uo CRo(duo/dt)) (8)
Type (8) in inferential reasoning supposition Re<<Ro. Because time stable state dil/dt=0, dUo/dt=0, by the type (8) the stable state equation is iL=Uo/Ro. This indicated when stable state the inductance average current winds through the load completely. (8) various variables attach the small signal undulating quantity to the type
The type (9) reduces the type (8)
iL ReC(dil/dt)=1/Ro(Uo CRo(dUo/dt)) (10)
-like (10) will carry on Laplace transform
iL(s)=(Uo(s)/Ro) · [(1 sCRo)/(1 sCRe)] (11)
(s)= (11) thought generally in turn-on frequency’s frequency band scope the input voltage is constant, then supposition =0 and its substitution type (5), -like (5) will carry on the Laplace transform
sLiL’(s) =d(s)Uin-Uo’(s) (12)
By type (11), type (12)
Uo’(s) /d(s)=Uin[(1 sCRe)/(s2LC s(ReC L/Ro) 1] (13)
iL’(s) /d(s)=[(1 sCRo)/s2LC s(ReC L/Ro) 1] · Uin/Ro (14)
Type (13), type (14) is the Buck electric circuit - outputs the small signal transfer function continuously in the inductance electric current time control.
2 voltage mode control (VMC)
The voltage mode control method only uses the univoltage link to carry on the adjustment, is quite simple, easy to realize, may satisfy in the majority situations the performance requirements, as shown in Figure 2.
In Figure 2, when the voltage error amplifier (E/A) increases is low, the band width is very narrow, the Vc profile approximate direct-current level, and has
D=Vc/Vs (15)
d=Vc’/Vs (16)
The type (16) is the type (15) small signal wave equation. Entire electric circuit’s ring circuit structure as shown in Figure 3. Figure 3 has not considered the input voltage the change, namely supposition Uin=0. In Figure 3, (is generally 0) and respectively assigns be the voltage with the voltage output small signal fluctuation; KFB=UREF/Uo, for feedback factor; Erroneous e is an output sampling value deviation stable state undulating quantity, after the voltage error amplifier KEA enlargement; KMOD is the pulse width modulator increases, KMOD=d/=1/Vs; The KPWR electric circuit increases primarily, KPWR=/d=Uin; KLC is outputs the filter transfer function, KLC=(1 sCRe)/[S2LC s(ReC L/Ro) 1].
After known ring circuit other part of transfer function expression, then designed the voltage error amplifier. Because KLC has provided a zero and two resonant extreme, therefore, designs generally E/The PI regulator then, KEA=KP (1 ωz/s). And ωz uses in eliminating the static error, generally takes is below KLC zero extreme 1/10; KP uses in enabling the cutting frequency place open-loop gain to pass through the 0dB line by - the 20dB/ten frequency multiplications, the phase angle allowance slightly is smaller than 90°.
The VMC method has the following shortcoming:
1) has not been possible to forecast that the input voltage influence the voltage forward feed mechanism, is slow to the transient input voltage response, needs the very high loop gain;
2) to has the second-order extreme which (produces 180° phase-shift) by L and C not to constitute the compensation, the dynamic response is slow.
The VMC shortcoming available under is going to introduce the CMC method overcomes.
3 mean current mode control (AverageCMC)
The mean current mode control includes the voltage outer ring and the electric current inner rim two ring circuits, as shown in Figure 4. The electricity pressure ring provides the inductance electric current assignment, the electric current link uses the inductance electric current which the erroneous amplifier sends gifts to each other into to assign (Vcv) and the feedback signal (iLRs) the difference carries on the comparison, the enlargement, obtains the erroneous amplifier outputs Vc to carry on the comparison again with triangular wave Vs, finally namely control dutyfactor switching signal. Figure 4 Rs is the sampling resistance. Regarding a design good current error amplifier, Vc will not be a direct-current quantity, when switch breakover, the inductance electric current rise, will cause Vc to drop; The switch shuts off, when inductance electric current drop, will cause Vc to rise. The electric current link’s principle of design is, cannot cause the Vc rising slope to surpass the triangular wave the rising slope, both slope is equal when is most superior. The reason is: If the Vc rising slope surpasses the triangular wave the rising slope, will cause the Vc peak value to surpass Vs the peak value, when next cycle Vc and Vs possibly will not intersect, create the subharmonic oscillation.
After using the pitch match the method carries on the most superior design, the PWM controller’s gain along with the dutyfactor D change will change, as shown in Figure 5.
When D is very big, small Vc will cause the D big change, but D will be small time, even if the Vc change will be very big, the D change will not be big, namely the gain will drop. Therefore has
d=DV’/Vs (17)
Might as well supposes the voltage annulus widely to be lower than the electric current link far, when analyzes the electric current link Vcv is a constant. When the Vc rising slope is equal to the triangular wave slope, in turn-on frequency fs place, current error amplifier’s gain GCA is
GCA[d(iLRs)/dt]=GCA(Vo/L)Rs=Vsfs (18)
GCA=Vc’/(iL’Rs) =VsfsL/(UoRs) (19)
High frequency, -like (14) in the member “1″ and in the denominator low-order item will neglect, and will simplify,
iL’(s) = [d(s)Uin]/sL (20)
(17) and the type (20) has by the type
(iL’Rs) /Vc’=[Rsd(s)Uin/(sL)]/[d(s)Vs/D]=(RsUinD)/(sLVs) (21)
-like (19) (21) will multiple with the type, the entire electric current link’s split-ring transfer function is
(RsUinD/sLVs) · (VsfsL)/(UoRs)=fs/s (22)
Figure 7
S=2πfc the substitution above equation, and is equal to the above equation 1:00, may result in ring circuit’s cutting frequency fc=fs/(2π). Therefore, may the electric current link equivalent for the delay time constant be a switching period pure inertial element, as shown in Figure 6. Obviously, when current error amplifier’s gain GCA being smaller than optimum value, the electric current response’s time delay will be longer.
In GCA must generally or high frequencier place forms a high frequency extreme in fs place, causes the fs later electric current link open-loop gain by - the 40dB/dec slope drop, although like this causes the phase angle allowance changes slightly is small, but may eliminate the current feedback profile the high frequency burr influence, sharpens the electric current link’s antijamming ability. Low frequency under must add a zero spot generally, causes the electric current link open-loop gain fill-out, reduces the static error.
Entire ring circuit’s structure as shown in Figure 7. Before and KEA, KFB definition like. Obviously relative VMC says (see also Figure 3), average CMC eliminated the extreme which caused by the filter inductance (additional extreme fs to be very originally big, was very small to electricity pressure ring influence), adjusted the first order system the ring circuit, the electricity pressure ring increased may maintain constant, did not change along with input voltage Vin, the outer ring design became easier.
4 summit current mode control (PeakCMC)
Because average CMC wants the sampling filter inductance the electric current, sometimes appears is not too convenient, therefore, in the practice uses the electric current mode control method which frequently one kind is accommodating, namely peak value CMC, as shown in Figure 8. The voltage outer ring output control quantity (Vc) and by the inductance electric current rise along pitch profile (Vs) which forms carries on the comparison after the voltage comparator, directly obtains switching valve’s shutdown signal (clear signal by clock automatically to give), therefore, the electricity pressure ring’s output control quantity is the inductance electric current peak value gives the quota, by inductance electric current peak control dutyfactor.
What peak value CMC control is the inductance electric current peak value, but is not the inductance electric current (after the filter is load current), but between the summit current and the mean current has the error, therefore, the peak value CMC performance is inferior to average CMC. Generally when full load the inductance electric current in the breakover period’s electric current increase design amounts decides the electric current about 10%, therefore, in the best situation between the peak value inductance electric current and the mean value error also has 5%, a load lighter error is bigger, after specially enters the uninterrupted current (DCM) the work area, the error will surpass 100%, the system possibly will sometimes present the oscillatory occurrences. Below cutting frequency fc, may know average CMC from Figure 6 the electric current link open-loop gain to be possible to rise very high to (may >1000), the electric current may be under the control completely, but the peak value CMC electric current link open-loop gain can only maintain in 10 invariable (between summit current and mean value error causes), therefore, peak value CMC is suitable for the full load situation.
The peak value CMC shortcoming also includes to the noise is sensitive, needs to carry on questions and so on pitch compensation solution subharmonic oscillation. But as a result of the peak value CMC existence by cycle current limiting and so on unique merit, and easily through pulse current mutual inductor and so on simple means reappearance inductance electric current peak value, therefore, it still obtained the widespread application in the Buck electric circuit.
5 conclusions
Uses the average equation of state the method to be possible to obtain the Buck electric circuit’s small signal frequency range model, and may depend on this to carry on the ring circuit design. The voltage mode control, the mean current mode control and the summit current mode control method may use for to carry on the ring circuit design, has its good and bad points respectively, the suitable scope is also different.
