Section 5 Dynamics and Control of DC-DC Converters

Transcription

1 Section 5 Dynamics and ontrol of D-D onverters 5.2. Recap on State-Space Theory x Ax Bu () (2) yxdu u v d ; y v x2 sx () s Ax() s Bu() s ignoring x (0) (3) ( si A) X( s) Bu( s) (4) X s si A BU s () ( ) () (5) Y() s X() s DU() s (6) Y() s ( si A) BU() s (7) Gs () si ( A) B : transfer function matrix (8) For stability: roots of ( si ) A must be in the LHS plane. Y s si A BU s () ( ) () (9) y(t) can be found from inverse Laplace transformation of (9). Y(s) can be modified using compensators in such a way that the transient response of y(t) is faster than normally achieved, with small over or undershoots, and Dynamics and ontrol F. Rahman of D D onverters

2 with adequate stability for the expected variations of system parameters, and d(t). i L v Note that y could be which are the feedback signals, 0, and these are continuously regulated. Small Signal Analysis of the Buck onverter We consider that D varies with time as an arbitrary signal. With continuous conduction, voi() t D() t Vd where v is oi regarded as an averaged D voltage. Thus v () s V D() s oi d R sl V 0 VDs () d v oi R s R Figure 5. voi () s v oi is a PWM waveform Total impedance Dynamics and ontrol 2 F. Rahman f

3 Z() s RL slrc // R s (0) I L () s VDs d () Z() s () I 0 () s IL() s R RR s s (2) V () s I () s R 0 0 (3) V0 () s sr Vd Ds () 2 R RL L s s R L L (4) A more appropriate modeling method that is easily applicable to all converter circuits is the State-Space Averaged modeling approach which is described below. Dynamics and ontrol 3 F. Rahman of D D onverters

4 The State-Space Averaged Model Any linear time invariant system can be represented by a state-space model like x Ax Bu (5) y x Du x, ignoring feed forward. (6) For single output v x 0 (7) Power electronic circuits during T ON and T states can be OFF described by two sets of differential equations. Thus, when T is ON x Ax Bu (8) v0 x (9) When T is OFF x A xb u (20) 2 2 v0 2x (2) Dynamics and ontrol 4 F. Rahman of D D onverters

5 T is ON for dt and OFF for ( s dt ) s. The lower case d represents continuously variable D(t). Averaging x over the interval T using s where Thus T s s 0 ( ) ( ) 2 2 of allstate var iables dt and x T gives x Ad A d x Bd B d u (22) v d ( d) x 0 2 (23) A Ad A ( d ) 2 (24) B Bd B ( d ) 2 (25) d ( d ) 2 (26) x Ax Bu v o x (26a) Dynamics and ontrol 5 F. Rahman of D D onverters

6 We assume that the states x vary, or have perturbations, around a steady-state X. So that x X x (27) v V v o o o ~ d D d (28) u U u (29) Note that V d (= U ) may also have perturbations. However, we assume that V d remains constant, and our task here is to find how v o varies with variations in d. During steady-state, perturbations have died down, so that 0 AX BU (30) X A BU (3) V0 A BU (32) A, B and matrices are weighted average of A & A 2, B & B 2 and & 2 respectively. Now x X x 0 x x (33) Dynamics and ontrol 6 F. Rahman of D D onverters

7 Thus ~ ~ x A D d A 2 ( D d) X x ~ ~ B D d B 2 ( D d) U u (34) Neglecting products of small perturbations, and assuming V to remain constant, d x AD A D X x BD B D U 2 2 A A2 X B B2 U d ~ Note that n the steady-state, AX BU 0, (34a) x Ax A A X B B Ud 2 2 ~ (35) Similarly, ~ Vo vo D2 D X x 2X d ~ ~ ~ o v x Xd 2 (36) Also, Vo X (36a) Dynamics and ontrol 7 F. Rahman of D D onverters

8 From 34a and 36a V V 0 d A B, where U Vd (37) From 35 x Ax ( A A2) X ( B B2) U d x Ax ( A A2) X ( B B2) V d d ~ ~ xs () SI A A A2XB B2V d d s (38) Taking Laplace transform of 36 and replacing x s from (38), the following is obtained. A A2 X B B2 V d v 0() s SI A d () s (39) X Bu 2 Dynamics and ontrol 8 F. Rahman of D D onverters

9 Examples. Average state equations and v () 0 s ds () of a Buck converter v s ds () 2. Average state equations and 0() converter of a Forward v s ds () 3. Average state equations and 0() converter v s ds () 4. Average state equations and 0() converter of a Boost of a Buck-Boost State-Space Averaged Representation of the Buck onverter (or Forward onverter with N N2) L i L RL i dv dt c V d N N 2 T R R v Assume ideal switch and diodes Dynamics and ontrol 9 F. Rahman f

10 Let il x ; vc x2 During T on for loop : di dvc Vd L RLiR( il ) dt dt Vd Lx RL R( x x 2) () Also, for loop 2 v R i ( i i ) R L dv dv v R ir R dt dt or, (2) dv From (2), ( RR) v ir dt ( R R ) x x Rx (2.) 2 2 From () and (2.) RR RR L RR L R x LR ( R) LR ( R) x L V x 2 R x 2 0 R ( R) R ( R) d Dynamics and ontrol 0 F. Rahman of D D onverters

11 A Therefore: RR RR L RR L R LR ( R) LR ( R) R R ( R) R ( R) B L 0 We have y v0 R( il i) R( x x 2) RR R yv x x 0 2 RR R R y v RR R x 0 R R R R x 2 RR R RR RR During t off, ie during ( DT ) s, the circuit remains the same, except that D 2 conducting makes the input voltage to the secondary equal to zero. Thus A A B Dynamics and ontrol F. Rahman of D D onverters

12 Now, A Ad A ( d) 2 A A and B Bd B ( d) 2 B Bd and, d ( d) 2 d ( d) In general, R R R and RR L 0. So that: R RL L L A R B L d 0 R Dynamics and ontrol 2 F. Rahman of D D onverters

13 For the 2x2 matrix a A a a 2 a 2 22 using the following formula:, the inverse can be found a22 a 2 a22 a 2 A det( A) a a a a a a a a R RL L L A R A R L R RL R RL RL L L A R L R RL R RL RL L L A RL R L RR RL R RL L Dynamics and ontrol 3 F. Rahman of D D onverters

14 A L R RL R L RR RL RR R L RR RL R RL RL R( R RL) L RR RL RR R L For the steady-state, from (37), we have V V 0 d V V 0 d A B L R RR RL RR R L d A BR L RL R( R RL) 0 R R RL R R R L V 0 RL RL R R R( R RL) d A B ; L Vd RR RL RR RL 0 V0 R L RL d V RR R L d L V0 R R d V RR R d L D From (39) v0() s ds () si A A A2 X B B2 V d 2 X Bu Dynamics and ontrol 4 F. Rahman of D D onverters

15 with I 0 0 ; R RL L L A R ; R ; A A2 0 Similarly ; 2 0 v0() s sr ds () 2 ( R RL) L s s R R L The denominator in the side brackets is of the form s 2 s Therefore, the transfer function Tp() s of the power stage and the output can be written as v () s s Tp() s V ds () s 2 s d z z where 0 L and ( R RL) R L 20 z. R Dynamics and ontrol 5 F. Rahman of D D onverters

16 20log0 db Vd d 0 z o 90 o 80 o Figure 3: Bode Plot of a Buck converter Dynamics and ontrol 6 F. Rahman of D D onverters

17 State-Space Averaged Model of the Buck-Boost onverter T RON D V d L i L i R V 0 R L R Figure 5.4 During T ON, ie for 0 t DTs, D is Open di dt L L ilrl ilron Vd or, di R R V i dt L L L L ON d L R R V L L L ON d or x x () Also dv dt R v R dv dt dv R R v dt Dynamics and ontrol 7 F. Rahman f

18 dv v dt R R x2 x2 R R (2) dv R R yv R v x 0 2 dt R R R R R A B L 0 L R L 0 ON ; u Vd 0 R R R 0 R R ; During T off, ie DTS t TS, D is ON di dt dv dt L L RLiLVDR v or dil RL R dv v VD il (3) dt L L dt L L Dynamics and ontrol 8 F. Rahman of D D onverters

19 Also, dv dt R v R dv dt dv ( RR) v dt dv v dt R R (4) From (3) R x R x v v V L D L L R R L L R L VD x x v L L R R L R L VD x x x2 L L R R L (3.) dv R y v0 R v dt R R R y v x 0 2 R R Dynamics and ontrol 9 F. Rahman of D D onverters

20 A B 2 2 RL R L R R L 0 R R V d L ; 0 ; 2 0 R R R Using (36), v 0 can be written. Laplace transform of v 0 will give v () 0 s ds () Boost type converters generally produce a transfer characteristic of the form: s/ s/ v0() s Vd f( D) d() s as bs c z z2 2 There is a zero, 2 z, in the right-half s plane. The location of the zero, ie, its value depends on R and L. The quantity L is related to L and D by a non-linear equation. The RH plane zero has stability implication for high gains in the compensators. Dynamics and ontrol 20 F. Rahman of D D onverters

21 Figure 5.5 Figure 5.6 Dynamics and ontrol 2 F. Rahman of D D onverters

22 Dynamics of converter circuits. Figure onverter small-signal gain = G c V D Dynamics and ontrol 22 F. Rahman of D D onverters

23 Model of a PWM Figure 5.8: Representation of the PWM ec e D V 2 Ts V tri, peak s tri, peak c Dynamics and ontrol 23 F. Rahman of D D onverters

24 Figure 5.9 Typical design of a forward converter using a lag-lead controller The controller G () s c K( s/ c ) ( s/ )( s/ ) c2 F Typically, it is desirable that 45 o m to o 60. Dynamics and ontrol 24 F. Rahman of D D onverters

25 c c 4 and 2 4 c c c is the bandwidth of the voltage controlled system. Normally, the existing system gain (PWM and converter) is not enough to guarantee the level of dynamic performance required of a power supply. The performance figure is roughly given by the 0 db crossover frequency oi in figure 5.6 for the buck converter. Additional gain is normally required to increase the 0 db cross-over frequency. This is supplied by the error amplifier of figure 5.5. Lag-lead, proportional + integral or other controllers may be used for the error amplifier. Figure 5.9 indicates the design of a lag-lead compensator for the outer voltage control loop. The Bode diagram of the uncompensated system is indicated by the dotted blue lines in Figure 5.9. c and c2 are the break-points of the zero and pole of this amplifier and F is the cut-off frequency of a filter (pole) which is normally located at a frequency which is lower than the switching frequency by a factor of 0 in order to filter out the switching frequency ripples. The 0 db cross-over frequency and the phase margin of the uncompensated system are not adequate. The 0 db crossover frequency c and the phase margin m2 of the compensated system are satisfactory. Note that in figure Dynamics and ontrol 25 F. Rahman of D D onverters

26 5.9, the error amplifier provides exactly the deficit in gain of the existing system at the desired cut-off frequency c. PWM with feed-forward The voltage controlled system (with output voltage feedback only) does not respond well (i.e., quick enough) to changes of V d in controlling V 0. Note that change in V0 must occur first before D can be changed by the error amplifier and the pulse-width modulator to remove the error in V 0 or to keep it at its reference value. PWM with feed-forward is often used to speed up the response to changes in V d. Figure 5.0 Dynamics and ontrol 26 F. Rahman of D D onverters

27 With the feed-forward scheme of figure 5.0, an increase (or decrease) in V d increases (or decreases) the peak value of the saw-tooth carrier. This in turn reduces (or increases) the duty cycle D appropriately, without waiting for V0 to change and show up in the feedback signal for V 0 at the input of the error amplifier. The delays in the feedback, controller and converter circuits are thus avoided. ontrol using inner current loop (urrent Mode control) ontrol of the output voltage using a single voltage loop does not fully protect the devices from over-current at start, when a large capacitor charging current may flow, when the load changes very fast or when the load is too high or short circuited. A better way is to control the current through the device(s), or the buck or boost inductor L, through a feedback loop inside the outer voltage loop. The inner loop normally acts much faster than the outer voltage loop, so that the inner loop predominates the outer loop in transient operation. In this way the inner loop establishes the inductor current according to the load demand. The inner current loop also keeps this current to a preset maximum value during transient operation, such as at start or when the load is short circuited by the uncharged capacitor. It thus protects the devices during starting and sudden abnormal working conditions. Because these currents also reflect the D Dynamics and ontrol 27 F. Rahman of D D onverters

28 level of the load current, the controlled level of current in the switch or the inductor L also represents the demanded current by the load. The structure of such a control system is indicated in Figure 5.. Figure 5. The switch, or the inductor current, can be sensed and fed back as indicated in Figure 5.2. Figure 5.2 Dynamics and ontrol 28 F. Rahman of D D onverters

29 Types of current controllers. Hysteresis band control: t on and t off according to hysteresis band settings. Figure onstant toff control; t on according to il max or i sw max. Schemes and 2 have varying switching frequency. 3. onstant f s control; t on according to i L max or i swmax, t off according to T s. Switch turn-on occurs at preset clock times. This scheme is widely used. 4. ontinuous current control with PWM also widely used. This scheme controls the average level of current in the switch or in the inductors. 5. Adaptive hysteresis band control. The adaptation tends to keep the switching frequency over a narrow Dynamics and ontrol 29 F. Rahman of D D onverters

30 range. Difficult for the hysteresis band to be made fully adaptive. With continuous current control, the two-stage converter controller design reduces to an inner current controller and an outer voltage controller which are de-coupled. This means that inner loop has a much higher band-width than the outer loop, so that designs for each can be approached separately or independently. With the fast inner current loop, the representation of the outer voltage loop is now simplified, as indicated below for a buck converter. The inductor current is regulated by the inner current controller. The outer voltage loop compensator output defines the current reference for this current loop. Assuming that the inductor current reaches steady-state much quicker than the outer voltage loop, we can represent v 0 dynamics as Figure 5.4 Dynamics and ontrol 30 F. Rahman of D D onverters

31 Rc 0() () s src I s IL s IL() s RR ( R Rc ) s c s R( sr c ) V0() s I0() s R IL() s ( R R ) s c V0 () s R( sr c ) I () s s( R R ) ( T s) L c F With high bandwidth inner current loop, the current control loop is represented by IL() s ds () Ts Note our assumption: ( RR ) R T i c c i Gain and phase plots (Bode diagram) of an uncompensated Buck converter, the inner current controller, the PI voltage controller and the open-loop transfer function of the compensated voltage controlled system are indicated in Figure 5.5. It should be noted that with the inner current controller, the PI controller for the outer voltage loop can be designed with more favourable phase margin. In fact, with the inner current Dynamics and ontrol 3 F. Rahman of D D onverters

32 controller, the phase plot may not be needed as the phase lag of the voltage control system may have adequate phase margin to start with. Thus, a PI controller which gives zero steady-state error can be designed easily. Figure 5.5: Bode diagrams for a Buck converter. Dynamics and ontrol 32 F. Rahman of D D onverters