buck quasi-resonant zvs converter with linear feedback control

For CMC, Fig. 2, the incremental switching frequencycan be given byf ˆ = H vˆ+ H vˆ+ H vˆ+ H iˆ(4)sEEIIOOL Lcontrolled oscillator (VCO), i.e., fˆK ( −βvˆv )with= ,s VCO O + ˆxv = −Kxˆ= −[ k k ]xˆ(6)ˆ x1 2Results QR converters were given in [4]. For the buckQR-ZVS converter, it isHE=2LFsA R K VlssIHIV= K I Fs−(4a)KO2sVIH =OKVO2sVIHL2LFs= K LFs−(4b)K VsIWith CMC in Fig. 2, vˆEis given byVˆ( s)= −βA( s)Vˆ( s)(5)EvOEquation (4) corresponds to feedback and feedforwardsignal paths to be added to (3). The result can berepresented by the block diagram in Fig. 3.Fig. 4: Linear feedback control for QR-ZVS converters.To minimise the steady-state error of the output voltage,v O , an integrator is inserted in the forward path, i.e.,A v (s)=k 3 /s. The output of the error amplifier, v E , is a thirdstate-variable, and the incremental value of the switchingfrequency is given by,fˆsa[ k 1 k 2 k 3 ] xˆa= −KKxˆ= −K(7)VCOVCOwhere x a =[i L v C v E ] t is state-vector of the augmentedsmall-signal model:⎡ A p 0⎤⎡Bp ⎤ ⎡Ep ⎤ 0ˆ.ˆ vˆI fˆ⎡ ⎤x & a = ⎢a + s + vˆref0⎥ x + ⎢0⎥ ⎢0⎥ ⎢1⎥ (8a)⎣−Cβ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦[ C 0] xˆav ˆ =(8b)OFig. 3: Small-signal model of a QR converter with CMC.Besides the feed-forward path, which gain is H I , Fig. 3resembles that of a linear feedback control, although thegains are fixed by the converter circuit and steady-stateDC values.3. LINEAR FEEDBACK CONTROLFig. 3, suggests that the QR converter can be controlledby the linear feedback control law in Fig. 4. Theincremental switching frequency in (3) is madeproportional to an error voltage by using a voltageModel (8) must be controllable and observable. Since î Land vˆO are easily measurable, the ESR of the outputcapacitor is neglected and thus, to simplify the circuit, astate estimator is not considered. The general model of aQR converter with either CMC, or state feedback controlcan be represented by Fig. 5, with the correspondentgains in Table I.With CMC, the line-to-output, T IO (s), and error-to-output,T EO (s), transfer functions can be derived from (3) and (4),and the closed-loop diagram in Fig. 6 is used to designA v (s) by classical control techniques. With state feedbackcontrol, k 1 , k 2 , and k 3 can be obtained by modern poleplacement design.3