7.2.5. Perturbation and linearization

1 fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!27! Perturbation and linearization !Ldi(t)Tsdt=d(t)vg(t)Ts+d'( t)v(t)TsCdv(t)Tsdt= d'(t)i(t)Ts v(t)TsRig(t)Ts=d(t)i(t)TsConverter averaged equations:! nonlinear because of multiplication of the time-varying quantity d(t) with other time-varying quantities such as i(t) and v(t).! fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!28!Construct small-signal model:"Linearize about quiescent operating point!If the converter is driven with some steady-state, or quiescent, inputs!d(t)=Dvg(t)Ts=Vgthen, from the analysis of Chapter 2, after transients have subsided the inductor current, capacitor voltage, and input current!i(t)Ts,v(t)Ts,ig(t)Tsreach the quiescent values I, V, and Ig, given by the steady-state analysis as!

2 V= DD'VgI= VD'RIg=DIFundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!29! Perturbation !So let us assume that the input voltage and duty cycle are equal to some given (dc) quiescent values, plus superimposed small ac variations:!vg(t)Ts=Vg+vg(t)d(t)=D+d(t)I n response, and after any transients have subsided, the converter dependent voltages and currents will be equal to the corresponding quiescent values, plus small ac variations:!i(t)Ts=I+i(t)v(t)Ts=V+v(t)ig (t)Ts=Ig+ig(t) fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!30!The small-signal assumption!vg(t)<<Vgd(t)<<Di(t)<<Iv(t)<< Vig(t)<<IgIf the ac variations are much smaller in magnitude than the respective quiescent values,!then the nonlinear converter equations can be linearized.

3 ! fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!31! Perturbation of inductor equation!Insert the perturbed expressions into the inductor differential equation:!LdI+i(t)dt=D+d(t)Vg+vg(t)+D' d(t)V+v(t)note that d (t) is given by!d'(t)= 1 d(t)=1 D+d(t)=D' d(t)with D = 1 D Multiply out and collect terms:!LdIdt 0+di(t)dt=DVg+D'V+Dvg(t)+D'v(t)+Vg Vd(t)+d(t)vg(t) v(t)Dc terms1storder ac terms2ndorder ac terms(linear)(nonlinear) fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!32!The perturbed inductor equation!LdIdt 0+di(t)dt=DVg+D'V+Dvg(t)+D'v(t)+Vg Vd(t)+d(t)vg(t) v(t)Dc terms1storder ac terms2ndorder ac terms(linear)(nonlinear)Since I is a constant (dc) term, its derivative is zero!The right-hand side contains three types of terms:!

4 Dc terms, containing only dc quantities! First-order ac terms, containing a single ac quantity, usually multiplied by a constant coefficient such as a dc term. These are linear functions of the ac variations! Second-order ac terms, containing products of ac quantities. These are nonlinear, because they involve multiplication of ac quantities! fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!33!Neglect of second-order terms!LdIdt 0+di(t)dt=DVg+D'V+Dvg(t)+D'v(t)+Vg Vd(t)+d(t)vg(t) v(t)Dc terms1storder ac terms2ndorder ac terms(linear)(nonlinear)vg(t)<<Vgd(t)<<D i(t)<<Iv(t)<<Vig(t)<<IgProvided!then the second-order ac terms are much smaller than the first-order terms. For example,!d(t)vg(t)<<Dvg(t)d(t)<<Dwhen!So neglect second-order terms.!Also, dc terms on each side of equation are equal.

5 ! fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!34!Linearized inductor equation!Upon discarding second-order terms, and removing dc terms (which add to zero), we are left with!Ldi(t)dt=Dvg(t)+D'v(t)+Vg Vd(t)This is the desired result: a linearized equation which describes small-signal ac variations.!Note that the quiescent values D, D , V, Vg, are treated as given constants in the equation.! fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!35!Capacitor equation! Perturbation leads to!CdVdt 0+dv(t)dt= D'I VR+ D'i(t) v(t)R+Id(t)+d(t)i(t)Dc terms1storder ac terms2ndorder ac term(linear)(nonlinear)Neglect second-order terms. Dc terms on both sides of equation are equal. The following terms remain:!Cdv(t)dt= D'i(t) v(t)R+Id(t)This is the desired small-signal linearized capacitor equation.

6 !CdV+v(t)dt= D' d(t)I+i(t) V+v(t)RCollect terms:! fundamentals of power Electronics!Chapter 7: AC equivalent circuit modeling!36!Average input current! Perturbation leads to!Ig+ig(t)=D+d(t)I+i(t)Collect terms:!Ig+ig(t)=DI+Di(t)+Id(t)+d(t)i(t)D c term1storder ac term Dc term1storder ac terms2ndorder ac term(linear)(nonlinear)Neglect second-order terms. Dc terms on both sides of equation are equal. The following first-order terms remain:!ig(t)=Di(t)+Id(t)This is the linearized small-signal equation which described the converter input port.!