I
Figure 5.5: The general structure of a PWM switch.
(a)
(b)
Figure 5.6: The buck-boost PWM converter (a) can be re-drawn as in (b) to fit the model of Fig. 5.5
valid only for frequencies far below the switching frequency, the resonant elements may often be neglected.
Under the assumption that the resonant inductor and capacitor may be neglected, the small-signal model of a quasi-resonant converter, drawn according to the form of the converter fragment of Fig. 5.5, is shown in Fig. 5.7. In this figure, the quantities υoff and ιon are shown appearing across a single capacitor and in an inductor, respectively.
These elements are used for convenience only, and the figure applies equally to cases where more than one element produces υθff or ton.
I I
Figure 5.7: The small-signal model of a zero-current quasi-resonant converter, following the structure of Fig. 5.5
Perhaps the most important feature of the small-signal model of a half-wave quasi- resonant converter is the introduction of damping. This damping is indicated in the small-signal model by resistors. Figure 5.8 shows how these resistors arise. In Fig. 5.8(a), the current source h,∙vυoff has been replaced by two current sources, one across the voltage sources of D% and one across υoff. Figure 5.8(a) is equivalent in every way to Fig. 5.7.
In Fig. 5.8(a), however, the current source across the voltage sources of D2 may be eliminated. The other current source, across υθff, is an element that generates a current proportional to the voltage across it. Such an element is a resistor, as shown in Fig. 5.4(b).
The value of the resistance across υoff is h÷υ1.
By a similar process, the voltage source Λt,,∙tθn may be replaced by two sources, one in the ion branch and one in series with current sources from the switch S, as in Fig. 5.8(c).
The voltage source in series with ideal current sources may be eliminated, while the source in series with ton becomes a resistor of value -hυi, as shown in Fig. 5.8(d).
Since Jιu,∙ is always negative and Zι,∙υ always positive, both resistors in Fig. 5.8(d) are positive. These positive resistors indicate damping in the converter dynamic behavior.
The model of Fig. 5.8(d) is not reciprocal, however, and it is not immediately clear
+ υoff -
Figure 5.8: The controlled sources in Fig. 5.7 may be moved to new, equivalent positions.
The current source hiυ, translated as in (a), becomes a resistor acrossV0tt (b), while the voltage source hυi, shifted as shown in (c)l is equivalent to a resistor in series with Ion (d).
whether the remaining sources generate or absorb power.
To demonstrate that the net effect is indeed damping, let the driving generators (those dependent on fs) be set to zero in Fig. 5.7. The power absorbed by the remaining A
sources is
P = vs*S ÷ vp⅛> (5.20)
= (voff - vd) ÎS + VD (is - »on) (5.21)
= Λ,vυoff + (Λ,∙,∙ huu)υoffton — Λυ,∙ton . (5.22) The first and second terms of Eq.(5.22) represent the power dissipated respectively by the resistors h~υ1 and -hυi of Fig. 5.8(d). The cross term, (Λ,∙,∙ — Λvv)υoffto∏, may add to or subtract from the power lost in the resistors. However, the quadratic form of Eq.(5.22) can be shown, using the expressions of Eqs.(5.14) through (5.19), to be positive semi- definite. The actual damping may therefore be greater or less than that expected from the resistors ∕ι^j1 and — hυi alone, but damping is always present.
Note that the switch S and diode D% do not actually dissipate any power. The apparent “damping” is instead a result of modulation of the switching instants which promotes decay of perturbations.
The damping introduced by the half-wave resonant switch is a feature shared with PWM converters operated either under current-mode programming or in the discontinu
ous conduction mode. In both of these cases, the inductor current is constrained to some fixed value during each switching period. The dependent inductor current is no longer a state variable, and the system order is reduced by one. In current-mode program
ming, this effect is never fully realized and instead one pole lies between one-sixth and two-thirds of the switching frequency [15]. The damping in current-mode programming is visible in the circuit model of [15] as a resistance in series with the inductor whose current is constrained. Damping in the discontinuous conduction mode of operation is
“complete” in the sense that the second pole completely disappears [16].
Control of power converters is difficult because resistive damping of the high-Q res
onances internal to the converter must be small for reasons of efficiency. The load rep
resents the only lossless element available for damping. The half-wave resonant switch
+ υoff ---II
h%ffs r≥
hiiton ' hvu⅝ff --- +^
hvjfs
voff
Figure 5.9: In a full-wave quasi-resonant converter, the controlled sources may be shifted to new positions (a), and combined into a transformer (b).
represents a new way to provide lossless damping in a converter. It joins the ranks of current-mode programming [15], discontinuous conduction mode [16], and storage-time modulation [17] as means of introducing the effects of damping by appropriate control of the switching process, without actually dissipating any power.
As p→ 1 in a half-wave quasi-resonant converter, ∕ι,∙u and Λu,∙ both approach zero and the damping disappears.
In converters with a full-wave switch, Λ,∙v and Aυt∙ are zero for all values of p (under the excellent approximation that G(p) = 1 for all p}, and Λ,∙,∙ and hυυ both equal Fs/Fq. For such converters, the small-signal model may be drawn as in Fig. 5.9(a). Both the current sources, h,ffs and haton, have been shifted to positions across υoff, and both voltage sources, hυffs and hυυvott, have been moved in series with ton. Since both Λ,∙,∙
and hvv are equal to Fs/Fq, the two controlled sources constitute a transformer, as
shown in Fig. 5.9(b). The circuit model is therefore reciprocal (with the exception of the independent sources) and lossless.