(a)
(b)
Figure 5.12: (a) A quasi-resonant buck converter with the resonant capacitor across the output filter inductor, (b) The small-signal model for this converter.
that neither Cr nor Lr can be state variables. These elements may introduce zeros in the transfer functions and slightly modify the positions of existing poles, but they cannot add any new poles.
The fact that the quasi-resonant converter has the same order as its parent PWM converter indicates that the resonant switch itself has no dynamics. Despite the highly
“dynamic” resonant waveforms present during each cycle, the switch waveforms is and v∑> must return to zero at the end of every cycle. The switch cannot carry any information from one cycle over to the next in the form of an initial condition. Having no “memory,”
therefore, the resonant switch has no dynamics.
Chapter 6
Extensions of Resonant Switches
Chapters 3 through 5 have considered only a limited class of quasi-resonant con
verters: those based on PWM converters with a single transistor and diode and no transformer. In addition, those chapters have considered only the zero-current resonant switches. Zero-voltage switches may be treated in exactly the same way as zero-current switches, however, and the restrictions of a single switch and no transformers may of
ten be circumvented. In Section 6.1 of this chapter the zero-voltage resonant switch is studied by application of duality principles to the zero-current switch. Section 6.2 next reveals the power of the general analysis method developed in previous chapters. A sim
ple generalization of the functions G‰(p) and Gfw(p) is all that is needed to account for another mode of operation of resonant switches—multiple conduction cycles. Finally, Section 6.3 gives an example of modeling a converter with a transformer and more than one transistor switch.
6.1 Zero-Voltage Resonant Switches
Resonant switches that turn ON and OFF into zero voltage were introduced in [8].
These switches are duals of the zero-current-switching (ZCS) switches previously ana
lyzed. Rather than consider each zero-voltage-switching (ZVS) quasi-resonant converter as a dual of some other zero-current converter, duality will be applied first to the PWM converter itself, deriving the duals of Definition 1 and Theorems 1 and 2 of Chapter 3.
Then, using the dual of Definition 2, the behavior of the zero-voltage resonant switch can be derived in as general a fashion as for the zero-current switch.
Voltage <—► Current Impedance <—> Admittance
Loop <—> Cut-set
R G
——∖ΛΛ--- <—► --- ∖ΛΛ---
—ΠΓδΊΟ— C
—
a ∖ b 4___ a ∖ b
HP — 5-κP
a b «___k
Table 6.1: In taking the dual of a network or theorem, the quantities above are inter
changed.
6.1.1 Duality
Forming the dual of a circuit is both a topological and a quantitative process.
“Meshes” and nodes are interchanged to obtain the topology of the new circuit. The voltage and current in each branch are interchanged, so that the circuit elements of each branch are replaced by their duals. Table 6.1 lists elements and quantities that are duals of each other.
Some networks do not have physical duals. For example, a network with coupled inductors would have coupled capacitors in its dual. The equations describing such a network may be written, but the circuit could not be built. Also, a network whose graph is nonplanar has no dual. We assume that all the converters considered have planar graphs and no coupled inductors, so that duals always exist.
6.1.2 Dual of a Resonant Switch Converter
As seen in Chapter 3, a quasi-resonant converter can be formed by starting with a PWM converter satisfying Definition 1 and then performing the modifications of Defini-
tion 2. The dual of the resulting converter can be developed by applying duality to each step of the transformation from PWM to quasi-resonant converter.
Definition 1, describing PWM converters, is its own dual when one considers that the input current source arising from the dual of item 3 of the definition will always be implemented in practice by a voltage source in series with a “stiff” inductor. The dual of a PWM converter is therefore another PWM converter, a well-known fact. The only change in a PWM converter introduced by duality is the interchanging of the ON and OFF times of the transistor switch as a result of the interchange of the switch current and voltage waveforms. The duty ratio D of a PWM converter is replaced by its complement, 1 — D, in the dual converter.
Application of duality to the next step in the PWM-to-resonant switch transfor
mation, the rules of Definition 2, yields the following topological rules governing ZVS resonant switches:
Rule 3: The diode D%, the resonant inductor Lr, and a (possibly empty) set of inductors form a cut-set *n a ZVS converter.
Rule 4: The two-quadrant switch S, the resonant capacitor Cr, and a (pos
sibly empty) set of capacitors and voltage sources form a loop in a ZVS converter.
Insertion of the resonant inductor Lr in series with the diode Dz and the resonant capacitor Cr in parallel with the two-quadrant switch S always forms a ZVS converter.
As with the zero-current case, severed other locations for Lr and Cr are possible for each initial topology.
Theorems 3 and 4 are duals of each other, so the pair is unchanged after application of duality. Continued application of the duality transformation to the waveforms derived in Chapter 3 reveals that only a few changes are necessary to transform the results of Chapters 3, 4, and 5 into ZVS converters. These changes are listed in Table 6.2.
For instance, the ZCS converter has average switch current
.S = ∕on⅛(p) . (6.1)
Zero-Current Zero-Voltage
•Ion -→ ⅛>ff
½rff —÷ Λ>n
Ro → ¾'1
R -→ R~1
P -→ P~1
is -→ vs
υD -→ *D
M -→ M~1
D -→ 1- D
Table 6.2: Transformations that change the equations for a ZCS quasi-resonant converter into those for a ZVS converter.
Application of the transformations of Table 6.2 yields the average switch voltage in the dual ZVS converter:
vs=V0ff^G(p^1) ∙ (6.2)
As a further example, where 0 < p < 1 for zero-current turn-OFF in the ZCS converter, the requirement for zero-current turn-OFF in the dual ZVS converter is 0 < p^^1 < 1, or 1 < p < ∞.
6.1.3 Dc and Ac Behavior of ZVS Converters
Performing the substitutions of Table 6.2 on Eq.(4.11), which related the switching frequency and conversion ratio for ZCS converters in Chapter 4, one obtains
Fs 1 — ∙Dp(Af)
⅞ G(∕,→)
This expression, valid for all ZVS quasi-resonant converters, allows plotting of the dc characteristics. Note that the function G is unchanged by the duality transformation;
the argument alone is inverted.
When duality is applied to the small-signal ac model, the subscripts of the h- parameters must be changed, in addition to the transformations of Table 6.2. The pa
rameter A1∙u, for example, indicates the value of a current source controlled by a voltage
hiffs
Figure 6.1: The emall-signal model of a zero-voltage-switching quasi-resonant converter.
(6∙4) (6∙5) (6∙6) (6∙7) (6∙8) (6∙9) (υoff)∙ When currents and voltages are interchanged, the source including Λ,∙u is now a voltage source controlled by the current ton, so the appropriate parameter of the voltage source is denoted Λu,∙. The general small-signal model of a ZVS quasi-resonant converter is shown in Fig. 6.1, and the corresponding A-parameters are given by
hυυ = [1 - Z>p(M)] (1 - m) l _ mE[l - Pp(M)]
Λ -· - μ
l „ [1 - Dj>(M)]
h"' ~ v°"--- ft--- , mΛf[l - Dp(M)]
hi- =---B--- hii = (l + m)[l-Dp(M)}
l τ [1 - Pp(M)]
λ>7 = Ion---JΓ--- ∙
Note that m, actually m(p) for the ZCS converter, is replaced by m(p-1) in the ZVS converter.
vt
. D2 Lr
i⅛—ΟΠΤΟ
(a)
Λt't,on
(b)
Figure 6.2: A zero-voltage-8witching boost converter (a) and its small-signal model (b).
Since G∕w(p) is nearly constant with p, so is G∕w(p~1). Without any analysis, then, one can surmise that full-wave ZVS resonant converters will behave like their PWM counterparts, both at dc and under low-frequency ac modulation, with duty ratio control D replaced by the “complementary” switching frequency control (1 — Fs/Fo).
Alternatively, one can think of the control variable F$ ∕Fq replacing the complementary duty ratio 1 — 2?. The small-signal model for the half-wave ZVS switch, like the model for the ZCS switch, exhibits lossless damping and a lack of reciprocity.
Figure 6.2 shows an example of a ZVS boost converter, along with its small-signal circuit model.
Figure 6.S: A four-quadrant switch that allows all combinations of forward and reverse current and voltage.