State-plane analysis of a constant-frequency clamped-mode parallel-resonant converter

364 IEEE TRANSACTIONS ON POWER LLECTRONICS. VOL 3 . NO 3. JULY 1988

State-Plane Analysis of a Constant-Frequency Clamped-Mode Parallel-Resonant Converter

Absfruct-A novel clamped-mode parallel-resonant converter which operates at a constant frequency and provides a wide output regulation range is proposed and analyzed. Employing graphical state-plane techniques, five circuit operating modes are identified and their mode boundaries defined. Regions for natural and forced commutation of power devices are specified. The dc control-to-output characteristics are derived to facilitate converter design. The predicted operating modes are experimentally verified using a 105-kHz prototype circuit.

I. INTRODUCTION PARALLEL-resonant converter (PRC) can be oper- A ated with either zero-voltage turn-on (above the res-

onant frequency) [ l ] or zero-current turn-off (below the resonant frequency) [2], 131 to eliminate the switches’ turn-on or turn-off losses. Due to the reduction of switching losses, the converter is particularly suited for high- power and high-frequency operation.

Conventionally, a PRC is controlled employing frequency-modulation techniques. Due to the line and load variations, the operating frequency of a PRC usually var- ies over a wide range to regulate the output. This results in a penalty in filter design and poor utilization of mag- netic components.

Recently, a phase-modulation scheme was proposed to operate a PRC at a constant frequency 141, [ 5 ] . Employ- ing two resonant inverters connected as in Fig. l(a), the output vo of a phase-controlled parallel-resonant converter (PC-PRC) can be regulated by controlling the phase displacement between the triggering signals to the inverters without varying the frequency.

One disadvantage of the proposed phase-control scheme is that the two resonant inverters see their common load with different power factors due to the phase displacement between them [ 5 ] . The inverters thus operate with unbal- anced currents and voltages in their tank circuits. The current imbalance results in excessive component stresses in one of the inverters (usually the inverter with lagging triggering signals). Another disadvantage of a PC-PRC is the high circulating currents in the inverters at light load. The

Manuscript received August 5 , 1987; revised April 8 , 1988. This work was originally presented at INTELEC’87. Stockholm. Sweden, June 14- 17, 1987.

F. S . Tsai and F. C . Y . Lee are with the Virginia Power Electronics Center, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061.

Y . Chin is with Digital Equipment Corporation, Maynard. M A 01754- 257 1 .

IEEE Log Number 8822282

circulating currents increase as the load decreases since the operating frequency of the inverters is unchanged.

To alleviate the aforementioned difficulties, a clamped- mode parallel-resonant converter (CM-PRC), which is functionally equivalent to a PC-PRC, is proposed [ 6 ] . As shown in Fig. l(d), a CM-PRC consists of four controlled switches, S1, S2, S3, and S4, and a single resonant tank. It is derived by combining the two resonant tanks in a PC- PRC in a procedure illustrated in Fig. 1. The four controlled switches are operated in a timing sequence shown in Fig. l(f) to generate a quasi-square-wave voltage vs across the resonant tank. The frequency of vs is fixed. The pulsewidth of us is controlled by varying the phase displacement between the conduction intervals of S1 (S3) and S2 (S4) . The output of the converter is regulated by controlling the pulsewidth of us.

Since there is only one resonant tank in a CM-PRC, the problem of current and voltage imbalance in a PC-PRC is eliminated. Nonetheless, the switching devices carry different peak and rms currents [7]. The tank current reduces as the load decreases since the pulsewidth of the exciting voltage vs is reduced.

In this paper, graphical state-plane techniques are em- ployed to analyze a CM-PRC. By directly viewing the circuit behaviors on the state plane, various circuit operating modes are easily identified. The mode boundaries are defined and operating regions for natural commutation and forced commutation of power switches are specified. The control-to-output characteristics are derived to facilitate a converter design. Finally, the analytical results are substantiated using a prototype circuit.

11. OPERATION OF A CM-PRC

The four controlled switches S1, S2, S3, and S4 in a CM-PRC are implemented by transistors Q1, Q2, Q3, and Q4 and their antiparallel diodes D1, D2, D3, and D4 in Fig. 2(a). The transistors operate at 50 percent duty cycle and are triggered in a timing sequence illustrated in Fig. 2(b). Transistors Q1 and Q3 are triggered according to a clock signal whose frequency determines the converter’s operating frequency. Transistors Q2 and Q4 are triggered with a controllable time delay + s / w s with respect to the triggering of Q1 and 4 3 . The time delay is the clumped interval during which the resonant tank is short-circuited by a transistor and a diode. By varying the clamped interval, the converter’s output voltage Vo is regulated.

0885-8993/88/0700-0364$01 .OO 0 1988 IEEE

TSAI el U / . : STATE-PLANE ANALYSIS OF PARALLEL-RESONANT CONVERTER 365

a (c l

Fig. 1. Derivation of CM-PRC from PC-PRC. (a)-(c) PC-PRC. (d)-(f) CM-PRC

A typical circuit operation of a CM-PRC is illustrated in Fig. 2(b). At t = 0, transistor Q1 turns on while diodes D3 and D4 are conducting. Diode D3 is commutated due to the conduction of Q1 and the inductor current i, resonates through Q1 and D4. At t = t l , transistor Q2 is triggered, commutating diode D4. The inductor current resonates through Q 1 and Q2. At t = t 2, i, decreases to zero due to resonance; transistors Q1 and Q2 turn ofnaturally and subsequently, diodes D l and D2 conduct. The inductor current resonates through D1 and D2. At the end of the first half of a switching cycle t = T s / 2 , transistar Q? turns on, commutating diode D 1, and a similar process occurs with the roles of Q1, Q2, D l , D2 and Q3, Q4, D3, D4 interchanged, respectively. The reflected load current iR changes polarity whenever the capacitor voltage vc crosses the zero axis.

In this example, all the transistors Q1, Q2, Q3, and Q4 are naturally commutated. As will be seen later, it is also possible that Q1 and Q3 or Ql-Q4 are force commutated under other operating conditions. In these cases, a simple lossless snubber C, can be used across the force-com-

mutated transistors, as illustrated in Fig. 2(a), to reduce their turn-off losses. This is possible since the force-commutated transistors always turn on at zero voltage and no energy in the snubbers is dumped through the transistors [ l l .

111. STATE-PLANE ANALYSIS State-plane techniques have been successfully em-

ployed in characterizing the steady-state and transient behaviors of resonant converters [6]-[lo]. In this section, various equilibrium trajectories characterizing different operating modes of a CM-PRC are constructed. The circuit behaviors of a CM-PRC are clearly portrayed in the state plane. To simplify the analysis, the converter's operating frequencyf, is limited between 50 and 100 percent of the resonant frequencyf, = 1 /27r C C , and the following conditions are assumed:

1) all the switches are ideal with zero switching time

2) the quality factor of the resonant tank is infinite, i .e., and no conduction drop;

no loss in the tank circuit:

366

3)

4)

IEFF TRANSACTIONS ON P O W k R FLFCTRONICS. VOL 3. NO 3. J U L Y 1988

(b)

Fig. 2 . Circuit operation of CM-PRC

the output filter is large enough that the output current I , can be assumed constant during several switching cycles; the transistors are driven with 50-percent duty cycle gating signals.

A . Circuit Topological Modes The one-cycle operation of a CM-PRC is composed of

a sequence of linear-circuit stages, each corresponding to a particular switching interval, as illustrated in Fig. 2(b). Eight possible circuit topologies exist, as shown in Fig. 3 , for a CM-PRC. These topologies are referred to as circuit topological modes. A steady-state operation of a CM-

(a) CM-PRC. (b) Circuit waveforms

PRC comprises a specific sequence of these topological modes.

The expressions for the inductor current iL and the capacitor voltage ziC for each topology can be easily derived by solving the corresponding circuit equations. These expressions are further manipulated to derive a set of trajectory equations which represent the relationship between uC and iL in the state plane ( z i C versus iL plane). The trajectory equations for all the topological modes are also shown in Fig. 3 , where

vcN = u c / E iLN = i L / ( E /Z , )

normalized capacitor voltage, normalized inductor current,

TSAI er a l . : STATE-PLANE ANALYSIS OF PARALLEL-RESONANT CONVERTER 367

( iLN - + (VCN - 1)2 = R 2 21

i L

“T -p-$+o - [iLN - + (vCN + 1)2 = R2

(h)

Fig. 3. Topological modes of CM-PRC and trajectory equations. (a) M1: uC < 0, Q1, Q2 or D1. D2 on. (b) M2: t i c > 0, Q I , 4 2 or D1, D2 on. (c) M3: uC > 0. Q1, D4 or 4 4 , D1 or Q2, D3 or 4 3 , D2 on. (d) M4: t ’<. > 0. Q3, 4 4 or D3, D4 on. (e) M5: t j C < 0, Q3, 4 4 or D3, D4 on. ( f ) M6: uc < 0, Q1, D4 or Q4, D1 or Q2, D3 or 4 3 . D2 on. (8) ML: I ! ( . = 0. Q1, Q2 or D1, D2 or Q3, Q4 or D3, D4 on. (h) MO: uC = 0, Q 1 , D4 or Q4, D1 or Q2, D3 or 43 , D2 on.

368 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 3 . NO 3 . J U L Y 19x8

VCoN = V c o / E normalized initial capacitor volt-

ILON Z L o / ( E / Z o ) normalized initial inductor cur-

I,, = ZO/( E / Z o ) normalized load current, 2, = JLTC characteristic impedance.

age,

rent,

The normalizing factors for voltages and currents are E , the input voltage, and E / Z o , respectively.

Topological modes M I , M2, M3, M4, M5, and M6, are called “resonant modes.” Their trajectories in the state plane are circular arcs with centers located at ( 1 ,

-I,,), and radii as derived in Fig. 3. Fig. 4 shows a family of the trajectories for the resonant topological modes, where the centers for Ml-M6 are indicated by ml-m6, respectively. Time is implicit in these trajectories. ’The time elapsed in a topological mode is measured by the angle subtended by the trajectory with respect to its center. For example, the time elapsed in topological mode M l from point a to point b in Fig. 4(a) is A t = y / w o , where wo = 2rfo is the angular resonant frequency. As time advances, the trajectory travels in the direction indicated by the arrow.

Topological mode ML is called ‘ ‘linear-charging mode. ” It occurs when the magnitude of the inductor current 1 i12, l is less than I,, as ucN = 0 crosses the zero axis and us = E or -E . In this topological mode, the resonant capacitor C is shorted through the output bridge and the inductor L is linearly charged by the source voltage [ lo] . The trajectory for ML is a line segment along the iLN axis, as illustrated in Fig. 4(g). The amount of time elapsed in this mode is proportional to the link 1 of the line segment, A t = l / w o .

Topological mode MO is called “freewheeling mode. ” It occurs at a similar condition as ML except that us is clamped at 0 V as uc crosses the zero axis. In this topological mode, C is shorted through the output bridge, and iLN freewheels through a shorted path in the resonant inverter. The trajectory for MO is a stationary point lying on the iLN axis with coordinate (0, I L O N ) or (0, as illustrated in Fig. 4(h). Finite time elapses at the stationary point since both iLN and ziCN are independent of time. A circle is used to indicate such a stationary point.

-IC”. (1 , b N ) , (0, I,,), ( - 1 , I O N ) , ( -1 , -Io,), (0,

B. Equilibrium Trajectories

An equilibrium trajectory is represented by a closed contour which is symmetric with respect to the origin. It is constructed via a particular sequence of circuit topological modes determined by the circuit operation. An equilibrium trajectory represents a steady-state operation and can be handily used to characterize the circuit behav- ior of a converter.

The equilibrium trajectories for a CM-PRC can be constructed using a composite diagram generated by overlap- ping Fig. 5(a)-(g) on the same axes. The rules for con- structing equilibrium trajectories in the composite diagram are as follows.

1) On the left half-plane ( vchr < 0) , triggering Q1 initiates M6, triggering Q2 initiates M1, and triggering Q4 initiates M5.

2) On the right half-plane (ucN > 0) , triggering Q3 initiates M3, triggering Q4 initiates M4, and triggering Q2 initiates M2.

3) On the iLN axis ( Z J ~ , = 0), a) if the magnitude of iLN is greater than ION,

triggering of Q2 initiates M2 and triggering of 4 4 initiates M5;

b) if the magnitude of iLN is greater gering event occurs,

M2 is initiated if the previous




MI ,

M6,

M4 >

M3 9

than ION and no trig-

topological mode is

topological mode is

topological mode is

topological mode is

(the switching of the circuit’s topological mode results from uc changing polarity); c) if the magnitude of iL,,, is equal to I,,,

MO is initiated if the previous topological mode is M3 or M6 (MO will be terminated when Q2 or Q4 is triggered, which initiates M2 or M5, respectively ), M2 is initiated if the previous topological mode is ML and iLN > 0, M5 is initiated if the previous topological mode is ML and iLN < 0;

and d) if the magnitude of iLN is less than ION,

MO is initiated if the previous topological mode is M3 or M6; MO will be terminated when Q2 or Q4 is triggered, which initiates ML; ML is initiated if the previous topological mode is MI or M4.

Fig. 5 illustrates the construction of the equilibrium trajectory representing the circuit operation discussed in Section 11. In Fig. 5 , M6 ( M3 ) is initiated when Q 1 ( Q3 ) is triggered at point a ( c ) and M1 (M4) is initiated when Q2(Q4) is triggered at point b ( d ) . Mode M2( M5) is initiated when Z I ~ ~ ~ reverses polarity.

A systematic way to construct equilibrium trajectories of a CM-PRC is illustrated in Fig. 6. Beginning with an equilibrium trajectory representing a conventional PRC operation (trajectory aoscotao in the left side of Fig. 6(a), or trajectory aO~s’cOtt’aO in the right side of Fig. 6(a)), the triggering point of Q1 ( a o , a,, a?, * * ) is gradually moved counterclockwise along the trajectory of M5. By doing so, the effective duty ratio, ps / ( os T s ) , of the trajectory is gradually reduced. Various equilibrium trajectories are obtained. Fig. 6(a) shows the equilibrium trajectories for the case when transistors Q1-Q4 are all

TSAI P I U / . STATE-PLANE ANALYSIS OF PARALLEL-RESONANT CONVERTER 369

ION t-- ---- I

- 1 -

“ 0 4

‘ON

Fig. 4. State trajectories corresponding to topological modes of Fig. 3

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~ L N

TRONICS. VOL. 3. NO. 3. JULY 1988

Fig. 5. Construction of equilibrium state trajectory in composite diagram

Fig. 6. Systematic way to constmct equilibrium trajectories of CM-PRC. (a) Natural-commutation trajectories. (b) Force-commutation trajectories: Q1, 4 3 force-commutated, and Q2, 4 4 naturally commutated. (c) Force-commutation trajectories: Q 1 , 02. 03. Q4 all force-commutated.

TSAI P I U / . : STATE-PLANE ANALYSIS OF PARALLEL-RESONANT CONVERTER 37 I

naturally commutated. Fig. 6(b) shows the equilibrium trajectories when transistors Q 1 , 4 3 are force-commutated and Q2, Q4 are naturally commutated. Fig. 6(c) shows the equilibrium trajectories when all the transistors, Ql-Q4, are force-commutated. The triggering points of Q2 in these trajectories are indicated by bo, b , , b2, . . . . Fig. 7 summarizes all the possible equilibrium tra-

jectories. The corresponding circuit waveforms for each trajectory are also shown. The circuit operation for each trajectory is briefly discussed in the following.

1) Natural-Commutation Trajectories: Trajectory I : As illustrated in Fig. 7(a), prior to t =

a , D3 and D4 are conducting. At t = a , Q1 is switched on, and D3 is commutated. The inductor current iL resonates through Q1 and D4(M6) . At t = b, Q2 turns on and diode D4 is Commutated. The inductor current resonates through Q1 and 4 2 (M1 ). At t = e , uc changes polarity. The inductor current continues to resonate through Q1 and 4 2 (M2) . At t = d , Q1 and Q2 turn off naturally, and subsequently, diodes D l and D2 conduct. The inductor current resonates through D1 and D2. At t = e , Q3 turns on, and a similar process occurs with the roles of Q 1, Q2, D 1, D2 and Q3, 4 4 , D3, D4 interchanged, respectively. The topological mode sequence of this trajectory is M6-Ml-M2-M3-M4-M5, which is defined as mode-I operation of CM-PRC.

Trajectory ZZ: As illustrated in Fig. 7(b), the circuit operation for this trajectory is similar to Trajectory I except that oc changes polarity before Q2 (Q4) is triggered at t = e ( g ) . The topological mode sequence of this trajectory is M6-M3-M2-M3-M6-M5, which is defined as mode-I1 operation.

Trajectory ZZZ: As illustrated in Fig. 7(c), the circuit operation for this trajectory is similar to Trajectory I ex- pect that when uc crosses the zero axis, the magnitude of iL is less than Zo. As a result, a linear-charging period (ML) of the inductor current exists. The topological mode sequence of this trajectory is M6-M 1 -ML-M2-M3-M4- ML-MS, which is defined as mode-I11 operation.

Trajectory ZV: As illustrated in Fig. 7(d), the circuit operation for this trajectory is similar to Trajectory I11 except that uC crosses the zero axis before Q2 or Q4 is triggered. Thus, in addition to the linear-charging period, a freewheeling period (MO) also exists. The topological mode sequence of this trajectory is M6-MO-ML-M2- M3-MO-ML-M5, which is defined as mode-IV operation.

2) Force Commutation Trajectories: The force-commutation trajectories can be divided into two categories. In the first, Q1, Q3 are force-commutated and Q2, Q4 are naturally commutated.

Trajectory Z’: As illustrated in Fig. 7(e), at t = a , Q1 is triggered and Q3 is forced off. Diode D1 conducts since iL is negative. The inductor current resonates through D1 and Q4 (M6) . At t = b, iL reverses polarity. Diode D1 and transistor 4 4 are commutated naturally and subsequently, transistor Q1 and diode D4 conduct. The inductor current resonates through Q1 and D4. At t = e ,

Q2 turns on commutating diode D4. The inductor current resonates through Q1 and 4 2 ( M 1 ). At t = d, uc changes polarity. The inductor current continues to resonate through Q1 and Q2 (M2) . At t = e , 4 3 is triggered, and Q1 is forced off. A similar process occurs with the roles of Q1, Q2, D1, D2 and 4 3 , 4 4 , D3, D4 interchanged, respectively. The topological mode sequence of this trajectory is the same as Trajectory I. Thus it belongs to the mode I operation.

Trajectory ZZ’: As illustrated in Fig. 7(f) , the circuit operation of this trajectory is similar to Trajectory I’ except that vc changes polarity before Q2 (Q4) is triggered. The topological mode sequence of this trajectory is the same as Trajectory 11.

Trajectory IZZ’: As illustrated in Fig. 7(g), the circuit operation for this trajectory is similar to Trajectory I’ ex- pect that when uc crosses the zero axis, the magnitude of iL is less than I,. As a result, a linear-charging period (ML) exists in the inductor current. The topological mode sequence of this trajectory is the same as Trajectory 111.

Trajectory ZV’: As illustrated in Fig. 7(h), the circuit operation for this trajectory is similar to Trajectory 111’ except that vc crosses the zero axis before 4 2 or 4 4 is triggered. Thus in addition to the linear-charging period, a freewheeling period (MO) also exists. The topological mode sequence of this trajectory is the same as Trajectory IV.

In the second category, Q1, Q2, Q3, Q4 are force-commutated.

Trajectory I ” : As illustrated in Fig. 7(i), prior to r = a , Q3 and Q4 are conducting. At t = a , Q3 is forced off, and Q1 is triggered. Diode D1 conducts since iL is negative. The inductor current resonates through Q4, D1 (M6) . At t = 6 , Q2 is triggered and Q4 is forced off. Diode D2 conducts since i, is still negative. The inductor current resonates through D 1, D2 ( M 1 ). At t = e , iL in- creases to zero due to resonance. Diodes D1, D2 turn off naturally, and subsequently, transistor Q 1, Q2 conduct. At t = d , uc changes polarity. The inductor current continues to resonate through Q 1, Q2 (M2) . At t = e , Q1 is forced off, and Q3 is triggered. A similar process occurs with the roles of Q1, Q2, D1, D2 and Q3, Q4, D3, D4 interchanged, respectively. The topological mode sequence of this trajectory is the same as Trajectory I.

No mode I1 operation is possible. The circuit operations for Trajectories 111” and IV”

are illustrated in Fig. 7(j) and (k), respectively, which can be easily described in a similar manner as in Trajec- tory I ” . The topological mode sequences of these trajectories are the same as Trajectories I11 and IV, respectively.

Trajectory V: As illustrated in Fig. 7(1), this trajectory is degenerated from either Trajectory 111” or Trajec- tory IV”. The capacitor voltage uc in this trajectory is always zero. In other words, the resonant capacitor C is always shorted through the output bridge rectifier. The circuit is either in the linear-charging mode or the freewheeling mode. No resonant action occurs, and no output

372 IEEE TRANSACTIONS O N POWEK tLtCIKONICS. V O L 3. N O 3 . J U L Y 1 9 X X

M6 . . . M1 M3.314

t jLN M3 M6

M6 M1 ML M3M4ML

(d)

Fig. 7 . Equilibrium trajectories and their corresponding circuit waveforms. (a ) Trajectory I . (b ) Trajectory 11. (c ) Tra.jectory 111. (d) Trajectory 1V. (e) Trajectory 1' . ( f ) Trajectory 11'. (g) Trajectory 111'. ( h ) Trajectory IV ' . ( i ) Trajectory I " . (,i) Trajectory 111". (k) Trajectory I V " . ( I ) Trajectory V .

TSAl er (I/. ' STATE-PLANE ANALYSE OF PARALLEL-RESONANT CONVERTER

t j L N

0 1 0 1 Q3 0 3 ' i E I D 4 i Q2 i :: i D 2 / Q4 i

MOML M5 M q M L M 2

~

3 74 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 3. NO 3. JULY 1988

t iLN

* "CN

t jLN

t iLN

I MO / M L : MO ~ M L :

TSAI e r a1 STATE-PLANE ANALYSIS OF PARALLEL-RESONANT CONVERTER

TABLE I CIRCUIT OPERATING MODES OF CM-PRC A N D THEIR CORRESPONDING

TOPOLOGICAL MODE SEQUENCES

Operating Modes Topological Mode Sequences

M6 + M1 + M2 + M 3 + M4 + M5 M6 + M3 + M2 + M3 + M6 + M5 M6 + M1 + ML + M2 + M3 + M4 + ML + M5 M6 + MO + ML + M2 + M3 + MO + ML + M5 MO + ML + MO + ML

I I1 111 IV V

8

315

m

Fig. 8. Mode boundaries. NC = natural commutation. FC = forced commutation. (a) us,\ = 0.6 . (b) u5,% = 0.7. (c) = 0.8. (d) wSN = 0.9.

voltage is generated. The topological mode sequence is MO-ML-MO-ML, which is defined as mode V operation. This operating mode only exists momentarily when the output is short-circuited and the output inductor energy discharges to the load through the bridge rectifier.

Table I summarizes the topological mode sequences for all the operating modes. Note that the trajectories shown

in Fig. 7 are used only to illustrate the existence of various circuit operating modes of a CM-PRC. Their corresponding frequencies are not necessarily the same.

C. Mode Boundaries and Operating Regions To characterize a CM-PRC, it is crucial to know the

circuit’s operating mode (topological-mode sequence) for

~

376 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 3 , NO 3. J U L Y lY88

a given circuit condition. Knowing the converter's operating mode enables one to construct the corresponding equilibrium trajectory according to the specific topological mode sequence. The radii and angles defining the trajectory can then be calculated from the given condition and used to calculate various salient features of the circuit such as the average output voltage, peak capacitor voltage, rms inductor current, rms switch currents, and average diode current [ 111. Thus defining the operating regions is the first step toward characterizing a CM-PRC. The boundaries between various operating modes have been derived numerically. Fig. 8 shows the mode boundaries as a function of ION and wSN, where wSN = w s / w o is the normalized switching frequency. A dotted line is used in each figure to indicate the boundary between natural and force commutation regions. Above the dotted line, all the transistors are naturally commutated. Below the dotted line, at least two of the transistors (Q1 , Q3 ) are force- commutated. Angle ps is measured in degrees. A ps = wsTs is equal to 180". Using this figure, the operating mode of a CM-PRC for a given ION and ps can be easily determined. For example, if I O N = 0.5, ps = 120", and wSN = 0.8 , the converter will operate in Mode I1 with natural commutation, as illustrated in Fig. 8(c) by an as- terisk.

D. Control to Output Characteristics Fig. 9 shows the average output voltage VON as a func-

tion of Os and wSN. Again, a dotted line is used in the figure to indicate the natural-commutation boundary. Above the dotted line, all the transistors are naturally commutated. Below the dotted line, at least two of the transistors are force-commutated. It can be seen from the figure that the natural-commutation region of a CM-PRC is limited especially at frequencies less than 80 percent of the resonant frequency. It is thus difficult to design a CM- PRC in the natural commutation region to accommodate a wide load range.

One altemative is to design a CM-PRC in the force- commutation region where two of the transistors, Q1 and Q3, are force-commutated while the other two transistors, Q2 and 4 4 , are naturally commutated. By doing so, the tum-on losses of Q1 and Q3 and the tum-off losses of Q2 and 4 4 are eliminated. A simple capacitor lossless snubber can be used across Q1 and Q3 to reduce their tum-off losses. A CM-PRC can be designed to accommodate a wide load range in this region. A simple example to illustrate the design of a CM-PRC in both natural and force commutation regions follows.

Given input voltage = 40 - 60 V output voltage = 5 V switching frequency = 80 percent of the resonant frequency,

and assuming a transformer with tum ratio n is used between the resonant capacitor and the output bridge, then

5n 5n 40 60

VON(max) = - Vo,,,(min) = -.

I ) Design in the Natural-Commutation Region: Case I: Choose VON (min) = 1.75, as shown in Fig.

10(a), to obtain maximum possible load current. The tum ratio of the transformer is then calculated as

= 21 1.75 x 60

5 n =

and 5n 40

From Fig. 10(a), this voltage level cannot be achieved at wSN = 0.8, thus the design is impossible.

Case ZZ: Choose VON (max) = 2.13, as shown in Fig. 10(b), to ensure output regulation for the entire input voltage range:

VoN(max) = - = 2.625.

2.13 X 40 5

= 17 n =

and vON (min) = - 5n = 1.42.

60 From Fig. 10(b), the load current magnitude ION cannot exceed 0.25. Otherwise, the converter will operate in the force-commutation region. This design can be achieved either by using a small characteristic impedance Z,, or by limiting the output load current magnitude IO to a small value. In either case, the circulating current in the inverter circuit will be high and the components will experience excessive stresses.

2 ) Design in the Force-Commutation Region: Choose VoN(max) = 1.63, as shown in Fig. lO(c), to obtain an ION up to 1.0:

1.63 X 40 5

= 13 n =

and 5n 60

VON (min) = - = 1.083.

From Fig. 1O(c), an ION range from 0.45 to 1.0 is obtained. When a smaller VoN (max) is chosen, a wider ION range results. The circuit, however, may begin to operate in the region where all the transistors are force-commutated. The range of VON and ION determines the range of ps angle, which can be used to determine the circuit's operating modes from Fig. 8 to predict the converter's behaviors.

Notice that a lossless snubber cannot be used across switches that may be naturally commutated during the operation. Therefore, it is recommended that the design be confined either in the natural-commutation region or in the force-commutation region. In the force-commutation region, the converter can be designed to accommodate a wide range of input and output variations. However, a minimum load current must be maintained.

IV. HARDWARE EXPERIMENTS A prototype CM-PRC operating at 105 kHz has been

built to verify the analytical predictions. The resonant frequency of the circuit is designed at 145 kHz and the char-

TSAI et al. : STATE-PLANE ANALYSIS OF PARALLEL-RESONANT CONVERTER 371

0

ps 8

Fig. 9. [IC conrrtrl-to-output charactenstlcs. ( a ) U,, = 0.6. (b) us\ = 0. (c ) w.~, = 0.8. (d) u5, = 0.9.

I.

0 ?

Fig. I O . Design examples.

acteristic impedance is designed at 50 Cl. A 50-V input voltage is used. The experimental results are shown in Fig. 11 where the state trajectories are obtained by exter- nally calibrating the inductor current so that the ratio of the y-axis scale to the x-axis scale is the same as the ratio of iLN to uCN. All the predicted operating modes have been illustrated in the oscillograms except operating mode V. Fig. l l (g) shows a trajectory close to operating mode V which is obtained by shorting the output. Slight asym- metry in the trajectories is observed due to the imbalance of gating signals.

Fig. I I Experimental rewlt\ .

V . CONCLUSION A comprehensive dc analysis of a clamped-mode par-

allcl-resonant converter is presented. The analysis em- ploys state-plane techniques to identify five circuit operating modes in the frequency range from 50 to 100 percent of the resonant frequency. The circuit behaviors in each operating mode are characterized by an equilibrium tra- jector! in the state plane, which is constructed via a particular sequence of topological modes. Regions for all the operating modes are defined. and the boundary between natural and force commutations is specified.

The con t ro I - t o -o u t put c h ;t rac t c r i s t i c s have been derived to facilitate a converter design. I t is shown that natural comniutation o f power switches can be achieved only with restrictt'cl load range. especially when the switching frequency is less than 80 percent of the resonant frequency. 7'0 acconiiiiodate wider ranges of line and load variations, a n altt'rnati\'c is suggested to operate a CM-PRC in the

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State-plane analysis of a constant-frequency clamped-mode parallel-resonant converter