Dynamic and Steady-State Models for the PRC-LCC Resonant Topology With a Capacitor as Output Filter

2262 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 4, AUGUST 2007

Dynamic and Steady-State Models forthe PRC-LCC Resonant TopologyWith a Capacitor as Output Filter

Juan A. Martín-Ramos, Member, IEEE, Juan Díaz, Member, IEEE, Alberto M. Pernía, Member, IEEE,Juan Manuel Lopera, Member, IEEE, and Fernando Nuño, Member, IEEE

Abstract—In high-voltage dc power supplies (hundreds of kilo-volts), avoiding the inductance in the output filter is an importantasset. Thus, a capacitor as a unique component of that filter ispreferable. On the other hand, a high-voltage step-up transformershows a high capacitance in its secondary, which can be includedin a resonant stage. Both capacitances, i.e., one from the filterand the other from the transformer, are separated by the outputrectifier. When this rectifier is on, they are in parallel; when it isoff, they are disconnected. A resonant stage where a very largecapacitor appears and disappears several times in one period isnot linear and has not been dynamically modeled before. Thispaper shows two simple models for the parallel–series (LCC-typeparallel resonant converter) resonant topology with a capacitoras output filter. Its dynamics and steady state are studied andconstitute an important help for any designer.

Index Terms—DC/DC converters, high voltage, modeling,resonant converters.

I. INTRODUCTION

NOWADAYS, resonant topologies are a very attractivesolution in high-power applications. They show quasi-

sinusoidal voltage and/or current waveforms that favor thereduction of losses and electromagnetic interference. Further-more, in some applications, the reactive effects in the powerstage are noticeable and unavoidable, and a resonant topologythus appears as a natural solution. An example of this is a num-ber of high-power (tens of kilowatts) and high-voltage (tens orhundreds of kilovolts) applications, where a step-up trans-former is needed. The stray capacitance and the leakage in-ductance of these transformers are always very important dueto their constructive restrictions, such as long isolation dis-tances and large turns ratio. In fact, their resonant frequencyis often lower than 100 kHz, even in the best transformerdesigns [1], and any simplified equivalent circuit must considerthem (Fig. 1).

Manuscript received May 31, 2006; revised December 22, 2006. This workwas supported by the Spanish Government through the Interministerial Scienceand Technology Commission (CICYT). Earlier versions of this paper werepresented at the IEEE 17th Applied Power Electronic Conference (IEEEAPEC ’02), Dallas, TX, March 2002, and at the IEEE Power ElectronicsSpecialist Conference (IEEE PESC ’02), Cairns, Australia, June 2002.

The authors are with the Electronic Technology Area, University of Oviedo,33204 Gijón, Spain (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2007.894763

Fig. 1. Equivalent circuit for a step-up transformer.

Fig. 2. Series–parallel resonant converter PRC-LCC with a capacitor asoutput filter. The resistor “r” models the losses in the resonant tank, and valueson the secondary of the transformer have been marked with an apostrophe.

Having a transformer such as that in Fig. 1, it appears asa good idea to include the parasitic components of parallelresonant converters (PRCs) into a resonant net. However, theseconverters are difficult to control if load and input voltage varywidely. As a consequence, series resonant converters (SRCs)are commonly used [1]–[3] because their characteristics arebetter. However, since the SRC does not cope with the trans-former stray capacitance, its use is not free from problems.In this paper, an alternative solution is considered, i.e., theseries–parallel PRC-LCC resonant topology.

The PRC-LCC topology, adding an inductance–capacitance(LC) filter at the output, has been compared to the PRC andSRC in [1], [4], and [11]. The aforementioned studies concludethat PRC-LCC shows a number of more desirable features ascompared with the series or parallel resonant inverters, whichare capable of operating with wide output voltage and outputpower ranges. In fact, dc/dc conversion with the PRC-LCC res-onant topology with LC filter is well known and has been wellexplored by a number of authors (e.g., [5]–[7]). Unfortunately,in kilovolt applications, the presence of an inductor in the filteris a serious drawback, and therefore, the use of the PRC-LCCresonant topology with only a capacitor as output filter (Fig. 2)and a frequency and duty cycle combined control was proposedin [8] and [11].

0278-0046/$25.00 © 2007 IEEE

MARTÍN-RAMOS et al.: DYNAMIC AND STEADY-STATE MODELS FOR THE PRC-LCC RESONANT TOPOLOGY 2263

Fig. 3. (Top) Oscilloscope showing the shape of the resonant current in aPRC-LCC prototype (with 2A/div). (Bottom) FFT analysis of the current. Themagnitude of the first harmonic of the current is 7.5 dBV, and the third harmoniccontent is −17.4 dBV (5% of the first-harmonic content).

Of course, avoiding the inductance in the output filter is animportant asset for any high-voltage converter, but the behaviorof the topology changes completely. In Fig. 2, it can be ob-served that the capacitances from the filter and transformer areonly separated by the output rectifier. When the output rectifieris on, they are connected in parallel; when the output rectifier isoff, they are disconnected. A resonant stage where a very largecapacitor (from the filter) appears and disappears several timesin one period is not linear and shows additional complexityfrom the modeling point of view. In fact, complex behaviorand lack of models have been traditionally the major hindranceto implement this topology as a solution [1], [10]. Certainly,some steady-state models have been shown [8], [11]–[13], butthese models are far too complicated. Most of them divide thetopology into several operation modes (up to nine modes), anda different model is given for each of them. In any case, therehas been no attempt at all in the literature that is directed atthe dynamic modeling of the PRC-LCC resonant topology withonly a capacitor as output filter [9]. In this paper, the differentsteps to obtain this dynamic model are shown. Moreover, avery simple steady-state solution is also described, introducinga useful and unique equivalent circuit for the whole topology(no modes). As a result, the understanding of the topologyis simplified, and a very useful tool is at the disposal of thedesigner.

II. NONLINEAR STATE EQUATIONS: FIRST VERSION

For the intended applications (high voltage), the PRC-LCCconverter is used near the resonance of the reactive elementsLS , CS , and CP . Thus, as a basic assumption for any model,the current iL is continuously flowing in the resonant stagewith at least a nearly sinusoidal shape. The fast Fourier trans-formation (FFT) of that current, as shown in Fig. 3 from aprototype, supports this first-harmonic approach. This currentalways flows through CS , which also shows a nearly sinusoidalvoltage. Notwithstanding, things are not the same in CP . Whilethe voltage VP is lower than that reflected for the primary outputvoltage V0, the output rectifier cannot be turned on, whichobliges the current iL to flow throughCP . During this lapse, VP

also has a nearly sinusoidal evolution. However, when VP triesto surpass V0, the output rectifier is automatically turned on,and VP is clamped to the value of V0. (Of course, this clampingoccurs because Cf is much larger than CP ). Only when thecurrent reaches zero is the output rectifier turned off again, andthe sequence is repeated with a different sign. Fig. 3 shows themain waveforms for the whole converter.

Once the waveforms of the topology are known, a mathe-matical formulation for its behavior can be done. Inspecting thecircuit for the topology (Fig. 2), three nonlinear state equationscan be easily obtained: The equation of the resonant loop (1a),the relationship between the series capacitor current and itsvoltage (1b), and the equation of the knot after the outputrectifier (1c) are as follows:

VAB(t) =VS(t) + VP (t) + LS · diL(t)dt

+ r · iL(t) (1a)

CS · dVS(t)dt

= iL(t) (1b)

iD =CfdV0(t)dt

+V0

R. (1c)

The definition of every variable and parameter in (1a)–(1c)is given in Fig. 2. Notice that r is intended to quantify lossesin the resonant tank and that the magnitudes in (1c) have beenreduced to primary (absence of apostrophe in the symbol) tosimplify this paper.

Equations (1a)–(1c) form a system where six variables areshown, i.e., 1) VS(t), 2) VP (t), 3) V0(t), 4) iL(t), 5) iD(t),and 6) VAB(t). The system cannot be solved unless some newrelationships between variables are found. In other words, onlythree variables can remain in this system. Space-state modelingtheory dubs them the state variables and provides a criterionto select them [16], [17]. They must be of energy-storingmagnitudes. After selecting them, the other variables must beexpressed as a function of the state variables and the controlparameters.

The control parameters are easy to define. They are the dcinput voltage VG, the frequency f , and the duty cycle d. In(1a)–(1c), they appear to be implicitly included in the inputvoltage to the resonant tank VAB.

The state variables are also easily identifiable. Paying atten-tion to the circuit behavior, it is concluded that energy is onlystored in its reactive elements LS , CS , and Cf . Besides, it iswell known that the electrical magnitudes that better representstored energy in reactive elements are voltage over capacitorsand current through inductors. Now, the three state variableshave been found, forming the state vector

x̄(t) = [iL(t), VS(t), V0(t)] . (2)

It is remarkable that the voltage VP over the parallel capac-itor CP has not been considered as a state variable in (2) andthat its differential relationship between current and voltagehas not been included in (1a)–(1c). This is due to the VPsingular behavior. It has neither a sinusoidal evolution nor evena differentiable evolution. In fact, its initial sinusoidal form isclamped to the output voltage V0 twice every period (Fig. 4)


Fig. 4. Main waveforms in a converter based on the PRC-LCC topology witha capacitor as output filter. The secondary waves and values have been reducedto the primary (absence of apostrophe).

when the rectifying diodes are on. Then, the much larger filtercapacitor Cf is connected in parallel with CP . In comparisonwith Cf , CP stores a very small amount of energy, which maybe ignored during this lapse: CP disappears. Therefore, thevalue of CP and the differential relationship between VP andiL are only relevant to establish the time required for sinusoidaltransitions between the clamping values +V ′

0 and −V ′0, and not

during the whole period. An appropriate value for the angleψ (Fig. 4) will adequately define the status in CP . Hence,the differential relationship between VP and iL will be onlyused, together with the value of V0, to calculate ψ dynamicallyevery period.

Now, once the control parameters and the state variables havebeen identified, the state equations (1a)–(1c) still cannot besolved. There are too many variables, and those related to thecontrol are not explicitly shown. Consequently, certain relation-ships must be found, and on the way, some simplifications willhave to be done.

III. STATE VARIABLES HARMONIC APPROACH

As expected in a resonant converter, the current through theinductor iL(t) and the voltage in the series capacitor VS(t)are predominantly sinusoidal, without any dc component andthe low presence of high-frequency harmonics (Figs. 3 and 4).Thus, it is accurate enough to approximate these waves bytheir fundamental terms in the Fourier series, resulting in thefollowing equations if the sinusoidal formulation is used:

iL(t) ≈ iLA(t) · sin(ωt) + iLB(t) · cos(ωt) (3)

VS(t) ≈VSA(t) · sin(ωt) + VSB(t) · cos(ωt). (4)

In this case, ω represents the pulsation for the switchingfrequency.

Once (3) has been assumed, and according to the definitionof Fourier series, the amplitude and phase for iL(t) can be

estimated as follows:

iLP =√

[iLA]2 + [iLB]2 sinϕ =−〈iLB〉1iLP

(5)

and a similar expression might be deduced easily for VS(t).Going on with the state variables, the output voltage V0(t)

is a continuous magnitude with a very low ripple. Therefore, itcan be approximated every period by its average value, whichcan be obtained again with the Fourier series

VO(t) ≈ V0avg(t). (6)

The absence of an apostrophe in the variables indicates that theyhave been reduced to the primary.

Now, the three initial state variables from (2) have been ap-proximated, and they have evolved into five new state variables,forming the new state vector, i.e.,

x̄(t) = [iLA(t), iLB(t), VSA(t), VSB(t), V0avg(t)] . (7)

It must be remembered that the state variables are timedependents. Consequently, their dynamic behavior can beinvestigated.

IV. EXTENDED DESCRIBING FUNCTION (EDF)

To simplify the equations of the model stated in (1a)–(1c),and to obtain a more useful one, it is necessary to expressVAB(t), VP (t), and iD(t) as a function of the control (f, d, VG)and/or the new state variables (7). The major hindrance is thepresence of two nonlinear stages (inverter and output rectifier).The EDF concept [14], [15] must be applied. This way, theequations for each stage can be rewritten in a proper manner.

A. Inverter

This stage generates a quasi-square voltage at its outputVAB(t), which is applied to the resonant tank (Figs. 2 and 4).The EDF method is applied to obtain a Fourier formulation forthis voltage. Since in a period, it only depends on d and VG(Fig. 4), the EDF will be a function of these control variables,as shown in the following:

VAB(t) = f1,0(d, VG)

+∑k

[f1A,k(d, VG) · sin(kωt)+f1B,k(d, VG) · cos(kωt)] (8)

where f1 is the EDF for this stage, and f1A,k and f1B,k are itssinusoidal and cosinusoidal kth harmonic components.

As the resonant tank acts as a bandpass filter and the inverteris tuned, the effect of any harmonic component of VAB, exceptthat of the first, can be neglected (first-harmonic prevalence).The sinusoidal appearance of the resonant current confirmsthis fact. Therefore, it is accurate enough to consider only thefirst harmonic for VAB in (8). Attending to the information


Fig. 5. Detail of the output rectifier connection.

Fig. 6. Simplified equivalent circuit for the topology. The state variableappearance is sinusoidal in the net on the right and continuous on the left.

contained in Fig. 4 and applying the Fourier series, thecoefficients in (8) can be integrated, resulting in

f1A,1(d, VG) =4 · VGπ

· sin(πd) (9a)

f1B,1(d, VG) = 0. (9b)

Due to this approximation, VAB(t) has been expressed as afunction of the control parameters (and the state variables).

B. Output Rectifier

The EDF is applied to this stage to obtain the describingfunctions f2(t) and f3(t) for the voltage in the parallel capacitorVP (t) and for the current through the rectifier diodes transferredto the primary iD(t).

To work out these functions, the behavior of the stage mustbe investigated using Figs. 2, 4, and 5. It can be observedthat when VP (t) is clamped to the reflected output voltage,the resonant current goes through the rectifying diodes andiD(t) = iL(t). Then, iL(t) changes its sign, and all the diodesare off. The current iL(t) flows entirely through CP , and itsvoltage experiences a sinusoidal evolution from −V0 to +V0

(or vice versa) and is clamped once more. From this clamping,the whole process is repeated. Hence, it can be affirmed thatthe conduction periods in the output rectifier are determinedexclusively by two of the state variables, i.e., V0(t) and iL(t).

Bearing this in mind, the following equations are written:

VP (t)=f2,0(iL, V0)+∑k

[f2A,k(iL, V0) · sin(kωt)

+ f2B,k(iL, V0) · cos(kωt)] (10a)

iD(t)=f3,0(iL, V0)+∑k

[f3A,k(iL, V0) · sin(kωt)

+ f3B,k(iL, V0) · cos(kωt)] . (10b)

Again, the resonant tank bandpass filtering action explains theprevalence of the first-harmonic effects on VP (t) (Fig. 6). Onthe contrary, the output filter has been designed to provide dc

voltage, and only the average value of iD(t) is relevant forthis magnitude (Fig. 6). Hence, (11a) and (11b) can easily beadmitted, i.e.,

VP (t)≈ f2A,1(iL, V0)·sin(ωt)+f2B,1(iL, V0)·cos(ωt) (11a)

iD(t)≈ f3,0(iL, V0). (11b)

The orders of the Fourier approaches have therefore beenestablished, and they can be calculated by paying attention toFig. 4. From Section III, the amplitudes of iL(t) and V0(t) areknown. Thus, the electrical angle ψ can be easily estimatedfrom the sinusoidal transition of VP (t). Bearing in mind theinitial and final voltages of CP , and that the transition occurswhen the whole resonant current passes through it, (12a) canbe written, and (12b) can be deduced from it, as shown in thefollowing:

VP (ϕ+ ψ) =VP (ϕ) +1CP

·ϕ+ψ∫ϕ

iL(t) · dωt

⇒ V0avg = −V0avg +iLP

CPω· [1 − cos(ψ)] (12a)

⇒ cosψ =1 − 2 · V0avg · CPω√i2LA + i2LB

. (12b)

Now, the coefficients of the Fourier series for f2 can becalculated. For that, the waveform of VP (t) in Fig. 4 and (12a)and (12b) must be taken into account. The result is shown in thefollowing:

f2A,1 =VPA =1

π · CPω[iLA · sin2 ψ + iLB · µ] (13a)

f2B,1 =VPB =1

π · CPω[iLB · sin2 ψ − iLA · µ] (13b)

where µ = [ψ − sinψ · cosψ]. (13c)

In the same way, the coefficient for f3 can be calculated using(12a) and (12b) and the waveform of iD(t), as shown in Fig. 4.The result is written as follows:

f3,0(iL, V0′) = iDavg =

√i2LA + i2LB

π· [1 + cosψ]. (14)

Now, iD(t) and VP (t) have been approximated by an expres-sion that only depends on the control parameters and the statevariables. Now that three variables from the system shown in(1a)–(1c) are eliminated, the solution is possible.

V. HARMONIC BALANCE

In the state equations (1a)–(1c), the state variables alsoappear in their first-derivative form. To conveniently simplifythe system, they must be expressed by their corresponding har-monic ,approximation. Assuming frequency as a time-invariantmagnitude, or at least with a slow enough variation, and


remembering (3), (4), and (6), these derivatives can be operated,as shown in the following:

d(iL(t))dt

≈ d

dt[iLA · sin(ωt) + iLB · cos(ωt)]

=[d[iLA]dt

− ωiLB

]· sin(ωt)

+[d[iLB]dt

+ ωiLA

]· cos(ωt) (15a)

d(VS(t))dt

≈ d

dt[VSA · sin(ωt) + VSB · cos(ωt)]

=[d[VSA]dt

− ωVSB

]· sin(ωt)

+[d[VSB]dt

+ ωVSA

]· cos(ωt) (15b)

dVOdt

≈ d[V0avg]dt

. (15c)

Now, it is possible to rewrite (1a)–(1c) using the results of theharmonic approximation in Section III, the EDF in Section IV,and (15a)–(15c). In the resulting expressions, the variables thathave been approximated by their first-harmonic component arealways multiplied by the sinusoidal terms sin(ωt) and cos(ωt).These terms must be collected and put into separate groups.This way, (1a) is divided into two parts: the one with theterms multiplied by sin(ωt) (16a) and the one with the termsmultiplied by cos(ωt) (16b). Identically, (1b) gives place to(16c) and (16d). Notwithstanding, (1c) is different, and it istransformed into (16e) because the average approach used forthe variables in the output rectifier avoids the presence of anysinusoidal coefficient, i.e.,

LSd[iLA]dt

=4 · VGπ

sin(πd) + ωLS · iLB

− VSA − VPA − r · iLA (16a)

LSd[iLB]dt

= −ωLS · iLA − VSB − VPB − r · iLB (16b)

CS · ddt

[VSA] = iLA + ωCS · VSB (16c)

CS · ddt

[VSB] = iLB − ωCS · VSA (16d)

Cfd[V0avg]dt

=(1 + cosψ)

√i2LA + i2LB

π− V0avg

R. (16e)

The system in (16a)–(16e), which is completed with (12b)and (13a)–(13c), constitutes a simplified model for the large-signal behavior of the topology. Moreover, as it is formed byfive equations and five variables (the state variables), this modelcan be solved without further transformation.

VI. NORMALIZATION

To generalize the model formulation of (16a)–(16e), it shouldbe normalized. The following base magnitudes have been

selected, wherein any other base magnitude can be defined(for instance, IB = VB/ZB):

Voltage: VB = VG (17a)

Impedance: ZB =

√LSC

(17b)

Frequency: fB =1

2π√LS · C

. (17c)

In (17a)–(17c), the capacitance C represents the series asso-ciation of CS and CP .

It is also interesting to define a ratio between the value of thetopology capacitors, i.e.,

α =C

CP=

CSCS + CP

(18a)

β =C

Cf. (18b)

Regarding the aforementioned relationships, the normalizedequations that describe the state space are

12π

· d i�

LA

d t� =

4π· sen(πd) −V

�

SA−V�

PA− r · i�LA

ZB+f

�

· i�LB

12π

· d i�

LB

d t� = −V

�

SB−V�

PB− r · i�LB

ZB−f

�

· i�LA

12π

· d V�

SA

d t� = (1 − α) · i�LA+f

�

· V�

SB

12π

· dV�

SB

d t� = (1 − α) · i�LB−f

�

· V�

SA

12π

· d V�

0avg

d t� = −V

�

0avg · βR� +

β ·√i�2LA+ i

�2LB

π· [1+cosψ]

(19)

where the variables with a hat represent the normalized magni-tudes. The system is completed with the following:

V�

P 1 =α

π · f� [ i

�

LA · sen2ψ + i�

LB · µ] (20a)

V�

P 2 =α

π · f� [ i

�

LB · sen2ψ − i�

LA · µ] (20b)

µ = [ψ − senψ · cos ψ] (20c)

cos ψ =1 − 2 · V�

0avg · f�

α ·√i�2LA + i

�2LB

. (20d)

VII. SOLUTION OF THE MODEL AND

OBSERVATION VARIABLES

Any of the differential equation systems in (16a)–(16e) or(19) constitutes a large-signal model of the topology by itself.The only difference is that the last one is normalized. Therefore,to obtain the large-signal evolution of the topology, any ofthem must be integrated. Numerical methods must be used with


this purpose. Based on the Runge–Kutta module 4 integratingmethod, an application in a mathematical environment hasbeen programmed. The predictions calculated in this way havebeen compared successfully with experimental measurements,as shown in Section X. The calculus procedure is shown in thefollowing.Initial data. As in any system of differential equations, it

is necessary to know the initial state. In this case, the initialstate of the topology is defined by the value of state variablesiLA(t0), iLB(t0), VSA(t0), VSB(t0), and V0avg(t0) at the initialtime t0. Besides, the temporal evolution of control parametersf(t), VG(t), and d(t) must be also completely known (open-loop condition) or calculable (extra feedback algorithm).VPA(t0), VPB(t0), and ψ(t0) can be calculated from

(15a)–(15c), respectively, using the initial data.Notwithstanding, in (12b), neither can the denominator be

zero (no resonant current), nor can the whole expression besmaller than −1 (the voltage in the output filter V0avg(t0) isvery high, and the resonant current cannot charge CP to thatvalue). Thus, any transition or initial state without resonantcurrent, or with low resonant current and large voltage in thefilter capacitor, will revert in convergence problems. Furtherrefinement of the model must be done. In fact, these two criticalcases have something in common: they can be electricallydescribed by an output rectifier that is never on. In the system,ψ(t) precisely describes the part of the period when the outputrectifier is off. If ψ(t) = 0, the output rectifier is on throughoutthe period. If ψ(t) = π, the output rectifier is always off. Thus,the following lines of code must be added to the algorithm fora generic instant tK :

if (iLA(tk) == 0 &&iLB(tk) == 0) , then z(tk) = −2

else z(tk) =V0avg(tk) · CP · ω(tk)√[iLA(tk)]

2 + [iLB(tk)]2

if (Abs (1 − z(tk)) > 1) , then ψ(tk) = π

else ψ(tk) = Arc cos (1 − z(tk)) .

Differential equations (16a)–(16e) are used to estimate thevalue of the derivatives at t0. As an example, the followingexpression can be written from (16a):

diLA

dt

∣∣∣∣t0

=4 · VG(t0)LS · π · sen [π · d(t0)]

− VSA(t0) + VPA(t0) + r · iLA(t0)LS

+ ω(t0) · iLB(t0). (21)

Once the state variables and their derivatives are known att0, a numeric integration method is applied to obtain the valueof the state variables at the next moment t1. For instance, usingthe Euler method (the simplest one), it is very easy to calculatethe following:

iLA(t1) ≈ iLA(t0) + (t1 − t0) ·diLA

dt

∣∣∣∣t0

. (22)

Euler method provides an error of (t1 − t0)2. Runge–Kuttamethod is slightly more complicated, but it diminishes theerror to (t1 − t0)4. In both cases, the smaller is the differencebetween contiguous instants, the smaller is the error. Goodresults have been obtained using 20 points per period.

Once the state variables are known at t1, they can be used tocalculate their values at t2 using the same procedure as before.

The numerical solution of the large-signal model thusprovides the dynamical evolution of any magnitude in theconverter, which is, in case of a sinusoidal waveform, thedynamical evolution of its amplitude (providing an envelopeover the real waveform) and its phase [18]–[20].

VIII. EXPERIMENTAL VERIFICATION OF THE

LARGE-SIGNAL MODEL

The accuracy of the mathematical model that is obtained ischecked after comparing its results with some experiments. Aprototype has been constructed to verify the model. The overallefficiency of the converter is estimated in 90% at full load.Several tests with sudden variations in the input voltage orthe load have been made with it. Experimental data have beencompared with the model predictions, confirming its accuracy.As an example, an input voltage step and a load variation arestudied in Figs. 7 and 8, respectively. To make the comparisoneasier, the envelopes from the model have been drawn twice,i.e., with negative and positive signs, since the representation ofsinusoidal waveforms in the oscilloscope is difficult to rectify.For both figures, the prototype and test parameters can be foundin Table I.

IX. STEADY-STATE ANALYSIS

Any large-signal model implicitly contains a steady-statesolution. In this case, the steady-state model is very simple andvery easy to solve. It is not necessary to divide the topologyinto several operating modes as in [9] and [11], defining adifferent set of equations for each mode. The steady-state modelshown in this paper describes the global topology behavior withonly a few equations. To obtain them, it is just necessary tosubstitute the derivatives in (16) by zero since the variablesdo not experience any change in steady state. The solutionof this system without derivatives is shown in Appendix A[(A1)–(A11)]. It can be demonstrated that the equations of thesteady-state solution correspond to the equivalent circuit for thetopology shown in Fig. 9. This circuit is divided into two parts.The first one models the behavior of the resonant tank, andthe second one is related to the output (reduced to primary).As it can be seen in Fig. 9, the effects in the resonant tankof the output rectifier, including the parallel capacitor CP andthe load R, are modeled by a capacitor CX and a resistor RXthat are connected in series. The power dissipated in RX is theoutput power of the global converter. As the input voltage in theresonant tank is sinusoidal (9), the current through the inductoriL has to be calculated by complex operation. The equationsobtained are those in the appendixes. To calculate the outputvoltage with the circuit of Fig. 9, the “output” section has beenincluded in it. In this section, the waveforms are continuous,


Fig. 7. (a) Voltage in the series capacitor. (b) Current in the series inductor.(c) Voltage in the output filter capacitor.

and the current source value is dependent of the peak value ofiL, which is dubbed as iLP.

To calculate the value ofCX andRX in Fig. 9, it is necessaryto know the switching frequency and the value of R and CP .This way, the angle ψ ∈ [0, π] is obtained using (A1), and µ is

Fig. 8. (a) Current through series inductor. (b) Voltage in the output filter.

known from (A2). Then, if the value of the other componentsis also known, the equivalent circuit is solved easily. It isremarkable that (A1) can be deduced by remembering that thepower dissipated by RX is equal to the power dissipated by R.

X. EXPERIMENTAL VERIFICATION OF THE

STEADY-STATE MODEL

At anytime that the steady-state model has been checked withexperiments, the agreement has been very good. In a wide setof experimental tests, the error is commonly below 5%, withabout 15% as the maximum error obtained. As an example,in Fig. 10, a set of experimental values is compared with thetheoretical values in permanent conditions. It must be pointedout that the data of the model are obtained instantaneously dueto the simplicity of the model.

XI. STEADY-STATE BEHAVIOR OF THE TOPOLOGY

Using the equivalent circuit in Fig. 9 and its equations in theappendixes, it is very simple to collect in graphics the operation


TABLE IPARAMETERS IN THE SIMULATION OF FIGS. 7 AND 8

Fig. 9. Equivalent circuit of the topology for steady-state condition. The inputvoltage is expressed by its peak value.

points of the topology. These graphics can adopt several formatsdepending on the magnitudes studied and on which of them aremaintained constant. In Figs. 11 and 12, some of these graphicsare presented as an example. In them, all the magnitudes havebeen normalized according to the base in Section VI, and CShas been kept equal to CP .

According to Fig. 11, the same power and voltage can beobtained at the output with different duty cycle and frequencycombinations in the control. One possibility is to drive thistopology as close as possible to the optimum switching linemaintaining ZVS [8]. In this mode of operation, a small amountof energy is returned from the resonant circuit to the inputvoltage source (minimizing processed energy and conductionlosses), and the four switches are forced off (minimizingswitching losses). Ideally, to work under this condition, it isnecessary to maintain null the phase between the resonantcurrent and the inverter output voltage (φ = 0). Therefore,resonant current should be sensed, and a feedback loop shouldbe implemented to synchronize the switching of one leg of theinverter with the resonant current zero crossing. (In practice, φis slightly below 0, but the difference is not that much.) Theduty cycle is determined this way using its value to assureright synchronization. The appearance of the resulting topologywaveforms is very similar to that described in Fig. 4, but φ = 0.This condition can be mathematically described as follows:

ϕ0 =π

2− π · d0

⇒ d0 =12− 1π· Arctg

[− iLB0

iLA0

]

⇒ d0 =12− 1π· Arctg

[− 1H0

](23)

where (5) and (A6) have been employed.Once the duty cycle has been calculated, the equivalent

circuit of Fig. 9 can be used to model the converter under thiscontrol fashion and to obtain some graphics, characterizing thebehavior of the topology on optimum switching line. Some ofthem are shown in Fig. 13. It can be observed that over theoptimum switching line, the maximum power transferable is

below 4 per unit (p.u.) when the output voltage varies between1.5 and 2 p.u. Again, all the magnitudes have been normalizedaccording to the base in Section VI, andCS has been kept equalto CP .

All magnitudes are normalized. These graphics represent thebehavior of the topology when constant duty cycle is selected ascontrol strategy. In many high-voltage applications, it would beinteresting that the converter itself shows a voltage conversionratio above 1. Thus, a maximum power of 4 p.u. is a naturallimit in the topology. This gives one of the equations (24) toselect the parameters in the resonant tank for certain applica-tion. The other relationship to select these resonant elementsis given by the maximum switching frequency (25), which islimited by the losses in the semiconductors and is related to fBin Fig. 13, i.e.,

PMAX < 4 · PB =4 · U2B

ZB=

U2E√

LS/C(24)

fMAX < 1.1 · fB =1.1 · 12 · π

√LS · C

. (25)

Of course, other relationships between capacities are possible,and the reaction of the converter can also be estimated underthese circumstances. It is even possible to compare how theoperation points are modified this way, defining which ratiobetween capacitances is the best. Fig. 14 shows an example ofthis possibility. From this figure, it is evident that the smaller theα, the bigger the resonant current and the voltage in the seriescapacitor for the same output power and voltage, and the biggerthe switching frequency. This is a matter of trade in the designprocess, but a good compromise is usually found with values ofα between 0.5 and 0.9.

XII. SMALL-SIGNAL MODEL

Once the large-signal model has been obtained, it can belinearized around an operating point to obtain the small-signalmodel. The well-known Taylor polinomium can be used tocomplete this operation. Once the differential equation systemhas been linearized, it must be solved. This step can be carriedout using Laplace transformation. Thus, the linearized systemcan be transformed in (26), and the transfer functions betweenthe state variables and the input parameters might be obtainedusing (27), i.e.,

s · [X(s)] = [A] · [X(s)] + [B] · [W (s)] (26)

[X(s)] = [s · I −A]−1 · [B] · [W (s)]

= [Q(s)] · [W (s)] . (27)


Fig. 10. Experiment versus model when duty cycle d0 is 0.3 and the load is constant. Several magnitudes have been measured in the prototype every 5 kHz andcompared with the model predictions.

Fig. 11. Steady-state relationships when the output power is constant and equal to 1 p.u., and several different output voltages V0 are selected. All the magnitudesare normalized. The phase is referred to angle φ in Fig. 4. The graphics demonstrate that the same output power and voltage can be achieved with differentcombinations of switching frequency and duty cycle and diverse effects on iL, VS , and their phases.

The coefficients of arrays [A] and [B] can be consulted inAppendix B. From the equations in Appendix B, the transferfunction between output voltage (for instance) and any control

parameter (frequency, duty cycle, or dc input voltage) could beinvestigated. Frequency and duty cycle are the natural controlparameters of the topology. As shown in the experimental


Fig. 12. Steady-state relationships for different loads R when the duty cycle is constant, i.e., d0 = 0.3.

Fig. 13. Steady-state relationships for different output power situations under optimum switching line condition. All magnitudes are normalized.

results, the transfer function of the output voltage usually showsa single pole below 3 kHz when the large current obligesus to use a switching frequency below 75 kHz. If the inputdc bus is obtained from the three-phase main line, an input

ripple of 300 Hz is too close to the dominant pole. A feedfor-ward control might be implemented. In this case, the transferfunction between output voltage and input voltage is alsointeresting.


Fig. 14. Steady-state curves when constant power is transferred to the output. The output voltage is maintained at 1 p.u., but the relationship between the resonantcapacities α (18a) changes. The optimum switching line mode of operation is observed.

Fig. 15. Comparison between Bode plots (module and phase) from the model and the experiment. In this case, the output variable was the output voltage, andthe control variable was the frequency (duty cycle and input voltage were constant). Switching frequency in the equilibrium point is 70 kHz.

With these two criteria, and considering CS equal to CP , thesuitable elements in the resonant tank have been selected.

XIII. EXPERIMENTAL VERIFICATION OF THE

SMALL-SIGNAL MODEL

To experimentally verify the accuracy of the small-signalmodel, a wide set of Bode plots has been measured in severalprototypes and compared to the model predictions. In all thecases, a good agreement between the calculation and experi-ment has been obtained. Figs. 15 and 16 show representativeexamples of the results obtained when the control variable isthe frequency or the duty cycle.

XIV. CONCLUSION

Using harmonic approach and extended description func-tions, steady-state, large-signal and small-signal models have

been deduced for an interesting topology such as the PRC-LCCwith a capacitor as output filter. In the case of the steady state,the model is extremely simple and can be expressed as an easy-to-use equivalent circuit. All the models predict all the electricalmagnitudes in the topology with really low calculus time and anexcellent accuracy, as confirmed by experimentation. As such,a complete study of any power source based on the topologycan be accomplished. Thus, the performance of the equipmentcan be predicted before assembly, helping the designer to selectvalues, components, and control strategy. This is of signifi-cant relevance in the cases of large output power and largeoutput voltage applications, where the prototypes are alwaysexpensive.

APPENDIX A

The equations for the steady-state model can be obtainedfrom Kirchoff analysis of the equivalent circuit in Fig. 7 or


Fig. 16. Comparison between Bode plots (module and phase) from the model and the experiment. In this case, the output variable was the output voltage, andthe control variable was the duty cycle (frequency and input voltage were constant). Switching frequency is 70 kHz.

from the large-signal model of (16) by setting their derivativesto zero, i.e.,

ψ0 = Arc cos(π − 2R · CP · ω0

π + 2R · CP · ω0

)(A1)

with 0 ≤ ψ0 ≤ π.

µ0 = [ψ0 − senψ0 · cosψ0] (A2)

g0 = [1 + π · r · CP · ω0 − cos2 ψ0] (A3)

H0 =CS · g0

π · CP + µ0 · CS − π · ω20 · LS · CS · CP

(A4)

iLA0 =H2

0

1 +H20

CP · ω0

g0· 4VG0 · sin(π · d0) (A5)

iLB0 =iLA0

H0(A6)

VSA0 =iLB0

CS · ω0(A7)

VSB0 =−iLA0

CS · ω0(A8)

VPA0 =1

π · ω0 · CP[iLA0 · sen2ψ0 + iLB0 · µ0] (A9)

VPB0 =1

π · ω0 · CP[iLB0 · sen2ψ0 − iLA0 · µ0] (A10)

V0avg,0 =2R

π + 2R · ω0 · CP

√i2LA0 + i2LB0. (A11)

The subindex “0” pertains to the steady-state magnitude.

APPENDIX B

[X(s)] =

iLA

iLB

VSA

VSB

V0avg

[W (s)] =

dωVG

[A] =

a11 a12 a13 0 a15

a21 a22 0 a24 a25

a31 0 0 a34 00 a42 a43 0 0a51 a52 0 0 a55

[B] =

b11 b12 b130 b22 00 b32 00 b42 00 b52 0

a11 =4V0avg,0 ·cos(φ0)·cos(θ0)

iLP0 ·π ·LS− πr·ω0 ·CP + sin2(ψ0)

π ·ω0 ·CP ·LS(B1)

a12 =ω0 −µ0

π ·ω0 ·CP ·LS− 4V0avg,0 ·sin(φ0)·cos(θ0)

iLP0 ·π ·LS(B2)

a13 = a24 =−1LS

(B3)

a15 =4·cos(θ0)π ·LS

(B4)

a21 = −ω0 +µ0

π ·ω0 ·CP ·LS− 4V0avg,0 ·cos(φ0)·sin(θ0)

iLP0 ·π ·LS(B5)

a22 =4V0avg,0 ·sin(φ0)·sin(θ0)

iLP0 ·π ·LS− πr·ω0 ·CP + sin2(ψ0)

π ·ω0 ·CP ·LS(B6)

a25 =4·sin(θ0)π ·LS

(B7)

a31 = a42 =1CS

(B8)

a34 = −a43 = −ω0 (B9)

a51 =2·cos(φ0)π ·Cf

(B10)

a52 = − 2·sin(φ0)π ·Cf

(B11)

a55 = − 1R·Cf

− 2·ω0 ·CPπ ·Cf

(B12)


b11 =4·cos(π ·d0)·VG0

LS(B13)

b12 = −iLP0 ·sin(φ0)(

1 +µ0

π ·(ω0)2 ·CP ·LS

)

+4V0avg,0 ·sin(φ0)·sin(ψ0)

π ·ω0 ·LS

+2V0avg,0 ·cos(φ0)·[1 − cos(ψ0)]

π ·ω0 ·LS(B14)

b13 =4·sin(π ·d0)

π ·LS(B15)

b22 = −iLP0 ·cos(φ0)(

1 +µ0

π ·(ω0)2 ·CP ·LS

)

+4V0avg,0 ·cos(φ0)·sin(ψ0)

π ·ω0 ·LS

+2V0avg,0 ·sen(φ0)·[1 − cos(ψ0)]

π ·ω0 ·LS(B16)

b32 =VSB0 (B17)

b42 = −VSA0 (B18)

b52 = − 2·CP ·V0avg,0

π ·Cf. (B19)

The subindex “0” pertains to the steady-state magnitude.

REFERENCES

[1] V. Vlatkovic, M. J. Schutten, and R. L. Steigerwald, “Auxiliary seriesresonant converter: A new converter for high-voltage, high-power appli-cations,” in Proc. IEEE APEC, 1996, pp. 493–499.

[2] M. J. Schutten, V. Vlatkovic, and R. L. Steigerwald, “Resonant converterwith wide load range,” U.S. Patent 5 546 294, Aug. 13, 1996.

[3] H. van der Broek, W. Rexhausen, B. Wagner, and N. Geerkens, “Powersupply unit including a PWM inverter, notably for an X-ray generator,”U.S. Patent 6 072 856, Jun. 6, 2000.

[4] R. L. Steigerwald, “A comparison of half-bridge resonant convertertopologies,” IEEE Trans. Ind. Electron., vol. 3, no. 2, pp. 174–182,Apr. 1988.

[5] S. D. Johnson and R. D. Erikson, “Steady state analysis and design of theparallel resonant converter,” IEEE Trans. Power Electron., vol. 3, no. 1,pp. 93–104, Jan. 1988.

[6] E. X. Yang, F. Lee, and M. M. Jovanovic, “Small-signal modeling of LCCresonant converter,” in Proc. IEEE PESC, 1992, pp. 941–948.

[7] Y. Zhang and P. C. Sen, “D–Q models for resonant converters,” in Proc.IEEE PESC, 2004, pp. 1749–1753.

[8] V. García, M. Rico, J. Sebastián, M. Hernando, and J. Uceda, “An opti-mized DC–DC converter topology for high voltage pulse loads applica-tions,” in Proc. IEEE PESC, 1994, pp. 1413–1421.

[9] V. García, M. Rico, J. Sebastián, and M. Hernando, “Using the hy-brid series parallel resonant converter with capacitive output filter andPWM phase-shifted control for high-voltage applications,” in Proc. IEEEIECON, 1994, pp. 1659–1664.

[10] J. M. Sun, S. P. Wang, T. Nishimura, and M. Nakaoka, “Resonant modePWM DC–DC converter with a high-voltage transformer link and itscontrol methods for medical-use X-ray power supply,” in Proc. EPE,Lausanne, Switzerland, 1999. CD-ROM.

[11] A. K. S. Bhat, “Analysis and design of a series–parallel resonant converterwith capacitive output filter,” IEEE Trans. Ind. Appl., vol. 27, no. 3,pp. 523–530, May/Jun. 1991.

[12] C. Loef, “Analysis of a full-bridge LCC-type parallel resonant converterwith capacitive output filter,” in Proc. IEEE PESC, 1997, pp. 1402–1407.

[13] G. Ivensky, A. Kats, and S. Ben-Yaakov, “An RC load model of paral-lel and series–parallel resonant DC–DC converters with capacitive out-put filter,” IEEE Trans. Power Electron., vol. 14, no. 3, pp. 515–521,May 1999.

[14] E. X. Yang, F. C. Lee, and M. M. Jovanovic, “Extended describing func-tion technique applied to the modelling of resonant converters,” in Proc.VPEC, 1991, pp. 179–191.

[15] E. X. Yang, F. C. Lee, and M. M. Jovanovic, “Small-signal modelling ofpower electronic circuits using extended describing function technique,”in Proc. VPEC, 1991, pp. 167–178.

[16] S. R. Sanders, J. M. Noworolski, X. Z. Liu, and G. Verghese, “Generalizedaveraging method for power conversion circuits,” IEEE Trans. PowerElectron., vol. 6, no. 2, pp. 251–259, Apr. 1991.

[17] J. G. Kassakian, M. F. Schlecht, and G. V. Verghese, Principles of PowerElectronics. Reading, MA: Addison-Wesley, Jul. 1991.

[18] J. Martín-Ramos, J. Díaz, A. M. Pernía, F. Nuño, and J. Sebastián, “Alarge signal model for the PRC-LCC topology with a capacitor as outputfilter,” in Proc. IEEE APEC, 2002, pp. 1120–1126.

[19] J. Martín-Ramos, A. M. Pernía, J. Díaz, F. Nuño, and J. M. Alonso, “Acircuit for the large and small signal dynamic modelling of the PRC-LCCresonant topology with a capacitor as output filter,” in Proc. IEEE PESC,2005, pp. 635–641.

[20] S. Lineykin and S. Ben-Yaakov, “Unified spice compatible modelfor large and small signal envelope simulation of linear circuitsexcited by modulated signals,” IEEE Trans. Ind. Electron., vol. 53, no. 3,pp. 745–751, Jun. 2006.

Juan A. Martín-Ramos (M’02) received the M.S.and Ph.D. degrees in industrial engineering fromthe University of Oviedo, Gijón, Spain, in 1996 and2001, respectively.

In 1997, he joined the Electronic TechnologyArea, University of Oviedo, where he is currentlyan Assistant Professor. His main research interestsinclude magnetism, piezoelectrics, dc–dc converters,and dc high-voltage applications.

Juan Díaz (M’02) was born in Mieres, Spain, in1966. He received the M.Sc. and Ph.D. degrees inelectrical engineering from the University of Oviedo,Gijón, Spain, in 1990 and 1995, respectively.

From 1990 to 1999, he was an Assistant Professorin the Department of Electric, Electronic, Computerand Systems Engineering, University of Oviedo.Since May 1999, he has been an Associate Professorin the Electronic Technology Area, University ofOviedo. His research interests include switching-mode power supplies, resonant power conversion,

piezoelectrics, dc high-voltage applications, and design of systems based onmicrocontrollers.

Alberto M. Pernía (M’97) was born in Cantabria,Spain, in 1964. He received the M.S. and Ph.D.degrees in electrical engineering from the Univer-sity of Oviedo, Gijón, Spain, in 1991 and 1996,respectively.

Since 1992, he has been an Associate Professorin the Electronic Technology Area, University ofOviedo. He has worked on different topics relatedto power electronics, particularly on high-frequencypower supply systems. He has published severalpapers in international conference proceedings and

international journals related to magnetic component integration and high-power-density dc/dc converters. At present, his work and research are orientedto voltage regulator modules, high-output-voltage converters, and modeling ofmagnetic structures.


Juan Manuel Lopera (M’96) was born in Avilés,Spain, in 1966. He received the M.Sc. and Ph.D.degrees in electrical engineering from the Univer-sity of Oviedo, Gijón, Spain, in 1990 and 1994,respectively.

He is currently an Associate Professor in theElectronic Technology Area, University of Oviedo.His research interests are high-frequency high-density power converters, electronic ballast, model-ing of magnetic elements, and magnetic sensors.

Fernando Nuño (M’96) was born in Siero, Spain,in 1963. He received the M.Sc. and Ph.D. degrees inelectrical engineering from the University of Oviedo,Gijón, Spain, in 1988 and 1991, respectively.

From 1988 to 1993, he was an Assistant Professorat the University of Oviedo. Since May 1993, hehas been an Associate Professor in the ElectronicTechnology Area, University of Oviedo. His researchinterests include switching-mode power supplies,resonant power conversion, modeling and design ofmagnetic devices, piezoelectric transformers, high-

power-factor rectifiers, and design of electronic systems based on microproces-sors and microcontrollers.

Dynamic and Steady-State Models for the PRC-LCC Resonant Topology With a Capacitor as Output Filter

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