Dual-Loop Controller for LLC Resonant Converters Using an Average Equivalent Model

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Dual-Loop Controller for LLC Resonant Converters Using an Average Equivalent Model

Franco Degioanni , Student Member, IEEE, Ignacio Galiano Zurbriggen , Student Member, IEEE, and Martin Ordonez , Member, IEEE

Abstract—LLC resonant converters have gained popularity in a wide number of industrial applications due to their high efficiency and power density. Common applications of these converters are battery chargers, electric vehicles, and high-efficiency power sup- plies, which require tight output voltage regulations. However, tra- ditional averaging techniques employed in pulse width modulation converters cannot be employed for LLCs. As a consequence, design- ing linear controllers for these types of converters requires com- plex analysis or applying empirical methods. This paper proposes a simple and straightforward methodology for designing linear con- trollers for LLC resonant converters. A dual-loop control scheme including an inner current loop and an outer voltage loop is intro- duced. A simplified second-order equivalent circuit is employed to derive all the relevant equations for designing the compensators.

Simulations and experimental results using a 150-W platform are employed to validate the theoretical analysis.

Index Terms—Closed loop control, dual-loop, linear control, LLC resonant converter.

I. INTRODUCTION

T

HE advancement of renewable energy applications have demanded dc–dc power converters with higher efficiency and power density levels. Resonant power converters meet these requirements by operating at high switching frequencies [1], while maintaining high efficiency [2]. Particularly, LLC resonant converters are usually preferred over other resonant topologies due to their higher efficiency [3], higher power density [4], and low electromagnetic interference [5]. This converter has been introduced in a wide number of applications as battery chargers [6], [7], photovoltaic applications [8], [9], fuel cells [10], and high-efficiency power supplies [11], among others [12], [13].

The LLC resonant converter can work in a wide output voltage range with a relatively small range variation of the switching frequency, offering wider output regulation than series- and parallel-resonant converters [14]–[16]. Moreover, zero-voltage switching and zero current switching conditions can be achieved at the input MOSFETs and the output rectifier diodes, respectively [17], achieving higher efficiency levels.

Manuscript received August 29, 2017; revised October 24, 2017; accepted December 5, 2017. Date of publication December 21, 2017; date of current version August 7, 2018. Recommended for publication by Associate Editor M.

Vitelli. (Corresponding author: Martin Ordonez.)

The authors are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: fdegi [email protected]; [email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TPEL.2017.2786044

The above-mentioned applications demand tight regulation at the output voltage and current, which is traditionally achieved by employing closed-loop controllers based on small-signal models [18], [19]. Obtaining these models is not straightforward, since conventional averaging methods [20] cannot be directly applied to the currents and voltages in the resonant tank because they have no dc component.

Several techniques have been proposed to derive small-signal models of resonant converters. Sample data and discrete-time analysis is used in [21], while Cheng et al. [22] propose employing communication theory to derive a model of the LLC resonant converter. The extended describing function (EDF) method was introduced in [23], and it has been widely adopted in different applications [24]–[27]. The above-mentioned methods show accurate approximations of the dynamic behavior of the converter.

However, the high mathematical complexity involved requires empirical methodologies, such as software simulation [28] and hardware measurements [29], as a mean to obtain the frequency response of LLC resonant converters.

These modeling methodologies are usually employed to design a single voltage control loop rather than a dual-loop control scheme. The implementation of current-mode controllers has been vastly developed for pulse width modulation buck and boost converters [30]–[33] but not for LLC resonant converters.

Prior conference papers [34], [35] have explored the benefits of employing the current mode in LLC converters with some inter- esting empirical methods. In order to augment the understand- ing of LLC converters, closed-form expressions that describe the system behavior for different parameters are needed. This is also discussed in this paper to successfully compensate such a complex topology as the LLC resonant converter.

This paper presents a comprehensive analysis, modeling, and control of LLC resonant converters using a dual-loop strategy.

The distinctive features of this paper include the development of a large-signal model of the converter operating at resonant frequency, the derivation of a linearized model to design closed- loop controllers, and the analysis and identification of the most challenging condition from a control point of view. As a re- sult, an effective and straightforward linear controller design methodology is achieved. The proposed controller, as shown in Fig. 1(a), is formed by an inner rectified current loop and an outer output voltage loop. The application of dual-loop schemes enables significant advantages, as it is detailed in this work. Those advantages include tight current regulation, overcurrent protection, and ample bandwidths. Moreover, the implementation of

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9876 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 2018

Fig. 1. Full-bridge LLC resonant converter. (a) Circuit schematic with the proposed dual-loop controller. (b) Voltage gain versus the switching frequency. The dynamics of the converter depends on the operating condition and oscillations increase at resonance.

an inner-current loop enables desired closed-loop performance over a wide range of operating conditions. In this way, this paper introduces an accurate linear model for LLC converters valid for the most challenging control condition, the application of dual-loop controllers to successfully compensate such a complex topology as the LLC resonant converter, and a useful tool that enables a simple and straightforward linear controller design.

The rest of this paper is organized as follows. Section II stud- ies the dynamic behavior of the LLC resonant converter. An equivalent simplified model of the LLC converter operating at resonant frequency is presented in Section III. In Section IV, the small-signal model of the resonant converter is derived obtaining a simple and straightforward second-order system. The proposed control strategy is presented in Section V and the feedback control loop is analyzed. A controller design example is presented in Section VI. Simulation and experimental verifica- tion is shown in Section VII. Finally, conclusions are drawn in Section VIII.

II. NUMERICALIDENTIFICATION OF THEMOSTCHALLENGING

OPERATINGCONDITION

Identifying the most challenging condition from a control point of view is a handy strategy to design linear compensators that guarantee stable operation at all operating conditions.

The dc gain of the converter under different load conditions is shown in Fig. 1(b). As illustrated in the figure, operating at different switching frequencies (fs), the converter shows output oscillations with variable overshoot values and settling time.

In particular, operation at resonant frequency (fr) resembles a second-order response and shows the longest settling time.

In order to analyze and identify the aforementioned conditions, the EDF method is applied to the LLC resonant converter.

A. Nonlinear Equations of the LLC Resonant Converter From the schematic circuit shown in Fig. 1(a), the nonlinear equations of the converter can be expressed as

diL r(t) dt = 1

L_rVin−vC r(t) L_r

− n

Lrsign(iL r(t)−iL m(t))vo(t) (1) dvc(t)

dt =iL r(t)

C_r (2)

diL m(t)

dt = n

L_mv_osign(iL r(t)−i_{L m}(t)) (3) dv_o(t)

dt = n

Co|i_{L r}(t)−i_{L m}(t)| − v_o(t) RL

. (4)

The state variables are defined in function of the series inductor current (iL r), the series capacitor voltage (vc), the parallel magnetizing inductor (iL m), and the output capacitor voltage (vo).

B. Harmonic Approximation

The current through the resonant inductor and the voltage in the resonant capacitor are nearly sinusoidal waveforms. They can be approximated by their fundamental components, whereas the current and voltage at the output can be approximated by their dc terms. By using first harmonic approximation, the ac state variables may be defined as a combination of sine and cosine components in function of the switching frequency. The resonant inductor current is approximated by

iL r(t) =ir s(t) sin (ωst)−ir c(t) cos (ωst) (5) di_{L r}(t)

dt =

di_{r s}(t)

dt +ωsir c(t)

sin (ωst)

−

dir c(t)

dt −ωsir s(t)

cos (ωst). (6) The termsi_{r s} and i_{r c} represent the sine and cosine components of the current in the resonant tank, respectively. The same concept can be applied fori_m(t)andv_c(t).

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C. EDF Method

The EDF method approximates the nonlinear terms by their fundamental harmonics components defined by

F₁(d, V_in) =4Vin

π (7)

F₂(iss, isc, Vo) = 4 π

iss

i_p Vo (8)

F₃(iss, isc, Vo) = 4 π

isc

i_p Vo (9)

F₄(iss, isc) = 2

πip (10)

ip =

(ir s−im s)²−(ir c−im c)² (11) where Vin is the input voltage and d is the duty cycle (fixed 50%), iss and isc are the sine and cosine components of the secondary of the transformer, respectively, andipis the current flowing in the primary of the transformer defined by (11). From (7)–(10), the nonlinear terms of the state equations (1)–(4) can be approximated by

Vin =F₁(d, Vin) sin (ωst) (12) sign(iL r−iL m)Vo(t) =F₂(iss, isc, Vo) sin (ωst)

−F₃(i_ss, i_sc, V_o) cos (ω_st) (13) i_s =F₄(i_ss, i_cs). (14) The currenti_sis the one flowing at the secondary of the transformer. Substituting the quasi-sinusoidal waveforms and the nonlinear terms of the state equations by their approximations, the continuous state equations are obtained as follows:

L_rdir s

dt =4Vin

π −L_rω_si_{r s}−v_cs−4n π

(ir s−im s) i_p V_o

(15) Lr

dir c

dt =Lrωsir c−vcc−4n π

(ir c−im c)

i_p Vo (16) C_rdv_cs

dt =−C_rω_sv_cc+i_{r s} (17) Cr

dv_cc

dt =Crωsvcs+ir c (18) Lm

di_{m s}

dt =−Lmωsim c+4n π

(i_{r s}−i_{m s}) ip

Vo (19)

Lm

dim c

dt =Lmωsim s+4n π

(ir c−im c)

i_p Vo (20)

C_odV_o dt =2n

π i_p− n RL

v_o. (21)

This set of equations describes the approximated large-signal model of the LLC resonant converter. This model continues having some nonlinear terms arising from the product of two or more time-varying quantities. However, the system can be linearized around one operating point. The linearized model can be expressed in a state-space representationx(t) =˙ Ax(t) +

TABLE I NORMALIZEDPARAMETERS

Parameter Value

Vi n 1

Lr 1

2π

Cr 1

2π

Lm 3Lr

fr 1

RL 1

n 1

Bu(t), whereAandBare the Jacobian matrices of the system given by

A_ij = ∂f(x(t), u(t))

∂fj(t)

xo,uo

(22)

Bij = ∂f(x(t), u(t))

∂uj(t)

xo,uo

. (23)

The aforementioned expressions and derivations are high- lighted in detail in the Appendix. Equation (22) and (23) are matrices of dimensions 7×7 and 7×1, respectively.

The steady-state parameters,x_o = [I_{r s}, I_{r c}, V_cs, V_cc, I_{m s}, I_{m c}, V_o]^T, can be obtained by making the derivative components in (15)–(21) equal to zero and solving for a givenΩs.

D. Eigenvalues Analysis

The eigenvalues (λi) of matrixArepresent the poles of the linearized model around a quiescent operating point, and their position in the complex plane describes the dynamic behavior of the system. A numerical analysis is employed to evaluate the dynamic behavior of the converter. To gain generality, the subsequent analysis is done using the normalized LLC resonant converter shown in Table I, employing the following base quantitiesv_o,Z_o =

Lr

Cr,i_n = _Z^v^o_o, andf_n = _2π^√¹_L_r_C_r. The location of the eigenvalues of the normalized converter is plotted in Fig. 2 for different switching frequencies. As shown in the figure, the eigenvalues go through different lines of constant damping (ζ) and natural frequency (ω_n) asf_schanges. In LLC resonant converters, the resonant tank elements are usually small compared with the output filter parameters in order to achieve high switching frequencies with low ripple components at the output voltage. Therefore, the dynamics of the resonant tank is much faster, and it is only defined by the resonant tank elements.

On the other hand, there is a low-frequency component deter- mined by the interaction between the output filter capacitor and the equivalent dynamic impedance of the resonant tank. From the figure, a pair of dominant poles can be identified. Since the modeling efforts focus on characterizing the converter’s low- frequency dynamics, the spotlight of this work is put in the dominant poles.

The displacement of the dominant poles for different load conditions is detailed in Fig. 2(b). As illustrated in the figure, the eigenvalues show the lowest damping ratio at resonant frequency. When the load is decreased, the poles move over lines of

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Fig. 2. Eigenvalues of the normalized LLC resonant converter of Table I. (a) Eigenvalues displacement for different operating conditions defined by thefs; a pair of dominant pole is identified. Note that the resonant frequency shows the less damped operating condition, and this is a critical point for control compensation.

(b) Displacement of the dominant poles for different load conditions; the damping factor decreases with the load. (c) Output voltage of the normalized LLC converter under a small frequency step (2%) around different operating points. When the converter operates around resonance, the output voltage presents oscillation with the larger overshoot and settling time.

constantω_n while crossing diverseζlines. At constant switching frequency, the minimum damping is obtained for minimum load and operating at resonant frequency.

The time-domain response of the output voltage around different operating conditions after applying a2%frequency step is illustrated in Fig. 2(c). The converter displays an oscillatory behavior below and around the resonant point. However, at resonant operation, the oscillations show the larger overshoot and settling time. Moreover, above resonance, the output has an overdamped response and no oscillations are appreciated.

The performed numerical analysis shows the minimum damping operation happened at the resonant frequency, making this

condition the most challenging from a control loop design point of view.

III. AVERAGEMODELLLC RESONANTCONVERTER

As shown in the previous section, the low-frequency dynamic behavior of the LLC resonant converter operating at resonant frequency is defined by a pair of dominant poles. A simplified second-order model is derived in this section.

At resonant frequency, the LLC converter shows two equivalent structures depending on the position of the switches, as shown in Fig. 3. When the switchesS₁ andS₄ in Fig. 1(a) are

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Fig. 3. Operating modes of the LLC resonant converter at resonant frequency.

(a) Structure I, the inverter isONand the applied voltage in the resonant tank is positive. (b) Structure II, the inverter isONand the applied voltage in the resonant tank is negative.

turnedON(structure I), the voltageV_in−nV_ois applied to the resonant tank and the converter behaves as shown in Fig. 3(a).

On the other hand, whenS₂andS₃are conducting (structure II), the resonant tank voltage is defined bynVo−Vin, as shown in Fig. 3(b).

Structure I is defined by the following differential equations:

Vin=Lr

diL r(t)

dt +vC r(t) +nvo(t) (24) nvo(t) =Lm

dim(t)

dt (25)

iL r(t) =Cr

dvC r(t)

dt (26)

ir ec(t) = Co

n nvo(t)

dt (27)

i_{r ec}(t) =n(i_{L r}(t)−i_{L m}(t)). (28) By solving (24)–(27), the solutions fori_{L r}(t),i_{L m}(t),v_{C r}(t), vo(t), andir ec(t) are obtained. Since in practice Co >> Cr, simplified current and voltage solutions including only one sinusoidal component of frequencyωr=√

LrCrcan be obtained as

iL m(t) =iL m(tk) +nvo(tk)

L_m t (29)

iL r(t) =iL r(tk) +

C_o Co+n²Cr

Vin−nvo(tk)−vC r(tk) Zo

sin (ωrt) +nvo(tk) Lm

t (30) vC r(t) =vC r(tk) +

C_o Co+n²Cr

(Vin−nvo(tk)−vC r(tk))(1−cos (wrt)) +nv_o(t_k) 2Lm

t² (31)

vo(t) =vo(tk) +

nCr

C_o+n²C_r

(V_in−nv_o(t_k)−v_{C r}(t_k))(1−cos (w_rt)) (32) ir ec(t) =n

Co

Co+n²Cr

(Vin−nvo(tk)−vC r(tk)) sin (wrt). (33) As the converter operates at resonant frequency, each time subinterval is defined by

tk =kπ

LrCr, withk= 0, 1, 2, 3, 4.... (34) The indexkrepresents the half switching resonant period, and it defines the operating mode of the LLC converter. The converter operates in Structure I during half cyclek(tk < t < tk+1), and it switches to Structure II during half cycle(k+ 1) (tk+1 < t <

tk+2).

On the other hand, the differential equations for structure II are given by

−V_in =L_rdiL r(t)

dt +v_{C r}(t)−nv_o(t) (35)

−nv_o(t) =L_mdi_m(t)

dt (36)

i_{L r}(t) =C_rdv_{C r}(t)

dt (37)

i_{r ec}(t) =−C_o n

nv_o(t)

dt (38)

ir ec(t) =−n(iL r(t)−iL m(t)). (39) By solving (35)–(39), the current and voltages in the converter when it is operating in structure II are given by

iL m(t) =iL m(tk+1)−nv_o(t_k+1) Lm

t (40)

i_{L r}(t) =i_{L r}(t_k+1) +

Co

C_o+n²C_r

−V_in+nv_o(t_k₊₁)−v_{C r}(t_k₊₁) Zo

sin (ω_rt)−nv_o(t_k+1) Lm

t (41) v_{C r}(t) =v_{C r}(t_k₊₁) +

C_o C_o+n²C_r

(−V_in+nv_o(t_k₊₁)

−v_{C r}(t_k₊₁))(1−cos (w_rt))−nvo(tk+1)

2L_m t² (42) vo(t) =vo(tk+1)−

nCr

Co+n²Cr

(−Vin+nvo(tk+1)−vC r(tk+1))(1−cos (wrt)) (43) ir ec(t) =−n

C_o Co+n²Cr

(−Vin+nvo(tk+1)

−vC r(tk+1)) sin (wrt). (44) The initial conditions for each subinterval are defined by the final state in the previous subinterval, and they can be solved

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Fig. 4. Start-up response of the LLC resonant converter at resonant frequency. (a) Discrete equations predict the dynamic response of the converter at each switching action. The low-frequency response can be modeled by a second-order system by neglecting the high-frequency dynamics. (b) Experimental start-up of an LLC resonant converter operating at resonant frequency. Parameters:Lr=81μH, Cr =32nF, Co=16.5 nF,andn= 5.33.

in a cycle-by-cycle manner. As the converter is operating at resonance, the length of each subinterval is defined byT_s =_2f¹_r, and (31)–(43) can be expressed in the discrete domain as v_{C r}(k+ 1) =v_{C r}(k) + 2

C_o Co+n²Cr

(V_in−nv_o(k)−v_{C r}(k)) +nv_o(k)

8Lmf_r² (45) v_o(k+ 1) =v_o(k) + 2

nC_r Co+n²Cr

(Vin−nvo(k)−vC r(k)) (46) v_{C r}(k+ 2) =v_{C r}(k+ 1) + 2

Co

C_o+n²C_r

(−Vin+nv_o(k+1)−v_{C r}(k+1))−nvo(k+1) 8L_mf_r²

(47) vo(k+ 2) =vo(k+ 1)−2

nCr

Co+n²Cr

(−Vin+nvo(k+ 1)−vC r(k+ 1)). (48) The start-up response of the LLC converter is illustrated in Fig. 4.

The response presented by the low-frequency components of the rectified current and output voltage resembles a sinusoidal waveform similar to those of a second-order LC circuit.

The discrete values of the output voltage at every switching action can be predicted by (45)–(48). This low-frequency response of the converter can be modeled by a second-order system by neglecting the high-frequency dynamics. In order to find the

closed-form time that includes the discrete solutions, the vari- ablesv_{C r}(k),v_o(k),v_{C r}(k+ 1), andv_o(k+ 1) are redefined asx₁(k),x₂(k),x₃(k), andx₄(k), respectively, andv_c andv_o can be rewritten in an matrix form as shown in (50). Defining

a= C_o Co+n²Cr

, b= C_r Co+n²Cr

, (49)

⎡

⎢⎢

⎣

x₁(k+ 1) x₂(k+ 1) x₃(k+ 1) x₄(k+ 1)

⎤

⎥⎥

⎦ =

⎡

⎢⎢

⎣

0 0 1−2a −2na

0 0 −2nb 1−2n²b

1−2a 2na 0 0

2nb 1−2n²b 0 0

⎤

⎥⎥

⎦

⎡

⎢⎢

⎣ x₁(k) x₂(k) x₃(k) x₄(k)

⎤

⎥⎥

⎦+

⎡

⎢⎢

⎣ a nb

−a nb

⎤

⎥⎥

⎦2V_in. (50)

The discrete expression (50) can be solved by applying theZ transform. Solving forvo, the output voltage expression in the Zdomain is given as

x₂(z) =vo(z) =Vin2nb

z(z+ 1)

(z−1) [z²+ 2 (n²b−a)z+ (2a−1)]. (51) TheZtransform for a cosine waveform of amplitudeAand periodTwith a dc component is given as

Z(A(1−cos (ωkT))) = (1−cos (ωT))(z+ 1)z (z−1)(z²−2zcos (ωT) + 1).

(52)

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Fig. 5. Simplified equivalent circuit of the LLC resonant converter that models the average behavior of the converter operating at resonance.

According to (51) and (52), and considering thatCo >> Cr, the output voltage can be expressed as

vo(k) =Vin

n (1−cos (ωeqTsk)) (53) with Ts= ₂¹_f_r. By comparing (53) and (51), the following expression is obtained:

cos

ωeq 1 2fr

= Co

Co+n²Cr

1−n²Cr

Co

. (54) The low-frequency behavior of the converter can be modeled by the equivalent circuit shown in Fig. 5. The inductor L_eq represents the impedance of the resonant tank at resonant frequency, whereas the diodesD₁andD₂model the output rectifier. The natural frequency of the average model is defined by ωeq =√ ⁿ

Le qCo

, andLeq can be derived from (54) as Leq =n²C_r

Co

π²L_r

cos⁻¹ Co

C_o+_n2Cr

1−n²Cr

C_o

₂ (55)

as in practice, C_o >> C_r and (55) can be approximated by Taylor series as

L_eq = 1

4 +n²Cr

6C_o −n⁴C_r⁴

60C_o² +8n⁶C_r² 945C_o −...

π²L_r ∼= π² 4 L_r.

(56) This equation shows the simplified mathematical expression synthesis of the equivalent inductor introduced in Fig. 5.

This equivalent inductance represents the dynamic equivalent impedance of the resonant tank described in Section II. The interaction betweenLeq andCocan be represented as a second- order circuit that enables modeling the low-frequency dynamic behavior of the LLC resonant converter operating at resonant frequency.

IV. SMALL-SIGNALMODEL ATRESONANTFREQUENCY

The derived equivalent circuit describes the average behavior of the LLC resonant converter operating at resonant frequency, and small-signal analysis can be employed to derive a linear model of the power plant.

The dynamic relationship between the output voltage and the switching frequency can be found as the slope of the dc gain voltage characteristics given by

M = nv_o

Vin = f_n²(m−1)

(m(fn²−1))²+f_n²(fn²−1)²(m−1)²Q² (57)

Fig. 6. Small-signal equivalent circuit of the LLC resonant converter at resonant frequency. This linearized model is employed to derive the transfer functions for designing the proposed dual-loop controller.

with Q=

L_r CrRac

, Rac = 8 π²

N_p² N_s²RL,

f_n =f_s fr

, m=L_r+L_m Lr

. (58) Taking the partial derivative, the slope is found as

∂M

∂fn =−Lrfn(2LmLrf_n²+ 2L²m(fn²−1)+LrQ²(fn⁶−f_n²)) L³_m

f_n²

Lr

Lm + 1

−1₂

+^L²^r^Q²^f_Lⁿ²2^(fⁿ²⁻¹⁾² m

³₂ .

(59) Assuming resonant operation, the dynamic small-signal gain is derived as

k_f = ∂Vo

∂f_s =−8 π

Vin

n Lm

L_r 1

f_r. (60)

Linearizing the average model shown in Fig. 5, the small- signal model around the resonant frequency can be obtained, as shown in Fig. 6. The output-voltage-to-control-frequency transfer function (G_{v f}) can be obtained from the linearized model as

G_{v f}(s) = vo(s)

f_s(s)= kf

1 +s_R^L^{e q}

Ln² +s²^L^{e q}_n₂^C^o. (61) The Bode plot of the transfer function Gv f is illustrated in Fig. 7. The dynamic response of the LLC resonant converter resembles the small-signal model of a Buck converter. The switching to natural frequency ratio of a buck converter is critical to achieve high-bandwidth controllers. In the case of LLC converters. the ratio between the resonant and the double-pole frequencies is defined by

rf c= fr

f_eq = π 2n

Co

C_r (62)

which is dependent on the converter parameters.

The control-to-output-voltage transfer function was derived in this section based on the small-signal model shown in Fig. 6.

The same methodology can be employed to obtain the control- to-rectified-current transfer function and the rectified-current- to-output-voltage transfer function required for designing a dual-loop controller.

V. DUAL-LOOPCURRENTCONTROLLER

In the presence of disturbances, the output voltage may show oscillations, overshoot, and steady-state error. A closed-loop

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Fig. 7. Control-frequency-to-output-voltage transfer function of the LLC resonant converter presents a second-order frequency response.

Fig. 8. Proposed control scheme for the LLC resonant converter. The control design procedure is simplified by employing the linearized model of the LLC power converter.

controller can be employed to minimize these issues. This section includes analysis and considerations required to design the closed-loop controller.

A dual-loop control scheme employing an inner current and an outer voltage loop is proposed and illustrated in Fig. 8. Im- plementing a dual-loop scheme enables significant advantages such as tight current regulation and overcurrent protection. As shown in the figure, the desired output voltage is employed as a set point for the outer loop, whereas the inner-loop current reference is commanded by the voltage loop compensator. The averaging of the rectified current is included in the feedback path of the inner loop, and the current loop compensator adjusts the switching frequency in order to track the average current reference.

The control-frequency-to-rectified-current transfer function is given by

G_if(s) =i_{r ec,av g}(s)

fs(s) = kf

1 RL +sCo

1 +s_n^L₂^{e q}_R

L +s²^L^{e q}_n₂^C^o . (63) As can be observed in the equation, the transfer function has a double pole located atωeq = √ ⁿ

Le qCo

and a zero defined by the interaction of the output capacitor and the load. The gaink_f is negative as a consequence of the negative on the dc

Fig. 9. Control-frequency-to-average-rectified-current transfer function considering two resonant frequencies. The ratio betweenfrandfe q is critical in the controller design.

Fig. 10. Average-rectified-current-to-output-voltage transfer function. The outer voltage loop has been simplified.

characteristics of the LLC converter, which translate into a

−180^◦phase delay of the dc component. The Bode plot of (63) shown in Fig. 9 includes the discretization effect given by the averaging block and controller sampling frequency (f_s= 2f_{N y}).

As the control loop bandwidth is desired to be betweenf_eq and f_{N y}, the ratio between these frequencies (^r^{f c}₂ ) is critical for the controller design, as shown in Fig. 9.

Provided enough bandwidth difference between the two control loops, the inner loop can be considered as a controlled current source simplifying the loop dynamics to a first-order system, as shown in Figs. 10 and 11. The rectified-current-to- output voltage transfer function is then given by

G_{v i} = v_o(s)

ir ec,av g(s) = 1

1

RL +sC_o. (64)

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Fig. 11. Simplified equivalent circuit of the outer loop for designing the voltage controller. (a) Circuit model. (b) Closed-loop block control diagram. The implementation of an inner current loop simplifies the voltage loop controller design.

TABLE II EXPERIMENTALPARAMETERS

Parameter Value

Vi n 120 V

Lr 81μH

Cr 32 nF

Lm 227μH

fr 98 kHz

Po 150 W

n 5.33

Co 660μF fe q 2.3 kHz

Once the transfer functions have been defined for both inner and outer loops, linear controllers can be implemented by employing conventional techniques. In this way, reducing the LLC resonant converter dynamics to a second-order system enables a simple and straightforward controller design procedure.

VI. CONTROLLERDESIGNPROCEDURE

A controller design example employing the values specified in Table II is provided in this section. The control-to-rectified- current transfer function (63) shown in Fig. 9 has, in this case, anf_eqof 2.3 kHz and ^f₂^r of 50 kHz. The compensator employed in the inner current loop must be designed to satisfy a loop bandwidth betweenfeq and ^f₂^r. A bandwidth of 10 kHz is selected to satisfy this condition. In the case of the rectified-current-to- output-voltage transfer function (64), the compensator must be designed to achieve a much lower bandwidth than the inner loop, which leads to a selection of a 1-kHz frequency.

Proportional–integral controllers are employed as compensators for each control loop. The compensated open-loop Bode plots are shown in Figs. 12 and 13 for the inner and outer loops.

As can be observed, the desired bandwidths are achieved with phase margins (φ_m) of 57^◦and 79^◦, respectively.

To verify the design of the current and voltage controllers, the dynamics of both inner and outer control loops are included in the closed-loop response of Figs. 14 and 15. Designing the outer- loop bandwidth ten times slower than the inner loop enables a

−3-dB bandwidth of 1.5-kHz closed-loop bandwidth, as shown

Fig. 12. Compensated control-frequency-to-rectified-current transfer function. The bandwidth loop is selected to attenuate the double pole atfe q.

Fig. 13. Compensated rectified-current-to-output-voltage transfer function.

The bandwidth of the voltage loop is ten times lower than the inner-loop bandwidth.

in Fig. 14. The relative distance between control loops is critical to guarantee closed-loop stability. A parametric analysis of the closed-loop Bode response is performed in Fig. 15 for different voltage control loop bandwidths. As shown in the figure, the closed-loop−3-dB bandwidth varies with the gain of the voltage controller. Larger bandwidth in the outer loop enables higher closed-loop bandwidths; however, designing the outer loop with large gains may create oscillations between both control loops.

VII. SIMULATION ANDEXPERIMENTALRESULTS

A. Simulation Results

An LLC resonant converter using the parameters of Table II is simulated under different operating conditions implementing the designed dual-loop controller.

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Fig. 14. Closed-loop Bode plot of the proposed dual-loop control scheme for the LLC resonant converter of Table II. The obtained closed-loop 3-dB bandwidth is 1.5 kHz. The outer voltage loop gain is adjusted to obtain a voltage bandwidth ten times slower than the inner current loop to prevent the interaction between control loops.

Fig. 15. Different closed-loop Bode plots of the proposed dual-loop scheme for different outer-loop bandwidths. Larger gains in the voltage loop enable higher bandwidth and faster responses in the closed-loop system. However, large gains may create oscillation between both control loops.

Simulation results of the proposed dual-loop scheme under different load steps and voltage reference set points are shown in Fig. 16 and compared with the open-loop responses (constant fs). The upper plot illustrates the response ofvo for a current load step from1to 7 A. As expected, the closed-loop system response shows no oscillations and zero steady-state error under the applied disturbances at different operating points.

The second part of Fig. 16 illustrates the variation of the switching frequency during the transients. As shown, the dual- loop controller is able to compensate the output over a wide range of frequencies above and below resonance. It can be observed that the controller continues operating properly for

Fig. 16. Simulation results for the proposed dual-loop controller for the LLC converter of Table II under a load step for different operating points. The closed- loop system is stable under a wide range of operating conditions below and above the resonance.

reference set points that reach 34 and 18 V with switching frequencies ranging from 70 to 150 kHz.

The output voltage behavior under a set-point step is illustrated in Fig. 17. The figure shows a comparison between different loading conditions for closed-loop operation and illustrates how the switching frequency must change in order to track the changing set point. As shown, the closed-loop system is stable under a wide range of operating points below and above the resonance.

B. Experimental Results

A 150–W prototype of a full-bridge LLC resonant converter was implemented employing the parameters given by Table II.

A dual-loop controller, inner current and outer voltage, is implemented based on the design of Section VI. The proposed

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Fig. 17. Simulation results for the proposed dual-loop controller for the LLC converter of Table II under a voltage reference step for different load conditions.

The controller operates in a wide range of switching frequencies.

Fig. 18. Experimental capture current load step from 1 to 7 A for a reference voltage of 18 V. C1: output voltage, C2: average current from the microcontroller, C3: rectified current, C4: load current.

controller is implemented in a Texas Instrument digital microcontroller for the series C2000.

Experimental results for the transient response of the 150-W prototype under a current load step are presented in Figs. 18–21.

The experimental captures show the dynamic response of the LLC converter under a current load step operating at different voltage set points. The response of the converter for a voltage reference of 18 V is shown in Fig. 18. As shown, the setting time is 2.2 and 1.2 ms for a load current step-down and step-up, while the overshoot is 1.2 V and 1.5 ms, respectively. For a set point of 22 V, the converter operates around the resonant point,

as shown in Fig 19, with settling time and overshoot values of 60 mV and 1.2 ms, respectively. The behavior of the closed- loop system with the converter operating belowf_ris illustrated in Figs. 20 and 21. As can be observed, both cases show similar settling time and overshot values.

The dynamic response of the experimental platform under a voltage set-point step is shown in Fig. 22. The capture illustrates the output voltage and switching frequency of the converter for three different loading conditions. As shown, the output voltage reaches the reference point with no overshot and no oscillations for all loading condition . During the step-up, the output shows similar responses for all loading conditions. However, during step-down, lighter loads take longer time to settle due to the converter entering in the discontinuous conduction mode.

A conventional voltage-mode controller has been implemented to compare the dynamic response of the dual-loop controller. The experimental results for load transient response of the 150-W prototype are shown in Figs. 23–26. As observed, the dual-loop control shows superior transient behavior over a wide range of operating conditions. Apart from reducing the order and simplifying the voltage loop, the inner current loop

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Fig. 22. Experimental results in the prototype LLC resonant converter under a voltage reference step for different loading conditions. C1, M1, and M3 show vo foril o a d 7, 3.5, and 1 A, respectively, while C2, M2, and M4 illustrate the variation offs. C3 shows the voltage reference step from 20 to 30 V. The closed-loop system operates in a wide range of operating points.

Fig. 23. Experimental capture current load step from 1 to 7 A for a reference voltage of 18 V employing a single voltage loop. C1: output voltage, C2: average current from the microcontroller, C3: rectified current, C4: load current. The transient response shows larger overshoot.