Series Resonant Converters with an Adaptive Secondary-side Digital Control for MHz 48V VRs:
Circuit Analysis and Modeling
Shangzhi Pan, member,  Majid Pahlevaninezhad, member,  Praveen. K. Jain, Fellow, IEEE
Dept. of Electrical and Computer Engineering
Queen’s University
19 Union St., Kingston, On, Canada K7L 3N6
Email: shangzhi.pan@queensu.ca
Abstract  - An adaptive secondary-side digital control is proposed for the series resonant converter to solve the loss of ZVS under certain input and load conditions, making it a good candidate for MHz 48V voltage regulators (VRs). By means of varying AC equivalent resistance, the output voltage regulation is achieved by controlling the duty cycle of the synchronous MOSFET. An additional control variable has added complexity to circuit analysis, modeling and controller design, since the state variables such as the inductor current and capacitor voltage are AC domain. A sine-cosine tran
sformation is applied to these AC resonant state variables, correspondingly, the resonant circuit can be decomposed into two subcircuits under sine and cosine axes at each harmonics frequency of interest. The transformed state space model can be used for the large signal and small signal analysis for the series resonant converter with the secondary-side control. A power circuit is designed based on the large signal analysis and a feedback controller is designed based on the derived small signal model. Simulation and experimental results verify the validation of the design.
I. INTRODUCTION
The 48V power distributed architecture (PDA) has been widely used in telecom and data centers for high performance power management and delivery. In such a PDA, power is delivered through a high voltage low current bus, thus reducing the power delivery loss, and points-of-use voltage regulators (VRs) are used to tightly regulate the voltage in close proximity to the end users. The high voltage distribution power system also benefits from the significantly reduced input filter and good decoupling between different loads on the same distribution bus [1].  Most of today’s high-performance multi-core
microprocessors for workstations and servers operate with voltages near 1V and employ power-man
agement strategies to minimize power consumption. High dynamic characteristics of microprocessors due to intelligent power-management strategies make it very difficult to maintain an accurate voltage regulation. As a result, the switching frequency should be increased to improve the transient response and to reduce the size of the costly filter inductance and capacitance. However, many 48V VR topologies, such as the symmetrical half-bridge topology and the push–pull forward topology can hardly operate at MHz operating frequency because of high switching losses [1]. Usually with
hard switching, when the switching frequency is from 100 kHz to 500 kHz, the efficiency may drops about10%.
Resonant converters are able to achieve soft-switching; thereby allowing it operate at MHz switching frequency, to reduce the size and improve transient performance without suffering big efficiency penalty [2-9]. However, series resonant converters may lose zero voltage switching (ZVS) under certain conditions such as high input voltage and light load [2-4]. The secondary-side control was proposed to solve the loss of ZVS in the series resonant converter [10]. Output voltage regulation is achieved by controlling AC equivalent resistance, which is dependent on the duty cycle of the rectifier switches. By taking advantage of the secondary-side control, the control of primary-side swit
ches is completely independent of the load and input conditions, therefore series resonant converters can operate under zero-voltage switching (ZVS) at any input voltage and load conditions.
Modeling of resonant power converters is important for circuit analysis and controller design. The resonant frequency of the resonant tank is designed to be in close proximity to the switching frequency, so that ZVS is achieved for switches. The energy circulating in the resonant tank is mainly in the form of the fundamental frequency. Hence, the widely-used state space averaging model approach is not valid in the resonant converter [14-17].
An additional control variable on the secondary-side has added complexity to circuit analysis, modeling and controller design, since the state variables such as the inductor current and capacitor voltage are AC domain. A sine-cosine
transformation is applied to these AC resonant state variables, correspondingly, the resonant circuit can be decomposed into  two subcircuits under sine and cosine axes at each harmonics frequency of interest. . By such a transformation, all state variables in the state space model are at dc frequency, no longer resonant AC. The large signal model and the small signal model for the series resonant converter with the secondary-side control have been successfully derived to guide the pow
er circuit and the feedback controller design. Simulation and experimental results verify the validation of design.
o
Fig. 1 A typical APWM seies resonant converter with synchronous MOSFET
current sensorless drivers
II. DESCRIPTION  OF  THE  SERIES  RESONANT  CONVERTER  WITH  THE  ADAPTIVE  SECNDARY-SIDE
CONTROL  An asymmetric PWM (APWM) series resonant converter
with the secondary-side control circuit is given in Fig.1. The primary-side switches (M 1 & M 2) operate in the complementary mode against input voltage variations. The series resonant tank consis
ts of a capacitor C s  and an inductor L s , which transforms the unipolar pulsed voltage into a high frequency sinusoidal current. A high frequency transformer T  is employed for the safety isolation and the voltage stepping-down. The AC current from the transformer is rectified by synchronous rectifier switches (S 1 & S
2) and filtered by the output filter capacitor C f .
The rectifier switches (S 1 & S 2) operate in the PWM control mode against load variations. The volta
ge regulation can be achieved by varying the AC equivalent resistance through the duty cycle of the rectifier switches (S 1 & S 2). The control of primary-side switches is independent of the load conditions, even independent of the input voltage. Such a characteristic allows the ZVS operation of primary-side switches under any load and input conditions, which is not achievable in a conventional series resonant converter [2-4].
Two high gain operational amplifiers (OP) with hysteresis are used to monitor the polarity of the voltage on the synchronous rectifier MOSFETs, and generate two signals V S1 and V S2 indicating current polarity of each MOSFET. As shown in Fig. 2, the logic ‘NOR’ of the signals Vs1 and Vs2
will have a positive pulse with the width, which is consistent with the cross-conduction duration. The phase adaptive controller is implemented to eliminate the cross-conduction by moving the phase Φ of the control signal on switches S1 and S2. Fig. 3 illustrates the basic flowchart of the adaptive control. Such an adaptive control eliminates the current sensor in the power path and automatically compensates the uncertain control delay.
Fig.2 Cross-conduction operation waveforms
Fig. 3 Basic flow chart of the adaptive control
V V
Fig. 4 The typical steady state operating waveforms
Voltage on
Voltage on
Cross-conduction
Fig. 4 gives the typical steady-state operating waveforms,
given that:  (1) The main switches and their diodes (M1 & M2) are ideal.
The duty cycle of switch M1 is D m  and the duty cycle of switch M2 is 1-D m . The dead time between the two switches is neglected;  (2) the control of synchronous MOSFETs is symmetrical, that’s, each MOSFET conducts in the middle of half of each period for half of desired duty cycle (D /2, total duty cycle is  D ), by applying dual-edge modulation;
(3) The rectifier switches (S1 & S2) are ideal, but their body diode forward voltage is V d .  III.  CIRCUIT  ANALYSIS  AND  THE  PHASOR
TRANSFORMED  STATE-SPACE  MODEL
A.  State-space equations
Assumed that the tank current is continuous, the series
resonant converter in Fig. 1can be described by state-space
equations (1):
⎪⎪⎪
⎩
⎪⎪
⎪⎨⎧+⋅=+⋅+==++⋅+o L Cf Cf f
L c s
AB p s s
i i n R V dt dV C R r i dt dv C v v v i r dt di L 1(  (1)
Where, r s  is the series equivalent resistance, including MOSFET on-state resistance and ESR of inductors, and i o  is a
small signal perturbation over the DC component. v AB  is generated by the voltage chopper (M1 and M2) operating in the APWM mode. When the secondary-side control is applied, as illustrated in Fig.4, the voltage v p  on the primary side of the transformer can be given by (2):
()()()()
⎪⎪⎪⎪
⎪⎪
⎩⎪⎪⎪⎪⎪⎪⎨⎧
+<<=+⋅−⋅++
=<<=⋅−⋅−+
=<<=+⋅−+
=<<=+⋅⋅++
=<<=⋅⋅−+=<<=+⋅=s
o d s
o s
o d s
o d s o
s o d p T t t t V V n T D t t t t V n T D t t t t V V n T t t t t V V n T D t t t t V n T D t t t t V V n t v 090960650540410104
)3(4)3(2
4
)1(4)1()((2)
The interested output variable is the output voltage, as given by:
L
c c c
Cf
c o c o R r r r V r i i n r v //'')('=⋅++⋅⋅=
adaptive(3)
Equation (3) describes the output rectifier stage, where r c  represents the equivalent series resistance of the filter
capacitor.
B. Transformation of state-space equations  To achieve ZVS for the switches, the resonant frequency of the resonant tank in the series resonant is chosen to be slightly smaller than the switching frequency. In the steady
state, some of the sate variables, the voltage and current of the energy storage components in the resonant are resonant
AC. Hence, the small ripple assumption over state variables in the state-space averaged model, does not stand in the
resonant converters [14-17]. The state-space averaging model
method [11-13] is not applicable to the resonant converter
although it has been widely used for pulse width modulation (PWM) DC–DC converters.  It is easily observed that the energy circulating in the resonant tank is in the resonant forms at all the harmonic
frequencies, k o , k  times the switching frequency o . In other
words, the energy is transferred in the harmonic components
from the input to the output, mainly in the fundamental component, not at dc frequency as in the other non-resonant
converters. So the circuit has to be studied in forms of both dc and all harmonic frequencies.  In the steady state, all state variables are periodical, i.e., x(t)=x(t+T). Fourier analysis can be applied to all state variables in state-space equations (1) and (3). The resonant current i  in the energy storage component L s  and the resonant voltage v  on the energy storage component C s  can be
approximated with a limited number of Fourier series:  ⎪
⎩⎪⎨⎧⋅+⋅+=⋅+⋅+=∑∑k o ck o sk dc k o ck o sk dc t k t v t k t v V t v t k t i t k t i I t i )]cos()()sin()([)()]
cos()()sin()([)(ωωωω      (4)
The voltage v p  can be represented in terms of Fourier series in one switching cycle, T s , as follow,  ∑
⋅⋅⋅=⎥⎥⎥
⎦
⎤⎢
⎢⎢⎣
⎡⋅⋅⋅⋅−+⋅=
,5,3,1)sin(2sin
2
sin 4)(4)(k d o d p t k k kD k V n V V n t v ωπ
ππ (5)
The chopped input voltage v AB , which is generated by the voltage chopper (M1 and M2) operating in the APWM mode, can be derived in terms of Fourier series:
∑⎥⎦
⎤⎢
⎣⎡⋅⋅−⋅⋅⋅⋅⋅⋅+⋅=k m o m g g m AB D k t k k D k V V D t v )cos()sin(2)(πωπ    (6)
⎪⎪⎪⎪⎪⎪
⎪⎩
⎪⎪⎪
⎪⎪⎪
⎪⎨⎧
++=+⋅+=+=−==+−+++⋅++=+−+++⋅+−o c s L Cf cf f L c c s o c s s
c o s s g
m dc m g c s c d d o c c s s o c s c s s d d o s s s c o s s i i i n R V dt dV C R r i
v dt
dv C i v dt dv
C V
D V D V i i i D V V V n v i r i dt di L i i i D V V V n v i r i dt di L 2222222)1()()()sin(22sin (4)(0)2sin (4)(ωωπππ
πωππω                    (8)All the state-space variables in (1) has been decomposed into two parts, sinusoidal and cosinusoidal parts which are orthogonal to each other. As a result, the original circuit will be decomposed into two dc subcircuits at any harmonic frequency , including dc frequency. Since the resonant tank filters out most of the harmonics, only the fundamental component will be considered in the model. The original circuit thus can be approximated with sets of two subcircuits at the switching frequency and the dc frequency. Then the resonant state variables i  and v  are simplified as:
⎩⎨
⎧⋅+⋅+=⋅+⋅+=)
cos()()sin()()()cos()()sin()()(t t v t t v V t v t t i t t i I t i o c o s dc o c o s dc ωωωω              (7)
Substituting (5) (6) and (7) into (1), the original state-space equations can be decomposed into three subcircuits, in DC, sinusoidal, and cosinusoidal terms, respectively. Since sinusoidal and cosinusoidal terms are orthogonal, we can equate coefficients of DC, sinusoidal, cosinusoidal terms respectively, the original state-space equations (1) is transformed into phasor frames ( sine and cosine), which is expressed by the state space equations (8).
In the orthogonally transformed state-space model, all the state variables are in DC, not resonant AC any more, which are defined as:
T cf c s c s v v v i i x ),,,,(=
T o g m i v d d u ),,,(=        o v y =
Therefore, all standard methods in a state-space model
can be applied. By simply letting derivatives in (8) be zeros,
steady state solutions are obtained, which can used as the
steady state operation point for deriving the small signal model:
22e
e e e c R Z R V I +=                      22e e e
e s R Z Z V I += s o e e e e s C R Z R V V ω122
+=        s
o e e e e c C R Z Z V V ω12
2+−= )2
sin
1(82
2πααπD R n R L
ac ⋅−+⋅⋅=
ac s e R r R +=
o
d V V =
α s
o s o e C L Z ω1
−
=              )sin(2πm g e D V V =
C. Circuit analysis
Therefore, if r s  is neglected, the voltage conversion gain can be found from steady state solutions:
2
2
2
2
)1(1)2
sin
1(2)
sin(22
ωωπααππ
π−⋅+⋅⋅−+⋅=
⋅+⋅=⋅⋅==o m g
L
s c g
L
g o Q D n D V R I I n V R I n V V M
ac s r o R L Q ⋅=
ω          r o ωωω=            s
s r C L ⋅=
1ω If the feed-forward control is applied, the modified
voltage conversion ratio M s  can be found with (9).
22
)1(1)2sin
1(21
)
sin(ωωπααπ−⋅+⋅⋅−+⋅=
⋅=
o m g o
s Q D n D V V M  (9)
Design of the resonant tank (C s  and L s ) is critical to achieve good converter performance. Fig. 5 illustrates the
normalized voltage conversion ratio M s  at different Q o  and ω according to (9). The gain (M s ) decreases with the ω or Q o
increasing. It is concluded that lower ω (ω >1) will help achieve good controls over the whole load range and higher ω
may cause losing controls at light load. With lower ω, the effective duty cycle range will be narrower, that’s, efficiency
at reduced load could be improved. Similarly, higher Q o  at the full load will extend the effective duty cycle and may cause losing controls at light load. Therefore, to improve efficiency at reduced loads, reasonable low Q o  and ω is preferred.
Duty Cycle (D)
V o l t a g e  C o n v e r s i o n  R a t i o  (M s )
Duty Cycle (D)
V o l t a g e  C o n v e r s i o n  R a t i o  (M s )
Fig. 5 the voltage conversion gain (M s ) at different Q o  and ω
Output Current I O  (A)
O u t p u t  V o l t a g e  V O  (V )
Fig. 6 the regulated output voltage with circuit design parameters
The design specification is given in Table 1.
Table 1 Design specification
Circuit parameters Input voltage range 43-56V Maximum load current  20A Regulated output voltage  1V Switching frequency
1MHz
ω=1.15 and  Q o  =1 are selected to have a reasonable duty cycle range, so a 3.3uH inductor L s  and a 10nF capacitor C s  can be determined for the resonant tank, and the transformer ratio (n ) is 20.
Fig. 6 gives the regulated output voltage with circuit design parameters. The effective duty cycle is ranging from 0.73 to 0.91.
D. The small signal model and controller design
By perturbing the large signal model (8) around the steady state point and making linearization under the small signal assumption, the small signal model is represented by:
∧
∧∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧∧∧
∧∧∧===⎪⎩⎪⎨⎧⋅+⋅=⋅+⋅=o
T
o g m T cf c s c s v y i v d d u v v v i i x u E x C y u B x A dt
x d ),,,(),,,,(
⎥⎥⎥⎥
⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢
⎢⎢
⎢⎢
⎢⎢⎢⎢
⎢⎣
⎡+−++−−−−−−−−=f c L f
c L L c f c L L s o s o
s s
c s
s c s c
s s
s s s s s C r R C r R R nK C r R R nK C C L nK L L R L Z L nK L L Z L R A )(10
)(2)(200100
001410401ωω
⎥⎥
⎥⎥⎥
⎥
⎥⎥
⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢⎢⎢
⎢
⎢⎢⎢
⎢
⎢⎣
⎡
+=f c L L
s
m s m g s
c d s
s d C r R R L D L D V L D K nV L D K nV B )(0000
00000000)sin(2)cos(22cos 20002cos 2πππππ
⎥⎦
⎤⎢
⎣⎡=c c c c c s r r r nK r nK C '0
0'
2'2π
π
[]'0
00
c r E =
222
e
e e ac s c Z R Z
R r R ++=      222
e e e ac s s Z R R R r R ++=
(a)
(b)

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