# Feedforward Inversion Control of DC/DC Dual-Bridge Series Resonant Converter
in Buck and Boost Modes

Alex Borisevich<EMAIL_ADDRESS>
Filipp Gleyzer<EMAIL_ADDRESS>
Valeria Gavrilenko<EMAIL_ADDRESS>
## Abstract

In the paper, a nonlinear inversion technique for the steady-state model of
the active dual-bridge series resonant converter is presented. The obtained
control strategy allows cycle averaged output current regulation and performs
waveform alignment for the controllable achievement of ZVS and synchronous
rectification. The control is valid both for voltage buck and boost operating
modes, as well as for low-power operation at a fixed frequency. Robustness of
the control is studied by simulations with external linear control loops.

##  1 Introduction

The dual-bridge (DB) series resonant converter (SRC) in comparison to many
well known isolated DC/DC topologies like LLC or non-resonant dual active
bridge has some unique features due to controlled secondary-side bridge and
the presence of resonant LC tank, such as close to sinusoidal transformer
current and a capability of bidirectional power flow and voltage boost (step-
up) operation [1]. Such type of converter topology is particularly promising
for electric vehicles chargers applications [1-3], including off-board
charging stations and on-board charger modules with an ability for
bidirectional vehicle-to-grid power flow. Additionally, many other power
electronics applications like battery energy storage and DC line power
transmission are target areas of DB SRC topology.

The modulation scheme for DB SRC consists of driving all the switches in the
two bridges work with 50

The modulation scheme for DB SRC usually enables pure ZVS on the primary side,
and combined ZVS/ZCS on the secondary side, and allows for efficient
utilization of the primary side switches with wide changes in load or supply
voltage, which is preferred in many applications [2]. The impedance of the
series resonant tank is determined by the value of the inductor and capacitor
together with the switching frequency. Therefore, it can operate at a higher
frequency under a high power level [3] than just dual active bridge topologies
without a resonant capacitor, which allows more compact magnetic designs.

The operating modes and controls principles of the DB SRC converters have been
extensively analyzed in [5, 6, 7]. Most of the previous works concern the
control of the series resonant converter loaded by a diode bridge.

One of the earliest approaches to the control of SRC is based on state-plane
trajectory analysis pioneered in the seminal work of R. Oruganti [8]. The key
idea is to represent the converter dynamics in the two-dimensional plane of
resonant inductor current and resonant capacitor voltage. The analysis is
greatly simplified by the fact that with a proper normalization the state
plane trajectory of voltage step response of an undamped LC circuit is
circular and is centered at the DC solution of the circuit. Thus, accurate and
purely geometric analysis can be performed for arbitrary operating point and
switching pattern. The downside of this method is that the closed form
solutions are becoming very complicated with less trivial PWM patterns.

The examples of implementation of this strategy [9, 10] for a corresponding
class of SRC converters are very simple circuits consist of few analog linear
components, logic gates, and flip-flops. Since the control circuit directly
drives transistors of the input MOSFET bridge without a PWM modulator these
circuits provide excellent transient performance. Some recent developments
following the geometric state-plane trajectory analysis are presented in [11,
12]. The disadvantage of the approach as mentioned that it is developed only
for converters without modulated secondary side, so the voltage gain is always
below 1 (i.e. buck mode).

Very closely related to the state-plane trajectory approach are self-sustained
oscillating [13, 14] and resonant capacitor charge bang-bang control [15].
Both of these approaches formulate the control problem of converter output
voltage using a quantity that directly affects the switching of the input
bridge. For the self-sustained oscillating, the voltage controller controls a
delay from positive zero crossing of resonant current to the rising edge of
input bridge voltage. The capacitor charge bang-bang control uses the resonant
capacitor voltage threshold as a control variable and switches the bridge by
reaching a threshold of it, defined by the output voltage controller. Both of
these techniques suffer from the same drawback as they able to control only
the input bridge, and the output secondary side bridge acts as an
uncontrollable rectifier.

The nonlinear control problem of SRC has inevitable nonlinearities and
intrinsic constraints. In power supply applications, the controller should be
fast enough to reject load transients. These problems are recognized by
control systems researchers, and some advanced techniques as passivity-based
controllers [16, 17], hybrid-flatness approach [18], and backstepping control
approach [19, 20] are presented. All of the works [16-20] consider SRC
converters without modulated secondary side, thus the same voltage gain
limitation applies.

In recent works, the advantages of the controlled rectifier stage were fully
understood and appreciated. The paper [21] introduced and analyzed a new
serial resonant converter derived from the classical SRC, with controlled
switches at the rectifier stage. The possibility of unlocking voltage gain
higher than 1 is emphasized.

In [22] the bridge shift phase is varied to keep the average power factor at
its peak meanwhile the frequency modulation is used to manipulate the power to
be transferred. Both bridges are fully driven, no duty cycle control is used,
the phase shift between bridges is a function of a measured voltage gain.

A detailed time-domain analysis of ZVS conditions is presented in [23, 24],
based on which the closed-form solution is formulated which guarantees
sufficient ZVS commutation current at switching instants, accounting for the
presence of switches parasitic and/or snubber capacitances, while maintains
the RMS tank current at near-to-minimum levels. The phase shift quantities are
open-loop based model calculations, without measuring the actual resonant tank
current and its position with respect to the bridge switching events.

A similar to [23, 24] control approach is presented in [25], which is based on
a detailed model with the multi-harmonics approximation of switching voltage
waveforms by Fourier series. The phase shift between bridges is also an open-
loop quantity based on the model calculations. Both bridges are fully driven
without duty cycle control.

In the paper [26] the resonant tank current is being measured, but a very
simple analog controller is proposed for the phase shift between bridges.
Also, both bridges are fully driven and the duty cycle control is not
explored.

Based on previous publications, the following conclusions could be drawn:

\- An attention to control problem of DB SRC is only recently given following
appreciation the advantages of this topologies. Most of the published works
are remain for SRC converter without a controlled secondary side.

\- The duty cycle control in conjunction with phase shift control between
bridges is not sufficiently reflected in the published works.

\- A closed-loop control of the duty cycles and phase shift between bridges
for maintaining ZVS and optimizing efficiency is never considered so far.

In this work, we will systematically analyze the control problem of DB SRC in
quasi-steady-state mode for efficiency optimization and output current
regulation. We are going to show that by using a simple resonant current zero-
crossing detector and measuring timing between resonant tank current and
bridge switching events, it is possible to actively control duty cycles of the
bridges and the phase shift between primary and secondary sides. This ensures
ZVS and maximum efficiency in all operating conditions subject to the model
uncertainties.

Our approach of directly inverting the static nonlinearity is technically
inspired by [16, 17]. A usage of model-based feedforward maps is based on
previous works [23-26], which are augmented by closed loop controllers in the
paper. While an idea using the resonant tank zero crossing events is based on
[26] and [28].

##  2 Background

This section is a recall of previously published results, and the model [29]
in particularly.

###  2.1 The converter topology

The basic electrical circuit of the series LC resonant DC/DC converter is
shown in Figure 1.

Figure 1: DC/DC circuit.

The circuit consists of two full (H) transistor bridges: input bridge with
switches $S_{1}$–$S_{4}$ and output bridge with switches $S_{5}$–$S_{8}$. The
output bridge is directly connected to the secondary side of transformer TX.
The input bridge is connected to the transformer TX through capacitor C.
Simple representation of transformer is used with magnetizing inductance
$L_{m}$, leakage inductance $L$ and an ideal transformer with turns ratio $n$.

If the magnetizing inductance $L_{m}$ is much larger than leakage inductance
$L$: $L_{m}\gg L$, then there is almost no circulating magnetizing current in
the circuit. This has an advantage because all the current from primary side
is flowing to the secondary side which increases the efficiency. Also the
current in secondary side of transformer is in phase with primary side, which
allows easy synchronous rectification by measuring current only in primary
side.

By eliminating $L_{m}$ from the circuit, we can spot that the stray inductance
$L$ together with capacitor $C$ forms LC resonant tank. Also ideal transformer
can be embedded into secondary bridge for simplification of analysis. Thus we
can obtain a black-box circuit presented in Figure 1.

The alternating current which flows through the leakage inductance and
resonant capacitor is called tank current $i_{t}(t)$. This current is being
rectified by secondary transistor bridge and assuming that all the ripple
components are blocked by output filter capacitors (not shown in Figure 1), a
direct current $I_{out}$ is induced through a load.

###  2.2 The voltage waveforms

Let’s consider voltages $v_{in}(t)$ and $v_{out}(t)$ at output terminals of
bridges in the circuit 1.

  * •

The voltage $v_{in}(t)$ corresponds to a voltage measured between points A–B
of input bridge.

  * •

The voltage $v_{out}(t)$ corresponds to scaled by $n$ voltage measured between
points C–D of output bridge.

Effectively, these voltage $v_{in}-v_{out}$ is applied to LC-circuit. Figure 2
shows the real and rectangular approximated voltages.

Figure 2: Real and approximated voltages applied to resonant LC circuit and
PWM parameters.

Let’s consider rectangular approximated voltages $u_{in}(t)$ and $u_{out}(t)$
for easier explanation of controllable switching parameters:

  * •

The input voltage source is producing rectangular pulses with controlled duty
cycle and frequency. The amplitude of pulse is $V_{in}$ and the on-time is $d$
(in radians, i.e. $d=\pi$ is full square wave). The angular frequency is
$\omega$, which is equivalent to frequency $F=1/T$ in Hz.

  * •

The output voltage source is producing rectangular pulses with amplitude
$nV_{out}$ with controlled duty cycle, frequency, and phase shift respect to
input bridge. The reciprocal of duty cycle, the off-time or short time is $s$
(defined in radians, $s=\pi$ means that secondary side of transformer is fully
shorted). The off-time is always located at the beginning of switching cycle.
The phase shift between the output bridge switching cycle and input bridge
switching cycle is $\beta$.

The output to input voltage ratio is defined as converter voltage gain:

$G=\frac{nV_{out}}{V_{in}}$ (1)

The real waveforms are looking differently than approximated ones because of
dead times, which are smoothing rising and falling edges of the switching
pulses. Since this DC/DC belongs to the class of resonant converters, energy
is also transferring during the dead time, and the dead time is an essential
phase of converter operation.

The duration of dead-time for input bridge is $T_{D}^{in}$, and the duration
of dead-time for output bridge is $T_{D}^{out}$.

###  2.3 The model

The converter model obtained in [29] by first harmonic approximation of tank
current can be summarized as:

$W=\frac{I_{out}}{V_{in}}=\frac{n}{2\pi^{2}}\frac{\sqrt{A^{2}+B^{2}}}{Z(\omega)}(\cos(s+\delta)+\cos\delta)\\\
$ (2)

where coefficients $A$ and $B$ along with timing quantities $\sigma$ and
$\delta$ are given as

$\begin{gathered}A=4\sin d+4G\sin(\beta+s)+4G\sin\beta\\\
B=4-4G\cos(\beta+s)-4G\cos\beta-4\cos d\\\
\sigma=\operatorname{atan2}(B,A),\;\;\delta=\beta-\sigma\end{gathered}$ (3)

and $Z(\omega)$ is resonant LC tank impedance:

$Z(\omega)=X_{L}-X_{C}=\omega L-\frac{1}{\omega C}$ (4)

The timing quantities $\sigma$, $\delta$ and $\beta$ are shown in Figure 3

Figure 3: Timing of input and output voltages respect to tank current zero
level crossing.

Namely:

  * •

The $\sigma$ is an angular time (in radians) from the rising edge of positive
input bridge voltage $v_{in}(t)$ to a moment when the tank current crosses 0
level: $i_{t}=0$. I.e. formally, $i_{t}(\sigma/\omega)=0$.

  * •

The $\delta$ is an angular time (in radians) from the moment when the tank
current crosses 0 level $i_{t}=0$ to the rising edge of positive output bridge
voltage $v_{out}(t)$.

It is pretty obvious from the Figure 3 that:

$\sigma+\delta=\beta$ (5)

Worth noting that both $\sigma$ and $\delta$ are signed.

###  2.4 Regulation problem and optimality metrics

In this particular work the DC/DC converter will be represented as a current
source. This particular representation covers EV battery charger applications,
which is main application area of such topologies.

Thus, an output DC current problem regulation can be formulated as follows:

Find PWM switching parameters: $d$, $s$, $\beta$, $\omega$ in order to achieve
desired steady-state output current:

$I_{out}(d,s,\beta,\omega)=I_{out}^{*}$ (6)

where $I_{out}^{*}$ is reference desired value of output current.

Sine the converter topology considered here is lossless, some additional facts
need to be considered for derivation of optimality conditions. The following
requirements can be fulfilled in order to operate the converter efficiently:

  * •

Conduction loss minimization.

In order to minimize conduction loss, the amplitude (or RMS) value $I_{t}$ of
tank current $i_{t}(t)$ needs to be minimized. Thus every set of switching
parameters $(d,s,\beta,\omega)$ can be evaluated for conduction loss
optimality:

$\frac{I_{t}}{I_{out}}\to\min$ (7)

  * •

Turn-on switching loss minimization.

The topology under consideration can be operating in soft-switching mode,
which minimizes turn-on losses down to zero. The conditions for soft-switching
discussed in many studies. Necessary conditions can be summarized in two
statements:

1\. A rising edge of positive $v_{in}(t)$ should be at negative tank current
$i_{t}(t)$.

2\. A rising edge of positive $v_{out}(t)$ should be at positive tank current
$i_{t}(t)$.

These two statements can be summarized with respect to the timing of switching
events. Let’s assume that $t=0$ during positive rising edge of $v_{in}(t)$.
Then the soft switching conditions for turn-off are:

$i_{t}(0)\leq 0,\;\;i_{t}(\beta/\omega)\geq 0$ (8)

or using angular quantities defined in Figure 3:

$\sigma\geq 0,\;\;\delta\geq 0$ (9)

  * •

Turn-off switching loss minimization.

For the turn-off losses, a true soft-switching can’t be achieved in this
topology. The turn-off losses can be only mitigated. One practical approach is
to use a capacitive snubber across the MOSFETs. Additionally, turning-off
close to zero current, i.e. achieving ZCS commutation is usually helps achieve
better efficiency.

Thus, we can introduce the following criteria:

$|i_{t}(0)|\to\min,\;\;|i_{t}(\beta/\omega)|\to\min$ (10)

It worth noting, that commutating at zero current, i.e. achieving an obvious
goal $i_{t}(0)=i_{t}(\beta/\omega)=0$ is not practical since ZVS can be
achieved only at specific current levels which allow discharge of internal
Coss and external snubber capacitors (if any) during a dead-time interval.

So generally speaking, the following constraints need to be satisfied:

$i_{t}(0)=-I_{th}^{in},\;\;i_{t}(\beta/\omega)=I_{th}^{out}$ (11)

where $I_{th}^{in},I_{th}^{out}\geq 0$ are threshold currents for commutation
of input and output bridges correspondingly.

###  2.5 The output control problem reformulation

Let’s follow the discussion from section 2.4 by taking into account model
equations (2).

First, achieving the desired output current $I_{out}^{*}$ is the same as
regulating to a prescribed transconductance $W^{*}=I_{out}^{*}/V_{in}$ for a
given input voltage $V_{in}$. So the reformulated output control problem is to
find switching parameters $d$, $s$, $\beta$, $\omega$ so that:

$W(d,s,\beta,\omega)=W^{*}$ (12)

where function $W(d,s,\beta,\omega)$ is given by (2).

Additionally, from [29] it is known that the amplitude of tank current is
given as:

$I_{t}=\frac{V_{in}}{2\pi Z}\sqrt{A^{2}+B^{2}}$ (13)

Using this, the ratio of $I_{t}/I_{out}$ can be expressed as:

$\frac{I_{t}}{I_{out}}=\frac{\pi}{n}\cdot\frac{1}{\cos(s+\delta)+\cos\delta}$
(14)

The ratio $I_{t}/I_{out}$ is minimized when denominator term
$\cos(s+\delta)+\cos\delta$ is maximized. Its maximal value 2 is achieved when
both $\delta=0$ and $s=0$.

So the optimality for tank current amplitude can be formulated as follows:

$\frac{I_{t}}{I_{out}}\to\min\;\Longleftrightarrow\;\begin{cases}\delta\to\min\\\
s\to\min\end{cases}$ (15)

This result can be physically interpreted as minimization of reactive power in
the resonant tank. Furthermore, the case of $\delta=0$ corresponds to ideal
synchronous rectification by the secondary output bridge, i.e. when voltage
waveform is fully aligned with tank current.

This type of control which uses phase shift control of the secondary side PWM
pulses to align the commutation of the secondary bridge with the sign of the
resonant tank is known as indirect or PLL-based synchronous rectification
[28]. In contrast with conventional direct synchronous rectification in which
the switching of the secondary side is driven by a sign of the resonant tank
current detected by a comparator.

The advantages of indirect synchronous rectification to the direct one are
twofold:

1\. It is possible to compensate comparator’s and gate driver’s delay by
switching the secondary side before the resonant tank current flips polarity,
which leads to efficiency increase by minimization of reactive power

2\. Since the output of the current sign comparator is no longer driving the
secondary side directly, the system is more prone to current measurement noise
and potentially can function even without the current sensor (in open loop
alignment control, assuming some margins on inefficiency).

##  3 Model inversion for commutation parameters $d,s,\beta$

###  3.1 Nonlinear inversion problem formulation

In order to regulate waveform quantities $\sigma$ and $\delta$ to desired
values that maximize efficiency, a control system needs to be implemented
around the plant which takes $d,s,\beta$ and outputs $\sigma,\delta$ as
measurable quantities.

A feedforward inversion approach is proposed below to linearize $\sigma$ and
$\delta$ as functions of controllables $d,s,\beta$. Thus, the following
nonlinear inversion problem can be formulated in form of a system of nonlinear
equations:

$\begin{gathered}\sigma(d,s,\beta,G)=\sigma^{*}\\\
\delta(d,s,\beta,G)=\delta^{*}\end{gathered}$ (16)

with respect to unknown variables $d\in[0,\pi]$, $s\in[0,\pi]$,
$\beta\in[-\pi,\pi]$ and for given referenced values
$\sigma^{*}\in[-\pi/2,\pi/2]$, $\delta^{*}\in[-\pi/2,\pi/2]$ and parameter
$G\geq 0$.

The concept of nonlinear inversion can be illustrated by a block diagram in
Figure 4, where nonlinear function $F$ is given by model equations (3) and
$F^{-1}$ is a solution of (16).

$\sigma$$\delta$$F$$F^{-1}$$\sigma^{*}$$\delta^{*}$$G$$G$$d$$s$$\beta$(3)
Figure 4: Nonlinear inversion with respect to $\sigma$ and $\delta$.

###  3.2 Nonlinear inversion condition

Let’s recall (3) as $\sigma^{*}=\operatorname{atan2}(B,A)$. In order to have
$\sigma^{*}\in[-\pi/2,\pi/2]$, the argument $A$ should not be negative:

$A=\sin d+G\sin(\beta+s)+G\sin\beta\geq 0$ (17)

If this holds, then $\operatorname{atan2}$ can be replaced by $\tan$, and the
nonlinear inversion problem can be rewritten as following two equations:

$\begin{gathered}\frac{1-G\cos(\beta+s)-G\cos\beta-\cos d}{\sin
d+G\sin(\beta+s)+G\sin\beta}=\tan\sigma^{*}\\\
\delta^{*}+\sigma^{*}=\beta\end{gathered}$ (18)

Note that $\beta=\delta^{*}+\sigma^{*}$ became constant.

This can be transformed as follows:

$\tan\sigma^{*}\cdot\left(G\sin(\beta+s)+G\sin\beta+\sin
d\right)+G\cos(\beta+s)+G\cos\beta+\cos d-1=0$ (19)

And the equation can be simplified further by using identity of harmonic
addition:

$\tan\sigma^{*}\sin(x)+\cos(x)=\sqrt{(\tan\sigma^{*})^{2}+1}\cdot\cos(x-\operatorname{atan}\tan\sigma^{*})=\sqrt{(\tan\sigma^{*})^{2}+1}\cdot\cos(x-\sigma^{*})$
(20)

where $x$ is an arbitrary angular argument.

Furthermore, taking into account the following trigonometric identity:

$\frac{1}{\sqrt{(\tan\sigma^{*})^{2}+1}}=|\cos\sigma^{*}|$ (21)

and using harmonic addition, the nonlinear inversion condition can be
rewritten as:

$G\cos(\beta+s-\sigma^{*})+G\cos(\beta-\sigma^{*})+\cos(d-\sigma^{*})-\cos\sigma^{*}=0$
(22)

Note that the absolute value of $\cos\sigma^{*}$ is safely omitted since the
range of $\sigma^{*}$ is $[-\pi/2,\pi/2]$.

Finally, using $\beta=\delta^{*}+\sigma^{*}$, it simplifies even further,
which we will refer as nonlinear inversion condition (for $d$ and $s$):

$G\cos(\delta^{*}+s)+G\cos\delta^{*}+\cos(d-\sigma^{*})-\cos\sigma^{*}=0$ (23)

It is important to note that (23) is underdetermined since it has two unknown
variables $d$ and $s$. So in order to find a unique solution, the minimization
of $s\to\min$ should be additionally imposed.

####  3.2.1 Buck mode with $s=0$

This is the simplest mode of operation and obviously satisfies minimization
condition since the $s=0$ is minimal value of $s$. Substituting this to (23)

$2G\cos\delta^{*}+\cos(d-\sigma^{*})-\cos\sigma^{*}=0$ (24)

thus:

$d=\operatorname{acos}\left(\cos\sigma^{*}-2G\cos\delta^{*}\right)+\sigma^{*}$
(25)

The domain of this buck operating mode is given by domain of
$\operatorname{acos}$ function and the range of $d\in[0,\pi]$

$\begin{gathered}-1\leq\cos\sigma^{*}-2G\cos\delta^{*}\leq 1\\\
0\leq\operatorname{acos}\left(\cos\sigma^{*}-2G\cos\delta^{*}\right)+\sigma^{*}\leq\pi\end{gathered}$
(26)

Practically, since the $\sigma^{*}$ is additive to result of
$\operatorname{acos}$ in (25), it is pretty much possible that the calculation
of $d$ could give greater than $\pi$ value. The boundary condition is

$\cos\sigma^{*}-2G\cos\delta^{*}=\cos\left(\pi-\sigma^{*}\right)=-\cos\sigma^{*}$
(27)

and the domain for $d\leq\pi$ is defined by inequality:

$\cos\sigma^{*}\geq G\cos\delta^{*}$ (28)

Another useful boundary case is when the argument of $\operatorname{acos}$ in
(25) is out of $[-1,1]$ range, particularly for high values of $G$, which
corresponds to a boost mode:

$\cos\sigma^{*}-2G\cos\delta^{*}=-1$ (29)

which translates to the following inequality:

$\cos\sigma^{*}\geq 2G\cos\delta^{*}-1$ (30)

####  3.2.2 Boost mode with $d=\pi$

Substituting $d=\pi$ to (23) gives:

$G\cos(\delta^{*}+s)+G\cos\delta^{*}-2\cos\sigma^{*}=0$ (31)

thus for variable $s$ it gives:

$s=\operatorname{acos}\left(2\cos\sigma^{*}/G-\cos\delta^{*}\right)-\delta^{*}$
(32)

The domain of this boost operating mode is given by domain of
$\operatorname{acos}$ function and the range of $s\in[0,\pi]$

$\begin{gathered}-1\leq 2\cos\sigma^{*}/G-\cos\delta^{*}\leq 1\\\
0\leq\operatorname{acos}\left(2\cos\sigma^{*}/G-\cos\delta^{*}\right)-\delta^{*}\leq\pi\end{gathered}$
(33)

Practically, since the $\delta^{*}$ is additive to result of
$\operatorname{acos}$ in (32), it is pretty much possible that the calculation
of $s$ could give negative value. The boundary condition is

$2\cos\sigma^{*}/G=2\cos\delta^{*}$ (34)

and the domain for $s\geq 0$ is defined by inequality:

$\cos\sigma^{*}\leq G\cos\delta^{*}$ (35)

Another useful boundary case is when the argument of $\operatorname{acos}$ in
(25) is out of $[-1,1]$ range, particularly for low values of $G$, which
corresponds to a buck mode:

$2\cos\sigma^{*}/G-\cos\delta^{*}=1$ (36)

which translates to the following inequality:

$2\cos\sigma^{*}/G\leq 1+\cos\delta^{*}$ (37)

####  3.2.3 Explicit additive reference of $s$

There are use cases when both $d<\pi$ and $s>0$ are required. In this case,
value of $s$ consists of two parts: a value required by (32) which will be
denoted as $s_{min}$ and an additive component $s_{add}\geq 0$ which is
specified externally:

$s=s_{min}+s_{add}$ (38)

It worth noting that just increasing $s$ by addition of $s_{min}$ without
corresponding adjustment of $d$ and $\beta$ will change resulting $\sigma$ and
$\delta$. So in order to maintain prescribed $\sigma^{*}$ and $\delta^{*}$,
the $d$ and $\beta$ need to be adjusted.

Thus, the nonlinear inversion equation (23) needs to be modified as follows:

$G\cos(\delta^{*}+s_{min}+s_{add})+G\cos\delta^{*}+\cos(d-\sigma^{*})-\cos\sigma^{*}=0$
(39)

where $s_{add}$ is a constant.

First consider the case of $s_{min}=0$, which corresponds to the buck mode.

$d=\operatorname{acos}\left(\cos\sigma^{*}-G\cos(\delta^{*}+s_{add})-G\cos\delta^{*}\right)+\sigma^{*}$
(40)

The buck mode is defined by a proper range of $d\leq\pi$ when no $s$ is needed
except the $s_{add}$ commanded externally, so that $s_{min}=0$, $s=s_{add}$.
Thus it is possible to generalize (28) as follows:

$2\cos\sigma\geq G\cos(\delta+s_{add})+G\cos\delta$ (41)

If the buck mode with $d\leq\pi$ and $s=s_{add}$ is not feasible, then the
boost mode is used. A minimal value of $s$ without $s_{add}$ is given by (32)
which is

$s_{min}=\operatorname{acos}\left(2\cos\sigma^{*}/G-\cos\delta^{*}\right)-\delta^{*}$
(42)

such that $s$ is sum of $s_{min}$ and $s_{add}$:

$s=\operatorname{acos}\left(2\cos\sigma^{*}/G-\cos\delta^{*}\right)-\delta^{*}+s_{add}$
(43)

Then in presence of $s_{add}>0$, the value of $d$ which satisfies inversion
equation (39) needs to be calculated as follows:

$d=\operatorname{acos}\left(\cos\sigma^{*}-G\cos(\delta^{*}+s_{min}+s_{add})-G\cos\delta^{*}\right)+\sigma^{*}$
(44)

The steps for obtaining the inversion map $F^{-1}$ are:

\- Check condition (28), if it holds, then this is a buck mode, use
$s_{min}=0$ and $s=s_{add}$.

\- If (28) doesn’t hold, then this is a boost mode and $s$ is calculated by
(43).

\- Calculate $d$ using (44).

\- Check inequality (17). If it is not satisfied, then the provided
$\delta^{*}$ and $\sigma^{*}$ are not feasible.

###  3.3 Examples of feed-forward characteristics

The solution (44),(43) of the inversion problem (16) is evaluated for
different $\sigma,\delta$ and $G$. The results are presented in this section
in form of contour plots.

####  3.3.1 Buck mode $G=0.5$

Figure 5: The switching waveform parameters $d$ and $s$ as functions of
required $\sigma$ and $\delta$ for buck mode $G=0.5$ and $s_{add}=0$
(resulting in $s=0$) Figure 6: The switching waveform parameters $d$ and $s$
as functions of required $\sigma$ and $\delta$ for buck mode $G=0.5$ and
$s_{add}=0.2$ (resulting in $s=0$) Figure 7: The switching waveform parameters
$d$ and $s$ as functions of required $\sigma$ and $\delta$ for $G=1$ and
$s_{add}=0$ Figure 8: The switching waveform parameters $d$ and $s$ as
functions of required $\sigma$ and $\delta$ for $G=1$ and $s_{add}=0.2$ Figure
9: The switching waveform parameters $d$ and $s$ as functions of required
$\sigma$ and $\delta$ for boost mode $G=1.5$ and $s_{add}=0$ Figure 10: The
switching waveform parameters $d$ and $s$ as functions of required $\sigma$
and $\delta$ for boost mode $G=1.5$ and $s_{add}=0.2$

###  3.4 Simulation of the feed-forward control

The proposed feed-forward control (43), (44) is simulating along with the
plant steady-state model (3) in order to illustrate the control trajectories
of PWM quantities $d,s,\beta$. In this simulation a sinisouidal reference
trajectories are generated for $\sigma^{*},\delta^{*},s_{add}$. The $G$ swept
from 0 to 2 in time so that $G=t$.

The results are presented in Figures below

Figure 11: Simulated trajectories of $\sigma$ and $\delta$: corresponding
reference and output curves are overlapping Figure 12: Simulated trajectories
of $s$ and $s_{add}$: $s_{add}=s$ for buck mode $G<1$ Figure 13: Simulated
trajectories of control inputs $s,d,\beta$

###  3.5 Special cases

####  3.5.1 Special case of $G=1$

It is important to discuss case of $G=1$, because this operating point has
unique properties. Furthermore, it is possible to show that $G=1$ is most
efficient operating point with respect to the active losses in the circuit.

Substituing $G=1$, $s_{add}=0$ and $\delta^{*}=0$ to (44) and (43) gives:

$\begin{gathered}s=\operatorname{acos}\left(2\cos\sigma^{*}-1\right)\\\
d=\operatorname{acos}\left(\cos\sigma^{*}-2\right)+\sigma^{*}\end{gathered}$
(45)

It is pretty obvious that $s=0$ and $d=\pi$ for $\sigma^{*}=0$, along with
$\beta=0$. Then calculating coefficients $A$ and $B$ by (3) gives: $A=0$ and
$B=0$. Thus, the amplitude of tank current is $I_{t}=0$ according to (13).
This leads to an important conclusion: in order to prevent collapse of tank
current amplitude, the refernce value of $\sigma^{*}$ should be non-zero:
$\sigma^{*}>0$.

####  3.5.2 Ideal synchronous rectification with $\delta=0$

Let’s consider second equation in (2). In order to satisfy the control goal
$\delta=0$ for synchronous rectification, the following should be true:

$\delta=\beta-\sigma=0$ (46)

which gives trivially

$\beta=\sigma$ (47)

or

$\tan\beta=\tan\sigma$ (48)

or expanding by using (3) and considering only $\tan\beta$ in the left hand
side:

$\frac{\sin\beta}{\cos\beta}=\frac{B}{A}=\frac{1-G\cos(\beta+s)-G\cos\beta-\cos
d}{\sin d+G\sin(\beta+s)+G\sin\beta}$ (49)

After some trivial simplification, the last equation can be simplified down to
the following, which we will call synchronous rectification condition:

$\cos\beta-G-\cos(\beta-d)-G\cos s=0$ (50)

####  3.5.3 Edge case of $\beta=0$

As discussed earlier, it is beneficial to bring primary and secondary voltage
waveforms close to each other in order to minimize commutation current. Such
approach is not always practical since the current might be not enough to
ensure ZVS commutation. However, this case is interesting in terms of
theoretical analysis of converter characteristics.

Substituting $\beta=0$ into (50) immediately gives:

$1-G-\cos d-G\cos s=0$ (51)

Since from the previous section, $\sigma=\beta$ then the equation (51)
describes a case when both input and output waveforms are brought together.

Figure 14: Voltage and current waveforms aligned with $\beta=0$ for buck
($G=0.7$) and boost ($G=1.3$) voltage ratios (under the same input voltage and
frequency)

It is interesting to study domain of arguments for (51). The equation can be
resolved with respect to $d$:

$d=\operatorname{acos}(1-G-G\cos s)$ (52)

and in the same time can be resolved with resprct to $s$:

$s=\operatorname{acos}\left(\frac{1-\cos d}{G}-1\right)$ (53)

Lets consider two cases:

  * •

$s=0$

From (52), the following is evident:

$d=\operatorname{acos}(1-2G)$ (54)

which is defined on interval $0\geq G\geq 1$.

If $G=1$, then $d=\pi$.

It is also easy to see that when $d=\pi$ or $1-G-G\cos s=-1$, then $\cos
s=2/G-1$ and $s$ increases with the increase of $G\geq 1$.

  * •

$d=\pi$

From (53), the following is evident:

$s=\operatorname{acos}\left(\frac{2}{G}-1\right)$ (55)

which is defined on interval $G\geq 1$.

The combined characteristics of $d(G)$ and $s(G)$ are shown in Figure 15

$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$2.2$$0$$1$$2$$3$$G$$d,\;s$,
rad$d(G)$$s(G)$ Figure 15: Maps of $d(G)$ and $s(G)$

It is possible to combine both feedforward control laws (54) and (55) into the
single one by introducing a new artificial variable $q$ piecewise parametrized
by $G$ and defined as follows:

$q(d,s)=\begin{cases}d,\;d<\pi\\\
s+\pi,\;d=\pi\end{cases}=\begin{cases}d,\;G\leq 1\\\ s+\pi,\;G>1\end{cases}$
(56)

with the following inverse mapping back to $d$ and $s$:

$d(q)=\begin{cases}q,\;G\leq 1\\\
\pi,\;G>1\end{cases},\;s(q)=\begin{cases}0,\;G\leq 1\\\
q-\pi,\;G>1\end{cases}$ (57)

$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$2.2$$0$$1$$2$$3$$4$$5$$G$$q$,
rad Figure 16: Map of $q(G)$

####  3.5.4 Primary side fully driven converter

In two subsections below a special case of the PWM modulation with fully
driven primary side will be discussed. The operating mode of primary input
bridge is the same as for the conventional LLC converter. However, with a
control of secondary side duty cycle $s$ it is possible to obtain a boost
operating mode and significantly increase voltage range of the converter.

Technically, driving the converter in this PWM type of modulation will give a
sligtly worse efficiency (because the primary bridge is being switched at
higher instantenous current), but this type if modulation might be desirable
for EMI purposes, because the switching voltage has fewer harmonics.

Everywhere in this section we will assume $d=\pi$.

##### Buck mode with $d=\pi$ and $s=0$.

Lets consider first a buck mode with $s=0$, which corresponds to a fully
driven converter on primary side and a synchronous rectification on the
secondary side. Obviously, buck mode is optimal in terms of $s\to\min$, since
0 is minimal possible value of $s$.

From (50) follows that since $\cos s=1$, then

$\beta=\sigma=\operatorname{acos}G$ (58)

It immediately follows that in order to have a real solution for $\beta$, the
voltage ratio should be $G\leq 1$, which coincides with the buck mode of
converter operation.

To satisfy control goal constraint $\sigma\geq\sigma_{min}$, additional
consideration should be imposed for voltage ratio $G$:

$\begin{gathered}\sigma\geq\sigma_{min}\\\
\operatorname{acos}G\geq\sigma_{min}\\\ G\leq\cos(\sigma_{min})\end{gathered}$
(59)

Thus, as long as $G\leq\cos(\sigma_{min})$, the buck mode control with $s=0$
can satisfy goal constraint $\sigma\geq\sigma_{min}$, and voltage ratio
$G=\cos(\sigma_{min})$ is a boundary between buck and boost modes of converter
operation.

##### Boost mode with $d=\pi$ and $s>0$, buck-boost transition.

The boost mode is activated when the buck mode with $s=0$ is no longer capable
to satisfy the control goal for $\sigma$, and as it was established in
previous section, for $G\geq\cos(\sigma_{min})$ a boost mode should be used.
In order to satisfy optimality condition $s\to\min$, the $\sigma$ should be
fixed in boost mode, and we have following control problem for angles:

$\begin{gathered}\delta=0\\\ \sigma=\sigma_{min}\end{gathered}$ (60)

or

$\sigma=\beta=\sigma_{min}$ (61)

From (50) the shorting time $s$ is determined by:

$s=\operatorname{acos}\left(2\cos(\sigma_{min})/G-1\right)$ (62)

Note for the particular voltage ratio $G=\cos(\sigma_{min})$, i.e. exactly at
the buck to boost transition border, the $s=0$ according to (62), which proofs
continuity of $s(G)$ function. The same is for $\beta$, since
$\beta=\operatorname{acos}G$ in buck mode, then at critical voltage ratio
$G=\cos(\sigma_{min})$ we are obtaining
$\beta=\operatorname{acos}\cos(\sigma_{min})=\sigma_{min}$, which coincides
with continuity of $\beta(G)$ function.

We can combine $\beta$ and $s$ maps for both buck and boost modes by stitching
(58), (61) and (62) using minimum and maximum functions (result is pictured in
Figure 17):

$\begin{gathered}\beta=\operatorname{acos}(\min\\{G,G^{*}\\})\\\
s=\operatorname{acos}\left(\frac{2G^{*}}{\max\\{G,G^{*}\\}}-1\right)\end{gathered}$
(63)

where

$G^{*}=\cos(\sigma_{min})$ (64)

$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$2.2$$0$$0.5$$1$$1.5$$G$$\beta,\;s$$\beta(G)$$s(G)$
Figure 17: Maps of $\beta(G)$ and $s(G)$ for particular $G^{*}=0.95$

####  3.5.5 Linearization around $\delta^{*}=0$ and $\sigma^{*}=0$

Even though the approach presented in this paper doesnt rely on any kind of
linearization (except harmonic truncation for the first harmonic
linearization), it is interesting to examine behaviour of the linearization
maps around steady-state point of $\delta^{*}=0$ and $\sigma^{*}=0$.

Lets assume $s_{add}=0$ for simplicity and lets linearize (25) and (32) around
$\delta^{*}=0$ and $\sigma^{*}=0$:

$\begin{gathered}d=d(\delta^{*},\sigma^{*})\approx d(0,0)+\frac{\partial
d}{\partial\delta}\biggr{\rvert}_{\delta=0}\cdot\delta^{*}+\frac{\partial
d}{\partial\sigma}\biggr{\rvert}_{\sigma=0}\cdot\sigma^{*}=\\\
=\operatorname{acos}(1-2G)-\frac{G\sin\delta}{\sqrt{G(1-G)}}\biggr{\rvert}_{\delta=0}\cdot\delta^{*}+\frac{\sin\sigma}{2\sqrt{G(1-G)}}\biggr{\rvert}_{\sigma=0}\cdot\sigma^{*}+\sigma^{*}\end{gathered}$
(65)

$\begin{gathered}s=s(\delta^{*},\sigma^{*})\approx
s(0,0)+\delta^{*}\frac{\partial
s}{\partial\delta}\biggr{\rvert}_{\delta=0}\cdot\delta^{*}+\sigma^{*}\frac{\partial
s}{\partial\sigma}\biggr{\rvert}_{\sigma=0}\cdot\sigma^{*}=\\\
=\pi-\operatorname{acos}(1-2/G)-\frac{G\sin\delta}{2\sqrt{G-1}}\biggr{\rvert}_{\delta=0}\cdot\delta^{*}+\frac{\sin\sigma}{\sqrt{G-1}}\biggr{\rvert}_{\sigma=0}\cdot\sigma^{*}-\delta^{*}\end{gathered}$
(66)

Thus, in trivial case, the decoupling transformation can be approximated as
follows:

$\begin{gathered}d=\begin{cases}\operatorname{acos}(1-2G)+\sigma^{*},\;G<1\\\
\pi,\;G>1\end{cases}\\\ s=\begin{cases}0,\;G<1\\\
\pi-\operatorname{acos}(1-2/G)-\delta^{*},\;G>1\end{cases}\\\
\beta=\sigma^{*}+\delta^{*}\end{gathered}$ (67)

This transformation is not defined at $G=1$, because according to section
3.5.1, in order to prevent the tank current from a collapse to zero the
following should be true $\sigma^{*}>0$. Also according to (65) and (66),
partial derivatives $\partial d/\partial\delta$, $\partial d/\partial\sigma$,
$\partial s/\partial\delta$, $\partial s/\partial\sigma$ are all singular near
$G=1$.

It worth noting that this (67) is only an approximation of exact inversion
(25) and (32). An external feedback controller has to be used in order to
achieve satisfactory results. This can be illustrated by simulation results
presented in Figures 18 and 19.

Figure 18: Simulated trajectories of $\sigma$ and $\delta$ without external
feedback controller, only using linear map (67) Figure 19: Simulated
trajectories of $\sigma$ and $\delta$ using individual PI controllers in
addition to feed-forward term (67)

The conclusions from the simulation results are: introduction of PI
controllers are enhancing regulation accuracy in comparision to purely feed-
forward control (67), also the control trajectories have a cusp in $G=1$,
which is expected since the control approximation is discontinuous there.

##  4 Output power control

###  4.1 Variable frequency control

According to (2) the output current control can be formulated in terms of
transconductance $W$ by simply multiplying it by input voltage $V_{in}$:
$I_{out}=WV_{in}$. Thus, the output current can be regulated by means of
transconductance regulation by satisfying the following control goal:

$W=W^{*}$ (68)

where $W^{*}=I_{out}^{*}/V_{in}$, and $I_{out}^{*}$ is referenced output
current.

Let’s assume that commutation parameters $d,s,\beta$ are at their steady-
states, defined by controls (25) and (32) or corresponding special cases,
discussed in the previous chapter. That means that coefficients $A$ and $B$
from (3) can be parametrized in terms of reference values $\sigma^{*}$ and
$\delta^{*}$, along with $s_{add}$ (if this feature is used).

If the nonlinear inversion condition (23) holds, then following trigonometric
manipulations from section 3.2, it is possible to conclude that
$B=A\tan\sigma^{*}$ and furthermore:

$\sqrt{A^{2}+B^{2}}=\frac{A}{\cos\sigma^{*}}$ (69)

The $A$ term is actually a function of reference values
$\sigma^{*},\delta^{*},s_{add}$ and $G$ if the values $d,s,\beta$ are
controlled to achieve the references:

$\begin{gathered}A=4\sin
d(\sigma^{*},\delta^{*},s_{add})+4G\sin(\sigma^{*}+\delta^{*}+s(\sigma^{*},\delta^{*},s_{add}))+4G\sin(\sigma^{*}+\delta^{*})=A(\sigma^{*},\delta^{*},s_{add},G)\end{gathered}$
(70)

Based on this, the transconductance (2) can be separated into two parts $H$
and $Z$: one is parametrized by reference commutation parameters
$\sigma^{*},\delta^{*},s_{add}$ and another one by switching frequency
$\omega$.

$W=\frac{I_{out}}{V_{in}}=\frac{n}{2\pi^{2}}\frac{H(\sigma^{*},\delta^{*},s_{add},G)}{Z(\omega)}\\\
$ (71)

where

$H=A(\sigma^{*},\delta^{*},s_{add},G)\cdot\frac{\cos(s(\sigma^{*},\delta^{*},s_{add})+\delta^{*})+\cos\delta^{*}}{\cos\sigma^{*}}$
(72)

Then the frequency control consists of calculation of $\omega$ which satisfies
(71) for a given referenced $W^{*}$. The (71) can be solved with respect to
$Z(\omega)$:

$Z=\frac{n}{2\pi^{2}}\frac{H(\sigma^{*},\delta^{*},s_{add},G)}{W^{*}}$ (73)

And in the same time $Z(\omega)=\omega L-1/(\omega C)$, from here the required
$\omega$ can be easilty obtained:

$\omega=\frac{1}{2LC}\left(\sqrt{C^{2}Z^{2}+4LC}+CZ\right)$ (74)

###  4.2 Special case for fully driven converter

The output transconductance with feed-forward maps $\cos\beta=G$ and
$\delta=0$ can be calculated as after all simplifications using (58) and
$s=0$:

$W=\frac{8n}{\pi^{2}}\frac{1}{Z(\omega)}\sqrt{1-G^{2}}$ (75)

Let’s calculate frequency $\omega$ which is needed to achieve desired output
$W^{*}$:

$\begin{gathered}Z=\frac{8n\sqrt{1-G^{2}}}{\pi^{2}W^{*}}\\\
\omega=\frac{1}{2LC}\left(\sqrt{C^{2}Z^{2}+4LC}+CZ\right)\end{gathered}$ (76)

The combined feed-forward control law (both for buck and boost modes) can be
formulated as follows:

For given $W^{*}$ and $G$, and also pre-calculated buck to boost threshold
$G^{*}=\cos(\sigma_{min})$:

1\. First calculate the control inputs $\beta$ and $s$

$\begin{gathered}\beta=\operatorname{acos}(\min\\{G,G^{*}\\})\\\
s=\operatorname{acos}\left(\frac{2G^{*}}{\max\\{G,G^{*}\\}}-1\right)\end{gathered}$
(77)

2\. Then calculate desired tank impedance:

$Z=4n\frac{\cos s+1}{\pi^{2}W^{*}}\sqrt{1-G\cos(\beta+s)}$ (78)

3\. The switching frequency will be determined by equation:

$\omega=\frac{1}{2LC}\left(\sqrt{C^{2}Z^{2}+4LC}+CZ\right)$ (79)

###  4.3 Low power mode with $\omega=\omega_{max}$

The switching frequency in control (74) is unbounded, i.e. in order to get
$W^{*}=0$ frequency should be $\omega\to\infty$. In real systems, the upper
frequency is always constrained at $\omega_{max}$. Thus if we need output
power even lesser than we can get at $\omega=\omega_{max}$, impedance can’t be
further increased and control of output power can be achieved only by
commutation parameters $s$ and $\beta$ with fixed $\omega=\omega_{max}$. This
mode we will call a low power mode.

Let’s define maximum impedance that we can get:

$Z_{max}=\omega_{max}L-\frac{1}{\omega_{max}C}$ (80)

The control law for low power can be formulated as follows: for a given
$W^{*}$ find commutation parameters $d,s,\beta$ which are satisfying following
conditions:

$W^{*}=\frac{n}{2\pi^{2}}\frac{H(\sigma^{*},\delta^{*},s_{add},G)}{Z_{max}}$
(81)

Let’s consider an idea to limit output power by additional increase of $s$,
which is equivalent to dimming the output down to zero power at $s=\pi$ when
secondary side is completely shorted. Just to mention that an another
possibility for output power regulation at constant frequency is to shift the
secondary side respect to primary side by introducing $\delta>0$.

For implementation of this idea the $s_{add}$ quantity was introduced in
section 3.2.3 which resulted in inversion maps (25) and (32). This mode of
operation can be described as a partial rectification, because only a part of
resonant tank current is being rectified by corresponding seconday side
commutation, another part is being recycled and adds to reactive power.

The examples of waveforms with and without of addition of $s_{add}$ are given
in Figure 20. The operating point under consideration is defined by $G=0.7$ (a
buck mode), $\omega=3\omega_{0}/2$, and $\delta^{*}=0$, $\sigma^{*}=0.1$. The
output current calculated by (2) was 2 times lower when introducing
$s_{add}=3/2$ rad.

Figure 20: Waveforms without (left) and with (right) an additional $s_{add}$
introduction for low power mode at $G=0.7$

To characterize dependency of $W$ from $s$ with a fixed frequency, lets try to
calculate $W$ by (71) (or model equation (2)) for different values of $G$ and
$s_{add}$. Let’s denote $W_{0}$ the output power obtained for fixed maximal
frequency $\omega_{max}$ without any additional secondary side shorting for
low power, i.e. $s_{add}=0$.

$W_{0}=\frac{n}{2\pi^{2}}\frac{H(\sigma^{*},\delta^{*},0,G)}{Z_{max}}$ (82)

The output power variation for different values of G is demonstrated in Figure
21 as ratio of $W/W_{0}$. As one can see from the picture, a zero power
operation $W=0$ is achieved for $s=\pi$ (fully shorted secondary side)
regardless of G. But an interesting property of the system is that $W$ doesn’t
decrease monotonically with increasing of $s_{add}$. That means that a
discontinuous control should be applied for $s_{add}$ when transitioning from
regular operation with $s_{add}=0$ to a low power operation with $s_{add}>0$
and $W<W_{0}$.

Figure 21: Output power variation with $s_{add}$ for different $G$ and
$\delta^{*}=0$, $\sigma^{*}=0.1$

When the low power control is requested, minimal $s_{add}^{0}>0$ needs to be
applied which satisfies the condition $W(s_{add}^{0})=W(0)$ or:

$H(\sigma^{*},\delta^{*},s_{add}^{0},G)=H(\sigma^{*},\delta^{*},0,G)$ (83)

This $s_{add}^{0}$ is a value of $s_{add}$ after which the output power will
decrease monotonically. The figure 22 illustrates the boundary of valid
$s_{add}$ values, i.e. the solution of (83).

Figure 22: Initial values of $s_{add}^{0}$ for monotonic decrease of $W$, the
solution of (83)

The full map of normilized $W$ to $W_{0}$ is given in the Figure 23. The white
space are areas of invalid $s_{add}$ values, where either $W>W_{0}$ or $W<0$.

Figure 23: Output power variation with $s_{add}$ for different $G$ and
$\delta^{*}=0$, $\sigma^{*}=0.1$

The inverse mapping algorithm of $W^{-1}$ for a given $\sigma^{*}$,
$\delta^{*}$ and $W^{*}$ can be formulated as follows:

1\. Calculate $\omega$ using (73) and (74) for $s_{add}=0$ and determined
commutation parameters $d,s,\beta$ obtained by (44) and (43).

2\. If $\omega>\omega_{max}$ by (74), then use $\omega=\omega_{max}$ and
iterate $s_{add}$ from $0$ to $\pi$ by recalculating the commutation
parameters $d,s,\beta$ by (44) and (43) in order to approximately solve (81)
with respect to $s_{add}$.

##  5 Controller architecture

###  5.1 Inner loop: commutation parameters control

####  5.1.1 Series nonlinear compensation

In order to overcome uncertainties of the model (3), additional external
(linear) feedback controllers should be used in conjunction to decoupling and
linearization as a solution of (16).

One approach would be to use a series nonlinear compensation, which consisting
of regulating inputs $\sigma^{*}$ and $\delta^{*}$ of the nonlinear map
$F^{-1}$ as it presented in Figure 24. These two controllers could have PID
form and regulate $\sigma$ and $\delta$ independently. The direct feedforward
links added from the $\sigma$ and $\delta$ setpoints to the corresponding
inputs of $F^{-1}$ map. Under an ideal model matching condition, the outputs
of the PID regulator keep at zero.

$\sigma$$\delta$$F$$F^{-1}$$\sigma^{*}$$\delta^{*}$$G$$G$$d$$s$$\beta$(3)$s_{add}$$+$$+$$PID$$PID$$-$$-$(43),(44)$\sigma_{ref}$$\delta_{ref}$$+$$+$$+$$+$
Figure 24: Architecture of control system for $\sigma$ and $\delta$ with
series nonlinear compensation.

####  5.1.2 Parallel nonlinear compensation

It is well known [30], that the performance of series nonlinear compensation
often susceptible to the plant uncertainties. In order to overcome this issue,
a parallel nonlinear compensation can be used. In this case the control action
from feedback controllers is added to the outputs the feedback map $F^{-1}$.

There are 3 outputs $d,s,\beta$ of the inversion map $F^{-1}$ in the
particular system under the consideration, but two controllers for $\sigma$
and $\delta$. In order to overcome this problem lets combine both $d$ and $s$
in a single variable $q$ by (56), using observation that if $s>0$ then $d=\pi$
and if $s=0$, then $d<\pi$ (assuming no low power mode, i.e. $s_{add}=0$). In
order to embed $s_{add}$ into the definition of $q$, the following
modification of (56) can be proposed:

$\begin{pmatrix}d\\\ s\end{pmatrix}=\begin{cases}\begin{pmatrix}q\\\
s_{add}\end{pmatrix}\;,\;q\leq\pi\\\ \begin{pmatrix}\pi\\\
q-\pi\end{pmatrix}\;,\;q>\pi\end{cases}$ (84)

Note that for $q>\pi$ in (84), the $s_{add}$ is added to $s$, so the equation
consists $s_{add}$ only in the buck case $q<\pi$.

Let’s consider two cases:

1\. The buck mode $q<\pi$: $d=q$, $s=s_{add}$.

By substituting $d:=q$ into (23), obtaining for $q\in[0,\pi]$:

$q=\pm\operatorname{acos}(\cos\sigma^{*}-G\cos\delta^{*}-G\cos(\delta^{*}+s_{add}))+2\pi
n+\sigma^{*}$ (85)

where $n$ is an integer number.

2\. The boost mode $q>\pi$: $d=\pi$, $s=q-\pi=s_{min}+s_{add}$.

Substituting $s:=s_{min}=q-\pi-s_{add}$ to (23) and using the fact that
$\cos(x-\pi)=\cos(\pi-x)=-\cos x$, then obtaining for $q\in[\pi,2\pi]$:

$q=\pm\operatorname{acos}(\cos\delta^{*}-2/G\cos\sigma^{*})+2\pi
n-\delta^{*}+s_{add}$ (86)

where $n$ is an integer number.

Following this approach, the nonlinear inversion map becomes square, it has
two inputs $\sigma^{*}$, $\delta^{*}$ and two outputs $q$, $\beta$. One can
observe that the $\sigma^{*}$ is a linear term of the $q$ for buck mode
($q<\pi$). So a feedback term which regulates $\sigma$ using additional linear
controller can be added to $q$. A feedback term that regulates $\delta$ can be
added to $\beta$ because it is linear for both $\delta$ and $\sigma$:
$\beta=\delta+\sigma$.

The remaining part is $q$ in the boost mode (86) which has $\delta^{*}$ as a
linear term. This can be changed substituting $\delta^{*}=\beta-\sigma^{*}$.

$q=\pm\operatorname{acos}(\cos\delta^{*}-2/G\cos\sigma^{*})+2\pi
n+s_{add}-\beta+\sigma^{*}$ (87)

The resulted control architecture for $\sigma$ and $\delta$ is given in Figure
25.

$\sigma$$\delta$$F$$\sigma^{*}$$\delta^{*}$$G$$G$$d$$s$$\beta$(3)$s_{add}$$+$$+$$PID$$PID$$-$$-$(84)$\sigma_{reg}$$\delta_{reg}$$d(q),s(q)$$+$$+$(85),(87)$+$$q$$\beta$$F_{q}^{-1}$$s_{add}$$+$
Figure 25: Architecture of control system for $\sigma$ and $\delta$ with
parallel nonlinear compensation.

###  5.2 Outer loop: output power control

####  5.2.1 Series nonlinear compensation

The series nonlinear compensation essentially extends the architecture
presented in section 5.1.1. The mappings $W^{-1}$ and $F^{-1}$ are merged
together since the calculation of frequency $\omega$ in accordance with the
algorithm from the end of section 4.3 requires actually calculated $d,s,\beta$
and iterating of $s_{add}$ which is an input of the mapping $F^{-1}$.

In a practical sense, the combined $W^{-1}$ and $F^{-1}$ map is a 3D look-up
table.

$\sigma$$\delta$$F$$F^{-1}$$\sigma^{*}$$\delta^{*}$$G$$G$$d$$s$$\beta$(3)$s_{add}$$+$$+$$PID$$PID$$-$$-$(43),(44)$\sigma_{ref}$$\delta_{ref}$$PID$$W^{-1}$(73),(74),(81)$W^{*}$$\omega$$W$$+$$-$$W_{ref}$$+$$+$$+$
Figure 26: Controller architecture with commutation parameters control
$\sigma$ and $\delta$, and output power control $W$.

####  5.2.2 Overall architecture

The final control architecture consists of parallel nonlinear compensation for
commutation parameters $\sigma$ and $\delta$ as described in 5.1.2 with series
nonlinear compensation of $\omega$ (and $s_{add}$) for regulation of $W$. As
it evident from (73),(74), the inverse map of $W$ is smooth and monotonic, so
no need to derive parallel compensation scheme. This hybrid approach allows
obtaining reasonably robust control, as it demonstrated in the next section
with simulation results.

$\sigma$$\delta$$F$$\sigma^{*}$$\delta^{*}$$G$$G$$d$$s$$\beta$(3)$+$$+$$PID$$PID$$-$$-$(84)$\delta_{reg}$$d(q),s(q)$$+$$+$(85),(87)$+$$q$$\beta$$F_{q}^{-1}$$s_{add}$$+$$+$$PID$$-$$\sigma_{reg}$$W^{-1}$$W$$W$(73),(74)$\omega$(81)$+$$W^{*}$Algorithm
1 Figure 27: Controller architecture with commutation parameters control
$\sigma$ and $\delta$, and output power control $W$.

The implementable algorithm of nonlinear inversion $F_{q}^{-1}$ and $W^{-1}$
can be formalized by the algorithm 1.

Algorithm 1 Calculation of inversion maps $F_{q}^{-1}$ and $W^{-1}$.

for all $s_{add}\in[0,\pi]$ do

$x:=\cos\sigma^{*}-G\cos\delta^{*}-G\cos(\delta^{*}+s_{add})$ $\triangleright$
trying to evaluate (85)

if $|x|\leq 1$ then $\triangleright$ if this is a buck mode?

$q:=\operatorname{acos}x+\sigma^{*}$ $\triangleright$ calculate $q$ for buck
mode

else$\triangleright$ else it is a boost mode

$q:=2\pi-\operatorname{acos}(\cos\delta^{*}-2/G\cos\sigma^{*})-\delta^{*}+s_{add}$
$\triangleright$ calculate $q$ for boost mode by (87)

end if

$\beta:=\sigma^{*}+\delta^{*}$ $\triangleright$ calculate $\beta$ by (5)

$\beta:=\beta+\delta_{reg}$ $\triangleright$ add external PID controller
action for $\delta$

$q:=q+\sigma_{reg}$ $\triangleright$ add external PID controller action for
$\sigma$

if $q\leq\pi$ then $\triangleright$ determining $d,s$ depending if this is
buck or boost

$d:=q$ $\triangleright$ it is a buck mode, so $d\leq\pi$

$s:=s_{add}$ $\triangleright$ in case if $s_{add}$ requested for low power

else

$d:=\pi$ $\triangleright$ it is a boost mode, so $d=\pi$

$s:=q-\pi$ $\triangleright$ $s_{add}$ is contained in $s$

end if

$A:=4\sin d+4G\sin(\beta+s)+4G\sin\beta$ $\triangleright$ by (3)

$H:=A\cdot\dfrac{\cos(s+\delta^{*})+\cos\delta^{*}}{\cos\sigma^{*}}$
$\triangleright$ by (72)

$Z:=\dfrac{n}{2\pi^{2}}\dfrac{H}{W^{*}}$ $\triangleright$ by (73)

$\omega:=\dfrac{1}{2LC}\left(\sqrt{C^{2}Z^{2}+4LC}+CZ\right)$ $\triangleright$
by (74)

if $\omega\leq\omega_{\max}$ then return $\triangleright$ switching parameters
are calculated

else

$s_{add}:=s_{add}+\Delta s_{add}$ $\triangleright$ increase $s_{add}$ by a
small step $\Delta s_{add}$

end if

end for

##  6 Simulation

A MATLAB Simulink model was developed in order to prototype the control
strategies proposed. A battery charger application is used as case study
example.

A simple model of 30Ah pack is used as a load of the resonant converter. The
battery model consists of an integrator for coulomb counting and a map which
maps Ah to battery voltage.

The charger is a power supply that consists of an outer voltage loop which
regulates the current and inner current loop with input limitation. The model
is presented in Figure 28. The submodel block Current source is a model of
closed loop resonant LC DC/DC converter that implements the control approach
developed in this paper.

Figure 28: Top view of EV charger Simulink model

The internals of Current source block is shown in Figure 29

Figure 29: The Simulink model for current source, the LC resonant DC/DC
converter

The block Inverse implements the algorithm 1. The block DC/DC models steady-
state model (3) of LC converter and reused from [23]. The sensor dynamics for
measurements of $W$, $\sigma$ and $\delta$ are modeled by first order transfer
functions.

The parameters of the resonant tank were given as follows: $C=47$ nF, $L=80$
uH, which gives the resonant frequency of about 82 kHz. The maximum frequency
is set to 165 kHz.

The transformer turns ration is chosen so that the point of $G=1$ is being
passed roughly in the middle of the charge. The charging profile consists of
constant current (CC) mode with 25 A setpoint followed by a constant voltage
(CV) mode with 400 V setpoint. The voltage varies during the charge from 240 V
to 400 V. The input voltage of the converter is fixed at 600 V.

In order to demonstrate robustness of the proposed control approach two model
uncertainties were introduced:

1\. Constant offset is applied to the $\beta$ switching parameter at the
plant’s (converter’s) side: $\beta:=\beta-0.1$.

2\. The series inductance of the resonant tank is increased by 5%.

The results of simulation presented in Figures 30, 31, 32 and 33. The time
axis has arbitrary scale and the dynamics is subject to PI tuning.

From the simulation it is evident that the controllable outputs are following
their reference values while the converter passes through lower power buck
mode, regular buck mode, then boost mode and finally ends up with low power
boost mode. The plant uncertainty introduced are correctable by external PI
controllers in accordance to the architecture proposed in the section 5.2.2.

Figure 30: Output and referenced current Figure 31: Variation of the voltage
ratio $G$ during the charge Figure 32: The control outputs for commutation PWM
parameters Figure 33: Controllable outputs of waveform alignment quantities
and their reference values

##  7 Conclusion

In this paper a control approach that consists of direct nonlinear inversion
of the steady-state model equations (3) under various operating conditions.

Namely, the following main results are obtained:

1\. The optimality of operating mode is characterized using angular quantities
(9) that measure alignment of the primary and secondary switching event with
respect to the tank current. The main novelty of the approach presented, that
these phase displacements are being measured and controlled using feedback
controllers, which maintains the efficiency of the converter in a wide
operating range and in presence of uncertainties.

2\. The nonlinear control problem is formulated (16) and solved (44), (43) to
actively control the waveform alignment quantities. As a result, a ZVS
commutation can be maintained across a wide range of operating conditions.

3\. An indirect synchronous rectification approach is a byproduct of the
active control of the alignment of the secondary side voltage waveform with
the resonant tank current. This approach increases the converter efficiency by
a compensation of gate and sensing circuit delays, and is more robust to a
noise of the zero current detection circuit.

4\. The nonlinear inversion is square, i.e. has the same number of inputs and
outputs. It is achieved by combining duty cycles of primary and secondary
sides into a single variable (84).

5\. The output current regulation problem (71) is formulated in terms of
transconductance, which normalizes the converter operation by the input
voltage.

6\. For the low power operation, an additional duty cycle modulation of the
secondary side under constant frequency is proposed (81) and corresponding
control strategy is derived.

7\. A parallel nonlinear compensation which consists of adding control actions
of external linear regulators to the manipulated plant inputs is developed in
order to achieve robustness of the control implementation.

The future work will be devoted to a cycle-to-cycle dynamics of the converter
and corresponding nonlinear control.

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