11institutetext: ALBA Synchrotron, Barcelona, Spain

# One-Quadrant Switched-Mode Power Converters

R. Petrocelli

###### Abstract

This article presents the main topics related to one-quadrant power
converters. The basic topologies are analysed and a simple methodology to
obtain the steady-state output–input voltage ratio is set out. A short
discussion of different methods to control one-quadrant power converters is
presented. Some of the reported derived topologies of one-quadrant power
converters are also considered. Some topics related to one-quadrant power
converters such as synchronous rectification, hard and soft commutation, and
interleaved converters are discussed. Finally, a brief introduction to
resonant converters is given.

Keywords
One-quadrant; converter; topology; switch-mode; magnet; energy.

##  1 Introduction

Switched-mode power converters are a very efficient way to transfer energy
from a source to a load. A switched-mode power converter is formed from
switches, inductors, and capacitors. If these components are ideal, they do
not dissipate energy, and the efficiency of the power converter is 100%. A
further advantage of switched-mode power converters is the increase in power
density that has occurred in recent years. The power density is defined as the
ratio between the nominal power of the converter and its volume. This power
density has been constantly increasing owing to an increasing switching
frequency.

As this is a broad topic, there are several different ways to classify
switched-mode power converters. The one-quadrant power converters are those
with the capability to provide an output in only one quadrant of a plot of
output voltage versus output current. These converters have the capability to
transfer energy in only one direction. Most one-quadrant converters can also
be classified as DC–DC converters. The main application of these converters is
in voltage regulation. In this application, a non-regulated voltage, which is
usually provided by a diode rectifier stage, is converted to a regulated
output voltage which is not affected by the grid or variations in the load.

This article gives an introduction to one-quadrant power converters. It
presents the basic, or direct, DC–DC converter topologies, showing typical
waveforms and the basic design equations. The basic concepts of regulation and
control of DC–DC converters are also discussed. Section 4 presents some of the
most typical derived converter topologies reported in the literature. Some
topics related to one-quadrant power converters, such as synchronous
rectification, interleaving, and soft commutation, are presented in Section 5.
Finally, resonant converters are briefly mentioned.

###  1.1 One-quadrant switched-mode power converters in particle accelerators

The main application of one-quadrant power converters is in storage ring
particle accelerators. These accelerators are the key components of
synchrotron radiation light source facilities. A storage ring is a particle
accelerator where the particles circulate around a ring for several hours
while their energy is maintained constant. As the energy of the particles is
constant, the magnetic fields used for bending and focusing the particle beam
are also constant. These magnetic fields are generated by currents provided by
high-precision, stable power supplies. The precision and stability of these
converters are of the order of several parts per million.

##  2 Basic one-quadrant power converter topologies

This section provides a short introduction to the basic topologies of one-
quadrant power converters. These basic topologies are also known as direct
DC–DC converters. One characteristic of these topologies is that they are
built using only one commutation cell. A brief description, a methodology for
the analysis of these topologies, and key design equations are presented.
First, the DC chopper converter is discussed as an introduction and to show
how the average output voltage of a converter can be modified using ‘on–off’
switches. The basic topologies considered are the buck, boost, and buck–boost
converters.

###  2.1 DC chopper

The simplest one-quadrant converter is the DC chopper, shown in fig:chopper01.
This consists of a DC input voltage source, a controllable switch, and a load
resistor. When the switch is closed, the input voltage is applied to the
resistor. This is defined as the ‘on’ state of the converter. When the switch
is open, the current through the resistor is zero and the voltage is zero.
This is called the ‘off’ state of the converter.

Figure 1: Simple DC chopper: (a) circuit diagram; (b) output voltage waveform

The switch can be operated with a duty ratio $D$, defined as the ratio of the
on time of the switch to the sum of the on and off times. For constant-
frequency operation,

$D=\frac{t_{\mathrm{on}}}{t_{\mathrm{on}}+t_{\mathrm{off}}}\frac{t_{\mathrm{on}}}{T},$
(1)

where $T=1/f$ is the period of the switching frequency $f$. The average value
of the output voltage is

$V_{\mathrm{O}}=DV_{\mathrm{S}}.$ (2)

This average output voltage can be regulated by adjusting the duty cycle $D$.

The circuit shown in fig:chopper01 is not a ‘practical’ circuit. The main
reason for this is that it is not possible to build pure resistive elements.
In practice, the connection of devices and circuit elements creates stray
inductances that have to be taken into account. A circuit including the stray
inductance of the load is shown in fig:chopper02.

Figure 2: DC chopper with inductive load

The inclusion of the inductance of the load leads to a need to provide a path
for the current in the inductor. This is to comply with the ‘rule’ of power
electronics that a current source that is like an inductor must not be open-
circuited. Two circuits are defined according to the state of the switch;
these circuits are shown in fig:chopper03. The current paths are shown in red.

Figure 3: DC chopper: (a) current path in on state; (b) current path in off
state

The load voltage and current are plotted in fig:chopper04 under the
assumptions that the load current never reaches zero and the time constant of
the load $\tau={L_{\mathrm{L}}}/{R_{\mathrm{L}}}$ is much greater than the
period $T$. The average values of the output voltage and current can be
adjusted by changing the duty ratio $D$. This is the principle of control of
switched-mode power converters.

Figure 4: DC chopper: output voltage and load current

###  2.2 Buck converter

The addition of a capacitor to the circuit of fig:chopper02 leads to the first
topology of basic, or direct, DC–DC converters. These are step-down converters
and are commonly known as buck converters. The basic circuit diagram of a buck
converter is shown in fig:buck01. It consists of a DC input voltage source
$V_{\mathrm{S}}$, a controlled switch $S$, a diode $D$, a filter inductor $L$,
a filter capacitor $C$, and a load resistance $R$.

Figure 5: Buck (step-down) converter

DC–DC converters can operate in two distinct modes with respect to the
inductor current $i_{L}$. Figure 6 depicts the continuous conduction mode
(CCM), in which the inductor current is always greater than zero. When the
average value of the input current is low (high $R$) and/or the switching
frequency $f$ is low, the converter may enter the discontinuous conduction
mode (DCM). In the DCM, the inductor current is zero during a portion of the
switching period.

Typical waveforms of the converter in the CCM are shown in fig:buckWaveforms.
It can be seen from the circuit that when the switch $S$ is commanded to the
on state, the diode $D$ is reverse biased. When the switch $S$ is off, the
diode conducts so as to support an uninterrupted current in the inductor.

Figure 6: Waveforms for buck converter

The relationships among the input voltage, the output voltage, and the switch
duty ratio $D$ can be derived from, for instance, the waveform of the inductor
voltage $V_{L}$ (fig:buckWaveforms). According to Faraday’s law, the volt-
second product for the inductor over a period of steady-state operation should
be zero. For the buck converter,

$(V_{\mathrm{S}}-V_{\mathrm{O}})DT=V_{\mathrm{O}}(1-D)T.$ (3)

From Eq. (3), the steady-state DC voltage transfer function, defined as the
ratio of the output voltage to the input voltage, is

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=D.$ (4)

It can be seen from Eq. (4) that the output voltage is always smaller than the
input voltage.

The CCM is preferred over the DCM because it provides higher efficiency and
makes better use of semiconductor switches and passive components. The DCM may
be used in applications with special control requirements, since the dynamic
order of the converter is reduced (the energy stored in the inductor is zero
at the beginning and end of each switching period). It is uncommon to mix
these two operating modes, because different control algorithms are needed.
For the buck converter, the critical value of the inductance which is needed
for CCM operation is

$L_{\mathrm{b}}=\frac{(1-D)R}{2f}.$ (5)

The current $i_{L}$ in the filter inductor in the CCM consists of a DC
component $I_{\mathrm{O}}$ with a superimposed triangular AC component. Almost
all of this AC component flows through the filter capacitor as a current
$i_{C}$. The current $i_{C}$ causes a small voltage ripple in the DC output
voltage $V_{\mathrm{O}}$. To limit the peak-to-peak value of the ripple
voltage below a certain value $V_{\mathrm{r}}$, the filter capacitance $C$
must be greater than

$C_{\mathrm{MIN}}=\frac{(1-D)V_{\mathrm{O}}}{8V_{\mathrm{r}}Lf^{2}}.$ (6)

Equations (5) and (6) are the key design equations for the buck converter. The
input and output DC voltages (and, hence, the duty ratio $D$) and the range of
the load resistance $R$ are usually determined by preliminary specifications.
The designer needs to determine the values $L$ and $C$ of the passive
components, and the switching frequency $f$. The value $L$ of the filter
inductor is calculated from the CCM/DCM condition using Eq. (5).

The value $C$ of the filter capacitor is obtained from the voltage ripple
condition (Eq. (6)). For compactness and low conduction losses in the
converter, it is desirable to use small passive components. Equations (5) and
(6) show that this can be accomplished by using a high switching frequency
$f$. The switching frequency is limited, however, by the type of semiconductor
switches used and by switching losses. It should also be noted that values of
$L$ and $C$ may be altered by the effects of parasitic components in the
converter, especially the equivalent series resistance of the capacitor.

####  2.2.1 Discontinuous current mode

Whether the converter operates in the CCM or the DCM depends on the load on
the converter. If the value of the output resistance increases beyond the
value used in Eq. (5), the converter starts to operate in the DCM. Figure 7(a)
shows the waveforms of the inductor voltage $v_{L}$ and inductor current
$i_{L}$ at the edge of the CCM. The value of the output current $i_{R}$ is
also shown in the plot. This value is the average of the inductor current over
a period $T$.

Figure 7: CCM and DCM: (a) boundary condition waveforms; (b) DCM waveforms

The output current $i_{R}$ for the condition at the boundary between the two
modes is given by

$i_{R\mathrm{B}}=\frac{DT}{2L}(V_{\mathrm{S}}-V_{\mathrm{O}})=\frac{TV_{\mathrm{S}}}{2L}(D-D^{2}).$
(7)

The output current required in the CCM is maximum for

$I_{R\mathrm{B_{MAX}}}=\frac{TV_{\mathrm{S}}}{8L}.$ (8)

Figure 8 shows a plot of the minimum output current for CCM operation as a
function of the duty cycle $D$.

Figure 8: Minimum output current for CCM versus duty cycle keeping
$V_{\mathrm{S}}$ constant

The output–input voltage ratio under DCM conditions can be calculated using
the waveforms in fig:buck_DCM01(b). The volt-second product over a period
should be zero for a steady-state condition. Therefore

$(V_{\mathrm{S}}-V_{\mathrm{O}})DT=V_{\mathrm{O}}\Delta T,$ (9)
$\frac{V_{\mathrm{O}}}{V_{\mathrm{O}}}=\frac{D}{D\Delta}.$ (10)

The output current is the average of the inductor current $I_{L}$ over the
period $T$:

$I_{R}=\frac{I_{L_{\mathrm{peak}}}(D+\Delta)T}{2T}.$ (11)

Using Eqs. (8) and (11), Eq. (10) can be rewritten as

$I_{R}=4I_{R\mathrm{B_{MAX}}}D\Delta.$ (12)

From Eqs. (10) and (13),

$\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=\frac{D^{2}}{\left(D^{2}+\frac{1}{4}\frac{I_{R}}{I_{R\mathrm{B_{MAX}}}}\right)}.$
(13)

Figure 9 shows the characteristics of the buck converter for both modes of
operation. The voltage ratio is plotted versus the output current for several
duty cycles. The boundary between the DCM and CCM is shown by the dashed
curve.

Figure 9: Buck converter: output–input voltage ratio in DCM and CCM

###  2.3 Boost converter

Figure 10: Boost converter: circuit diagram

The second topology to be analysed is the step-up, or boost, converter. It
receives this name because the output voltage is always higher than the input
voltage. Figure 10 depicts a simple circuit diagram of a boost converter. This
comprises a DC input voltage source $V_{\mathrm{S}}$, a boost inductor $L$, a
controlled switch $S$, a diode $D$, a filter capacitor $C$, and a load
resistance $R$. The converter waveforms in the CCM are presented in
fig:boost02. When the switch $S$ is in the ‘on’ state, the current in the
boost inductor increases linearly. The diode $D$ is off at this time. When the
switch $S$ is turned off, the energy stored in the inductor is released
through the diode into the output RC circuit.

Figure 11: Waveforms for boost converter

Using Faraday’s law for the boost inductor,

$V_{\mathrm{S}}DT=(V_{\mathrm{O}}-V_{\mathrm{S}})(1-D)T,$ (14)

from which the DC voltage transfer function turns out to be

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=\frac{1}{(1-D)}.$
(15)

The boost converter operates in the CCM for $L>L_{\mathrm{b}}$, where

$L_{\mathrm{b}}=\frac{(1-D)^{2}DR}{2f}.$ (16)

The main waveforms of the boost converter are shown in fig:boost02. The
current supplied to the RC output filter is discontinuous. Thus, a larger
output filter capacitor is required than in the buck converter in order to
limit the output voltage ripple. The output filter capacitor must provide the
DC current to the load when the diode $D$ is off. The minimum value of the
filter capacitance that results in a voltage ripple $V_{\mathrm{r}}$ is given
by

$C_{\mathrm{min}}=\frac{DV_{\mathrm{O}}}{V_{\mathrm{r}}Rf}.$ (17)

###  2.4 Buck–boost converter

The last of the basic converter topologies is the buck–boost converter. A
circuit diagram is shown in fig:buckboost01. This converter consists of a DC
input voltage source $V_{\mathrm{S}}$, a controlled switch $S$, an inductor
$L$, a diode $D$, a filter capacitor $C$, and a load resistance $R$. While the
switch $S$ is ‘on’, the inductor current increases and the diode is maintained
‘off’. When the switch is turned off, the diode provides a path for the
inductor current. Note the polarity of the diode, which results in its current
being drawn from the output. For this reason, the output voltage polarity is
negative.

Figure 12: Buck–boost converter: circuit diagram Figure 13: Waveforms for
buck–boost converter

The waveforms for the buck–boost converter are depicted in fig:buckboost02.
The condition for a zero volt-second product for the inductor in the steady
state and the steady-state DC transfer function are given by the following
equations:

$V_{\mathrm{S}}TD=-V_{\mathrm{O}}(1-D)T,$ (18)
$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=\frac{D}{(1-D)}.$
(19)

The critical value of the inductor that fixes the boundary between the CCM and
DCM is given by

$L_{\mathrm{b}}=\frac{(1-D)^{2}DR}{2f}.$ (20)

The current that feeds the RC output filter is the same as that for the boost
converter but in the reverse direction. Hence the value of the capacitor is
given by Eq. (17).

##  3 Control of DC–DC converters

One of the common features of the power converters presented in the previous
section is that the output voltage depends on the duty cycle $D$, which was
defined as the ratio between $t_{\mathrm{on}}$ and the total period of
switching $T$. Changes in the duty cycle produce variations in the output
voltage of the converter. In other words, the output voltage can be controlled
by changing the duty cycle.

The analysis of a converter from the point of view of control can be performed
using the switching function. The switching function is a mathematical tool to
represent the state of a power converter. For the basic power converters
described above, the switching function has two values. When the switch is on,
the value of the switching function is unity, and when the switch is off, the
value is zero:

$s(t)=\left\\{\begin{tabular}[]{cr}1&when the switch is on,\\\ 0&when the
switch is off.\\\ \end{tabular}\right.$ (21)

A block diagram of the control of a power converter is shown in
fig:modulation01. The block labelled “Modulator” transforms an input signal
$v_{D}$ into the switching function $s(t)$, which controls the switch in the
power converter. The modulator block can be built in several different ways.
It can be implemented using a few analog components or using sophisticated
digital circuits.

Figure 14: Block diagram of power converter

One of the most common applications of DC–DC converters is to provide a
regulated output regardless of variations in the load and the input voltage.
Another source of variation in the output voltage is changes in the components
of the power converter; components can be affected by external factors such as
temperature or ageing. The most common schemes used to regulate the output of
power converters are presented in fig:modulation02.

The feedback scheme (fig:modulation02(a)) uses information from the output to
provide the signal $v_{D}$ to the modulator. The main advantage of this scheme
is that a feedback controller receives information directly from the output
voltage, which is the value intended to be regulated, and therefore it can
compensate for all variations regardless of their source. A disadvantage is
that feedback regulation can lead to instability in the converter. Another
disadvantage of the feedback scheme is that the output has to be affected in
order to provide information to the feedback controller. In other words, the
controller may not be fast enough to see a fluctuation and compensate for it.
For example, a variation in the input voltage $V_{\mathrm{S}}$ such as 100 Hz
ripple from the rectification stage could have an amplitude such that the
feedback scheme cannot completely reject it. In such cases a feedforward
scheme, as shown in fig:modulation02(b), can be used. If it is known how the
input voltage affects the output voltage (the DC transfer function), a
feedforward controller can provide a signal $v_{D}$ to the modulator that
gives the correct value of the output voltage.

Figure 15: Block diagrams of regulation schemes: (a) feedback regulation; (b)
feedforward regulation

The various ways of obtaining the switching function to be applied to the
power converter are referred as control, or modulation, methods. A list and
brief discussion of the most common methods are given in the following
sections.

###  3.1 Constant-frequency pulse width modulation (PWM)

This is the most popular method of controlling power converters. In this
modulation method, the period of the switching signal $T$ is constant and the
information is contained in the width of the pulse, i.e., the period of time
for which the switching signal is on, $t_{\mathrm{on}}$. This method is also
known as carrier-based pulse width modulation. The switching function for
constant-frequency PWM can be obtained by comparing the signal $v_{D}$ with a
carrier signal $c(t)$ as depicted in fig:pwm01.

Figure 16: PWM generation: (a) comparator; (b) waveforms

Figure 16 shows an analog implementation of a PWM modulator. In digital
implementations, the carrier is replaced by a counter/timer block. In both
types of implementation, a way to avoid multiple transitions within a carrier
period needs to be added.

###  3.2 Variable-frequency PWM

Variable-frequency PWM, although not as popular as constant-frequency PWM, is
also used in practice. The common variations of variable-frequency PWM are the
constant-off-time/variable-on-time and constant-on-time/variable-off-time
versions.

Figure 17 shows the waveform for constant-off-time/variable-on-time PWM using
a sawtooth-like carrier signal. The switch is turned on after a fixed time; at
this point, the sawtooth signal starts to rise at a constant rate. The switch
is turned off again when the sawtooth signal intersects the signal $v_{D}$.
After the turn-off of the switch, the carrier is reset to zero for a fixed
period of time. The figure shows that the on time increases and the switching
frequency decreases when the signal $v_{D}$ is increased, resulting in
variable-frequency operation. Constant-on-time/variable-off-time PWM can be
implemented in a similar way.

Figure 17: PWM generation: constant-off-time/variable-on-time PWM

Variable-frequency PWM is used in light-load operation of power converters in
order to increase the efficiency of the converter under these conditions. The
main disadvantage of this is the difficulty associated with the design of the
input and output filters. The filter cut-off frequency has to be selected
based on the lowest possible switching frequency in order to provide the
required attenuation of ripple and electromagnetic interference (EMI) under
all possible operation conditions. This usually leads to a conservative design
with significant volume and cost penalties.

###  3.3 Current mode control

Current mode control uses direct measurement of the inductor current to
generate the switching function. One way of generating the switching function
is to replace the sawtooth carrier waveform used to generate the PWM by the
inductor current, as shown in fig:CM01(b). The switching function is set to
unity by a clock signal of period $T$. When the inductor current reaches the
value of the signal $v_{D}$, the switching function is reset. Figure 18(a)
shows a simple circuit for current mode PWM generation.

Figure 18: Current mode: (a) circuit diagram; (b) waveforms

####  3.3.1 Slope compensation

The peak current mode method is inherently unstable at duty cycles higher than
0.5, resulting in subharmonic oscillations. A compensating ramp (with slope
equal to the inductor current down-slope) is usually applied to the comparator
input to eliminate this instability.

Note that in current mode control, the output voltage is not directly
controlled. An addition loop is needed in order to obtain a regulated output
voltage.

###  3.4 Variable-structure control

Variable-structure control offers an alternative way to implement a switching
function, which exploits the inherently variable nature of the structure of
DC–DC converters. In practice, the converter switches are driven as a function
of the instantaneous values of the voltages and currents. The most important
feature of variable-structure control is its ability to provide robust
control. Although the study of variable-structure control is far beyond on the
scope of this article, hysteresis control provides a very simple and intuitive
example. Sliding-mode control is other variable-structure control method for
power converters reported in the literature.

####  3.4.1 Hysteresis control

Hysteresis control is based on comparing the output voltage $V_{R}$ with a
reference voltage $v_{D}$ using a hysteresis comparator. If the voltage is
lower than the upper limit of the comparator, the switch is turned off until
the output voltage reaches the lower limit of the hysteresis comparator. When
the voltage reaches this limit, the switch is turned on. Figure 19 shows a
simple circuit diagram for hysteresis control, and typical waveforms.

Figure 19: Hysteresis control: (a) circuit diagram; (b) waveforms

##  4 Derived converter topologies

Several different topologies have been derived from the basic DC–DC
converters. In this section, three different groups of converters will be
presented. The first group of converter topologies are isolated versions of
the basic converters. The use of a transformer provides galvanic isolation
between the input and output voltages. An additional advantage of the use of a
transformer is an increase in the output voltage range of the converter due to
the turns ratio of the transformer. The second group is formed by the flyback
converter. This converter has galvanic isolation provided by coupled
inductors. The difference between a transformer and coupled inductors is that
in the latter case there is no direct energy flow from the input to the
output. This fact limits the use of these converters to low-power
applications. Finally, a converter with continuous input and output currents
is presented.

###  4.1 Half-bridge converter

The circuit diagram of a half-bridge converter is shown in fig:half01. The
output filter part of the circuit is the same as that in a buck converter. The
square waveform of the transformer secondary is rectified and then filtered by
the output stage. The voltage at the primary of the transformer is shown in
fig:half02. The voltage per second on the primary and secondary sides during
the switching period has to be zero in order to avoid saturation of the
transformer.

Figure 20: Half-bridge converter: circuit diagram

In this converter, two capacitors act as an input voltage divider, which is
connected to one of the inputs of the transformer. A two-polarity voltage is
obtained by connecting the other input to $V_{\mathrm{S}}$ or a reference
voltage using the switches $S_{1}$ and $S_{2}$. The switches operate shifted
in phase by $T$/2 with the same duty ratio $D$. The duty ratio must be smaller
than 0.5. When the switch $S_{1}$ is on, the diode $D_{1}$ conducts and the
diode $D_{2}$ is off. The diode states are reversed when the switch $S_{2}$ is
on. When both controllable switches are off, both diodes are on and share the
filter inductor current equally. This short-circuits the transformer and
forces the transformer input voltage to zero. The DC voltage transfer function
of the half-bridge converter is given by the following equation:

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=2D\frac{n_{2}}{n_{1}}.$
(22)

Figure 21: Half-bridge converter: transformer primary voltage

###  4.2 Push–pull converter

Push–pull converters, like half-bridge converters, can be regarded as two
single-switch buck converters running out of phase. A circuit diagram is shown
in fig:push01. The volts per second are balanced by alternating the operation
of the switches $S_{1}$ and $S_{2}$. Similarly to the half-bridge converter,
the output filter stage is the same as that in the buck converter. The main
advantage of this topology is that both switches are connected to the
reference voltage, which avoids the use of a floating driver for the switches.
The main disadvantage is that differences in the period $t_{\mathrm{on}}$ of
the switches could lead to saturation of the transformer.

Figure 22: Push–pull converter: circuit diagram

The DC voltage transfer function of the push–pull converter is

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=2D\frac{n_{2}}{n_{1}}.$
(23)

###  4.3 Full-bridge converter

The circuit diagram of a full-bridge converter is shown in fig:full01. In this
topology, the controllable switches are operated in pairs. When $S_{2}$ and
$S_{3}$ are on, a positive voltage $V_{\mathrm{S}}$ is applied to the primary
side of the transformer. In this condition, the diode $D_{1}$ conducts the
inductor current. With $S_{1}$ and $S_{4}$ on, the voltage applied to the
transformer is $-V_{\mathrm{S}}$ and the the diode $D_{2}$ is on. If all the
switches are off, both diodes conduct the output current and the secondary of
the transformer is short-circuited.

Figure 23: Full-bridge converter: circuit diagram

The steady-state DC voltage transfer function is given by

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=2D\frac{n_{2}}{n_{1}}.$
(24)

###  4.4 Forward converter

The forward converter is another topology that uses a transformer in order to
provide galvanic isolation. The voltage output can be lower or higher than the
input voltage depending on the transformer ratio. A circuit diagram is shown
in fig:forward01. As in the previous isolated topologies, the output filter
stage is identical to that in the buck converter. When the switch $S$ is
closed, the diode $D_{1}$ conducts and transfers energy from the input to the
output stage. During the period when the switch $S$ is open, the diode $D_{2}$
carries the current in the inductor $L$ and the diode $D_{3}$ connects the
third transformer winding to the input voltage to decrease the magnetizing
current to zero.

Figure 24: Forward converter: circuit diagram

The steady-state DC voltage transfer function is given by

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=2D\frac{n_{2}}{n_{1}}.$
(25)

The condition to avoid saturation of the transformer is

$n_{1}D\leq n_{3}(1-D).$ (26)

###  4.5 Flyback converter

The flyback converter is derived from the buck–boost topology. The inductor is
split into two coupled inductors. As a result of this, galvanic isolation is
achieved and the DC voltage transfer function is multiplied by the turns ratio
of the inductors. The basic circuit diagram is shown in fig:flyback01.

Figure 25: Flyback converter: circuit diagram

The magnetic components of the flyback converter act more like two inductors
sharing a common magnetic core than as a transformer. Energy is stored in the
common magnetic core during the time $t_{\mathrm{on}}$ of the switch $S$ and
is then transferred to the output stage during the time $t_{\mathrm{off}}$.
There is no direct energy transfer between the primary and secondary windings,
unlike the converters mentioned above. This limits the application of this
converter to low-power applications.

###  4.6 Ćuk converter

The Ćuk (pronounced ‘chook’) converter was introduced by Slobodan Ćuk of the
California Institute of Technology. The circuit diagram of this converter is
shown in fig:cuk01. The main advantage of this converter is the continuous
currents at the input and output of the converter. The main disadvantage is
the high current stress on the switch.

Figure 26: Ćuk converter: circuit diagram

The DC voltage transfer function of the converter is obtained using the
average current in the capacitor $C_{1}$, which should be zero during a
switching period when there is power balance between the input and output of
the converter:

$I_{L_{2}}DT=I_{L_{1}}(1-D)T,$ (27)
$P_{\mathrm{IN}}=V_{\mathrm{S}}I_{L_{1}}=-V_{\mathrm{O}}I_{L_{2}}=P_{\mathrm{OUT}}.$
(28)

Using these two equations,

$M_{\mathrm{V}}\equiv\frac{V_{\mathrm{O}}}{V_{\mathrm{S}}}=\frac{D}{(1-D)}.$
(29)

Isolated versions of the Ćuk converter can be found in the literature. There
are also versions of this converter in which the two inductors $L_{1}$ and
$L_{2}$ are integrated into one magnetic element in order to reduce the cost
and volume of the converter.

The list of converters presented in this section is not intended to be
exhaustive. The converter topologies presented are just a small sample of the
many topologies that have been reported in the literature. The right topology
for each application depends on many factors, which include the stress on the
semiconductors and passive components, the level of power to be transferred,
and the switching frequency, among others.

##  5 Additional topics

This section presents some additional topics related to one-quadrant power
converters that are worth mentioning. Firstly, synchronous rectification is
addressed as a method to reduce losses, especially in low-voltage power
converters. Interleaved converters are discussed as a way to increase the
power handled by converters while reducing the stress on the input and output
filters. Finally, a brief introduction to hard switching, snubbers, and soft
switching is given.

###  5.1 Synchronous rectification

One component of most one-quadrant converter topologies is a diode. This diode
is used to provide a path for the current when the main switch is turned off,
or for rectification and providing a DC output. The conduction losses of the
diode make an important contribution to the overall power loss of a converter,
especially for low-output-voltage, high-output-current power converters.

The conduction loss of a diode in a buck converter is given by the following
equation:

$P_{\mathrm{D}}=V_{\mathrm{F}}I_{\mathrm{O}}(1-D).$ (30)

The overall efficiency due to the diode conduction loss can be expressed as

$\nu=\frac{V_{\mathrm{O}}I_{\mathrm{O}}}{V_{\mathrm{O}}I_{\mathrm{O}}+V_{\mathrm{F}}I_{\mathrm{O}}(1-D)}=\frac{V_{\mathrm{O}}}{V_{\mathrm{O}}+V_{\mathrm{F}}(1-D)}.$
(31)

For low-voltage power converters, this equation shows that the overall
efficiency of the converter can drop to unacceptable levels. One way to
increase the overall efficiency under these conditions is to replace the diode
by a MOSFET, as shown in fig:synch01. The MOSFET introduces an almost linear
resistance with a lower voltage drop. Figure 28 shows the power loss in
$S_{2}$ for a diode and for a MOSFET.

Figure 27: Synchronous rectification: circuit diagram Figure 28: Power loss in
$S_{2}$: diode (solid line) and MOSFET (dashed line)

In addition to minimizing conduction losses, MOSFETs offer the additional
benefit that they can easily be paralleled because their
$R_{\mathrm{DS_{on}}}$ has a positive temperature coefficient.

###  5.2 Interleaved converters

The concept of paralleling several DC–DC power converters in order to increase
the output power is well known in power electronics. A further step is to
group the components together and control the resulting converter as a unit.
Interleaving is one of the techniques most often used, and provides several
advantages such as an increase in the resulting switching frequency and a
reduction in the component size. Figure 29 shows the circuit diagram of an
interleaved buck converter. The switching functions for the switches
$S_{1},S_{2},\ldots,S_{N}$ have the same switching frequency but there is a
phase shift between them equal to $2\pi/N$, where $N$ is the number of
individual stages that form the converter.

Figure 29: Interleaved buck converter

The first plot in fig:inter02 shows the output currents of the individual
stages and the overall output current formed by addition of the inductor
currents for each phase. The ripple frequency of the overall output current is
$N$ times the frequency of the inductor phase currents. This multiplication of
the ripple frequency reduces the requirements on the output filter. Figure 31
shows the normalized output ripple current. This shows how the ripple is
reduced; for particular values of $D$, the ripple current is zero.

Figure 30: Output and input currents for a three-stage interleaved buck
converter Figure 31: Normalized output ripple current

A similar effect is produced in the input current. The requirements on the
input capacitor are also reduced by the interleaving of the converters. Figure
30(b) shows the input current for each branch and the total input current. An
increase in the frequency and a reduction in the RMS value can be observed.
Figure 32 shows the normalized input capacitor RMS current for different
values of $N$.

Figure 32: Normalized input capacitor RMS current

###  5.3 Hard switching, snubbers, and soft switching

One of the big advantages of operating a power converter in switched mode is
that voltages across and currents through the switches are not present
simultaneously during the ‘on’ and ‘off’ states. This cannot be true, however,
during the transitions between the states of the converter. The trajectories
during the transitions determine whether the switches operate in a hard or
soft switching mode. The switching trajectory of a hard-switched power device
is shown in fig:hardSW. During the turn-on and turn-off processes, the power
device has to withstand high voltages and currents simultaneously, resulting
in high switching losses and stress.

Figure 33: Hard switching: (a) single switch without snubber; (b) switch
voltage and current during transition; (c) transition diagram.

The switching losses are proportional to the operating frequency of the
converter and are usually the main factor that limits the operating frequency.
Another disadvantage of hard-switching operation is an increase in high-
frequency emissions, which produce EMI.

Dissipative passive snubbers are usually added to power circuits in order to
reduce the $\mathrm{d}v/\mathrm{d}t$ and $\mathrm{d}i/\mathrm{d}t$ of the
power devices, and the switching losses and stresses can be diverted to these
passive snubber circuits. Figure 34 shows how the transition trajectory is
modified by the presence of an RC snubber. The $\mathrm{d}v/\mathrm{d}t$ of
the switch is reduced during the turn-off and part of the transition energy is
transferred to the dissipative element of the snubber.

Figure 34: Snubbers: (a) circuit diagram of RC snubber; (b) switch voltage and
current during transition; (c) transition diagram.

The snubber design has to be done carefully in order to correctly dimension
the components and ensure proper operation. More complex snubber circuits have
the ability to recover the transition energy and increase the overall
efficiency of the converter.

A further step can be taken to reduce the commutation losses by operating the
switches in a soft switching mode. The basic idea of soft switching can be
explained using the plots in fig:SoftSwitching01. The voltage across the
switch is forced to be zero while the current is dropping. When the current is
zero, the switch voltage starts to rise until the switch reaches the turn-off
state. The losses during the transition are zero.

Figure 35: Soft switching: (a) switch voltage and current during transition;
(b) transition diagram

A simple circuit operating in a soft switching mode is shown in
fig:SoftSwitching02. The switch $M_{1}$ opens with zero voltage across its
terminals because this voltage is maintained by the capacitor $C_{1}$. The
voltage $v_{\mathrm{A}}$ drops because the inductor current charges and
discharges the capacitors $C_{1}$ and $C_{2}$. Once the voltage reaches zero,
the diode $D_{2}$ starts to carry current and k-eps-converted-to.pdf the
voltage close to zero. During the period of time that the diode is conducting,
the switch $M_{2}$ is turned on. The positive transition of voltage
$v_{\mathrm{A}}$ is produced in a similar way.

Figure 36: Soft switching: (a) circuit diagram; (b) waveforms

##  6 DC–DC resonant converters

The trend in power electronics is towards an increase in efficiency, power,
and component density. For this reason, resonant converters have received
special interest in recent years. These converters have the potential to
operate at higher frequencies and with lower switching losses than hard-
switching converters. The design of resonant converters and their control
presents challenges different from those for other converters. In particular,
the control of these converters is usually done by frequency modulation
instead of pulse width modulation. Although a comprehensive analysis of DC
resonant converters is far beyond the scope of this article, a brief
description of their functional principles is given here.

All of the resonant-converter topologies operate in essentially the same way.
The power switches generate a square wave voltage or current, which is applied
to a resonant circuit. Energy circulates in the resonant circuit, and part of
this energy is transferred to the output.

The two basic types of resonant converter are the series resonant converter,
shown in fig:Resonant01(a), and the parallel resonant converter, shown in
fig:Resonant01(b). In both topologies, the energy transferred to the output is
regulated by changing the frequency of the driving voltage. The resonant
circuit forms a voltage divider with the resistor $R$. By changing the
frequency, the impedance of the resonant circuit and therefore the output
voltage can be varied.

Figure 37: (a) Series resonant converter; (b) parallel resonant converter

The circuits shown in fig:Resonant01 have limitations. The series resonant
converter cannot work under very light load conditions, because the operating
frequency would have to be very high in order to regulate the output voltage.
The parallel resonant converter requires large amounts of circulating current
when operating under heavy load conditions.

###  6.1 LCC and LLC resonant converters

To overcome the limitations of the resonant converters shown in
fig:Resonant01, converters combining series and parallel topologies have been
proposed. Figure 38(a) shows a topology which includes two capacitors and one
inductor. This topology is named the LCC resonant converter.

Figure 38: (a) LCC resonant converter; (b) LLC resonant converter

The topology formed by two inductors and one capacitor (the LLC resonant
converter) is shown in fig:Resonant02(b). This topology has several advantages
over the LCC topology. For instance, the two inductors can be integrated into
one physical magnetic component, and it also provides galvanic isolation.

A simplified circuit diagram of an LLC resonant converter is shown in
fig:Resonant03. The two inductors are the magnetizing and leakage inductances
of a transformer. The LLC topology has the additional advantage of being able
to operate in a soft switching mode. This allows the switching frequency and
power density of the converter to be increased.

Figure 39: LLC resonant converter: circuit diagram

The typical waveforms for an LLC resonant converter are shown in
fig:Resonant04. Soft commutation is achieved in a similar way to that
described in fig:SoftSwitching02.

Figure 40: Waveforms for LLC resonant converter

##  7 Conclusions

One-quadrant power converters play an important role in industrial and
consumer electronics applications. Almost every electronic circuit is supplied
with a DC voltage provided by some kind of power converter. If a regulated
voltage is needed, this converter usually consists of a rectifier stage
followed by a regulated DC–DC power converter. The field of power converters
for particle accelerators is no exception to this trend. As was mentioned in
the introduction, one-quadrant power converters are used to generate the high-
precision, stable currents needed for storage ring accelerator magnets. The
topologies used in this application depend on the power level required.

This article was intended to be an introduction to this broad subject. All the
topics discussed here are covered in depth in books on power converters and in
journal and conference papers. The references provided at the end of this
article should be a good starting point for further studies.

## References

  * [1] M.H. Rashid, Power Electronics Handbook (Academic Press, San Diego, 2001).
  * [2] N. Mohan and T.M. Undeland, Power Electronics: Converters, Applications, and Design (Wiley, 2007).
  * [3] F. Bordry, Power converters: definitions, classification and converter topologies, Proc. CAS–CERN Accelerator School and CLRC Daresbury Laboratory, 2004.
  * [4] J.Y. Hung, W. Gao, and J.C. Hung, IEEE Trans. Ind. Electron. 40(1) (1993) 2.
  * [5] S. Cuk, IEEE Trans. Magn. 19(2) (1983) 57.
  * [6] R. Sheehan and H. Anatole, Understanding and applying current-mode control theory, Power Electronics Technology Exhibition and Conf., Dallas, TX, 2007, vol. 30, p. 1.
  * [7] H. Huang, Designing an llc resonant half-bridge power converter, TI Power Supply Design Seminar, SEM1900, 2010.
  * [8] S. Mappus, Synchronous rectification for forward converters, Fairchild Semiconductor Power Seminar 2010–2011 (Fairchild Semiconductor, 2010).
  * [9] T. Hegarty, Benefits of multi-phasing buck converters, EE Times India (2007) 1.
  * [10] S.-C. Tan, Y.-M. Lai, and C.K. Tse, IEEE Trans. Ind. Electron. 55(3) (2008) 1160.


# Entanglement fidelity for electron-electron interaction in strongly coupled
semiclassical plasma and under external fields

B. J. Falaye<EMAIL_ADDRESS>Applied Theoretical Physics
Division, Department of Physics, Federal University Lafia, P. M. B. 146,
Lafia, Nigeria.  K. J. Oyewumi  Theoretical Physics Section, Department of
Physics, University of Ilorin, P. M. B. 1515, Ilorin, Nigeria.  O. A. Falaiye
Theoretical Physics Section, Department of Physics, University of Ilorin, P.
M. B. 1515, Ilorin, Nigeria.  C. A. Onate  Physics Programme, Department of
Physical Sciences, Landmark Univerisity, Omu-Aran, Nigeria  O. J. Oluwadare
Department of Physics, Federal University Oye-Ekiti, P. M. B. 373, Ekiti
State, Nigeria  W. A. Yahya  Department of Physics and Material Science, Kwara
State University, P. M. B. 1530, Ilorin, Nigeria

###### Abstract

This paper presents the effects of AB-flux field and electric field on
electron-electron interaction, encircled by a strongly coupled semiclassical
plasma. We found that weak external fields are required to perpetuate a low-
energy elastic electron-electron interaction in a strongly coupled
semiclassical plasma. The entanglement fidelity in the interaction process has
been examined. We have used partial wave analysis to derive the entanglement
fidelity. We found that for a weak electric field, the fidelity ratio for
electron-electron interaction increase as projectile energy increase but
remains constant or almost zero for a strong electric field. Our results
provide an invaluable information on how the efficiency of entanglement
fidelity for a low-energy elastic electron-electron interaction in a strongly
coupled semiclassical plasma can be influenced by the presence of external
fields.

Semiclassical plasma; Entanglement fidelity; AB-flux field

###### pacs:

52.25.Kn, 52.27.Gr, 52.27.Lw, 52.25.Fi.

††preprint: APS/123-QED

##  I Introduction

There has been a ceaseless interest (A1 ; A2 ; A3  and Refs. therein) in
studying electron-ion and electron-electron interaction process in plasmas
environment due to its application in diagnosing various plasma and also
providing passable knowledge on collision dynamics. It was shown in Ref. A3
that the optically allowed but 1s-2s cross section is less sensitive to the
plasma effect but 1s-2p and 2s-2p cross sections for electron impact
excitation are substantially reduced by the plasma screening. The effect of
the carrier reservoir dimensionality on electron-electron scattering in
quantum dot materials was investigated in Ref. ER1 . It was found that 3D
scattering benefits from its additional degree of freedom in the momentum
space. Jung ER2  studied quantum-mechanical effects on electron-electron
scattering in dense high-temperature plasmas. Suppression of inelastic
electron-electron scattering in Anderson insulators was explored by Ovadyahu
ER3 .

The study of entanglement has been a subject of active research in physics,
chemistry and related areas. Quantum entanglement lies in the heart of quantum
mechanics (TE1 ; TE2 ; TE3  and refs. therein). Entanglement has become very
useful in information processing and quantum communication. Really, it plays a
crucial role in potential realization of quantum computers. It also represent
a quantitative measure for electron-electron correlation in many body system.
A very useful tool to distinguish between two quantum states (say $\rho_{1}$
and $\rho_{2}$) is known as the fidelity between $\rho_{1}$ and $\rho_{2}$,
i.e.,
$F(\rho_{1},\rho_{2}):=$Max$|\left\langle\phi_{1}\right.|\left.\phi_{2}\right\rangle|$,
where $\left.|\phi_{1}\right\rangle$ and $\left.|\phi_{2}\right\rangle$
correspond to the state of $\rho_{1}$ and $\rho_{2}$.

The interest in studying entanglement fidelity in the electron-electron
scattering process has received a considerable attention in the last few
years. This is due to the key role that quantum correlation plays in
understanding information processing. The entanglement in scattering processes
was investigated by Mishima et al. JA11 . It has been found that, a low energy
elastic scattering is desirable for entanglement enhancement. The quantum
effect on the entanglement fidelity in elastic electron-ion scattering was
investigated under strongly coupled semiclassical plasmas in Ref. A7 . It was
shown that the quantum effect significantly augments the entanglement fidelity
in a strongly coupled semiclassical plasmas. In the present work, we study the
conditions to obtain a low-energy elastic electron-electron scattering by
taking into account the effect of electric field, AB-flux field and uniform
magnetic field directed along $z$-axis and surrounded by strongly coupled
semiclassical plasmas. Once the conditions to obtain a low energy level are
established, then entanglement fidelity will be examined.

##  II Theory and Calculations

The Debye-Hückel model which provides a modern treatment of non-ideality in
plasma via the screening effect is given by
$\chi(r)=Z_{a}Z_{b}/r\exp(-r/\lambda_{D})$, where $\lambda_{D}$ represent
Debye length given by $\lambda_{D}=\sqrt{k_{\beta}T/4\pi
e^{2}\sum_{i}n_{i}Z_{i}^{2}}$ with $Z_{a}$ and $Z_{b}$ being the $a-b$
interaction (here, $a=b=e$) . This model accounts for pair correlations.
However, besides correlation, the quantum mechanical effects of diffraction
also take place in semi classical plasma. Hence, the effective model which
accounts for both screening and diffraction in strongly coupled semiclassical
plasmas can be written as A9 :

$\displaystyle\chi(r)$ $\displaystyle=$
$\displaystyle-\frac{Z_{a}Z_{b}e^{2}}{\sqrt{1-4\frac{\lambda_{ab}^{2}}{\lambda_{D}^{2}}}}\left[\mathcal{B}^{2}(\lambda_{ab},\lambda_{D})\frac{\exp[-\mathcal{A}(\lambda_{ab},\lambda_{D})r]}{r}\right.$
(1)
$\displaystyle\left.-\mathcal{A}^{2}(\lambda_{ab},\lambda_{D})\frac{\exp[-\mathcal{B}(\lambda_{ab},\lambda_{D})r]}{r}\right],$

where $\mathcal{A}$ and $\mathcal{B}$ are quantum screening parameters given
by
$\mathcal{A}^{2}\equiv\left(1-\sqrt{1-4\lambda^{2}_{ab}/\lambda_{D}^{2}}\right)/2$,
$\mathcal{B}^{2}\equiv\left(1+\sqrt{1-4\lambda_{ab}^{2}/\lambda_{D}^{2}}\right)/2$
and the de Broglie thermal wavelength is denoted by $\lambda_{ab}$.

On this foundation, we can build the model equation for electron-electron
interaction under the influence of electric field, AB-flux field and uniform
magnetic field directed along $z$-axis and surrounded by strongly coupled
semiclassical plasmas, in spherical polar coordinates as:

$\left[\frac{-\hbar^{2}}{2\mu}{\nabla}^{2}-\underbrace{\frac{\left(Z_{e}e\right)^{2}}{\sqrt{1-4\frac{\lambda_{ee}^{2}}{\lambda_{D}^{2}}}}\left[\mathcal{B}^{2}\frac{e^{-\mathcal{A}(\lambda_{ee},\lambda_{D})r}}{r}-\mathcal{A}^{2}\frac{e^{-\mathcal{B}(\lambda_{ee},\lambda_{D})r}}{r}\right]}_{\mbox{screened
effective pseudopotential
model}}-Fr\cos(\theta)\right]\psi(r,\theta)=E_{nm}\psi(r,\theta),$ (2)

where $E$ denotes the eigenvalues, $\mu$ is the effective mass of an electron,
vector potential $\vec{A}=\frac{\phi_{AB}}{2\pi r}\hat{\phi}$ represents the
magnetic flux $\phi_{AB}$ created by a solenoid inserted inside the antidot
with $\vec{\nabla}\times\vec{A}=0$. $\lambda_{D}$ is the screening parameter.
$Z$ denotes the atomic number which is found useful in describing energy
levels of light to heavy neutral atoms. The thermal de Broglie wavelength of
the electron-electron pair is denoted by $\lambda_{ee}=\hbar/\sqrt{\pi\mu
k_{\beta}T_{e}}$ where $T_{e}$ is the temperature of the plasma electrons,
Planck constant is denoted as $\hbar$ and $k_{\beta}$ is the Boltzmann
constant. Moreover, the characteristic properties of the plasma are denoted by
the coupling parameter $\alpha k_{\beta}T_{e}\Gamma_{ee}=(Ze)^{2}$ (where
$\alpha$ is the average distance between particles).
$\lambda_{D}=\sqrt{K_{\beta}T_{e}/4\pi n_{e}e^{2}}$ is the Debye radius where
the electron density is denoted by $n_{e}$. The ranges of electron density and
temperature, $T_{e}$ are known as $~{}10^{20}-10^{24}$cm-3 and $~{}5\times
10^{4}-10^{6}K$, respectively in dense classical plasma.

Furthermore, $F$ represents electric field strength with angle $\theta$
between $F$ and $r$. With $\theta=0$, then $Fr\cos(\theta)$ becomes $Fr$. The
variation of the effective potential energy as a function of various model
parameters have been displayed in figure 1. Now, let us take a wavefunction in
cylindrical coordinates as
$\psi(r,\phi)=\frac{1}{\sqrt{2r\pi}}e^{im\phi}\mathcal{H}_{nm}(r)$, where
$m=0,\pm 1,\pm 2,...$ denotes the magnetic quantum number. Inserting this
wavefunction into equation (2), we find a second order differential equation
${d^{2}\mathcal{H}_{nm}(r)}/{dr^{2}}+{2\mu}/{\hbar^{2}}\left[E_{nm}-\chi_{\rm
eff.}\right]\mathcal{H}_{nm}(r)=0$, where the effective potential Veff. is

$\displaystyle\chi_{\rm eff.}$ $\displaystyle=$
$\displaystyle-\frac{\alpha\Gamma_{ee}}{\sqrt{1-\frac{24\Gamma_{ee}^{2}}{\pi
r_{s}\left(1+\delta^{2}\right)}}}\left[A_{-}^{2}\frac{e^{-A_{+}r}}{r}-A_{+}^{2}\frac{e^{-A_{-}r}}{r}\right]-Fr+\frac{\hbar^{2}}{2\mu}\left[\frac{\left(\sigma_{0m}-\frac{1}{2}\right)^{2}-\frac{1}{4}}{r^{2}}\right]$
(3) $\displaystyle=\chi_{\rm
eff.}^{(1)}+\frac{\hbar^{2}}{2\mu}\left[\frac{\left(\sigma_{0m}-\frac{1}{2}\right)^{2}-\frac{1}{4}}{r^{2}}\right],\
\ \mbox{with}\ \ \ A_{+}(\lambda_{ee},\lambda_{D})^{2}\equiv\frac{\pi
r_{s}}{4\Gamma_{ee}}\left(1+\sqrt{1-\frac{24\Gamma_{ee}^{2}}{\pi
r_{s}(1+\delta^{2})}}\right)\ \ \ \mbox{,}$ $\displaystyle
A_{-}(\lambda_{ee},\lambda_{D})^{2}\equiv\frac{\pi
r_{s}}{4\Gamma_{ee}}\left(1-\sqrt{1-\frac{24\Gamma_{ee}^{2}}{\pi
r_{s}(1+\delta^{2})}}\right)\ \ \mbox{and}\ \ \sigma_{nm}=n+m+\xi+\frac{1}{2}$

where we have also incorporate the effect of dynamic screening KJ1 .
$\xi=\phi_{AB}/\phi_{0}$ is taken as integer with the flux quantum
$\phi_{0}=hc/e$. The relative velocity is denoted by $\delta$ and the density
parameter is $r_{s}=\alpha/a_{0}$ (where $a_{0}$ is the Bohr radius).

The purpose of this research is to study entanglement fidelity for low-energy
elastic electron-electron interaction in a weak electric field and a strongly
coupled semiclassical plasma. Moreover, how can we enhance or generate or
simulate a low-energy elastic electron-electron interaction? To answer this
question, we need to solve the radial schrödinger equation with the effective
model (3) using perturbation theory SUS1 ; SUS2 , since the equation does not
admit an exact solution with the model. Thus, we obtain the $n$-th state
energy as

$\displaystyle E_{nm}$ $\displaystyle\approx$
$\displaystyle-\frac{\mu}{2\hbar^{2}}\frac{\mathcal{M}^{2}\left(A_{+}^{2}-A_{-}^{2}\right)^{2}}{\sigma_{nm}^{2}}-\frac{\hbar^{2}F}{2\mu\mathcal{M}\left(A_{+}^{2}-A_{-}^{2}\right)}\left[3\sigma_{nm}^{2}-\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{1}{4}\right]$
(4)
$\displaystyle\mathcal{M}\left(A_{+}^{2}A_{-}-A_{+}A_{-}^{2}\right)+\left\\{\left[\frac{\mathcal{M}}{6}\left(A_{+}^{2}A_{-}^{3}-A_{+}^{3}A_{-}^{2}\right)\right]-\frac{\sigma_{nm}^{2}\hbar^{2}F^{2}}{{2\mu\mathcal{M}^{2}\left(A_{+}^{2}-A_{-}^{2}\right)^{2}}}\right\\}$
$\displaystyle\times\frac{\sigma_{nm}^{2}\hbar^{4}}{{\mu^{2}\mathcal{M}^{2}\left(A_{+}^{2}-A_{-}^{2}\right)^{2}}}\left\\{5\sigma_{nm}^{2}-3\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{7}{4}\right\\}-\frac{\sigma_{nm}^{2}\hbar^{6}\left(A_{+}^{2}A_{-}^{4}-A_{+}^{4}A_{-}^{2}\right)}{96\mu^{3}\mathcal{M}^{2}\left(A_{+}^{2}-A_{-}^{2}\right)^{3}}$
$\displaystyle\times\left[5\sigma_{nm}^{2}-3\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{3}{4}\right]\left[5\sigma_{nm}^{2}-3\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{7}{4}\right]+\frac{\sigma_{nm}^{4}\hbar^{8}}{8\mu^{4}\mathcal{M}^{5}\left(A_{+}^{2}-A_{-}^{2}\right)^{5}}$
$\displaystyle\times\left[5\sigma_{nm}^{2}-3\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{7}{4}\right]\left[\frac{\mathcal{M}}{6}\left(A_{+}^{2}A_{-}^{3}-A_{+}^{3}A_{-}^{2}\right)\right]\left[9\sigma_{nm}^{2}-5\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{5}{4}\right]$
$\displaystyle-\frac{\sigma_{nm}^{6}\hbar^{10}}{8\mu^{5}\mathcal{M}^{7}\left(A_{+}^{2}-A_{-}^{2}\right)^{7}}\left[9\sigma_{nm}^{2}-5\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{5}{4}\right]\left[5\sigma_{nm}^{2}-3\left(\sigma_{0m}-\frac{1}{2}\right)^{2}+\frac{7}{4}\right],$

where $\mathcal{M}=-{\alpha\Gamma_{ee}}/{\sqrt{1-{24\Gamma_{ee}^{2}}/({\pi
r_{s}\left(1+\delta^{2}\right)})}}$. Table 1 displays eigenvalues for
electron-electron interaction in semiclassical plasma under the influence of
external fields (AB-flux field and electric field) in atomic units and in low
vibrational $n$ and rotational $m$ states. When there are no external fields,
(i.e., when $\xi=F=0$), the spacing between the energy levels of the effective
potential is narrow and decreases as $n$ increases. We found that, there exist
degeneracy among some states $(n,m)$, for instance $(0,1)$ and $(2,-1)$;
$(1,1)$ and $(3,-1)$. There also exist quasi-degeneracy of the energy levels
in $(2,0)$ and $(1,1)$; $(3,0)$ and $(2,1)$. Not only does the energy levels
of the effective potential and spacings between states increases by exposing
the system to external fields, but also the degeneracies are removed and
become split up and down.

Table 1: The energy values for electron-electron interaction under the influence of AB-flux and external electric fields with various values of magnetic quantum numbers. The following fitting parameters have been employed: $\alpha=r_{s}=5$, $\Gamma_{ee}=1$, $\delta=8$ and $\hbar=\mu=c=1$ . m | n |  $F=0$, $\xi=0$ |  $F=0$, $\xi=2$, |  $F=0.01$, $\xi=0$ |  $F=0.01$, $\xi=2$
---|---|---|---|---|---
0 | 0 | -3092.11 | -131.152 | -3092.11 | -131.150
| 1 | -350.535 | -70.4042 | -350.534 | -70.4000
| 2 | -131.104 | -44.5892 | -131.102 | -44.5819
| 3 | -70.2885 | -29.7294 | -70.2838 | -29.7183
1 | 0 | -350.538 | -70.5367 | -350.538 | -70.5331
| 1 | -131.117 | -44.8667 | -131.114 | -44.8601
| 2 | -70.3183 | -30.2518 | -70.3137 | -30.2413
| 3 | -44.4121 | -18.6248 | -44.4044 | -18.6098
-1 | 0 | -3092.11 | -350.538 | -3092.11 | -350.538
| 1 | -3092.11 | -131.117 | -3092.11 | -131.114
| 2 | -350.538 | -70.3183 | -350.538 | -70.3137
| 3 | -131.117 | -44.4121 | -131.114 | -44.4044

It has been shown that increasing the strenght of AB-flux field would lead to
a substantial shift in the energies. Its important to note that the dominance
and confinement effects of AB-flux field on electron-electron interaction in
semiclassical plasma is stronger than the electric field and as consequence
the localizations of the quantum states can be manipulated via application of
AB-flux field

It can also be seen from this table that the lowest energy can be obtained
when $n=0|1$, $m=-1$, $\xi=0$ and $F=0|0.01$. However we can do a more
thorough analysis. Suppose $\xi=0.4$ and $F=1$, we have
$E_{0,-1}=-77114.1$a.u. For $\xi=2$ and $F=1$, we have $E_{0,-1}=-350.462$a.u.
Such a large discrepancy in the energies reveal that the effect of AB flux
field is more dominance on the system than the electric field. Consequently,
we focus on $\xi$ and we found that for $\xi\rightarrow 0.5$, for any $F$,
$E_{0,-1}\rightarrow-\infty$. In addition, for $\xi=0.7$ and $F=1$, we have
$E_{0,-1}=-19284.4$a.u. This indicates that the lowest $E_{nm}$ would be
obtained when $m=-1$ and $\xi\rightarrow 0.5$.

Now, to study the entanglement fidelity in the interaction process under the
influence of external electric field and in a strongly coupled semiclassical
plasmas, let’s define the entanglement fidelity (which is the overlap between
maximally entangled state) integrated over the internal coordinates $\vec{R}$
and $\vec{r}$ as JA11
$f(\tau)=\left|\left.\left\langle\psi_{me}(\vec{R},\vec{r})\right.\right.\right|\left.\left.\psi_{ee}(\vec{R},\vec{r},\tau)\right\rangle\right|^{2}$
where

$\displaystyle\psi_{ee}(\vec{R},\vec{r},\tau)$ $\displaystyle=$
$\displaystyle(2\pi)^{-3/2}\exp\left[-i\left(\frac{P^{2}}{2M}+E\right)\tau\right]$
(5) $\displaystyle\times\exp(i\vec{P}\cdot\vec{R})\Psi(\vec{r})$

denotes the wavefunction of the bipartite interaction process, $\vec{P}$ is
the momentum of the electron-electron interaction and

$\psi_{me}(\vec{R},\vec{r})=(2\pi)^{-3/2}\int
d^{3}k\exp\left[i\vec{k}\cdot\vec{r}_{1}\right]\exp(i\vec{k}\cdot\vec{r}_{2}).$
(6)

With these expressions, the entanglement fidelity can be written as square of
the scattered wave function for a given model; i.e.,
$f_{\vec{k}}\propto\left|d^{3}{\bf r}\Psi_{\vec{k}}({\bf r})\right|^{2}$ JA11
, where $\Psi_{\vec{k}}({\bf r})$ is the solution of the scattered wave
function which can be expressed in terms of partial wave expansion as

$\Psi_{\vec{k}}({\bf
r})=(2\pi)^{-3/2}\sum_{m+\xi=\frac{1}{2}}^{\infty}(2m+2\xi)i^{g}\mathcal{T}^{k}_{g}R_{g}^{k}(r)P_{g}(cos\theta),$
(7)

where the wave number is $k=\sqrt{2\mu E/\hbar^{2}}$, $g=m+\xi+1/2$,
$\mathcal{T}^{k}_{g}$ represents the expansion coefficient, $R_{g}^{k}(r)$
denotes the solution of the radial wave equation and the Legendre polynomial
is denoted as $P_{g}(cos\theta)$. For a spherical symmetric potential
($\chi_{\rm eff.}^{(1)}$), the following expressions

$\displaystyle\mathcal{T}^{k}_{g}=\left[1+\frac{2i\mu
k}{\hbar^{2}}\int_{0}^{\infty}j_{g}(kr)\chi_{\rm
eff.}^{(1)}R_{g}^{k}(r)r^{2}dr\right]^{-1}\ \ \ \mbox{and}$
$\displaystyle\left[\frac{d^{2}}{dr^{2}}+\frac{2}{r}\frac{d}{dr}+k^{2}-\frac{2\mu}{\hbar^{2}}\left(\chi_{\rm
eff.}^{(1)}+\frac{\hbar^{2}}{2\mu}\left[\frac{g(g-1)}{r^{2}}\right]\right)\right]R_{g}^{k}(r)$
$\displaystyle=0,$ (8)

hold for the expansion coefficient and the radial wave equation respectively.
The eigenfunction $R_{g}^{k}(r)$ can be expressed in terms of spherical
Neumann function and spherical Bessel function as follows:

$\displaystyle R_{g}^{k}(r)$ $\displaystyle=$ $\displaystyle
j_{g}(kr)+\frac{2\pi
k}{\hbar^{2}}\left[n_{g}(kr)\int_{0}^{r}drr^{2}j_{g}(kr)\chi_{\rm
eff.}^{(1)}R_{g}^{k}(r)\right.$ (9)
$\displaystyle\left.+j_{g}(r)\int_{r}^{\infty}drr^{2}n_{g}(kr)\chi_{\rm
eff.}^{(1)}R_{g}^{g}(r)\right].$

As earlier stated, the lowest energy can be obtained by setting $m=-1$ and
$\xi\approx 0.5$, hence, we write the entanglement fidelity for low-energy
elastic interaction by using equations (8) and (9) to find

$\displaystyle f_{\vec{k}}$ $\displaystyle\propto$
$\displaystyle\frac{\left|\int_{0}^{\infty}d{r}r^{2}R_{0}^{k}(r)\right|^{2}}{1+\left|\frac{2\mu
k}{\hbar^{2}}\int_{0}^{\infty}d{r}r^{2}\chi_{\rm
eff.}^{(1)}R_{0}^{k}(r)\right|^{2}}$ (10) $\displaystyle\propto$
$\displaystyle\frac{\left|\int_{0}^{\infty}d{r}r^{2}\frac{\sin(kr)}{kr}\right|^{2}}{1+\left|\frac{2\mu
k}{\hbar^{2}}\int_{0}^{\infty}d{r}r^{2}\chi_{\rm
eff.}^{(1)}\frac{\sin(kr)}{kr}\right|^{2}}.$

Figure 1: Variation of fidelity ratio for electron-electron interaction under
the influence of electric field in atomic units using the fitting parameters
$\alpha=2=r_{s}=2$ $\Gamma_{ee}=1$, $\delta=5$, $\Gamma_{ee}=1$ (a) as a
function of the scaled projectile energy for various values of F. (b) as a
function of the scaled projectile energy for various values of the density
parameter and with $F=0.02$(c) Same as (b) but for $F=0.2$.

Thus, we express the entanglement fidelity for electron-electron interaction
under the influence of electric field and surrounded by strongly coupled
semiclassical plasmas, in terms of the ratio of entanglement fidelity for
model 3 to that of pure Coulomb potential as:

$\displaystyle R_{K}^{(\lambda_{ee},\lambda_{D})}$ $\displaystyle=$
$\displaystyle\frac{1+\left|-\frac{2\mu
Z^{2}e^{2}k}{\hbar^{2}}\int_{0}^{\infty}drr\frac{\sin(kr)}{kr}\right|^{2}}{1+\left|{-\frac{2\mu\mathcal{M}k}{\hbar^{2}}\int_{0}^{\infty}drr^{2}\left[\left(A_{-}^{2}\frac{e^{-Ar}}{r}-A_{+}^{2}\frac{e^{-Br}}{r}\right)-Fr+\frac{\omega_{c}\hbar}{2}\left(\sigma_{0m}-\frac{1}{2}\right)+\left(\frac{\mu\omega_{c}^{2}}{8}\right)r^{2}\right]\frac{\sin(kr)}{kr}}\right|^{2}}$
(11) $\displaystyle=$
$\displaystyle\left(\epsilon+1\right)\left({\epsilon+\epsilon^{2}\tilde{\mathcal{M}}^{2}\left[\frac{A_{-}^{2}}{A_{+}^{2}+k^{2}}-\frac{A_{+}^{2}}{A_{-}^{2}+k^{2}}-\frac{2F}{k^{4}}\right]^{2}}\right)^{-1},$

where $\epsilon$ is the projectile energy and it is given by
$\epsilon=E/(13.6Z_{e}^{4})$ and
$\tilde{\mathcal{M}}=2\mu^{2}z^{2}e^{2}\mathcal{M}/\hbar^{4}$. In figure 1, we
have plotted the variation of $R_{K}^{(\lambda_{ee},\lambda_{D})}$ as a
function of projectile energy. In (a) we found that for a weak electric field,
the fidelity ratio for electron-electron interaction increase as projectile
energy increase but remains constant or almost zero for a strong electric
field. That is to say, a weak electric field intensifies entanglement fidelity
in a strong coupled semiclassical plasmas. We use the result of (a) to proceed
to (b) (i.e. we fixed $F$ at $0.02$) by studying the fidelity ratio as a
function of projectile energy for various values of $\alpha$ which is average
distance between the particles. As it can be seen, increasing the distance
between the electrons has infinitesimal effect on the fidelity ratio. Moreover
distorting the intensity electric field (say, $F=0.02\rightarrow 0.2$) will
results into a large discrepancy. These results provide us a valuable
information on how the efficiency of entanglement fidelity for a low-energy
elastic electron-electron interaction in a strong semiclassical plasma can be
influenced by the presence of external fields.

##  III Concluding Remarks

We found that to maintain a low-energy elastic electron-electron interaction
in strongly semiclassical plasma, weak external fields are prerequisite. The
entanglement fidelity in the interaction process has been explored. We have
used partial wave analysis to derive the entanglement fidelity. We found that
for a low electric field, the entanglement fidelity varies with projectile
energy. Our results provide a valuable information on how the efficiency of
entanglement fidelity for a low-energy elastic electron-electron interaction
in a strong semiclassical plasma can be influenced by the presence of external
field.

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