Download Analysis of Series-Parallel Resonant Inductive Coupling Circuit

IECON2015-Yokohama
November 9-12, 2015
Analysis of Series-Parallel Resonant Inductive Coupling
Circuit using the Two-port Network Theory
Kunwar Aditya, Student Member, IEEE, Mohamed Youssef, Senior Member, IEEE,
and Sheldon S. Williamson, Senior Member, IEEE
Smart Transportation Electrification and Energy Research (STEER) Group
Advanced Storage Systems and Electric Transportation (ASSET) Laboratory
UOIT-Automotive Center of Excellence (UOIT-ACE)
Department of Electrical, Computer, and Software Engineering
Faculty of Engineering and Applied Science
University of Ontario-Institute of Technology
ACE-2025, 2000 Simcoe Street North
Oshawa, ON L1H 7K4, Canada
Tel: +1/ (905) 721-8668, ext. 5744
Fax: +1(905) 721-3178
EML: [email protected]
URL: http://www.engineering.uoit.ca/; http://ace.uoit.ca/
Abstract²In this paper thorough analysis of Series-Parallel
resonant inductive coupling (SPRIC) has been presented using
two-port network theory. Characteristic of SPRIC has been
derived and plotted against load variation and frequency
variation in MATLAB. Concept of forced resonance and natural
resonance has been discussed. Important expressions such as
current transfer ratio, voltage transfer ratio, maximum efficiency
etc. has been derived and presented. To study the circuit
behavior of the SPRIC a prototype has been built in the lab.
Experimental results along with simulation results has been
presented to verify the derived expression and characteristics of
series-parallel topology.
Keywords²Circuit theory, electric vehicles, inductive energy
storage, magnetic resonance, quality factor.
I. INTRODUCTION
Inductive power transfer (IPT), which while theorized in
the 1800s, has only seen relatively recent use in the last 15-20
years most notably for EVs battery charging application [1, 2].
Inductive coupling can provide power to the battery and at the
same time provide galvanic isolation and hence are
advantageous than conventional energy transmission
techniques using wires and connectors from the point of safety,
reliability, low maintenance, and long product life [3, 4]. It is
EDVHG RQ IXQGDPHQWDO SULQFLSOH RI DPSHUH¶V circuital law and
IDUDGD\¶V ODZ of electromagnetic induction on which
conventional transformers and induction motor works.
However, in case of transformer and induction motor primary
and secondary are placed in close proximity therefore coupling
coefficient is high usually 90-98%. For battery charging
application via IPT system the large air gap is desired to
minimise protrusion from either side and to allow good
clearance between the vehicle base and the ground therefore
leakage flux is more and hence is low (usually 10-20%) [5].
Poor coupling in loosely coupled system leads to poor transfer
of power. To improve coupling and compensate leakage
inductance, capacitive compensation in primary and secondary
978-1-4799-1762-4/15/$31.00 ©2015 IEEE
windings is required. For this purpose capacitor can be
connected in either series or parallel of both winding giving
total of four topology namely Series-Series (SS) topology,
Series-Parallel (SP) topology, Parallel-Series (PS) topology
and Parallel-Parallel (PP) topology. Since capacitor is made to
resonate with self-inductance of primary and secondary coil it
is more appropriate to call it resonant inductive coupling rather
than merely inductive coupling.
All topology has certain advantage and disadvantage and
their choice mainly depends on type of application. There has
been research to find out most suitable topology for particular
application such as battery charging [6-10]. While there are
other topologies that may be used, the parallel secondary
architecture is beneficial for battery charging because of its
constant current source characteristics, which occurs if primary
current is maintained constant [11-13]. In this paper analysis of
Series-Parallel resonant inductive coupling has been done
using two port network with the aim of understanding its
circuit behavior when fed from constant input voltage. For the
analysis a prototype of SP resonant coils has been built in the
lab. Absence of core makes it similar to linear transformer and
therefore linear circuit theory can be applied for the analysis.
Paper has been arranged in following manner: Section II
explains the concept of forced resonance and natural
resonance, in section III characteristics of SP topology has
been derived, In section IV efficiency and condition for
maximum efficiency has been discussed, in section V
simulation and experimental results has been presented,
Section VI concludes the paper.
II. FORCED RESONANCE AND NATURAL RESONANCE
Secondary compensation is done to improve the power
transfer capability of the system and Primary capacitor is so
chosen that the impedance as seen from the source side is
purely resistive in nature so as to ensure that the input current
and voltage are in phase, and therefore it reduces the VA
005402
rating of supply [14]. Fig. 1. Shows the equivalent circuit of
series-parallel topology.
Ip
Rp
Cp
Is
Rs
Ls
z12Is
Ic
z21Ip
Cs RL
D
R1
Fig.2. Series-Parallel RC equivalent circuits
Here,
߱௢ ൌ
ଵ
(7)
ඥ஼ೞ ௅ೄ
(1)
ଵାఠమ ோಽమ ஼ೞమ
ଵାఠమ ோಽమ ஼ೞమ
Resonant frequency for loop ABCD is given by [15]:
߱௢ is the natural resonant frequency of the secondary. This
resonant frequency is independent of load and is fixed for
selected frequency. To analyze the behavior of this circuit it
can be seen as a series RLC circuit operating at resonance in
which output is picked off the secondary capacitance, Cs in
order to take advantage of the fact that at resonance the
amplitude of the voltage across the capacitor is Q (= ఠோ೚௅ೞ )
ೞ
times the amplitude of the source voltage . Fig. 4 (a) and (b)
gives the phasor diagram of Fig. 5 with and without capacitor
Cs for same load. Secondary resistance has been neglected in
drawing these phasor for clear understanding of circuit.
Rs
ோಽ
C
Fig. 3. Secondary equivalent circuit of SP topology
C1
Cs
Vs
Cs RL
Vs
Parallel Cs and RL can be replaced by its series equivalent of C1
and R1 as shown in fig.2:
ఠమ ோಽమ ஼ೞమ
Io
IR
Fig.1. Equivalent circuit of Series-Parallel Topology
‫ܥ‬ଵ ൌ
Is
Rs
j߱MIp
Vp
ܴଵ ൌ
B
A
j߱IsLs
‫ܥ‬௦
Ic
Is
V
(2)
Then total impedance of secondary is given by:
‫ ்ݖ‬ൌ
ଵ
௝ఠ஼భ
൅ ݆߱‫ܮ‬௦ ൅ ܴ௦ ൅ ܴଵ
(3)
At forced resonance frequency ߱oF, ‫ ்ݖ‬should be purely
resistive. µ)¶ VXEscript is used to denote values at forced
resonant frequency. Equating imaginary terms in eq. (3) gives
the forced resonant frequency as:
߱௢ி ൌ ට
ଵ
௅ೞ ஼ೞ
െ
ଵ
ோಽమ ஼ೞమ
‫ݖ‬ோିி ൌ
ோభ ାோೞ
(5)
Since reflected impedance is purely resistive, capacitive
compensation in primary, Cp-F is independent of either Mutual
inductance or load and is given by:
‫ܥ‬௣ିி ൌ
ଵ
మ ௅
ఠ೚ಷ
೛
(a)
j߱IsLs
V
(4)
This type of tuning can be used in case of fixed load system.
With ߱oF, reflected impedance is purely resistive and is given
by:
మ
ఠ೚ಷ
ெమ
Vs
Io
(6)
Therefore tuning in primary is not disrupted with the variation
of mutual inductance or load and system remains perfectly
tuned. However, one can observe from eq. (4) ߱௢ி is
dependent on load so value of Cs need to be adjusted each time
load changes and therefore to eliminate this problem, instead of
tuning entire secondary circuit we tune the loop ABCD as
shown in fig.3.
Io
Vs
(b)
Fig. 4. Phasor diagram of Fig. 3, (a) with capacitor Cs (b) without capacitor Cs
V is input voltage of circuit i.e. j߱MIp in this case. From phasor
diagram one can observe that for given load ȁܸȁ ൏ ȁܸ௦ ȁ for
compensated secondary (with Cs) and ȁܸȁ ൐ ȁܸ௦ ȁ for
uncompensated secondary (without Cs). This means that for a
given load, addition of Cs reduces the input voltage. From
practical point of view this means if load increases parallel
compensation can be done to maintain the same load voltage
without increasing the stress on input voltage. Value of primary
capacitance at natural resonance frequency, ߱௢ is given as:
‫ܥ‬௣ ൌ
௅మೞ ஼ೞ
൫௅೛ ௅ೞ ିெమ ൯
ൌ
ଵ
మ ௅ ሺଵି௞ మ ሻ
ఠ೚
ು
(8)
In the derivation of ‫ܥ‬௣ effect of primary and secondary coil
resistance has been neglected.
005403
However since
ଵ
ඥ௅ೞ ஼ೞ
൐
ଵ
ට௅ೞ ஼ೞ ିோಽమ ஼ೞమ
secondary will not operate
at unity factor.
ோಽ
߱௢ is selected as resonant frequency rather than ߱௢ி .
Values of Primary and Secondary Capacitance
Natural
Forced resonance
Resonance
ͳ
߱௢ி ‫ܮ‬௣
ͳ
߱௢ଶ ‫ܮ‬௉ ሺͳ െ ݇ ଶ ሻ
ଶ ଶ
ͳ
‫ܮ‬௦
Ͷ߱௢ி
ቌͳ ൅ ඨͳ െ
ቍ
ଶ
ܴ௅ଶ
ʹ߱௢ி ‫ܮ‬௦
ͳ
߱௢ଶ ‫ܮ‬௦
Primary
Capacitance
Secondary
Capacitance
III. CHARACTERISITCS OF SERIES-PARALLEL TOPOLOGY
Fig.5 gives the two-port model for SP topology shown in Fig.1
in terms of z parameters.
Ip
z11
ܴଵ ൌ
‫ܥ‬ଵ ൌ
ோಽ
ଵାఠమ ோಽమ ஼ೞమ
ଵାఠమ ோಽమ ஼ೞమ
ఠమ ோಽమ ஼ೞమ
z12Is'
Vp
z21Ip
Vs
R1
Two Port Network
Fig.5. Two-Port model of SP topology
ଵ
௝ఠ஼೛
൅ ݆߱‫ܮ‬௣ ൅ ܴ௣
(9)
‫ݖ‬ଵଶ ൌ ‫ݖ‬ଶଵ ൌ ݆߱‫ܯ‬
(10)
TABLE II.
(14)
‫ܥ‬௦
(15)
Using z-parameters, equations that describes the circuit in
Fig.5 can be written as:
(16)
ܸ௣ ൌ ‫ݖ‬ଵଵ ‫ܫ‬௣ ൅ ‫ݖ‬ଵଶ ‫ܫ‬௦ᇱ
(17)
ܸ௦ ൌ ‫ݖ‬ଶଵ ‫ܫ‬௣ ൅ ‫ݖ‬ଶଶ ‫ܫ‬௦ᇱ
(18)
ܸ௦ ൌ െ‫ܫ‬௦ᇱ ܼ௅
Solving equations (16) to (18) gives the characteristics of SP
topology. These characteristics at any frequency ߱ and at
resonance frequency ߱o has been arranged in Table 2. To
minimize the losses, the IPT coils are designed to have
winding resistance as low as possible and are usually in the
milliohm range. If that is the case then their product ܴ௦ ܴ௣ ̱Ͳ
and can be neglected. So from table 2 one can write:
‫ݖ‬௜௡ ൌ
‫ݖ‬௜௡ ൌ
Is'
z22
Here, C1 and R1 are series equivalent of parallel Cs and RL and
is given by:
ଵ
൅ ݆߱௢ ‫ܮ‬௣ ൅
௝ఠ೚ ஼೛
ெమ ோಽ
C1
Here,
‫ݖ‬ଵଵ ൌ
(11)
(12)
(13)
௝ఠ஼భ
Table 1 gives the value of primary and secondary capacitance
at forced resonance frequency ߱௢ி and natural resonance
frequency, ߱௢ . From Table 1 one can observe that at forced
resonance frequency value of primary capacitance can become
imaginary at certain combination of frequency and load
ఠ ௅
resistance i.e. at certain load quality factor ( ೚ ೞ ). Therefore
TABLE I.
Values of
Capacitors
‫ݖ‬ଶଶ ൌ ݆߱‫ܮ‬௦ ൅ ܴ௦
‫ܫ‬௦ᇱ ൌ െ‫ܫ‬௦
ଵ
‫ݖ‬௅ ൌ ܴଵ ൅
௅మೞ
ൌ ݇ଶ
௅ು
௅ೄ
ெమ ோಽ
௅మೞ
െ
௝ఠ೚ ெమ
ܴ௅
(19(a))
௝ఠ ெమ
RHIOHFWHG LPSHGDQFHDW UHVRQDQFH LH µ- ೚ Ԣ component in
௅ೞ
Zin is capacitive component. 0RUHRYHU WKLV FDSDFLWLYH
UHDFWDQFH ZLOO RSSRVH WKH SULPDU\ LQGXFWDQFH /S FDXVLQJ D
UHGXFWLRQLQ=LQWKLVPHDQVDODUJHUFDSDFLWRUZLOOEHUHTXLUHG
WR WXQH RXW WKH FDSDFLWLYH 9$5 ORDGLQJ RQ WKH SULPDU\ E\
VHFRQGDU\ FRLO DV FRPSDUHG WR WKH YDOXH UHTXLUHG LQ VHULHV
VHULHVFRPSHQVDWLRQ>@
After neglecting winding resistance and using value of
primary and secondary capacitance at resonance voltagetransfer characteristics ݄௩ି௦௣ of SP topology at resonance can
Characteristics of Series-Parallel topology
Characteristic
At generalized frequency, ߱
At resonance frequency, ߱o
hi-sp = Is/Ip
‫ݖ‬ଶଵ
‫ݖ‬௅ ൅ ‫ݖ‬ଶଶ
݆߱௢ ‫ܯ‬ሺͳ ൅ ݆߱௢ ܴ௅ ‫ܥ‬௦ ሻ
ܴ௅ ൅ ሺܴ௦ ൅ ݆߱௢ ‫ܮ‬௦ ሻሺͳ ൅ ݆߱௢ ܴ௅ ‫ܥ‬௦ ሻ
hv-p=Vs/Vp
‫ݖ‬ଶଵ ܼ௅
‫ݖ‬௅ ‫ݖ‬ଵଵ ൅ ‫ݖ‬ଶଶ ‫ݖ‬ଵଵ െ ‫ݖ‬ଵଶ ‫ݖ‬ଶଵ
Zin=Vp/Ip
Gsp=Is/Vp
‫ݖ‬ଵଵ െ
‫ݖ‬ଵଶ ‫ݖ‬ଶଵ
‫ݖ‬௅ ൅ ‫ݖ‬ଶଶ
‫ݖ‬ଶଵ
‫ݖ‬௅ ‫ݖ‬ଵଵ ൅ ‫ݖ‬ଶଶ ‫ݖ‬ଵଵ െ ‫ݖ‬ଵଶ ‫ݖ‬ଶଵ
(19)
௅ೞ
݆߱௢ ‫ܯ‬
ሺ݆߱௢ ‫ܮ‬௣ ൅
ଵ
௝ఠ೚ ஼೛
ܴ௣ ൅ ݆߱௢ ‫ܮ‬௣ ൅
൅ ܴ௣ ሻሺ
ோಽ
ଵା௝ఠ೚ ஼ೞ ோಽ
ோಽ
ଵା௝ఠ೚ ஼ೞ ோಽ
൅ ܴ௦ ൅ ݆߱௢ ‫ܮ‬௦ ሻ ൅ ߱௢ଶ ‫ܯ‬ଶ
ͳ
߱௢ଶ ‫ܯ‬ଶ ሺͳ ൅ ݆߱௢ ‫ܥ‬௦ ܴ௅ ሻ
൅
݆߱௢ ‫ܥ‬௣ ܴ௅ ൅ ሺܴ௦ ൅ ݆߱௢ ‫ܮ‬௦ ሻሺͳ ൅ ݆߱௢ ‫ܥ‬௦ ܴ௅ ሻ
݆߱௢ ‫ܯ‬
ሺ݆߱௢ ‫ܮ‬௣ ൅
005404
ଵ
௝ఠ೚ ஼೛
൅ ܴ௣ ሻሺ
ோಽ
ଵା௝ఠ೚ ஼ೞ ோಽ
൅ ܴ௦ ൅ ݆߱௢ ‫ܮ‬௦ ሻ ൅ ߱௢ଶ ‫ܯ‬ଶ
be simplified as:
௏
௅
݄௩ି௦௣ ൌ ೞ ൌ ೞ
௏೛
ெ
(20)
From equation (20) one can observe that output voltage is
independent of load resistance RL and will be constant as long
as mutual inductance M is maintained constant and therefore
SP topology acts as ideal voltage source. Constant voltage
characteristics is useful in multiple outlets system and those
system which has intermediate dc bus such as modern variable
speed ac drives or for multiple outlet [12]. As opposed to SS
topology SP topology is not short circuit proof.
A. Characteristics with respect to frequency:
This subsection study the variation of characteristics w.r.t.
frequency for different load. As load resistance increases (i.e.
load decreases) load quality factor QL increases and primary
quality factor QP decreases as seen in eq. (21) and (22) [12].
ܳ௅ ൌ
ܳ௉ ൌ
ோಽ
(21)
ఠ೚ ௅ ೞ
ఠ೚ ௅ು ௅మೄ
ெమ ோಽ
(22)
Fig.6 shows the voltage transfer characteristics plotted w.r.t.
frequency. For low value of QL, characteristics has only one
peak since primary quality factor dominates and SP topology
behaves as single tuned circuit and is more sensitive to
frequency change. As the load quality factor increases
characteristics acquires U shape because topology behaves as
double tuned circuit. In the middle of U region, characteristics
becomes more sensitive to variation in load but less sensitive to
variation in frequency. If resonant tank are properly designed it
can give stable gain over a large frequency band (bandwidth)
as is evident from curve for QL =1. This also shows that at
resonant frequency voltage transfer characteristics is almost
insensitive to load variation which was established earlier
through equation (20).
Fig.7. Current Transfer Characteristics Vs Frequency
Fig.7 shows the plot of current transfer characteristics w.r.t.
frequency. It shows that at resonance frequency, hi-sp has
maximum value and is sensitive to load variation however as
frequency increases away from resonant frequency hi-sp
becomes independent of load variation. Fig.8 shows the plot of
total input impedance w.r.t. frequency. From the plot one can
observe that for low value of QL plot is similar to any series
RLC circuit where total impedance decreases as frequency
increases and becomes minimum at resonance frequency, and
henceforth increases as frequency increases. This is due to the
fact that at low QL primary quality factor dominates and circuit
behaves as single tuned circuit. However as QL increases Qp
decreases and circuit behaves as double-tuned circuit which
explain the nonlinear nature of plot around resonant frequency
for high QL.
Fig.8. Input impedance Vs Frequency
Fig.6. Voltage transfer characteristics Vs Frequency
B. Characteristics with respect to load:
In this subsection characteristics variation w.r.t. load variation
has been discussed.
005405
M (μH)
31.4
Rp Ÿ
0.2244
Rs Ÿ
0.1174
Cp (μF)
0.56026
Cs (μF)
2.468
fo(kHz)
21.5
Air-gap(mm)
25
Fig.9. Characteristics Vs load
Ip
From Fig.9 one can observe that hv-p is almost constant
irrespective of change in load resistance as was established
earlier in eq. (20). Zin and hi-sp increases as load increases.
Vdc
Efficiency of Series-Parallel resonant inductive coupling is
given by [3]:
ߟൌ
ூ೚మ
ோಽ
(21)
మ ሺ௥௘௦௜௦௧௔௡௖௘௢௙௣௥௜௠௔௥௬ା௥௘௙௟௘௖௧௘ௗ௥௘௦௜௦௧௔௡௖௘ሻ
ூು
ோಽ
మ
మ
మ మ
(22)
మ
ೃು ೃ
ಽ ೃು ೃು ೃ ೃ
ೃ ೃ
మೃ ೃ ೃ
ோೄ ାோಽ ା మ ೄమ ା ೄ మ ା ర మೄ ಽమ ା ಽమ ೄ మ ು ା ಽమ ೄ
ಾ
ഘ ಾ
ഘ ಾ
ഘ ಽ ಾ
ഘ ಽమ
೚
೚ ೄ
೚
೚ ೄ
Value of load resistance which gives maximum coupling
efficiency and value of maximum efficiency is given by eq.
(23) and (24) respectively.
ܴ௅ି௠௔௫ ൌ ߱௢ ‫ܮ‬ௌ ට
ଵାொమమ
ோಽష೘ೌೣ
ೃು ೃమ
ಽమ
ೃು
మೃ ೃ
ೄ
ೄ
ଶቆ మ మ ା మ ቇା൬ మೄ మು ାଵ൰ோಽష೘ೌೣ ାோೄ
ಾ
ഘ೚ ಾ
ഘ೚ ಾ
In eq. (23) K =
Lp
M
Ls
Cs
RL
Vs
ெ
ඥ௅ೄ ௅ು
, Q1 =
Fig. 10. Circuit diagram for simulation and experiment.
To verify the voltage-source characteristics of Series-Parallel
topology the SP topology was simulated in MATLAB and
Simulation results were experimentally verified. Fig. 10 gives
the circuit diagram for the circuit used in Simulation and
experiment and Fig.11 shows the experimental setup.
(23)
ଵା௄ொమ ொభ
ߟ௠௔௫ ൌ
Cp
VP
IR
Rs
EFFICIENCY OF SP TOPOLOGY
IV.
ߟ ൌ
Is
Rp
ఠ೚ ௅ು
ோು
(24)
and Q2 =
ఠ೚ ௅ೞ
ோೄ
are coupling
coefficient, intrinsic quality factor of primary circuit and
intrinsic quality factor of secondary circuit respectively.
V. SIMULATION AND EXPERIMENTAL RESULTS
For the proof of concept of theories presented in this paper
a prototype of air cored IPT coils to deliver 30 watt at 20 kHz
and 20mm air gap, has been built in the lab. At high frequency
operation equivalent series resistance (ESR) of wire and
therefore copper loss increases due to the skin effect and
proximity effects. To mitigate this loss, braided enamelled
conductors known as Litz wire must be used for making coils.
Therefore to build coils, Litz wire 90/38 SPN SN (90 strands,
38 awg each strand) has been used. For the experiment,
compensation capacitors with very low ESR Polypropylene
film type have been selected. Parameters of the coils along with
compensating capacitors has been shown in table 3.
TABLE III.
Fig.11. Experimental setup.
To show that SP topology gives load independent output
voltage when fed from constant input voltage and fixed
frequency supply, load was varied in the sequence [1.2, 2.21,
3.45, 4.57, 5.34, and 6.23] ohm while dc link of inverter was
fixed at 15 volts and switching frequency was kept 15 kHz.
Due to practical availability, value of capacitor used in
experimental setup were adjusted to Cp=0.56 μF and Cs=2.7
μF. Output voltage by input voltage (Vs/Vp) for simulation and
experiment has been plotted in fig.13.
Parameters of SP Topology
Parameters
Values
Lp (μH)
142.12
Ls (μH)
22.25
005406
Simulation
Experiment
Ideal
0,7
0,6
Vs/VP
0,5
0,4
0,3
0,2
0,1
0
Ϭ
Ϯ
ϰ
ϲ
ϴ
Load Resistance RL (ȍ)
Fig. 12. Voltage transfer characteristics of SP topology
In Fig.12 Ideal plot shows the voltage transfer
characteristics for the case when winding resistance has been
neglected. It shows that when winding resistance are neglected
SP topology behaves as ideal voltage source. However due to
coil resistance characteristics deviates from ideal condition as
is evident from simulation and experimental plot in Fig. 13.
There is difference in magnitude of voltage transfer
characteristics for simulation and experiment. This is due to
the fact that in simulation tuning was perfect in primary and
secondary coil however in experiment ideal tuning was not
possible due to practical availability of capacitors. Moreover
switching time delay, parasitic elements such as switches and
connecting wires inductance were not included in simulations
however they are unavoidable during experiment.
VI. CONCLUSIONS
In this paper characteristics of Series-Parallel topology has
been studied when primary is fed from constant input voltage.
To simplify the analysis concept of two-port network has been
utilized for deriving the characteristics of circuits. All the
characteristics has been plotted and discussed with respect to
load as well as frequency variations. One of the important
characteristics of Series-parallel topology has been found to
give constant-voltage output. This particular characteristics
has been validated using MATLAB simulation and hardware
implementation. Simulation and experimental results were
found to be in close agreement. Limiting factor to the theory
was found to be parasitic resistance such as coil resistance and
connecting wire resistance, as well as deviation form ideal
tuning condition. Characteristics not only establish the concept
about circuit but also help in establishing the behaviour of the
topology which in turn help in selecting the topology for a
particular application.
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APPENDIX A
NOMENCLATURE :
Vp= Primary voltage
LP= Primary Self inductance
CP= Primary capacitor
LS= Secondary Self inductance
CS= Secondary capacitor
RL= Load resistance
Ip= Primary current
Is= Secondary current
Io= Output current
Vs= Secondary voltage
hi-sp= Current transfer ratio of series-parallel topology
hv-sp= Voltage Transfer ratio of Series-Parallel topology
Zin= Total input impedance seen by source
fo = Resonant frequency
fsw = Switching frequency
߱ = Angular frequency
߱o= Angular resonant frequency
k=Coupling Coefficient
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