Linearization and Control of Series-Series Compensated ... - MDPI

energies Article

Linearization and Control of Series-Series Compensated Inductive Power Transfer System Based on Extended Describing Function Concept Kunwar Aditya and Sheldon Williamson * Department of Electrical, Computer and Software Engineering, University of Ontario Institute of Technology, Oshawa, ON L1H 7K4, Canada; [email protected] * Correspondence: [email protected]; Tel.: +1-905-721-8668 (ext. 5744) Academic Editor: Jin Ye Received: 9 August 2016; Accepted: 11 November 2016; Published: 17 November 2016

Abstract: The extended describing function (EDF) is a well-known method for modelling resonant converters due to its high accuracy. However, it requires complex mathematical formulation effort. This paper presents a simplified non-linear mathematical model of series-series (SS) compensated inductive power transfer (IPT) system, considering zero-voltage switching in the inverter. This simplified mathematical model permits the user to derive the small-signal model using the EDF method, with less computational effort, while maintaining the accuracy of an actual physical model. The derived model has been verified using a frequency sweep method in PLECS. The small-signal model has been used to design the voltage loop controller for a SS compensated IPT system. The designed controller was implemented on a 3.6 kW experimental setup, to test its robustness. Keywords: chargers; energy storage; inductive energy storage; power electronics; resonant power conversion; and transportation

1. Introduction Inductive Power Transfer (IPT) is receiving wide interest in wireless charging of electric vehicles (EVs), due to advantages such as: opportunity charging; battery volume reduction; safety; visual appeal due to removal of cord; and relieving the user from handling a bulky and heavy charging chord [1]. An IPT system consists of two coils: a primary coil placed on the surface, connected to a power supply; and a secondary coil, placed underneath the vehicle, connected to the load. Since power transfer takes place due to the mutual coupling of coils, an optimum coupling between both coils is required for an efficient power transfer. This requires the vehicle to be parked in a specific position [2,3]. However, while parking the vehicle over the primary coil, certain misalignments could always occur. Due to misalignments, coils could deviate from the optimal coupling scenario. Moreover, the parameters of the system could vary due to reasons such as: fluctuation in supply voltage; and variation of load due to varying state of the charge of the battery pack. These issues can lead to the deviation of output voltage and current from the desired operating points. A suitably designed closed-loop controller is consistently needed, in order to improve tolerance to misalignments and parameter variations. Recent papers mainly present steady-state models and experiments [4,5]. Steady-state models cannot accurately predict the dynamic behavior of a system; therefore, they cannot be used as a tool for controller and physical system design. For the design of control loops, dynamic analysis of the IPT system is essential [6]. An IPT system is a typical higher-order resonant circuit, and consists of one slow moving pole, due to the output filter (rectifier + capacitor) and fast moving poles, and due to the resonant tank elements. This causes a difference in frequency between the resonant network and the filter network. This

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relegates usage of techniques modelling techniques typically used for PWM converters DC/DC converters [7]. Therefore, usage ofthe modelling typically used for PWM DC/DC [7]. Therefore, in due in due course of various time, various specialized techniques have developed been developed for dynamic analysis of course of time, specialized techniques have been for dynamic analysis of such such resonant systems. the popular techniques include: Generalized state-space averaging resonant systems. SomeSome of theof popular techniques include: Generalized state-space averaging (GSSA) (GSSA) [7], sampled modelling (SDM) technique extendeddescribing describingfunction function (EDF) methodmethod [7], sampled data data modelling (SDM) technique [8],[8], extended (EDF) method method [9], [9], and and general general unified unifiedphasor phasortransformation transformation(GUPT) (GUPT)[10]. [10]. While While each each method method has has its its own own advantages advantagesand andlimitations, limitations,ititisisnot notthe theintention intentionof ofthis thispaper paperto tocompare comparethem. them.The Theaim aim of of the the paper paper merelyisistotoderive deriveananaccurate accurate simplified small-signal model of compensated SS compensated system using merely simplified small-signal model of SS IPT IPT system using the the EDF method. derived model be used to design a voltage control EDF method. The The derived model will will be used to design a voltage control loop.loop. TheEDF EDFmethod methodhas has been preferred other methods due its reputation of generating The been preferred overover other methods due to its to reputation of generating highly highly accurate mathematical of resonant converters. The EDF technique introduced accurate mathematical model model of resonant converters. The EDF technique was was firstfirst introduced by by Yang in [11]. 1991The [11].most Therecent most work recentand work and advancement on method the EDFhas method has been Yang et al.etinal. 1991 advancement on the EDF been presented presented in [12,13]. Thistechnique modelingworks technique on the basic principle of approximation first harmonic in [12,13]. This modeling on theworks basic principle of first harmonic approximation of This state in variables. This ina turn a setfunctions of modulation functions that variables relate the of state variables. turn provides set ofprovides modulation that relate the state state the variables the input and control variables, which are both in time-domain and frequencywith input with and control variables, which are both in time-domain and frequency-domain [13]. domain The isEDF method is highly accurate but requires higher orderand representation and The EDF [13]. method highly accurate but requires higher order representation therefore more thereforeformulation more complex formulation effort. For instance, a small signal model of IPT SS compensated IPT complex effort. For instance, a small signal model of SS compensated system has been system has been derived inthe [14–17] the ninth order system. derived in [14–17] which is ninthwhich order is system. In [18], authors model of SS IPT IPT system. The In authors derived derivedaasimplified simplifiedmathematical mathematical model of compensated SS compensated system. derived model is aisfifth order system asas compared The derived model a fifth order system comparedtotothe theninth ninthorder order model model derived derived by others. However,only onlysimulation simulationresults resultswere werepresented presentedand andthe thepaper paper lacks experimental verification. However, lacks experimental verification. AsAs a a follow-up to that paper, the authors present the detailed analysis of the derived model in this paper. follow-up to that paper, the authors present the detailed analysis of the derived model in Detailedanalysis analysisincludes includesverification verificationof ofsmall smallsignal signalmodel modelusing usingAC AC sweep sweep method method and and verification verification Detailed of designed designed controller controller for for its its robustness robustness using usingexperimental experimental results. results. Targeted Targeted application application is is static static of chargingofofEVs. EVs.For For of application, thesystem IPT system hassecondary single secondary coil charging thisthis typetype of application, the IPT usuallyusually has single coil drawing drawing power from the singlecoil. primary to the coupled nature of primary the circuit, power from the single primary Due tocoil. the Due coupled nature of the circuit, sideprimary control side has control has been preferred to avoid side DC-DC converterlosses and associated with it. been preferred to avoid secondary sidesecondary DC-DC converter and associated with it. Anlosses experimental An experimental setup ofcharger a 3.6 kW charger built in lab to operate at frequency a nominal setup of a 3.6 kW wireless haswireless been built in thehas labbeen to operate at the a nominal switching switching frequency was of 40 kHz. based Frequency was selected on DSP module and switching of 40 kHz. Frequency selected on DSP module andbased switching elements available in the lab elements available in the lab at the time. However, the idea presented in the paper is applicable to at the time. However, the idea presented in the paper is applicable to the entire power and frequency theinterest entire related power and frequency of interest related to IPT system for EV battery charging. of to IPT system for EV battery charging. 2. Equivalent Circuit Circuit Derivation Derivation for for an an SS SS Compensated Compensated IPT System 2. Equivalent A compensated IPT IPTsystem systemunder underthe theprimary primary side control is shown in Figure 1. A typical typical SS compensated side control is shown in Figure 1. An An H-Bridge inverter in the primary side converts input into high frequency voltage/current H-Bridge inverter in the primary side converts thethe DCDC input into high frequency voltage/current for for resonant tank. diodebridge bridgerectifier rectifieron onthe the secondary secondary side side converts thethe resonant tank. AA diode converts the high high frequency frequency voltage/current back to to the the DC DC value value required by the load resistance R Roo.. voltage/current back

Figure 1. 1. Series-series Series-series compensated compensated inductive inductive power powertransfer transfersystem. system. Figure

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Figure 1 will be referred to as the ‘full model’ of SS compensated IPT system in this paper from Figure will be referred to as theof ‘full of SS compensated system dynamic in this paper from now on. To1 simplify the derivation themodel’ small-signal model, first IPT a reduced model is now on. To simplify the derivation of the small-signal model, first a reduced dynamic model is obtained from the full model. From the reduced dynamic model a small signal model has been obtained full model. From the reduced dynamic modeland a small signal model has required, been derived. derived. from This the two-step derivation reduces the complexity computation effort for This two-step derivation reduces the complexity and computation effort required, for applying EDF applying EDF for the derivation of the small-signal model. The following discussion entails the for the derivation of the small-signal model. The following discussion entails the derivation of the derivation of the reduced dynamic model as follows: reduced as follows: It isdynamic assumedmodel that the secondary coil is resonating at the switching frequency = and It is assumed that the secondary coil is resonating at the switching frequency ωs = √ L1 C and the s s . should be less than to the primary coil resonating at the resonant frequency = primary coil resonating at the resonant frequency ωo = √ L1 C . ωo should be less than ωs to maintain P P maintain ZVS ininverter. H-bridge inverter. Under this assumption, equivalent circuit of be Figure 1 can ZVS in H-bridge Under this assumption, equivalent circuit of Figure 1 can redrawn as be in redrawn as in Figure 2. Figure 2.

Figure 2. 2. Equivalent circuit of of series-series series-series topology. topology. Figure Equivalent circuit

Applying Kirchhoff’s voltage Law (KVL) to circuit shown in Figure 2: Applying Kirchhoff’s voltage Law (KVL) to circuit shown in Figure 2:  + = di + + + ( ) VAB = L P P + Vm + R p + n2 RS i P + sgn(i P )nV0 dt = dVm CP = iP Applying Kirchhoff’s current law (KCL) todt the rectifier network of Figure 1: Applying Kirchhoff’s current law (KCL) to the rectifier network of Figure 1: + = | | dVo Vo cf + = |iS | dt RL Output equation is given by:

(1) (1) (2) (2)

(3) (3)

= (4) Vo = Vc f (4) Equations (1)–(4) gives the ‘reduced dynamic model’ of the SS compensated IPT system. Here, . Meaning of gives symbols are given in Nomenclature = Equations (1)–(4) theused ‘reduced dynamic model’ of thesection. SS compensated IPT system. Here, ωs M n = R + R L . Meaning of symbols used are given in Nomenclature section. Output equation is given by:

S

3. Small-Signal Modeling 3. Small-Signal Modeling The equivalent circuit shown in Figure 2 is a greater simplification over the equivalent circuit The in equivalent Figure 2 is a greater simplification over the equivalent circuit derived [14–17]. circuit This isshown similarinto the equivalent circuit of a series-loaded resonant DC/DC derived inSince, [14–17]. and This is are similar to sinusoidal the equivalent circuit of aprovided series-loaded resonant DC/DC almost (due to filtering by resonant circuit) they converter. converter. Since, i and V are almost sinusoidal (due to filtering provided by resonant circuit) they m P can be approximated to their fundamental component using harmonic approximation. Hence, can be approximated to their fundamental component using harmonic approximation. Hence, = cos( ) + sin( ) (5) Vm ==Vx cos( cos(ωs t)) + sin(ωs t)) + Vysin(

(5) (6)

VAB can be modelled using the ifundamental ) + Iy sin(ωsof t) the inverter output waveform since (6) P = Ix cos( ωs tcomponent in EDF only the first harmonic component is of prime importance [12]. There are mainly three narrow VAB can be modelled using the fundamental component of the inverter output waveform since in frequency-range control techniques used for the control of resonant converters, namely asymmetrical EDF only the first harmonic component is of prime importance [12]. There are mainly three duty cycle (ADC), asymmetrical clamped mode (ACM), and symmetrical clamped modenarrow (SCM) frequency-range control techniques used for the control of resonant converters, namely asymmetrical control. It has been shown in [19–21] that only ACM control has the best performance in terms of the least losses in the inverter switches as compared to other two control strategies and has the lowest

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duty cycle (ADC), asymmetrical clamped mode (ACM), and symmetrical clamped mode (SCM) control. It has been in [19–21] that only ACM control has the best performance in terms of the4least Energies 2016, 9,shown 962 of 16 losses in the inverter switches as compared to other two control strategies and has the lowest switching frequency frequency for obtaining under ZVS all load conditions. controlACM has been considered in switching forZVS obtaining under all load Therefore, conditions.ACM Therefore, control has been this paper for the inverter control. considered in this paper for the inverter control. Figure 3 shows a typical switching logic and the inverter output voltage for ACM control. control.

Figure Figure 3. 3. Switching Switching scheme scheme and and voltage voltage waveform waveform of of the the ACM ACM control. control.

VAB in term of duty cycle d (= VAB in term of duty cycle d (=

ton) is given by Equation (7). T ) is given by Equation (7).

(7) = − V sin 2 cos + V (3 + cos 2 ) sin VAB = − dc sin2πdcosωs t + dc (3 + cos2πd) sinωs t (7) π π Using EDF non-linear terms = ( ) is written as function of and . Hence, Using EDF non-linear terms VCD = sgn(i P )nV0 is written as function of i P and Vc f . Hence, 4 4 = sin + cos (8) Iy 4n Ix 4n VCD = V V sinωs t + cosωs t (8) π c f IP π c f IP Here, =q + = (9) 2 2 (9) IP = Ix + Iy = i p Put Equations (5)–(8) in Equations (1) and (2), and by equating the coefficients of DC, sine, and cosine terms respectively, can get: (1) and (2), and by equating the coefficients of DC, sine, and Put Equations (5)–(8)one in Equations cosine terms respectively, one can get: 4 ) − = − −( + + (3 + cos 2 ) (10)   dIy I 4n V y LP = L P Ix ωs − Vy − R P + n2 RS Iy − 4Vc f + dc (3 + cos2πd) (10) dt π IP −π sin 2 ) − =− − −( + (11) Here,

LP

  dIx 4n Ix V = − L P Iy ωs − Vx − R P + n2 RS Ix − Vc f − DC sin2πd dt π IP π = −

(11) (12)

CP

dVx = Ix − Vy CP ωs dt = +

(12) (13)

CP

dVy = Iy2+ Vx CP ωs − dt =

(13) (14)

dVc f 2nI Vo C f model = of Pthe − SS compensated IPT system in terms of (14) Equations (10)–(14) give the large-signal the dt π Ro state variables. It contains both the steady-state model and the small-signal model. One can observe from Equations (10)–(14) the large-signal of the SSascompensated IPT system in terms of the large signal model thatgive it is constructed of fivemodel state variables opposed to the nine variables used in the state variables. It contains both the steady-state model and the small-signal model. One can } and have been mentioned in Appendix A. [14–17]. The operating point is determined by { observe from the large signal model that it is constructed state variables The state variable vector and control/input variables vector of canfive be expressed as: as opposed to the nine

=

(15) =

(16)

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variables used in [14–17]. The operating point is determined by {Ix Iy Vx Vy } and have been mentioned in Appendix A. The state variable vector and control/input variables vector can be expressed as: →

X=

h

Ix

→

U=

Iy h

Vx

Vdc

Vy

Vc f

iT

(15)

iT

d

(16)

ωs

The small-signal model can be derived by introducing ac perturbation, represented by ‘~’ in input variables vector and state variables resulting in: →

e ωs = ωs + ω edc , d = d + d, es x = X + xe, vdc = Vdc + V Separating perturbations from the DC and very small signals gives the small-signal model of SS topology: de x = Ae x + Bue (17) dt ye = Ce x + D ue

(18)

Here,      A=   

−1 LP



4n π IP Vc f

+ R P + n2 R s



− ωs −1 LP

ωs 1 CP



4n π IP Vc f

+ R P + n2 R s



0

0 nIP π Ix C f



1 CP nIP πIy C f

dc − sin2πd − 2V πL P L P cos2πd  1 2V dc   πLP (3 + cos2πd) − LP sin2πd B= 0 0   0 0  0 0   ei x    ei  edc V  y   e    ue =  d  xe =  vex     vey  es ω vec f h i ec f , C = 0 0 0 0 1 & D ye = V

− Iy Ix −Vy Vx 0

=0

−1 LP

0

0

−1 LP

0 ωs 0

− ωs 0 0

−4nIx π IP L P −4nIy π IP L P

0 0 −1 Ro C f

        

       

(19)

4. Verification of Small-Signal Model and Design of Control Loops A prototype of the SS compensated IPT system was built in the lab. Table 1 gives the parameters of the experimental setup. Using theories presented in [19–21], and depending on availability of capacitors in the market, fS was calculated to be 41.426 kHz for ensuring ZVS for any variation of α, and the primary coil was tuned to a resonant frequency of 39.031 kHz. The controller was designed based on fixed frequency and variable duty ratio control. Some of the latest work, presented in the literature, related to the controller design for an IPT system can be read in [22–25]. Following subsection discusses a step by step procedure for derivation of a voltage controller.

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Table 1. Parameters of Series-Series Compensated IPT System. Parameters

Values

Vdc (V) 340 Lp (µH) 400.65 LS (µH) 101.10 M (µH) 40.23 Rp (Ω) 0.13 Rs (Ω) 0.06 Cp (nF) 41.5 Energies 2016, 9, 962 6 of 16 CS (nF) 146 fo (kHz) 39.031 The controller was designed based on fixed frequency and variable duty ratio control. Some of fs (kHz) 41.426 the latest work, presented in the literature, related to the controller Cf (uF) 220 design for an IPT system can be Rated load, Ro (Ω) a step by step7.84 read in [22–25]. Following subsection discusses procedure for derivation of a voltage Air-gap (cm) 16 controller. Rated output (kW) 3.6

4.1. Bode Plot of Open-Loop System 4.1. Bode Plot of Open-Loop System Transfer function of output variable to the control variable (i.e., plant) can be obtained using Transfer Equation (20).function of output variable to the control variable (i.e., plant) can be obtained using Equation (20). (20) (s) −A A))−1BB + GGP (s) ==CC(sI (sI − + DD (20) Here, GP(s) represents plant transfer function, Ve0. Figure 4 compares the open loop Bode plot Here, GP (s) represents plant transfer function, e . Figure 4 compares the open loop Bode plot d obtained small-signal model, model of of the the SS SS compensated compensated obtained from from small-signal model, reduced reduced dynamic dynamic model model and and full full model IPT system. The Bode plot of the full model and the reduced dynamic model in open loop have have been IPT system. The Bode plot of the full model and the reduced dynamic model in open loop obtained by simulating these models in PLECS 3.7.5 and applying an AC sweep for a few discrete been obtained by simulating these models in PLECS 3.7.5 and applying an AC sweep for a few points. discrete points.

Figure Open loop Figure 4. 4. Open loop Bode Bode plot plot of of

e0 V de

for 0.8. for dd = = 0.8.

In Figure 4, one can observe that the AC sweep result for the reduced dynamic model closely follows the Bode plot of the small-signal small-signal model. Both 12°◦ at the Both models models have have the phase margin of 12 crossover frequency of 10.2 kHz. The gain cross-over frequency lies to the left of the phase crossover frequency which indicates indicatesaastable stableopen-loop open-loopsystem. system.The Thegain gaincrossover crossoverfrequency frequency the full model frequency which ofof the full model is is also near to kHz 10 kHz the phase crossover frequency less the than thecrossover gain crossover frequency also near to 10 but but the phase crossover frequency is lessisthan gain frequency which which the system actual system unstable fordisturbances small disturbances in the dutyload, cycle, load, and input DC makes makes the actual unstable for small in the duty cycle, and input DC voltage. voltage. The Bode plot of full model closely follows the small-signal model up to 4 kHz frequency. Here the aim is to design the control loop for output voltage control which is DC. As a rule of thumb, the bandwidth of the closed-loop system is selected 5 to 10 times the highest frequency being controlled. Therefore, the derived small-signal model is useful for the design of voltage control loop.

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The Bode plot of full model closely follows the small-signal model up to 4 kHz frequency. Here the aim is to design the control loop for output voltage control which is DC. As a rule of thumb, the bandwidth of the closed-loop system is selected 5 to 10 times the highest frequency being controlled. Therefore, the derived model is useful for the design of voltage control loop. Energies 2016, 9,small-signal 962 7 of 16 4.2. Derivation Derivation of of Closed-Loop Closed-Loop Voltage Voltage Controller Controller 4.2. The design design of of voltage voltage loop, loop, shown shown in in Figure Figure 5, 5, involves involves defining defining the the voltage voltage loop loop quantitatively quantitatively The and should meet the design criteria of phase margin (PM) and bandwidth (BW) or crossover crossover and should meet the design criteria of phase margin (PM) and bandwidth (BW) or frequency, ffcc.. frequency,

Figure 5. 5. Voltage Voltage control control loop. Figure

Here, Here, G Gcc(s) (s) isis the the transfer transfer function functionof of PI PI regulator. regulator. Voltage Voltage loop loop transfer transfer function function (LTF) (LTF) is is then then given by, LTF(s) = G c(s) GP(s). The following steps were followed in deriving the transfer function of given by, LTF(s) = Gc (s) GP (s). The following steps were followed in deriving the transfer function of the the controller: controller: 1. and type type of of controller: controller: 1. Select Select phase phase margin, margin, cross-over cross-over frequency, frequency, and A A bandwidth bandwidth (f (fcc))of of40 40Hz Hzisisselected. selected.The Thephase phasemargin margin(PM) (PM) of of85° 85◦isischosen chosento toprovide provideadequate adequate damping to the the closed-loop closed-loopsystem systemfor fora asudden sudden change duty cycle. Since slope of open the open damping to change in in duty cycle. Since the the slope of the loop loop plant is 0 dB/dec at the selected crossover frequency, therefore, a simple PI regulator is sufficient plant is 0 dB/dec at the selected crossover frequency, therefore, a simple PI regulator is sufficient to to regulate output voltage. The transfer function controller is given Equation (21): regulate thethe output voltage. The transfer function of of thethe PI PI controller is given byby Equation (21): 1+s G (s) = K 1 + sτ Gc (s) = KPI s sτ

(21) (21)

2. Calculate the needed Phase boost: 2. Calculate the needed Phase boost: From the Bode plot of the open loop plant GP(s): | ( )| = 45 dB and ∠ ( ) = −11° . From PI thecontroller Bode plot openaloop plant f c )| = 45 dB. dB Here, and ∠Gs ( f c ) = −11◦ . | GPa(gain Therefore, hasoftothe present phase lead G ofP (s):and of 45 Therefore, PI controller has to present a phase lead of φ and a gain of 45 dB. Here, = PM − (180 + ∠G (f )) = −84° (22) ◦ = PM −frequency (180 + ∠Giss (given fc ) = by −84Equation The required phase boost at φ crossover (23):

boost = + 90 6° by Equation (23): The required phase boost at crossover frequency is = given 3. Calculate the time constant, :

boost = φ + 90 = 6◦

3. Calculate the time constant, τ:

tan

2 f

= boost

(22)

(23) (23)

(24)

∴ = 4.182 × 10−4. tan−1 2πfc τ = boost 4. Calculate the controller gain at the crossover frequency: ∴ τ =At4.182 × 10−4 . crossover frequency the LTF should cross 0 dB line, i.e., the desired

20log (LTF)fc = 0 dB

(24)

(25)

∴ KPI = 0.00054. Therefore PI controller becomes: G (s) = 0.00054

1 + 0.0004182 0.0004182

Figure 6 shows the Bode diagram of the designed PI controller.

(26)

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4. Calculate the controller gain at the crossover frequency: At the desired crossover frequency the LTF should cross 0 dB line, i.e., 20log (LTF)fc = 0 dB

(25)

∴ KPI = 0.00054. Therefore PI controller becomes: Gc (s) = 0.00054 shows EnergiesFigure 2016,9,9,6 962 Energies 2016, 962

1 + 0.0004182s 0.0004182s

the Bode diagram of the designed PI controller.

(26) of16 16 88of

Figure 6. Bode plot of designedPI PIcontroller, controller,G Gcc(s). (s). Figure6. 6.Bode Bodeplot plotof ofdesigned designed PI controller, Figure G c

FromFigure Figure 6, onecan can observethat thatat at40 40Hz, Hz,the thephase phaseboost boostprovided providedby bythe thePI PIcontroller controllerisis6°. 6°. From From Figure6,6,one one canobserve observe that at 40 Hz, the phase boost provided by the PI controller is Moreover, a gain of –45 dB provided by the PI controller will make the gain of LTF zero at 40 Hz, i.e., Moreover, a gain of –45 by the will will make the gain of LTF zerozero at 40atHz, i.e., 6◦ . Moreover, a gain ofdB –45provided dB provided byPI thecontroller PI controller make the gain of LTF 40 Hz, crossover frequency of 40 Hz is achieved. Figure 7 shows the Bode plot of the loop gain obtained crossover frequency of 40ofHz is achieved. Figure 7 shows the gain obtained obtained i.e., crossover frequency 40 Hz is achieved. Figure 7 shows theBode Bodeplot plotof ofthe the loop loop gain usingthe thesmall-signal small-signalmodel model(i.e., (i.e.,LTF(s)) LTF(s))as aswell wellas asthe theloop loopgain gainobtained obtainedusing usingthe theac acsweep sweepof offull full using using the small-signal model (i.e., LTF(s)) as well as the loop gain obtained using the ac sweep of full modelsimulated simulated in PLECS3.7.5. 3.7.5. model model simulatedininPLECS PLECS 3.7.5.

Figure 7. Bode plots forloop looptransfer transferfunction. function. Figure Figure7. 7.Bode Bodeplots plotsfor for loop transfer

From Figure Figure 7, 7, one one can can observe observe that that the the gain gain crossover crossover frequency frequency of of 40 40 Hz Hz and and PM PM of of 85° 85° has has From been achieved for the derived small-signal model as well as for the full model (precisely 87° in ac been achieved for the derived small-signal model as well as for the full model (precisely 87° in ac sweep). Moreover, Moreover,for for the theclosed-loop closed-loopsystem, system, the thelocation location of of gain gaincross-over cross-overfrequency frequency isis to to the the left left sweep). of the phase crossover frequency for both the derived model as well as the full model. Therefore, the of the phase crossover frequency for both the derived model as well as the full model. Therefore, the closedloop loopsystem systemisisstable. stable. closed

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From Figure 7, one can observe that the gain crossover frequency of 40 Hz and PM of 85◦ has been achieved for the derived small-signal model as well as for the full model (precisely 87◦ in ac sweep). Moreover, for the closed-loop system, the location of gain cross-over frequency is to the left of the phase crossover frequency for both the derived model as well as the full model. Therefore, the closed loop system is stable. 5. Simulation and Experimental Results In this section, simulation results and experimental results for the closed-loop voltage control have been compared for different cases. Designed controller was implemented on the IPT system whose parameters have been mentioned in Table 1. Simulation results have been obtained from PLECS3.7.5 standalone. For experimental results, the closed-loop voltage controller was implemented in real-time using dSPACE® ControlDesk 5.3. LEM sensor LV20-P was used to sense the output voltage. A 63800 series (63804) programmable AC/DC electronic load from Chroma system solutions was Energies 2016, 9, 962 9 of 16 used as the DC load. XR Series power supplies from Magna-Power Electronics were used as the programmable programmable DC DC power power supply. supply. The The inverter inverter was was built built using using full-bridge full-bridge MOSFET MOSFET power power module module (APTM120H29FG) from Microsemi (Aliso Viejo, CA, USA). For the rectifier, low loss (APTM120H29FG) from Microsemi (Aliso Viejo, CA, USA). For the rectifier, low loss fast fast recovery recovery diode diode module module from from IXYS IXYS (Milpitas, (Milpitas, CA, CA, USA) USA) (DSEI2X101-06A) (DSEI2X101-06A) was was used. used. To To build build the the coils, coils, Litz Litz wire wire Type II 8 AWG 5 × 5 × 42/32 has been used. Outer diameter of both the primary and secondary coil Type II 8 AWG 5 × 5 × 42/32 has been used. Outer diameter of both the primary and secondary coil is is 47 The inner inner diameter diameter of of primary primary coil coil is is 22 22 cm cm whereas whereas inner inner diameter diameter of of secondary secondary coil coil is is 36 36 cm. cm. 47 cm. cm. The The The experimental experimental setup setup is is shown shown in in Figure Figure 8. 8. Different Different test test cases cases are arediscussed discussedbelow. below.

Figure 8. 8. Experimental Experimental setup. setup. Figure

Case I—Change in load at fixed reference voltage: Case I—Change in load at fixed reference voltage: To test the dynamic performance of the designed controller, a step change in the DC load was To test the dynamic performance of the designed controller, a step change in the DC load was performed at the fixed reference voltage. For this purpose, the reference voltage was set at 168V and performed at the fixed reference voltage. For this purpose, the reference voltage was set at 168 V and load was stepped down from 8.84 Ω (3192.76 W) to 11.56 Ω (2441.5 W). Figure 9 shows the transient load was stepped down from 8.84 Ω (3192.76 W) to 11.56 Ω (2441.5 W). Figure 9 shows the transient performance of simulated results, while experimental results are shown in Figure 10. From the results, it can be observed that when the step increase in load is applied, output voltage overshoots and the controller takes corrective action to bring it back to 168 V in about 12 ms in experimental result and 11 ms in simulation result. Voltage overshoot in simulation and experimental results is 9% and 13% respectively.

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performance of simulated results, while experimental results are shown in Figure 10. From the results, it can be observed that when the step increase in load is applied, output voltage overshoots and the controller takes corrective action to bring it back to 168 V in about 12 ms in experimental result and 11 ms in simulation result. Voltage overshoot in simulation and experimental results is 9% and 13% respectively. Energies 2016, 9, 962 10 of 16 Energies 2016, 9, 962

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Figure Figure 9. 9. Simulation Simulation results results for for step step change change in in load. load. Figure 9.

Figure in load. Figure 10. 10. Experimental Experimental results results for for step step change change in load.

Case load: II—Change in in reference reference voltage voltage at fixed Case II—Change voltage at at fixed fixed load: load: Reference voltage was was stepped stepped down down from from 168 168 V to 92 7.84Ω Ωload loadresistance. resistance. Reference voltage V (rated (rated Voltage) Voltage) to 92 V V at at 7.84 7.84 Ω load resistance. Figure 11 shows Figure 11 shows the the transient transient performance performance of of simulated simulated results, results, while while experimental experimental results results are are shown shown in Figure 12. in Figure 12.

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Figure Figure 11. 11. Simulation Simulationresults resultsfor for step step change change in in reference reference voltage. voltage. Figure 11. Simulation results for step change in reference voltage.

Figure 12. Experimental results for step change in reference voltage. Figure Figure 12. 12. Experimental Experimentalresults results for for step step change change in in reference reference voltage. voltage.

From From the the results, results, it it can can be be observed observed that that the the controller controller takes takes approximately approximately 14 14 ms ms to to reach reach steady steady From the results, it can be observed that the controller takes approximately 14 ms to reach steady state for the simulation and 18 ms for the experimental results. No transient undershoot/overshoot state for the simulation and 18 ms for the experimental results. No transient undershoot/overshoot state for thetosimulation and 18 ms for the experimental results. No transient undershoot/overshoot occurs of occurs due due to damping damping provided provided by by the the high high PM PM (85°) (85°) of controller. controller. ◦ occurs due to damping provided by the high PM (85 ) of controller. Case III—Tracking performance of controller for variation in DC input voltage: Case Case III—Tracking III—Tracking performance of controller for variation in DC input voltage: To test the robustness of the designed controller for the variations in DC input voltage, the To To test the robustness of the designed controller controller for for the the variations variations in in DC DC input input voltage, voltage, the the reference voltage was kept 160 V at 10 Ω load and ±10% variation in DC input voltage (340 V) was reference ±10% reference voltage voltage was was kept kept 160 160 V at 10 Ω load and ± 10%variation variationin inDC DC input input voltage voltage (340 (340 V) V) was was introduced manually. Results obtained are shown in Figure 13. introduced introduced manually. manually. Results obtained are shown in Figure 13.

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Figure 13. Experimental results for variation variationin inDC DCinput inputvoltage. voltage. Figure13. 13.Experimental Experimental results results for Figure variation in DC input voltage.

From Figure Figure 13, it it can be be seen that that the controller controller keeps keeps the the output output voltage voltage fixed fixed at at a reference reference From From Figure13, 13, itcan can be seen seen that the the controller keeps the output voltage fixed at aareference voltage in in spite of of variations in in the DC DC input voltage. voltage. Since Since the the load load is is fixed, fixed, the the output output current current also also voltage voltage inspite spite ofvariations variations in the the DC input input voltage. Since the load is fixed, the output current also remains constant. remains remainsconstant. constant. CaseIV—Tracking IV—Tracking performance performance of Case IV—Tracking performance controllerfor forvariation variationinin inmutual mutualcoupling: coupling: Case of controller controller for variation mutual coupling: Totest testthe thecontroller controllerperformance performanceduring during variation mutual coupling between primary To variation in in mutual coupling between thethe primary and To test the controller performance during variation in mutual coupling between the primary and and secondary coils, alignment of secondary with respect to primary wasmanually varied manually in the secondary coils, alignment of secondary with respect to primary was varied in the sequence secondary coils, alignment of secondary with respect to primary was varied manually in the sequence [0 cm–5 cm–10 cm–15 The reference voltage load was168 kept andΩ [0sequence cm–5 cm–10 cm–10 cm–15 cm–0 cm].cm–0 The cm]. reference voltage and and the and loadthe was kept V 168 andV7.84 7.84 [0 cm–5 cm–15 cm–0 cm]. The reference voltage the load was kept 168 V and Ω 7.84 Ω respectively. obtained are shown in Figure 14. respectively. Results Results obtained are shown shown in Figure Figure 14. respectively. Results obtained are in 14.

Figure 14. Experimental results for change changein inmutual mutualcoupling. coupling. Figure14. 14.Experimental Experimental results results for Figure for change in mutual coupling.

From Figure Figure 14, one one can observe observe that that the the output output voltage voltage remains remains fixed fixed in in spite spite of of the variation variation From From Figure14, 14, one can observe that the output voltage remains fixed in spite of the the variation in in the mutual inductance between the primary and the secondary coil. inthe themutual mutualinductance inductance between primary and secondary between thethe primary and thethe secondary coil.coil. 6. Verification Verification of of ZVS ZVS in in INVERTER INVERTER Switches Switches 6. To verify verify for for the the ZVS ZVS in in the the inverter inverter switches, switches, the the method method described described in in [21] [21] has has been been used. used. To According to [19], polarity of current at four switching instance (t 0, t1, t2, t3) shown in Figure 3 should According to [19], polarity of current at four switching instance (t0, t1, t2, t3) shown in Figure 3 should be according according to to the the Equation Equation (27) (27) for for the the switches switches to to operate operate under under ZVS ZVS condition. condition. be

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6. Verification of ZVS in INVERTER Switches To verify for the ZVS in the inverter switches, the method described in [21] has been used. According to [19], polarity of current at four switching instance (t0 , t1 , t2 , t3 ) shown in Figure 3 should be according to the Equation (27) for the switches to operate under ZVS condition. Energies 2016, 9, 962

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IP (t0 ) < 0 For S1 ( ) < 0 For S1 IP ((t1)) > > 00 For For S3 S3 For S2 S2 IP ((t2)) > > 00 For

(27) (27)

IP ((t3)) < < 00 For For S4 S4

Figure 15a 15a shows the primary voltage and the primary current waveform for 168 V, 7.84 Ω Ω load, Figure (i.e., 3.6 kW) thethe primary voltage andand primary current for 92 Ω, (i.e., (i.e., kW) output. output.Figure Figure15b 15bshows shows primary voltage primary current forV,927.84 V, 7.84 Ω, 1.08 kW). (i.e., 1.08 kW).

(a)

(b)

Figure 15. 15. (a) (a) Waveform Waveform for for 168 168 V V reference reference voltage, voltage, 7.84 7.84 Ω Ω load; load; (b) (b) Waveform Waveform for for 92 92 V V reference reference Figure voltage, 7.84 7.84 Ω Ω load. load. voltage,

From Figure 15, one can observe that current full fills the ZVS condition defined by Equation From Figure 15, one can observe that current full fills the ZVS condition defined by Equation (27) (27) for the rated load (3.6 kW) condition, as well as, for the partial load (1.08 kW) condition. Table 2 for the rated load (3.6 kW) condition, as well as, for the partial load (1.08 kW) condition. Table 2 gives gives the DC to DC efficiency for all the cases discussed previously. the DC to DC efficiency for all the cases discussed previously. It should be noted that, although inverter output voltage is asymmetric, the inverter output is being applied to a series resonant which acts for as sharply tuned band pass filter. The primary Table 2. tank DC-DC Efficiency All the Test Cases. capacitor will block the DC component from flowing into primary coil inductor. Therefore, primary current and hence the secondary current beIDC almost can be observed in Cases Pin =will VDC × (Watts)sinusoidal. Pout = VoThis × Io (Watts) η (%) experimental result as shown the×primary current is sinusoidal. RO = 8.84inΩFigure 15 that 340 10.32 168 × 19 90.97 I. Vref = 168 V

RO = 11.56 Ω

168 × 14.53

89.76

II. RO = 7.84 Ω

for All the Test Cases. Vref = Table 168 V 2. DC-DC Efficiency 340 × 11.6 168 × 21.42 Vref = 92 V 340 × 3.54 92 × 11.73

91.2 89.66

340 × 8

Pout = Vo × Io (Watts) η (%) Pin = VDC × IDC (Watts) 306 × 9.24 160 × 16 90.54 340 × 10.32 168 × 19 90.97 374 × 7.54 160 × 16 90.78 340 × 8 168 × 14.53 89.76 340 × 8.3 160 × 16 90.71 Vref = 168 V 340 × 11.6 168 × 21.42 91.2 II. RO = 7.84 Ω Misalignment = 5 cm 340 × 11.72 168 × 21.42 90.30 IV. Vref = 168 V Vref = 92 V 340 × 3.54 92 × 11.73 89.66 Misalignment = 10 cm 340 × 11.92 168 × 21.42 88.79 RO = 7.84 Ω VDC ==306 V 306 × 9.24 160 × 16 90.54 Misalignment 15 cm 340 × 12.52 168 × 21.42 84.53 III. RO = 10 Ω VDC = 374 V 374 × 7.54 160 × 16 90.78 Vref = 160 V VDC = 374 V 340 × 8.3 160 × 16 90.71 It should be noted that, although inverter output voltage is asymmetric, the inverter output is Misalignment = 5 cm 340 × 11.72 168 × 21.42 90.30 IV. Vref = 168 V being applied to a series resonant=tank as sharply pass filter. The primary Misalignment 10 cmwhich acts340 × 11.92 tuned band168 × 21.42 88.79 Ω block the DC component from flowing into primary coil inductor. Therefore, primary RO = 7.84will capacitor Misalignment = 15 cm 340 × 12.52 168 × 21.42 84.53 Cases

III. R = 10 Ω I. VrefO = 168 V Vref = 160 V

VDC = 306 V RO = 8.84 Ω VDC = 374 V R O = 11.56 Ω VDC = 374 V

7. Conclusions In this paper, a simplified small-signal model of an SS compensated IPT system has been presented. To derive the small signal model, a reduced dynamic model was first derived considering

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current and hence the secondary current will be almost sinusoidal. This can be observed in experimental result as shown in Figure 15 that the primary current is sinusoidal. 7. Conclusions In this paper, a simplified small-signal model of an SS compensated IPT system has been presented. To derive the small signal model, a reduced dynamic model was first derived considering the ZVS in the inverter switches. Although the ZVS has been considered, a fairly accurate mathematical model can also be derived for the zero current switching (ZCS) conditions. AC sweep results show that the frequency response of the derived small-signal model follows the frequency response of the actual full model with proximity in low-frequency region. Therefore, a derived model can be used for the design of voltage control loop. From the derived model an output voltage controller was designed. The voltage controller was implemented on a 3.6 kW IPT system built in the lab. The IPT system was designed for the frequency and voltage level depending on the equipment’s (load, power supply) ratings available in the lab. However, the idea presented in this paper applies to the standard defined by SAE J2954. A PI controller was designed using the derived small-signal model. Although the controller was designed considering 0.8 duty cycle operation at 7.84 Ω load resistance, it shows good tracking capability for different tests conditions such as: different load resistance; change in DC input voltage; change in reference voltage; and change in coupling coefficient. Therefore, it can be said that the derived controller is very robust against parameter variations. Results obtained show that the derived small signal-model is useful for the design of control loops. Author Contributions: Kunwar Aditya did the analysis, build the hardware, performed the experiments and wrote the paper. Sheldon Williamson as research supervisor provided guidance as well as funding for the research and key suggestions for writing this paper. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature Vdc VAB VCD LP CP LS CS M Cf Ro IP IS iP Vm |iS | Io Vcf fs fo ωs ωo α

DC input to inverter VP = Output of inverter/Primary Voltage Input voltage to rectifier Primary Self inductance Primary capacitor Secondary Self inductance Secondary capacitor Mutual Inductance Filter capacitor Load resistance Primary Peak current Secondary Peak current Primary current Peak Voltage across capacitor Rectified current flowing into output filter network output current Vo = Output voltage switching frequency, Hz resonant frequency in Hz switching frequency in rad/s Resonant frequency in rad/s Conduction angle

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Appendix A Operating Points: Ix = −

Beta Vdc (3 + cos2πd) − al pha sin2πd al pha π ( al pha2 + Beta2 )

Ix = CP Vy ωs Iy = −CP Vx ωs Iy al pha = −

Vdc sin2πd − Ix Beta π

R P + n2 R s +

8n2 Ro = Beta π2

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