an1477, digital compensator design for llc resonant converter · · 2013-06-022012 microchip...

AN1477Digital Compensator Design for LLC Resonant Converter

ABSTRACT

A half-bridge LLC resonant converter with Zero VoltageSwitching (ZVS) and Pulse Frequency Modulation(PFM) is a lucrative topology for DC/DC conversion. ADigital Signal Controller (DSC) provides componentcost reduction, flexible design, and the ability to monitorand process the system conditions to achieve greaterstability. The dynamics of the LLC resonant converterare investigated using the small signal modeling tech-nique based on Extended Describing Functions (EDF)methodology. Also, a comprehensive description of thedesign for the compensator for control of the LLCconverter is presented.

INTRODUCTION

The LLC resonant converter topology, illustrated inFigure 1, allows ZVS for half-bridge MOSFETs, therebyconsiderably lowering the switching losses and improv-ing the converter efficiency. The control system designof resonant converters is different from the conven-tional fixed frequency Pulse-Width Modulation (PWM)converters. In order to design a suitable digitalcompensator, the large signal and small signal modelsof the LLC resonant converter are derived using theEDF technique.

FIGURE 1: LLC RESONANT CONVERTER SCHEMATIC

Author: Meeravali Shaik and Ramesh Kankanala Microchip Technology Inc.

D4

Q1

Q2

PWMOut put

DigitalCompensator

D1

A

Vin

B

C2

C1

D2

D3

Q1

Q2 dsPIC33FJXXGSXXX

V0R

irLs

Cs im

Lmnp

ns

ns

isp Io

rc

Cf

–

+

ADCDriverTransformers

R1

R2

2012 Microchip Technology Inc. DS01477A-page 1

AN1477

Conventional methods, such as State-Space Averaging(SSA), have been successfully applied to PWM switch-ing converters. In PWM switching converters, theswitch network is replaced by an average circuit modeland only low-frequency (DC) components are consid-ered while ignoring switching harmonics. In general,the large and small signal modeling of PWM switchingconverters is done by considering the output LC filter.Typically, the natural frequency (fo) of the output LCfilter is much lower than the switching frequency (fs).

In frequency controlled resonant converters, switchingfrequency is close to the natural frequency of the LCresonant tank. The inductor current and capacitor volt-age of the LC resonant tank, magnetizing current andprimary voltage of the transformer, contain switchingfrequency harmonics which must be considered toobtain an accurate model. Therefore, modeling is doneby considering magnetizing inductance (Lm), leakageinductance (Ls) and resonant capacitance (Cs). The Ls,Lm and Cs constitute the primary resonant components.

The small signal modeling approach, based on theEDF method, is generally applied to model LLC reso-nant converters as this method considers all switchingfrequency harmonics for accuracy. Using the EDF, it iseasy to obtain the commonly used transfer functions,such as control-to-output transfer function (Gvω) andline-to-output transfer function (Gvg).

SMALL SIGNAL MODELING OF LLC RESONANT CONVERTER

Resonant DC/DC converters are nonlinear systemsand a dynamic model is helpful to determine the linear-ized small signal model, and thereby, the systemtransfer functions for the Pulse Frequency ModulatedDC/DC converters.

The following seven-step process describes how toobtain the plant transfer functions for the PFM DC/DCconverters.

1. Time Variant Nonlinear State Equations

State equations are obtained by writing the circuitequations using Kirchhoff's Laws for each statevariable.

2. Harmonic Approximation

Quasi-sinusoidal current and voltage waveformsof the LLC resonant tank are resonant current(ir(t)), magnetizing current (im(t)) and voltageacross resonant capacitor (vcr(t)). Theseparameters are approximated to their fundamentalcomponents. The current and voltage of the outputfilter are approximated to their DC components.

3. Extended Describing Function (EDF)

A linear, stationary system responds to a sinusoidwith another sinusoid of the same frequency, butwith modified amplitude and phase. Thedescribing function method is used to represent anonlinear function in a linear manner byconsidering only the fundamental component ofthe response of the nonlinear system.

In this application note, higher order harmonicsare ignored as they are considered to benegligible. This principle of describing functions isextended to model resonant converters and it islabelled as EDF.

Using the EDF method, the discontinuous terms inthe nonlinear state equations are approximated totheir fundamental or DC components.

4. Harmonic Balance

The quasi-sinusoidal terms and the nonlineardiscontinuous terms obtained from the harmonicapproximation and EDF are substituted in thestate equations. The coefficients of DC, sine andcosine components are then separated to obtainthe modulation equations (an approximate largesignal model).

5. Obtaining the Steady-State Operating Point

A large signal model from the harmonic balance isused to obtain the steady-state operating point bysetting the derivative terms of harmonic balanceequations to zero. This is because the statevariables do not change with time in steady state.

6. Perturbation and Linearization of HarmonicBalance Equations

The large signal model obtained from theharmonic balance has nonlinear terms arisingfrom the product of two or more time varyingquantities. The linearized model is obtained byperturbing the large signal model equations abouta chosen operating point, and by eliminating thehigher order (nonlinear) terms.

7. State-Space Model

The state-space model of a continuous timedynamic system can be obtained from theperturbed and linearized model of the harmonicbalance equations, described in Step 6, to derivethe control-to-output transfer function.

DS01477A-page 2 2012 Microchip Technology Inc.

AN1477

Derivation of Nonlinear State Equations

A quasi-square wave voltage (vAB ), generated from theactive half-bridge network, is applied to the resonanttank of the LLC resonant converter, as illustrated inFigure 2.

FIGURE 2: EQUIVALENT CIRCUIT OF LLC RESONANT CONVERTER

The state equations are obtained in Continuous TankCurrent mode by using Kirchhoff’s Circuit Laws (KCL),as shown in Equation 1 through Equation 4.

EQUATION 1: RESONANT TANK VOLTAGE

In this application, the LLC resonant converter outputvoltage is regulated by modulating the switchingfrequency (ωs).

EQUATION 2: RESONANT TANK CURRENT

EQUATION 3: TRANSFORMER PRIMARY VOLTAGE

EQUATION 4: TRANSFORMER SECONDARY CURRENT

The output voltage (v0) is shown in Equation 5.

EQUATION 5: OUTPUT VOLTAGE

ir

V'cf

A

B

VAB

ipLs Cs

+ Vcr –

Lm

np:ns :ns

D3

D4

Io

VcfR Vo

Ts

Vin

rs

mi

Cf

rc

+

–+–

isp

vAB Ls

dirdt------- irrs vcr ip( )v'cf

sgn+ + +=

Where:

sgn(ip) = -1, if v’cf < 0

+1, if v’cf ≥ 0

ir Cs

dvcr

dt----------=

v'cfLm

dim

dt--------=

isp 1 rcR----+

Cf

dvcf

dt--------- 1

R--- vcf

+=

Where:

vo r'c abs× isp( )r'crc----- vcf

+=

r'c rc R=


AN1477

Applying Harmonic Approximation

The Fourier series decomposes periodic functions orperiodic signals into a sum of (possibly infinite) simpleoscillating functions (sines and cosines, or complexexponentials). Expressing the function (f(x)) as an infi-nite series of sine and cosine functions is shown inEquation 6.

EQUATION 6: GENERAL FOURIER EXPANSION

The primary side resonant tank parameters, ir(t), vc(t)and im(t), provided in Equation 7, are approximated totheir fundamental harmonics, and the output filter volt-age (vcf) is approximated to the DC component. Thederivatives of ir(t), vcr(t) and im(t) are shown inEquation 7.

EQUATION 7: FUNDAMENTAL APPROXIMATION OF PRIMARY TANK PARAMETERS

The parameters, sine component of resonantcurrent (is), cosine component of resonant current (ic),sine component of resonant capacitor voltage (vs),cosine component of resonant capacitor voltage (vc),sine component of magnetizing current (ims) and cosinecomponent of magnetizing current (imc) are slow timevarying components. Therefore, the dynamic behaviorof these parameters can be analyzed.

Figure 3 and Figure 4 illustrate the simulation wave-forms of the LLC resonant converter operating belowthe resonant frequency and continuous tank currentmode.

f x( ) a0 an nxsin bn nxcos+( )n 1=

∞

±=

f x( ) a1 xsin b1 xcos±=

Expressing f(x) by considering only the fundamental components and ignoring the DC component, and otherharmonic terms is :

a0 a1 xsin± a2 2xsin± a3 3x b1 x b2 2x b3 3xcos±cos±cos±sin±( )=

ir t( ) is t( ) ωstsin ic t( ) ωstcos–=

vcrt( ) vs t( ) ωstsin vc t( ) ωstcos–=

im t( ) ims

t( ) ωstsin imc

t( ) ωstcos–=

dirdt-------

dis

dt------- ωsic+ ωstsin

dicdt------- ωsis– ωscos t–=

dvcr

dt-----------

dvs

dt-------- ωsvc+ ωstsin

dvcdt

-------- ωsvs– ωscos t–=

dimdt

---------dims

dt----------- ωs imc+ ωs tsin

dimcdt

------------ ωsims–

ωscos t–=

Where:

ωs = switching frequency in radians/second


AN1477

FIGURE 3: SIMULATION WAVEFORMS OF LLC RESONANT CONVERTER

Time (ms)

Time (ms)

Time (ms)

Resonant Inductor Current

Resonant Capacitor Voltage

Input Voltage


AN1477

FIGURE 4: SIMULATION WAVEFORMS OF LLC RESONANT CONVERTER

Magnetizing Inductor Current

Output Filter Capacitor Voltage

Time (ms)

Time (ms)


AN1477

Applying Extended Describing Function (EDF)

Extended Describing Function is a powerful mathemat-ical approach for understanding, analyzing, improvingand designing the behavior of nonlinear systems.Every system is nonlinear, except in limited operatingregions.

The nonlinear terms provided in Equation 1 throughEquation 5, sgn(ip) * v’

cf and abs(isp) can be approxi-mated to their fundamental harmonic terms and DCterms.

The functions, f1(d, vin ), f2(iss, isp,v’cf ), f3(isc, isp, v’cf ) andf4(iss, isc ), are called EDFs. Where, iss, isc are the sine andcosine components of the transformer secondary cur-rent, and isp is the resultant current flowing in secondary.f1, f2, f3 and f4 are functions of the harmonic coefficientsof state variables at chosen operating conditions. TheEDF terms can be calculated by using the Fourierexpansion of nonlinear terms. The EDF approximationto nonlinear states is shown in Equation 8.

EQUATION 8: EDF APPROXIMATION

Figure 5 illustrates a typical switching waveform of ahalf-bridge inverter which is the input to the LLCresonant tank (θ = Dead Time, d = Duty Cycle).

FIGURE 5: OUTPUT SWITCHING WAVEFORM OF HALF-BRIDGE INVERTER

The fundamental output voltage of a half-bridgeinverter is shown in Equation 9.

EQUATION 9: OUTPUT VOLTAGE OF HALF-BRIDGE INVERTER

The switching waveform has an odd symmetry. There-fore, there is no cosine component (vec = 0, where vecis the cosine component of the output voltage of thehalf-bridge inverter) in the switching waveform and thesine component (ves) forms the fundamentalcomponent of vAB .

isp( ) v'cfsgn f2 iss isp v'cf

, , ωstsin f3 isc isp v'cf

, , ωstcos–=

vAB t( ) f1 d vin,( ) ωstsin=

isp f4 iss isc,( )=

2π0

d π

π

θ θ

Vin

ωt

f1 d vin,( ) 22π------ vin ωt( )sin× ωtd

θ

π θ–( )

=

f1 d vin,( )2vin2π

----------- ωt( )cos θπ θ–( )–=

f1 d vin,( )2vin

π----------- θcos

2vinπ

----------- π2--- dπ

2------–

cos×= =

f1 d vin,( )2vin2π

----------- θcos π θ–( )cos–[ ]=

Where:

ves = Sine component of the output voltageof half-bridge inverter

f1 d vin,( )2vin

π----------- π

2---d sin ves= =

θ π2--- dπ

2------–=


AN1477

The EDF approximation to the nonlinear transformerprimary voltage is shown in Equation 10.

EQUATION 10: EDF APPROXIMATION TO TRANSFORMER PRIMARY VOLTAGE

Harmonic Balance

Harmonic balance is a frequency domain method usedto calculate the steady-state response of nonlineardifferential equations. The term, “harmonic balance”, isdescriptive of the method, which uses the Kirchhoff'sCurrent Laws (KCL) written in the frequency domainand a chosen number of harmonics. Effectively, themethod assumes that the solution can be representedby a linear combination of sinusoids, and then balancescurrent and voltage sinusoids to satisfy the Kirchhoff'sLaws. The harmonic balance method is commonlyused to simulate circuits which include nonlinearelements.

Substituting Equation 9 and Equation 10 intoEquation 1 through Equation 5, and separating the DC,sine and cosine terms, Equation 11 throughEquation 13 are obtained.

EQUATION 11: SINE AND COSINE COMPONENTS OF TANK VOLTAGE

EQUATION 12: SINE AND COSINE COMPONENTS OF TANK CURRENT

EQUATION 13: SINE AND COSINE COMPONENT OF TRANSFORMER PRIMARY VOLTAGE

Only the DC term is considered for the output capacitorvoltage, as shown in Equation 14.

EQUATION 14: OUTPUT FILTER CAPACITOR VOLTAGE

The output voltage equation is shown in Equation 15.

EQUATION 15: OUTPUT VOLTAGE

4nπ

------ipcipp-------- vcf

vpc==

f2 iss isp

v'cf, ,

4π---

iss

isp------ v'cf

4π---

ips

ipp-------- v'cf

==

f3 isc isp v'cf, ,

4π---

isc

isp------ v'cf

4π---

ipcipp-------- v'cf

==

ipp ips

2ipc

2+=

4nπ

------ipsipp-------- vcf

vps==

Where:

vps, vpc = sine, cosine components of the

transformer primary voltage

ips, ipc = sine, cosine components of the

transformer primary current

ipp = resultant transformer primary current

iss, isc = sine, cosine components of the

transformer secondary current

isp = resultant current flowing in secondary

n = np/ns = transformer turns ratio

L= s

dis

dt------- ωsic+ rsis vs

4nπ

------ipsipp--------vcf

+ + +

ves Ls

dis

dt------- ωsic+ rsis vs vps+ + +=

vec Ls

dicdt------- ωsis– rsic vc vpc+ + +=

L= s

dicdt------- ωsis– rsic vc

4nπ

------ipc

ipp--------vcf

+ + +

is Cs

dvs

dt-------- ωsvc+ =

ic Cs

dvcdt

-------- ωsvs– =

Lm

dims

dt----------- ωsimc+ 4n

π------

ips

ipp-------- vcf

vps= =

Lm

dimcdt

------------ ωsims– 4n

π------

ipcipp-------- vcf

vpc= =

1rcR-----+

Cf

dvcfdt

---------- 1R---vcf

+ 2π--- isp=

v02π--- r'c i

sp

r'crc------

vcf+=


AN1477

Equation 11 through Equation 15 are the nonlinearlarge signal model of the LLC resonant converterpower stage and are illustrated in Figure 6. It is impor-tant to note that the input of Equation 12 through

Equation 15, vg, ωs, d, is slow varying quantities withrespect to the switching frequency. Therefore, themodulation equations can be easily perturbed andlinearized at chosen operating points.

FIGURE 6: LARGE SIGNAL MODEL OF LLC RESONANT CONVERTER

Deriving Steady-State Operating Point

Under steady-state conditions, the state variables ofthe modulation equations, Equation 12 throughEquation 14, do not change with time. For a chosenoperating point, the time derivatives in Equation 12through Equation 14 are set to zero and the steady-state values are obtained (shown in upper case letters).The transformer currents on the primary and secondarysides are shown in Equation 16.

EQUATION 16: TRANSFORMER CURRENTS

The output filter capacitor voltage can be calculated bysubstituting Equation 16 into Equation 14, as shown inEquation 17.

EQUATION 17: FILTER CAPACITOR VOLTAGE

The steady-state analysis for the tank current, resonantcapacitor voltage and magnetizing current are providedin Equation 18 through Equation 22.

ΩsLmims

imsCs

–+

+ –

+–+

–

–+

– +is rs Ls + Vs -

Cs

ips

LmVpsVes

–+

–+

Vpc

Lmi

Ω sCsV s

+ Vc –

Ω s LsisLsrsic

Vec

Cf2/p isp

rc

Vcf

VoR

ΩsLmimc

mc ipc

Io+

–

Ωs Ls icΩs Cs Vc

ipp ips

2ipc

2+=Primary Current:

isp

iss2

isc2+ n ips

2ipc

2+ nipp= = =

Where:n = np/ns = transformer turns ratio

Secondary Current:

vcfR

------ 2nπ

------ ipp2π--- i

sp= =

vcf 2n

π------ ippR=

vcf π

4n------ ippRe=

Re8π2-----n

2R=

V 'cfnVcf

π4--- IppRe= =

Where:

= reflected voltage of secondary on the primary

V 'cf

= equivalent load resistance referred to primary side


AN1477

Substituting the value of Equation 17 into the sinecomponent of tank voltage, the result obtained isshown in Equation 18.

EQUATION 18: SINE COMPONENT OF TANK VOLTAGE

Substituting the value of Equation 17 into the cosinecomponent of the tank voltage, the result obtained isshown in Equation 19.

EQUATION 19: COSINE COMPONENT OF TANK VOLTAGE

The steady-state values of sine and cosine compo-nents of the tank current can be obtained by equatingdvs/dt and dvc/dt to zero. The result is shown inEquation 20.

EQUATION 20: SINE AND COSINE COMPONENTS OF TANK CURRENT

Substituting the value of Equation 17 into the sinecomponent of magnetizing current, the result is shownin Equation 21.

EQUATION 21: SINE COMPONENT OF MAGNETIZING CURRENT

Substituting the value of Equation 17 into the cosinecomponent of magnetizing current, the result is shownin Equation 22.

EQUATION 22: COSINE COMPONENT OF MAGNETIZING CURRENT

Equation 19 through Equation 22 are arranged, asshown in Equation 23.

EQUATION 23: ARRANGEMENT OF STEADY-STATE EQUATIONS

vesLs

dis

dt------- ωsic+ rsis vs

4π---

ips

ipp--------v'cf

+ + +=

LsΩsIc rsIs Vs

4π---

Ips

Ipp-------- π

4--- Ipp Re+ + + Ves

2π---Vin= =

rsIs LsΩsIc Vs ReIps

+ + + 2π--- Vin=

IpsIs Ims

–=Where:

rs Re+( )Is LsΩsIc Vs ReIms

2π---Vin=–+ +

LsΩsIs– rsIc Vc4π---

IpcIpp-------- π

4--- Ipp Re Vec 0==+ + +

vec Ls

dicdt------- ωsis– rsic vc

4π---

ipcipp-------- v'cf

+ + +=

LsΩsIs– rsIc Vc IpcRe 0=+ + +

Ipc Ic Imc–=

LsΩsIs– rs Re+( )Ic Vc ImcRe–+ + 0=

Where:

Cs

dvsdt

-------- ωsvc+ is=

Is CsΩsVc 0=–

Ic CsΩsVs+ 0=

Cs

dvc

dt-------- ωsvs– ic=

Lmdims

dt---------- ωs imc+ 4n

π------

ips

ipp-------- vcf

××=

LmΩsImc ReIps0=–

ReIs LmΩsImc– ReIms– 0=

Lm

dimcdt

------------ ωsims– 4n

π------

ipcipp-------- vcf

=

L– mΩsIms( ) ReIpc 0=–

LmΩsIms ReIc ReImc–+ 0=

rs Re+( )Is LsΩsIc Vs ReIms–+ + 2

π---Vin Ves= =

Is CsΩsVc– 0=

Ic CsΩsVs+ 0=

ReIs LmΩsImc– ReIms– 0=

LmΩsImsReIc ReImc–+ 0=

LsΩsIs– rs Re+( )Ic Vc ImcRe–+ + 0 Vec= =


AN1477

To obtain the tank current, capacitor voltage and mag-netizing current from the steady-state equations,Equation 23 is formulated in the matrix form, as shownin Equation 24.

EQUATION 24: STEADY-STATE OPERATING POINT

Perturbation and Linearization of Harmonic Balance Equations

The nonlinear system equations, Equation 12 throughEquation 14, are in the form of: x′ = f(x(t), u(t));x(t) = state of the nonlinear system and u(t) = input tothe system.

The function (x′) can be linearized about an operatingpoint and is expressed in the form of: x′ = Ax + Bu,where A and B are the Jacobian matrices of the systemwith respect to x(t) and u(t), as shown in Equation 25.

EQUATION 25: JACOBIAN MATRICES

In the perturbation and linearization step, it is assumedthat the averaged state variables and the input vari-ables consist of the constant DC component and asmall signal AC variation about the DC component.Perturbed signals are shown in Equation 26.

EQUATION 26: PERTURBED SIGNALS

Where:

X

rs Re+ LsΩs 1 0 Re– 0

LsΩ s– rs Re+ 0 1 0 Re–

1 0 0 Cs Ω s– 0 0

0 1 Cs Ω s 0 0 0

Re 0 0 0 Re– Lm Ω s–

0 Re 0 0 Lm Ωs Re–

=

U0

Ves

00000

= Y

Is

Ic

Vs

Vc

Ims

Imc

=

X Y× U0=

Y X1–

U0×=

Bijδf x t( ) u t( )( , )i

uj t( )∂--------------------------------

x0 u0,

=

Aijδf x t( ) u t( )( , )i

xj t( )∂--------------------------------

x0 u0,

=

Where:x0 and u0 represent the steady-state operating

points.

vin Vin vin+= d D d+= ωs Ωs ωs+= vpsVps

vps+= vpc Vpc vpc+= ipsIps

ips+=

ipc Ipc ipc+= ipp Ipp ipp+= vcfVcf

vcf+= ims

Imsims+= imc Imc imc+= is Is is

ˆ+=

ic Ic ic+= vs Vs vs+= vc Vc vc+= vesVes

ves+= v0 V0 v0+=

, , , , , ,

, , , , , ,

, , , ,


AN1477

The sine component of the transformer primaryvoltage (vps) is linearized around the steady-stateoperating point, as shown in Equation 27.

EQUATION 27: LINEARIZATION OF SINE COMPONENT OF TRANSFORMER PRIMARY VOLTAGE

vps4nπ

------ips

ipp--------vcf

4nπ

------ips

ips2

ipc2+

--------------------------------vcf= =

vps

4nVcfπ

--------------

Ips

2Ipc

2+ 1

Ips

2Ipc

2+--------------------------------–

Ips2

Ipc2+

-----------------------------------------------------------------------

× ips

4nVcfπ

--------------Ips

Ipc

Ips2

Ipc2+

32---

-------------------------------------× ipc– 4nπ

------Ips

Ips2 Ipc

2+------------------------------× vcf

×+=

vps

4nVcfπ

--------------Ipc

2

Ipp3

--------- ips

4nVcfπ

--------------Ips Ipc

Ipp3

--------------- ipc– 4nπ

------Ips

Ipp------- vcf

+=

vps Hip ips Hic ipc Hvcf vcf+ +=

vps Hipis Hicic Hipi'ms– Hicimc– Hvcf vcf+ +=

Where:

Hip

4nVcfπ

--------------Ipc

2

Ipp3

---------=

Hic

4nVcfπ

--------------Ips Ipc

Ipp3

---------------–=

Hvcf4nπ

------Ips

Ipp-------=

ims


AN1477

The cosine component of the transformer primaryvoltage (vpc) is linearized around the steady-stateoperating point, as shown in Equation 28.

EQUATION 28: LINEARIZATION OF COSINE COMPONENT OF TRANSFORMER PRIMARY VOLTAGE

The linearization of the input voltage (ves) is shown inEquation 29.

EQUATION 29: LINEARIZATION OF HALF-BRIDGE INVERTER OUTPUT VOLTAGE

Removing the steady-state terms and other higherorder perturbed terms in Equation 29 to get thelinearized input voltage is shown in Equation 30.

EQUATION 30: LINEARIZATION OF HALF-BRIDGE INVERTER OUTPUT VOLTAGE

vpc4nπ

------ipc

ipp------ vcf

×× 4nπ

------ipc

ips2

ipc2+

------------------------------ vcf××= =

vpc

4nVcfπ

--------------– Ips Ipc

Ips2

Ipc2+( )

32---

----------------------------------

ips

4nVcfπ

--------------

Ips2

Ipc2+ 1

Ips2

Ipc2+

------------------------------–

Ips2

Ipc2+

2

-------------------------------------------------------------------

ipc4nπ

------Ipc

Ips2 Ipc

2+------------------------------

vcf+ +=

vpc Gip is Gicic Gip ims– Gicimc– Gvcf vcf+ +=

vpc

4nVcfπ

--------------Ips Ipc

Ipp3

---------------- ips–4nVcf

π--------------

Ipc2

Ipp3

------------ ipc4nπ

------Ipc

Ipp-------- vcf

+ +=

Where:

Gip

4nVcfπ

--------------Ips Ipc

Ipp3

----------------–=

Gic

4nVcfπ

--------------Ipsc

2

Ipp3

------------=

Gvcf4nπ

------Ipc

Ipp-------=

ves

2vin

π---------- π

2---d sin=

ves2π--- Vin vin+( )× π

2--- D d+( ) sin=

Expandingπ2--- D d+( ) sin

π2---D π

2---d cossin= π

2---D cos π

2--- d sin+

π2---D sin= π

2--- π

2---D dcos+

ves2π--- Vin vin+( ) π

2---D sin π

2--- π

2---D dcos+

=

ves2π--- π

2---D vinsin Vin

π2---D dcos+=

ves K1vin K2d+=

Where:

K12π--- π

2---D sin=

K2 Vinπ2---D cos=


AN1477

The linearization and perturbation of the tank current,capacitor voltage, transformer primary voltage, outputvoltage and output filter capacitor voltage, after remov-ing the second order and DC terms, are provided inEquation 31 through Equation 42.

In resonant converters, the poles and zeroes arethe functions of normalized switching frequency(ωsn = ωs/ω0), where ωs is the switching frequencyand ω0 is the resonant frequency.

The linearization and perturbation of the sine componentof the tank voltage is provided in Equation 31.

EQUATION 31: LINEARIZATION OF SINE COMPONENT OF TANK VOLTAGE

Substitute the values of Equation 27 and Equation 30into the sine component of the tank voltage, as shownin Equation 32.

EQUATION 32: LINEARIZATION OF SINE COMPONENT OF TANK VOLTAGE

The linearization and perturbation of cosine componentof tank voltage is provided in Equation 33.

EQUATION 33: LINEARIZATION OF COSINE COMPONENT OF TANK VOLTAGE

Substituting the values of Equation 28 into the cosinecomponent of the tank voltage, the result obtained isshown in Equation 34.

EQUATION 34: LINEARIZATION OF COSINE COMPONENT OF TANK VOLTAGE

Ls

d Is is+( )dt

---------------------- rs Is is+( ) Ls Ic ic+( ) Ωs ω0ωs

ω0-------+

Vs vs+( ) Vps vps+( )+ + + + Ves ves+( )=

Lsdis

dt------- rsis ΩsLsic Lsω0Icωsn vs vps+ + + + + ves=

Ls

disdt------- rsis ΩsLsic Lsω0Icωsn vs Hipis Hicic Hipims– Hicimc– HvcfVcf+ + + + + + + K1vin K2d+=

Ls

disdt------- Hip rs+( ) is– ΩsLs Hic+( ) ic– vs– Hipims Hicimc HvcfVcf

– K1vin K2d Lsω0Icωsn–+ + + +=

disdt-------

Hip rs+

Ls--------------------

is–ΩsLs Hic+

Ls-----------------------------

ic– 1Ls----- vs–

HipLs

--------- ims

HicLs

--------- imc

HvcfLs

------------Vcf–

K1Ls------- vin

K2Ls------- d

Lsω0IcLs

------------------ωsn–+ + + +=

Ls

d Ic ic+( )dt

---------------------- rs Ic ic+( ) Ls Is is+( ) Ωs ω0ωs

ω0------+

– Vc vc+( ) Vpc vpc+( )+ + + 0=

Ls

dicdt------- rsic+

ΩsLsis Lsω0Isωsn vc vpc+ + + + 0=– –

Ls

dicdt------- rs ic+

ΩsLsis Lsω0 Isωsn vc Gipis Gicic Gipims– Gicimc– Gvcfvcf+ + + + + + 0=

Ls

dicdt------- ΩsLs Gip–( ) is Gic rs+( ) ic– vc– G

ipims Gicimc Gvcfvcf

– Lsω0Isωsn+ + +=

dicdt-------

ΩsLs Gip–( )Ls

------------------------------- isGic rs+( )

Ls----------------------- ic– 1

Ls----- vc–

Gip

Ls-------- ims

Gic

Ls-------- imc

Gvcf

Ls----------- vcf

–Lsω0Is

Ls-----------------ωsn+ + +=

– –

ic


AN1477

The linearization and perturbation of the sine componentof the tank current is provided in Equation 35.

EQUATION 35: LINEARIZATION OF SINE COMPONENT OF TANK CURRENT

The linearization and perturbation of the cosinecomponent of the tank current is provided inEquation 36.

EQUATION 36: LINEARIZATION OF COSINE COMPONENT OF TANK CURRENT

The linearization and perturbation of the sine componentof the magnetizing current is provided in Equation 37.

EQUATION 37: LINEARIZATION OF SINE COMPONENT OF MAGNETIZING CURRENT

Substituting the value of Equation 27 into the sine com-ponent of the transformer primary voltage, the resultsare shown in Equation 38.

EQUATION 38: LINEARIZATION OF SINE COMPONENT OF MAGNETIZING CURRENT

Cs

dvs

dt-------- CsΩsvc Csω0Vcωsn is=+ +

Cs

d Vs vs+( )

dt------------------------- Cs Vc vc+( ) Ωs ω0

ωs

ω0-------+

+ Is is+( )=

dvs

dt-------- 1

Cs------ is

CsΩs

Cs------------- vc–

Csω0VcCs

--------------------ωsn–=

Cs

dvcdt

-------- CsΩsvs– Csω0Vcωsn– ic=

dvcdt

-------- 1Cs------ is

CsΩs

Cs------------- vs

Csω0VsCs

--------------------ωsn+ +=

Cs

d Vc vc+( )

dt-------------------------- Cs Vs vs+( ) Ωs ω0

ωs

ω0-------+

– Ic ic+( )=

ic

Lm

dimsdt

------------ LmΩsimc LmImcω0ωsn+ + vps=

Lm

d Ims ims+( )

dt-------------------------------- Lm Imc imc+( ) Ωs ω0

ωs

ω0-------+

+ Vps vps+( )=

Lm

dimsdt

------------ LmΩsimc LmImcω0ωsn+ + Hipis Hicic Hipims– Hicimc– Hvcf vcf+ +=

Lm

dimsdt

------------ Hipis Hicic Hipims– Hic LmΩs+( ) imc– Hvcf vcf LmImcω0ωsn–+ +=

dimsdt

------------HipLm--------- is

HicLm--------- ic

HipLm--------- ims–

Hic LmΩs+( )

Lm----------------------------------- imc–

HvcfLm

------------ vcf

LmImcω0Lm

------------------------- ωsn–+ +=


AN1477

The linearization and perturbation of the cosinecomponent of the tank voltage is provided inEquation 39.

EQUATION 39: LINEARIZATION OF COSINE COMPONENT OF MAGNETIZING CURRENT

Substituting Equation 28 into the cosine component ofthe magnetizing current, the result is shown inEquation 40.

EQUATION 40: LINEARIZATION OF COSINE COMPONENT OF MAGNETIZING CURRENT

The linearization and perturbation of the output filtercapacitor voltage is provided in Equation 41.

EQUATION 41: LINEARIZATION OF OUTPUT CAPACITOR VOLTAGE

Lm

dimcdt

------------ LmΩsims– LmImsω0ωsn– vpc=

Lm

d Imc imc+( )

dt--------------------------------- Lm Ims ims+( ) Ωs ω0

ωs

ω0-------+

– Vpc vpc+( )=

dimcdt

------------GipLm--------- is

GicLm--------- ic

Gip LmΩs–( )

Lm----------------------------------- ims–

GicLm--------- imsc–

Gvcf

Lm---------- vcf

LmImsω0Lm

------------------------- ωsn+ + +=

Lmdimc

dt----------- LmΩsims– LmImsω0ωsn– Gip is Gicic Gip ims– Gic imc– Gvcf vcf+ +=

ic

1rcR-----+

Cf

dvcfdt

---------- 1R---+× vcf

2π--- isp=

1rcR-----+

Cf

d Vcfvcf

+

dt------------------------------ 1

R--- Vcf

vcf+

+ 2π--- Isp isp+( )=

From Equation 16:

isp2nπ

------ ips2

ipc2+=

isp2nπ

------Ips

Ips2

Ipc2+

--------------------------------- ips2nπ

------Ipc

Ips2

Ipc2+

--------------------------------- ipc+=

Kisips Kicipc+

isp Kisis Kicic Kisims– Kicimc–+=

1rcR-----+

Cf

dvcfdt

---------- Kisis=× Kicic Kisims– Kicimc– 1R--- vcf

–+

ips is ims–= ipc ic imc–=

Where:

rcr'c------Cf

dvcfdt

----------+ Kisis Kicic Kisims– Kicimc– 1R--- vcf

–+=

dvcfdt

----------Kis r'cCf

rc--------------- is

Kis r'cCf

rc--------------- ic

Kis r'cCf

rc--------------- ims–

Kicr'cCf rc

--------------- imc–r'c

RCfrc

--------------- vcf–+=

Kis2nπ

------Ips

Ips2 Ipc

2+---------------------------------=

and

Kic2nπ

------Ipc

Ips2

Ipc2+

---------------------------------=and

ic


AN1477

The linearization and perturbation of the output voltageis provided in Equation 42.

EQUATION 42: LINEARIZATION OF OUTPUT VOLTAGE

Equation 31 through Equation 42 are arranged, asshown in Equation 43.

EQUATION 43: LINEARIZED SMALL SIGNAL MODEL OF LLC RESONANT CONVERTER

V0 v0+ 2π---r'c Isp isp+( )

r'crc------

Vcfvcf

+ +=

v0 2π---r'cisp

r'crc------

vcf+=

v0 r'c Kisis Kicic Kisims– Kicimc–+( )r'crc------

vcf+=

v0 Kisr'cis Kicr'cic Kisr'cims– Kicr'ciˆmc–+( )

r'crc------

vcf+=

disdt-------

Hip rs+

Ls--------------------

is–ΩsLs Hic+

Ls-----------------------------

ic– 1Ls----- vs–

HipLs

--------- ims

HicLs

--------- imc

HvcfLs

------------ vcf–K1Ls------- vin

K2Ls------- d

Lsω0IcLs

------------------ ωsn–+ + + +=

dicdt-------

ΩsLs Gip–( )

Ls--------------------------------- is

Gic rs+( )

Ls------------------------- ic– 1

Ls----- vc–

GipLs

--------- ims

GicLs

--------- imc

GvcfLs

------------ vcf–Lsω0Is

Ls------------------ωsn+ + +=

dvsdt

-------- 1Cs------ is

CsΩs

Cs-------------- vc–

Csω0

Vc

Cs--------------------ωsn–=

dvcdt

-------- 1Cs------ ic

CsΩs

Cs-------------- vs

Csω0

Vs

Cs--------------------ωsn+ +=

dimsdt

------------HipLm--------- is

HicLm--------- ic

HipLm--------- ims–

Hic LmΩs+

Lm------------------------------- imc–

HvcfLm

------------ vcf

LmImcω0

Lm------------------------- ωsn–+ +=

dimcdt

------------GipLm--------- is

GicLm--------- ic

Gip LmΩs–( )

Lm----------------------------------- ims–

GicLm--------- imc–

GvcfLm

------------ vcf

LmImsω0

Lm------------------------- ωsn+ + +=

The output equation is:

v0 Kis r'c is Kic r'c ic Kis r'c ims– Kic r'c imc–r'crc------

vcf+ +=

dVCfdt

-------------Kis r'cCf rc

--------------- is

Kic r'cCf rc

--------------- ic

Kis r'c

Cf rc--------------- ims–

Kis r'c

Cf rc--------------- imc–

r'cRCf rc--------------- vcf–+=


AN1477

Formation of State-Space Model

State-space representation is a mathematical model ofa physical system as a set of input, output and statevariables, related by first order differential equations.

Additionally, if the dynamic system is linear and timeinvariant, the differential and algebraic equations maybe written in matrix form.

The state-space representation (known as time domainapproach) provides a convenient and compact way tomodel and analyze systems with multiple inputs andoutputs.

Equation 44 provides the state-space representation ofthe LLC resonant converter.

EQUATION 44: STATE-SPACE MODEL OF LLC RESONANT CONVERTER

A

Hip rs+

Ls--------------------–

ΩsLs Hic+( )

Ls---------------------------------– 1

Ls-----– 0

HipLs

---------HicLs

---------Hvcf

Ls------------–

ΩsLs Gip–( )

Ls---------------------------------

Gic rs+

Ls--------------------– 0 1

Ls-----–

GipLs

---------GicLs

---------Gvcf

Ls------------–

1Cs------ 0 0

CsΩsCs

--------------– 0 0 0

0 1Cs------

CsΩsCs

-------------- 0 0 0 0

HipLm---------

HicLm--------- 0 0

HipLm---------–

Hic LmΩs+

Lm-------------------------------–

HvcfLm

------------

GipLm---------

GicLm--------- 0 0

Gip LmΩs–

Lm-------------------------------–

GicLm---------–

GvcfLm

------------

Kisr'cCf

rc---------------

Kicr'cCf

rc--------------- 0 0

Kisr'cCf

rc---------------–

Kicr'cCf

rc---------------–

r'cRCf

rc---------------–

=

BLsω0Ic

Ls-----------------–

Lsω0Is

Ls---------------- Csω0Vc

Cs-------------------–

Csω0Vs

Cs------------------- Lmω0Imc

Lm----------------------–

Lmω0Ims

Lm----------------------

0 T

=

D 0=

C Kisr'

c( ) Kicr'c( ) 0( ) 0( ) Kisr'c–( ) Kicr'c–( )

r'crc------

=

For the linearized system, the required control-to-output voltage transfer function is:

v0

ωsn

---------- C SI A–( ) 1–B D+ Gp s( )= =

dxdt------ Ax Bu+=

y Cx Du+=

Where:

x is ic vs vc ims imc vcf T

= States of the system

u fsn or ωsn( )= Control inputs and all other disturbance inputs are ignored

y v0( )= Output


AN1477

HARDWARE DESIGN SPECIFICATIONS

Series resonant inductor (Ls) = 62 µH

Series resonant capacitance (Cs) = 9.4 nF

Magnetizing inductor (Lm) = 268 µH

Input voltage (Vin) = 400V (DC)

Output filter capacitance (Cf ) = 2000 µF

Output power = 200W

Switching frequency (fs) = 200 kHz

DCR of resonant inductor (rs) = 15 mΩ

ESR of output capacitor (rc) = 15 mΩ

Equation 44 can be solved using MATLAB® to obtainthe control-to-output (plant) transfer function,sys = ss(A, B, C, D). The ss command arrangesthe A, B, C and D matrices in a state-space model. TheGp(s) = tf(sys) command gives the transfer functionof the system, where sys indicates the system. Theplant transfer function (Gp(s)), along with the designvalues, are shown in Equation 45.

EQUATION 45: PLANT TRANSFER FUNCTION

The general form of Gp(s) is shown in Equation 46.

EQUATION 46: GENERALIZED FORM OF PLANT TRANSFER FUNCTION

The [p, z] = pzmap (Gp (s)) command gives the polesand zeros of the plant transfer function. Figure 7 illus-trates the pole-zero plot for the Gp(s), which is obtainedfrom the MATLAB command, pzmap (Gp(s)). Figure 8illustrates the bode plot obtained from the hardwareusing the network analyzer. Figure 9 illustrates thebode plot obtained using MATLAB.

As illustrated in Figure 8 and Figure 9, the bode plotcaptured, using the network analyzer, matches theanalytical bode plot obtained in MATLAB, thereby,confirming the veracity of the mathematical model.

Gp s( )224213315399 s 3.314 104×+( )× s 8.262 105×–( )×

s2 973.6s 8.949+ 108×+( ) s

2 2.76+ 105s 1.227 1012×+×( )×

--------------------------------------------------------------------------------------------------------------------------------------------------------------=

Gp s( )

5.5895 1 s

3.314 104×-----------------------------+

s

8.262 105×----------------------------- 1–

×

s2

8.949 108×----------------------------- 973.6s

8.949 108×----------------------------- 1+ +

s

2

1.227 1012×-------------------------------- 2.75 105

s×

1.227 1012×-------------------------------- 1+ +

×

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Gp s( )

Gpo 1 sωesr------------+

s

ωRHP---------------- 1–

×

s2

ωp12

---------- sQ1 ωp1×------------------------ 1+ +

s2

ωp22

---------- sQ2 ωp2×------------------------ 1+ +

×

------------------------------------------------------------------------------------------------------------------------=


AN1477

FIGURE 7: POLE-ZERO MAP OF PLANT TRANSFER FUNCTION

FIGURE 8: MEASURED BODE DIAGRAM OF PLANT TRANSFER FUNCTION


AN1477

FIGURE 9: SIMULATED BODE DIAGRAM OF PLANT TRANSFER FUNCTION

Digital Compensator Design for LLC Resonant Converter

The plant model is derived as shown in Equation 45. Inorder to attain the desired gain margin, phase marginand crossover frequency, a digital 3P3Z compensatoris designed. The digital 3P3Z compensator is derivedusing the design by emulation or digital redesignapproach.

In this method, an analog compensator is first designedin the continuous time domain and then converted todiscrete time domain using bilinear or tustin transfor-mation. Figure 10 illustrates the block diagram of theLLC resonant converter with a digital compensator.

FIGURE 10: BLOCK DIAGRAM OF LLC RESONANT CONVERTER

As seen from Equation 46, plant transfer function con-sists of an ESR zero and a pair of dominant complexpoles. In order to compensate for the effect of ESR zero(increased high-frequency gain, and thereby,increased ripple), a pole (ωp) is included in the compen-sator. In order to minimize the steady-state error, anintegrator (Kc) is also added to the compensator.

Furthermore, in order to compensate for the effect ofthe complex dominant poles (reduction in systemdamping, and hence, increased overshoots and set-tling time), two zeros, (s+a+jb) and (s+a-jb), are added.Also, to achieve sufficient attenuation at switchingfrequency, a pole is added to the compensator at halfthe switching frequency.

–

3P3ZCompensator

DPWM

KA/D GPFc

A/D

VREF[n] + Vc[n] F(t)

Resonant Converter

Vo[t]

Voltage Sensor

Vm[n]

e[n]


AN1477

Effectively, the system will have a 3-Pole 2-Zero (3P2Z)compensator in continuous domain, as shown inEquation 47.

EQUATION 47: COMPENSATOR GC(s) IN CONTINUOUS TIME DOMAIN

One of the digital compensator poles (ωp = 2πfp) isplaced at fp to cancel the ESR zero due to output filtercapacitor ESR (fesr = ωesr/2π). The compensator sec-ond pole (ωpc) is placed at half the switching frequency(fs) to obtain sufficient attenuation at the switchingfrequency. Therefore, ωpc = 2πfs/2.

Kc represents the integral gain of the compensator andis adjusted to achieve the desired crossover frequencyof the system.

A pair of complex zeros of the compensator, on thecomplex s-plane, is at s1 = –α + jβ and s2 = –α – jβ. Thecompensator zero frequency magnitude (ωz) is 2πfz.The frequency (fz) is chosen slightly below or equal tothe corner frequency of the dominant resonantpoles (ωp1) to provide the necessary phase lead. Thecompensator quality factor (Qc) is chosen to be compa-rable or equal to the Q1 of the dominant complex polepair, of the plant transfer function, at the maximum loadcurrent. In this analysis, the computation delay isassumed to be unity.

If the desired crossover frequency is denoted as (fc),then ωc = j2πfc.

At crossover frequency, the loop gain of the systemshould be 0 dB or one on linear scale, as shown inEquation 48.

EQUATION 48: COMPENSATOR GAIN CALCULATION

The compensator first pole (ωp) is placed at 37k radi-ans/second, the second pole is placed at 100k radians/second and the complex pair of zeros is placed at 30kradians/second. The resulting compensator for acrossover frequency of 2000 Hz is shown inEquation 49.

EQUATION 49: COMPENSATOR TRANSFER FUNCTION

Gc s( )

Kcs2

ωz2

------- sQc ωz×-------------------- 1+ +

×

s sωp------- 1+ s

ωpc---------- 1+

××

--------------------------------------------------------------=

Gc s( )Kc ωz

2s α jβ+ +( ) s α jβ–+( )××⁄

s sωp------- 1+ s

ωpc---------- 1+

××

----------------------------------------------------------------------------------------=

Gp s( )s ωc=

Gc s( )s ωc=

× 1=

Kc1

Gp s( )s ωc=

--------------------------------- 1Gc s( )

s ωc=

---------------------------------×=

The required gain of the compensator is:

Gc371249.6041 s

2 973.6s 8.949 108×+ +( )×

s s 3.314 104×+( ) s 1.03 106×+( )××----------------------------------------------------------------------------------------------------------=


AN1477

The [p, z] = pzmap (Gc(s)) command gives the polesand zeros of the compensator. Figure 11 throughFigure 13 illustrate the pole-zero plot for a Gc, practical

bode plot (loop gain) obtained using the networkanalyzer, and the bode plot (loop gain) obtained usingMATLAB.

FIGURE 11: POLE-ZERO MAP OF COMPENSATOR

FIGURE 12: SIMULATION BODE DIAGRAM OF LOOP GAIN


AN1477

FIGURE 13: MEASURED LOOP GAIN

The discrete compensator transfer function (Gc_d) isobtained using the tustin or bilinear transformation witha sampling frequency of 50 kHz, as shown inEquation 50.

EQUATION 50: COMPENSATOR TRANSFER FUNCTION IN DISCRETE DOMAIN

0.2711 z3 0.178 z

2 0.1828 z 0.2663+×–×–×

z3 0.6791 z

2 0.7342 z 0.4133+×–×–-----------------------------------------------------------------------------------------------------------------Gc_d =


AN1477

CONCLUSION

Pulse Frequency Modulated LLC resonant converterplant transfer function is derived by employing the EDF.A digital compensator is designed to meet the specifi-cations of phase margin, gain margin and bandwidthfor the control system. The hardware results or wave-forms are in conformity to the developed analyticalmodel and also meet the target specifications.

REFERENCES

• “Topology Investigation for Front End DC/DC Power Conversion for Distributed Power Systems”, by Bo Yang, Dissertation, Virginia Polytechnic Institute and State University, 2003.

• “Small-Signal Analysis for LLC Resonant Converter”, by Bo Yang and F.C. Lee, CPES Seminar, 2003, S7.3, Pages: 144-149.

• “Small-Signal Modeling of Series and Parallel Resonant Converters”, by Yang, E.X.; Lee, F.C.; Jovanovich, M.M., Applied Power Electronics Conference and Exposition, 1992. APEC' 92. Conference Proceedings 1992, Seventh Annual, 1992, Page(s): 785-792.

• “Approximate Small-Signal Analysis of the Series and the Parallel Resonant Converters”, by Vorperian, V., Power Electronics, IEEE Transactions on, Vol. 4, Issue 1, January 1989, Page(s): 15-24.

• “DC/DC LLC Reference Design Using the dsPIC® DSC” (AN1336)

LIST OF PARAMETERS

TABLE 1: LIST OF PARAMETERS AND DESCRIPTION

Parameter Description

Ir Resonant tank current

Vc Resonant tank capacitor voltage

Im Magnetizing current

Vcf Output capacitor voltage

v’cf Reflected output capacitor voltage on primary side

Isp Transformer secondary current

Iis Sine component of resonant tank current

Iic Cosine component of resonant tank current

Vcs Sine component of resonant tank capacitor voltage

Vcc Cosine component of resonant tank capacitor voltage

Ims Sine component of magnetizing current

Imc Cosine component of magnetizing current

Iss Sine component of transformer secondary current

Isc Cosine component of transformer secondary current

Ves Sine component of half-bridge inverter output voltage

Vec Cosine component of half-bridge inverter output voltage

Ips Sine component of transformer primary current

Ipc Cosine component of transformer primary current

Ipp Total primary current of transformer

Vps Sine component of transformer primary voltage

Vpc Cosine component of transformer primary voltage

n Transformer turns ratio


AN1477

NOTES:


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== ISO/TS 16949 ==

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ISBN: 978-1-62076-690-3

DS01477A-page 27

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