1Dynamic Modeling and Analysisof Resonant InvertersIssues for design of closed-loopresonant converter system:• Need to model loop gain, closed-looptransfer functions• How does control-to-output transferfunction depend on the tank transferfunction H(s)?Closed-loop controlsystem to regulateamplitude of ac output(Lamp ballast exampleshown, but otherapplications have similarneeds) (frequencymodulation control shown)+–+v–vgPowerinputLoad(lamp)CompensatorvrefReferenceinputK || iload ||Frequencymodulator(VCO)vcGatedriversGc(s)veiloadSwitchnetwork+vs–fs–+FeedbackloopTanknetworkH(s)Peakdetector2Control via frequency modulation (FM)and amplitude modulation (AM)tvs(t)Fundamental componentVg– Vg4πVgvs1(t)TsFrequency ModulationVary switching period TsChanges frequency of vs1 butnot amplitudetvs(t)Fundamental componentVg– Vgvs1(t)TsDTs2Amplitude ModulationVary duty cycle D with constantswitching period TsChanges amplitude offundamental component of vs1but not frequency3The control-to-output-envelopetransfer function Genv(s)For frequency modulation, the control-to-output-envelope transfer functionis the small-signal transfer function from variations in the switchingfrequency fs to variations in the envelope of the output voltage v (orother output quantity):Genvs =vfstv(t)Envelope || v(t) ||Steady-state VPerturbation vvt = V + v t cos φ t +θ tThe output v(t) is a sinusoidcontaining both frequencyand amplitude modulation.We want to model how theoutput amplitude varies.4Spectrum of v(t)Spectrumof v(t)frequencyfsfs + fmfs – fmCarrier (switching)frequencySidebandSidebandThe control input is varied atmodulation frequency fmThis leads to sidebands atfrequencies fs ± fm (true for bothAM and FM)The spectrum of v(t) containsno component at f = fm.Effect of the tank transfer function H(s) on the output:• Changing the amplitude of the carrier affects the steady-state output amplitude• Changing the amplitudes of the sidebands affects the ac variations of theoutput amplitude— i.e., the envelope• The control-to-output-envelope transfer function Genv(s) depends on the tanktransfer function H(s) at the sideband frequencies fs ± fm. It doesnt depend onH(s) at f = fm.5Poles of Genv(s)Spectrumof v(t)frequencyfsfs + fmfs – fmCarrier (switching)frequencySidebandSidebandTank|| H(s) ||foExample:H(s) has resonant poles at f = foThese poles affect the sidebandswhen fs ± fm = foHence poles are observed inGenv(s) at modulation frequenciesof fm = | fs – fo |fs – fo|| Genv(s) ||fm6Outline of discussion1. How small-signal variations in the switching frequency affect thespectrum of the switch network output voltage vs1(t)2. Passing the frequency-modulated voltage vs1(t) through the tanktransfer function H(s) leads to amplitude modulation of the outputvoltage v(t)3. How to recover the envelope of the output voltage and determine thesmall-signal control-to-output-envelope transfer function Genv(s)4. Equivalent circuit modeling via the phasor transform5. PSPICE simulation of Genv(s) using the phasor transform71. The small-signal spectrum of vs1Steady-state switch network waveformsis(t)+–vgvs(t)+–Switch networkNS1212tvs(t)Fundamental componentVg– Vg4πVgvs1(t)Actual square wave:Fundamental component:vs(t)=4VgπΣn = 1, 3, 5,...1ncos (nωst)vs1(t)=4Vgπcos (ωst)=Vs1cos (ωst)Angular switching frequency s = 2fs8Modulation of the switching frequencyDefine the phase variable (t) as the integral of the instantaneous switchingfrequency s(t):φ t = ωst dtωst =2π fst =dφ tdtHence we can express the switch network output voltage asvs1t =4Vgπcos φ t = Vs1cos φ t9Modulation of the switching frequency, p. 2Next, we introduce a small variation in the control input. In response, thefrequency modulator (VCO) varies the switching frequency fs as follows:fst = Fs+ fstwithfst=∆ f cos ωmtωst = ωs0+ ∆ω cos ωmtwhere∆ω=2π∆ fωs0=2πFsThen,From standard communicationstheory: this FM expression forvs1(t) can be expanded into a“carrier” at frequency s0, andsidebands at s0 ± nm, usingBessel functionsφ t = ωst dt =ωs0t + β sin ωmtandvs1t = Vs1cosωs0t + β sin ωmtwhereβ =∆ωωm10The small-signal assumptionWe next make the usual small signal assumption, that the variations in thecontrol input are much smaller than the steady-state value. For frequencymodulation, this means that the variations in the switching frequency fs aremuch smaller than the steady-state switching frequency Fs. This coincideswith the “narrowband” assumption in communications theory:fs< Fs∆ω <ωs0This implies also that  < 1. It can be shown that, under this small-signalcondition, the switch voltage vs1(t) can be expressed as:Spectrumof vs1(t)frequencyfsfs + fmfs – fmCarrier (switching)frequencySidebandSidebandwhich includes the carriersignal and two sidebands(see notes for derivation)vs1t ≈ Vs1cosωs0t+β2cosωs0+ωmt−β2cosωs0−ωmt112. Effect of the tank transfer function H(s)vin(t)Frequencymodulatorfs(t)φ(t)SwitchnetworkUpper sideband (ωs0 + ωm)Carrier (ωs0)Lower sideband (ωs0 – ωm)Tank transfer functionH( j(ωs0 + ωm))H( jωs0)H( j(ωs0 – ωm))+–+v(t)vs1(t){From previous slide: under small-signal conditions, the switch voltagewaveform includes sinusoidal components at the switching (carrier)frequency and at upper and lower sidebands:The tank transfer function H(s) operates on each of these frequencycomponents, and we can compute the resulting output voltage v(t) usingsuperposition as illustrated below:vs1t ≈ Vs1cosωs0t+β2cosωs0+ωmt−β2cosωs0−ωmt12Phasor analysisComponent at the carrier frequency s0: the switch voltage isVs1cos ωs0t =Vs12ejωs0t+ e−jωs0twhich has a phasor representation of Vs1. This component passes throughthe tank transfer function and leads to a component of the output voltagev(t) equal to:A0cos ωs0t + ∠A0=12A0ejωs0t+ A0*e−jωs0twhere A0 = Vs1 H(js0) is the phasor representing the output carriercomponent, and A0* is the complex conjugate of A0.13Phasor analysis, p. 2Component at the upper sideband frequency s0 + m : the switch voltage isNote that the frequency differs from the carrier frequency used in thephasors of the previous slide. Lets nonetheless define a phasor Vs1/2 forthis component. Upon passing through the tank transfer function, thiscomponent leads to a component of the output voltage v(t) equal to:where Au = Vs1 (/2) H(j(s0 + m)) is the phasor representing the