Dynamical Investigation Of Output Capacitor ESR In Fixed Off-Time Controlled Buck Converter

With simple control circuit and fast transient response, fixed off-time(FOT) control technique is widely used. In FOT controlled buck converter,the equivalent series resistance (ESR) of the output capacitor has a greateffect on operation state and stability of the system. In this paper, regardingthe output capacitor ESR as main circuit parameter, the stability anddynamical performance of FOT controlled buck converter are studied.Based on the PSIM simulation, the reason of the phase of outputvoltage lagging of the phase of inductor current is analyzed qualitatively,and the mechanism to cause the pulse bursting phenomena in FOTcontrolled buck converter are explained. The way to choose the outputcapacitor ESR to eliminate these complex nonlinear phenomena is discussed.The critical ESR to ensure the FOT controlled Buck converter stableoperation is obtained. The research results indicate that when the ESR issmaller than the critical value, the output voltage phase lags behind theinductor current phase, resulting in the pulse bursting phenomenon; whilewhen the ESR is larger than the critical value, the output voltage variationkeeps in phase with the inductor current variation, the pulse burstingphenomenon disappearing. In addition, the describing function method isused to obtain the transfer function from the reference voltage to the outputvoltage. By using Routh-Hurwitz criterion, it is indicated that the criticalESR is the condition of instability in FOT controlled buck converter.Based on the analysis of the switching states of Buck converter, adiscrete-time model of FOT controlled buck converter is established with fiveoperation modes and four borderlines by sampling at the end of each FOTcontrol pulse. Taking output capacitor ESR, output capacitor and fixedoff-time as bifurcation parameters, the bifurcation diagrams are depictedrespectively，from which the complex dynamical behaviors of the system areanalyzed. The corresponding maximal Lyapunov exponents are given byderiving the eigenvalues of the Jacobian around the fixed point are derived,which have the same dynamical behaviors with the bifurcation diagrams. Asetup of FOT controlled buck converter is implemented and the experimentalresults well verify the validity of theoretical analysis. FOT technique with ramp compensation is presented. Based on thePSIM simulation, the effect of sample resistor on the stability of FOTcontrolled buck converter with ramp compensation under low outputcapacitor ESR is analyzed, and the critical sample resistor value to ensurethe system stable operation is obtained. Based on the discrete-time model ofFOT controlled buck converter with ramp compensation, the bifurcationdiagrams are constructed. The research results indicate that FOT techniquewith ramp compensation is a valid way to reduce the dependence of theconverter on the output capacitor ESR and can expand the stability region ofoutput capacitor ESR in FOT controlled buck converter with rampcompensation.