Dynamic Analysis of Switching Converters

Chapter 6Dynamic analysis of switching converters

Dynamic analysis of switching converters

Power switching convertersDynamic analysis of switching converters*OverviewContinuous-Time Linear Models Switching converter analysis using classical control techniquesAveraged switching converter modelsReview of negative feedback using classical-control techniquesFeedback compensation State-space representation of switching converters Input EMI filters


Power switching convertersDynamic analysis of switching converters*OverviewDiscrete-time models Continuous-time and discrete-time domains Continuous-time state-space model Discrete-time model of the switching converter Design of a discrete control system with complete state feedback


Power switching convertersDynamic analysis of switching converters*Dynamic analysisDynamic or small-signal analysis of the switching converter enables designers to predict the dynamic performance of the switching converter to reduce prototyping cost and design cycle time Dynamic analysis can be either numerical or analytical


Power switching convertersDynamic analysis of switching converters*Dynamic analysisSwitching converters are non-linear time-variant circuits Nevertheless, it is possible to derive a continuous time-invariant linear model to represent a switching converter Continuous-time models are easier to handle, but not very accurate Since a switching converter is a sampled system, a discrete model gives a higher level of accuracy


Power switching convertersDynamic analysis of switching converters*Linear model of a switching converter


Power switching convertersDynamic analysis of switching converters*PWM modulator model Sensitivity of the duty cycle with respect to vref Voltage-mode control


Power switching convertersDynamic analysis of switching converters*PWM modulator modelVariation of the duty cycle due to a perturbation in the inductor current Current- mode control


Power switching convertersDynamic analysis of switching converters*PWM modulator modelVariation of the duty cycle due to a perturbation in the output voltage Current- mode control


Power switching convertersDynamic analysis of switching converters*PWM modulator modelVariation of the duty cycle due to a perturbation on the peak current Current- mode control


Power switching convertersDynamic analysis of switching converters*Averaged switching converter models Three-terminal averaged-switch model Averaged-switch model for voltage-mode control


Power switching convertersDynamic analysis of switching converters*Averaged switching converter models Examples of switching converters with an averaged switch


Power switching convertersDynamic analysis of switching converters*Averaged switching converter models Small-signal averaged-switch model for the discontinuous mode


Power switching convertersDynamic analysis of switching converters*Averaged switching converter models Small-signal model for current-mode control


Power switching convertersDynamic analysis of switching converters*Output filter model Output filter of a switching converter


Power switching convertersDynamic analysis of switching converters*Output filter model Magnitude response of the output filter for several values of the output resistance Ro


Power switching convertersDynamic analysis of switching converters*Output filter model Phase response of the output filter for several values of the output resistance Ro


Power switching convertersDynamic analysis of switching converters*Output filter model Output filter with a capacitor Resr


Power switching convertersDynamic analysis of switching converters*Output filter model Magnitude response of an output filter with a capacitor having a Resr for several values of the output resistance Ro


Power switching convertersDynamic analysis of switching converters*Output filter model Phase response of an output filter with a capacitor having a Resr for several values of the output resistance Ro


Power switching convertersDynamic analysis of switching converters*Example 6.4The boost converter shown in Figure 2.10 has the following parameters: Vin = 10 V, Vo = 20 V, fs = 1 kHz, L = 10 mH, C = 100 F and RL = 20 . The reference voltage is 5 V. The converter operates in the continuous-conduction mode under the voltage-mode. Using (a) the averaged-switch model, calculate the output-to-control transfer function, and (b) Matlab to draw the Bode plot of the transfer function found in (a) .


Power switching convertersDynamic analysis of switching converters*Example 6.4Small-signal model of the boost converter (a)The nominal duty cycle can be calculated as

for the given input and output voltages, we have D=0.5.


Power switching convertersDynamic analysis of switching converters*Example 6.4


Power switching convertersDynamic analysis of switching converters*Example 6.4Bode plot of the small-signal transfer function of the boost converter


Power switching convertersDynamic analysis of switching converters*Small-signal models of switching converters


Power switching convertersDynamic analysis of switching converters*Review of negative feedbackBlock diagram representation for a closed-loop system


Power switching convertersDynamic analysis of switching converters*Review of negative feedbackClosed-loop gain Loop gain For TL>>1 Stability analysis


Power switching convertersDynamic analysis of switching converters*Relative stability Definitions of gain and phase margins


Power switching convertersDynamic analysis of switching converters*Relative stabilityLoop gain of a system with three poles


Power switching convertersDynamic analysis of switching converters*Closed-loop switching converter


Power switching convertersDynamic analysis of switching converters*Feedback network


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks PI Compensation networkThe total phase lag


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks Frequency response of the PI compensation network


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks Phase response of the PI compensation network


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks PID Compensation network


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks Magnitude response of the PID compensation network


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks Phase response of the PID compensation network


Power switching convertersDynamic analysis of switching converters*Error amplifier compensation networks Asymptotic approximated magnitude response of the PID compensation network


Power switching convertersDynamic analysis of switching converters*Compensation in a buck converter with output capacitor ESRaverage output voltage: 5 V input voltage: 12 V load resistance RL = 5

Design the compensation to shape the closed-loop magnitude response of the switching converter to achieve a -20 dB/decade roll-off rate at the unity-gain crossover frequency with a sufficient phase margin for stability


Power switching convertersDynamic analysis of switching converters*Compensation in a buck converter with output capacitor ESRf1, is chosen to be one-fifth of the switching frequency


Power switching convertersDynamic analysis of switching converters*Compensation in a buck converter with output capacitor ESRMagnitude response of the buck converteropen-loop (ABCD) closed-loop (JKLMNO) error amplifier EFGH


Power switching convertersDynamic analysis of switching converters*Compensation in a buck converter with output capacitor ESR


Power switching convertersDynamic analysis of switching converters*Compensation in a buck converter with no output capacitor ESR


Power switching convertersDynamic analysis of switching converters*Compensation in a buck converter with no output capacitor ESR Magnitude response of the buck converteropen-loop ABCclosed-loop HIJKLerror amplifier DEFG


Power switching convertersDynamic analysis of switching converters*Linear model of a voltage regulator including external perturbancesaudio susceptibility output impedance


Power switching convertersDynamic analysis of switching converters*Output impedance and stability

Output impedance


Power switching convertersDynamic analysis of switching converters*State-space representation of switching converters Review of Linear System AnalysisA simple second-order low-pass circuit


Power switching convertersDynamic analysis of switching converters*State-Space Averaging

approximates the switching converter as a continuous linear system requires that the effective output filter corner frequency to be much smaller than the switching frequency



Step 1: Identify switched models over a switching cycle. Draw the linear switched circuit model for each state of the switching converter (e.g., currents through inductors and voltages across capacitors). Step 2: Identify state variables of the switching converter. Write state equations for each switched circuit model using Kirchoff's voltage and current laws.Step 3: Perform state-space averaging using the duty cycle as a weighting factor and combine state equations into a single averaged state equation. The state-space averaged equation is Procedures for state-space averaging



Step 4: Perturb the averaged state equation to yield steady-state (DC) and dynamic (AC) terms and eliminate the product of any AC terms. Step 5: Draw the linearized equivalent circuit model. Step 6: Perform hybrid modeling using a DC transformer, if desired.


Power switching convertersDynamic analysis of switching converters*State-Space Averaged Model for an Ideal Buck Converter


Power switching convertersDynamic analysis of switching converters*A nonlinear continuous equivalent circuit of the ideal buck converter


Power switching convertersDynamic analysis of switching converters*A linear equivalent circuit of the ideal buck converter


Power switching convertersDynamic analysis of switching converters*A source-reflected linearized equivalent circuit of the ideal buck converter


Power switching convertersDynamic analysis of switching converters*A linearized equivalent circuit of the ideal buck converter using a DC transformer


Power switching convertersDynamic analysis of switching converters*State-space averaged model for the discontinuous-mode buck converter


Power switching convertersDynamic analysis of switching converters*A nonlinear continuous equivalent circuit for the discontinuous-mode buck converter


Power switching convertersDynamic analysis of switching converters*A linearized equivalent circuit for the discontinuous-mode buck converter


Power switching convertersDynamic analysis of switching converters*State-Space Averaged Model for a Buck Converter with a Capacitor ESR


Power switching convertersDynamic analysis of switching converters*Switched models for the buck converter with a Resr


Power switching convertersDynamic analysis of switching converters*A nonlinear continuous equivalent circuit for the buck converter with a Resr


Power switching convertersDynamic analysis of switching converters*A linearized continuous equivalent circuit for the buck converter with a ResrThe DC terms areThe AC terms are


Power switching convertersDynamic analysis of switching converters*A linearized equivalent circuit using DC transformer with a turns-ratio of D


Power switching convertersDynamic analysis of switching converters*State-Space Averaged Model for an Ideal Boost Converter


Power switching convertersDynamic analysis of switching converters*Nonlinear continuous equivalent circuit of the ideal boost converter


Power switching convertersDynamic analysis of switching converters*Linearized equivalent circuit of the ideal boost converter


Power switching convertersDynamic analysis of switching converters*Linearized equivalent circuit of the ideal boost converterDC solutions


Power switching convertersDynamic analysis of switching converters*Linearized equivalent circuit of the ideal boost converterAC solutions small-signal averaged state-space equation


Power switching convertersDynamic analysis of switching converters*Linearized equivalent circuit of the ideal boost converter


Power switching convertersDynamic analysis of switching converters*Source-reflected linearized equivalent circuit for the ideal boost converter


Power switching convertersDynamic analysis of switching converters*Load-reflected linearized circuit for the ideal boost converter


Power switching convertersDynamic analysis of switching converters*DC transformer equivalent circuit for the ideal boost converter


Power switching convertersDynamic analysis of switching converters*Switching Converter Transfer Functions Source-to-State Transfer Functions


Power switching convertersDynamic analysis of switching converters*Switching Converter Transfer Functions Source-to-State Transfer Functions linearized control law


Power switching convertersDynamic analysis of switching converters*Switching Converter Transfer Functions BUCK CONVERTER


Power switching convertersDynamic analysis of switching converters*Switching Converter Transfer Functions BOOST CONVERTER


Power switching convertersDynamic analysis of switching converters*Complete state feedback This technique allows us to calculate the gains of the feedback vector required to place the closed-loop poles at a desired location All the states of the converter are sensed and multiplied by a feedback gain


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback control strategy closed-loop poles The closed-loop poles can be arbitrarily placed by choosing the elements of F


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback Pole selection Feedback gains One way of choosing the closed-loop poles is to select an ith order low-pass Bessel filter for the transfer function, where i is the order of the system that is being designed


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback A buck converter designed to operate in the continuous conduction mode has the following parameters: R = 4 , L = 1.330 mH, C = 94 F, Vs = 42 V, and Va = 12 V. Calculate (a) the open-loop poles, (b) the feedback gains to locate the closed loop poles at P = 1000 * {-0.3298 + 0.10i -0.3298 - 0.10i}, (c) the closed loop system matrix ACL.Example


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback Solution


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback polesOL = eig(A)

polesOL = 1000 * { -1.3298 + 2.4961i, -1.3298 - 2.4961i}


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback Step response of the linearized buck converter sysOL=ss(A,B,C,0)step(sysOL)


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback design the control strategy for voltage-mode control If we apply complete state feedback


Power switching convertersDynamic analysis of switching converters*Design of a control system with complete state feedback we calculate the feedback gains asP=1000 *[-0.3298 + 0.10i -0.3298 - 0.10i]'

Then, F = {-2.6600 -0.3202}.check the locations of the closed loop poles eig(ACL); which givesans = 1e+2 * [ -3.2980 + 1.0000i -3.2980 - 1.0000i]


Power switching convertersDynamic analysis of switching converters*PSpice schematic


Power switching convertersDynamic analysis of switching converters*Transient response of the open-loop and closed-loop converters


Power switching convertersDynamic analysis of switching converters*Expanded view of the transient at 5 ms


Power switching convertersDynamic analysis of switching converters*Input EMI filters An input EMI filter placed between the power source and the switching converter is often required to preserve the integrity of the power source The major purpose of the input EMI filter is to prevent the input current waveform of the switching converter from interfering with the power source As such, the major role of the input EMI filter is to optimize the mismatch between the power source and switching converter impedances


Power switching convertersDynamic analysis of switching converters*Input EMI filters Circuit model of a buck converter with an input EMI filter


Power switching convertersDynamic analysis of switching converters*Input EMI filters The stability of a closed-loop switching converter with an input EMI filter can be found by comparing the output impedance of the input EMI filter to the input impedance of the switching converter The closed-loop switching converter exhibits a negative input impedance Stability Considerations


Power switching convertersDynamic analysis of switching converters*Input EMI filters Input impedance versus frequency for a buck converter Output impedance of the EMI filter At the resonant frequency Above the resonant frequency


Power switching convertersDynamic analysis of switching converters*Input EMI filters The maximum output impedance of the input EMI filter, ZEMI,max, must be less than the magnitude of the input impedance of the switching converter to avoid instability The switching converter negative input impedance in combination with the input EMI filter can under certain conditions constitute a negative resistance oscillator To ensure stability, however, the poles of should lie in the left-hand plane Stability Considerations


Power switching convertersDynamic analysis of switching converters*Input EMI filters A resistance in series with the input EMI filter inductor can be added to improve stability However, it is undesirable to increase the series resistance of the input EMI filter to improve stability since it increases conduction losses Stability Considerations


Power switching convertersDynamic analysis of switching converters*Input EMI filtersInput EMI filter with LR reactive damping


Power switching convertersDynamic analysis of switching converters*Input EMI filtersInput EMI filter with RC reactive damping


Power switching convertersDynamic analysis of switching converters*Input EMI filters It should be noted that high core losses in the input EMI filter inductor is desirable to dissipate the energy at the EMI frequency so as to prevent it from being reflected back to the power source Stability Considerations


Power switching convertersDynamic analysis of switching converters*Input EMI filtersA fourth-order input EMI filter with LR reactive damping


Power switching convertersDynamic analysis of switching converters*Input EMI filtersInput impedance, Zin(f), of the buck converter and output impedance, ZEMI(f), of the input EMI filter


Power switching convertersDynamic analysis of switching converters*Part 2Discrete-time models


Power switching convertersDynamic analysis of switching converters*Continuous-time and discrete-time domains continuous-time system The solution for the differential equation


Power switching convertersDynamic analysis of switching converters*Continuous-time and discrete-time domains the discrete-time expression


Power switching convertersDynamic analysis of switching converters*Continuous-time state-space modelEquivalent circuit during ton: A1


Power switching convertersDynamic analysis of switching converters*Continuous-time state-space modelEquivalent circuit during toff: A2


Power switching convertersDynamic analysis of switching converters*Continuous-time state-space modelswitching functions nonlinear model


Power switching convertersDynamic analysis of switching converters*Continuous-time state-space modelsmall-signal model


Power switching convertersDynamic analysis of switching converters*Continuous-time state-space modelsteady-state equation perturbation in the state vector


Power switching convertersDynamic analysis of switching converters*Discrete-time model of the switching converter


Power switching convertersDynamic analysis of switching converters*Design of a discrete control system with complete state feedback The closed-loop poles can be arbitrarily placed by choosing the elements of F


Power switching convertersDynamic analysis of switching converters*Design of a discrete control system with complete state feedback Pole selection One way of choosing the closed-loop poles is to design a low-pass Bessel filter of the same orderThe step response of a Bessel filter has no overshoot, thus it is suitable for a voltage regulator The desired filter can then be selected for a step response that meets a specified settling time Feedback gains


Power switching convertersDynamic analysis of switching converters*Design of a discrete control system with complete state feedback Voltage mode control


Power switching convertersDynamic analysis of switching converters*Extended-state model for a tracking regulator Digital tracking system with full-state feedback


Power switching convertersDynamic analysis of switching converters*Current mode control

Sensitivities of the duty cycle


Power switching convertersDynamic analysis of switching converters*Current mode control

With complete state feedback


Power switching convertersDynamic analysis of switching converters*Extended-state model for a tracking regulatorDigital tracking system with full-state feedback


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Dynamic Analysis of Switching Converters