Dynamic analysis of switching converters Chapter 6 Dynamic analysis of switching converters

Dynamic analysis of switching converters Overview Continuous-Time Linear Models Switching converter analysis using classical control techniques Averaged switching converter models Review of negative feedback using classical-control techniques Feedback compensation State-space representation of switching converters Input EMI filters Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Overview Discrete-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 converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Dynamic 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 converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Switching 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 converters Dynamic analysis of switching converters

Linear model of a switching converter Power switching converters Dynamic analysis of switching converters

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

Dynamic analysis of switching converters PWM modulator model Current- mode control Variation of the duty cycle due to a perturbation in the inductor current Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters PWM modulator model Current- mode control Variation of the duty cycle due to a perturbation in the output voltage Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters PWM modulator model Current- mode control Variation of the duty cycle due to a perturbation on the peak current Power switching converters Dynamic analysis of switching converters

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

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

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

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

Dynamic analysis of switching converters Output filter model Output filter of a switching converter Power switching converters Dynamic analysis of switching converters

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

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

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

Dynamic 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 converters Dynamic analysis of switching converters

Dynamic 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 converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Example 6.4 The 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 µF and RL = 20 Ω. 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 converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Example 6.4 (a) The nominal duty cycle can be calculated as for the given input and output voltages, we have D=0.5. Small-signal model of the boost converter Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Example 6.4 Power switching converters Dynamic analysis of switching converters

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

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Small-signal models of switching converters Power switching converters Dynamic analysis of switching converters

Review of negative feedback Block diagram representation for a closed-loop system Power switching converters Dynamic analysis of switching converters

Review of negative feedback Closed-loop gain Loop gain For TL>>1 Stability analysis Power switching converters Dynamic analysis of switching converters

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

Dynamic analysis of switching converters Relative stability Loop gain of a system with three poles Power switching converters Dynamic analysis of switching converters

Closed-loop switching converter Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Feedback network Power switching converters Dynamic analysis of switching converters

Error amplifier compensation networks PI Compensation network The total phase lag Power switching converters Dynamic analysis of switching converters

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

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

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

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

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

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

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

Compensation in a buck converter with output capacitor ESR average output voltage: 5 V input voltage: 12 V load resistance RL = 5 Ω 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 converters Dynamic analysis of switching converters

Compensation in a buck converter with output capacitor ESR f1, is chosen to be one-fifth of the switching frequency Power switching converters Dynamic analysis of switching converters

Compensation in a buck converter with output capacitor ESR Magnitude response of the buck converter open-loop (ABCD) closed-loop (JKLMNO) error amplifier EFGH Power switching converters Dynamic analysis of switching converters

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

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

Compensation in a buck converter with no output capacitor ESR Magnitude response of the buck converter open-loop ABC closed-loop HIJKL error amplifier DEFG Power switching converters Dynamic analysis of switching converters

Linear model of a voltage regulator including external perturbances audio susceptibility output impedance Power switching converters Dynamic analysis of switching converters

Output impedance and stability Power switching converters Dynamic analysis of switching converters

State-space representation of switching converters Review of Linear System Analysis A simple second-order low-pass circuit Power switching converters Dynamic analysis of switching converters

State-space representation of switching converters Review of Linear System Analysis A simple second-order low-pass circuit Power switching converters Dynamic 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 Power switching converters Dynamic analysis of switching converters

State-Space Averaging Procedures for state-space averaging 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 Power switching converters Dynamic analysis of switching converters

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 converters Dynamic analysis of switching converters

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Dynamic analysis of switching converters A linearized continuous equivalent circuit for the buck converter with a Resr The DC terms are The AC terms are Power switching converters Dynamic analysis of switching converters

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

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

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

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

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

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

Linearized equivalent circuit of the ideal boost converter DC solutions Power switching converters Dynamic analysis of switching converters

Linearized equivalent circuit of the ideal boost converter AC solutions small-signal averaged state-space equation Power switching converters Dynamic analysis of switching converters

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

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

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

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

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

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

Switching Converter Transfer Functions BUCK CONVERTER Power switching converters Dynamic analysis of switching converters

Switching Converter Transfer Functions BUCK CONVERTER Power switching converters Dynamic analysis of switching converters

Switching Converter Transfer Functions BOOST CONVERTER Power switching converters Dynamic analysis of switching converters

Switching Converter Transfer Functions BOOST CONVERTER Power switching converters Dynamic 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 converters Dynamic 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 converters Dynamic analysis of switching converters

Design of a control system with complete state feedback Pole selection 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 Feedback gains Power switching converters Dynamic analysis of switching converters

Design of a control system with complete state feedback Example A buck converter designed to operate in the continuous conduction mode has the following parameters: R = 4 Ω, L = 1.330 mH, C = 94 µF, 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. Power switching converters Dynamic analysis of switching converters

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

Design of a control system with complete state feedback we calculate the feedback gains as P=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 gives ans = 1e+2 * [ -3.2980 + 1.0000i -3.2980 - 1.0000i] Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters PSpice schematic Power switching converters Dynamic analysis of switching converters

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

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

Dynamic 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 converters Dynamic analysis of switching converters

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

Input EMI filters Stability Considerations 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 Power switching converters Dynamic analysis of switching converters

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

Input EMI filters Stability Considerations 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 Power switching converters Dynamic analysis of switching converters

Input EMI filters Stability Considerations 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 Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Input EMI filters Input EMI filter with LR reactive damping Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Input EMI filters Input EMI filter with RC reactive damping Power switching converters Dynamic analysis of switching converters

Input EMI filters Stability Considerations 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 Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Input EMI filters A fourth-order input EMI filter with LR reactive damping Power switching converters Dynamic analysis of switching converters

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

Part 2 Discrete-time models Power switching converters Dynamic analysis of switching converters

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

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

Continuous-time state-space model Equivalent circuit during ton: A1 Power switching converters Dynamic analysis of switching converters

Continuous-time state-space model Equivalent circuit during toff: A2 Power switching converters Dynamic analysis of switching converters

Continuous-time state-space model switching functions nonlinear model Power switching converters Dynamic analysis of switching converters

Continuous-time state-space model small-signal model Power switching converters Dynamic analysis of switching converters

Continuous-time state-space model steady-state equation perturbation in the state vector Power switching converters Dynamic analysis of switching converters

Discrete-time model of the switching converter Power switching converters Dynamic 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 converters Dynamic 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 order The 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 converters Dynamic analysis of switching converters

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

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

Dynamic analysis of switching converters Current mode control Sensitivities of the duty cycle Power switching converters Dynamic analysis of switching converters

Dynamic analysis of switching converters Current mode control With complete state feedback Power switching converters Dynamic analysis of switching converters

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