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Chapter 6 Dynamic analysis of switching converters

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Power switching convertersDynamic analysis of switching converters2 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

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Power switching convertersDynamic analysis of switching converters3 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

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Power switching convertersDynamic analysis of switching converters4 Dynamic analysis 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

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Power switching convertersDynamic analysis of switching converters5 Dynamic analysis 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

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Power switching convertersDynamic analysis of switching converters6 Linear model of a switching converter

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Power switching convertersDynamic analysis of switching converters7 PWM modulator model Sensitivity of the duty cycle with respect to v ref Voltage-mode control

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Power switching convertersDynamic analysis of switching converters8 PWM modulator model Variation of the duty cycle due to a perturbation in the inductor current Current- mode control

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Power switching convertersDynamic analysis of switching converters9 PWM modulator model Variation of the duty cycle due to a perturbation in the output voltage Current- mode control

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Power switching convertersDynamic analysis of switching converters10 PWM modulator model Variation of the duty cycle due to a perturbation on the peak current Current- mode control

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Power switching convertersDynamic analysis of switching converters11 Averaged switching converter models Three-terminal averaged-switch model Averaged-switch model for voltage-mode control

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Power switching convertersDynamic analysis of switching converters12 Averaged switching converter models Examples of switching converters with an averaged switch

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Power switching convertersDynamic analysis of switching converters13 Averaged switching converter models Small-signal averaged-switch model for the discontinuous mode

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Power switching convertersDynamic analysis of switching converters14 Averaged switching converter models Small-signal model for current-mode control

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Power switching convertersDynamic analysis of switching converters15 Output filter model Output filter of a switching converter

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Power switching convertersDynamic analysis of switching converters16 Output filter model Magnitude response of the output filter for several values of the output resistance R o

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Power switching convertersDynamic analysis of switching converters17 Output filter model Phase response of the output filter for several values of the output resistance R o

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Power switching convertersDynamic analysis of switching converters18 Output filter model Output filter with a capacitor Resr

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Power switching convertersDynamic analysis of switching converters19 Output filter model Magnitude response of an output filter with a capacitor having a R esr for several values of the output resistance R o

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Power switching convertersDynamic analysis of switching converters20 Output filter model Phase response of an output filter with a capacitor having a R esr for several values of the output resistance R o

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Power switching convertersDynamic analysis of switching converters21 Example 6.4 The boost converter shown in Figure 2.10 has the following parameters: V in = 10 V, V o = 20 V, f s = 1 kHz, L = 10 mH, C = 100 F and R L = 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).

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Power switching convertersDynamic analysis of switching converters22 Example 6.4 Small-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.

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Power switching convertersDynamic analysis of switching converters23 Example 6.4

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Power switching convertersDynamic analysis of switching converters24 Example 6.4 Bode plot of the small-signal transfer function of the boost converter

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Power switching convertersDynamic analysis of switching converters25 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters26 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters27 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters28 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters29 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters30 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters31 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters32 Small-signal models of switching converters

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Power switching convertersDynamic analysis of switching converters33 Review of negative feedback Block diagram representation for a closed-loop system

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Power switching convertersDynamic analysis of switching converters34 Review of negative feedback Closed-loop gain Loop gain For T L >>1 Stability analysis

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Power switching convertersDynamic analysis of switching converters35 Relative stability Definitions of gain and phase margins

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Power switching convertersDynamic analysis of switching converters36 Relative stability Loop gain of a system with three poles

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Power switching convertersDynamic analysis of switching converters37 Closed-loop switching converter

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Power switching convertersDynamic analysis of switching converters38 Feedback network

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Power switching convertersDynamic analysis of switching converters39 Error amplifier compensation networks PI Compensation network The total phase lag

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Power switching convertersDynamic analysis of switching converters40 Error amplifier compensation networks Frequency response of the PI compensation network

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Power switching convertersDynamic analysis of switching converters41 Error amplifier compensation networks Phase response of the PI compensation network

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Power switching convertersDynamic analysis of switching converters42 Error amplifier compensation networks PID Compensation network

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Power switching convertersDynamic analysis of switching converters43 Error amplifier compensation networks Magnitude response of the PID compensation network

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Power switching convertersDynamic analysis of switching converters44 Error amplifier compensation networks Magnitude response of the PID compensation network

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Power switching convertersDynamic analysis of switching converters45 Error amplifier compensation networks Phase response of the PID compensation network

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Power switching convertersDynamic analysis of switching converters46 Error amplifier compensation networks Asymptotic approximated magnitude response of the PID compensation network

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Power switching convertersDynamic analysis of switching converters47 Compensation in a buck converter with output capacitor ESR average output voltage: 5 V input voltage: 12 V load resistance R L = 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

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Power switching convertersDynamic analysis of switching converters48 Compensation in a buck converter with output capacitor ESR f 1, is chosen to be one-fifth of the switching frequency

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Power switching convertersDynamic analysis of switching converters49 Compensation in a buck converter with output capacitor ESR Magnitude response of the buck converter open-loop (ABCD) closed-loop (JKLMNO) error amplifier EFGH

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Power switching convertersDynamic analysis of switching converters50 Compensation in a buck converter with output capacitor ESR

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Power switching convertersDynamic analysis of switching converters51 Compensation in a buck converter with no output capacitor ESR

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Power switching convertersDynamic analysis of switching converters52 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

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Power switching convertersDynamic analysis of switching converters53 Linear model of a voltage regulator including external perturbances audio susceptibility output impedance

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Power switching convertersDynamic analysis of switching converters54 Output impedance and stability Output impedance

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Power switching convertersDynamic analysis of switching converters55 State-space representation of switching converters Review of Linear System Analysis A simple second-order low-pass circuit

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Power switching convertersDynamic analysis of switching converters56 State-space representation of switching converters Review of Linear System Analysis A simple second-order low-pass circuit

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Power switching convertersDynamic analysis of switching converters57 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

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Power switching convertersDynamic analysis of switching converters58 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 Procedures for state-space averaging

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Power switching convertersDynamic analysis of switching converters59 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.

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Power switching convertersDynamic analysis of switching converters60 State-Space Averaged Model for an Ideal Buck Converter

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Power switching convertersDynamic analysis of switching converters61 State-Space Averaged Model for an Ideal Buck Converter

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Power switching convertersDynamic analysis of switching converters62 State-Space Averaged Model for an Ideal Buck Converter

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Power switching convertersDynamic analysis of switching converters63 A nonlinear continuous equivalent circuit of the ideal buck converter

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Power switching convertersDynamic analysis of switching converters64 A linear equivalent circuit of the ideal buck converter

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Power switching convertersDynamic analysis of switching converters65 A linear equivalent circuit of the ideal buck converter

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Power switching convertersDynamic analysis of switching converters66 A source-reflected linearized equivalent circuit of the ideal buck converter

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Power switching convertersDynamic analysis of switching converters67 A linearized equivalent circuit of the ideal buck converter using a DC transformer

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Power switching convertersDynamic analysis of switching converters68 State-space averaged model for the discontinuous-mode buck converter

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Power switching convertersDynamic analysis of switching converters69 State-space averaged model for the discontinuous-mode buck converter

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Power switching convertersDynamic analysis of switching converters70 A nonlinear continuous equivalent circuit for the discontinuous-mode buck converter

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Power switching convertersDynamic analysis of switching converters71 A nonlinear continuous equivalent circuit for the discontinuous-mode buck converter

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Power switching convertersDynamic analysis of switching converters72 A linearized equivalent circuit for the discontinuous-mode buck converter

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Power switching convertersDynamic analysis of switching converters73 State-Space Averaged Model for a Buck Converter with a Capacitor ESR

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Power switching convertersDynamic analysis of switching converters74 Switched models for the buck converter with a R esr

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Power switching convertersDynamic analysis of switching converters75 Switched models for the buck converter with a R esr

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Power switching convertersDynamic analysis of switching converters76 A nonlinear continuous equivalent circuit for the buck converter with a R esr

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Power switching convertersDynamic analysis of switching converters77 A linearized continuous equivalent circuit for the buck converter with a R esr The DC terms are The AC terms are

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Power switching convertersDynamic analysis of switching converters78 A linearized equivalent circuit using DC transformer with a turns-ratio of D

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Power switching convertersDynamic analysis of switching converters79 State-Space Averaged Model for an Ideal Boost Converter

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Power switching convertersDynamic analysis of switching converters80 State-Space Averaged Model for an Ideal Boost Converter

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Power switching convertersDynamic analysis of switching converters81 State-Space Averaged Model for an Ideal Boost Converter

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Power switching convertersDynamic analysis of switching converters82 Nonlinear continuous equivalent circuit of the ideal boost converter

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Power switching convertersDynamic analysis of switching converters83 Linearized equivalent circuit of the ideal boost converter

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Power switching convertersDynamic analysis of switching converters84 Linearized equivalent circuit of the ideal boost converter DC solutions

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Power switching convertersDynamic analysis of switching converters85 Linearized equivalent circuit of the ideal boost converter AC solutions small-signal averaged state-space equation

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Power switching convertersDynamic analysis of switching converters86 Linearized equivalent circuit of the ideal boost converter

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Power switching convertersDynamic analysis of switching converters87 Source-reflected linearized equivalent circuit for the ideal boost converter

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Power switching convertersDynamic analysis of switching converters88 Load-reflected linearized circuit for the ideal boost converter

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Power switching convertersDynamic analysis of switching converters89 DC transformer equivalent circuit for the ideal boost converter

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Power switching convertersDynamic analysis of switching converters90 Switching Converter Transfer Functions Source-to-State Transfer Functions

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Power switching convertersDynamic analysis of switching converters91 Switching Converter Transfer Functions Source-to-State Transfer Functions linearized control law

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Power switching convertersDynamic analysis of switching converters92 Switching Converter Transfer Functions BUCK CONVERTER

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Power switching convertersDynamic analysis of switching converters93 Switching Converter Transfer Functions BUCK CONVERTER

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Power switching convertersDynamic analysis of switching converters94 Switching Converter Transfer Functions BOOST CONVERTER

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Power switching convertersDynamic analysis of switching converters95 Switching Converter Transfer Functions BOOST CONVERTER

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Power switching convertersDynamic analysis of switching converters96 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

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Power switching convertersDynamic analysis of switching converters97 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

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Power switching convertersDynamic analysis of switching converters98 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 i th order low-pass Bessel filter for the transfer function, where i is the order of the system that is being designed

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Power switching convertersDynamic analysis of switching converters99 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

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Power switching convertersDynamic analysis of switching converters100 Design of a control system with complete state feedback Solution

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Power switching convertersDynamic analysis of switching converters101 Design of a control system with complete state feedback polesOL = eig(A) polesOL = 1000 * { -1.3298 + 2.4961i, -1.3298 - 2.4961i}

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Power switching convertersDynamic analysis of switching converters102 Design of a control system with complete state feedback Step response of the linearized buck converter sysOL=ss(A,B,C,0) step(sysOL)

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Power switching convertersDynamic analysis of switching converters103 Design of a control system with complete state feedback design the control strategy for voltage-mode control If we apply complete state feedback

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Power switching convertersDynamic analysis of switching converters104 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]

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Power switching convertersDynamic analysis of switching converters105 PSpice schematic

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Power switching convertersDynamic analysis of switching converters106 Transient response of the open-loop and closed-loop converters

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Power switching convertersDynamic analysis of switching converters107 Expanded view of the transient at 5 ms

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Power switching convertersDynamic analysis of switching converters108 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

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Power switching convertersDynamic analysis of switching converters109 Input EMI filters Circuit model of a buck converter with an input EMI filter

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Power switching convertersDynamic analysis of switching converters110 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

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Power switching convertersDynamic analysis of switching converters111 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

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Power switching convertersDynamic analysis of switching converters112 Input EMI filters The maximum output impedance of the input EMI filter, Z EMI,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

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Power switching convertersDynamic analysis of switching converters113 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

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Power switching convertersDynamic analysis of switching converters114 Input EMI filters Input EMI filter with LR reactive damping

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Power switching convertersDynamic analysis of switching converters115 Input EMI filters Input EMI filter with RC reactive damping

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Power switching convertersDynamic analysis of switching converters116 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

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Power switching convertersDynamic analysis of switching converters117 Input EMI filters A fourth-order input EMI filter with LR reactive damping

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Power switching convertersDynamic analysis of switching converters118 Input EMI filters Input impedance, Z in (f), of the buck converter and output impedance, Z EMI (f), of the input EMI filter

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Power switching convertersDynamic analysis of switching converters119 Part 2 Discrete-time models

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Power switching convertersDynamic analysis of switching converters120 Continuous-time and discrete-time domains continuous-time system The solution for the differential equation

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Power switching convertersDynamic analysis of switching converters121 Continuous-time and discrete-time domains the discrete-time expression

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Power switching convertersDynamic analysis of switching converters122 Continuous-time state-space model Equivalent circuit during t on : A 1

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Power switching convertersDynamic analysis of switching converters123 Continuous-time state-space model Equivalent circuit during t off : A 2

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Power switching convertersDynamic analysis of switching converters124 Continuous-time state-space model switching functions nonlinear model

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Power switching convertersDynamic analysis of switching converters125 Continuous-time state-space model small-signal model

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Power switching convertersDynamic analysis of switching converters126 Continuous-time state-space model steady-state equation perturbation in the state vector

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Power switching convertersDynamic analysis of switching converters127 Discrete-time model of the switching converter

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Power switching convertersDynamic analysis of switching converters128 Design of a discrete control system with complete state feedback The closed-loop poles can be arbitrarily placed by choosing the elements of F

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Power switching convertersDynamic analysis of switching converters129 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

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Power switching convertersDynamic analysis of switching converters130 Design of a discrete control system with complete state feedback Voltage mode control

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Power switching convertersDynamic analysis of switching converters131 Extended-state model for a tracking regulator Digital tracking system with full-state feedback

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Power switching convertersDynamic analysis of switching converters132 Current mode control Sensitivities of the duty cycle

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Power switching convertersDynamic analysis of switching converters133 Current mode control With complete state feedback

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Power switching convertersDynamic analysis of switching converters134 Extended-state model for a tracking regulator Digital tracking system with full-state feedback