Chapter 6 Dynamic analysis of switching converters
Slide 2
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
Slide 3
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
Slide 4
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
Slide 5
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
Slide 6
Power switching convertersDynamic analysis of switching
converters6 Linear model of a switching converter
Slide 7
Power switching convertersDynamic analysis of switching
converters7 PWM modulator model Sensitivity of the duty cycle with
respect to v ref Voltage-mode control
Slide 8
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
Slide 9
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
Slide 10
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
Slide 11
Power switching convertersDynamic analysis of switching
converters11 Averaged switching converter models Three-terminal
averaged-switch model Averaged-switch model for voltage-mode
control
Slide 12
Power switching convertersDynamic analysis of switching
converters12 Averaged switching converter models Examples of
switching converters with an averaged switch
Slide 13
Power switching convertersDynamic analysis of switching
converters13 Averaged switching converter models Small-signal
averaged-switch model for the discontinuous mode
Slide 14
Power switching convertersDynamic analysis of switching
converters14 Averaged switching converter models Small-signal model
for current-mode control
Slide 15
Power switching convertersDynamic analysis of switching
converters15 Output filter model Output filter of a switching
converter
Slide 16
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
Slide 17
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
Slide 18
Power switching convertersDynamic analysis of switching
converters18 Output filter model Output filter with a capacitor
Resr
Slide 19
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
Slide 20
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
Slide 21
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).
Slide 22
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.
Slide 23
Power switching convertersDynamic analysis of switching
converters23 Example 6.4
Slide 24
Power switching convertersDynamic analysis of switching
converters24 Example 6.4 Bode plot of the small-signal transfer
function of the boost converter
Slide 25
Power switching convertersDynamic analysis of switching
converters25 Small-signal models of switching converters
Slide 26
Power switching convertersDynamic analysis of switching
converters26 Small-signal models of switching converters
Slide 27
Power switching convertersDynamic analysis of switching
converters27 Small-signal models of switching converters
Slide 28
Power switching convertersDynamic analysis of switching
converters28 Small-signal models of switching converters
Slide 29
Power switching convertersDynamic analysis of switching
converters29 Small-signal models of switching converters
Slide 30
Power switching convertersDynamic analysis of switching
converters30 Small-signal models of switching converters
Slide 31
Power switching convertersDynamic analysis of switching
converters31 Small-signal models of switching converters
Slide 32
Power switching convertersDynamic analysis of switching
converters32 Small-signal models of switching converters
Slide 33
Power switching convertersDynamic analysis of switching
converters33 Review of negative feedback Block diagram
representation for a closed-loop system
Slide 34
Power switching convertersDynamic analysis of switching
converters34 Review of negative feedback Closed-loop gain Loop gain
For T L >>1 Stability analysis
Slide 35
Power switching convertersDynamic analysis of switching
converters35 Relative stability Definitions of gain and phase
margins
Slide 36
Power switching convertersDynamic analysis of switching
converters36 Relative stability Loop gain of a system with three
poles
Slide 37
Power switching convertersDynamic analysis of switching
converters37 Closed-loop switching converter
Slide 38
Power switching convertersDynamic analysis of switching
converters38 Feedback network
Slide 39
Power switching convertersDynamic analysis of switching
converters39 Error amplifier compensation networks PI Compensation
network The total phase lag
Slide 40
Power switching convertersDynamic analysis of switching
converters40 Error amplifier compensation networks Frequency
response of the PI compensation network
Slide 41
Power switching convertersDynamic analysis of switching
converters41 Error amplifier compensation networks Phase response
of the PI compensation network
Slide 42
Power switching convertersDynamic analysis of switching
converters42 Error amplifier compensation networks PID Compensation
network
Slide 43
Power switching convertersDynamic analysis of switching
converters43 Error amplifier compensation networks Magnitude
response of the PID compensation network
Slide 44
Power switching convertersDynamic analysis of switching
converters44 Error amplifier compensation networks Magnitude
response of the PID compensation network
Slide 45
Power switching convertersDynamic analysis of switching
converters45 Error amplifier compensation networks Phase response
of the PID compensation network
Slide 46
Power switching convertersDynamic analysis of switching
converters46 Error amplifier compensation networks Asymptotic
approximated magnitude response of the PID compensation
network
Slide 47
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
Slide 48
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
Slide 49
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
Slide 50
Power switching convertersDynamic analysis of switching
converters50 Compensation in a buck converter with output capacitor
ESR
Slide 51
Power switching convertersDynamic analysis of switching
converters51 Compensation in a buck converter with no output
capacitor ESR
Slide 52
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
Slide 53
Power switching convertersDynamic analysis of switching
converters53 Linear model of a voltage regulator including external
perturbances audio susceptibility output impedance
Slide 54
Power switching convertersDynamic analysis of switching
converters54 Output impedance and stability Output impedance
Slide 55
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
Slide 56
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
Slide 57
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
Slide 58
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
Slide 59
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.
Slide 60
Power switching convertersDynamic analysis of switching
converters60 State-Space Averaged Model for an Ideal Buck
Converter
Slide 61
Power switching convertersDynamic analysis of switching
converters61 State-Space Averaged Model for an Ideal Buck
Converter
Slide 62
Power switching convertersDynamic analysis of switching
converters62 State-Space Averaged Model for an Ideal Buck
Converter
Slide 63
Power switching convertersDynamic analysis of switching
converters63 A nonlinear continuous equivalent circuit of the ideal
buck converter
Slide 64
Power switching convertersDynamic analysis of switching
converters64 A linear equivalent circuit of the ideal buck
converter
Slide 65
Power switching convertersDynamic analysis of switching
converters65 A linear equivalent circuit of the ideal buck
converter
Slide 66
Power switching convertersDynamic analysis of switching
converters66 A source-reflected linearized equivalent circuit of
the ideal buck converter
Slide 67
Power switching convertersDynamic analysis of switching
converters67 A linearized equivalent circuit of the ideal buck
converter using a DC transformer
Slide 68
Power switching convertersDynamic analysis of switching
converters68 State-space averaged model for the discontinuous-mode
buck converter
Slide 69
Power switching convertersDynamic analysis of switching
converters69 State-space averaged model for the discontinuous-mode
buck converter
Slide 70
Power switching convertersDynamic analysis of switching
converters70 A nonlinear continuous equivalent circuit for the
discontinuous-mode buck converter
Slide 71
Power switching convertersDynamic analysis of switching
converters71 A nonlinear continuous equivalent circuit for the
discontinuous-mode buck converter
Slide 72
Power switching convertersDynamic analysis of switching
converters72 A linearized equivalent circuit for the
discontinuous-mode buck converter
Slide 73
Power switching convertersDynamic analysis of switching
converters73 State-Space Averaged Model for a Buck Converter with a
Capacitor ESR
Slide 74
Power switching convertersDynamic analysis of switching
converters74 Switched models for the buck converter with a R
esr
Slide 75
Power switching convertersDynamic analysis of switching
converters75 Switched models for the buck converter with a R
esr
Slide 76
Power switching convertersDynamic analysis of switching
converters76 A nonlinear continuous equivalent circuit for the buck
converter with a R esr
Slide 77
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
Slide 78
Power switching convertersDynamic analysis of switching
converters78 A linearized equivalent circuit using DC transformer
with a turns-ratio of D
Slide 79
Power switching convertersDynamic analysis of switching
converters79 State-Space Averaged Model for an Ideal Boost
Converter
Slide 80
Power switching convertersDynamic analysis of switching
converters80 State-Space Averaged Model for an Ideal Boost
Converter
Slide 81
Power switching convertersDynamic analysis of switching
converters81 State-Space Averaged Model for an Ideal Boost
Converter
Slide 82
Power switching convertersDynamic analysis of switching
converters82 Nonlinear continuous equivalent circuit of the ideal
boost converter
Slide 83
Power switching convertersDynamic analysis of switching
converters83 Linearized equivalent circuit of the ideal boost
converter
Slide 84
Power switching convertersDynamic analysis of switching
converters84 Linearized equivalent circuit of the ideal boost
converter DC solutions
Slide 85
Power switching convertersDynamic analysis of switching
converters85 Linearized equivalent circuit of the ideal boost
converter AC solutions small-signal averaged state-space
equation
Slide 86
Power switching convertersDynamic analysis of switching
converters86 Linearized equivalent circuit of the ideal boost
converter
Slide 87
Power switching convertersDynamic analysis of switching
converters87 Source-reflected linearized equivalent circuit for the
ideal boost converter
Slide 88
Power switching convertersDynamic analysis of switching
converters88 Load-reflected linearized circuit for the ideal boost
converter
Slide 89
Power switching convertersDynamic analysis of switching
converters89 DC transformer equivalent circuit for the ideal boost
converter
Slide 90
Power switching convertersDynamic analysis of switching
converters90 Switching Converter Transfer Functions Source-to-State
Transfer Functions
Slide 91
Power switching convertersDynamic analysis of switching
converters91 Switching Converter Transfer Functions Source-to-State
Transfer Functions linearized control law
Slide 92
Power switching convertersDynamic analysis of switching
converters92 Switching Converter Transfer Functions BUCK
CONVERTER
Slide 93
Power switching convertersDynamic analysis of switching
converters93 Switching Converter Transfer Functions BUCK
CONVERTER
Slide 94
Power switching convertersDynamic analysis of switching
converters94 Switching Converter Transfer Functions BOOST
CONVERTER
Slide 95
Power switching convertersDynamic analysis of switching
converters95 Switching Converter Transfer Functions BOOST
CONVERTER
Slide 96
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
Slide 97
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
Slide 98
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
Slide 99
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
Slide 100
Power switching convertersDynamic analysis of switching
converters100 Design of a control system with complete state
feedback Solution
Slide 101
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}
Slide 102
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)
Slide 103
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
Slide 104
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]
Slide 105
Power switching convertersDynamic analysis of switching
converters105 PSpice schematic
Slide 106
Power switching convertersDynamic analysis of switching
converters106 Transient response of the open-loop and closed-loop
converters
Slide 107
Power switching convertersDynamic analysis of switching
converters107 Expanded view of the transient at 5 ms
Slide 108
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
Slide 109
Power switching convertersDynamic analysis of switching
converters109 Input EMI filters Circuit model of a buck converter
with an input EMI filter
Slide 110
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
Slide 111
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
Slide 112
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
Slide 113
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
Slide 114
Power switching convertersDynamic analysis of switching
converters114 Input EMI filters Input EMI filter with LR reactive
damping
Slide 115
Power switching convertersDynamic analysis of switching
converters115 Input EMI filters Input EMI filter with RC reactive
damping
Slide 116
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
Slide 117
Power switching convertersDynamic analysis of switching
converters117 Input EMI filters A fourth-order input EMI filter
with LR reactive damping
Slide 118
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
Slide 119
Power switching convertersDynamic analysis of switching
converters119 Part 2 Discrete-time models
Slide 120
Power switching convertersDynamic analysis of switching
converters120 Continuous-time and discrete-time domains
continuous-time system The solution for the differential
equation
Slide 121
Power switching convertersDynamic analysis of switching
converters121 Continuous-time and discrete-time domains the
discrete-time expression
Slide 122
Power switching convertersDynamic analysis of switching
converters122 Continuous-time state-space model Equivalent circuit
during t on : A 1
Slide 123
Power switching convertersDynamic analysis of switching
converters123 Continuous-time state-space model Equivalent circuit
during t off : A 2
Slide 124
Power switching convertersDynamic analysis of switching
converters124 Continuous-time state-space model switching functions
nonlinear model
Slide 125
Power switching convertersDynamic analysis of switching
converters125 Continuous-time state-space model small-signal
model
Slide 126
Power switching convertersDynamic analysis of switching
converters126 Continuous-time state-space model steady-state
equation perturbation in the state vector
Slide 127
Power switching convertersDynamic analysis of switching
converters127 Discrete-time model of the switching converter
Slide 128
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
Slide 129
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
Slide 130
Power switching convertersDynamic analysis of switching
converters130 Design of a discrete control system with complete
state feedback Voltage mode control
Slide 131
Power switching convertersDynamic analysis of switching
converters131 Extended-state model for a tracking regulator Digital
tracking system with full-state feedback
Slide 132
Power switching convertersDynamic analysis of switching
converters132 Current mode control Sensitivities of the duty
cycle
Slide 133
Power switching convertersDynamic analysis of switching
converters133 Current mode control With complete state
feedback
Slide 134
Power switching convertersDynamic analysis of switching
converters134 Extended-state model for a tracking regulator Digital
tracking system with full-state feedback