Transcript of Chapter 6 Dynamic analysis of switching converters.
- Slide 1
- 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