PowerPoint-prжsentation

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Power Electronics
Small-signal converter modeling and frequency
dependant behavior in controller synthesis
by Dr. Carsten Nesgaard
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Agenda

• Small-signal approximation

• Voltage-mode controlled BUCK

• Converter transfer functions ? dynamics of
  switching networks

• Controller design (voltage-mode control)

• Discrete time systems

• Measurements

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Small-signal approximation
Advantages of small-signal approximation of
complex networks

• An analytical evaluation of equipment performance

• An analysis of equipment dynamics

• Stability

• Bandwidth

• A design oriented equipment synthesis

The linearization of basic AC equivalent circuit
modeling corresponds to the mathematical concept
of series expansion.
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Small-signal approximation
Drawbacks of small-signal approximation of
complex networks

• Limited to rather low frequencies (roughly fS/10)

• Inability to predict large-signal behavior

• Transients

• High frequency load steps

• Calculation complexity increases quite rapidly

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Voltage-mode controlled BUCK
Basic BUCK topology with closed feedback loop
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Voltage-mode controlled BUCK
Converter waveforms
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AC modeling
Converter states
0 lt t lt d?T d?T lt t lt (1 d)?T
   
KCL KVL
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AC modeling
Averaging and linearization (in terms of input
and output variables)
Inductor equation
 
Capacitor equation (same for both intervals)
Input current equation
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AC modeling
Resulting AC equivalent circuit
DC transformer relating input voltage and
inductor current, thus behaving almost like a
real transformer.
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Canonical AC model
Rearranging the AC equivalent circuit found on
the previous slide by the use of traditional
circuit theory a universal model can be
established
A similar model applies to a wide variety of
other converter topologies. In Fundamentals of
Power Electronics SE a table containing
coefficients for the different sources can be
found.
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Converter transfer functions
Basic control system
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Converter transfer functions
Opening the loop and rewriting the system
equations the following trans-fer functions can
be obtained
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State-space averaging
State variables
By definition the following apply
Output variable y (dependent)
Source variable u (independent)
In order to contain past information all
variables are functions of time
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State-space averaging
Averaging the equations previously found results
in the following non-linearized matrices
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State-space averaging
The averaged and linearized matrices can now be
identified
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State-space averaging
Averaging and linearizing the control variable d
(PWM controller) in terms of state variables
gives the following relation
is the sawtooth peak voltage is the EA gain, a
factor and comp.
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State-space averaging
Summarizing the voltage-mode controlled BUCK
matrices
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Stability
Nyquist stability requirement for closed loop
systems Prh(GCL(s)) Prh(GOL(s)) ?
0 Where Prh number of right half-plane
poles ? number of times the Nyquist contour
of the open- loop transfer function circles
the point (-1,0) GCL Closed-loop transfer
function GOL Open-loop transfer function
Minimum open-loop transfer function gain
margin 6 - 8 dB Minimum open-loop transfer
function phase margin 30? - 60?
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Voltage-mode controlled BUCK
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Voltage-mode controlled BUCK
A plot of the open-loop transfer function is
shown below (K 1)
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Voltage-mode controlled BUCK
A 3D plot of the converter filter transfer
function is shown to the right.
Note The zero caused by RESR
increases the phase (green curve) as a function
of frequency and RESR. Unfortunately due to the
same zero the filter attenuation drops (red
curve).
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Compensation
PI-comp. Lag-comp. PD comp. Lead-comp. A
PI-Lead-comp. (PID) will be used in this
presentation
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Compensation
Widely accepted error amplifier configuration
Pole at f 0 for increased DC gain Pole at fESR
for compensation Double zero at resonance peak
for increased phase margin
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Compensation
Compensator and converter transfer functions
Amplitude Phase
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Voltage-mode controlled BUCK
Using the previously derived matrices an
expression for the input impedance Zin can be
established
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Voltage-mode controlled BUCK
A plot of the open-loop transfer function during
Discontinuous Conduction Mode (red curve) and EA
compensation (blue curve)
DCM reduces the converter transfer function to a
first order system, since the time derivative of
the small-signal inductor current is zero and
thus disqualifies the inductor current as a state
variable.
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Discrete time systems
Transient response and the relationship between
the s-plane and the z-plane
Discrete time Continuous time
Inserting into the expression to the left, it can
be seen that the continuous time stability
requirement maps onto the z-plane in form of the
unit circle.
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Discrete time systems
Arithmetic and operations

• Integration and differentiation
• Plotting the frequency response
• Tustins rule
• Sampling rate

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Discrete time systems
Plot of the discrete compensation transfer
function
Cont Continuous time Disc_1 Discrete time with
sample frequency 50 kHz (no prewarping) Disc_2 D
iscrete time with sample frequency 100 kHz (no
prewarping)
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Measurements
Below is a comparison of the predicted continuous
time loop gain, predicted discrete time loop gain
and an actual measurement of the loop gain
GH Continuous time GD_2 Discrete time with sample
frequency 50 kHz (no prewarping) Meas Actual
measurement
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Measurements
The same transfer function as before, but during
Discontinuous Conduction Mode
GH Continuous time Meas Actual measurement