Different Degrees of Simulation Detail 1.
Power Power circuit modeled as linear transfer function Small signal behavior No switching, no harmonics !
2.
Controller design
Power Power circuit modeled with ideal components Large signal behavior, voltage and current waveforms Overall system performance !
3.
Circuit design and controller verification
Power Power circuit with manufacturer specific components Parasitic effects (magnetic hysteresis) Switching transitions (diode reverse recovery) Component stress (electrical or thermal) !
Choice of components
Power input
v i
ii
Power converter
Power output
io
Control signals
Controller 3
Comparison - SPICE and PLECS Passive Passive component models are similar R, L, C
The main difference is the semiconductor model MOSFET model - SPICE Detailed physical device model Has 47 parameters
MOSFET model - PLECS Simplified behavioral model Two parameters: R on and T
4
v o
Load
Measurement Reference
High Speed Simulations with Ideal Switches Conventional continuous diode mode Arbitrary static and dynamic characteristic Snubber often required
Ideal diode model in PLECS Instantaneous on/off characteristic Optional on-resistance and forward for ward voltage 5
Comparison: Diode Rectifier Simulation with conventional and ideal switches
Simulation steps: 1160 153 153 Computation time: 0.6s 0.08s "
"
6
State Space Model: Buck Converter State space description
Switch conducting
Diode conducting
7
Variable Time-Step Simulation: Buck Converter
Transistor Transistor conducts Diode blocks
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Variable Time-Step Simulation: Buck Converter
Transistor Transistor opens Impulsive voltage across inductor
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Variable Time-Step Simulation: Buck Converter
Impulsive voltage closes diode
10
Variable Time-Step Simulation: Buck Converter
Transistor Transistor open Diode conducts
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Variable Time-Step Simulation: Buck Converter
Switch timing Problem: Diode opens too late Impulsive voltage across inductor
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Variable Time-Step Simulation: Buck Converter
Zero-Crossing Detection: Time-step is reduced Diode opens exactly at the zero-crossing
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Operating Principle of PLECS Circuit transformed into state-variable system One set of matrices per switch combination Solver
1 s
y u
B
A
C
D g
PLECS S-function 14
r e g a n a m h c t i w S
Solver Overview
Continuous solvers Taylor series polynomial Step size control Acceptable error Relative, absolute tolerance Output refining
Discrete solvers Trapezoidal rule Step size selection 15
Taylor Series Ser ies Expansion Approximate a continuous function with a higher order polynomial The higher the order, the more accurate the solution
Taylor series:
1st order
5th order y=sin(x)
Source: Wikipedia
16
Continuous Solver Operation y(t) can be constructed using in a piecewise fashion using p1(t) and p2(t), which are Taylor series polynomials A continuous solver calculates the point y n+1 by calculating the equivalent Taylor series for p1(t) An nth order solver has the same accuracy as an nth order Taylor series. '#
%$"#
%$ !&#
%$ '()"
'(
*()"
*( $ ()"
$ ( 17
Solver Settings
18
$
Acceptable Error Local error Difference between 4th and 5th order solutions
Acceptable error Defines the local error limit Determined by tol rel except for small state values
Result valid if:
Local error
Acceptable error
19
Step Size Control Step size automatically controlled by the solver (variable step) Goal: Keep error within acceptable error limits Key advantage: Accuracy directly specified by the user
Step size calculated using Relative error, ! Relative tolerance, tol rel Previous time step, h old
hold
20
Tolerances Relative tolerance Determines acceptable error limit when x > 0 Start with 10-3 (0.1%) -16 Numerical limit is 10 -16
Absolute tolerance Best to set to auto
21
Example LC circuit Analytical solution:
Result: tolrel = 1e-3
22
Example LC circuit Analytical solution:
Result: tolrel = 1e-6
23
Refining the Output Option 1: Reduce tolrel or time step Solver must recalculate polynomial coefficients at each time step Less efficient
Option 2: Increase refine factor Solver uses existing polynomial coefficients to calculate additional points. More efficient
24
Solver Families Non-stiff Inherently more efficient (no iteration it eration required) Smaller stability domain Forward Euler - 1st order ode45/Dopri - 5th order
Stiff Less efficient (iteration required) Larger stability domain Backward Euler - 1st order Radau/ode23tb - 5th order
25
Forward Euler Truncate the Taylor Series after the first term:
1st order accurate Explicit integration integration algorithm
26
Numerical Experiment Scalar system Analytical solution Forward Euler
Unstable for ah<-2
27
Stability Analysis Autonomous LTI system Analytical solution Analytically stable if
28
Stability Domain of Forward For ward Euler Integration algorithm
Stable if the eigenvalues of are inside the unit circle F are I.e. if the eigenvalues of Ah are inside a unit circle around (-1, 0) 29
Analysis of RLC Example
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Floating Point Arithmetic
What is 0.3 - 0.2 - 0.1 0.1 ?
What MATLAB calculates: 0.299999999999999989 -
0.200000000000000011
-
0.100000000000000006
= -0.000000000000 -0.000000000000000028 000028
31
Floating Point Arithmetic (2)
Summation of floating point numbers a +
b
+
c
a+b+c
=
32
Backward Euler Develop Taylor Series around a time in the future:
Must be solved iteratively for the unknown x (t*+h) Implicit integration integration algorithm
33
Numerical Experiment (Revisited) Scalar system Analytical solution Backward Euler
34
Stability Domain of Backward Euler Integration algorithm
Stable if the eigenvalues of Ah are outside a unit circle around (1, 0)
35
Stability Domain Dopri
Tustin
Radau
36
Higher Order Integration Methods E.g. Explicit Midpoint Rule
Two stages (one intermediate, one final) 2nd order accurate More accurate than two FE steps of half size!
37
Event Detection Define auxiliary functions that help locate discontinuities. Function zeros must coincide with the discontinuities.
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Missed Events - Problem
A
B
0 Time step1
Time step2
Output 0
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Missed Events - Solution
A
0
1 Output 0
40
B
PWM Generation
A
B
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Trapezoidal Rule
"&'( "& "#$% ! $ &
$ &'(
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Time Step Selection Accuracy is indirectly determined by the time step To ensure accuracy, reduce the time step and observe any changes in the output Or: Compare with a continuous simulation
Continuous waveform Highest transient frequency constrains sample time Set tsample < ttransient /10 /10 Integration underestimate approx 3%
Switched system Switches turned on at sample instants Set tsample < tsw /100 /100
43
Comparison - Continuous and Discrete solver Time steps: Underdamped RLC circuit
Continuous: tolrel = 10-6
Simulated current
Discrete: ts = 50" s
44
Conclusion Variable Variable step solver operation Causes of model stiffness Solutions to stiffness Explicit (non-stiff) vs. Implicit (stiff) solvers Limits of stability Limits of numerical accuracy Event detection
45
Tips to Achieve a Fast Simulation
Plexim GmbH
46
Introduction The problem Large simulation models can become slow Many states, detailed component models
Speeding up the simulation Model simplification Model tuning Real-time accelerator Averaged modeling Efficient PWM generation
Example - Three phase inverter
47
Simplification of PV Cell Model Exact model: accounts for non-linear VI characteristic Needed for implementing MPPT simulation Temperature Temperature and insolation dependency also impor tant
Insolation = 1000 W/m2 T = 25 °C MPP occurs at 17.6 V
•
•
•
48
Diode PV Cell Model Current = fn(voltage, insolation, temp) Cell model Shockley diode eq.
I
Reference: 49
PV Lookup Table Implementation Simplified implementation Voltage-controlled current source Capacitor included to add a state and avoid an algebraic loop Precalculated PV data
50
Model Tuning Simulation settings. Choose the correct solver. Blockset
Standalone
Non-stiff
ode45
Dopri
Stiff
ode23tb/ode15s
Radau
Variable time step solver. solver. Reducing the switching frequency reduces the computational overhead.
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Real Time Accelerator Converts model into c-code. Compiles executable. executable. Runs as native rather than interpreted program.
PLECS circuit converted into C-code. Automatically integrated with Simulink c-code.
Speed gain = 2 - 5 times. Generate and compile takes 20 seconds.
52
Averaged Av eraged Converter Modeling Averaged Averaged converter models Retain the low frequency dynamics. Averages switching action over one cycle. Requires specialized knowledge of topology. Speed gains are an order of magnitude.
Procedure Average inverter model. Average space vector modulation. Break algebraic loops.
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Averaging Av eraging a Three Phase Converter Full switching model Simulates exact electrical dynamics. Switching action slows down the simulation where slower dynamics are studied.
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Inverter Leg
d.T
Vdc
d.Vdc
ia
d.ia
d=0.3
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Averaged Av eraged Inverter Leg Va modeled using as a controlled voltage source idc modeled as a controlled current source
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Averaged Av eraged Inverter Model Assumes AC voltages are ideally imposed Used for rotor-side and grid-side converter
Averaging of inverter legs A and B
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Averaging Space Vector Modulation Converter Vdq* to ma, mb, mc Use 3 phase PWM. Extend operating range using overmodulation. overmodulation.
Approximation of space vector modulation with a continuous modulation index
58
Overmodulation Add zero-sequence signal to modulation index
Reference: Vector Control of a Double-Sided PWM Converter and Induction Machine Drive, R. Ottersten
59
Algebraic Loops What? Multiple direct feedthrough blocks connected in a loop. PLECS continuous inputs appear as a direct feedthrough system to Simulink. In feedback control systems, this will cause an algebraic loop.
Problem: Algebraic loops must be solved iteratively The simulation time of each step is increased
60
Breaking Algebraic Loops Option 1. Use memory block or similar Time delay causes instability. instability. Step size may need to be limited.
Option 2. Use a low pass filter The extra state breaks the algebraic loop. Time constant must be chosen with care. Provides more deterministic behavior with variable step simulation.
Breaking an algebraic loop with a low-pass filter. 61
Average Av erage DC-DC Converter Modeling Average Average switch cell modeling Replace switch/diode combination with a voltage current source. Ref: Fundamentals of Power Power Electronics, Erickson & Maksimovic
Equations I2 = (1 - d)/I1 V1 = (1 - d)/V 2
62
Simplification of Current Controlled Converters Boost converter is current controlled Can replace the inductor with a current source.
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Efficient PWM Generation in PLECS Sampled PWM
Continuous PWM
Doubly fed induction generator example, 0.1 0.1 s simulation, PLECS Blockset Sampled PWM: 29.6 s => 52 % faster Continuous PWM: 45.1 s 64
Example - Three Phase Boost Rectifier Control aims Sinusoidal input currents Regulated dc bus voltage
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Block Diagram
PWM
•Stiff
system •Simulated for 0.2 s in PLECS Blockset & Standalone 66
Results - PLECS Blockset Controls in Simulink
•ode23tb
solver used (stiff) 67
Results - PLECS Standalone
•Radau
solver used (stiff) 68
Conclusion Choose the right solver for the job (stiff or non-stiff) Use sampled PWM Make appropriate simplifications For extremely large models: PLECS Standalone Real-time workshop Averaged converter modeling
69
An Introduction to Thermal Modeling and Simulation
Plexim GmbH
70
Why Thermal Simulation?
Reduce development time Predict performance perf ormance and investigate investigate key tradeoffs early in the t he design process Thermal measurements can be difficult and time consuming
71
Sources of Thermal Losses Passive Passive components compo nents Resistive power loss: p loss(t) = vR(t) iR(t) Loads e.g. break resistors Filters Winding resistance
Power Power semiconductors Conduction loss Switching loss
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Semiconductor Losses Gate signal
Switching loss Conduction loss
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Switching Losses Switching energy loss dependent on: Blocking voltage, device current, junction temperature, gate drive Eon = f(Vce, Ic, Tj, Rg)
Turn on
Turn off 74
Example IGCT Turn-of Turn-off: f: Varying Varying Stray Inductance kV
kA 3.0
300 nH (10.5 Ws)
4.5
800 nH (12 Ws)
VPK = 3800V
1500 nH (13.5 Ws)
3.0
2.0
VDC = 2 kV
1.5
1.0
TJ = 125°C
0.0
0.0 5
Courtesy ABB
10
15
µs
tf ! 2.5µs, ttail ! 7µs
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Switching Loss Calculation from Transients Transients Accurate physical device models required Often unavailable
Physical parameters often unknown during design phase. Stray inductance of PCB
Small simulation steps required Large computation times
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Lookup Table Table Approach for Switching Losses Instantaneous switching maintained for speed. Switching losses are read from a database after switching event. Esw = f(Tj, vblock, ion) (Rg = const)
77
Example Lookup Table Turn-off loss a function of: Current before switching Voltage after switching Temperature at switching Rg is assumed constant
Exact loss found using interpolation Note the voltage and current polarities!
78
Semiconductor Conduction Losses On-state loss Conduction profile is nonlinear: von = f(ion, Tj).
Conduction profile stored in lookup table Exact voltage found using interpolation Conduction power loss: ploss(t) = von(t) ion(t)
Off-state loss Negligible - low leakage current
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Simulation of an Electrical-Thermal Model
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Combined Electrical-Thermal Simulation Thermal and electrical domains not coupled Semiconductor losses don’t appear in electrical circuit Energy conservation can be maintained by subtracting thermal losses from electrical circuit
Switching losses added over zero time Filtered by large capacitance of thermal circuit 81
Semiconductor Thermal Behavior
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Verification of Lookup Table Approach IGBT loss modeling using an ideal switch and lookup table Simulation speed increased by a factor of 20 Good agreement with measurements measurements
Pt - simulated IGBT loss Ptm - measured IGBT loss
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Thermal Modeling of a Pulsed Resonant Convert Converter er Simulation results
"t
"t
= 1.4 !
= 1.4 !
Experimental results Acknowledments: Dr. Fabio Carastro, University of Nottingham 84
A combined electrical-thermal model Losses, heatsink, ambient temperature. How are these represented using PLECS?
85
Thermal Domain Thermal circuit analogous to electrical circuit. Thermal and electrical circuits solved simultaneously.
86
A complete electrical-thermal model The heatsink is the interface between the two domains Automatically absorbs component losses Propagates temperature temperature back to semiconductors
Thermal impedance modeled with RC elements
87
Hierarchical Modeling of Thermal Structures Junctions
IGBT Plate
Heatsink
Dual IGBT module
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Different Thermal Equivalent Networks Cauer equivalent
Foster equivalent Curve fitting approach based on heating and cooling characteristics. No correspondence between R th,n resp. Cth,n and the physical structure!
Physics based thermal equivalent circuit Each Rth and Cth pair represents a physical layer in the thermal circuit.
Any modification of the system requires recalculation of all values
89
Junction-Case Thermal Impedance Define in thermal editor to observe junction temp fluctuations Foster coefficients often given in datasheet
Foster network coefficients
Example junction-case thermal impedance
90
Foster Network Pitfalls Only accurate if reference point x is a constant temperature Cannot be arbitrarily extended beyond point x Tjunc immediately affected by temperature changes at x
Reference: M. März and P. Nance, “Thermal modeli ng of power electronic systems,” www.iisb.fraunhofer.de 91
Solution 1 - Use First Order Cauer Network Calculate # from 63% R value C = # /R
Vc reaches 63.2% Vfinal after #
92
Solution 2 - Use a Constant-Temperatur Constant-Temperature e Heatsink Set Cheatsink to 0 Set heatsink temperature with a const temperature source Foster network can be used for junction thermal impedance
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Measuring Average Device Losses Concept Calculate total switching and conduction energy lost during a sw cycle Output as an average power pulse during the next cycle
Implementation Based on a C-Script block Conduction and switching losses measure with a Probe
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Challenge: Large Thermal Time Constants Thermal time constants: 0.1 … 100 s Switching frequency: 1 … 100 kHz Long simulation time until thermal steady-state is reached Example
95
Newton Raphson Analysis Approach: Finding the roots of
: initial state vector : final state vector after a time T Iterative solution
Jacobian J calculated numerically requires n+1 simulation runs (for n state variables) 96
Newton iteration
Newton Raphson: Convergen Convergence ce Typically converges converges after af ter <10 iterations
97
Newton Raphson: Requirements for Conver Convergence gence The system must be convergent Example problem PLL model Angle is a ramp signal towards infinity !
Solution: create a periodic signal with a self-resetting integrator
98
Thermal Loss Feedback 1 P1,Pi=100W
Po=100W
Efficiency:
99
Thermal Loss Feedback 2 P1=105W
Pi=100W
Po,P2=100W
Efficiency:
100
P2=95W
Thermal Loss Feedback 3 P1=105.3W
Pi=100W
Po,P2=100W
Thermal feedback: Efficiency: 101 101
Physical Rds Implementation Rds constant during switching cycle Resistance resolution:
102
Obtaining Switching Loss Data Experimental measurements Switching losses highly dependent on gate drive dr ive circuit and stray parameters Use a switching loss setup to characterize loss dependency on voltage and current for two temperatures
Datasheets Given for a specific gate resistance and stray inductance Good approximations can be made by extrapolating manufacturers data (or asking for complete loss measurements)
103
Conclusion Fast & accurate thermal simulation using lookup tables Operation of a combined electrical-thermal simulation Calculating average device losses Steady state analysis using Newton Raphson analysis
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Advanced PLECS Tools
Plexim GmbH
105
Advanced Tools C-Script block Control design tools Steady-state analysis AC sweep Impulse response analysis Loop gain analysis Other Subsystem - custom components
106
Small Signal Analysis Tools Possible Possible workflow for controller design using PLECS tools: Converter Open Loop Transfer Function
AC Sweep Analysis Impulse Response Analysis
Voltage controller design Freq domain
Calculate loop gain
Loop Gain Analysis plbode function
Gain/phase margins met?
Verify with simulation
107
Frequency Sweep Algorithm: For each frequency: 1. Run Run steady steady-st -state ate anal analysi ysiss 2. Apply sinusoidal perturbation 3. Extract system response using Fourier analysis
Caveats: Period length: least common multiple of system period and perturbation period Computationally expensive expensive
108
Impulse Response Analysis Impulse response of a buck converter
Transfer Transfer function: 109
Impulse Response Analysis The original Laplace transform:
Apply some math …
Reference: D.Maksimovic, „Computer-Aided Small-Signal Analysis Based on Impulse Response of DC/DC Switching Power Converters“, IEEE Trans. On Power Electronics, Vol. 15, No. 6, Nov. 2000, pp. 1183-1191 110
Buck Converter: Tr Transfer ansfer Function
111
Stability Verification Loop Gain Use Loop Gain Analysis Tool (Blockset) AC Sweep (Standalone), or Calculate using plbode function Measure gain and phase margins to check chec k stability
112
Custom Components Why? Support the top-down design approach approach Easy to reuse, configure and measure
Custom component features Formed using the subsystem User defined parameter, icon, initialisation commands and probe signals !
All machines in PLECS are custom components
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Example - PV String Model
4 3.5 3 2.5
) A ( t 2 n e r r u1.5 C
Current surface data in lookup table
1 0.5 0 1
0.5
0 Insolation (kW/m2)
25
20
15 Voltage (V)
114
10
5
0
Masked Subsystem Implementation
115
C-script Block
Plexim GmbH
116
What can it do? C-control code emulation Model custom components Efficient sequence generation State machine modeling
117
Function call interface Solver operation (PLECS or Simulink) Some function calls dependent on continuous or discrete states
118
Function calls in C-script block Main loop Always executed Output() called at least once Update() called after Output if discrete state variables exist
Integration loop Executed if continuous states defined in settings Used for solving continuous differential diff erential equations Continuous states used for declaring DEs
Event detection loop Executed if a zero crossing signal defined in settings For detecting the exact instant of a discontinuity
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C-Script parameter window
120
Code editor window
121
Sample time setting Discrete Value: +ve Fixed sample time
Continuous Value: 0 Inherited from solver
Variable Value: -2 At each time step the NextSampleHit must be specified.
122
Hybrid fixed and variable step setting Fixed and variable time Format: [1/fs,0; -2, 0] First row element is the time setting Second element is the offset time
Continuous, only major time steps Format: [0, -1] C-Script not called during integration or event detection loop Eliminates unnecessary calls for certain blocks. Use for blocks with discrete output. Example: Lookup table
123
Pitfalls - cascading C-Script blocks Controller cycle hit time - fixed step: Cycle time obtained by multiplication: tn = 1/fs*n
PWM cycle hit time - variable step: Cycle time is obtained by addition: tn = 1/fs + 1/fs
+ ...
n
Synchronization may be lost due to accumulated rounding error
Solution Use a hybrid fixed/variable step setting for PWM block
124
Macros For interacting with model or solver Examples Input(0) Output(0) NextSampleHit CurrentTime
125
Example - Space Vector Control Three phase boost rectifier Sinusoidal input currents Regulated dc bus voltage
126
Control Strategy
•Outer
voltage control loop decoupled current control loop •Space vector modulation •Inner
127
Space vector modulation Timing calculation Switch signal generation
C-script block2 C-script block1 128
Efficient sequence generation Typical switching sequence Space vector modulation Option blanking delay
Method 1: Fixed time step Test if t > t hit, apply new switch signal Requires small steps for accuracy => increased computational overhead
Method 2: Variable time step Calculate hit times at beginning of switching sequence Fewer simulation steps – more efficient
129
Example: PWM with blanking Calculate transition times, t1,t2,t3,t4 at beginning of cycle State machine program executes at each transition: Switching signal is updated NextSampleHit is updated
Bipolar switching signal
1: S1, S4 on -1: S2, S3 on
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Efficient state machine implementation Event driven Useful for controller mode sequencing. Example: Startup, overload, shutdown. shutdown. Relational operator - generates major step External event
> v1
External event
C-Script State machine
Time setting: Simulation model: variable C-Script: continuous
> i1
if (IsMajorStep) (IsMajorStep) test_input(v1,i1); update_state_machine();
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Conclusion Implement complex nonlinear and/or piecewise functions without complex block diagrams Model custom components and controls Generate efficient sequencing with wit h exact but flexible time step control State machine modeling Incorporate external C code for hardware controllers
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