an introduction to solvers plecs

7/25/2019 An Introduction to Solvers PLECS

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1

An Introduction to Solvers

Plexim GmbH

Outline

Variable-step simulation

Ideal switches

Piecewise linear simulation

Non-stiff, stiff solvers

Stability domain

Event detection

2


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3

Different Degrees of Simulation Detail

1. Power circuit modeled as linear transfer function

Small signal behavior

No switching, no harmonics

! Controller design

2. Power circuit modeled with ideal components

Large signal behavior, voltage and current waveforms

Overall system performance

! Circuit design and controller verification

3. Power circuit with manufacturer specific components

Parasitic effects (magnetic hysteresis)

Switching transitions (diode reverse recovery)

Component stress (electrical or thermal)

! Choice of components

Powerconverter

Controller

LoadPower input Power output

Controlsignals

Reference

Measurement

vi ii io vo

4

Comparison - SPICE and PLECS

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 - PLECSSimplified behavioral model

Two parameters: Ronand T


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High Speed Simulations with Ideal Switches

Conventional continuous diode mode

Arbitrary static and

dynamic characteristicSnubber often required

Ideal diode model in PLECS

Instantaneous on/offcharacteristic

Optional on-resistanceand forward voltage

6

Comparison: Diode Rectifier

Simulation with conventional and ideal switches

Simulation steps:1160"153

Computation time:0.6s "0.08s


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State Space Model: Buck Converter

State space description

Switch conducting Diode conducting

8

Variable Time-Step Simulation: Buck Converter

Transistor conductsDiode blocks


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Transistor opens

Impulsive voltage acrossinductor

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Impulsive voltage closesdiode


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Transistor open

Diode conducts

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Switch timing Problem:Diode opens too late

Impulsive voltage acrossinductor


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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

AB C

D

1s

Switchmanag

er

PLECS S-function

Solver

u

g

y


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Solver Overview

Continuous solvers

Taylor series polynomial

Step size control

Acceptable error

Relative, absolute tolerance

Output refining

Discrete solvers

Trapezoidal rule

Step size selection

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Taylor Series Expansion

Approximate a continuousfunction with a higher orderpolynomial

The higher the order, themore accurate the solution

16

y=sin(x)

1storder

5thorder

Source: WikipediaTaylor series:


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Continuous Solver Operation

y(t) can be constructed using in a piecewise fashion usingp1(t) and p2(t), which are Taylor series polynomials

A continuous solver calculates the point yn+1by calculatingthe equivalent Taylor series for p1(t)

An nthorder solver has the same accuracy as an nthorderTaylor series.

"#$%!$%

'#$%

$$( $()"

'('()"

*( *()"

Solver Settings

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Acceptable Error

Local error

Difference between 4th and 5th order solutions

Acceptable errorDefines the local error limitDetermined by tolrelexcept for small state values

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Acceptable errorLocal error

Result valid if:

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, tolrelPrevious time step, hold

20

hold


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Tolerances

Relative tolerance

Determines acceptable error limit when x > 0

Start with 10-3

(0.1%)Numerical limit is 10-16

Absolute tolerance

Best to set to auto

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Example LC circuit

22

Analytical solution:

Result: tolrel= 1e-3


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Example LC circuit

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Analytical solution:

Result: tolrel= 1e-6

Refining the Output

Option 1: Reduce tolrelor time step

Solver must recalculate polynomial coefficients at each timestep

Less efficient

Option 2: Increase refine factor

Solver uses existing polynomial coefficients to calculate

additional points.More efficient

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Solver Families

Non-stiff

Inherently more efficient (no iteration required)

Smaller stability domainForward Euler - 1st order

ode45/Dopri - 5th order

Stiff

Less efficient (iteration required)

Larger stability domain

Backward Euler - 1st order

Radau/ode23tb - 5th order

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Forward Euler

Truncate the Taylor Seriesafter the first term:

1st order accurate

Explicitintegrationalgorithm

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Numerical Experiment

Scalar system

Analytical solution

Forward Euler

Unstable for ah


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Stability Domain of Forward Euler

Integration algorithm

Stable if the eigenvalues of

Fare inside the unit circle

I.e. if the eigenvalues of Ah

are inside a unit circlearound (-1, 0)

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Analysis of RLC Example

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Floating Point Arithmetic

What is 0.3 - 0.2 - 0.1 ?

0.299999999999999989

- 0.200000000000000011

- 0.100000000000000006

= -0.000000000000000028

What MATLAB calculates:

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Floating Point Arithmetic (2)

a

b+

c+

a + b + c=

Summation of floating point numbers


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Backward Euler

Develop Taylor Seriesaround a time in the future:

Must be solved iterativelyfor the unknown x(t*+h)

Implicitintegrationalgorithm

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Numerical Experiment (Revisited)

Scalar system

Analytical solution

Backward Euler

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Stability Domain of Backward Euler

Integration algorithm

Stable if the eigenvalues ofAhare outside a unit circle

around (1, 0)

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Stability Domain

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Dopri

RadauTustin


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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!

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Event Detection

Define auxiliary functions that help locate discontinuities.

Function zeros must coincide with the discontinuities.

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Missed Events - Problem

39

B

A

Output

0

0

Time step1 Time step2

Missed Events - Solution

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B

A

Output

0

0

1


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PWM Generation

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A

B

!

"#$%

$& $&'(

"&

"&'(

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 outputOr: Compare with a continuous simulation

Continuous waveform

Highest transient frequency constrains sample time

Set tsample< ttransient/10

Integration underestimate approx 3%

Switched system

Switches turned on at sample instants

Set tsample< tsw/100

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Comparison - Continuous and Discrete solver

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Simulated current

Time steps:

Continuous: tolrel= 10-6

Discrete: ts = 50"s

Underdamped RLC circuit


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Conclusion

Variable step solver operation

Causes of model stiffness

Solutions to stiffnessExplicit (non-stiff) vs. Implicit (stiff) solvers

Limits of stability

Limits of numerical accuracy

Event detection

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Plexim GmbH

Tips to Achieve a Fast Simulation


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Introduction

The problemLarge simulation models can become slow

Many states, detailed component modelsSpeeding up the simulation

Model simplification

Model tuning

Real-time accelerator

Averaged modeling

Efficient PWM generation

Example - Three phase inverter

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Simplification of PV Cell Model

Exact model: accounts for non-linear VI characteristicNeeded for implementing MPPT simulation

Temperature and insolation dependency also important

Insolation = 1000 W/m2

T = 25 CMPP occurs at 17.6 V


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Diode PV Cell Model

Current = fn(voltage, insolation,temp)

Cell model

Shockley diode eq.

I

Reference:

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PV Lookup Table Implementation

Simplified implementation

Voltage-controlled current source

Capacitor included to add a state and avoid an algebraicloop

Precalculated PV data


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Model Tuning

Simulation settings.

Choose the correct solver.

Variable time step solver.

Reducing the switching frequency reduces the computationaloverhead.

Blockset StandaloneNon-stiff ode45 Dopri

Stiff ode23tb/ode15s Radau

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Real Time Accelerator

Converts model into c-code.

Compiles 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.


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Averaged Converter Modeling

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 a Three Phase Converter

Full switching model

Simulates exact electrical dynamics.

Switching action slows down the simulation where slower dynamics arestudied.


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Inverter Leg

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d=0.3

d.T

Vdc

d.Vdc

d.ia

ia

Averaged Inverter Leg

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Va modeled using as a controlled voltage source

idc modeled as a controlled current source


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Averaged 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, mcUse 3 phase PWM.

Extend operating range using overmodulation.

Approximation of space vector modulation with a continuous modulation index


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Overmodulation

Add zero-sequence signal to modulation index

Reference: Vector Control of a Double-Sided PWM Converter and Induction Machine Drive, R. Ottersten

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Algebraic Loops

What?

Multiple direct feedthrough blocks connected in a loop.

PLECS continuous inputs appear as a direct feedthrough system toSimulink.

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


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Breaking Algebraic Loops

Option 1. Use memory block or similar

Time delay causes 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 stepsimulation.

Breaking an algebraic loop with a low-pass filter.

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Average DC-DC Converter Modeling

Average switch cell modeling

Replace switch/diode combination with a voltage current source.

Ref: Fundamentals of Power Electronics, Erickson & Maksimovic

Equations

I2= (1 - d)/I1

V1= (1 - d)/V2


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Simplification of Current Controlled Converters

Boost converter is current controlled

Can replace the inductor with a current source.

Efficient PWM Generation in PLECS

Doubly fed induction generator example, 0.1 s simulation, PLECS Blockset

Sampled PWM: 29.6 s

Continuous PWM: 45.1 s

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Sampled PWM Continuous PWM

=> 52 % faster


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Control aims

Sinusoidal input currents

Regulated dc bus voltage

Example - Three Phase Boost Rectifier

Block Diagram

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PWM

Stiff systemSimulated for 0.2 s in PLECS Blockset & Standalone


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Results - PLECS Blockset

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Controls in Simulink

ode23tb solver used (stiff)

Results - PLECS Standalone

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Radau solver used (stiff)


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Conclusion

Choose the right solver for the job (stiff or non-stiff)

Use sampled PWM

Make appropriate simplificationsFor extremely large models:

PLECS Standalone

Real-time workshop

Averaged converter modeling

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An Introduction to Thermal Modelingand Simulation

Plexim GmbH


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Why Thermal Simulation?

Reduce development time

Predict performance and investigate keytradeoffs early in the design process

Thermal measurements can be difficult andtime consuming

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Sources of Thermal Losses

Passive components

Resistive power loss: ploss(t) = vR(t) iR(t)

Loads e.g. break resistors

Filters

Winding resistance

Power semiconductors

Conduction lossSwitching loss


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Semiconductor Losses

Conduction loss

Switching loss

Gate signal

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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


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Example IGCT Turn-off: Varying Stray Inductance

Courtesy ABB

0.0

1.5

3.0

4.5kV

0.0

1.0

2.0

3.0kA

VPK= 3800V

VDC = 2 kV

TJ = 125C

5 10 15 s

300 nH (10.5 Ws)

800 nH (12 Ws)

1500 nH (13.5 Ws)

tf!2.5s, ttail!7s

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Switching Loss Calculation from 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 Approach for Switching Losses

Instantaneous switching maintained for speed.

Switching losses are read from a database after switchingevent.

Esw= f(Tj, vblock, ion) (Rg= const)

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Example Lookup Table

Turn-off loss a function of:

Current before switching

Voltage after switching

Temperature at switching

Rgis assumed constant

Exact loss found using

interpolationNote the voltage andcurrent polarities!


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Semiconductor Conduction Losses

On-state loss

Conduction profile is nonlinear:

von= f(ion, Tj).Conduction profile stored inlookup table

Exact voltage found usinginterpolation

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 electricaldomains not coupled

Semiconductor losses dontappear in electrical circuit

Energy conservation can bemaintained by subtractingthermal losses from electricalcircuit

Switching losses added overzero time

Filtered by large capacitanceof thermal circuit

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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

Pt - simulated IGBT lossPtm - measured IGBT loss

Thermal Modeling of a Pulsed Resonant Converter

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Acknowledments: Dr. Fabio Carastro, University of Nottingham

Simulation results

Experimental results

"t = 1.4!

"t = 1.4!


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A combined electrical-thermal model

Losses, heatsink, ambient temperature.

How are these represented using PLECS?

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Thermal Domain

Thermal circuit analogous to electrical circuit.

Thermal and electrical circuits solved simultaneously.


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A complete electrical-thermal model

The heatsink is the interface between the two domains

Automatically absorbs component losses

Propagates temperature back to semiconductorsThermal impedance modeled with RC elements

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Hierarchical Modeling of Thermal Structures

Junctions

Dual IGBTmodule

Heatsink

IGBT Plate


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Different Thermal Equivalent Networks

Cauer equivalent

Physics based thermal equivalentcircuit

Each Rthand Cthpair represents a

physical layer in the thermal circuit.

Foster equivalentCurve fitting approach based onheating and cooling characteristics.

No correspondence between Rth,nresp. Cth,n and the physical structure!

Any modification of the systemrequires recalculation of all values

Junction-Case Thermal Impedance

Define in thermal editor to observe junction temp fluctuations

Foster coefficients often given in datasheet

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Example junction-case thermal impedance

Foster network coefficients


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Foster Network Pitfalls

Only accurate if reference point x is a constant temperature

Cannot be arbitrarily extended beyond point x

Tjuncimmediately affected by temperature changes at x

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Reference: M. Mrz and P. Nance, Thermal modeling of power electronic systems, www.iisb.fraunhofer.de

Solution 1 - Use First Order Cauer Network

Calculate #from 63% R value

C = #/R

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Vcreaches 63.2% Vfinalafter #


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Solution 2 - Use a Constant-Temperature Heatsink

Set Cheatsinkto 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

ConceptCalculate total switching

and conduction energy lostduring a sw cycle

Output as an average

power pulse during thenext cycle

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ImplementationBased 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

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Newton Raphson Analysis

Newton iterationApproach: 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 statevariables)


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Newton Raphson: Convergence

Typically converges after


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Thermal Loss Feedback 1

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Efficiency:

P1,Pi=100W Po=100WP2=95W


100

Efficiency:

P1=105W Po,P2=100WPi=100W


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Efficiency:

Thermal feedback:

P1=105.3W Po,P2=100WPi=100W

Physical RdsImplementation

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Rdsconstant during switching cycle

Resistance resolution:


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Obtaining Switching Loss Data

Experimental measurements

Switching losses highly dependent on gate drive circuit andstray parameters

Use a switching loss setup to characterize loss dependency onvoltage and current for two temperatures

Datasheets

Given for a specific gate resistance and stray inductance

Good approximations can be made by extrapolatingmanufacturers data (or asking for complete lossmeasurements)

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

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Advanced Tools

C-Script block

Control design tools

Steady-state analysis

AC sweep

Impulse response analysis

Loop gain analysis

Other

Subsystem - custom components


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Small Signal Analysis Tools

Possible workflow for controller design using PLECS tools:

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AC Sweep Analysis

Impulse Response Analysis

Converter Open LoopTransfer Function

Voltage controller designFreq domain

Loop Gain Analysisplbodefunction

Calculate loop gain

Gain/phasemargins met?

Verify with simulation

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Frequency Sweep

Algorithm:

For each frequency:

1. Run steady-state analysis

2. Apply sinusoidal perturbation

3. Extract system response using Fourier analysis

Caveats:Period length: least common multiple of system period andperturbation period

Computationally expensive


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Impulse responseof a buck converter

Transfer function:

110


The original Laplace transform:

Apply some math

Reference: D.Maksimovic, Computer-Aided Small-Signal Analysis Based on Impulse Response of DC/DCSwitching Power Converters, IEEE Trans. On Power Electronics, Vol. 15, No. 6, Nov. 2000, pp. 1183-1191


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Buck Converter: Transfer Function

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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 stability


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Custom Components

Why?

Support the top-down design approach

Easy to reuse, configure and measureCustom component features

Formed using the subsystem

User defined parameter, icon, initialisation commands andprobe signals

! All machines in PLECS are custom components

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0

0.5

1

0510152025

0

0.5

1

1.5

2

2.5

3

3.5

4

Voltage (V)Insolation (kW/m2)

Current(A)

Example - PV String Model

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Current surface datain lookup table


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Masked Subsystem Implementation

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C-script Block

Plexim GmbH


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What can it do?

C-control code emulation

Model custom components

Efficient sequence generationState machine modeling

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Function call interface

Solver operation (PLECS or Simulink)

Some function calls dependent on continuous or discrete states


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Main loop

Always executed

Output() called at least onceUpdate() called after Output if discrete state variables exist

Integration loop

Executed if continuous states defined in settings

Used for solving continuous differential 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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Function calls in C-script block

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C-Script parameter window


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Code editor window

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Sample time setting

Discrete

Value: +ve

Fixed sample time

Continuous

Value: 0

Inherited from solver

VariableValue: -2

At each time step the NextSampleHit must be specified.


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Hybrid fixed and variable step setting

Fixed and variable time

Format: [1/fs,0; -2, 0]

First row element is the time settingSecond 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

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Pitfalls - cascading C-Script blocks

124

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


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Macros

For interacting with model or solver

Examples

Input(0)Output(0)

NextSampleHit

CurrentTime

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Example - Space Vector Control

Three phase boost rectifier

Sinusoidal input currents

Regulated dc bus voltage


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Control Strategy

Outer voltage control loopInner decoupled current control loop

Space vector modulation

128


Timing calculationSwitch signal generation

C-script block1

C-script block2


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Efficient sequence generation

Typical switching sequence


Option blanking delayMethod 1: Fixed time step

Test if t > thit, apply new switch signal

Requires small steps for accuracy => increased computationaloverhead

Method 2: Variable time stepCalculate hit times at beginning of switching sequence

Fewer simulation steps more efficient

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Example: PWM with blanking

Calculate transition times, t1,t2,t3,t4at 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.

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External event

C-ScriptState machine

>

Relational operator- generates major step

Time setting:Simulation model: variableC-Script: continuous

if (IsMajorStep) test_input(v1,i1);

update_state_machine();

v1

External event >i1

Conclusion

Implement complex nonlinear and/or piecewisefunctions without complex block diagrams

Model custom components and controls

Generate efficient sequencing with exact but flexibletime step control

State machine modeling

Incorporate external C code for hardware controllers

an introduction to solvers plecs

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