write equations and waveforms for Variable Frequency convert

write equations and waveforms for Variable Frequency converter and verify them by using Pspice software and give output waveforms?

Solution

Overview of Computer Simulation of Switched Converter Waveforms 1. General PWM Converter simulation goals 2. Special challenges to power electronic simulations a. Non-linear switch conditions b. Long time scale of the simulation c. Need to Model Feedback loops d. Case for Use of Simplified Models e. Verification of Simulation with Experiment 3. A computer efficient simulation sequence is complex and artful 4. Two generic simulators for power electronics a. Equation Solvers b. Circuit simulators B. Solid State Switch Models for PSPICE 1. Overview 2. Diode 3. MOSFET 4. IGBT C. PSPICE BUCK PWM CONVERTER SIMULATION EXAMPLES 2 A. Overview of Computer Simulation of Switched Converter Waveforms/Circuits Computer aided calculation programs like Mathematica (Wolfram Research) and Matlab (Mathworks) are useful in power electronics simulation. But the most useful circuit waveform analysis is PSPICE (Microsim Corp.), as we shall see below. The idea is to fully simulate the converter BEFORE starting construction and testing. Fixing problems in a model that we invest some time and effort in, may pay for themselve’s later by spotting problems before circuit construction begins. That is the promise of modeling, but not always the end result because of difficulties outlined herein. 1. Goals of Converter Operation Simulation a) Calculate both dynamic and steady state performances of the whole PWM converter system, for all voltages and currents versus time. Power processor Controller load Measurements Reference Power input vi i i vo i o Control signals ·Model power flow circuits ·Model controller response ·Model total system response with feedback loops included b) Obtain required Vmax and Imax for all switching elements employed, so we can better specify required switches and avoid switch failure by better knowing the worst case switch stress conditions and planning for them. c) Estimate power loss in various components: 3 1. Switches are analyzed both from the two final DC switch states and the two dynamic switching transitions: from off to on and from on to off. Losses effect the heat generation and associated cooling requirements that we must meet. d). Magnetics Losses also contribute to heat flow requirements in addition to switch losses. (a) Inductor’s have core losses we can estimate (b) Transformers also have core losses e) Snubbers and Control Electronics can be simulated to anticipate possible SOA problems of power switches. f) We can estimate thermal cooling for critical components using data from c and d above. Both solid state devices and magnetic cores want to remain below 100 °C to operate properly. So the cooling must result in ambient temperatures BELOW 100 °C in the converter. This must be addressed. g) We aim to speed-up converter and construction by COMPLEMENTARY simulation together with a proof-ofconcept hardware prototype. We use both to iterate to a final design faster with fewer mistakes and a more reliable final product. That’s the goal of simulation. 2. Challenges in Power Electronic Simulation Unlike VLSI design we are simulating a discrete component system that has a variety of unique switching devices with MANY parasitic elements that often dominate PWM converter operation. a) Accurate non-linear switch models are not always easily available, even from the manufacturers. If available from the manufacturer’s data sheets, the data may not always be as accurate as needed. Nevertheless, the WWW page of various manufacturers. e.g. Motorola, Harris, 4 Siemans, IXYS, etc offers a wealth of information. b) The switching time tsw is usually msec but the converter system response time is seconds to minutes Þ lots of computer time is required to simulate PWM converter waveforms well. c) Chosen feedback controller models needs to be placed into the closed loop system model. Usually both current and voltage feedback loops are employed for PWM dc-dc converters. This complicates analysis. d) Early on we need to choose proper simplification steps so as to model only the immediate objectives. Limited simulation in early stages of design employs only a few switching cycles rather than millions or more to crudely estimate voltages and currents, which repeat with each cycle. This simplification step is very important. Often it must include only known to be dominant parasitic elements from prior experience! ! Include expected parasitic and circuit impedance’s on the input and output as well as line inductance 100 - 500mH for 208 ac mains. Also, include filter capacitors, snubbers, etc. e) Verify any model prediction by prototype PWM converter construction and testing. f) Include the control loop model and any surprises that may arise. Choose to model only portions of the power processor circuit rather than the full circuit in initial simulations to save time and simulation costs. 5 3. The converter simulation sequence is complex The switch model replaces a time varying circuit topology with a single time invariant two port equivalent circuit by averaging over the switch cycle. Switch models are given by time independent averaged characteristics over Ts. Yet, the model changes with changing choice of duty cycle. Clearly, the frequency response of any such approximate model is limited to f < fsw. The fact that we can even make an AC model of the PWM converter is suprising to some. Second semester we will make just such models by small signal expansion of the DC models we have made to date. The f < fsw open loop power processor model A(w) Results from such an exercise. For now just consider this small signal function as given: Power processor: each switching is represented; simple component models load vo Prespecified control signal i o Power output Obtain transfer functions: AOL = Vout control = AOL(w ) In practice converters all employ feedback to stabilize the output to within the user’s specifications. The chosen controller model employed is usually available from: 1. Control chip manufacturer 2. Controller software 6 The closed loop system for the controller is then: Power processor (small signal, linearized) model Controller D input D Load D Reference Control signals ACL AOL 1+ A º b ACL Vout control variable º One employs feedback because one expects big improvements in converter performance by using feedback including: ·Improved system stability to circuit component or switch device changes with both temperature and aging. ·Higher Zin, Lower Zout for the closed loop system ·A low frequency model allows the use of Bode plots for determining AOL(w). From open loop gain and phase plots versus frequency we can predict system stability as well as instability at all operating frequencies. One can also estimate system dynamical response to transient changes in the load or input by behavior of Ab the loop gain near 1 I180o . 3. Two generic types of simulators are available for power electronics based simulators: Circuit oriented and differential equation based a) Differential Equation Based Simulators Give total control of the solution to the non-linear differential equations that describe the PWM circuit and then choose: ·The Integration method. 7 ·Step time of simulation. ·First choose only those important terms in algebraic and differential equations. Neglect as many elements as possible for the first simulation to save computer time. ·Use C language to write the program steps. ·Choose your own graphical plotting tools. b) Equation solver solution sequence 1. Circuit topology choice and switching sequence both determine loop equations we must solve. L i L r L vC C R + - Voi (t) on off on t on Ts t (a) (b) Voi rLiL + L diL dt + vc = voi (KVL) iL - C dvc dt - vc R = 0 (KCL) or matrix form t c L L c oi di dt dv dt - r L - 1 L 1 C - 1 CR i v + 1 L 0 v (t) é ë ê ê ê ê ê ù û ú ú ú ú ú é ë ê ê ê ê ê ù û ú ú ú ú ú é ë ê ê ù û ú ú é ë ê ê ê ù û ú ú ú 2. The State variable matrix format introduced in Erickson Chapter 7 is useful. For now consider the circuit equations just in the standard state variable or matrix form: dx(t) dt = Ax(t) + bg(t) x(t) = iL vc and g(t) = voi é ë ê ê ù û ú ú A = - r L 1 C - 1 L 1 CR and b = 1 L 0 é L ë ê ê ê ê ê ù û ú ú ú ú ú é ë ê ê ê ù û ú ú ú 8 3. The numerical solution: time step Dt, uses a linear interpolation x(t) = x(t- t) + [A( ) x( ) + b( )g( )]d t- t t D D ò z z z z z implicitly known are the small signal equations from which we can start the process of calculating the required integral ¯ ¯ x(t) = x(t - Dt) + 1/2 Dt [A(t - Dt)x(t - Dt) + A(t)x(t)] Þ Linear + 1/2 Dt [b(t - Dt)g(t - Dt) + b(t)g(t)] Þ Algebra Trapezoidal area approximation may be employed: Use Matlab or Mathematica to do the numerical integration. You have complete freedom to employ other integration algorithms as well as to change the time sequences to suit your desire for a complete or just a crude model as we show below on page 9. The point to ponder is whether or not you have the interest in becoming a computer programmer of power electronics problems as this is a very time intensive endeavor. Sample results of simple models are given on page 9 when using the Matlab simulator. 9 Illustrative lines of code in Matlab The results are plotted below for a buck converter, which will later to be modeled also by Pspice which is a circuit simulator With this equation based approach we see both the fast fsw waveform and the slower averaged output waveforms for the PWM converter evolve during power up of the circuit. In this way we could estimate any peak excursions that might cause switch failure.