Power Electronics Conversion 2 ECE 3795 Power Electronics Conversion 2 Fall 2017 © A. Kwasinski, 2017

Power electronic applications Dynamic: Variable speed drives Arguably for wind generation Electric and hybrid electric cars Stationary: UPS Energy storage integration Information and communication technologies power plants Power supplies Solar power Micro-grids

Power electronics basics Types of interfaces: dc-dc: dc-dc converter ac-dc: rectifier dc-ac: inverter ac-ac: cycloconverter (used less often) Power electronic converters components: Semiconductor switches: Diodes MOSFETs IGBTs SCRs Energy storage elements Inductors Capacitors Other components: Transformer Control circuit Diode MOSFET SCR IGBT © A. Kwasinski, 2017

Power electronics basics dc-dc converters Buck converter Boost converter Buck-boost converter © A. Kwasinski, 2017

Power electronics basics Rectifiers v v v t t t Rectifier Filter © A. Kwasinski, 2017

Power electronics basic concepts Inverters dc to ac conversion Several control techniques. The simplest technique is square wave modulation (seen below). The most widespread control technique is Pulse-Width-Modulation (PWM). © A. Kwasinski, 2017

Power electronics basic concepts Inverters dc to ac conversion Several control techniques. The simplest technique is square wave modulation (seen below). The most widespread control technique is Pulse-Width-Modulation (PWM). © A. Kwasinski, 2017

Power electronics basic concepts Inverters dc to ac conversion Several control techniques. The simplest technique is square wave modulation (seen below). The most widespread control technique is Pulse-Width-Modulation (PWM). © A. Kwasinski, 2017

Power electronics basic concepts Energy storage When analyzing the circuit, the state of each energy storage element contributes to the overall system’s state. Hence, there is one state variable associated to each energy storage element. In an electric circuit, energy is stored in two fields: Electric fields (created by charges or variable magnetic fields and related with a voltage difference between two points in the space) Magnetic fields (created by magnetic dipoles or electric currents) Energy storage elements: Capacitors: Inductors: L C © A. Kwasinski, 2017

Power electronics basic concepts Capacitors: state variable: voltage Fundamental circuit equation: The capacitance gives an indication of electric inertia. Compare the above equation with Newton’s Capacitors will tend to hold its voltage fixed. For a finite current with an infinite capacitance, the voltage must be constant. Hence, capacitors tend to behave like voltage sources (the larger the capacitance, the closer they resemble a voltage source) A capacitor’s energy is © A. Kwasinski, 2017

Power electronics basic concepts Inductors state variable: current Fundamental circuit equation: The inductance gives an indication of electric inertia. Inductors will tend to hold its current fixed. Any attempt to change the current in an inductor will be answered with an opposing voltage by the inductor. If the current tends to drop, the voltage generated will tend to act as an electromotive force. If the current tends to increase, the voltage across the inductor will drop, like a resistance. For a finite voltage with an infinite inductance, the current must be constant. Hence, inductors tend to behave like current sources (the larger the inductance, the closer they resemble a current source) An inductor’s energy is © A. Kwasinski, 2017

Power electronics basic concepts Since capacitors behave like constant voltage sources you shall never connect a switch in parallel with a capacitor. Any attempt to violate this load will lead to high currents. Likewise, you shall never connect a switch in series with an inductor. Any attempt to violate this rule will lead to high voltages. Steady state: In between steady states there are transient periods, In steady state: That is, in steady state the energy in each of the energy storage elements is the same at the beginning and end of the cycle T. Of course, during the transient periods (if they could be called “periods”) there is a difference between the initial and final energy. or © A. Kwasinski, 2017

Power electronics basic concepts The average voltage across an inductor operating in periodic steady state is zero. Likewise, the average current through a capacitor operating in periodic steady state is zero. Hence, Both KCL and KVL apply in the average sense. © A. Kwasinski, 2017

Power electronics basic concepts Switch matrix It is a very useful tool to represent a power electronics circuit operation and to related (input) variables and (output) signals. Analysis with a switch matrix involves: 1) Identify and define all possible states. States are defined based on all possible combinations of the switches in the matrix. Switches have two possible states: ON (1) or OFF (0). 2) For each possible state relate (output) signals to (input) variables by taken into consideration the time at each state (i.e. the portion of the time with respect to the switching period). 3) Combine the previous relationship in order to calculate average values for the (output) signals. © A. Kwasinski, 2017

Switching function and duty cycle The logic signal used to represent the control signal in power electronic switches is called switching signal q(t). When q(t) =1 the switch (e.g. a MOSFET or an IGBT) is closed When q(t) =0 the switch is open. When power electronic switches are operated at a constant frequency in steady state q(t) is usually a periodic signal that may look like this: Then the duty cycle D is the portion of the time the switch is conducting current so, mathematically it equals the average of the switching function.

Power electronics basic concepts Switch selection There are two criteria: Current conduction direction There are two possible directions: Forward – Usually from source to load Bi directional – Both directions (if current only circulates in the reverse direction, just reverse the switch and make it a forward conducting switch). Voltage present at the switch when it is blocking the current flow. The definition relies on the voltage polarity off the switch when it is blocking current flow and with respect to the forward current direction convention. Can be reverse blocking (RB - diode), forward blocking (FB – BJT or MOSFET), or bi-directional blocking (BB - GTO). Switches power rating is significantly higher than their losses. + - © A. Kwasinski, 2017

Average value of periodic instantaneous power p(t)

Two-wire sinusoidal case zero average Displacement power factor Average power

Root-mean squared value of a periodic waveform with period T Compare to the average power expression compare The average value of the squared voltage Apply v(t) to a resistor rms is based on a power concept, describing the equivalent voltage that will produce a given average power to a resistor

Root-mean squared value of a periodic waveform with period T For the sinusoidal case

RMS of some common periodic waveforms Duty cycle controller DT T V 0 < D < 1 By inspection, this is the average value of the squared waveform

RMS of common periodic waveforms, cont. Sawtooth V T

RMS of common periodic waveforms, cont. Using the power concept, it is easy to reason that the following waveforms would all produce the same average power to a resistor, and thus their rms values are identical and equal to the previous example V V -V V V V V

RMS of common periodic waveforms, cont. Now, consider a useful example, based upon a waveform that is often seen in DC-DC converter currents. Decompose the waveform into its ripple, plus its minimum value. the ripple + = the minimum value

RMS of common periodic waveforms, cont. Define

RMS of common periodic waveforms, cont. Recognize that

RMS of segmented waveforms Consider a modification of the previous example. A constant value exists during D of the cycle, and a sawtooth exists during (1-D) of the cycle. In this example, is defined as the average value of the sawtooth portion DT (1-D)T

RMS of segmented waveforms, cont. a weighted average So, the squared rms value of a segmented waveform can be computed by finding the squared rms values of each segment, weighting each by its fraction of T, and adding

Practice Problem The periodic waveform shown is applied to a 100Ω resistor. What value of α yields 50W average power to the resistor? Since α represents the duty cycle and the rms value of a triangular waveform is the peak value divided by the square root of 3:

Fourier Analysis Harmonics Concept: periodic functions can be represented by combining sinusoidal functions Underlying assumption: the system is linear (superposition principle is valid.) e.g. square-wave generation Due to the presence of harmonics power electronic signals usually have components in the order of MHz.

Fourier Analysis Additional definitions related with Fourier analysis

Fourier series for any physically realizable periodic waveform with period T When using arctan, be careful to get the correct quadrant

Square wave V –V T T/2

Triangle wave V –V T T/2

Half-wave rectified cosine wave

Power electronics basic concepts In power electronic circuits, signals usually have harmonics added to the desired (fundamental) signal. Energy storage elements are used to Provide intermediate energy transfer buffers. Filter undesired harmonics There are two approaches: Linear approximation (based on time constants considerations). I.e., current and voltage ripples) Harmonic superposition © A. Kwasinski, 2017

RMS in terms of Fourier Coefficients which means that and that for any k

More on power in ac circuits Instantaneous power Average power Power in linear single frequency ac single phase circuits operating in steady state. if and Energy exchanges with energy storage components Displacement power factor Average power

More on power in (single phase) ac circuits if and Their phasors are So complex power is defined as where is the reactive power representing steady state energy exchanges with energy storage components.

Power in steady state balanced three phase ac circuits Now, if phase voltages and their respective phasors are Then, line voltage phasors are and

Power in steady state balanced three phase ac circuits Since the phase currents are: In 3-phase circuits with Y configuration Then, the power in this type of circuits is

Power in steady state balanced three phase ac circuits Since, Other definitions applicable to 3-phase power in balanced circuits are

Average power in terms of Fourier coefficients Messy!

Average power in terms of Fourier coefficients, cont. Cross products disappear because the product of unlike harmonics are themselves harmonics whose averages are zero over T! Not wanted in an AC system Harmonic power – usually small wrt. P1 Due to the DC Due to the 1st harmonic Due to the 2nd harmonic Due to the 3rd harmonic

Power Factor In the restricted case in which we have a linear circuit operating at a single frequency, the power factor is the displacement power factor: In a more general definition when a circuit operates with multiple frequencies, power factor is In both cases the power factor provides an idea related with efficiency because the power factor is a ratio of real power (usually the useful component of power) divided by the total power which includes the power associated to the energy necessary to build fields or to conversion processes.

Power Factor In a linear circuit with a single frequency excitation the power factor is not 1 when the reactance of the equivalent circuit is not zero (so the circuit is inductive or capacitive). The fact that the power factor is not 1, represents the fact that we are storing energy in magnetic and/or electric fields. In a circuit in which the voltage and/or the current signal has multiple harmonics, the power factor is not 1. In a power electronic circuit the fact that the power factor is not 1 represents the fact that power conversion processes usually create harmonic content in the signals.

Total harmonic distortion − THD (for voltage or current)

Some measured current waveforms Refrigerator THDi = 6.3% 240V residential air conditioner THDi = 10.5% 277V fluorescent light (electronic ballast) THDi = 11.6% 277V fluorescent light (magnetic ballast) THDi = 18.5%

Some measured current waveforms, cont. Vacuum cleaner THDi = 25.9% Microwave oven THDi = 31.9% PC THDi = 134%

Resulting voltage waveform at the service panel for a room filled with PCs THDV = 5.1% (2.2% of 3rd, 3.9% of 5th, 1.4% of 7th) -200 -150 -100 -50 50 100 150 200 Volts THDV = 5% considered to be the upper limit before problems are noticed THDV = 10% considered to be terrible

Some measured current waveforms, cont. Bad enough to cause many power electronic loads to malfunction 5000HP, three-phase, motor drive (locomotive-size)

Consider a special case where one single harmonic is superimposed on a fundamental frequency sine wave Fund. freq Combined + Harmonic Using the combined waveform, Determine the order of the harmonic Estimate the magnitude of the harmonic From the above, estimate the RMS value of the waveform, and the THD of the waveform

Single harmonic case, cont. Determine the order of the harmonic Count the number of cycles of the harmonic, or the number of peaks of the harmonic 17 T

Single harmonic case, cont. Estimate the magnitude of the harmonic Estimate the peak-to-peak value of the harmonic where the fundamental is approximately constant Viewed near the peak of the underlying fundamental (where the fundamental is reasonably constant), the peak-to-peak value of the harmonic appears to be about 30 Imagining the underlying fundamental, the peak value of the fundamental appears to be about 100 Thus, the peak value of the harmonic is about 15

Single harmonic case, cont. Estimate the RMS value of the waveform Note – without the harmonic, the rms value would have been 70.7V (almost as large!)

Single harmonic case, cont. Estimate the THD of the waveform

KVL and KCL in periodic steady-state Since KVL and KCL apply at any instance, then they must also be valid in averages. Consider KVL, KVL applies in the average sense The same reasoning applies to KCL KCL applies in the average sense

KVL and KCL in the average sense Consider the circuit shown that has a constant duty cycle switch + VSavg − + VRavg − 0 A R1 Iavg + VLavg = 0 − V L R2 Iavg A DC multimeter (i.e., averaging) would show and would show V = VSavg + VRavg

KVL and KCL in the average sense, cont. Consider the circuit shown that has a constant duty cycle switch + VSavg − + VRavg − R1 Iavg Iavg + VCavg − V C R2 A DC multimeter (i.e., averaging) would show and would show V = VSavg + VRavg + VCavg

Energy in steady state conditions Consider: Then,

Energy in steady state conditions Also Then, for any inductor, including one with infinitely large L: and since Then, if L is infinitely large but vL , the current is constant so…..

Energy in steady state conditions and, Then, Notice that VL,discharge is negative “Green” areas = “Orange” areas So the average inductor voltage is zero

Energy in steady state conditions So for an inductor Since capacitors are duals of inductors: “Green” areas = “Orange” areas So the average inductor voltage is zero “Green” areas = “Orange” areas So the average capacitor current is zero

Other figures of merit used in power electronics Efficiency: % Line regulation: % Load regulation System

Filtering In most power electronic circuits filtering is necessary in order to eliminate undesirable harmonics that are a result of the power conversion process. High-pass filters are rarely used because they tend to amplify high frequency components that are the result of the switching action, thus, increasing the noise in the circuit. Low pass filters are used both in ac and dc circuits.

Effects of “fast” switching on waveforms Switching is “fast” when the period of the switching function is much shorter than the circuit time constants. As a result, charging or discharging times are much shorter than the circuit time constants In these conditions, exponential and other waveforms representing the actual response of circuit variables can be approximated to lines. Remember the Taylor Series Expansion of a function f(x) about a point xo:

Power electronics basic concepts Time constants: In power electronics we tend to work in many circuits with “large” capacitances and inductances which leads to “large” time constants. What does “large” means? Large means time constants much larger than the period (whatever the period is. For example, a switching period. If you look close and for a short time interval, exponentials look like lines Time constant time scale Period time scale © A. Kwasinski, 2017

Power electronics basics Additional definitions Average RMS value Instantaneous power (Average) power Total harmonic distortion © A. Kwasinski, 2017

Power electronics basics Additional definitions Power factor Line regulation Load regulation © A. Kwasinski, 2017