Fullscreen HTML5Med Res (??MB)HTML4

By Andrew Levido Power Electronics Part 6: DC-to-AC Converters Having covered DC-DC and AC-DC converters in earlier articles in this series, we will now move on to DC-AC converters. They have been around for a long time, but their usage has become widespread over the last couple of decades. T here was a time when the applications for DC-to-AC converters were limited to industrial motor drives and commercial UPS systems. However, the widespread adoption of domestic solar power systems and electric/ hybrid vehicles means that DC-AC converters have become extremely common. As has become the custom for this series, we will start by analysing the simplest possible converter and build from there. Fig.1 shows four switches arranged in a H-bridge, fed from a DC voltage source. If we close S1 and S4 for a period equal to a half-cycle of the desired output frequency and S2 and S3 for the other half cycle, we can synthesise a square wave with amplitude Vsrc and frequency ω, shown in red. Recall that ω (omega) is just a frequency expressed in radians per second; 2π radians is one cycle, so 2π radians per second is the same as 1Hz. There are a couple of important things to note about this circuit. First, energy passes from the DC side to the AC side, so this is an inverter; in a rectifier, it goes the other way. Secondly, the switches create an AC voltage source at the output terminals, so this is a voltage-source inverter. The need for these seemingly obvious observations will become apparent soon. The RMS value of the output voltage is just Vsrc, so the only way to change vload, or the power delivered to the load, is to adjust Vsrc. We can change this by introducing a third switching state into the cycle, one where all switches are off and the output is zero, as shown in the blue trace. Here, the switches are off for a phase angle of δ (delta) at the beginning and end of each half-cycle. We can now use δ to adjust the RMS output voltage, and hence the power delivered to the load, according to the expression vload(rms) = Vsrc √1–2δ ÷ π. This kind of inverter is sometimes known as a tri-state inverter because each output terminal can be connected to +Vsrc, –Vsrc or zero (another term you might see referring to similar systems is ‘modified sinewave inverter’). Fig.1: the simplest single-phase voltage-source DC-AC converter uses four switches to create a square wave output. By leaving all switches open for a phase angle of δ at the beginning and end of each half-cycle, we can control the output voltage and power. 44 Silicon Chip Australia's electronics magazine Most of the interesting loads we want to drive are not purely resistive, so we should see what happens if we add an inductor to the load (Fig.2). We will assume that the L/R time constant is large compared to the output frequency so that the AC current can be approximated by its fundamental component. There is a phase shift between the voltage and current of Ø radians, as we would expect with an inductive load. This angle is fixed by the relationship between the load inductance and resistance, so it does not help us to change the power delivered to the load. Things are different if the load includes an AC voltage source, as shown in Fig.3. This scenario is pretty common; for example, in a solar inverter feeding the power grid or an inverter driving a synchronous motor with its sinusoidal back-EMF. This circuit gives us another variable to play with, as we can control the phase angle between vload and vac. With this circuit, you can even dictate the direction of power flow – from source to load (inversion) or load to source (rectification). We effectively have two AC sources: the inverter output vload, and the AC source, vac, with an inductor between them. The inductor acts as a ‘buffer’ between the two, absorbing the instantaneous voltage difference. The greater this difference is, and the longer it has to be sustained, the larger the inductor required. Still, a large inductor reduces the power factor, limiting the real power that can be transferred. The solution is to move the inductor to the DC side of the H-bridge, where it won’t impact the power factor, no matter how large it is. This arrangement (Fig.4) means that the output of the H-bridge is an AC current, so this is now a current-source inverter. The siliconchip.com.au DC voltage source and inductor are equivalent to the current source Isrc. We can’t use the same switching strategy to achieve the ‘zero-state’ output as we did in the voltage-source inverter. If we turned all the switches off, the source current would have nowhere to go and the voltage across the H-bridge would rise uncontrollably. Instead, we achieve the zero state by turning both S1 and S2 or S3 and S4 on at the same time, diverting the current away from the load. I have shown the current-source inverter switching pattern in the figure. This ‘shoot-through’ would be catastrophic in a voltage-sourced inverter, as would a short on the output, because the source would be short-circuited and the resulting current uncontrolled. Current-source inverters are inherently current-limited, making them very robust compared to their voltage-source counterparts, which require additional circuitry to protect against short circuits. I won’t go through the maths, but it is possible to show that the power delivered to the load by this current-­ source converter is given by the equation P = (2vac Isrc ÷ π)cos(δ)cos(Ø). We can control the power flowing to or from the load by changing either δ (the time spent in the zero state) or Ø (the phase angle between the inverter output and vac), or both. These are dictated by switch timing, so they can be nicely varied by a microcontroller-based implementation. Full control of Ø requires true bidirectional switches – switches that can conduct or block in both directions. This is somewhat difficult to implement, since Mosfets and IGBTs both normally have an anti-parallel diode that means they can never block reverse current. In the case of Mosfets, this is the inherent body diode, and in the case of IGBTs, it is usually a separate diode integrated into the package. Typical applications therefore tend to leave Ø at or very near zero and manipulate δ to control power. We have used single-phase DC-AC converters so far, but everything we have covered is equally applicable to three-phase converters. Fig.5 shows that these are just a single-phase converter with an extra pair of switches. Each phase-to-phase voltage is created by manipulating the switches just like siliconchip.com.au Fig.2: with an inductive load, there will be a phase angle difference between the voltage output and load current that is determined by the load inductance and resistance. Fig.3: if there is an AC voltage source associated with the load, such as a photovoltaic inverter feeding the grid, we can control the phase angle between it and the inverter output, allowing us to control the power flow in either direction. Fig.4: if the inductor in Fig.3 is moved to the DC side, it results in a currentsource inverter. The zero-state output occurs when switches S1 and S2 or S3 and S4 are on. Fig.5: a three-phase DC-AC converter is just like its single-phase counterpart with an extra pair of switches. The phase shift between switching is 2π/3 radians (120°) instead of the π radians (180°) in the single-phase case. Australia's electronics magazine April 2026  45 the single-phase example, but shifted by 2π/3 radians (120°) instead of π radians (180°) in the single-­phase case. You could, in theory, build an n-phase converter with n switch pairs (or two pairs for single-phase) with a phase shift of 2π ÷ n between them. However, diminishing returns means we rarely deal with more than three phases in reality. The exception is primarily in aircraft and ship motor drive systems, plus some large-scale mining equipment, where more phases (eg, six or 12) can result in lower torque ripple, a lower current per phase, reduced harmonics and EMI, plus better fault tolerance. Pulse-width modulation The examples we have seen so far have all produced more-or-less squarewave outputs, which we know have a terrible power factor, so are not very efficient at transferring power. Practical DC-AC converters therefore employ some kind of modulation scheme to reduce the level of harmonics in the output, or at least push them up to high frequencies where they are easier to filter. One common method used for small to medium power converters (up to a few hundred kilowatts) is sinusoidal pulse-width modulation. The switch drive is modulated at a carrier frequency (fpwm) much higher than the desired output frequency. The on-time in each modulation period is proportional to the instantaneous amplitude of the desired sinusoidal output waveform, as in Fig.6(a). I have shown only one half-cycle for clarity; the other is identical but reflected in the horizontal axis. In that diagram, the carrier frequency is an integral multiple of the output frequency, but this does not have to be the case if the carrier frequency is high enough – say a couple of hundred times higher than the maximum output frequency. This means the carrier frequency can be fixed, making synthesis a lot simpler. Textbooks generally show the switching signals derived from analog circuits that compare a triangle-shaped carrier wave to a sinusoidal reference, but in reality these days, they will almost certainly be produced by software running in a microcontroller. Pre-computed sine values are typically stored in a lookup table, and each carrier period, the appropriate one is extracted, multiplied by a scaling factor (the modulation depth) to set the amplitude, and loaded into a timer configured as a PWM generator. In the case of three-phase inverters, you have to add some third harmonic to the reference sinusoid or the output amplitude will be limited. For a fuller explanation, see the “Variable Speed Drive for Induction Motors” article we published in the November 2024 issue (siliconchip.au/Series/430). It provides a good overview of how this type of modulation can be implemented in a microcontroller. Current control In the case of current-source converters, such as in photovoltaic systems, it is preferable to control the inverter’s output current rather than its voltage. There are a couple of ways to do this, both shown in Figs.6(c) & (d). Both of these assume there is a current transducer measuring iload. The compensator-based current Fig.6: this figure shows four different modulation schemes for DC-AC converters. (a) & (b) are suitable for voltage-source converters, while (c) & (d) are for current-source converters. 46 Silicon Chip Australia's electronics magazine controller shown in Fig.6(c) uses a PWM modulator identical to the one discussed above, except that the reference input is not a fixed sinusoid. Instead, the PWM modulator is ‘wrapped’ in a current control loop that compares the load current to a sinusoidal reference and drives the modulator via a loop compensator to reduce the current error to zero. This control method has the advantage of a fixed switching frequency, but the current ripple can vary widely. You can also use hysteretic modulation, like in Fig.6(d), analogous to current-mode control in DC-DC converters. The switches are controlled in such a way that the load current always remains within a hysteresis band ±ih around the reference. This control method provides fixed current ripple and built-in current limiting, but it does mean that the switching frequency is variable. This might not be a problem for solar inverters, for example, which operate over a very narrow range of output frequencies. Harmonic elimination Really large (multi-megawatt) or high-voltage DC-AC converters generally don’t use Mosfet or IGBT switches. Instead, they use gate-turnoff (GTO) thyristors, which come with voltage ratings up to 4.5kV and can handle currents up to 4kA or more. Their switching frequency is limited to something less than 1kHz, so pulsewidth modulation is not generally practical. Instead, a technique known as harmonic elimination is used. The ‘tri-state’ waveform shown in blue in Fig.1, like all of the waveforms we have seen, is ‘half-wave symmetric’. This just means that the second halfwave looks like the first reflected in the horizontal axis. This type of waveform only has odd harmonics, so the largest amplitude harmonic after the fundamental is the third. Its amplitude will be vload(3) = (4Vsrc ÷ 3π)cos(3δ). If we set δ to π/6, the amplitude of the third harmonic drops to zero. In fact, all multiple-of-three harmonics (n = 3, 9, 15...) become zero. Furthermore, you can add additional zerostate pairs of specific length at specific angles to eliminate other harmonics and their multiples. With three such sets of zero-state ‘notches’ in the right places, you can eliminate the 3rd, 5th, 7th and 9th siliconchip.com.au Fig.7: space vector modulation maps a set of three-phase sinusoidal waveforms to a single rotating vector in the αβ plane. You can synthesise a threephase voltage set by determining how much time to spend in each state. harmonics with a switching frequency just five times the fundamental. The largest harmonic present in the output will be the 11th. This is shown graphically for one half-cycle in Fig.6(b). We already know that in a balanced three-phase system, the 3rd harmonic and its multiples are already zero, making the 5th harmonic the highest one present in the waveforms shown in Fig.5. In this case, the first harmonic elimination zero-state pair is therefore placed to eliminate the 5th harmonic and its multiples. Three pairs of zero-state notches can eliminate all harmonics below the 13th in three-phase systems. The main downside of harmonic elimination is that by fixing the zerostate angles, you can’t use δ to control the output voltage. Converters in the multi-megawatt range are normally driven from a dedicated transformer and use tap changers to regulate voltage. Space vector modulation The science-fiction sounding space vector modulation (SVM) is a modulation technique used in balanced threephase inverters, especially in motor drives. It takes advantage of the symmetry in these systems to simplify the calculations necessary to synthesise a desired set of sinusoidal waveforms. While it may simplify the modulator, understanding it can be a bit of a mind-bender, so we are going to need some background. It is possible to represent any set of three variables, like three-phase voltages at some instant in time, as a point in 3D space. Each of the variables defines a length along one of the three orthogonal axes. This set of three values is called a vector because it describes a point in 3D space that is some specific direction and distance from the origin. For physical space, we usually label siliconchip.com.au the coordinates x, y and z, but for our three-phase system, we will name the axes a, b and c, representing each phase. Three values (va, vb, vc) define a vector vabc that represents all threephase voltages at any instant in time. We use a bolded v to indicate that this is a vector. In the case of balanced three-phase systems, we have a further constraint. At any moment in time, the sum of the phase voltages must be zero. This severely limits the range of values that the vector describing those variables can take. In fact, vabc must lie on a plane passing through the origin and tilted such that its normal vector (a vector standing perpendicular to the plane) passes through the point (1, 1, 1). Don’t worry if this is hard to picture; we are about to simplify matters. With quantities confined to this plane, it is natural to assign a new pair of orthogonal axes as if ‘looking down’ the normal vector. These axes are traditionally labelled α and β, and the plane is unsurprisingly called the alpha-beta plane. On this plane, we can describe a three-value three-phase vector as a point with coordinates (xα, xβ). The transformation between the two systems is known as a Clarke transform. This is named for its inventor, Edith Clarke, who among her many distinctions was the first female American professor of electrical engineering when she joined the University of Texas in 1947. If we take a set of three-phase sinusoids and map them to the αβ plane, they correspond to a point rotating in a circle around the origin. The speed of rotation is related to the frequency, and the radius is related to the amplitude. We now have all the tools we need to explore space-vector modulation. A three-phase inverter like that shown in Fig.7(a) can produce eight possible switch states, which produce the three-phase voltages shown in the Australia's electronics magazine adjacent table. The phase voltages are referenced to an imaginary ‘Neutral’ point at half of the source voltage and shown in the table normalised to Vsrc. The last two columns in the table show how each of these switch states maps onto the αβ plane using the Clarke transformation. Don’t worry too much about the weird √6 values. What is important is that if you plot these points on the αβ plane, you get the result shown in Fig.7(b). Each non-zero state maps to a vertex of a hexagon, and the two zero states map to the origin. Any point inside the hexagon represents a set of balanced phase voltages that we could synthesise. Fig.7(c) shows how, using an example from the triangle with vertices x1, x2 and x0x7. To synthesise the voltage represented by the red vector, we need to be in each of these three states for an April 2026  47 appropriate proportion of time, so the result ‘averages out’ to the desired voltage set. The length of the blue vector relative to x1 defines the proportion of time we need to spend in state x1, while the length of the green vector relative to x2 defines it for the x2 state. The remaining time is spent in one or the other of the zero-states. In the example, the fraction of x1 is around 0.3 and the fraction in x2 is around 0.4, leaving 0.3 of the time spent in the zero states. If we want to synthesise a set of three-phase sinusoidal voltages, we just rotate the vector around the origin, tracing out the red circle/arc. Over each switching period (Tsw), the red vector advances by some angle proportional to the output frequency. Typical implementations start and end each switching period in a zero-state to minimise output harmonics. There will therefore be three transitions per switching period: zero to state A, state A to state B and state B to zero. Each transition only requires one switch to be changed if you select the right zero-states. The sequence of states is always the same as you rotate around the circle, so it can be programmed in advance. Only the time spent in each state has to be calculated (or looked up in a table) in real-time. This is more efficient code-wise than threephase sinusoidal PWM, although the speed and capability of today’s microprocessors makes this advantage less important than it once was. Like every modulation technique, space vector modulation has plenty of variations and options. It’s a pretty deep rabbit-hole if you want to go exploring. Getting practical I thought it would be interesting to dig into the design of a real DC-AC converter by looking at an aspect of converter design that is often overlooked: calculating the thermal losses in the switching elements. This is far from simple in the case of DC-AC converters with pulse-width modulation and inductive loads, as you will see. I will use the IGBT bridge from the Variable Speed Drive Induction Motors article mentioned earlier as an example. This three-phase converter uses DGTD65T15H2TF 650V 30A IGBTs with integral freewheeling diodes, switching at 15.625kHz. There are two kinds of losses that we have to consider: conduction losses that occur when the IGBT or diode is on, and switching losses that occur in the IGBT as it switches on and off. Conduction losses are the product of the current through, and the voltage across, the IGBT and diode when turned on. The manufacturer provides graphs to show how these quantities are related, reproduced in Fig.8. These characteristics are not linear, and in the case of the IGBT, vary with gate drive voltage. Conduction losses get worse with higher temperature, so I have assumed the maximum 175°C junction temperature to be safe. Superimposed on the charts is a dotted piecewise linear approximation that we will use to calculate conduction losses. We will assume that the IGBTs’ Vce is a combination of a fixed drop, Vto, plus a voltage that increases linearly with current – effectively an on-resistance, Rce. This converter uses a 15V gate drive, so I have set this resistance to match the appropriate curve. In this case, we can read off Vto ≈ 1V and Rce ≈ 90mW. We can do a similar thing with the diode characteristic, giving Vfo ≈ 1.2V and Rak ≈ 40mW. If the current through the IGBT and diode is ic, the instantaneous conduction losses will be Pi(cond) = Vto ic + Rce ic2 and Pd(cond) = Vfo if + Rak if 2. The current is an AC quantity, so we have to find the average power over one cycle. This requires us to integrate these expressions over one cycle and divide by 2π. Since the switching frequency is high and the load is inductive, the inverter current will be sinusoidal, given by the equation i = Ipk cos(θ – Ø), where Ø is the phase angle between voltage and current. This current will flow in the IGBT when the switch is on, and in the diode when the switch is off. The proportion flowing in the IGBT is determined by the duty cycle, which is in turn defined by the phase angle and the modulation depth m, Fig.8: calculating the conduction losses in an inverter bridge is not a simple undertaking. It starts with a piece-wise linear approximation of the forward characteristics and ends with the expressions below the graphs. 48 Silicon Chip Australia's electronics magazine siliconchip.com.au according to the expression D = ½(1 + mcos[θ]). The proportion flowing in the diode is determined by 1 – D. Plugging all this together makes for a pretty nasty integral, but I have shown the results below the graphs in Fig.8. The total conduction loss for one IGBT/diode is the sum of these two equations. The terms involving the modulation depth m and load power factor cos(Ø) are interesting. Both can vary from zero to one, and the effect of reducing either of them is to move losses between the IGBT and the diode, while the total stays roughly constant. This makes intuitive sense because in each case, there is less IGBT on-time and more diode freewheeling time Switching losses are (thankfully) a little easier to calculate. Switching loss depends mainly on capacitance, so the energy expended each transition is related to the square of the voltage. The data sheet provides the switching energies for each transition, helpfully at 175°C, and at 400V, a little above the maximum we can expect, so comfortably conservative. These energies are Eon = 342µJ and Eoff = 288µJ. With sinusoidal PWM, you can show the switching losses will be Pi(sw) = (Eon + Eoff) fsw ÷ π. Using values from the variable speed drive in single-phase mode (Ipk = 10A, PF = 0.95, m = 1, fsw = 15.625kHz) gives conduction losses of 4.8W for the IGBT and 0.6W for the diode, and switching losses of 3.1W, for a total worst-case loss of 8.5W per device. There are four active devices, so the total is 34W. For the three-phase case (Ipk = 6A, PF = 0.85, m = 1, fsw = 15.625kHz), the power dissipation works out to be 5.8W per device and, coincidentally, 34W for the six active devices. That’s close to 98% efficiency for the DC-AC converter part (there are additional losses in the rectifier). You will notice that the IGBT switching losses are approaching the conduction losses in magnitude at a switching frequency of just over 15kHz. In many applications, we want switching frequencies much higher than this, so switching losses can become a huge concern in high-power converters. The next instalment RP2350B Computer A Fully-assembled general-use computer The RP2350B Computer runs BASIC and is excellent for creating your own programs, games, tinkering with external circuits and more. And we are selling it pre-assembled, with little to no soldering required to have it up and running. It supports a keyboard, mouse or even a SNES controller. Video output: DVI via an HDMI connector <at> 640 × 480, 720 × 400, 800 × 600, 848 × 480, 1280 × 720 or 1024 × 768 pixels Removable file storage: microSD Card, FAT16/FAT32, up to 32GiB Clock Speed: 252-375MHz Non-volatile program memory: 184kiB General usage RAM: 220kiB (expandable to over 6MiB) Internal File Storage: 14MiB Audio formats: single-frequency tones, stereo WAV, FLAC, MP3 & MOD USB ports: four Type-A for peripherals, one Type-C for power/console and one micro Type-B for firmware loading Clock: battery-backed real-time clock & calendar External console: serial over USB <at> 115,200 baud via the USB Type-C socket External I/O connector: 30 pins with 22 GPIOs, including 7 with analog input ability, plus ground, 3.3V and 5V outputs Power supply: 5V <at> 220mA RP2350B Computer Assembled Module [ SC7531 | $90.00 + post ] fully-assembled PCB, except for the optional components (instrument case, mounting screws, 3-pin header for serial wire debugging and APS6404L PSRAM IC [SC7530 | $5]) Front & Rear Panels [ SC7532 | $7.50 + postage ] pre-cut panels, white silkscreen and black solder mask; not included with the kit above In part seven, we will take a look at resonant or soft-switching converters, which can minimise or even eliminate SC switching losses. For all the details on how to build it, check out the article in the November 2025 issue of Silicon Chip (siliconchip.au/Article/19220). siliconchip.com.au Australia's electronics magazine April 2026  49