Inverters

An inverter converts DC into AC of controlled voltage and frequency. Inverters connect solar panels and batteries to the grid, drive motors at variable speed (Lesson 9), produce clean AC in uninterruptible power supplies, power electric vehicles, and form the converter stations of modern HVDC links. This lesson builds up from the basic switching leg to single-phase and three-phase bridges, explains pulse-width modulation and its harmonic spectrum, introduces multilevel inverters and output filters, and covers two applications central to energy and data centre engineering: grid-connected inverters and UPS systems.

The switching leg

The building block of most inverters is a leg (or phase leg): two switches in series across the DC bus, each with an anti-parallel diode, with the output taken from the midpoint. Turning on the upper switch connects the output to \(+V_{dc}/2\) (measured from the DC midpoint); turning on the lower switch connects it to \(-V_{dc}/2\). The diodes allow current to flow in either direction regardless of which switch is on, which is essential with inductive loads. Dead time between the two gate signals prevents shoot-through (Lesson 1).

A half-bridge inverter is one leg with a split DC capacitor; its output swings between \(\pm V_{dc}/2\). A full-bridge (H-bridge) inverter uses two legs with the load between their midpoints, so the output can be \(+V_{dc}\), \(0\) or \(-V_{dc}\). A three-phase inverter uses three legs, one per phase.

Square-wave operation

The simplest control turns each leg's switches on for half the output period, producing a square wave. A square wave of amplitude \(V_{dc}/2\) has a fundamental of peak value \(\frac{4}{\pi} \cdot \frac{V_{dc}}{2}\) and odd harmonics with amplitudes falling as \(1/h\). Square-wave operation gives the maximum possible fundamental voltage and the lowest switching loss, but its voltage cannot be controlled except by varying \(V_{dc}\), and its large low-order harmonics (3rd, 5th, 7th) cause losses and torque ripple in motors. It is used in some high-power traction drives at top speed and in low-cost converters.

Sinusoidal pulse-width modulation

Pulse-width modulation (PWM) switches each leg at a high frequency and varies the duty cycle so that the average output over each switching period follows a sinusoidal reference. In sine-triangle PWM, a sinusoidal reference of amplitude \(m\) (the modulation index) is compared with a triangular carrier of amplitude 1 at the switching frequency; the upper switch is on whenever the reference exceeds the carrier. For \(0 \le m \le 1\) (linear modulation), the fundamental of each leg voltage has peak \(m\, V_{dc}/2\), and for a three-phase inverter the rms line-to-line fundamental is:

The harmonics are moved to frequencies around the switching frequency and its multiples, where they are easily filtered by the inductance of a motor or an output filter. The ratio of carrier to fundamental frequency is the frequency modulation ratio \(m_f\).

import numpy as np

Vdc, f1, mf = 650.0, 60.0, 39
N = 200000
t = np.arange(N) / (N * f1)
tri = 2 * np.abs(2 * ((t * f1 * mf) % 1) - 1) - 1


def leg(ref):
    return np.where(ref >= tri, Vdc / 2, -Vdc / 2)


for m, inject in [(0.8, False), (1.0, False), (1.15, True), (1.3, False)]:
    refs = [m * np.sin(2 * np.pi * f1 * t - k * 2 * np.pi / 3) for k in range(3)]
    if inject:
        third = m / 6 * np.sin(3 * 2 * np.pi * f1 * t)
        refs = [r + third for r in refs]
    va, vb = leg(refs[0]), leg(refs[1])
    vab = va - vb
    spec = np.abs(np.fft.rfft(vab)) * 2 / N / np.sqrt(2)
    fund = spec[1]
    thd = np.sqrt(np.sum(spec[2:]**2)) / fund
    low = np.sqrt(np.sum(spec[2:20] ** 2)) / fund
    tag = " with third-harmonic injection" if inject else ""
    print(f"m = {m:.2f}{tag}: VLL fundamental {fund:6.1f} V rms (linear theory {0.612 * m * Vdc:6.1f})  THD {100 * thd:5.1f}%  low-order {100 * low:4.1f}%")

m = 0.80: VLL fundamental  318.4 V rms (linear theory  318.2)  THD  91.5%  low-order  0.0%
m = 1.00: VLL fundamental  398.0 V rms (linear theory  397.8)  THD  68.6%  low-order  0.0%
m = 1.15 with third-harmonic injection: VLL fundamental  457.8 V rms (linear theory  457.5)  THD  52.8%  low-order  0.0%
m = 1.30: VLL fundamental  451.1 V rms (linear theory  517.1)  THD  55.4%  low-order  3.5%

The results show three regimes. In the linear range, the fundamental equals the theory exactly and there are no low-order harmonics; the high THD consists of harmonics near the carrier frequency, which the load inductance filters. Adding a third harmonic to all three references (which cancels in the line-to-line voltage) extends the linear range to \(m = 2/\sqrt{3} \approx 1.155\), giving 15.5% more output voltage from the same DC bus; space vector modulation, commonly used in digital controllers, achieves the same result. Beyond that, overmodulation increases the fundamental only slightly and introduces low-order harmonics.

Real inverters deviate from ideal PWM in one important way: the dead time inserted between the two switches of each leg. During the dead time, the output voltage is set by whichever diode carries the load current, so each switching period loses or gains a small volt-second area depending on the current direction. The result is a voltage error proportional to the dead time multiplied by the switching frequency, which distorts the output near current zero crossings and reduces voltage at low speeds. Drives measure the current direction and add compensating pulses to cancel most of this effect.

Multilevel inverters

A two-level inverter switches each output between two voltages, so every transition is a full \(V_{dc}\) step. Multilevel inverters synthesize the output from more levels, giving smaller steps, lower harmonics, lower \(dv/dt\) and the ability to use lower-voltage devices for higher total voltage. Common types are:

• Neutral-point-clamped (NPC) three-level inverters, with four switches and two clamping diodes per leg, widely used in medium-voltage drives, solar inverters and high-efficiency UPS systems.
• Flying capacitor inverters, which use capacitors to create intermediate levels.
• Cascaded H-bridge inverters, with several H-bridge cells in series per phase, each with its own isolated DC source, used in medium-voltage drives and static compensators.
• Modular multilevel converters (MMC), with hundreds of submodules per arm, now the standard topology for voltage-source HVDC links.

Output filters

Motor loads filter PWM harmonics with their own inductance, although fast edges cause other problems (Lesson 9). Grid-connected inverters and UPS outputs need explicit filters: an L filter (a series inductor), an LC filter for voltage-source outputs such as UPS, or an LCL filter for grid connection, which attenuates switching harmonics much more strongly with smaller components but has a resonance that must be damped passively or by the control. Filter design balances attenuation of switching-frequency ripple, size and cost, reactive current drawn by capacitors, and control bandwidth.

Grid-connected inverters

Solar, wind (through full or partial converters), battery storage and fuel cells connect to the grid through inverters. A grid-following inverter measures the grid voltage, tracks its phase with a phase-locked loop (PLL), and controls its output current to inject the desired active and reactive power. Grid-forming inverters instead behave like voltage sources that set their own voltage and frequency, sharing load through droop control; they can support weak grids and island operation and are increasingly required as inverter-based resources replace synchronous generators.

Interconnection standards specify how inverters must behave: voltage and frequency ranges, ride-through of disturbances rather than tripping, reactive power and voltage regulation functions, frequency response, power quality, and anti-islanding protection (detecting loss of the utility and stopping within about 2 seconds so that utility workers are not endangered by an energized island).

In Canada, distributed resources connecting to distribution systems follow CSA C22.3 No. 9 (Interconnection of distributed energy resources and electricity supply systems) and the requirements of the local utility; inverters are certified to CSA C22.2 No. 107.1 (and often UL 1741 as well). Some utilities have adopted requirements based on IEEE 1547-2018 for ride-through and grid support functions.

Uninterruptible power supplies

A UPS keeps critical loads running through utility disturbances and outages until generators start or the load shuts down safely. IEC 62040-3 classifies UPS by how the output depends on the input:

• Passive standby (offline): the load is normally fed from the utility; on failure, a switch transfers it to an inverter running from batteries. Cheap, but with a transfer gap of a few milliseconds and no conditioning; used for small loads.
• Line interactive: an inverter connected in parallel with the line regulates voltage and charges batteries; it takes over during failures. Used up to a few kilovolt-amperes and in some larger designs.
• Double conversion (online): a rectifier converts AC to DC, feeding batteries and an inverter that continuously supplies the load. The output is fully regenerated and isolated from input disturbances, with no transfer time. This is the standard for data centres.

Double conversion costs efficiency because every watt passes through two converters. Modern transformerless UPS designs with three-level IGBT or SiC inverters reach 96% to 97.5%. Eco modes run the load on the bypass during normal conditions, with the inverter ready to take over within milliseconds, reaching 99% efficiency; advanced modes keep the inverter connected to compensate harmonics and reactive power while the bypass carries the real power. Operators weigh the energy savings against the small risk of exposing loads to utility disturbances.

Lesson 10 applies inverters to solar, wind, storage and vehicle charging, and Lesson 11 scales the same principles up to HVDC and FACTS converters in the transmission grid.