Chapter 4 · Converter Circuits

DC-DC Converters

With one switch, one diode, one inductor and one capacitor you can build a converter that steps a DC voltage down, steps it up, or both. These are the workhorses of renewable electronics: the boost stage between a PV string and an inverter is one of them. This chapter derives all three basic converters from a single principle, sizes their inductors and capacitors, and finishes with a complete PV boost design.

One principle: volt-second balance

An inductor's voltage sets how fast its current changes: v_L = L \, di/dt. In steady state the inductor current must end each switching period where it started, otherwise it would grow without limit. That means the inductor voltage must average to zero over a period. The volts × seconds applied while the switch is on must be cancelled by the volts × seconds applied while it is off. Every conversion ratio in this chapter comes from writing down that balance.

The analysis below assumes continuous conduction mode (CCM): the inductor current never falls to zero. It also uses the small-ripple approximation: the output voltage is treated as constant within a period because the capacitor is large.

The buck converter

When S is on, the inductor sits between the input and the output: v_L = V_{in} - V_{out}, and its current ramps up. When S is off, the inductor current continues through D, the inductor sits across the output in reverse, v_L = -V_{out}, and its current ramps down. Balancing the two:

Worked example: a 48 V to 24 V buck

Convert 48 V to 24 V at 10 A, switching at 100 kHz, with inductor ripple of 30 % (3 A) and output ripple of 0.1 V.

1. Duty cycle: D = 24/48 = 0.5.
2. Inductor: L = V_{out}(1-D)/(\Delta I_L f_s) = 24 \times 0.5 / (3 \times 100\,\text{kHz}) = 40 µH.
3. Capacitor: C = \Delta I_L/(8 f_s \Delta V) = 3 / (8 \times 100\,\text{kHz} \times 0.1) = 37.5 µF, plus margin and a low-ESR type, since 3 A of ripple across just 33 mΩ of ESR would already add 0.1 V.
4. Switch: blocks 48 V and peaks at 11.5 A, so a 100 V MOSFET is a comfortable choice.

Replacing D with a second MOSFET gives a synchronous buck. The MOSFET's small R_{DS(on)} drop is much lower than a diode's forward voltage, which is why almost every low-voltage buck is synchronous.

The boost converter

When S is on, the inductor is connected straight across the input: v_L = V_{in}, its current ramps up and it stores energy, while the capacitor alone feeds the load. When S is off, the inductor's current is forced through D into the output, adding the inductor's voltage to the input: v_L = V_{in} - V_{out}, which is negative, so the current ramps down.

Real gain: why a boost can't step up forever

The ideal equation says the gain 1/(1-D) rises without limit as D approaches 1. In a real converter, the inductor's winding resistance r_L carries the large input current, and its loss grows faster than the output. Including r_L in the volt-second balance gives:

In practice a single boost stage is kept below a gain of roughly 3–4, where losses are still reasonable. Larger step-ups use isolated converters with a transformer (Chapter 7), or coupled-inductor and cascaded designs.

The buck-boost converter

Swap the positions of the inductor and the diode and you get a converter that can step up or down. While S is on, the inductor charges from the input; while S is off, it discharges into the output through D, with the output polarity reversed:

The inverted output is inconvenient, so non-inverting variants are often used instead: the SEPIC and Ćuk converters add a second inductor and a coupling capacitor, and the four-switch buck-boost combines a buck and a boost sharing one inductor.

When the inductor current reaches zero: DCM

At light load the average inductor current falls, while the ripple \Delta I_L stays the same. Once the average drops below half the ripple, the current reaches zero before the end of the period and the diode stops conducting. The converter enters discontinuous conduction mode (DCM), where the conversion ratio depends on the load as well as on D:

DCM is not a fault. Some small converters are deliberately designed for it, but most PV converters are designed to stay in CCM over most of their power range, and their control handles the transition at very low sun.

Design example: a PV boost stage

A PV string with its maximum power point at 300 V must be boosted to a 400 V DC link for a single-phase inverter. Rated power is 3 kW, the switching frequency is 50 kHz, and we want an inductor ripple of 20 % of the input current and a DC-link ripple from the boost stage of no more than 1 % (4 V).

1. Duty cycle: D = 1 - V_{in}/V_{out} = 1 - 300/400 = 0.25. A low duty cycle, so the gain of 1.33 is well inside the efficient region of Figure 4.2.
2. Currents: input (inductor) current I_{in} = 3000/300 = 10 A; output current I_{out} = 3000/400 = 7.5 A.
3. Inductor: \Delta I_L = 0.2 \times 10 = 2 A, so L = V_{in} D/(\Delta I_L f_s) = 300 \times 0.25 / (2 \times 50\,\text{kHz}) = 750 µH, rated for a peak current of 11 A without saturating.
4. Output capacitor (from the boost alone): C = I_{out} D/(\Delta V f_s) = 7.5 \times 0.25 / (4 \times 50\,\text{kHz}) \approx 9.4 µF. In a real inverter the DC link is far larger, because the single-phase inverter draws power pulsating at twice the grid frequency; that sizing belongs to the inverter (Chapter 6).
5. Switch and diode: both block 400 V plus turn-off spikes, so 600–650 V devices: a superjunction or SiC MOSFET and a SiC Schottky diode. The switch's RMS current is about 10\sqrt{0.25} = 5 A and the diode's average current is 7.5 A.
6. Light load: the converter stays in CCM while I_{in} > \Delta I_L/2 = 1 A, that is down to about 300 W, or 10 % of rated power.

The topologies side by side

Interleaving runs two or more boost phases in parallel, shifted in time. Their ripple currents partly cancel, so each inductor is smaller and the combined input and output ripple is lower.