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9.5 — Rectifiers, Converters and Inverters
A linear regulator taking 12 V down to 5 V at 1 A wastes 7 W to deliver 5. A switching converter doing the same job wastes half a watt. That difference is why every power supply built since about 1990 switches, and it is why a phone charger fits in a plug.
The mechanism is simple to state and full of consequences: do not drop the excess voltage across a resistance — switch the input on and off rapidly and use an inductor to smooth the result.
1. Why switching wins
A linear regulator's transistor operates in its active region, with voltage across it and current through it:
P_{loss}=(V_{in}-V_{out})\times I, \qquad \eta=\frac{V_{out}}{V_{in}}
Efficiency is fixed by the voltage ratio and nothing else can change it.
A switching converter's transistor is either fully on or fully off.
- Fully on: large current, near-zero voltage. P=I^2R_{DS(on)}, which is small.
- Fully off: full voltage, near-zero current. P\approx0.
Loss occurs only in the on-state resistance and during the transitions, which is the calculation Chapter 2.4 worked through.
Efficiency of 85 to 98%, and — the important part — it is nearly independent of the voltage ratio. A converter taking 400 V down to 12 V can be 94% efficient, which a linear regulator could never approach.
A switching converter looks like it should not work. It chops the supply into a square wave and yet delivers smooth DC at a different voltage, with almost none of the energy wasted. The picture below is the whole mechanism.
Switch closed. The supply is connected to the inductor, and current builds up through it — an inductor resists a change in current, so it ramps rather than jumping. Energy is being stored in the inductor's magnetic field while the load is fed.
Switch open. Now the interesting half. The inductor still has current flowing and it will not let that stop suddenly, so it drives the current onward, pulling it round through the diode. The load keeps receiving current from a supply that is disconnected, paid for out of the inductor's stored energy.
Switch fast enough and the load never sees the gap; it sees the average. And the average is the supply voltage times the fraction of time the switch was closed:
V_{out} = D\,V_{in}, \qquad D = \frac{t_{on}}{t_{on}+t_{off}}
where D is the duty cycle. That is the formula the rest of the section derives properly.
Why this beats a linear regulator is the point of the whole topology. A linear regulator drops the excess voltage across a transistor and turns it into heat, so 12 V down to 5 V at 1 A throws away 7 W. The switch here is either fully on, where it has almost no voltage across it, or fully off, where it carries almost no current. Power is voltage times current, and in both states one of the two is nearly zero, so the switch dissipates almost nothing. That is why a switching converter reaches 90–95% efficiency and a linear regulator in the same job reaches 40%.
2. The buck converter — stepping down
A switch, a diode, an inductor and a capacitor.
When the switch is on: the input drives the inductor, and the voltage across it is V_{in}-V_{out}. From v=L\,di/dt (Chapter 1.5), the current ramps up.
When the switch is off: the inductor's current cannot stop instantly, so it finds a path through the diode. The voltage across the inductor is now -V_{out}, so the current ramps down.
In steady state, the current must return to where it started each cycle, so the rise and the fall must be equal:
(V_{in}-V_{out})t_{on}=V_{out}\,t_{off}
Rearranging with D = t_{on}/T:
\boxed{V_{out}=D\,V_{in}}
The output is the input times the duty cycle. That is the entire buck converter, and the derivation used nothing but the inductor equation and the requirement that the current be periodic.
The principle behind it — "volt-seconds must balance across an inductor over a cycle" — is the single most useful idea in switching power supplies, and every converter topology is derived from it in the same two lines.
Ripple current:
\Delta I_L=\frac{(V_{in}-V_{out})D}{Lf}
Design target: 20 to 40% of the output current. Less ripple needs a bigger inductor; more ripple raises the losses and the output ripple voltage.
Output ripple voltage:
\Delta V_{out}=\frac{\Delta I_L}{8fC}+\Delta I_L\times ESR
The second term usually dominates. A capacitor's equivalent series resistance turns the ripple current directly into ripple voltage, which is why low-ESR capacitors matter far more than large ones in a switching supply, and why ceramic capacitors displaced electrolytics in this role.
Worked design
12 V in, 5 V out, 2 A, switching at 500 kHz, targeting 30% ripple.
D=\frac{5}{12}=0.417
\Delta I_L=0.3\times2=0.6\ \text{A}
L=\frac{(12-5)\times0.417}{0.6\times500{,}000}=\frac{2.92}{300{,}000}=9.7\ \mu\text{H} \;\to\; 10\ \mu\text{H}
Capacitor for 50 mV of ripple, assuming 5 mΩ of ESR:
\Delta V_{ESR}=0.6\times0.005=3\ \text{mV}
leaving 47 mV for the capacitive term:
C=\frac{0.6}{8\times500{,}000\times0.047}=3.2\ \mu\text{F} \;\to\; 10\ \mu\text{F}
Efficiency estimate. With a 20 mΩ MOSFET and a 0.3 V Schottky diode:
- Conduction in the switch: 2^2\times0.02\times0.417=33 mW
- Conduction in the diode: 0.3\times2\times0.583=350 mW
- Switching, at 20 nC of gate charge and 20 ns transitions: about 240 mW
- Inductor resistance, 30 mΩ: 2^2\times0.03=120 mW
P_{loss}=0.74\ \text{W}, \qquad \eta=\frac{10}{10.74}=93\%
The diode dominates. Replacing it with a MOSFET — synchronous rectification — drops that 350 mW to 2^2\times0.02\times0.583=47 mW.
\eta=\frac{10}{10.44}=96\%
Three percentage points from one component change, and it is why every modern converter uses synchronous rectification.
The catch is shoot-through: if both MOSFETs conduct simultaneously, even for nanoseconds, they short the input to ground. The gate driver therefore inserts dead time between turning one off and the other on, and during that gap the body diode conducts, which is a small loss deliberately accepted to prevent a large failure.
3. Boost — stepping up
Swap the positions of the inductor and the switch.
Switch on: the inductor is connected directly across the input, so its current ramps up and it stores energy. The output is supported by the capacitor alone.
Switch off: the inductor's current must continue, so it flows through the diode into the output — and to keep the current flowing, the inductor's voltage adds to the input.
Volt-second balance:
V_{in}t_{on}=(V_{out}-V_{in})t_{off}
\boxed{V_{out}=\frac{V_{in}}{1-D}}
As D approaches 1, the output voltage rises without limit — in theory. In practice the parasitic resistances impose a ceiling, and the efficiency collapses above about D = 0.8, so a practical boost ratio is limited to about 5:1.
Where it is used: LED backlights driven from a battery, photovoltaic maximum power point trackers, and the power factor correction stage at the front of every mains supply above 75 W.
The safety note that matters: a boost converter has a direct path from input to output through the inductor and diode, even with the switch permanently off. There is no way to disconnect the load by stopping switching, so a separate disconnect is needed for protection.
4. The other topologies
Buck-boost. Output either above or below the input, and inverted in polarity.
V_{out}=-\frac{D}{1-D}V_{in}
Non-inverting buck-boost uses four switches and operates as a buck when the input is high and a boost when it is low. This is what a USB Power Delivery supply needs, since it must produce anything from 5 to 20 V from a variable input.
Flyback. A buck-boost with the inductor replaced by a coupled pair of windings.
V_{out}=\frac{N_2}{N_1}\cdot\frac{D}{1-D}V_{in}
Two things it gains: galvanic isolation, and any voltage ratio from the turns ratio.
This is the topology inside almost every phone charger and laptop adapter, because mains-to-low-voltage requires isolation for safety and a large ratio for the voltage.
The name comes from the operation: energy is stored in the transformer's magnetising inductance while the switch is on, and "flies back" into the secondary when it turns off. Strictly it is not a transformer at all but a coupled inductor, since the two windings never conduct simultaneously.
Forward, half-bridge, full-bridge. True transformer operation, transferring energy while the switch is on. More efficient at higher power because the transformer is used properly rather than as an energy store, and correspondingly more complex. Above about 200 W these dominate.
Resonant converters (LLC). Shape the current into a sinusoid so that switches turn on and off at zero voltage or zero current. This eliminates most of the switching loss, allowing frequencies of 500 kHz to several megahertz and therefore very small magnetics.
The LLC converter is why modern GaN chargers are so small — high frequency shrinks the transformer, and zero-voltage switching is what makes high frequency affordable.
5. Control
Voltage mode. Compare the output with a reference, and the error signal sets the duty cycle. Simple, and slow to respond to input voltage changes because it must wait for the output to move.
Current mode. Two loops: an outer voltage loop sets a current target, and an inner loop terminates each switching cycle when the inductor current reaches it.
Three advantages, and all are real:
- Faster response to input changes, since the inductor current reacts immediately.
- Inherent cycle-by-cycle current limiting, which is a free protection feature.
- The inductor's pole is removed from the control loop, turning a second-order system into a first-order one and making compensation far easier — which is Chapter 6.2's argument applied.
The complication is subharmonic oscillation above 50% duty cycle, where a small perturbation in the inductor current grows rather than decays from cycle to cycle. The fix is slope compensation — subtracting a ramp from the current sense signal — and it is present in every current-mode controller.
Compensation of the feedback loop is exactly Chapter 6.5's problem. Target: 45 to 60° of phase margin, with the crossover at a fifth to a tenth of the switching frequency. Too fast and the loop responds to switching ripple; too slow and the output droops badly on a load step.
Pulse skipping at light load. At very low current, a converter running continuously wastes more in switching than it delivers. Skipping cycles keeps efficiency high at the cost of increased output ripple and an audible frequency component — which is the source of the faint whine some chargers make when idle.
6. Inverters — DC to AC
A full bridge of four switches connects the load to the supply in either polarity, or disconnects it.
Square wave output is the crudest form: switch fully one way then the other. It contains all odd harmonics with amplitude 1/n (Chapter 4.4), so the third harmonic is 33% of the fundamental. Motors run hot on it and electronics misbehave, so it is now found only in the cheapest equipment.
Modified sine wave — a stepped waveform with a zero period between the positive and negative pulses. Choosing the step width to be 130° eliminates the third and fifth harmonics entirely, which is a genuine improvement, and it is still not a sine wave.
Pulse width modulation is the real answer. Switch at 5 to 20 kHz with a duty cycle that varies sinusoidally over the output cycle:
D(t)=0.5+0.5\,m\sin(\omega t)
with m the modulation index. The switching frequency components are far above the output frequency, so a small filter removes them and leaves a clean sine.
V_{out(rms)}=\frac{m\,V_{dc}}{2\sqrt2}
Worked example. A 350 V DC bus with m=0.9:
V_{out}=\frac{0.9\times350}{2.83}=111\ \text{V rms}
For 230 V output you need a 730 V bus, which is why a European solar inverter runs its DC link near 400 V and uses a boost stage, or uses a transformer.
Third harmonic injection is a widely used trick. Adding 1/6 of a third harmonic to the modulation reference flattens the peaks of the reference waveform without changing the line-to-line output of a three-phase system, because the third harmonic is common to all three phases and cancels between them.
\text{usable output rises by } \frac{2}{\sqrt3}=15.5\%
Fifteen percent more output from the same DC bus, for free, and it is standard in every three-phase drive.
Space vector modulation achieves the same result by a different route, computing the switching pattern directly from the desired output vector, and is what most modern motor drives use.
7. Where these appear
The mains power supply. Bridge rectifier → power factor correction boost → isolated flyback or LLC → output.
The power factor correction stage deserves its own note. A plain rectifier and capacitor (Chapter 2.2) draws current only at the peaks of the mains waveform, giving a power factor around 0.6 and injecting substantial harmonic current into the supply. Regulations require correction above 75 W.
An active PFC boost stage shapes the input current to follow the input voltage sinusoidally, achieving a power factor above 0.98 and a nearly sinusoidal input current. It also conveniently produces a regulated 400 V bus for the next stage.
The motor drive. Rectifier → DC link capacitor → three-phase inverter, with the output frequency and voltage varied together to keep the flux constant:
\frac{V}{f}=\text{constant}
Read why: from Chapter 9.1, B\propto V/f. Reducing frequency without reducing voltage saturates the motor's core, so the two must move together — which is the fundamental control law of every variable-frequency drive.
Regeneration. A motor being decelerated acts as a generator, and the energy flows back into the DC link, raising its voltage. Three options: dump it in a braking resistor, store it, or feed it back into the mains with an active front end. A lift or a crane recovers a large fraction of its energy this way, and an electric vehicle's regenerative braking is the same mechanism.
The solar inverter. Panels → maximum power point tracking boost → inverter → grid.
Maximum power point tracking exists because a panel's power output has a peak partway along its current-voltage curve, and that peak moves with light and temperature. The tracker continually adjusts the operating point to sit on it, typically by the "perturb and observe" method: change the voltage slightly, see whether the power rose, and continue in that direction.
Grid-tied inverters must also disconnect within milliseconds if the grid fails — anti-islanding — because an inverter continuing to energise a supposedly dead line would endanger anyone working on it.
8. What actually limits a design
Switching loss rises linearly with frequency, and magnetic component size falls with frequency. The optimum is where the two meet.
P_{sw}\propto f, \qquad \text{size}\propto\frac{1}{f}
Silicon converters sit at 100 to 500 kHz. GaN and SiC devices, with far lower switching loss, push to 1 to 3 MHz — and that is entirely why GaN chargers are a third the size.
Thermal management is usually the binding constraint. A converter is limited by how much heat it can remove, not by how much power the semiconductors could theoretically handle.
T_j=T_{ambient}+P_{loss}\times R_{th(j-a)}
Worked example. 3 W of loss, 40 °C/W thermal resistance to ambient, 40 °C ambient:
T_j=40+120=160\ \text{°C}
Above the 150 °C limit. Either reduce the loss or improve the cooling — and a heatsink taking the thermal resistance to 10 °C/W gives 40+30=70 °C, which is comfortable.
Electromagnetic interference. Fast switching edges radiate and conduct noise, and every product must pass a conducted and radiated emissions test.
The mitigations, and each costs something:
- Slower edges — less noise, more switching loss.
- Input and output filters — cost, size and a possible interaction with the control loop.
- Spread spectrum switching, dithering the frequency by a few percent so the emission is smeared across a band rather than concentrated in a peak. This reduces the measured peak without reducing the total energy, which is a legitimate technique and also, honestly, a way of passing a test that measures peaks.
- Careful layout — minimising the area of the loops carrying high di/dt current, which is the single most effective measure and costs nothing but attention.
The layout point is worth emphasising. The loop containing the switch, the diode and the input capacitor carries a current that switches from zero to several amps in nanoseconds. The magnetic field from that loop is proportional to its area, so keeping it a few millimetres across rather than a few centimetres is the difference between passing and failing an emissions test — and it cannot be fixed afterwards by adding filters.
Part 10 puts everything in this volume into the objects that contain it, and takes a modern phone apart to find every chapter of this book inside it.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Switching converters
The master principle — volt-second balance. Over one steady-state cycle the inductor current must return to its starting value, so
\int_0^T v_L\,dt=0
Every topology below follows from that one line.
Buck
(V_{in}-V_{out})t_{on}=V_{out}t_{off} \;\Longrightarrow\; V_{out}=DV_{in}
\Delta I_L=\frac{(V_{in}-V_{out})D}{Lf}, \qquad L=\frac{(V_{in}-V_{out})D}{\Delta I_Lf}
\Delta V_{out}=\frac{\Delta I_L}{8fC}+\Delta I_L\times ESR
The ESR term usually dominates, which is why low-ESR matters more than large capacitance.
Design target: \Delta I_L = 20–40% of I_{out}.
Boost
V_{in}t_{on}=(V_{out}-V_{in})t_{off} \;\Longrightarrow\; V_{out}=\frac{V_{in}}{1-D}
Practical ratio limited to about 5:1 by parasitic resistance.
Buck-boost and flyback
V_{out}=-\frac{D}{1-D}V_{in} \qquad\text{(buck-boost)}
V_{out}=\frac{N_2}{N_1}\cdot\frac{D}{1-D}V_{in} \qquad\text{(flyback, with isolation)}
Losses
P_{cond}=I^2R_{DS(on)}D
P_{sw}=\tfrac12V_{DS}I_D(t_{on}+t_{off})f
P_{gate}=Q_GV_{GS}f
P_{diode}=V_FI_D(1-D)
Synchronous rectification replaces V_FI_D with I^2R_{DS(on)}, typically worth 3 percentage points of efficiency.
Dead time is inserted between the two switches to prevent shoot-through, at the cost of body-diode conduction during the gap.
Control
Crossover frequency: f_c = f_{sw}/10 to f_{sw}/5. Phase margin 45–60° (Chapter 6.5).
Current mode removes the inductor's pole from the loop, turning a second-order plant into a first-order one, and gives free cycle-by-cycle current limiting. Slope compensation is required above 50% duty to prevent subharmonic oscillation.
Inverters
Square wave: odd harmonics of amplitude V/n, so 33% third harmonic.
Sinusoidal PWM:
D(t)=0.5+0.5\,m\sin\omega t
V_{out(rms)}=\frac{m\,V_{dc}}{2\sqrt2} \quad\text{(single phase)}
V_{L(rms)}=\frac{\sqrt3\,m\,V_{dc}}{2\sqrt2} \quad\text{(three phase)}
Third harmonic injection:
\text{output rises by } \frac{2}{\sqrt3}=15.5\%
Why it works: the third harmonic is identical in all three phases, so it cancels in the line-to-line voltage while flattening the reference peaks and allowing a larger fundamental.
Constant volts per hertz:
\frac{V}{f}=\text{constant}
Because B\propto V/f — reducing frequency without reducing voltage saturates the motor's core.
Total harmonic distortion:
\text{THD}=\frac{\sqrt{\sum_{n\ge2}V_n^2}}{V_1}
Thermal
T_j=T_a+P\times R_{th(j-a)}
R_{th(j-a)}=R_{th(j-c)}+R_{th(c-s)}+R_{th(s-a)}
Worked: 3 W with 40 °C/W in 40 °C ambient gives 160 °C — over the limit. A heatsink taking it to 10 °C/W gives 70 °C.
Derating: the rated power dissipation is quoted at 25 °C case temperature and falls linearly to zero at the maximum junction temperature.
P_{max}(T)=\frac{T_{j(max)}-T_a}{R_{th(j-a)}}
Fourteen worked problems next.
What the next chapter fixes
Parts 1 to 9 have covered circuits, devices, logic, signals, control, radio and power as separate subjects. Part 10 puts them in one object: a microcontroller reading a sensor and driving something, then a smartphone, which contains essentially every technology in this volume in a package that fits in a pocket.