The motor that could only run at one speed
In rung 4 you met the induction motor and the rotating magnetic field it lives by. Recall the master fact: the field sweeps around the stator at the synchronous speed, set entirely by the supply frequency and the number of pole pairs. Feed it the grid's fixed 50 or 60 Hz and the field — and the rotor chasing it — turns at one nailed-down speed. A four-pole motor on a 50 Hz grid spins at almost exactly 1500 rpm, every day of its life. That rigidity is wonderful for a clock and a disaster for a fan.
Synchronous speed of an induction motor:
N_s = (120 · f) / p (rpm) f = supply frequency (Hz)
p = number of magnetic poles
4-pole motor, f = 50 Hz: N_s = 120·50 / 4 = 1500 rpm
4-pole motor, f = 60 Hz: N_s = 120·60 / 4 = 1800 rpm
The rotor runs a touch slower — that gap is the SLIP —
but speed still rides almost entirely on f.
Want half speed? → you need HALF the frequency. ← the whole idea.For most of the 20th century there was simply no cheap way to make a frequency other than the grid's. So industry cheated. To move less air, you partly closed a damper in front of the fan; to move less water, you throttled a valve downstream of the pump. The motor kept galloping at full speed against an artificial obstruction, like driving with one foot flat on the gas and the other on the brake. Up to half the energy could vanish as wasted pressure drop across a half-shut valve. The whole world ran this way because the alternative — a variable frequency — did not yet exist on the shelf.
Inside the box: rectifier, DC link, inverter
How do you build a frequency that isn't the grid's? You can't bend the incoming AC directly — its 50 Hz is baked in. So a variable-frequency drive (VFD) does something clever and a little brutal: it throws the grid's frequency away entirely, parks the energy in a DC reservoir, and then builds a fresh AC waveform from scratch at whatever frequency it likes. Three stages, always the same three, in every drive from a 100 W ceiling fan to a 10 MW steel-mill roller.
- Rectifier (AC → DC). A bank of diodes turns the incoming three-phase grid into rough DC — the same step-down-to-DC idea you met with rectifiers, just on three phases. The grid's frequency dies here; it has done its only job of delivering raw energy.
- DC link (the reservoir). A big capacitor (and often a small inductor) smooths that rough DC into a steady, stiff voltage rail — the DC bus, typically around 565 V for a 400 V grid, or 325 V for 230 V. This calm pool of energy is the launchpad for the new waveform.
- Inverter (DC → AC). Six fast transistors — MOSFETs for low power, IGBTs for high power — chop the DC bus into a three-phase AC output. By choosing *when* each switch turns on, the inverter sculpts an output of any frequency and voltage it wants. This is where the variable frequency is born.
RECTIFIER DC LINK INVERTER (6 switches)
(AC -> DC) (reservoir) (DC -> AC, variable f)
grid ┌──S1 ┌──S3 ┌──S5
L1 ──┐ ╲ ╱ ╲ ╱ +───┬──────┬───+ │ │ │
L2 ──┤ diode ═══╪══ C │ ├──U ├──V ├──W ──► to motor
L3 ──┘ bridge −───┴──────┴───− │ │ │
╲ ╱ ╲ ╱ DC bus └──S2 └──S4 └──S6
50/60 Hz in ~565 V (stiff DC) 0–120+ Hz out, V adjustable
Each motor phase (U,V,W) is tied to a HALF-BRIDGE: top switch ties it
to +DC, bottom switch ties it to −DC. Flip them fast = synthesize AC.PWM: painting a sine wave with a chopped square
Here is the apparent impossibility. Each inverter switch can only do two things: connect the motor terminal to +DC or to −DC. There is no in-between. Yet the motor needs a smooth sine wave that glides through every value in between. How do you paint a curve using a brush that only knows black and white? The same trick that ran the whole power-electronics track: pulse-width modulation. Chop fast, and vary *how long* each pulse stays high to track the sine you want.
The controller runs a high-frequency triangle wave — the carrier, typically 4 to 16 kHz — and compares it, instant by instant, to the slow sine wave it wants the output to follow (the reference, the actual motor frequency, say 30 Hz). Whenever the sine sits above the triangle, the top switch goes on; below, the bottom switch goes on. Where the sine is near its peak, the pulses come out fat; near the zero-crossing, thin. The *average* of those chopped pulses traces the sine exactly — and the motor's own winding inductance smooths the chop into a near-perfect sinusoidal current, just as the L-C filter did in the buck converter.
SINUSOIDAL PWM (one motor phase):
reference sine (slow, e.g. 30 Hz) vs triangle carrier (fast, e.g. 8 kHz)
sine ⌒‾‾⌒ /\/\/\/\/\/\/\/\/\/\ ← carrier
⌒ ⌒ compare every instant
────────────────────────────────────────────────────────────
switch output → +DC ▐ ▐▐ ▐▐▐ ▐▐▐▐ ▐▐▐ ▐▐ ▐ (wide pulses at the peak)
−DC ▌ ▌▌ ▌ ▌ ▌ ▌▌ ▌▌▌ (narrow pulses near zero)
motor winding inductance averages the chop → smooth sine CURRENT:
___ motor sees an effective
/ \ /‾\ sinusoidal current at the
─────/───────\──────/───\──── REFERENCE frequency, not the
\___/ carrier frequency.
Lower the reference frequency → motor turns slower. Done.The V/f law: why voltage must follow frequency
There is a trap waiting if you simply lower the frequency and leave the voltage alone. The reason reaches back to Faraday. The stator winding is an inductor, and the magnetic flux it builds depends on voltage divided by frequency. Lower f while holding V high and the flux balloons — the iron core saturates, magnetizing current explodes, and the motor draws a huge current that does no useful work and cooks the windings. Raise f while V can't keep up and the flux collapses, the field goes weak, and the motor loses torque. To keep the magnetic field at its healthy design strength, voltage must rise and fall in step with frequency.
Faraday inside the stator: flux Φ ∝ V / f (must stay ~constant)
Keep V/f constant → Φ stays at design value → full torque, cool iron.
CONSTANT-V/f operating line (a 400 V, 50 Hz motor):
V ▲
400 ┤ •──────────── field-weakening region
│ ╱ (V maxed out, only f rises → Φ falls)
│ constant V/f ╱
│ (V rises ╱
│ with f) ╱
│ ╱
0 └──────────────•──────────────► f
0 50 Hz (base) 100 Hz
V/f ratio here = 400 V / 50 Hz = 8 V per Hz.Hold V/f constant and the motor delivers the same full torque at every speed from a crawl up to base speed — exactly what a conveyor or a hoist needs. But the voltage can only climb until it hits the DC-bus ceiling. At base speed (the motor's rated 50 or 60 Hz) the inverter is already pushing out maximum voltage. Ask for more speed beyond that and V simply *can't* rise any further. So above base speed you keep raising f while V stays pinned at maximum — which means V/f, and the flux, now *falls*. This is the field-weakening region, and it has a personality of its own.
A worked V/f operating point
Let's put numbers on it. Take a standard four-pole induction motor rated 400 V, 50 Hz, driving a conveyor. Its nameplate fixes the V/f ratio: 400 V ÷ 50 Hz = 8 volts per hertz. We want to run the belt at 60% speed for a light load, then later push it 20% past rated for a rush job. The drive computes the voltage and the synchronous speed at each setpoint.
MOTOR: 4-pole, rated 400 V / 50 Hz. V/f ratio = 400/50 = 8 V/Hz
BASE synchronous speed: N_s = 120·50/4 = 1500 rpm
(1) Run at 60% speed → set f = 0.60 × 50 = 30 Hz (below base)
Voltage (constant V/f): V = 8 V/Hz × 30 Hz = 240 V
Speed: N_s = 120·30/4 = 900 rpm
Flux Φ ∝ V/f = 240/30 = 8 → UNCHANGED → FULL torque available ✔
(2) Run at 20% over rated → set f = 1.20 × 50 = 60 Hz (field-weakening)
Voltage is CLAMPED at the 400 V ceiling (inverter can't make more).
Flux Φ ∝ V/f = 400/60 = 6.67 ( < 8 ) → field weakened to ~83%
Speed: N_s = 120·60/4 = 1800 rpm
Torque capability falls in proportion to flux (~83% of rated). ✔
Note: at 30 Hz the SAME 8 V/Hz holds the motor at full pulling power;
at 60 Hz the voltage can't follow, so we trade torque for extra speed.- Read the nameplate to get the V/f ratio: rated voltage ÷ rated frequency (here 400/50 = 8 V/Hz). This is the motor's magnetic 'recipe'.
- Pick the target frequency from the desired speed: f = (target speed fraction) × rated f. Speed and frequency are proportional.
- If at or below base frequency, set V = (V/f ratio) × f to hold the flux constant — you keep full torque.
- If above base frequency, clamp V at maximum; the flux and therefore the available torque fall as 1/f. You're in field-weakening: more speed, less torque.
Gate drivers, and the energy bills drives demolish
One unglamorous part makes the whole inverter possible: the gate driver. The six power switches don't turn on by being politely asked — a power MOSFET or IGBT has a gate that behaves like a small capacitor, and to flip it on or off in tens of nanoseconds you must shove amps of current into that gate, then yank them out, thousands of times a second. The controller chip can't supply that punch; the gate driver is the muscle that does, translating a feather-light logic command into a brutal, perfectly-timed gate kick. Get it wrong and the switch turns on slowly, lingers in its lossy half-on state, and burns up.
Now the reason drives took over the world. For a fan or a centrifugal pump, the physics is merciless in your favour: the power needed to move fluid scales with the cube of the speed. Run a fan at 80% speed and it moves 80% of the air — but draws only 0.8³ ≈ 51% of the power. At 50% speed it sips just 12.5%. The old throttle-and-damper method kept the motor at 100% and threw the surplus away as wasted pressure; a VFD simply slows the motor and the power plummets down the cube curve.
AFFINITY LAW for fans & centrifugal pumps: Power ∝ (speed)³ Speed Flow (∝ N) Power (∝ N³) vs. throttling at full speed ----- ---------- ------------ -------------------------------- 100% 100% 100% baseline 90% 90% ~73% VFD saves ~27% 80% 80% ~51% VFD saves ~49% 70% 70% ~34% VFD saves ~66% 50% 50% ~12.5% VFD saves ~87% A pump that runs at 80% flow most of the day on a VFD instead of a throttling valve can cut its energy use roughly IN HALF — paying back the drive's cost in months, then saving for its entire 15-year life.
Motors consume a staggering share of all the electricity generated on Earth — well over a third — and pumps, fans and compressors are the lion's share of that. Slapping a drive on the biggest of them is one of the highest-leverage energy moves in all of engineering. A VFD also starts a motor *softly*, ramping the frequency up from zero instead of slamming it across the line and pulling six-times-rated inrush current, which spares the windings, the gearbox, and the grid. Better efficiency, gentler starts, exact speed — all from learning to forge a frequency.
Step back and the whole machine is three honest stages. A rectifier kills the grid's frequency and pours its energy into a DC link reservoir; six transistors driven by gate drivers chop that DC with PWM to synthesize a fresh three-phase output; and the constant-V/f law keeps voltage marching in step with frequency so the motor's magnetic field — and its torque — stay healthy, right up to base speed where field-weakening trades torque for extra rpm. Every elevator, electric car, washing machine and factory line you'll ever meet runs some version of these few ideas.