AC Voltage Controllers

There is no reactive power in the classical sense. The power factor is poor because of distortion, which the simple \( \cos\varphi \) picture does not capture at all.

The definition that always holds is \( \text{PF} = P/S \), where \( S = V_{rms}I_{rms} \). With a chopped waveform, \( I_{rms} \) contains a great deal of harmonic current that carries no real power — because the supply voltage is a pure sinusoid, and a harmonic current at a frequency the voltage does not contain transfers nothing.

The correct decomposition:

\[ \text{PF} = \underbrace{\frac{I_1}{I_{rms}}}_{\text{distortion}} \times \underbrace{\cos\varphi_1}_{\text{displacement}} \]

Both terms are degraded here, by the same mechanism:

• Chopping produces 3rd, 5th and 7th harmonic current, so \( I_1 < I_{rms} \) — the distortion factor falls.
• Delaying the firing shifts the fundamental component later in the cycle, so even into a resistor the fundamental current lags — the displacement factor falls too.

Why the distinction has practical teeth. Power-factor correction capacitors correct displacement only. Fit one here and you improve the second factor slightly while doing nothing about the first — and you create a resonant circuit with the supply inductance that the 5th and 7th harmonics may excite, producing overvoltages and capacitor failure.

Correcting distortion requires different equipment: a passive harmonic filter tuned to the offending orders, an active filter that injects cancelling current, or — best — a converter that does not distort in the first place. This is the argument that pushes larger installations away from phase control entirely.

Because for \( \alpha < \varphi \) the load current never reaches zero, so the thyristors conduct continuously and the load sees the full supply.

Trace it. The load current is

\[ i(\theta) \propto \sin(\theta - \varphi) - \sin(\alpha - \varphi)e^{-(\theta-\alpha)/\tan\varphi} \]

At \( \alpha = \varphi \) the second term vanishes, leaving a pure sinusoid \( \sin(\theta - \varphi) \) that reaches zero at \( \theta = \pi + \varphi \). But the opposite thyristor is fired at \( \pi + \alpha = \pi + \varphi \) — the same instant. Conduction passes seamlessly from one device to the other and never breaks.

For any \( \alpha < \varphi \) the situation is worse still: the outgoing device is still conducting when the incoming one is fired, so the transition is seamless with margin to spare. The output is a complete, undistorted sine at full amplitude.

Put numbers on a real fan. A shaded-pole motor has \( \varphi \approx 65^\circ \). So:

• \( \alpha = 0 \) to \( 65^\circ \): no effect at all — 36% of the control's travel.
• \( \alpha = 65^\circ \) to \( 180^\circ \): output falls from 100% to 0.

And the remaining range is not linear either, so the useful settings are compressed into a narrow band. This is why such controllers feel coarse and often have detented steps rather than a continuous dial — the manufacturer has picked a handful of angles that give distinguishable speeds.

The proper fix is not a better controller. Reducing an induction motor's voltage increases its slip and its rotor loss, so voltage control of a running motor is thermally poor as well as coarse. Modern fans use an electronically commutated (BLDC) motor with a small inverter, giving genuine speed control at high efficiency — Chapter 27.

Because it produces sub-harmonics instead, and for most loads those are worse than the harmonics it avoided.

The mechanism. With \( n \) cycles on and \( m \) off, the pattern repeats every \( n+m \) supply cycles. The waveform's true fundamental is therefore \( f_s/(n+m) \), and Fourier places energy there and at its multiples.

With \( n = 3 \), \( m = 7 \) on 50 Hz, that is 5 Hz.

Why 5 Hz is a serious problem:

1. Flicker. Human vision is most sensitive to luminance modulation between about 5 and 15 Hz. Every lamp on the same feeder will visibly pulse, and standards such as EN 61000-3-3 limit this explicitly.
2. Network disturbance. The load switches on and off at 5 Hz across the supply impedance, so the voltage dips and recovers at 5 Hz — affecting other customers, not just the equipment concerned.
3. Torque pulsation. On a rotating load, full torque then no torque is a violent excitation. Integral-cycle control is never used for motors for this reason alone.
4. Mechanical resonance. Transformers and machines have resonances in the single-digit hertz range, and exciting one produces audible drumming and fatigue.

Where it is right: loads whose thermal time constant far exceeds \( n+m \) cycles — ovens, furnaces, process and barrel heaters. There the modulation is invisible in the controlled variable, and avoiding harmonics and switching EMI is a real gain.

The modern improvement is distributed firing: spread the on-cycles evenly rather than in a block. The average is identical but the lowest sub-harmonic moves up several fold, often enough to pass the flicker limits. It costs nothing but firmware, and is worth specifying.

One thing it does not fix. The power factor is still \( \sqrt{n/(n+m)} \) — 0.55 at 30% power — because the current is simply absent most of the time. No firing pattern changes that.

Because a three-phase star load with no neutral needs two devices conducting at once to complete a circuit, and beyond 150° no two can ever overlap.

The single-phase case. One device conducting completes the circuit, because the return path — neutral or the other line — is permanently connected. Current can flow from \( \alpha \) right up to \( 180^\circ \).

The three-phase star case. Current entering one line must leave through another, so two devices in different lines must conduct simultaneously. Each is fired at \( \alpha \) after its own phase's zero crossing, and the phases are 120° apart.

What the driving voltage is. When two devices conduct, the voltage available is the line-to-line voltage between those two phases, which leads the phase voltage by 30°. Beyond \( \alpha = 150^\circ \), every firing instant falls after the relevant line voltage has already passed through zero and reversed — so no pair of devices is ever forward-biased together. Nothing conducts, regardless of the gate signal.

The delta-connected exception. If the controller sits inside a delta-connected load, each device sees one load branch directly with a complete circuit of its own. The single-phase analysis applies and the full 0°–180° range is available.

This is why large soft starters offer an "inside-delta" connection where the motor's six terminals are brought out: the extra 30° of range genuinely matters when starting a stiff load, and the device currents are lower by \( \sqrt3 \) as well.

A related benefit worth noting. With an isolated neutral, triplen harmonics cannot flow — the same cancellation as Chapters 17 and 18. So a three-phase controller's line current contains 5th, 7th, 11th and 13th but no 3rd, which is considerably better than three independent single-phase controllers would produce.

Almost certainly a single gate pulse rather than a pulse train, leaving one of the two thyristors unfired.

The failure sequence, on an inductive load:

1. \( T_1 \) is fired at \( \alpha \) and, because the current lags, keeps conducting past \( 180^\circ \).
2. At \( 180^\circ + \alpha \) the controller issues \( T_2 \)'s pulse — but \( T_1 \) is still conducting, so \( T_2 \) is held reverse-biased. The pulse achieves nothing.
3. \( T_1 \) stops at \( \beta \), by which time \( T_2 \)'s pulse has ended.
4. \( T_2 \) never conducts. Only positive half cycles reach the load.

Why a transformer suffers so badly. Its winding resistance is a fraction of an ohm, so even a few volts of DC drives a large unidirectional current. That current biases the core towards saturation on one half cycle, the magnetising current rises sharply, and the winding heats with no useful output. Transformers fed this way fail in hours to days.

The fix: a pulse train. Apply gate pulses continuously (or as a burst at a few kilohertz) from \( \alpha \) until the end of the half cycle. Whenever \( T_1 \) finally stops, a pulse is present to turn \( T_2 \) on. Every competent phase-control circuit does this.

Two related causes worth checking:

• Asymmetric firing. If the two devices are fired at slightly different angles — a component tolerance in an analogue trigger — a smaller but persistent DC component appears. Measure both firing instants on a scope.
• One device failed open. Rare, but it produces the same symptom exactly. Check the voltage across each device.

And a design note. Feeding a transformer from a phase-controlled supply is difficult even when done correctly: the chopped waveform has a DC-free average only if perfectly symmetric, and the inrush at each firing is severe. Where a transformer must be voltage-controlled, integral-cycle control or a tap changer is usually the better choice.

Because for a load that starts occasionally and then runs at full speed, the VFD's advantages are never used and its disadvantages run continuously.

What the soft starter actually costs and does:

• About a fifth the price of a VFD of the same rating — six thyristors against six IGBTs, a rectifier, a DC link and a control platform.
• About a third the size, with no DC link capacitors and much less heatsinking.
• Bypassed once running. A contactor shorts out the thyristors after the ramp, so the motor runs directly on the mains with zero converter loss and no harmonics whatever. A VFD's 2–3% conversion loss runs 8,760 hours a year.
• No harmonics in normal operation, so no supply pollution and no input filter.
• No motor derating. A VFD's PWM output causes extra iron loss and insulation stress; a bypassed soft starter causes neither.

Do the arithmetic for a 30 kW pump running 6,000 hours a year. A VFD losing 2.5% wastes 750 W continuously — 4,500 kWh a year. Over a ten-year life that is a substantial figure, and it buys speed control the pump never uses.

When the VFD is nevertheless right:

• Variable speed is genuinely needed. On a pump or fan, throttling by speed rather than by valve saves energy as the cube of the flow — an enormous gain that dwarfs the conversion loss.
• The load starts hard. A soft starter's torque falls as \( V^2 \), so 50% voltage gives 25% torque. A conveyor or compressor may simply not move. A VFD produces full torque at any speed.
• Frequent starting. Many starts per hour heat the motor; a VFD's controlled acceleration is gentler.
• Precise control of speed, position or torque.

The engineering lesson, which recurs throughout this book: the technically superior converter is not automatically the correct one. Match the capability to the requirement, and count the losses over the whole operating life rather than at the rated point.