Q2Advanced Power Electronics

(a) Principle of 'On-Off' (Integral Cycle) Control

On-off control (also called integral-cycle control or burst-firing control) is an alternative method of controlling average power delivered to a resistive load, distinct from phase-angle control. Instead of delaying the firing instant within each half-cycle (as in phase control), on-off control connects the load to the full, undelayed supply voltage for a certain integral number of complete half-cycles (m half-cycles), and then completely disconnects the load (blocking all thyristors) for the following n complete half-cycles, repeating this m-on, n-off pattern continuously.

Since the thyristors are fired exactly at each half-cycle's natural zero-crossing (α=0 effectively, during the 'on' periods) and are simply not fired at all during the 'off' periods, the switching always occurs at zero voltage/current crossings, dramatically reducing the harmonic distortion, radio-frequency interference (RFI), and switching transients that phase-angle control produces (since phase control switches the thyristor on abruptly at a non-zero point on the voltage waveform, creating steep di/dt and generating higher-order harmonics).

The average power delivered to the load is controlled by adjusting the duty ratio k = m/(m+n): Pavg = k×(Vrms²/R), where Vrms is the full, undelayed supply rms voltage, since during 'on' periods the load receives the complete, unclipped sine wave. On-off control is best suited to loads with a relatively long thermal time constant (such as resistive heating elements, furnaces), where the load's thermal inertia naturally smooths out the on/off power pulses into an effectively continuous, averaged heating effect, without noticeable flicker or output ripple at the load itself, but it is unsuitable for loads (lighting, some motor loads) that would show objectionable flicker or speed variation at the relatively slow on-off cycling rate (typically several to tens of cycles per on/off period) compared to the much finer, continuous control granularity phase-angle control provides.

(b) Single-Phase Voltage Controller with Resistive Load (General α Expressions)

Given: R = 3Ω, Vs = 230V (rms), f = 50Hz. Peak voltage Vm = √2×230 = 325.27 V. Since firing angle α is left general (not specified), the requested quantities are derived as functions of α.

(i) Maximum average and rms thyristor current for any firing angle α: for a full-wave controller with a purely resistive load, each thyristor conducts for exactly one half-cycle (from α to π), so the average current through a single thyristor is:

Substituting Vm=325.27V and R=3Ω:

This is maximum when α=0 (no phase delay, thyristor conducts the entire half-cycle), giving ITavg(max) = 17.25×2 = 34.5 A, and decreases toward zero as α approaches 180°.

The RMS current through a single thyristor is:

Substituting Vm=325.27V and R=3Ω:

This is maximum at α=0, giving ITrms(max) = 54.21×1 = 54.21 A (since at α=0 the bracket evaluates to [(π+0)/π]^0.5 = 1), and decreases as α increases toward 180°, where it approaches zero.

(ii) Minimum circuit turn-off time for any firing angle α: the circuit turn-off time tq (also called tc) is the time interval, following natural commutation (current zero-crossing) of the outgoing thyristor at the end of its conduction period, during which the thyristor is subjected to reverse (or, at minimum, non-forward) voltage before the next forward voltage half-cycle would otherwise attempt to re-forward-bias it — this available time interval must exceed the thyristor's own rated turn-off time tq(device) for reliable commutation without spurious re-triggering.

For a resistive load, the outgoing thyristor naturally stops conducting exactly at the voltage zero crossing (ωt=π, since current follows voltage instantaneously for pure R), and the incoming (opposite-polarity) thyristor is fired at angle α into the next half-cycle (i.e., at ωt=π+α). Therefore, the reverse-bias interval available for turn-off spans from ωt=π to ωt=π+α, giving a circuit turn-off angle equal to α itself, and correspondingly:

This result shows that the available circuit turn-off time increases directly in proportion to the firing angle α — at small firing angles (α close to 0, near-maximum output), very little reverse-bias time is available for commutation, imposing a stricter requirement on the thyristor's turn-off characteristics, whereas at larger firing angles (reduced output), considerably more turn-off time is available; this is an important practical constraint in the selection of thyristors for a given AC voltage controller application, since the device's rated tq must be safely less than the minimum tq the circuit will provide across its full intended range of firing angles (i.e., particularly at the smallest α the application requires), otherwise the outgoing thyristor risks failing to fully turn off before being re-forward-biased, causing an unwanted, uncontrolled 'shoot-through' conduction condition.

Numerical illustration at specific firing angles: to make the general α-dependent expressions concrete, consider α = 30° (π/6 rad), α = 60° (π/3 rad) and α = 90° (π/2 rad) for the same R = 3Ω, 230V/50Hz circuit:

These numbers confirm the expected trend: as α increases from 30° to 90°, the average thyristor current very nearly halves (32.15 A down to 17.25 A) while the available turn-off time triples (1.667 ms up to 5.0 ms) — in other words, the operating condition demanding the largest thyristor current rating (small α, near-full output) is exactly the same condition that provides the least commutation margin, so device selection must always be checked against the smallest α the application is specified to reach, not against some average or typical operating point.

Relationship between on-off control and phase-angle control — quantitative comparison: for the same average power demand, on-off control's rms load current (and hence rms thyristor current) over one full on/off cycle is:

which for a duty ratio m/(m+n) matched to give the same average power as a phase-controlled unit at a particular α will generally show a different rms-to-average current ratio (crest/form factor) than the phase-controlled case, because on-off control always switches full, undelayed half-cycles rather than partial ones — this is directly relevant to thyristor rating selection, since a device sized purely from its average current rating (a common simplified approach) can be under-rated for a given rms/thermal duty if the form factor of the actual current waveform (which differs between phase control and on-off control for the 'same' average power level) is not separately checked, and RTU numericals on this topic frequently require both average and rms current to be reported precisely for this reason.

Additional practical note on turn-off time margin: in an actual design, a safety margin (commonly 2 to 3 times the manufacturer's rated tq) is added beyond the theoretical minimum circuit turn-off time computed above, to account for temperature-dependent variation in the thyristor's actual turn-off time (tq typically increases with junction temperature), du/dt-induced re-triggering risk if a fast-rising reapplied forward voltage is presented immediately after the nominal tq interval, and tolerance/drift in the firing-angle control circuit itself; this is why practical AC voltage controller designs are frequently specified with a guaranteed minimum firing angle (a lower limit on α) below which the controller will not be allowed to operate, precisely to preserve this necessary commutation margin under all rated operating conditions.