I've made a few triac-based incandescent lighting controllers and never used a resistor from gate to neutral. Triacs are current-controlled devices, so they will stop being forced on as soon as you remove the drive current.
I'm sure you know that they latch on until the load current stops.
As for the size of the resistor, it will only see a short spike of current when the triac turns on. After that, the entire triac is a short-circuit except for a couple of diode drops. So R1 should be sized primarily to handle the voltage because the average power is still quite low.
As per the application notes of BTA08-600CW, to control a three phase induction motor, for triggering the triac gate, the value of resistor is shown as 510 ohm. How is this value arrived at?
Fairchild application note AN-3004 page 3 reads as follows:
Resistor R (shown in Figure 7) is not mandatory when RL is
a resistive load since the current is limited by the gate trigger
current (IGT) of the power triac. However, resistor R (in combination
with R-C snubber networks that are described in the
section "Inductive and Resistive Loads") prevents possible
destruction of the triac drive in applications where the load is
highly inductive.
Unintentional phase control of the main triac may happen if
the current limiting resistor R is too high in value. The function
of this resistor is to limit the current through the triac
driver in case the main triac is forced into the non-conductive
state close to the peak of the line voltage and the energy
stored in a "snubber" capacitor is discharged into the triac
driver. A calculation for the current limiting resistor R is
shown below for a typical 220 volt application: Assume the
line voltage is 220 volts RMS. Also assume the maximum
peak repetitive drive current (normally for a 10 micro second
maximum time interval is 1 ampere. Then
$$ R = \frac {V_{peak}}{I_{peak}} = \frac {220 \sqrt {2}}{1} = 311~\Omega $$
One should select a standard resistor value >311 ohms →
330 ohms.
The gate resistor RG (also shown in Figure 7) is only necessary
when the internal gate impedance of the triac or SCR is
very high which is the case with sensitive gate thyristors.
These devices display very poor noise immunity and thermal
stability without RG. The value of the gate resistor in this
case should be between 100 and 500. The circuit designer
should be aware that use of a gate resistor increases the
required trigger current (IGT) since RG drains off part of IGT.
Use of a gate resistor combined with the current limiting
resistor R can result in an unintended delay or phase shift
between the zero-cross point and the time the power triac triggers.
What would be the resistor wattage?
I'm not going to work this out for you but it only carries current for the very short time between turning on of the opto triac until the big triac switches on.
From the data sheet, which quadrant to be considered to know the gate trigger current?
I and III. Since the trigger voltage is derived from the supply voltage they always match.
Also for three phase supply, how is the voltage shown as 220 V instead of 440 V (in the schematic diagram)?
Why not? That's a 60 Hz supply and 220 V may be available.
Can the gate be triggered with either AC or DC? If so what is the impact of calculating the resistor value in both these cases?
No. This system is using zero-cross opto-isolated trigger devices. You feed about 5 to 20 mA into the opto LED and it will only light when forward biased. See the datasheet (which I didn't check). The LED has to be on at zero-cross so if you tried to trigger from AC it would be off (unless the trigger AC was from another phase).
For further reading, see my answer to Using AC current to trigger Triac where I explain the operation of the zero-cross circuit.
Best Answer
The datasheet for the triac says the average gate power is .5 watts. You need a resistor rated for at least that. Also, If we look at the Triac wikipedia it discusses that 3Q triacs are designed to operate heavily inductive loads. This would explain it functioning better. Normal TRIACs or 4Q triacs have problems due to high voltage and current angles when they turn off at zero crossing. They experience a voltage spike which can turn them back on again. Commutations (▲v/▲t) can be improved with the use of snubber networks. They are designed to provide at need to help balance the circuit. With the introduction of 3Q triacs this problem was avoided. If shopping for a device like this you can simply search for "Snubberless Triacs" which is what they are often called.