Use a high power resistor. For any given temperature, resistors will be more reliable than semiconductors, so there is no advantage in dissipating that power in a transistor.
This is common in power circuit design, for example dimmers and switched mode power supplies, you will need high power rated components because you are working at high voltages so even moderate currents produce high power dissipation.
I assume you are using a hard-fired technique, where you leave the gate current on continuously. In that case, the only way to reduce dissipation in the triac gate resistor is to use a more sensitive triac. But there is also the soft fired technique, where you only pulse the gate at the zero crossing and the triac stays latched on until the main current falls to zero. This is not suitable for all loads.
This kind of design assignment should not be too difficult to break down into constituent parts, so you can replace one big problem with a bunch of smaller problems. Hopefully you already know how to solve some, or all, of the smaller problems.
If you start at the output and work backwards- you want 3 LEDs driven by some kind of circuit that controls their illumination. Let's say you have a "DC" voltage that represents the RMS AC input voltage that you've been asked to measure. By "DC" I mean rectified and low-pass filtered so that it has little ripple. Say the voltage is 10V for 240V RMS, 9.5V for 5% low, and 10.5V for 5% high. So you need to design a circuit that will illuminate the Red, Green or Yellow LED based on that voltage (3 states, so it can be defined with two comparison bits). That's one smaller problem.
A second problem is how to power the circuit. You know you have a step-down transformer, so you should be able to design a power supply. But wait- there's an issue here with the specifications. You're told to illuminate a yellow LED if the voltage is 5% or more below nominal, but it's going to be hard to do that at 0V. You may have to make a reasonable assumption here- say it will work down to 30% under nominal. So your power supply has to work with as low as 160V in, and still provide (say) 15V regulated for the circuit to work.
That's smaller problem number two.
The third problem is how to get a voltage representing the RMS voltage into a DC voltage. One approach is to use a rectifier or precision rectifier circuit and rectify and filter the output voltage of the transformer. It's easier to measure the average value of the rectified voltage than the RMS value and assume it's a sine wave (this is where your AC analysis might come in, there is a constant factor between the two for a sine wave). This is really three even smaller problems- rectify the voltage, filter the voltage and (perhaps) scale the voltage so that it meets our requirements in the first problem of 230VAC->10.0V output.
So, a total of five smaller design problems, and we've detected a deficiency in the specifications. This is a fairly representative assignment in terms of what you'll run into, in miniature, but all the elements are there.
One little enhancement I'll recommend- keep the current draw (especially of the LED circuit since it will draw the most) constant regardless of which LEDs are illuminated. If you can describe why that's a good thing, you may get bonus marks.
Best Answer
No. The DC link voltage will be \$ 230 \sqrt 2 \ \text V \$.
simulate this circuit – Schematic created using CircuitLab
Figure 1. Possible test layout.
LAMP1 acts as a current limiter. Open SW1 while powering up and on initial unloaded trials. When you're happy that nothing bad is happening and the lamp doesn't glow full brightness then you can close the switch. Many variable frequency supplies use a similar arrangement to limit the power-on surge current due to the discharged capacitor as this might damage the rectifier. They'll use a resistor and a relay rather than a lamp and switch.
XMFR1 isolates the L + N so now you can ground your DC- and safely attach the scope earth clip there. The DC LINK+ will be dangerous. Take care.