A 3 kW device what ever it is, draws around 13 amp, appendix 15 on BS 7671:2008 says
The load current in any part of the circuit should be unlikely to exceed for long periods the current-carrying capacity of the cable (Regulation 433.1.5 refers). This can generally be achieved by:
(i) locating socket-outlets to provide reasonable sharing of the load around the ring
(ii) not supplying immersion heaters, comprehensive electric space heating or loads of a similar profile frog the ring circuit
(iii) connecting cookers, ovens and hobs with a rated power exceeding 2 kW on their own dedicated radial circuit
(iv) taking account of the total floor area being served. (Historically, limit of 100 m² has been adopted.)
So in the main a 3 kW device should be on their own dedicated circuit, however if we look at most homes, we have washing machines, tumble driers, immersion heaters, dish washers, ovens, all over 2 kW and only the immersion heater tends to have a dedicated supply.
Two things we look at, one is how long it draws the power for, and two how close to the centre of a ring final.
At the centre of a ring final any load is even on both legs, but as we get closer to the consumer unit (CU) the load on one leg exceeds the load on the other.
However ring final is the odd one out, we would hope most radial circuits are protected with an overload which will disconnect before the cable over heats. So a 16A or 20A radial even if you over load it, worst case is the overload disconnects it, assuming installed correct to start with.
We split the installation into circuits to avoid hazards and minimize inconvenience in the event of a fault, and to take account of danger that may arise from the failure of a single circuit such as a lighting circuit and to reduce the possibility of unwanted tripping of RCDs due to excessive protective conductor currents produced by equipment in normal operation.
A circuit is an assembly of electrical equipment supplied from the same origin and protected against overcurrent by
the same protective device(s), and with a RCD it still measures current imbalance so fitting a RCD creates a circuit. Although not quite in the same way as a MCB or fuse, and even if the RCD is rated 60/83/100 amp it will not disconnect with that load, it needs a fuse or MCB as well.
In the main we install RCD's in series with a increase/decrease of x3 or 1/3, so a board with a 300 mA may feed a few boards with 100 mA and they may in turn feed a few 30 mA which can feed a few 10 mA RCD's. With domestic today we start at 30 mA. The 10 mA RCD's are not very common.
In days gone by we used 100 mA which rarely tripped, so inconvenience was minimized, but when we went to 30 mA it was found often these tripped for what seemed no reason. We should test with the load applied the leakage does not exceed 9 mA, well my clamp on ammeter starts at 100 mA so could not do that if I wanted to. Until amendment 3 we tested at ½ and 5 times rated tripping current, and if the ½ was done with circuits connected then reasonable sure less than 9 mA leakage, also testing with circuits connected would identify if DC was freezing the RCD. However we were told to test with load removed, so this forum is full of people complaining their RCD trips.
The single width RCBO (that's a RCD and MCB combined) removed the problem of nuisance trips. It divided the installation into circuits for both overload and earth leakage.
However this all means when writing in answer on a forum mistakes are easy made, we have active and passive, built into FCU or socket, or part of a plug, built into a consumer unit (CU) either protecting many circuits, or one for each circuit (RCBO) and we have type AC, A, F, B, and S.
Then add to that personal preferences, I have no problem was a type AC RCD with a TN supply as secondary protection, I do with a TT supply however, but that is my personal opinion not what the rule book says.
Years ago the RCD socket was common, I have one to hand a 10 mA version, but as the regulations changed and we had to protect near enough every circuit, these have slowly vanished, but there are still some homes with no RCD protection, so you need to explain what you have so we don't make errors, it is so easy to assume all homes are like your own, but likely few homes have 16 RCBO's and SPD installed, so explain what you have.