What exactly is so difficult about using C-band for missile guidance? Doesn't it roughly just boil down to allowable antenna size? Given how large the HQ-9 missile is, fitting a C-band antenna is not an issue. I think the main reason why we see so many X-band guidance systems is that it facilitates reuse of technology.
The real question is that even if you are able to use C-band for guidance, what makes you think its better to use than X-band? The case is really why use C-band when X-band is superior for this purpose. The HQ-9 seeker is still fairly small and X-band will still give you better angular accuracy. It should not be any bigger than the S-300 missiles like the 48N6 and 5V55 that it is very similar to, and it should be slightly smaller. Yet the 48N6 and 5V55 are both guided on X-band.
If the HT-233 is anything like the Raytheon AN/MPQ-53 then it also has search, in addition to track and illuminate capability. Scale that up a bit, modify the waveguides for naval installation and what you have is a very capable PESA system on 052C. Wild speculation, of course.
AN/MPQ-53 is intended to be able to operate as a self contained all in one unit. Backed by an all search and acquisition radar is an option, although that option is always used. That's probably why it uses C-band it needs to do search. The MPQ-53 has over 5,000 elements while the HT-233 only has over 4,000. Its likely the MPQ-53 is either bigger or the array has a more densely arranged elements due to the use of a higher frequency in the C-band that allows it to be used for fire control.
HT-233 may have search, but it would be secondary search as it is not meant to operate alone. The array itself doesn't have enough range and its why the HT-233 has to work as part of a battery that includes a number of dedicated of search and acquisition radars such as the Type 305A, Type 305B, and Type 120 radars. Type 305A is a 3D AESA. What HT-233 has in relation to these radars the command guidance system and datalink to the HQ-9, which the acquisition radars lack.
This is the radar that you should be comparing the Type 346 with. Not only is it an AESA, but like Type 346, its an S-band AESA.
Now here is something that makes the HT233 difficult for naval use. HT233 happens to be a Spaced PESA.
That round thing way in the back of the array and sitting on the truck happens to the array's feed. If you don't get how it works, the feed projects the signal optically with a lens to the back of the array, and the array itself acts like a second lens that narrows the beam and directs it. This type of PESA isn't an uncommon one, it's typical of PESA supporting SAM, including MPQ-53, Flaplid, and Tombstone. The radars that support Shtil and HQ-16 are also of this nature.
This is different from the kind of PESA like SPY-1 or BARS used on the Su-30M. That kind of PESA has connecting cables from a central amp to each transmitting element in the array. This kind of array is very quality intensive, because the signal needs to reach the elements at the exact same time. A difference in latency or time delay among the elements will cause a frequency shift in the elements, which results in frequency steer. That affects the efficiency of the array. Quite frankly what it costs to build such arrays is probably one of the reasons the Chinese steered directly into AESA in the first place, because it is much easier to calibrate the array electronically if there are any time delays among the elements.
With a spaced phased array, you need to bring the big feed along, and putting it within the superstructure takes a large amount of space. This is why you don't see this type of PESA used in ships unless it is placed outside of the superstructure.
Studying the 051C, and ships that have similar radars like the Kirov and Slava class, tells you another thing. These radars are insufficient for search, and the ship in question has to bring another set of radars used for search and acquisition. You can see it on the Type 051C here --- the FREGAT MAE Top Plate radar.
With the Type 052C/D, you don't see anything used like such search radars. If it is accompanied by a Type 382 Top Plate, then you have a stronger reason to suggest the arrays would be on C-band.
The 052C/D does have the Type 517 metric wave radar, but the resolution of that isn't very good, and in the ships that use them, like the Jiangwei frigates, there is still an S-band Type 360S search radar.
The array on the 052C when first seen gave thought it might be a PESA, but this photo suggests otherwise, as it points to liquid and directed air cooling towards the array itself, which points to an AESA.
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