Hi David Rose,
Thank you for you code, and thank you for your patience.
I have got all the LPC802, LPC824, LPC845 board on my side, today, I test it with your code in all the lpc802, LPC824, LPC845 board, the same uart configuration:
LPC_USART0->CFG = (UART_CFG_ENABLE) // Enable UART operation - Specify parameters
| (UART_CFG_DATALEN_8)
| (UART_CFG_PARITY_NONE)
| (UART_CFG_STOPLEN_1)
| (UART_CFG_OESEL |UART_CFG_OEPOL |UART_CFG_OETA);
and check the user manual again.
I highly agree with your point, the LPC824 RTS wave is really abnormal.
Because, when OESEL=1, OETA=1, should like this in the user manual:the Output Enable signal remains asserted for 1 character time after then end the last stop bit of a transmission.
But from this WAVE:

After stop bit, the RTS should still wait for 1 character, but in LPC845, the RTS is desserted.
Anyway, I will report this problem to our according department, and thanks a lot for your good question.
Here, I post the LPC802, lpc824, lpc845 test wave, just for the post completeness.
1. LPC802
2,LPC824

3. LPC845

If I get any updated information, I will let you know.
Have a great day,
Kerry
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