25 mOhm ESR — the ripple-current limiter
The TPMD337M004R0025: The 25 mOhm ESR is the spec that sets the AC ripple current ceiling in a 330 µF tantalum. At 4 V rated, the cap's self-heating from ripple is I² × ESR — keep the RMS current below the thermal limit in the datasheet to avoid derating the lifetime.
330 µF, 4 V — bulk decoupling for a 1.8–3.3 V core rail
330 µF at 4 V gives enough charge reservoir for a moderate-load digital core (FPGA core, MCU rail, or POL output). The 4 V rating leaves zero headroom for a 3.3 V rail — the cap sees the full rail voltage, so a 6.3 V or 10 V part is safer if the rail has transients.
Listed as Active — no last-time-buy deadline to chase. KYOCERA AVX still runs the TPM series.