Protectio of lectrical Networks hristophe Preve opyright 0 006, IST td. Appedix B alculatio of Irush urret Durig apacitor Bak ergizatio Fixed bak The equivalet stream etwork sigle-phase diagram durig eergizatio of the fixed bak is show i Figure B-. (t) (): t sigle-phase voltage : stream etwork iductace : iductace of the coectio likig the switchig device to the capacitor bak Figure B-: equivalet diagram durig fixed bak eergizatio
494 Protectio of lectrical Networks We shall demostrate that the frequecy of the trasiet curret occurrig o eergizatio is very high (see sectio 0.6., example ; f0, 58 Hz ). This results i justificatio of eglect of the etwork resistace i relatio to the iductace: R f0, sice f0 50 Hz. Similarily, the resistace of the coectio likig the switchig device to the capacitor is egligible. The etwork frequecy (50 Hz) is egligible i relatio to the trasiet curret frequecy. We might therefore cosider that we have a voltage step throughout the duratio of the trasiet curret. The value of the step, at worst, is the peak value of the siusoidal voltage: U U : phase-to-phase voltage The curret it is determied by the followig differetial equatio: t di dt i d where: () t 0 for t 0 () t ˆ for t 0 We shall solve this equatio usig aplace trasforms. As a aplace trasform, the differetial equatio becomes: ˆ V t 0 0 s I s i t I s s s s The curret is zero before eergizatio ad it is assumed that the voltage at the capacitor termials is zero (worst case). Hece: it 0 0 ad V t 0 0
Appedix B 495 ˆ s s thus givig us: s Is Is hece: I s ˆ ˆ s s s s et us take: I s s Usig the aplace trasform tables, we ca deduce it : it ˆ si t it U si t The maximum peak irush curret is thus: Iˆ rush U A ad its frequecy: f 0 Switched steps bak The equivalet sigle-phase diagram durig switched steps bak eergizatio is show i Figure B-.
496 Protectio of lectrical Networks + U : stream etwork iductace : iductace of the coectio likig the switchig device to the bak Figure B-: equivalet diagram durig switched steps bak eergizatio The peak irush curret Irush is maximum whe baks are i service ad the th oe is eergized. The baks i service off load ito the bak that has just bee eergized. The stream iductace is very high i relatio to iductace (see sectio 0.6., example : 85 H ad example :.5 H). The curret splied by the stream part (etwork) is therefore eglected. It is assumed that, at worst, o eergizatio the voltage at the termials of U V t 0. each capacitor is ˆ The equivalet diagram is thus show i Figure B-. The diagram comprises parallel-coected braches with a impedace of Z j. j The equivalet impedace is therefore: Z eq Z j j
Appedix B 497 + : iitial voltage coditio at the capacitor termials Figure B- The diagram thus becomes that of Figure B-4. Figure B-4
498 Protectio of lectrical Networks We have two series-coected iductaces: We have two series-coected capacitaces: The equivalet diagram is thus that i Figure B-5. Figure B-5 The equivalet diagram i Figure B-5 is the same as that of a fixed bak. If we re-use the formula for a fixed bak, we immediately obtai:
Appedix B 499 U I rush ˆ U I rush ˆ f