When the current in an individual capacitor reaches a natural zero-crossing, the thyristors can be left unbated and no further current will flow. The reactive power supplied to the power system ceases abruptly. The capacitor, however, is left with a trapped charge (Figure 6.15(a)). Because of this charge, the voltage across the thyristors subsequently alternates between zero and twice the peak-phase voltage. The only instant when the thyristors can be gated again without transients is when the voltage across them is zero (Figure 6.15(b)). This coincides with peak-phase voltage.

**Ideal transient-free switching**

The simple case of a switched capacitor, with no other circuit elements than the voltage supply, is used first to describe the important concept of transient-free switching. Figure 6.16 shows the circuit.

With sinusoidal AC supply voltage v = v̂ sin (ω

_{0}t + α), the thyristors can be gated into conduction only at a peak value of voltage, that is, when
(6.8)

Gating at any other instant would require the current

*i*=*C*dv/dt to have a discontinuous step change at*t*= 0+. Such a step is impossible in practice because of inductance, which is considered in the next section. To permit analysis of Figure 6.16, the gating must occur at a voltage peak, and with this restriction the current is given by
Fig. 6.15 Ideal transient-free switching waveforms. (a) switching on; and (b) switching off.

Fig. 6.16 Circuit for analysis of transient-free switching.

(6.9)

where α = ±π/2. Now ω

_{0}C = B_{C}is the fundamental-frequency susceptance of the capacitor, and X_{C}= 1/B_{C}its reactance, so that with α = ±π/2
(6.10)

where

*î*is the peak value of tha AC current,_{AC}*î*_{AC}=v̂B_{C}= v̂/X_{C }.
In the absence of other circuit elements, we must also specify that the capacitor be precharged to the voltage

*V*_{C0}= ±*v̂*, that is, it must hold the prior charge ±*v̂/C.*This is because any prior DC voltage on the capacitor cannot be accounted for in the simple circuit of Figure 6.16. In practice this voltage would appear distributed across series inductance and resistance with a portion across the thyristor switch.
With these restrictions, that is, dv/dt = 0 and V

_{C0}= ±*v̂*at t = 0, we have the ideal case of transient-free switching, as illustrated in Figure 6.15. This concept is the basis for switching control in the TSC. In principle, once each capacitor is charged to either the positive or the negative system peak voltage, it is possible to switch any or all of the capacitors on or off for any integral number of half-cycles without transients.
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