1. Fig. below shows a single phase fully controlled converter with source inductance. For simplicity it has been assumed that the converter operates in the continuous conduction mode. Further, it has been assumed that the load current ripple is negligible and the load can be replaced by a dc current source the magnitude of which equals the average load current. Fig. shows the corresponding waveforms

2. It is assumed that the thyristors T3 and T4 were conducting at t = 0. T1 and T2 are fired at ωt = α. If there were no source inductance T3 and T4 would have commutated as soon as T1 and T2 are turned ON.

3. The input current polarity would have changed instantaneously. However, if a source inductance is present the commutation and change of input current polarity cannot be instantaneous. s. Therefore, when T1 and T2 are turned ON T3 T4 does not commutate immediately. Instead, for some interval all four thyristors continue to conduct as shown in Fig. 2(b). This interval is called “overlap” interval.

Fig current and voltage waveforms

1. During overlap interval the load current freewheels through the thyristors and the output voltage is clamped to zero. On the other hand, the input current starts changing polarity as the current through T1 and T2 increases and T3 T4 current decreases. At the end of the overlap interval the current through T3 and T4 becomes zero and they commutate, T1 and T2 starts conducting the full load current

2. The same process repeats during commutation from T1 T2 to T3T4 at ωt = π + α . From Fig. 2(b) it is clear that, commutation overlap not only reduces average output dc voltage but also reduces the extinction angle γ which may cause commutation failure in the inverting mode of operation if α is very close to 180º.

3. In the following analysis an expression of the overlap angle “μ” will be determined. From the equivalent circuit of the converter during overlap period.

$$L\frac{di_i}{dt}=v_i \ \ for \ \ α ≤ ωt + μ \\ \; \\ i_i(ωt=α)=-I_0 \\ \; \\ i_i=I-\frac{\sqrt2V_i}{ωL}cosωt \\ \; \\ \therefore i_i|_{ωt-α}=I-\frac{\sqrt2V_i}{ωL}cosα=-I_0 $$

$$ \therefore V_0=2\sqrt2\frac{v_i}{\pi}[cosα-cos(α+\mu)] \\ \; \\ \therefore V_0=\frac{2\sqrt2}{\pi}v_icosα-\frac{2}{\pi}ωLI_0---(eq1) $$

Equation 1 can be represented by the following equivalent circuit

Equivalent circuit representation of the single phase fully controlled rectifier with source inductance

The simple equivalent circuit of Fig. 3 represents the single phase fully controlled converter with source inductance as a practical dc source as far as its average behavior is concerned. The open circuit voltage of this practical source equals the average dc output voltage of an ideal converter (without source inductance) operating at a firing angle of α. The voltage drop across the internal resistance “RC” represents the voltage lost due to overlap shown in Fig. 1(b) by the hatched portion of the Vo waveform. Therefore, this is called the “Commutation resistance”. Although this resistance accounts for the voltage drop correctly there is no power loss associated with this resistance since the physical process of overlap does not involve any power loss. Therefore this resistance should be used carefully where power calculation is involved.