Cosine firing scheme for thyristors in single-phase converters is shown in Fig. The synchronizing transformer steps down the supply voltage to an appropriate level. The input to this transformer is taken from the same source from which converter circuit is energized.

The output voltage $v_{1}$ of synchronizing transformer is integrated to get cosine-wave $v_{2}$ . The d.c. control voltage $E_{c}$ varies from maximum positive $E_{c m}$ to maximum negative $E_{c m}$ so that firing angle can be varied from zero to $180^{\circ} .$ The cosine wave $v_{2}$ is compared in comparators 1 and 2 with $E_{c}$ and $-E_{c}$ . When $E_{c}$ is high as compared to $v_{2},$ output voltage $v_{3}$ is available from comparator 1 . Same is true for comparator 2 . So the comparators 1 and 2 give output pulses $v_{3}$ and $v_{4}$ respectively as shown in Fig. It is seen from this figure that firing angle is governed by the intersection of $v_{2}$ and $E_{e} .$ When $E_{c}$ is maximum, firing angle is zero. Thus, firing angle $\alpha$ in terms of $V_{2 m}$ and $E_{c}$ can be expressed as

$$V_{2 m} \cos \alpha=E_{c}$$

OR

$$\alpha=\cos ^{-1}\left(\frac{E_{c}}{V_{2 m}}\right)-----(1)$$

where $V_{2 m}=$ maximum value of cosine signal $v_{2}$

The signals $v_{3}, v_{4}$ obtained from comparators are fed to clock-pulse generators $1,2$ to get clock pulses $v_{5}, v_{6}$ as shown in Fig. These signals $v_{5}, v_{6}$ energies a $J K$ flip flop to generate output signals $v_{i}$ and $v_{j} .$ The signal $v_{i}$ is amplified through the circuit and is then employed to turn on the SCRs in the positive half cycle. Signal $v_{j},$ after amplification. is used to trigger SCRs in the negative half cycle.

For a single-phase full converter, average output voltage is given by

$$V_{0}=\frac{2 V_{m}}{\pi} \cos \alpha-----(2)$$

Substituting the value of $\alpha$ from Eq. 1 in Eq. 2, we get,

$$V_{0}=\frac{2 V_{m}}{\pi} \cos \left[\cos ^{-1} \frac{E_{c}}{V_{2 m}}\right]=\left[\frac{2 V_{m}}{\pi} \cdot \frac{1}{V_{2 m}}\right] \cdot E_{c}$$

$$V_{0}=k E_{c}$$

This shows that cosine firing scheme provides a linear transfer characteristic between the average output voltage $V_{0}$ and the control voltage $E_{\mathrm{e}} .$ This scheme, on account of its linear transfer characteristic, improves the closed-loop response of the converter system.