Introduction:
A single-phase full-wave controlled rectifier is a circuit that converts an alternating current (AC) voltage into a pulsating direct current (DC) voltage. It is commonly used in various applications such as power supplies, battery chargers, and motor drives. In this circuit, a centre-tap transformer is utilized to provide the required voltage for rectification.

Explanation:
The given information states that the voltage across each half of the secondary winding of the centre-tap transformer is Vm sin ωt. Let's understand the concept behind this and calculate the peak inverse voltage.

Understanding the Centre-Tap Transformer:
A centre-tap transformer consists of a secondary winding with a center tap, which is connected to the ground or common terminal. The primary winding of the transformer is connected to an AC voltage source. The secondary winding generates two equal and opposite voltages with respect to the center tap.

Voltage across Each Half of the Secondary Winding:
The voltage across each half of the secondary winding can be represented as Vm sin ωt. Here, Vm represents the peak value of the voltage and ωt represents the angular frequency of the AC voltage source.

Peak Inverse Voltage (PIV):
The peak inverse voltage is the maximum voltage that appears across the blocking diode or the thyristor during the reverse bias condition. In a full-wave controlled rectifier, this occurs when the diode or the thyristor is non-conducting.

Calculating the Peak Inverse Voltage:
In a single-phase full-wave controlled rectifier with a centre-tap transformer, the peak inverse voltage can be calculated as follows:

- During the positive half-cycle of the AC voltage, the upper diode conducts and the lower diode is non-conducting. Therefore, the voltage across the lower diode is Vm sin ωt.

- During the negative half-cycle of the AC voltage, the lower diode conducts and the upper diode is non-conducting. Therefore, the voltage across the upper diode is also Vm sin ωt.

- Since the voltage across each diode is Vm sin ωt, the peak inverse voltage is twice this value, i.e., 2 Vm sin ωt.

Therefore, the correct answer is option A) 2 Vm.

Explanation:
The single-phase full-wave controlled rectifier using a center-tap transformer is a circuit arrangement that converts alternating current (AC) to direct current (DC). It consists of a center-tapped transformer, two diodes, and a load resistor.

Working of the Circuit:
1. The center-tapped transformer has a primary winding connected to the AC power supply and a secondary winding with a center tap.
2. The AC input voltage is given by Vm sin ωt, where Vm is the peak voltage and ωt is the angular frequency.
3. When the AC voltage is positive, the diode connected to the upper half of the secondary winding conducts, allowing current to flow through the load resistor in one direction.
4. When the AC voltage is negative, the diode connected to the lower half of the secondary winding conducts, allowing current to flow through the load resistor in the opposite direction.
5. This process repeats for each half-cycle of the AC input, resulting in a pulsating DC output across the load resistor.

Peak Inverse Voltage (PIV):
The peak inverse voltage (PIV) refers to the maximum voltage that appears across the diode when it is reverse biased (not conducting). In the given circuit, we need to determine the PIV.

Analysis:
1. During the positive half-cycle of the AC voltage, the diode connected to the lower half of the secondary winding is reverse biased and does not conduct. Hence, the voltage across the lower half of the secondary winding is zero.
2. During the negative half-cycle of the AC voltage, the diode connected to the upper half of the secondary winding is reverse biased and does not conduct. Hence, the voltage across the upper half of the secondary winding is zero.
3. Therefore, the PIV is the maximum voltage across each half of the secondary winding, which is Vm sin ωt.
4. Since the peak value of sin ωt is 1, the PIV is Vm * 1 = Vm.

Conclusion:
The peak inverse voltage (PIV) in a single-phase full-wave controlled rectifier using a center-tap transformer is equal to the peak voltage (Vm). Therefore, the correct answer is option 'A' (2 Vm).