In this article, we will see the analysis of **Single Phase Half Wave Controlled Rectifier** with Resistive (R) Load as shown in Figure 1. V_{s} is supply and ‘i_{s}‘ is the source current. V_{T} and ‘i_{T}‘ is the SCR voltage and current respectively. V_{o} and ‘i_{o}‘ is the load voltage and current respectively.

V_{s} = V_{m}sin(ωt)

**Step-1: Write KVL in the given circuit**

v_{s} − v_{T} − v_{o} = 0

v_{ab} = v_{s} = v_{T} + v_{o}

v_{ab} = v_{s} = V_{m}sin(ωt)

v_{ba} = − v_{ab} = − V_{m}sin(ωt)

**Step-2: Device working status**

In the positive half cycle, SCR is forward biased and is turned ON at ωt = α by giving a firing pulse (triggering pulse). The angle **α** is called **delay angle** or **firing angle**. It is measured from the instant the SCR has become forward biased.

At ωt = π, the current through the SCR falls to zero i.e. below zero and simultaneously a reverse voltage appears across SCR and it turns OFF. Since the line voltage is used for commutation, it is also known as **line commutation converter**.

**1). 0 < ωt < α**

SCR is in forward blocking mode and SCR is OFF.

i_{o} = i_{s} = i_{T} = 0 Amp

v_{o} = 0 V

v_{T} = v_{ab} = V_{m}sin(ωt)

**2). α < ωt < π**

SCR is in forward conduction mode and SCR is ON.

v_{T} = 0 V

i_{o} = i_{s} = i_{T} = (V_{m}sin(ωt))/R

v_{o} = v_{ab} = V_{m}sin(ωt)

**3). π < ωt < 2π**

SCR is in forward blocking mode and SCR is OFF.

i_{o} = i_{s} = i_{T} = 0 Amp

v_{o} = 0 V

v_{T} = v_{ab} = V_{m}sin(ωt)

**4). 2π + α < ωt < 3π**

SCR is in forward conduction mode and SCR is ON.

v_{T} = 0 V

i_{o} = i_{s} = i_{T} = (V_{m}sin(ωt))/R

v_{o} = v_{ab} = V_{m}sin(ωt)

The waveforms for load voltage(v_{o}), and current(i_{o}), SCR voltage(v_{T}) is shown in figure 2. The waveforms of supply current and SCR current is same as load current.

**Note:** For a resistive load, v_{o} and i_{o} waveform will be same in nature except in magnitude.

**Step-3:**

1). Firing angle (α) = α

2). Extinction angle (β) = π

3). Conduction angle (γ) = β − α = π − α

4). Conduction time (t_{c}) = γ/ω = (π − α)/ω

5). Circuit turn off time (t_{ckt-off}) = (2π − π)/ω = π/ω

6). Peak Inverse voltage (PIV) = V_{m}

**Step-4:**

**Average Values**

The average output voltage V_{o(avg)} across load R is given by

The average output current I_{o(avg)} through the load R is given by

**RMS values**

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