Half wave rectified alternating current is one which flows for half the time during one cycle. It is illustrated in figure 1 where suppressed half cycle is shown dotted.

As mentioned earlier, for determining rms and average value of such an alternating current summation would be carried over the period for which current actually flows i.e. 0 to π but would be average for the whole cycle i.e. from 0 to 2π.

RMS value of current,

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Average value of current,

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Peak Factor =

Form factor =

*Example 1: An alternating voltage e=200 sin 314t is applied to a device which offers an ohmic resistance of 20 ohm to the flow of current in one direction, while preventing the flow of current in opposite direction. Calculate rms value, average value and form factor for the current over one cycle.*

**Solution: **

The instantaneous voltage applied to the rectifying device is given by the expression

e = 200 sin 314 *t*

Maximum value of applied voltage,

E_{max} = Coefficient of the sine of time angle = 200 volts

Resistance of rectifying device, R = 20 ohm

Maximum value of half-wave rectified alternating current,

RMS value of half-wave rectified alternating current,

**Ans.**

Average value of the half wave rectified alternating current,

**Ans.**

Form factor of the half-wave rectified alternating current

**Ans.**

*Example 2: A voltage is applied to the circuit shown. What is the rms value of current through the resistor R of 100-ohm? Derive the formula used.*

**Solution:**

The maximum value of voltage applied to the circuit

RMS value of circuit current,

**Ans**.

*Example 3: Find this value for the waveform shown in figure 3.*

**Solution:**

Maximum value of alternating current, I_{max} = 10 A.

Average value, **Ans.**

*Example 4: A circuit carries a current which is the resultant of direct of 20A, and a sinusoidal alternating current having a peak value of 20A. Find the rms value of current in the circuit.*

**Solution: **

The resultant current waveform is shown in figure 4

The instantaneous value of the resultant current wave is given by the expression

And

Or

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considering complete cycle

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or **Ans.**

*Example 5: A transmission line carries a dc voltage of 50 V and a half-wave rectified sinusoidal voltage as shown in figure 5.*

*Calculate:*

*RMS value**The average value**Form factor*

**Solution: **

The instantaneous value of the voltage wave is given by the expression

(1)

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**Ans.**

(2)

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= **Ans.**

(3) Form factor = **Ans.**

*Example 6: Calculate the RMS value of current given by i = 10 + 5 cos (628t + 30 ^{0})*

**Solution:**

The waveform of the given current is shown in figure 6.

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considering complete cycle.

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Or **Ans.**

*Example 7: The half cycle of an alternating signal is as follows:*

*It increases uniformly from zero at 0 ^{0} to F_{m} at α^{0}, remains constant from α^{0} to (180 – α^{0}), decreases uniformly from F_{m} at (180 – α^{0}) to zero at 180^{0}. Calculate the average and effective value of the signal.*

**Solution:**

The half cycle of given alternating signal is shown in figure 7 and may be expressed as

Average value, F_{av}

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= **Ans.**

RMS value is given by

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Or **Ans.**

*Example 8: Find the average and effective values of voltage for sinusoidal waveform shown in figure 8.*

**Solution:**

The equation of the given sinusoidal waveform is

= **Ans.**

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Or **Ans.**

*Example 9: Determine the form factor and peak factor for the unshaded waveform in figure 9.*

**Solution: **

Area of unshaded portion of the curve

= Area OAF + area FABE + area EBCD

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RMS value of unshaded waveform

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**Ans.**

**Ans.**