AC Generator, RMS Value and Average Value for Half Cycle

The basic principle of the ac generator is a direct consequence of Faraday's law of induction. When a conducting loop is rotated in a magnetic field at constant angular frequency ω a sinusoidal voltage (emf) is induced in the loop. This instantaneous voltage is,

            V = V₀ sin ωt                                       ... (i)

AC generator in action

The usual circuit diagram symbol for an ac source or generator is shown in Figure.

Another video to explain how AC Generator Works

In Equation (i) V₀ is the maximum output voltage of the ac generator, or the voltage amplitude and w is the angular frequency, equal to 2π times the frequency f.

            ω = 2πf

The frequency of ac in India is 50 Hz, i.e.,

            f = 50 Hz

So,       ω = 2πf » 314 rad/s

The time of one cycle is known as time period T, the number of cycles per second the frequency f.

            T = 1/f     or  T = 2π/ω

A sinusoidal current might be described as,

            i = i₀ sin ωt

If an alternating current is passed through an ordinary ammeter or voltmeter, it will record the mean value for the complete cycle, as the quantity to be measured varies with time. The average value of current for one cycle is,

Thus,    _{one cycle} = 0

Similarly, the average value of the voltage (or emf) for one cycle is zero.

         _{one cycle} = 0

Since, these averages for the whole cycle are zero, the dc instrument will indicate zero deflection. In ac, the average value of current is defined as its average taken over half the cycle. Hence,

This is sometimes simply written as, i_av. Hence,

               i_av = _{Half Cycle} = 2/π i₀ ≈ 0.637 i0

Similarly, V_av = (2/π)V₀ ≈ 0.637 V₀

A dc meter can be used in an ac circuit if it is connected in the full wave rectifier circuit. The average value of the rectified current is the same as the average current in any half cycle, i.e., 2/π time the maximum current i₀. A more useful way to describe a quantity is the root mean square (rms) value. We square the instantaneous current, take the average (mean) value of i² and finally take the square root of that average. This procedure defines the root-mean-square current denoted as irms. Even when i is negative, i² is always positive so irms is never zero (unless i is zero at every instant). Hence,

Similarly, we get RMS value V_rms = V₀/√2 ≈ 0.707 V₀

The square root of the mean square value is called the virtual value and is the value give by ac instruments.

Thus, when we speak of our house hold power supply as 220 volts ac, this means that the rms voltage is 220 volts and its voltage amplitude is,

            V₀ = √2V_rms = 311 volt

Form Factor

The ratio, rms value/average value = (V₀/√2)/(2V₀/π)

                                   = π/2√2 = 1.111

is known as form factor,

Note:

1. The average value of sin wt, cos wt, sin2wt, cos2wt, etc. is zero because it is positive half of the time and negative rest half of the time. Thus,

2. The average value of sin² wt and cos² wt is 1/2

3. Like SHM, general expressions of current/voltage in an sinusoidal ac are,

     = i₀ sin (ωt ± f),                   V = V₀ sin (ωt ± f)

or i = i₀ cos (ωt ± f),        and     V = V₀ cos (ωt ± f)