Alternating Current IIT JEE Study Material

An Alternating Current (AC) changes its direction periodically, whereas if the direction of the current is constant, then it is called a direct current (DC). Let us consider a sinusoidal varying function:

\(\begin\mathbf\;sin\;(\omega t\;+\;\Phi )>\end \)

Here, the maximum current (peak current) is denoted by Ι_m and ‘i’ is the instantaneous current. The factor ( ωt + Φ) is called phase, and ω is termed angular frequency. In terms of frequency,

\(\begin\mathbf<\omega \;=\;2\;\pi \;f>\end \) \(\begin\text\ \mathbf>\end \)

Average Value

It is the average of all the instantaneous values of an AC (alternating current) and alternating voltages over 1 full cycle. The +ve half cycle will be exactly equal to the -ve half cycle in symmetrical waves like voltage or sinusoidal current waveform. Therefore, the average value of a complete cycle will always be 0. The work is done by both the + ve and -ve cycles. Therefore, the average value is calculated without taking its sign into consideration. Therefore, the only +ve half cycle is taken into consideration in determining the average value of alternating quantities (sinusoidal waves).

The average value for one alternation 0 → π is

\(\begin\mathbf\;=\;\frac\;+\;i_\;+\;i_\;+\;.\;.\;.\;.\;+\;i_>>\end \)

The mean or average value of a given function over a specified time interval between t₁ and t₂ is determined by:

\(\begin\mathbf\;=\;\frac\;-\;t_>\;\;\int_^>\;\;f(t)\;dt>\end \)

The cycle of a periodic function is continuously repeated, irrespective of its shape. For example, sinusoidal function. The average value of such periodic function is determined by:

\(\begin\mathbf\;=\;\frac\;\int_^\;\;f(t)\;dt>\end \)

Where, T = time period of the periodic function.

For a symmetrical AC or alternating current (either sinusoidal or nonsinusoidal), the average value is calculated by taking only the mean or average of one alternation or one-half cycle. Therefore, for all the symmetrical waveforms,

\(\begin\mathbf\;=\;\frac>\;\int_^\;\;f(t)\;dt>\end \)

Root Mean Square (RMS) Value

Split the base of 1 alternation into ‘n’ equal parts. Let i₁, i₂,…., i_n be the ordinates of ‘n’ equal parts, then,

Heat generated during first interval

\(\begin=\mathbf^\;R\; \times \;\frac\;\;J>\end \)

Heat generated during second interval

\(\begin=\mathbf^\;R\; \times \;\frac\;\;J>\end \)

Heat generated during third interval

\(\begin=\mathbf^\;R\; \times \;\frac\;\;J>\end \)

Heat generated during nth interval

\(\begin=\mathbf^\;R\; \times \;\frac\;\;J>\end \)

Total heat generated in time T

\(\begin=\mathbf^\;+\;i_^\;+\;i_^\;+\;.\;.\;.\;+\;i_^> \right ]\;\;J>\end \)

If I_eff is the effective current, then the heat generated by this current in time

\(\begin\mathbf^\;\;R\;T >\end \) Joules. Also, by definition, both these expressions are equal. \(\begin\mathbf^\;=\;\left [ \frac^\;+\;i_^\;+\;i_^\;+\;.\;.\;.\;+\;i_^> \right ] >\end \) \(\begin\mathbf\;=\;\sqrt< \frac^\;+\;i_^\;+\;i_^\;+\;.\;.\;.\;+\;i_^>>>\end \)

The virtual or effective value of an alternating voltage or current is equal to the square root of the mean or average of the squares of the successive ordinates. Therefore, it is termed the root mean square or RMS value.

The effective or RMS (root mean square) value over a time period (T) of an alternating quantity is determined by:

\(\begin\mathbf\;=\;\sqrt\;\;\int_^\;F^\;(t)\;dt>>\end \)

Power Consumed

In an LCR circuit connected in series,

Voltage, V = V₀sinωt

Let the electric current flowing through it be,

Instantaneous power, P = VI

The average value of cos (2ωt + Φ) over a complete cycle is 0

The average power consumed in a cycle: