ALTERNATING CURRENT [ A.C ]

Refers to the *flow of electric charge that reverses periodically*( opposite to direct current ) . It starts from zero, grows to a maximum, decreases to zero, reverses, reaches a maximum in the opposite direction, returns again to zero, and repeats the cycle indefinitely. Here , period is the time taken to complete one cycle and Frequency is the number of cycles per second .The maximum value in either direction is the current’s amplitude. While low frequencies (50 – 60 cycles per second) are used for domestic and commercial power, frequencies of around 100 million cycles per second (100 megahertz) are used in television and of several thousand megahertz in radar and microwave communication. An important advantage of alternating current is that the voltage can be increased and decreased by a transformer for more efficient transmission over long distances.

A current or voltage is called alternating if –

- its amplitude is constant and
- its half cycle alternates between positive and negative

In case the current or voltage *changes periodically as the sin or cos function of time , its said to be sinusoidal*. The functions are –

**v = V sin ω t** or **v = V cos ω t** . Here , v – instantaneous potential difference , V – maximum potential difference [voltage amplitude ] and ω is the angular frequency [ = 2π x frequency ].

Also, ** i = I cos ω t ** or ** i = I sin ω t** , where i – instantaneous current and I – maximum current [ current amplitude ] .

Points to note –

- A rectifier is used to convert AC into a DC . For the reverse process , [ DC to AC ] an inverter is used.
- AC current is not useful for chemical processes like electroplating and electrolysis as large ions cannot follow the frequency of AC current.

**PHASOR DIAGRAMS** –

Rotating vector diagrams used to represent sinusoidally varying voltages / currents are referred to as phasor diagrams . Here , the instantaneous value of a quantity that varies sinusoidally with time is represented by projecting onto a horizontal axis of a vector with a length equal to the amplitude of the quantity and its angle with the *x*-axis at any instant representing the phase .

The term ‘ **phasor** ‘ refers to the vector which rotates counter-clockwise with a constant angular speed ω. As the projection of the phasor into the horizontal axis at time is I cos ω t , it’s the cosine function which is used. *The ‘vector’ term related to phasor , is not a real vector like velocity or momentum*. It’s basically a geometric method to evaluate physical quantities that changes sinusoidally with time. In the case of a simple harmonic motion , we deal with a single phasor. In this sense , sinusoidal quantities can be combined with phase differences by means of vector addition .

**Average / Mean of A.C** –

For an alternating current / voltage , if the average or mean value is taken for a full cycle , it would be zero as ∫^{γ}_{0} sin ωt dt or ∫^{γ}_{0} cos ωt dt is zero. Hence the values are taken for a half cycle , which is either negative or positive.

I_{av} = ∫^{γ/2}_{0} I dt / ∫^{γ/2}_{0} dt = ∫^{π/ω}_{0} I0 sin ωt dt / ∫^{π/ω}_{0} dt = **I _{av} = 2 I_{0}/ π**

**ROOT MEAN SQUARE ( RMS )** values of AC Current –

A quantity having either positive or negative values is better expressed by its rms values . This is because , *even if i is negative , i² would be positive and hence I _{rms} will never be zero* . Here , the average or mean value of i is taken after squaring it , and thereafter the square root of that average is taken . Now , i or instantaneous current = I cos ωt . Therefore , –

i² = I² cos² ωt = I² ½ ( 1 + cos 2 ωt ) [ as cos² A = ½ ( 1 + cos 2A) ] =

i² = ½ I² + ½ I² cos 2 ωt . As cos 2 ωt represents the positive half and the negative half of the time , its average is zero . Hence the average of i² is I² /2 .

Therefore , ** I _{rms} = √ I² /2 = I / √2**

the same can be applied for a sinusoidal voltage , i.e , **V _{rms} = V /√2** .

**AC Circuit Constituents** –

**Resistance**– Consider a sinusoidal current i = I cos ωt passing through a resistor with a resistance R . The current amplitude or the maximum current will be I , and the positive direction of the current will be counter-clockwise around the circuit . Now , the instantaneous voltage across the resistor , from Ohm’s law will be V_{R}= iR = (IR) cos ωt . Now , the voltage amplitude or the maximum voltage will be the coefficient of the cosine function,i.e, V_{R}= I_{R}. This means the equation can also be written as V_{R}= V_{R}cos ωt . It indicates that the current is in phase with the voltage as both the current i and the voltage V_{R }are proportional to cos ωt .

a) Resistor connected to ac circuit and b) graphs of i and VR as the functions of time

On the left is the phasor diagram for a resistor in an ac circuit . Note that as the i and V_{R }are in phase and hsve the same frequency , their ( current and voltage ) phasors rotate together. In addition , they are parallel at each instant .

**Inductance**– Consider an AC ciruit having an inductor of self inductance L and no resistance through which a current i = I cos ωt flows. Here again , the positive direction of the current is counter-clockwise around the circuit. Eventhough there is no resistance , a potential difference ( V_{L}) exists between the inductor terminals a and b as the current changes with time . The electro-motive force ,**ε =**is not equal to V*– L di/dt*_{L.}as the inductor current ( in the counter-clockwise direction , i.e from a to b ) is increasing , and hence the di/dt is positive and the induced emf acts in the left to oppose the increase of current . The net result is that the point a is at a higher potential than point b . Hence the potential at a is V_{L}= L di/dt .

V_{L} = L di/dt = L d( I cos ωt ) /dt = Hence , **V _{L} = – I ωL sin ωt**