Alternating Current

Alternating Current and Alternating Voltage

Alternating current is the current whose magnitude changes with time and the direction reverses periodically. 

Instantaneous value of and AC is given by

I=Im sin ωt or I = Im cost ωt

here Im is the peak value of current and ω = 2πν ..

Similarly, an alternating emf changes in magnitude continuously with time and reverses its direction periodically. 

Instantaneous value of an alternating emf is given by 

E=Em sin ωt or E = Em cost ωt

Here Em is the peak value of alternating voltage

ω = 2πν .

The Average, RMS & Peak Value of Alternating Current and Alternating Voltage

Average value of AC: it is that value of steady current, which when passed through a circuit for half the time period of the alternating current sends the same amount of charge as is done by the alternating current in the same time through the same circuit.

Average value of alternating current is zero over a cycle. 

Iav = 2Im/π = 0.636Im.

Root Mean Square value of AC: 

It is that value of steady current, which when passed through a resistance for a given time would produce the same amount of heat as is produced by the alternating current in the same resistance and in the same time. 

Iv = I0/√2  = 0.707 Im

Iv = virtual current or rms current

Im is the peak value of current

Em= 2Em/ pie = 0.636 Em

Ev = Em/√2 = 0.707 Em

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AC through a resistor: 

Here in alternating current and an alternating emf are in phase. 

V=IR

This relation is true for instantaneous, peak and rms values of V and I.

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AC through an Inductor

Here, an alternating current lags behind the alternating emf by a phase angle of π /2 

V = L dI/ dt

Vrms/ Irms = XL

XL = ω L = 2πν L

Here XL is called the inductive reactance and is measured in ohm.

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AC through a capacitor:

Here, an alternating current leads an alternating emf by a phase angle of π/2 

I = C dV/ dt

Vrms/ I rms = Xc

Xc = 1/ ωC = 1 / 2πνC

Xc is called the capacitive reactance and is measured in ohm. 

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Phasor Diagram: 

A diagram representing the alternating voltages and the alternating currents (of same frequency) as vectors along with the phase angle between them is called a phasor diagram. 

Phasors: Phasors are rotating vectors which are used to represent peak or rms values of alternating currents and voltages showing phase relationship between them. The peak or rms value of each alternating quantity is represented by an arrow rotating anticlockwise about the origin at the angular frequency ω . The projection of this vector on the vertical axis gives the instantaneous value. 

Phasor Diagram: 

a. Resistor in AC

b. A pure inductor in AC

c. A pure Capacitor in AC

d. A series LCR circuit. 

V0 = Peak voltage

V = RMS Voltage

V0 = V√2 

V0 /√2= V 

I0 = Peak current

I = RMS Current

I0 = I√2 

I0 /√2= I