Alternating Current Concept Page - 1

Result

Advantages of AC over DC

For low transmission loss in power lines from the generators to the loads, the voltage level needs to be increased to a larger value so that current is reduced. This feature is provided by transformer for alternating currents only. Hence, AC is preferred over DC for power generation and transmission purposes.

Definition

Amplitude of an alternating current

The current amplitude im$i_m$ of an alternating current source across a resistor R is given as
im=vmR$i_m=\frac{v_m}{R}$ where vm$v_m$ is the amplitude of the oscillating potential difference.

Definition

Define and calculate the time period and frequency of an alternating current

For an alternating current I=I0sinωt$I=I_{0}sin\omega t$

ω$\omega$ is frequency of the alternating current and 2πT${\displaystyle \frac{2\pi }{T}}$ is time period.

Definition

Alternating current

Alternating current (AC) is an electric current in which the flow of electric charge periodically reverses direction. 
i=iosin(ωt)$i=i_{o}sin(\omega t)$ is an example of AC current.
Frequency, f=1T$f={\displaystyle \frac{1}{T}}$ where T$T$ is the time period as shown in the figure.

Example

Calculate root mean square value of alternating current

Example: The equation of an alternating current is I=502√sin(400πt)$I=50\sqrt{2}sin(400\pi t)$ A, then find the frequency and the root mean square value of the current.

Solution:
i=502√sin(400πt)A$i=50\sqrt{2}sin(400\pi t)A$
=imaxsin(wt) A$=i_{max}sin(wt)A$
So, w=400π$w=400\pi$
2πf=400π$2\pi f=400\pi$
f=200Hz$f=200Hz$
imax=502√$i_{max}=50\sqrt{2}$
irms=imax2√$i_{rms}={\displaystyle \frac{i_{max}}{\sqrt{2}}}$
=502√2√$={\displaystyle \frac{50\sqrt{2}}{\sqrt{2}}}$
=50A$=50A$

Definition

Phasor Diagram

A phasor is a scaled line whose length represents an AC quantity that has both magnitude (peak amplitude) and direction (phase) which is frozen at some point in time. A phasor diagram is used to show the phase relationships between two or more sine waves having the same frequency. A phasor diagram is one in which the phasors, represented by open arrows, rotate counterclockwise, with an angular frequency of ω$\omega$ about the origin. The pasors have the following properties :

1. The length of a phasor is proportional to the maximum value of the alternating quantity involved.
2. The projection of a phasor on the vertical axis gives the  instantaneous value of the alternating quantity involved.

Formula

Rms current for a purely resistive circuit

Irms=12√im=0.707im$I_{rms}=\frac{1}{\sqrt{2}}{i}_m=0.707i_m$ where im$i_m$ is the current amplitude.

Formula

Rms voltage for a purely resistive circuit

v=12√vm$v=\frac{1}{\sqrt{2}}{v}_m$ where v$v$ is the rms voltage, vm$v_m$ is the amplitude of the oscillating potential difference

Definition

Instantaneous and average power

The instantaneous power dissipated in the resistor is
P=i2R=i2mR(sin(ωt))2$P=i^{2}R=i_m^{2}R{(sin(\omega t))}^2$ and
average power is P¯=1T∫T0P(t)dt=12i2mR$\overline{P}=\frac{1}{T}{\int }_0^{T}P(t)dt=\frac{1}{2}{i}_m^{2}R$
where ω$\omega$ is the angular frequency.

Definition

Voltage and current in same phase for a purely resistive circuit

For a purely resistive circuit,
I=imsin(ωt−ϕ)$I=i_{m}sin(\omega t-\varphi )$
V=imRsin(ωt−ϕ)$V=i_{m}Rsin(\omega t-\varphi )$
where I$I$ and V$V$ are current and voltage respectively which reach their maximum and minimum values at the same time i.e. they are in phase.

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