AC Circuit

AC Circuit Containing Resistance only

Resistance is the opposition that an element offers to the flow of electric current.

It is represented by the uppercase letter R. The standard unit of resistance is

the ohm, sometimes written out as a word, and sometimes symbolized by the

uppercase Greek letter omega: Ω. The behavior of an ideal resistor is dictated by

the relationship specified by Ohm's law.

Resistor is passive element. Passive element is an electrical component that does

not generate power, but instead dissipates, stores, and/or releases it.

Consider a circuit consisting of an ac source and a resistor. The instantaneous

voltage across the resistor

vR = v = Vmax sin wt

From Ohm's law

i = v / R = (Vmax sin wt) / R= Imax sin wt, the instantaneous current flowing through

the resistor .

If Vmax and Imax be the maximum values of the voltage and current respectively, it

follows that: Imax = Vmax / R……… (1)

But the r.m.s. value of a sine wave is 0.707 times the maximum value so that:

r.m.s. value of voltage = V = 0.707 Vmax

r.m.s. value of current = I =0.707 Imax

Substituting for Vmax and Imax in equation (1)

I/0.707 = V / 0.707R, and I = V / R.

Hence Ohm’s law can be applied without any modification to an a.c. circuit

possessing resistance only.

Phase Relationship between applied voltage and current in Resistor

  The graph shows the current through and the voltage across the resistor.
 The current and the voltage reach their maximum values at the same time.
 The current and the voltage are said to be in phase.
 The direction of the current has no effect on the behavior of the resistor.
 Resistors behave essentially the same way in both DC and AC circuits.

This “in-phase” effect can also be represented by a phasor diagram. Therefore, as

the voltage and current are both in-phase with each other, there will be no phase
difference ( θ = 0 ) between them, so the vectors of each quantity are drawn super-

imposed upon one another along the same reference axis.

Power in a resistive circuit

The instantaneous power in a resistive circuit is given by the product of

instantaneous voltage and instantaneous current. The instantaneous power is given

by

p = vi = Vmax sin ωt * Imax sin ωt

Writing Vmax= Vm and Imax= Im

The average power consumed in the circuit over a complete cycle is given by

Paverage = =

= = Vr.m.s.Ir.m.s. = VI

In summary Power to a Resistive Load

• p is always positive. From the above equation it is clear that whatever may

be the value of ωt the value of cos2ωt cannot be greater than 1 hence the

value of p cannot be negative. The value of p is always positive irrespective

of the instantaneous direction of voltage v and current i.

 • Power flows only from source to load and p is the rate of energy

consumption by the load

– All of the power delivered by the source is absorbed by the load.

• This power is known as active power. Power to a pure resistance consists of

active power only.

• Average value of power is halfway between zero and peak value of VmIm

• P = VmIm/2

• If V and I are in RMS values

– Then P = VI

• Also, P = I2R and P = V2/R

• Active power relationships for resistive circuits are the same for ac as for dc.

AC Circuit Containing Inductance only

Inductance is the property of an electrical conductor by which a change

in current through it induces an electromotive force in the conductor. It consists of

a conductor such as a wire, usually wound into a coil.

An "ideal inductor" has inductance, but no resistance or capacitance.

When the current flowing through an inductor changes, the time-varying magnetic

field induces an “e.m.f.” (eL) in the coil, according to Faraday’s law of

electromagnetic induction. According to Lenz's law the direction of induced

"e.m.f." is always such that it opposes the change in current that created it. As a

result, inductors always oppose a change in current. The instantaneous value of the

induced e.m.f. is given by

Since the resistance of the coil is assumed to be negligible, the whole of the

applied voltage is absorbed in neutralizing the induced e.m.f.,

where

Imax= Vmax/ωL.

Thus the current in an inductor lags the applied voltage by an angle π/2 or 90°.

Also from the expression it follows that maximum value of the current is Vmax/ωL,

i.e. Imax= Vmax/ωL, so that Vmax/ Imax = ωL= 2πfL.

If V and I are the r.m.s. values, then V/I = 0.707V max/0.707Imax = 2πfL = inductive
reactance. The term inductive reactance is denoted by the symbol X L. Hence I =

V/2πfL = V/ XL = V/ ωL. This is similar to I = V / R.

The inductive reactance is the opposition that an inductor (or coil) offers to the

alternating current. Therefore, ωL plays the same role as that of a resistor. The

inductor impedes the flow of alternating current in the circuit. Unit of X L is also

ohm.

To have a large reactance the coil

(i) Should have many turn as L N.

(ii) Should have an iron-core as L μrμo.

 (iii) Length and area of the coil as L area / length.

(iv) Also the frequency of a.c should be high.

 This is consistent with Faraday’s Law:

 The larger the rate of change of the current in the inductor, the

larger the back emf, giving an increase in the reactance and a

decrease in the current.

iv) XL in case of DC (direct current), is zero.

The inductive reactance of an inductor increases as the frequency increases. Also

as the frequency increases the current flowing through the inductor also reduces in

value.

The effect of very low and very high frequencies on the reactance of a pure AC

Inductance as follows:

In an AC circuit containing pure inductance the following formula applies:

Phase Relationship between applied voltage and current in Inductor

Phasor Diagram for an Inductor

Power in a Inductive circuit

The instantaneous power delivered to the purely inductive circuit is obtained by

In the above expression, it is found that the power is flowing in alternative

directions. From 0o to 90o it will have negative half cycle, from 90o to 180o it will

have positive half cycle, from 180oto 270o it will have again negative half cycle

and from 270o to 360o it will have again positive half cycle. Therefore this power

is alternating in nature with a frequency, double of supply frequency. As the

power is flowing in alternating direction i.e. from source to load in one quater
cycle and from load to source in next half cycle the average value of this power

is zero. The implication is that the inductive element receives energy from the

source during one-quarter of a cycle of the applied voltage and returns exactly

the same amount of energy to the driving source during the next quarter of a

cycle. Therefore this power does not do any useful work.

The power associated with an inductance is reactive power.

Energy Stored in an Inductor

If the circuit is purely inductive, energy will be stored in the magnetic field

during quarter of a cycle and is obtained by integrating power wave p between

limits of t = T/4 and t = T/2,

=
If L in henrys and Im in amperes respectively, WL is given in joule.

AC Circuit Containing Capacitance only

A capacitor (originally known as a condenser) is a passive two-terminal electrical

component used to store electrical energy temporarily in an electric field. The

forms of practical capacitors vary widely, but all contain at least two electrical

conductors (plates) separated by a dielectric (i.e. an insulator that can store

energy).

An ideal capacitor is wholly characterized by a constant capacitance C, defined as

the ratio of charge Q on each conductor to the voltage V between them.

The circuit contains a capacitor and an AC source. An inductor opposes a change

in current. A capacitor does the opposite. It opposes a change in voltage. Pure

capacitor has zero resistance. When an alternating voltage applied across the
capacitor, the capacitor first charged in one direction and then in another direction.

The charge q is given by

q = Cv = CVmaxsinωt

The flow of electrons “through” a capacitor (i.e. the charging current) is directly

proportional to the rate of change of voltage across the capacitor.

Expressed mathematically, the relationship between the current “through” the

capacitor and rate of voltage change across the capacitor is as such:

Thus the current in a pure capacitor leads the applied voltage by π/2 radian or 90°.

From the above expression it follows that the maximum value of the current is

ωCVmax or 2πfCVmax.

Vmax / Imax = 1 / 2πfC. If V and I are the r.m.s. values of voltage and current then
 = capacitive reactance. The impeding effect of a

capacitor on the current in an AC circuit is called the capacitive reactance. The

capacitive reactance is expressed in ohms and is represented by XC.

★ Capacitive reactance decreases with increasing frequency. In other words,

the higher the frequency, the less it opposes (the more it “conducts”) the AC

flow of electrons and the current increases.

★ As the frequency approaches zero, XC approaches infinity and the current

approaches zero.

o The capacitor would act as an open circuit, and that is why capacitor

blocks DC.

Capacitive Reactance against Frequency

The effect of very low and very high frequencies on the reactance of a pure AC

Capacitance as follows:

Phase Relationship of applied voltage and current in Capacitor

Pure capacitive circuit waveforms

Phasor Diagram for AC Capacitance

Power in a purely Capacitive circuit

In the first-quarter cycle both v and i are positive, therefore the power is also

positive (since p = vi, at any instant). In the second quarter-cycle v stays positive

while i has gone negative, therefore p is negative. In the third-quadrant

both i and v are negative and so p is positive. Finally, in the fourth-quadrant i is

positive and v is still negative resulting in p being negative. The power wave is

thus a series of identical positive and negative pulses whose average value over

an half-cycle of voltage is zero, also note that its frequency is twice the

frequency of the voltage.

During the first and third quarter-cycles the power is positive meaning that power

is supplied by the circuit to charge the capacitor. In the second and fourth quarter-

cycles the capacitor is discharging and thus supplies the energy stored in it back to

the circuit, thus p has a negative value. The minus or plus signs simply indicate the

direction in which the power is flowing. Since this interchange of energy dissipates

no average power no heating will occur and no power is lost.

The capacitive power does not do any useful work. This power is also a reactive

power.

Energy Stored in an Capacitor

The amount of energy received by the capacitor during quarter of a cycle and is

obtained by integrating power wave p between limits of t = 0 and t = T/4,

Since Im=ωCVm,

Summary

Resistance, Reactance

The following is a summary of the relationship between voltage and current in

circuits:

★ Resistance is the special case when φ = 0.

★ Reactance the special case when φ = ± 90°.

Component Resistor Inductor Capacitor

Difference of Voltage and Current lags Voltage lags
Phase between Current are in behind Voltage behind Current by
Voltage and phase by π/2 π/2
Current

Ohm’s Law R =V / I XL = V / I = ωL Xc =V / I = 1 /ωC

Memory Aid for Passive Elements in AC

An old, but very effective, way to remember the phase differences for inductors

and capacitors is:

“E L I” the “I C E”

E.m.f E is before current I in inductors L;

Emf E is after current I in capacitors C.