AMET UNIVERSITY

AMET UNIVERSITY8.1.22 Production of an Alternating E.M.F. Let a single turn coil be rotated with constant angular velocity in a uniform

magnetic field, its axis of revolution being perpendicular to the magnetic lines.

Let l m. = length of coil side Parallel to axis.

r m = radius of path.

= angular velocity of coil (i.e., number of radians/ sec.) B Wb/m2 = flux density Reckon time from the instant when The coil lies in the plane of reference XOX.

Linear velocity of coil sides v = r m/ sec. Angle turned through in t sec. = t

Velocity of coil side at right angles to field = AD = v sin . If the conductor continued its motion in the direction AC, then

Flux cut per sec. = B l v sin .

E.M.F. generated in coil side at time t = B l v sin volts. E.M.F. generated in coil at time t, = 2 B l v sin volts.

= 2 B l r sin t volts.

= sin t volts. Where = 2 B l r = maximum flux linking coil.

Max. value of e.m.f., Em = volts, since the max. value

of sine t is 1.

E.M.F. equation may be written e = Em sin t or e = Em sin,

where e = instantaneous e.m.f. after t secs.

Such an e.m.f. is called a Sinusoidal or Simple harmonic E.M.F. This is the ideal wave Form. If a wave deviates from this, it is regarded as a distorted wave. The sine wave has the advantage that it greatly simplifies the theory and calculation of the a.c.circuit. since = 2 f , the e.m.f. equation may be written e = Em sin 2 f t.

In the same way an alternating current may be written i = Im sin t or I = Im sin 2 f t, where I is the instantaneous

value and Im the maximum value.

Again, since = 2 f, f =

2

8-1-23 An Alternating quantity, either current or e.m.f., is one which periodically passes through a definite cycle of changes, each cycle consisting of two half cycles, during one of which the quantity acts in one direction around the circuit and during the other, in the opposite direction E.M.F. generated in coil side at time t = B l v sin volts.

E.M.F. generated in coil at time t, = 2 B l v sin volts.

= 2 B l r sin t volts.

= sin t volts.

Where = 2 B l r = maximum flux linking coil.

Max. value of e.m.f., Em = volts, since the max. value

of sine t is 1.

E.M.F equation may be written e = Em sin t, or e = Em sin,

where e = instantaneous e.m.f. after t secs.

Short Notes:

i) Alternating e.m.f.: An Alternating e.m.f., is one which

periodically passes through a definite cycle of changes, each cycle consisting of two half cycles, during one of which the quantity acts in one direction around the circuit and during the other, in the opposite direction

ii) Frequency or cycle: frequency is the number of cycles passed through per sec. symbol ~

iii) Amplitude: The maximum value of positive or negative which an

alternating quantity attains during one complete cycle is called Amplitude or Peak value or Maximum value. The amplitude of alternating voltage and current is represented by Em or Im.iv) RMS value (Root Mean Square Value): The RMS value of an alternating current is defined as that steady current which when flowing through a given resistance for a given time produces the same amount of heat as produced by the alternating current when flowing through the same resistance for the same time. It is also known as effective value of an alternating current.

The RMS value is given thus:

I RMS = area under the squared wave Period RMS value of voltage, Erms = 0.707 Em

RMS value of current, I rms = 0.707 Im

v) Average Value: The Arithmatical average of all the valuesof an alternating quantity over one cycle is called Average value.

Average value = Area under the curve Base (period)For alternating voltage or current varying sinusoidally is given by:

E av = 0.637 Em

I av = 0.637 Imvi) Form factor: The ratio of RMS value to Average value of an

alternating quantity is called Form factor.

Form factor = RMS value = 1.11 Average value8-1-24 a) A.C.circuit containing Resistance only: A pure resistive circuit is

one which has negligible inductance compared to its resistance. Ordinary filament lamps, water heater etc.are the examples.

Following are the Important properties of Resistance circuits:

(i) Current is in phase with the voltage.

(ii) Current I = V/R, where V and I are RMS values.

(iii) Power consumed in the circuit = I2 R = V x I. (b) A.C.circuit containing Inductance only: A circuit which has neither resistance nor capacitance is called pure Inductive circuit. Whenever an alternating voltage is applied to a circuit, an alternating magnetic field is set up. Its effect is to produce back e.m.f. due to self-induction of the coil. There is no ohmic resistance. So, the applied voltage has to overcome the self induced e.m.f.only. The applied e.m.f. is equal and opposite to-self induced e.m.f. at every instant.The effective resistance of the inductive coil is called Inductive

Reactance and is denoted by XL and its unit is Ohms.

XL = 2 f L, where f is the frequency and L is the Inductance

in Henry.

Following are the Important points of Inductive circuit.

(i) Current I = V amps. = V amps.

X L 2 f L

(ii) Current lags behind the voltage by 90.(iii) Power consumed in the circuit is 0. Variation of XL with f. As XL = L = 2 f l Where L is a constant. X L f. (c) A.C.circuit containing Capacitance only: A circuit which

has neither resistance nor inductance is called pure capacitive

circuit.Whenever a capacitor is connected to an a.c.supply, the capacitor charges and discharges during alternate quarter cycle and their charge and discharge give rise to a.c.currents in the

circuit. The effective resistance of capacitor is denoted by Xc = 1 ohms. 2 f C Where f is the frequency of the a.c. supply and C is the

Capacitance of the condenser in farad.

Following are the Important points of capacitive circuit: (i) I = V = V x 2 f C amps. Xc

(ii) Current leads the voltage by 90

(iii) Power consumed in the circuit is 0.8.1.25. a) Impedance: Total opposition offered by a series circuit

containing resistance, Inductance and capacitance is called Impedance. It is measured in Ohms and is denoted by Z. Z = R 2 + ( XL ~ Xc ) 2 ohms.

In D.C. circuit, I = V/R

In A.C. circuit, I = V/Z b) Relationship between Impedance, Voltage, Current and

Power Factor:The Power factor ( P.F.) relationship of a circuit:

(i) The ratio of resistance and Impedance = R/Z

(ii) The ratio = Watts = True power = V x I x cos Volt amperes Apparent power V x I(iii) The cosine of the angle of lead or lag of the current from applied voltage.

8.1.26. Ship board installations produce poor demand with a lagging power factor is due to running many induction motors below full load, lighting of fluorescent tubes and using transformers.

Power factor improvement is necessary in an installation because Low power factor has the following adverse effects :(i) For given power to be supplied, the current is increased due to low p.f., it causes increase in copper losses and decreases efficiency of both apparatus and the supply system.

(ii) With low p.f. generators, transformers, switches, transmission lines become over loaded.

(iii) Low p.f. makes the voltage regulation of generators, transformers and transmission lines greater.

(iv) With low p.f. cost of generation and transmission increases due to the increase in current and use of thicker wires and bigger switches etc.

8.1.27. a) The Power factor may be defined in any of the following ways:

i) The ratio of resistance and Impedance = R/Z

ii) The ratio = Watts = True power = V x I x cos Volt amperes Apparent power V x I

iii) The cosine of the angle of lead or lag of the current from applied

voltage.

b) P = I 2 R175 = 2.5 2 R therefore, R = 175 = 28 ohms. 6.25

Z = V/ I = 100 = 40 ohms. 2.5

Z = R 2 + X 2 therefore, X = 402 282 = 816 = 28.566 ohms.

Therefore, 2 f L = 28.566L = 28.566 = 0.28566 henry = 0.0909 henry. 2 x 50

P = V x I x cos 175 = 100 x 2.5 x cos Cos = 175 / 50 = 0.7. Therefore, Power Factor = 0.7