This document discusses AC circuits and polyphase systems. It begins by introducing AC quantities like instantaneous value, cycle, time period, frequency, amplitude, average value, RMS value, form factor and peak factor. It then covers phasor representation and analysis of RL, RC and RLC circuits. The document also discusses apparent power, true power, reactive power and power factor. Finally, it covers polyphase systems including phase and line voltages/currents, balanced and unbalanced systems, and the relationships between phase and line values for balanced delta and star connections.

1. AC Circuits

2. Introduction to AC quantities
Phasor representation of alternating quantities
Analysis of series RL circuit, RC circuit, RLC circuit
Parallel and series-parallel AC circuits
phasor method, admittance method

3. The use of direct currents is limited to a few applications e.g. charging of batteries,
electroplating, electric traction etc.
For large scale power distribution there are, however, many advantages in using alternating
current (a.c.).
The a.c. system has offered so many advantages that at present electrical energy is
universally generated, transmitted and used in the form of alternating current.
Even when d.c. energy is necessary, it is a common practice to convert a.c. into d.c. by
rectifiers.
Three principal advantages are claimed for a.c. system over the d.c. system. First, alternating
voltages can be stepped up or stepped down efficiently by means of a transformer.
The transmission of electric power at high voltages to achieve economy and distribute the
power at utilisation voltages.
Introduction

4. Generation of Alternating Voltages and
Currents
(i) by rotating a coil at constant angular velocity in a uniform magnetic field.
or
(ii) by rotating a magnetic field at a constant angular velocity within a stationary coil.
 The first method is used for small a.c.
generators while the second method
is employed for large a.c. generators.

5. Equation of Alternating Voltage and Current
Consider a rectangular coil of n turns rotating in anticlockwise direction with an angular velocity
of w rad/sec in a uniform magnetic field.

10. Instantaneous value.
The value of an alternating quantity at any instant is called instantaneous value. The instantaneous values of
alternating voltage and current are represented by v and i respectively.
Cycle
One complete set of positive and negative values of an alternating quantity is known as a cycle.
Time period
The time taken in seconds to complete one cycle of an alternating quantity is called its time period. It is generally
represented by T.

11. Frequency
The number of cycles that occur in one second is called the frequency (f) of the alternating quantity.
It is Hertz (Hz).
Amplitude/Peak value/Crest value/Maximum value
The maximum value (positive or negative) attained by an alternating quantity is called its amplitude
or peak value. The amplitude of an alternating voltage or current is designated by Vm (or Em) or Im.

12. Example: The maximum current in a sinusoidal a.c. circuit is 10A. What is the instantaneous current at 45º ?

13. Example: An alternating current i is given by ; i = 141·4 sin 314 t
Find (i) the maximum value (ii) frequency (iii) time period and (iv) the instantaneous value when t is 3 ms.

14. Values of Alternating Voltage and Current
In a d.c. system, the voltage and current are constant so that there is no problem of
specifying their magnitudes. However, an alternating voltage or current varies from
instant to instant. A natural question arises how to express the magnitude of an
alternating voltage or current.
There are four ways of expressing it, namely ;
(i) Peak value (ii) Average value or mean value
(iii) R.M.S. value or effective value (iv) Peak-to-peak value
Although peak, average and peak-to-peak values may be important in some engineering
applications, it is the r. m.s. or effective value which is used to express the magnitude of
an alternating voltage or current.

15. Peak Value
 It is the maximum value attained by an alternating quantity. The peak or maximum value of an
alternating voltage or current is represented by Vm or Im. The knowledge of peak value is important
in case of testing materials. However, peak value is not used to specify the magnitude of alternating
voltage or current. Instead, we generally use r.m.s. values to specify alternating voltages and
currents.

16. Average Value
 The average value of a waveform is the average of all its values over a period of time. In
performing such a computation, we regard the area above the time axis as positive area and area
below the time axis as negative area. The algebraic signs of the areas must be taken into account
when computing the total (net) area. The time interval over which the net area is computed is the
period T of the waveform.
 (i) In case of *symmetrical waves (e.g. sinusoidal voltage or current), the average value over one
cycle is zero. It is because positive half is exactly equal to the negative half so that net area is zero.
However, the average value of positive or negative half is not zero. Hence in case of symmetrical
waves, average value means the average value of half-cycle or one alternation.

18. Average Value of Sinusoidal Current

24. Form Factor and Peak Factor
 There exists a definite relation among the peak value, average value and r.m.s. value of an alternating quantity.
The relationship is expressed by two factors, namely ; form factor and peak factor.
 (i) Form factor: The ratio of r.m.s. value to the average value of an alternating quantity is known as form factor
i.e.

25. (ii) Peak factor: The ratio of maximum value to the r.m.s. value of an alternating quantity is known as peak
factor i.e.

26. Example: Find the average value, r.m.s. value, form factor and peak
factor for (i) halfwave rectified alternating current and (ii) full-wave
rectified alternating current.
(i) Half-wave rectified a.c.

28. Example: An alternating voltage v = 200 sin 314t is applied to a device which offers an ohmic resistance
of 20 Ω to the flow of current in one direction while entirely preventing the flow of current in the opposite
direction. Calculate the r.m.s. value, average value and form factor.

29. Phasor representation of alternating quantities
 An alternating voltage or current may be represented in the form of (i) waves and (ii) equations. The waveform
presents to the eye a very definite picture of what is happening at every instant. But it is difficult to draw the wave
accurately. No doubt the current flowing at any instant can be determined from the equation form i = Im sin ωt but
this equation presents no picture to the eye of what is happening in the circuit.
 The above difficulty has been overcome by representing sinusoidal alternating voltage or current by a line of
definite length rotating in *anticlockwise direction at a constant angular velocity (ω). Such a rotating line is called a
phasor. The length of the phasor is taken equal to the maximum value (on a suitable scale) of the alternating
quantity and angular velocity equal to the angular velocity of the alternating quantity. As we shall see presently, this
phasor (i.e. rotating line) will generate a sine

30. Phasor Representation of Sinusoidal Quantities
 Consider an alternating current represented by the equation i = Im sin ωt. Take a line OP to represent to scale
the maximum value Im. Imagine the line OP (or **phasor, as it is called) to be rotating in anticlockwise
direction at an angular velocity ω rad/sec about the point O. Measuring the time from the instant when OP is
horizontal, let OP rotate through an angle θ (= ωt) in the anticlockwise direction. The projection of OP on the
Y-axis is OM.
OM = OP sin θ
= Im sin ωt
= i, the value of current at that instant

32. Addition of alternating quantities
Following methods are used for addition of two or more sinusoidal quantities of the same kind.
1. Method of components
2. Analytical addition method
3. Parallelogram method
4. Waveform addition method

33.  Method of Components. This method provides a very convenient means to add two or more phasors. Each
phasor is resolved into horizontal and vertical components. The horizontals are summed up algebraically to
give the resultant horizontal component X. The verticals are likewise summed up algebraically to give the
resultant vertical component Y.

34. Example: A circuit consists of four loads in series ; the voltage across these loads are given by the
following relations measured in volts :
v1 = 50 sin ω t ; v2 = 25 sin (ω t + 60º)
v3 = 40 cos ω t ; v4 = 30 sin (ω t − 45º)
Calculate the supply voltage giving the relation in similar form.

52. R-L Series A.C. Circuit

53. If the applied voltage is v = Vm sin ωt, then equation for the
circuit current will be

57. Apparent, True and Reactive Powers
 Consider an inductive circuit in which circuit current I lags behind the
applied voltage V by φ°.
 The phasor diagram of the circuit is shown in Fig.
 The current I can be resolved into two rectangular components:
(i) I cos φ in phase with V.
(ii) I sin φ ; 90° out of phase with V.

58. 1. Apparent power
 The total power that appears to be transferred between the source and
load is called apparent power..
Apparent power, S = V × I = VI
 It is measured in volt-ampers (VA).
 Apparent power has two components: true power and reactive power.
2. True power
 The power which is actually consumed in the circuit is called true
power or active power.
 It is the useful component of apparent power.
 The product of voltage (V) and component of total current in phase
with voltage (I cos φ) is equal to true power.

59.  It is measured in watts (W). The component I cos φ is called in-phase
component or wattful component.
3. Reactive power
 The component of apparent power which is neither consumed nor does any
useful work in the circuit is called reactive power.
 The power consumed (or true power) in L and C is zero because all the
power received from the source in one quarter-cycle is returned to the
source in the next quarter-cycle. This circulating power is called reactive
power.

60.  The product of voltage (V) and component of total current 90° out of
phase with voltage (I sin φ) is equal to reactive power
 It is measured in volt-amperes reactive (VAR). The component I sin φ is
called the reactive component (or wattless component)

61. R-C Series A.C. Circuit

62. Power angle:

63. R-L-C Series A.C. Circuit

66. A coil having a resistance of 7  and an inductance of 31.8 mH
is connected to 230 V, 50 Hz supply.
Calculate :
(i) the circuit current
(ii) phase angle
(iii)power factor
(iv) power consumed and
(v) voltage drop across resistor and inductor.

68. A capacitor of capacitance 79.5 μ F is connected in series with a
non-inductive resistance of 30  across 100 V, 50 Hz supply.
Find:
(i) impedance
(ii) current
(iii)phase angle and
(iv) equation for the instantaneous value of current.

70. A 230 V, 50 Hz a.c. supply is applied to a coil of 0.06 H
inductance and 2.5 resistance connected in series with a 6.8 μF
capacitor.
Calculate:
(i) impedance
(ii) current
(iii)phase angle between current and voltage
(iv) power factor and
(v) power consumed.

72. Methods of Solving Parallel A.C. Circuits
1. By phasor diagram
2. Admittance method

73. (1)By phasor diagram:
 In this method, we find the magnitude and phase angle of each branch
current.
 We then draw the phasor diagram taking voltage as the reference phasor.
 The circuit or line current is the phasor sum of the branch currents and can
be determined either
(i) by parallelogram method or
(ii) by the method of components.

74. Branch 1,
Branch 2,
Consider a parallel circuit consisting of two branches and connected to an
alternating voltage of V volts (r.m.s.).

75.  The current I1 in branch 1 leads the applied
voltage V by φ1 as shown in the phasor
diagram.
 The current I2 in branch 2 lags behind the
applied voltage V by φ2 as shown in the
phasor diagram.
 The line current I is the phasor sum of I1
and I2. Suppose its phase angle is φ as
shown in Fig.
 The values of I and f can be determined by
resolving the currents into rectangular
components.

77. Admittance Triangle
 For an inductive circuit
 Admittance angle is equal to the impedance angle but is negative. For this
reason, BL will be along OY′-axis and hence negative.

79.  For Capacitive circuit
 admittance angle is equal to the impedance angle but of opposite sign. For
this reason, BC will lie along OY-axis and hence positive.

80. Admittance Method for Parallel A.C. Circuit Solution
 Fig. shows two impedances Z1 = R1 – j XC1 and Z2 = R2 + jXL2 in parallel
across an a.c. supply of V volts. We can convert the impedances into
equivalent admittances as under :

81. Admittance Method for Parallel A.C. Circuit Solution
 Fig. shows Y1 and Y2 resolved into conductances and
suceptances. It may be noted that conductance and suceptance
components of each admittance are paralleled elements.
where G = G1 + G2 and B = B1 – B2

82. Department of
Electrical Engineering
Polyphase
System
Basic of Electrical & Electronics
Engineering
(01EE1101)

83. Poly Phase System
A polyphase alternator has two or more separate but identical windings (called phases)
displaced from each other by **equal electrical angle and acted upon by the common uniform
magnetic field.

85. Phase Voltage
It is defined as the voltage across either phase winding or load terminal. It is denoted by
Vph. Phase voltage VRN, VYN and VBN are measured between R-N, Y-N, B-N for star
connection and between R-Y, Y-B, B-R in delta connection.
Line voltage
It is defined as the voltage across any two-line terminal. It is denoted by VL. Line voltage
VRY, VYB, VBR measure between R-Y, Y-B, B-R terminal for star and delta connection
both.

86. Phase current
It is defined as the current flowing through each phase winding or load. It is denoted by
Iph. Phase current IR(ph), IY(ph) and IB(Ph) measured in each phase of star and delta
connection. respectively.
Line current
It is defined as the current flowing through each line conductor. It denoted by IL. Line
current IR(line), IY(line), and IB((line) are measured in each line of star and delta
connection.

87. Phase sequence
The order in which three coil emf or currents attain their peak values is called the phase
sequence. It is customary to denoted the 3 phases by the three colours. i.e. red (R),
yellow (Y), blue (B).
Balance System
A system is said to be balance if the voltages and currents in all phase are equal in
magnitude and displaced from each other by equal angles.
Unbalance System
A system is said to be unbalance if the voltages and currents in all phase are unequal in
magnitude and displaced from each other by unequal angles.

88. Balance load
In this type the load in all phase are equal in magnitude. It means that the load
will have the same power factor equal currents in them.
Unbalance load
In this type the load in all phase have unequal power factor and currents.

89. Relation between line and phase values for voltage and current in case of
balanced delta connection.

92. Relation between line and phase values for voltage and current in case of balanced
star connection.