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# Notes - Alternating Current Theory Notes | EduRev

## : Notes - Alternating Current Theory Notes | EduRev

``` Page 1

AC Theory  â€“ Professor J R Lucas 1  November 2001
Alternating Current Theory - J R Lucas
An alternating waveform is a periodic waveform which alternate between positive and
negative  values.  Unlike direct waveforms, they cannot be characterised by one magnitude as
their amplitude is continuously varying from instant  to instant.  Thus various forms of
magnitudes are defined for such waveforms.
The advantage of the alternating waveform for electric power is that it can be stepped up or
stepped down in potential easily for transmission and utilisation.  Alternating waveforms can
be of many shapes.  The one that is used with electric power is the sinusoidal waveform.  This
has an equation of the form
v(t) = V
m
sin(? t + f )
If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p.
(a) Instantaneous value:   The instantaneous value of a
waveform is the value of the waveform at any given instant of
time. It is a time variable a(t).
For a sinusoid,  Instantaneous value  a(t)  = A
m
sin(? t+ f )
(b) Peak value:   The peak value,  or maximum value, of a
waveform is the maximum instantaneous value of the
waveform.
For a sinusoid, Peak value = A
m

(c) Mean value:   The mean value of a waveform is equal
to the mean value over a complete cycle of the waveform.
It also corresponds to the direct component of the
waveform.
Mean value  A
mean
=
?
+T t
t
o
o
dt t a
T
). (
1

The mean value of a waveform which has equal positive and negative half cycles must thus be
always zero.
For a sinusoid, Mean value =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
f ?  =  0
(d) Average value (rectified):   The full-wave rectified
average value or average value of a waveform is defined
as the mean value of the rectified waveform over a
complete cycle.
Average value  A
avg
=
?
?
?
?
?
?
?
?
?
?
-
? ?
fcycle hal
negative
fcycle hal
positive
dt t a dt t a
T
). ( ). (
1

For a sinusoid,
Average value   =
?
?
?
?
?
?
?
?
-
? ?
T
T
m
T
m
dt t A dt t A
T
2
2
0
. sin . sin
1
? ?
=
m m
A A
T p ?
2
2
1
= ·
The average value is defined in the above manner in electrical engineering as otherwise all
alternating waveforms would have zero value and would be indistinguishable.
a(t)
-A
m
A
mean
v(t)
t
T
T
V
m

f/?
A
avg
v
rect
t
T
T
Page 2

AC Theory  â€“ Professor J R Lucas 1  November 2001
Alternating Current Theory - J R Lucas
An alternating waveform is a periodic waveform which alternate between positive and
negative  values.  Unlike direct waveforms, they cannot be characterised by one magnitude as
their amplitude is continuously varying from instant  to instant.  Thus various forms of
magnitudes are defined for such waveforms.
The advantage of the alternating waveform for electric power is that it can be stepped up or
stepped down in potential easily for transmission and utilisation.  Alternating waveforms can
be of many shapes.  The one that is used with electric power is the sinusoidal waveform.  This
has an equation of the form
v(t) = V
m
sin(? t + f )
If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p.
(a) Instantaneous value:   The instantaneous value of a
waveform is the value of the waveform at any given instant of
time. It is a time variable a(t).
For a sinusoid,  Instantaneous value  a(t)  = A
m
sin(? t+ f )
(b) Peak value:   The peak value,  or maximum value, of a
waveform is the maximum instantaneous value of the
waveform.
For a sinusoid, Peak value = A
m

(c) Mean value:   The mean value of a waveform is equal
to the mean value over a complete cycle of the waveform.
It also corresponds to the direct component of the
waveform.
Mean value  A
mean
=
?
+T t
t
o
o
dt t a
T
). (
1

The mean value of a waveform which has equal positive and negative half cycles must thus be
always zero.
For a sinusoid, Mean value =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
f ?  =  0
(d) Average value (rectified):   The full-wave rectified
average value or average value of a waveform is defined
as the mean value of the rectified waveform over a
complete cycle.
Average value  A
avg
=
?
?
?
?
?
?
?
?
?
?
-
? ?
fcycle hal
negative
fcycle hal
positive
dt t a dt t a
T
). ( ). (
1

For a sinusoid,
Average value   =
?
?
?
?
?
?
?
?
-
? ?
T
T
m
T
m
dt t A dt t A
T
2
2
0
. sin . sin
1
? ?
=
m m
A A
T p ?
2
2
1
= ·
The average value is defined in the above manner in electrical engineering as otherwise all
alternating waveforms would have zero value and would be indistinguishable.
a(t)
-A
m
A
mean
v(t)
t
T
T
V
m

f/?
A
avg
v
rect
t
T
T
AC Theory  â€“ Professor J R Lucas 2  November 2001
(e) Effective value or r.m.s. value:
Neither the peak value, nor the mean value, nor the average value defines the useful value of
the waveform with regard to the power or energy.  Thus the effective value is defined based
on the power equivalent of the quantity.
The effective value is thus defined as the constant value which would cause the same power
dissipation as the original quantity over one complete period.
Thus considering current   Power dissipation  =  I
eff
2
. R  =
?
+T t
t
o
o
dt R t i
T
. ). (
1
2

giving I
eff
=
?
+T t
t
o
o
dt t i
T
). (
1
2

or in general Effective value  A
eff
=
?
+T t
t
o
o
dt t a
T
). (
1
2

It is to be noted that the method of calculating the effective value involves the following
processes. Taking the root of the mean of the squared waveform over one complete cycle.
Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform.
i.e.  r.m.s. value  A
rms
=
?
+T t
t
o
o
dt t a
T
). (
1
2

This is defined for both current as well as voltage.
For a sinusoid,
r.m.s. value   =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
2 2
f ?  =
2
m
A

Unless otherwise specified, the rms value is the value that is always specified for ac
waveforms, whether it be a voltage or a current.  For example, 230 V in the mains supply is
an rms value of the voltage.  Similarly when we talk about a 5 A, 13 A or 15A socket outlet
(plug point), we are again talking about the rms value of the rated current of the socket outlet.
For a given waveform, such as the sinusoid, the peak value, average value and the rms value
are dependant on each other.  The peak factor and the form factor are the two factors that are
most commonly defined.

value average
value rms
= Factor Form
and for a sinusoidal waveform,  Form Factor = 111 . 1 1107 . 1
2
2
? =
p
m m
V V

The form factor is useful such as when the average value has been measured using a rectifier
type moving coil meter and the rms value is required to be found. [Note: You will be studying
about these meters later]
Peak Factor =
peak value
rms value
and for a sinusoidal waveform,
Peak Factor =

The peak factor is useful when defining highly distorted
waveforms such as the current waveform of compact
fluorescent lamps.
V
V
m
m
2
2 1 4142 == .

v
2
(t)
t
T
T
t
i
Page 3

AC Theory  â€“ Professor J R Lucas 1  November 2001
Alternating Current Theory - J R Lucas
An alternating waveform is a periodic waveform which alternate between positive and
negative  values.  Unlike direct waveforms, they cannot be characterised by one magnitude as
their amplitude is continuously varying from instant  to instant.  Thus various forms of
magnitudes are defined for such waveforms.
The advantage of the alternating waveform for electric power is that it can be stepped up or
stepped down in potential easily for transmission and utilisation.  Alternating waveforms can
be of many shapes.  The one that is used with electric power is the sinusoidal waveform.  This
has an equation of the form
v(t) = V
m
sin(? t + f )
If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p.
(a) Instantaneous value:   The instantaneous value of a
waveform is the value of the waveform at any given instant of
time. It is a time variable a(t).
For a sinusoid,  Instantaneous value  a(t)  = A
m
sin(? t+ f )
(b) Peak value:   The peak value,  or maximum value, of a
waveform is the maximum instantaneous value of the
waveform.
For a sinusoid, Peak value = A
m

(c) Mean value:   The mean value of a waveform is equal
to the mean value over a complete cycle of the waveform.
It also corresponds to the direct component of the
waveform.
Mean value  A
mean
=
?
+T t
t
o
o
dt t a
T
). (
1

The mean value of a waveform which has equal positive and negative half cycles must thus be
always zero.
For a sinusoid, Mean value =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
f ?  =  0
(d) Average value (rectified):   The full-wave rectified
average value or average value of a waveform is defined
as the mean value of the rectified waveform over a
complete cycle.
Average value  A
avg
=
?
?
?
?
?
?
?
?
?
?
-
? ?
fcycle hal
negative
fcycle hal
positive
dt t a dt t a
T
). ( ). (
1

For a sinusoid,
Average value   =
?
?
?
?
?
?
?
?
-
? ?
T
T
m
T
m
dt t A dt t A
T
2
2
0
. sin . sin
1
? ?
=
m m
A A
T p ?
2
2
1
= ·
The average value is defined in the above manner in electrical engineering as otherwise all
alternating waveforms would have zero value and would be indistinguishable.
a(t)
-A
m
A
mean
v(t)
t
T
T
V
m

f/?
A
avg
v
rect
t
T
T
AC Theory  â€“ Professor J R Lucas 2  November 2001
(e) Effective value or r.m.s. value:
Neither the peak value, nor the mean value, nor the average value defines the useful value of
the waveform with regard to the power or energy.  Thus the effective value is defined based
on the power equivalent of the quantity.
The effective value is thus defined as the constant value which would cause the same power
dissipation as the original quantity over one complete period.
Thus considering current   Power dissipation  =  I
eff
2
. R  =
?
+T t
t
o
o
dt R t i
T
. ). (
1
2

giving I
eff
=
?
+T t
t
o
o
dt t i
T
). (
1
2

or in general Effective value  A
eff
=
?
+T t
t
o
o
dt t a
T
). (
1
2

It is to be noted that the method of calculating the effective value involves the following
processes. Taking the root of the mean of the squared waveform over one complete cycle.
Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform.
i.e.  r.m.s. value  A
rms
=
?
+T t
t
o
o
dt t a
T
). (
1
2

This is defined for both current as well as voltage.
For a sinusoid,
r.m.s. value   =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
2 2
f ?  =
2
m
A

Unless otherwise specified, the rms value is the value that is always specified for ac
waveforms, whether it be a voltage or a current.  For example, 230 V in the mains supply is
an rms value of the voltage.  Similarly when we talk about a 5 A, 13 A or 15A socket outlet
(plug point), we are again talking about the rms value of the rated current of the socket outlet.
For a given waveform, such as the sinusoid, the peak value, average value and the rms value
are dependant on each other.  The peak factor and the form factor are the two factors that are
most commonly defined.

value average
value rms
= Factor Form
and for a sinusoidal waveform,  Form Factor = 111 . 1 1107 . 1
2
2
? =
p
m m
V V

The form factor is useful such as when the average value has been measured using a rectifier
type moving coil meter and the rms value is required to be found. [Note: You will be studying
about these meters later]
Peak Factor =
peak value
rms value
and for a sinusoidal waveform,
Peak Factor =

The peak factor is useful when defining highly distorted
waveforms such as the current waveform of compact
fluorescent lamps.
V
V
m
m
2
2 1 4142 == .

v
2
(t)
t
T
T
t
i
AC Theory  â€“ Professor J R Lucas 3  November 2001
Some advantages of the sinusoidal waveform for electrical power applications
a. Sinusoidally varying voltages are easily generated by rotating machines
b. Differentiation or integration of a sinusoidal waveform produces a sinusoidal waveform of
the same frequency, differing only in magnitude and phase angle.  Thus when a sinusoidal
current is passed through (or a sinusoidal voltage applied across) a resistor, inductor or a
capacitor a sinusoidal voltage waveform (or current waveform) of the same frequency,
differing only in magnitude and phase angle, is obtained.
If  i(t) = I
m
sin (?t+f),
for a resistor,   v(t) = R.i(t) = R.I
m
sin (?t+f) = V
m
sin (?t+f)
-  magnitude changed by R but no phase shift
for an inductor,
vt L
di
dt
L
d
dt
It LI t LIt
mm m
() ..(sin()) ..cos() ..sin( /) == + = + = ++ ?f ? ? f ? ?f p 2

- magnitude changed by L? and phase angle changed by p/2
for a capacitor,
vt
C
idt
C
It dt
C
It
C
It
mm m
() . sin() .cos() .sin( /) =·= +·=
-
·
+=
·
+-
??
11 1 1
2 ?f
?
?f
?
?f p
- magnitude changed by 1/C? and phase angle changed by -p/2
c. Sinusoidal waveforms have the property of remaining unaltered in shape and frequency
when other sinusoids having the same frequency but different in magnitude and phase are
A sin (?t+a) + B sin (?t+ß)
= A sin ?t. cos a + ? cos ?t. sin a  + B sin ?t. cos ß + B cos ?t. sin ß
= (A. cos a + B. cos ß) sin ?t + (A. sin a + B. sin ß) cos ?t
= C sin (?t + ?),
where C = ) cos( 2
2 2
ß a - + + AB B A , and ? =
? ?
? ?
cos sin
sin cos
tan
1
B A
B A
+
+
-
are constants.
d. Periodic, but non-sinusoidal waveforms can be broken up to direct terms, its fundamental
and harmonics.
f(t) = F
o
+ F
1
sin(?t + ?
1
) + F
2
sin(2?t + ?
2
) + F
3
sin(3?t + ?
3
) + F
4
sin(4?t + ?
4
) + ........
where F
n
and ?
n
are constants dependant on the function  f(t).
e. Sinusoidal waveforms can be represented by the projections of a rotating phasor.
Phasor Representation of Sinusoids
You may be aware that sin ? can be
written in terms of exponentials and
complex numbers.

i.e.  ej
j?
?? =+ cos sin

or   etj t
jt ?
?? =+ cos sin

a(t)
t
T
A
m
sin? t
?
?
O
O
P
X
Page 4

AC Theory  â€“ Professor J R Lucas 1  November 2001
Alternating Current Theory - J R Lucas
An alternating waveform is a periodic waveform which alternate between positive and
negative  values.  Unlike direct waveforms, they cannot be characterised by one magnitude as
their amplitude is continuously varying from instant  to instant.  Thus various forms of
magnitudes are defined for such waveforms.
The advantage of the alternating waveform for electric power is that it can be stepped up or
stepped down in potential easily for transmission and utilisation.  Alternating waveforms can
be of many shapes.  The one that is used with electric power is the sinusoidal waveform.  This
has an equation of the form
v(t) = V
m
sin(? t + f )
If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p.
(a) Instantaneous value:   The instantaneous value of a
waveform is the value of the waveform at any given instant of
time. It is a time variable a(t).
For a sinusoid,  Instantaneous value  a(t)  = A
m
sin(? t+ f )
(b) Peak value:   The peak value,  or maximum value, of a
waveform is the maximum instantaneous value of the
waveform.
For a sinusoid, Peak value = A
m

(c) Mean value:   The mean value of a waveform is equal
to the mean value over a complete cycle of the waveform.
It also corresponds to the direct component of the
waveform.
Mean value  A
mean
=
?
+T t
t
o
o
dt t a
T
). (
1

The mean value of a waveform which has equal positive and negative half cycles must thus be
always zero.
For a sinusoid, Mean value =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
f ?  =  0
(d) Average value (rectified):   The full-wave rectified
average value or average value of a waveform is defined
as the mean value of the rectified waveform over a
complete cycle.
Average value  A
avg
=
?
?
?
?
?
?
?
?
?
?
-
? ?
fcycle hal
negative
fcycle hal
positive
dt t a dt t a
T
). ( ). (
1

For a sinusoid,
Average value   =
?
?
?
?
?
?
?
?
-
? ?
T
T
m
T
m
dt t A dt t A
T
2
2
0
. sin . sin
1
? ?
=
m m
A A
T p ?
2
2
1
= ·
The average value is defined in the above manner in electrical engineering as otherwise all
alternating waveforms would have zero value and would be indistinguishable.
a(t)
-A
m
A
mean
v(t)
t
T
T
V
m

f/?
A
avg
v
rect
t
T
T
AC Theory  â€“ Professor J R Lucas 2  November 2001
(e) Effective value or r.m.s. value:
Neither the peak value, nor the mean value, nor the average value defines the useful value of
the waveform with regard to the power or energy.  Thus the effective value is defined based
on the power equivalent of the quantity.
The effective value is thus defined as the constant value which would cause the same power
dissipation as the original quantity over one complete period.
Thus considering current   Power dissipation  =  I
eff
2
. R  =
?
+T t
t
o
o
dt R t i
T
. ). (
1
2

giving I
eff
=
?
+T t
t
o
o
dt t i
T
). (
1
2

or in general Effective value  A
eff
=
?
+T t
t
o
o
dt t a
T
). (
1
2

It is to be noted that the method of calculating the effective value involves the following
processes. Taking the root of the mean of the squared waveform over one complete cycle.
Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform.
i.e.  r.m.s. value  A
rms
=
?
+T t
t
o
o
dt t a
T
). (
1
2

This is defined for both current as well as voltage.
For a sinusoid,
r.m.s. value   =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
2 2
f ?  =
2
m
A

Unless otherwise specified, the rms value is the value that is always specified for ac
waveforms, whether it be a voltage or a current.  For example, 230 V in the mains supply is
an rms value of the voltage.  Similarly when we talk about a 5 A, 13 A or 15A socket outlet
(plug point), we are again talking about the rms value of the rated current of the socket outlet.
For a given waveform, such as the sinusoid, the peak value, average value and the rms value
are dependant on each other.  The peak factor and the form factor are the two factors that are
most commonly defined.

value average
value rms
= Factor Form
and for a sinusoidal waveform,  Form Factor = 111 . 1 1107 . 1
2
2
? =
p
m m
V V

The form factor is useful such as when the average value has been measured using a rectifier
type moving coil meter and the rms value is required to be found. [Note: You will be studying
about these meters later]
Peak Factor =
peak value
rms value
and for a sinusoidal waveform,
Peak Factor =

The peak factor is useful when defining highly distorted
waveforms such as the current waveform of compact
fluorescent lamps.
V
V
m
m
2
2 1 4142 == .

v
2
(t)
t
T
T
t
i
AC Theory  â€“ Professor J R Lucas 3  November 2001
Some advantages of the sinusoidal waveform for electrical power applications
a. Sinusoidally varying voltages are easily generated by rotating machines
b. Differentiation or integration of a sinusoidal waveform produces a sinusoidal waveform of
the same frequency, differing only in magnitude and phase angle.  Thus when a sinusoidal
current is passed through (or a sinusoidal voltage applied across) a resistor, inductor or a
capacitor a sinusoidal voltage waveform (or current waveform) of the same frequency,
differing only in magnitude and phase angle, is obtained.
If  i(t) = I
m
sin (?t+f),
for a resistor,   v(t) = R.i(t) = R.I
m
sin (?t+f) = V
m
sin (?t+f)
-  magnitude changed by R but no phase shift
for an inductor,
vt L
di
dt
L
d
dt
It LI t LIt
mm m
() ..(sin()) ..cos() ..sin( /) == + = + = ++ ?f ? ? f ? ?f p 2

- magnitude changed by L? and phase angle changed by p/2
for a capacitor,
vt
C
idt
C
It dt
C
It
C
It
mm m
() . sin() .cos() .sin( /) =·= +·=
-
·
+=
·
+-
??
11 1 1
2 ?f
?
?f
?
?f p
- magnitude changed by 1/C? and phase angle changed by -p/2
c. Sinusoidal waveforms have the property of remaining unaltered in shape and frequency
when other sinusoids having the same frequency but different in magnitude and phase are
A sin (?t+a) + B sin (?t+ß)
= A sin ?t. cos a + ? cos ?t. sin a  + B sin ?t. cos ß + B cos ?t. sin ß
= (A. cos a + B. cos ß) sin ?t + (A. sin a + B. sin ß) cos ?t
= C sin (?t + ?),
where C = ) cos( 2
2 2
ß a - + + AB B A , and ? =
? ?
? ?
cos sin
sin cos
tan
1
B A
B A
+
+
-
are constants.
d. Periodic, but non-sinusoidal waveforms can be broken up to direct terms, its fundamental
and harmonics.
f(t) = F
o
+ F
1
sin(?t + ?
1
) + F
2
sin(2?t + ?
2
) + F
3
sin(3?t + ?
3
) + F
4
sin(4?t + ?
4
) + ........
where F
n
and ?
n
are constants dependant on the function  f(t).
e. Sinusoidal waveforms can be represented by the projections of a rotating phasor.
Phasor Representation of Sinusoids
You may be aware that sin ? can be
written in terms of exponentials and
complex numbers.

i.e.  ej
j?
?? =+ cos sin

or   etj t
jt ?
?? =+ cos sin

a(t)
t
T
A
m
sin? t
?
?
O
O
P
X
AC Theory  â€“ Professor J R Lucas 4  November 2001
Consider a line OP  of length A
m
which is in the horizontal direction OX at time t=0.
If OP rotates at an angular velocity ? , then in time t its position would correspond to an
angle of ? t.
The projection of this rotating phasor OP (a phasor is somewhat similar to a vector, except
that it does not have a physical direction in space but a phase angle) on the y-axis would
correspond to OP sin ?t or  A
m
sin ? t and on the x-axis would correspond to A
m
cos ? t.  Thus
the sinusoidal waveform can be thought of being the projection on a particular direction of the
complex exponential  e
j?t
.

If we consider more than one phasor, and each phasor rotates at the same angular frequency,
then there is no relative motion between the phasors.  Thus if we fix the reference phasor OR
in a particular reference direction (without showing its rotation), then all others phasors
moving at the same angular frequency would  also be fixed at a relative position.  Usually this
reference direction is chosen as horizontal on the diagram for convenience.

It is also usual to draw the Phasor diagram using the rms value A of the sinusoidal
waveform, rather than with the peak value A
m
. This is shown on an enlarged diagram.  Thus
unless otherwise specified it is the rms value that is drawn on a phasor diagram.
It should be noted that the values on the phasor diagram are no longer time variables.  The
phasor A is characterised by its magnitude ?A? and its phase angle f.  These are also the polar
co-ordinates of the phasor and is commonly written as  ?A?  f .  The phasor A  can also be
characterised by its cartesian co-ordinates A
x
and A
y
and usually written using complex
numbers as A = A
x
+ j A
y
.
Note:  In electrical engineering, the letter  j  is always used for the complex operator - 1
because the letter  i is regularly used for electric current.
It is worth noting that
AA A
xy
=+
22
and that
tanf =
A
A
y
x
or
f =
?
?
?
?
?
?
-
tan
1
A
A
y
x

Also,   A
x
= ?A? cos f,    A
y
= ?A? sin f    and  ?A? e
jf
= ?A? cos f + j?A? sin f  = A
x
+ j A
y

Note: If the period of a sinusoidal waveform is T, then the corresponding angle would be ? T.
Also, the period of a waveform corresponds to 1 complete cycle or  2 p radians or 360
0
.
? ? T  = 2p

R
P
f
0
A
m

R
P
f
0
A
a(t)
t
T
A
m
sin (?  t+f)
?
?
0
0
R
X
P
f
Rotating Phasor diagram
reference direction
f
0
A
A
m
=
2
Phasor diagram
A
A
Page 5

AC Theory  â€“ Professor J R Lucas 1  November 2001
Alternating Current Theory - J R Lucas
An alternating waveform is a periodic waveform which alternate between positive and
negative  values.  Unlike direct waveforms, they cannot be characterised by one magnitude as
their amplitude is continuously varying from instant  to instant.  Thus various forms of
magnitudes are defined for such waveforms.
The advantage of the alternating waveform for electric power is that it can be stepped up or
stepped down in potential easily for transmission and utilisation.  Alternating waveforms can
be of many shapes.  The one that is used with electric power is the sinusoidal waveform.  This
has an equation of the form
v(t) = V
m
sin(? t + f )
If the period of the waveform is T, then its angular frequency ? corresponds to ?T = 2p.
(a) Instantaneous value:   The instantaneous value of a
waveform is the value of the waveform at any given instant of
time. It is a time variable a(t).
For a sinusoid,  Instantaneous value  a(t)  = A
m
sin(? t+ f )
(b) Peak value:   The peak value,  or maximum value, of a
waveform is the maximum instantaneous value of the
waveform.
For a sinusoid, Peak value = A
m

(c) Mean value:   The mean value of a waveform is equal
to the mean value over a complete cycle of the waveform.
It also corresponds to the direct component of the
waveform.
Mean value  A
mean
=
?
+T t
t
o
o
dt t a
T
). (
1

The mean value of a waveform which has equal positive and negative half cycles must thus be
always zero.
For a sinusoid, Mean value =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
f ?  =  0
(d) Average value (rectified):   The full-wave rectified
average value or average value of a waveform is defined
as the mean value of the rectified waveform over a
complete cycle.
Average value  A
avg
=
?
?
?
?
?
?
?
?
?
?
-
? ?
fcycle hal
negative
fcycle hal
positive
dt t a dt t a
T
). ( ). (
1

For a sinusoid,
Average value   =
?
?
?
?
?
?
?
?
-
? ?
T
T
m
T
m
dt t A dt t A
T
2
2
0
. sin . sin
1
? ?
=
m m
A A
T p ?
2
2
1
= ·
The average value is defined in the above manner in electrical engineering as otherwise all
alternating waveforms would have zero value and would be indistinguishable.
a(t)
-A
m
A
mean
v(t)
t
T
T
V
m

f/?
A
avg
v
rect
t
T
T
AC Theory  â€“ Professor J R Lucas 2  November 2001
(e) Effective value or r.m.s. value:
Neither the peak value, nor the mean value, nor the average value defines the useful value of
the waveform with regard to the power or energy.  Thus the effective value is defined based
on the power equivalent of the quantity.
The effective value is thus defined as the constant value which would cause the same power
dissipation as the original quantity over one complete period.
Thus considering current   Power dissipation  =  I
eff
2
. R  =
?
+T t
t
o
o
dt R t i
T
. ). (
1
2

giving I
eff
=
?
+T t
t
o
o
dt t i
T
). (
1
2

or in general Effective value  A
eff
=
?
+T t
t
o
o
dt t a
T
). (
1
2

It is to be noted that the method of calculating the effective value involves the following
processes. Taking the root of the mean of the squared waveform over one complete cycle.
Thus it is also designated as the root mean square value, or the r.m.s. value of the waveform.
i.e.  r.m.s. value  A
rms
=
?
+T t
t
o
o
dt t a
T
). (
1
2

This is defined for both current as well as voltage.
For a sinusoid,
r.m.s. value   =
?
+
+
T t
t
m
o
o
dt t A
T
). ( sin
1
2 2
f ?  =
2
m
A

Unless otherwise specified, the rms value is the value that is always specified for ac
waveforms, whether it be a voltage or a current.  For example, 230 V in the mains supply is
an rms value of the voltage.  Similarly when we talk about a 5 A, 13 A or 15A socket outlet
(plug point), we are again talking about the rms value of the rated current of the socket outlet.
For a given waveform, such as the sinusoid, the peak value, average value and the rms value
are dependant on each other.  The peak factor and the form factor are the two factors that are
most commonly defined.

value average
value rms
= Factor Form
and for a sinusoidal waveform,  Form Factor = 111 . 1 1107 . 1
2
2
? =
p
m m
V V

The form factor is useful such as when the average value has been measured using a rectifier
type moving coil meter and the rms value is required to be found. [Note: You will be studying
about these meters later]
Peak Factor =
peak value
rms value
and for a sinusoidal waveform,
Peak Factor =

The peak factor is useful when defining highly distorted
waveforms such as the current waveform of compact
fluorescent lamps.
V
V
m
m
2
2 1 4142 == .

v
2
(t)
t
T
T
t
i
AC Theory  â€“ Professor J R Lucas 3  November 2001
Some advantages of the sinusoidal waveform for electrical power applications
a. Sinusoidally varying voltages are easily generated by rotating machines
b. Differentiation or integration of a sinusoidal waveform produces a sinusoidal waveform of
the same frequency, differing only in magnitude and phase angle.  Thus when a sinusoidal
current is passed through (or a sinusoidal voltage applied across) a resistor, inductor or a
capacitor a sinusoidal voltage waveform (or current waveform) of the same frequency,
differing only in magnitude and phase angle, is obtained.
If  i(t) = I
m
sin (?t+f),
for a resistor,   v(t) = R.i(t) = R.I
m
sin (?t+f) = V
m
sin (?t+f)
-  magnitude changed by R but no phase shift
for an inductor,
vt L
di
dt
L
d
dt
It LI t LIt
mm m
() ..(sin()) ..cos() ..sin( /) == + = + = ++ ?f ? ? f ? ?f p 2

- magnitude changed by L? and phase angle changed by p/2
for a capacitor,
vt
C
idt
C
It dt
C
It
C
It
mm m
() . sin() .cos() .sin( /) =·= +·=
-
·
+=
·
+-
??
11 1 1
2 ?f
?
?f
?
?f p
- magnitude changed by 1/C? and phase angle changed by -p/2
c. Sinusoidal waveforms have the property of remaining unaltered in shape and frequency
when other sinusoids having the same frequency but different in magnitude and phase are
A sin (?t+a) + B sin (?t+ß)
= A sin ?t. cos a + ? cos ?t. sin a  + B sin ?t. cos ß + B cos ?t. sin ß
= (A. cos a + B. cos ß) sin ?t + (A. sin a + B. sin ß) cos ?t
= C sin (?t + ?),
where C = ) cos( 2
2 2
ß a - + + AB B A , and ? =
? ?
? ?
cos sin
sin cos
tan
1
B A
B A
+
+
-
are constants.
d. Periodic, but non-sinusoidal waveforms can be broken up to direct terms, its fundamental
and harmonics.
f(t) = F
o
+ F
1
sin(?t + ?
1
) + F
2
sin(2?t + ?
2
) + F
3
sin(3?t + ?
3
) + F
4
sin(4?t + ?
4
) + ........
where F
n
and ?
n
are constants dependant on the function  f(t).
e. Sinusoidal waveforms can be represented by the projections of a rotating phasor.
Phasor Representation of Sinusoids
You may be aware that sin ? can be
written in terms of exponentials and
complex numbers.

i.e.  ej
j?
?? =+ cos sin

or   etj t
jt ?
?? =+ cos sin

a(t)
t
T
A
m
sin? t
?
?
O
O
P
X
AC Theory  â€“ Professor J R Lucas 4  November 2001
Consider a line OP  of length A
m
which is in the horizontal direction OX at time t=0.
If OP rotates at an angular velocity ? , then in time t its position would correspond to an
angle of ? t.
The projection of this rotating phasor OP (a phasor is somewhat similar to a vector, except
that it does not have a physical direction in space but a phase angle) on the y-axis would
correspond to OP sin ?t or  A
m
sin ? t and on the x-axis would correspond to A
m
cos ? t.  Thus
the sinusoidal waveform can be thought of being the projection on a particular direction of the
complex exponential  e
j?t
.

If we consider more than one phasor, and each phasor rotates at the same angular frequency,
then there is no relative motion between the phasors.  Thus if we fix the reference phasor OR
in a particular reference direction (without showing its rotation), then all others phasors
moving at the same angular frequency would  also be fixed at a relative position.  Usually this
reference direction is chosen as horizontal on the diagram for convenience.

It is also usual to draw the Phasor diagram using the rms value A of the sinusoidal
waveform, rather than with the peak value A
m
. This is shown on an enlarged diagram.  Thus
unless otherwise specified it is the rms value that is drawn on a phasor diagram.
It should be noted that the values on the phasor diagram are no longer time variables.  The
phasor A is characterised by its magnitude ?A? and its phase angle f.  These are also the polar
co-ordinates of the phasor and is commonly written as  ?A?  f .  The phasor A  can also be
characterised by its cartesian co-ordinates A
x
and A
y
and usually written using complex
numbers as A = A
x
+ j A
y
.
Note:  In electrical engineering, the letter  j  is always used for the complex operator - 1
because the letter  i is regularly used for electric current.
It is worth noting that
AA A
xy
=+
22
and that
tanf =
A
A
y
x
or
f =
?
?
?
?
?
?
-
tan
1
A
A
y
x

Also,   A
x
= ?A? cos f,    A
y
= ?A? sin f    and  ?A? e
jf
= ?A? cos f + j?A? sin f  = A
x
+ j A
y

Note: If the period of a sinusoidal waveform is T, then the corresponding angle would be ? T.
Also, the period of a waveform corresponds to 1 complete cycle or  2 p radians or 360
0
.
? ? T  = 2p

R
P
f
0
A
m

R
P
f
0
A
a(t)
t
T
A
m
sin (?  t+f)
?
?
0
0
R
X
P
f
Rotating Phasor diagram
reference direction
f
0
A
A
m
=
2
Phasor diagram
A
A
AC Theory  â€“ Professor J R Lucas 5  November 2001
Phase difference
Consider the two waveforms  A
m
sin (? t+f
1
) and B
m
sin (? t+f
2
) as shown in the figure.  It can
be seen that they have different amplitudes and different phase angles with respect to a
common reference.

These two waveforms can also be represented by either rotating
phasors A
m
e
j(?t+f
1
)
and B
m
e
j (?t+f
2
)
with peak amplitudes A
m
and  B
m
,
or  by  a normal phasor diagram with complex values A and B  with
polar co-ordinates  ?A?  f
1
and   ?B?  f
2
as shown .
Any particular value (such as positive peak, or zero) of a(t) is seen to occur at a time T after
the corresponding value of b(t).  i.e. the positive peak A
m
occurs after an angle (f
2
-f
1
) after
the positive peak B
m
.  Similarly the zero of a(t) occurs after an angle (f
2
-f
1
) after the
corresponding zero of b(t).  In such a case we say that the waveform  b(t)  leads the waveform
a(t) by a phase angle of (f
2
-f
1
).  Similarly we could say that the waveform a(t)  lags the
waveform b(t) by a phase angle of (f
2
-f
1
). [Note: Only the angle less than 180
o
is used to
specify whether a waveform leads or lags another waveform].
We could also define, lead and lag by simply referring to the phasor diagram.  Since angles
are always measured anticlockwise (convention), we can see from the phasor diagram, that B
leads  A by an angle of (f
2
-f
1
) anticlockwise or that A  lags  B  by an angle (f
2
-f
1
).
Addition and subtraction of phasors can be done
using the same parallelogram and triangle laws as for
vectors, generally using complex numbers.  Thus the
addition of phasor A and phasor B would be
A + B = (A cos f
1
+  j A sin f
1
)  + (B cos f
2
+  j B sin f
2
)
= (A cos f
1
+ B cos f
2
) + j (A sin f
1
+ B sin f
2
)
= C
x
+ jC
y
=  ?C?  f
c
=  C
where
CC C A B A B
xy
=+ = + + +
22
12
2
12
2
(cos cos ) ( sin sin ) f f f f

and
?
?
?
?
?
?
+
+
= ?
?
?
?
?
?
=
- -
) cos cos (
) sin sin (
tan tan
2 1
2 1
1 1
f f
f f
f
B A
B A
C
C
x
y
c

Example 1
Find the addition and subtraction of the 2 complex numbers given by 10?30
o
and 25? 48
o
.
Addition =   10 ?30
o
+ 25 ?48
o
=  10(0.8660 + j 0.5000) + 25(0.6691 + j 0.7431)
= (8.660 + 16.728) + j (5.000 + 18.577)  =  25.388 + j 23.577  =  34.647 ? 42.9
o

Subtraction  =   10?30
o
- 25?48
o
= (8.660 - 16.728) + j (5.000 - 18.577)
= - 8.068 - j 13.577 = 15.793?239.3
o

f
1

0
A
A
m
=
2
f
2
-f
1

B
B
m
=
2
f
1

0
f
2

B
C
A
f
1
O
f
2

y(t)
? t
A
m
sin (? t+f
1
)
B
m
sin (? t+f
2
)
?T
f
2
-f
1

f
2
- f
1

A
m

B
m

?
```
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