Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

6.3 
Instantaneous Power In PerIodIc waveforms
A waveform v(t) is said to be periodic with a periodicity of T s if v(t
+
nT) 
=
v(t) for all integer values 
of n and for all t. This implies that it must be possible to identify a basic section of the waveform 
that lasts for T s and that repeats to infinite extent into the past and into the future. Thus, a waveform 
is strictly periodic only if it is ever-existent. But, in practice, waveforms are switched on at some 
definite time instant. Such switched waveforms cannot be called periodic waveforms in the strict 
sense of definition of periodicity. However, we can 
view them as periodic waveforms for circuit analysis 
purposes provided we focus our attention to time 
instants located far away from the instant at which 
the waveform was switched on.
Consider a two-terminal electrical element with 
the current and voltage variables marked as per 
passive sign convention in Fig. 6.3-1.
v
(
t
)
i
(
t
)
+
–
Fig. 6.3-1 
A 
two-terminal 
element 
with voltage and current 
marked as per passive sign 
convention

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6.14
Power and Energy in Periodic Waveforms 
The voltage difference v
AB
between two points A and B is the work to be done in moving 
+
1 C of 
charge from B to A. Energy has to be spent in carrying charge from a lower potential point to higher 
potential point. Similarly, energy is released when a charge is allowed to fall through a higher potential 
point to lower potential point. The amount of charge that went through the element from a higher 
potential point to lower potential point in one second is given by i(t). Therefore, the product of v(t) and 
i(t) must be the energy released into element in one second. The rate of change of energy is defined as 
instantaneous power and is denoted by p(t). 
Therefore, instantaneous power is 
delivered to
a two-terminal element, 
p 
(
t
) 
=
v 
(
t
) 
i 
(
t
), 
where 
v 
(
t
) and 
i 
(
t
) are the element variables defined as per passive sign convention.
Then, energy delivered to a two-terminal element is obviously given by 
E t
p t dt
v t i t dt
E
v t i t dt
t
t
t
( )
( )
( ) ( )
( )
( ) ( )
=
=
=
+
−∞
−∞
∫
∫
∫
0
0
where E(0) is the total energy dissipated in the element from infinite past to t 
=
0.
And the relation between the energy function E(t) and the instantaneous power p(t) is given by 
p t
dE t
dt
( )
( )
=
. Let E
i
(t) be the energy dissipation function (i.e., the net energy delivered from 
-∞
to t) 
of the i 
th
element in a b element electrical circuit. Then the total energy dissipation function of the 
circuit is E
T
(t) 
=
E t
i
i
b
( )
=
∑
1
. The circuit considered as a whole is an isolated system and the total energy 
in an isolated system is a constant by Law of Conservation of Energy. Therefore,
E t
i
i
b
( )
=
=
∑
1
A Constant.
This implies that if some elements of the circuit are receiving energy, then some other elements 
must be losing energy.
Since the statement of conservation of energy is true on an instant-to-instant basis, we can 
differentiate both sides of the above equation with respect to time:
dE t
dt
dE t
dt
dE t
dt
dE t
dt
T
b
( )
( )
( )
( )
= ⇒
+
+ +
=
0
0
1
2
But each term of the type 
dE t
dt
i
( )
can be interpreted as the instantaneous power delivered to the i 
th
circuit element.
p t
p t
p t
v t i t
v t i t
v t i t
b
b
b
1
2
1
1
2
2
0
0
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
+
+ +
= ⇒
+
+
=
Therefore, sum of instantaneous power delivered to all elements in a circuit will be zero. Equivalently, 
sum of instantaneous power delivered by all elements in a circuit will be zero. This implies that in any 
circuit some elements will be delivering positive power and the remaining elements will be receiving 
positive power at all instants of time. The sum of positive powers delivered will be equal to the sum of 
positive powers received (or consumed) at all t. This is the principle of conservation of instantaneous 

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Instantaneous Power in Periodic Waveforms 
6.15
power which is valid for all isolated circuits containing arbitrary kind of elements. Isolated circuit is 
one that has no interaction with environment. 
Principle of Conservation of Instantaneous Power – The sum of instantaneous power 
delivered to all elements in an isolated circuit will be zero, 
i.e.,
p t
p t
p t
b
1
2
0
( )
( )
( )
+
+
+
=
where 
b
is the total number of elements in the circuit.