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1.7.2 
linear and non-linear Elements
Two-terminal elements are classified as linear or nonlinear based on whether the voltage–current 
relationship of the element satisfies the linearity property.
Two variables of time – x(t) and y(t) – satisfy the property of linearity if the relation between them 
is homogeneous and additive at all t.
Let the relation between the variables be represented by y(t) 
=
f [x(t)].
y(t) 
=
f [x(t)] is homogeneous if f [ax(t)] 
=
af[x(t)] for any t where a is any real number. That is, 
scaling the variable x(t) by a real number a results in the scaling of the variable y(t) by the same real 
number a.
y(t) 
=
f [x(t)] is additive if f [x
1
(t) 
+
x
2
(t)] 
=
f [x
1
(t)] 
+
f [x
2
(t)] for any t. That is, y(t) corresponding 
to sum of two variables x
1
(t) and x
2
(t) is equal to the sum of y(t) corresponding to x
1
(t) and y(t) 
corresponding to x
2
(t) at any time instant.
We may combine the requirements of homogeneity and additivity into a composite requirement 
called superposition property.
y(t) 
=
f [x(t)] satisfies superposition property if
f [a
1
x
1
(t) 
+
a
2
x
2
(t)] 
=
a
1
f [x
1
(t)] 
+
a
2
f [x
2
(t)] for any combination of real numbers a
1
and a
2
and for 
any t.
Thus, a two-terminal element is linear if its v – i relationship satisfies the principle of superposition.
The simplest case of a linear relationship between two variables occurs when y(t) is proportional 
to x(t).
Let y(t) 
=
k x(t) where k is a real number.
Then, f [a
1
x
1
(t) 
+
a
2
x
2
(t)] 
=
k 
×
[a
1
x
1
(t) 
+
a
2
x
2
(t)] 
=
k a
1
x
1
(t) 
+
ka
2
x
2
(t) 
=
a
1
×
k x
1
(t) 
+
a
2
×
k x
2
(t) 
=
a
1
f [x
1
(t)] 
+
a
2
f [x
2
(t)].
Therefore, y(t) 
=
k x(t) is a linear relation for any real k.
But, a relation does not have to be algebraic for it to be a linear relation. Consider the relation 
y t
k
dx t
dt
( )
( )
.
=
Then,
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ClassificationofTwo-TerminalElements
 
 
1.39
f a x t
a x t
k
d a x t
a x t
dt
a k
dx t
d
1 1
2 2
1 1
2 2
1
1
( )
( )
(
( )
( ))
( )
+
[
]
= ×
+




=
tt
a k
dx t
dt
a f x t
a f x t



 +




=
[
]
+
[
]
2
2
1
1
2
2
( )
( )
( )
Therefore, 
y t
k
dx t
dt
( )
( )
=
is a linear relation.
Similarly, it can be shown that 
y t
k x t dt
t
( )
( )
=
∞
∫
−
too is a linear relation.
Beginners in Circuit Analysis often tend to equate the property of linearity to straight-line nature of 
functional relationship between the concerned variables. Consider the following relationship.
y(t) 
=
mx(t) 
+
c where m and c are two real numbers. Obviously, the graph of this function will be a 
straight-line with c as its vertical-axis intercept. But this is not a linear relation in the sense of linearity 
as defined in Circuit Theory.
f a x t
a x t
m a x t
a x t
c
a mx t
mx t
1 1
2 2
1 1
2 2
1
1
2
2
( )
( )
( )
( )]
( )
( )]
+
[
]
=
+
+
=
+
[
a
++
+
=
+
+
+
≠
c
a f x t
a f x t
a mx t
a mx t
a
a c
f a x
1
1
2
2
1
1
2
2
1
2
1
[ ( )]
[ ( )]
( )
( )] (
)
[
11
2 2
( )
( )]
t
a x t
+
Therefore, y(t) 
=
mx(t) 
+
c is not a linear relation in Circuit Theory. It does not satisfy the property 
of homogeneity. It does not satisfy the property of additivity too.
Let us examine the linearity property of various two-terminal elements we have discussed so far.
Consider a two-terminal resistance element. Its v – i relation is v(t) 
=
Ri(t). It is a linear element if 
the R parameter is a real constant or a function of time alone. The resistance of a piece of conductor is 
temperature dependent. It may depend on current level in certain cases. Thus a two-terminal resistance 
is linear if the temperature is constant and the R parameter is either a constant or is an independent 
function of time alone.
A two-terminal inductance is described by 
v t
d Li t
dt
( )
[ ( )]
=
 
in general. If the inductance parameter 
L is a constant, then two-terminal inductance is a linear element. L can vary with time if the physical 
geometry of the device changes with time (but independent of electrical variables). The element is 
linear in that case too. But if L varies as a function of the current in it, then, the element is a nonlinear 
one.
A two-terminal capacitance is described by 
i t
d Cv t
dt
( )
[ ( )]
=
in general. If the inductance parameter 
C is a constant, then two-terminal inductance is a linear element. C can vary with time if the physical 
geometry of the device changes with time (but independent of electrical variables). The element is 
linear in that case too. A tuning capacitor in a radio receiver is an example. But if C varies as a function 
of the charge in it, then, the element is a nonlinear one.
A two-terminal ideal independent voltage source is described by the relations v(t) 
=
E(t) (an 
independently specified function of time) and i(t) 
=
arbitrary. Obviously, this is a non-linear 
relationship. Thus an ideal independent voltage source is a nonlinear element. Similarly, an ideal 
independent current source is a nonlinear element.
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1.40


CircuitVariablesandCircuitElements