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5.5 
thevenIn’S theorem and norton’S theorem
The problem of solving a circuit with different load networks connected to same delivery network 
occurs in electrical and electronics engineering quite often. We do not want to write the same node 
equations or mesh equations of the delivery network whenever the load network undergoes some 
change and solve the circuit in its entirety repeatedly. Thevenin’s Theorem and Norton’s Theorem help 
us to avoid this kind of wasted effort and become efficient in solving circuits. They help us to conduct 
node analysis or mesh analysis of the delivery network and replace it with a simple equivalent circuit 
for further analysis when different load networks are connected to it. They are two tools indispensable 
to a circuit analyst.
Consider a memoryless network shown in Fig. 5.5-1 containing linear resistors, linear dependent 
sources and independent sources with a pair of terminals identified as the output terminals of the 

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5.22
Circuit Theorems
network. The network interacts with the external 
world only through this pair of terminals. No 
parameter inside the circuit changes, but different 
load networks may get connected to the circuit at 
its output terminals. Assume that an independent 
voltage source of source function v(t) is connected 
across the output terminals of the network. With 
no loss of generality, we assume further that the 
voltage source negative terminal is taken, i.e., 
a´
, as the reference node for writing the node equations 
of the circuit. We are interested in the behaviour of the current i(t) delivered by the circuit to the 
terminating voltage source versus the source function v(t).
We remember that any circuit variable in a linear circuit can be expressed as a linear combination 
of all the independent source functions in the circuit. Hence, i(t) in this circuit can be expressed as
i t
a v t
a v t
a v
t
b i t
b i t
s
s
n
sn
s
s
v
v
( ) [(
( )
( )
( )) (
( )
( )
=
+
+ +
+
+
+
1
1
2
2
1 1
2 2
+
+
b i
t
a v t
n sn
o
i
i
( ))]
( )
This current has two components – one contributed by all independent current and voltage sources 
within the circuit and the second contributed by the independent voltage source connected from 
outside, i.e., v(t). The functions v
s1
(t) … represent the source functions of independent voltage sources 
within the circuit and the functions i
s1
(t) … represent the source functions of independent current 
sources within the circuit. n
v
and n
i
are the number of independent voltage and current sources within 
the circuit. The contribution coefficients a
o
, a
1
, a
2
, … and b
1
, b
2
, … are determined by the circuit 
parameters. They may be found by node analysis or mesh analysis. 
The source functions and contribution coefficients are fixed once and for all by the circuit and the 
only aspect of the circuit that can change is the network that gets connected at the output terminals. 
Hence we may represent the terms within the square brackets in the expression for i(t) as a fixed 
function of time that does not depend on what is connected at the output and term it as i
sc
(t).
∴
=
+
=
+
=
=
∑
i t
i t
a v t
i t
a v t
b i t
sc
o
sc
i si
i
n
i si
i
v
( )
( )
( )
( )
( )
( )
where
1
11
n
i
∑
(5.5-1)
This equation can be interpreted in an interesting manner if a
o
can be written as –G
o
. The number 
a
o
can be obtained by finding out the current delivered to the voltage source when all the independent 
sources are set to zero. If the circuit contains only resistors, then the current will actually be delivered 
to the circuit, and hence a
o 
will be a negative number, making G
o
a positive number. The possibility 
of a
o
assuming a positive value does exist if there are dependent sources within the circuit. Therefore, 
G
o
is positive for a purely resistive network, whereas it could be negative for a circuit containing 
dependent sources.
∴
=
−
i t
i t
G v t
sc
o
( )
( )
( ) 
(5.5-2) 
This equation suggests that i(t) behaves as if it is 
coming from an independent current source of source 
function i
sc
(t) that is in parallel with a resistance of R
o
=
1/G
o
(see Fig. 5.5-2)
How do we get the source function i
sc
(t)? i(t) 
=
i
sc
(t) 
when v(t) 
=
0. Therefore, we can find i
sc
(t) by finding out 
Fig. 5.5-2 
A circuit that follows 
Eqn. 5.5-2
R
o
v
(
t
)
i
(
t
)
G
o
v
(
t
)
i
sc
(
t
)
a
a
–
+
Fig. 5.5-1 
A memoryless network 
terminated in a voltage source 
at its output terminals
Linear memoryless 
circuit with many 
independent and 
dependent sources
i
(
t
)
v
(
t
)
a
a
–
+

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Thevenin’s Theorem and Norton’s Theorem 
5.23
the current that flows out into a short-circuit that is put across its output. This is the reason why we 
used ‘sc’ as the subscript for this current source function.
Further, how do we find out the value of R
o
? If we can reduce i
sc
(t) to zero and apply a non-zero v(t), 
the ratio of current drawn from v(t) to v(t) will be R
o
. We can reduce i
sc
(t) to zero by deactivating all the 
independent sources within the circuit. Thus, we see that, R
o
is nothing but the equivalent resistance 
of the deactivated network from terminals