1E6 Electrical Engineering
DC Circuit Analysis
Lecture 6: Thevenin’s and Norton’s Theorems
6.1 Introduction
Electric circuits can sometimes become extensive in the number of
elements and branches or loops which they contain. This can make their analysis
unwieldy on occasions, though there are systematic methods which can be
applied for the purpose. Dc circuits contain only three primitive elements
namely: constant voltage sources, constant current sources and resistors.
Therefore any means of reducing the complexity of the circuit to contain fewer
of these primitive elements will greatly help the task of analysing a circuit. Two
theorems which allow such reduction are Thevenin’s Theorem and Norton’s
Theorem. However, before examining these, there are some other formal
concepts which need to be understood as they are used in the application of these
theorems.
6.2 Open-Circuit Load
Consider the non-ideal voltage source and current source shown in Fig. 1
below under open circuit load conditions. The subscript o/c is used to designate
this particular condition. In effect, the load has been disconnected so that RL→∞
and the output is referred to as open-circuit.
voltage source
RS
E
IO/C = 0
current source
IS = I
RL→∞
I
VO/C = E
RS
IO/C = 0
RL→∞
VO/C = IRS
Fig. 1 Voltage Source and Current Source under Open-Circuit Load Conditions
In this case no current flows into the load so that IO/C = 0 for both the voltage
source and the current source. The output voltage on the other hand is not zero.
In the case of the voltage source the output voltage with no load connected will
be equal to the cell voltage so that VO/C = E. In the case of the current source the
open circuit voltage will be determined by the current I flowing through the
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internal resistance of the non-ideal source so that VO/C = IRS. The use of an open
circuit load allows the characteristics of a circuit to be stipulated for infinite load
resistance. It is therefore particularly useful for determining the value of the
ideal cell voltage associated with a non-ideal voltage source.
6.2 Short-Circuit Load
Consider the non-ideal voltage source and current source shown in Fig. 2
below under short-circuit load conditions. The subscript s/c is used to designate
this particular condition. In effect, the load is set to its theoretical minimum with
RL→0 and the output is referred to as short-circuit.
voltage source
current source
IS/C = E / RS
RS
IS = 0
RL= 0
E
IS/C = I
RL= 0
I
VS/C = 0
RS
VS/C = 0
Fig. 2 Voltage Source and Current Source under Short-Circuit Load Conditions
In this case the voltage across the load is forced to zero so that VS/C = 0 for both
the voltage source and the current source. The output current on the other hand
is at a maximum. In the case of the voltage source the output current is
determined by the cell voltage and the internal resistance so that IS/C = E / RS. In
the case of the current source the short circuit provides no resistance to current
flow so that all of the current provided by the source flows into it and no current
flows through the internal resistance, RS, so that the output current IS/C = I. The
use of a short-circuit load allows the characteristics of a circuit to be stipulated
for zero load resistance. It is therefore particularly useful for determining the
ideal value of the current associated with a non-ideal current source.
6.2 Thevenin’s Theorem
This theorem was formally proposed by the French telegraph engineer
Léon Charles Thévenin (1857-1926), in 1883, though similar discoveries had
been made previously. In modern terms specific to our analysis:
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Thevenin’s Theorem states that: ‘as seen by a resistive load connected to it, any
linear electric circuit consisting of a combination of voltage or current sources and
resistors can be replaced by a single voltage source with the Thevenin voltage, VTH
and a single internal resistance equal to the Thevenin resistance, RTH.’
Consider the scenario shown below in Fig. 3 and Fig. 4.
IL
R3
R1
R4
R2
VL
I
E1
RL
E2
Fig. 3 An Electric Circuit Having a Combination of Sources and Resistors
Thevenin equivalent
voltage source
IL
RTH
VL
VTH
RL
Fig. 4 The Thevenin Equivalent of the Circuit of Fig. 3
The circuit of Fig. 3 arbitrarily contains two voltage sources, a current source
and several resistors. A load resistance, RL is connected to the terminals of the
circuit on the right hand side, and a voltage, VL appears across the load while a
current, IL flows through it. Thevenin’s Theorem states that this can be replaced
by the circuit of Fig. 4 where there is a single non-ideal voltage source of voltage,
VTH and internal resistance, RTH. In the second circuit the same voltage, VL is
developed across the load when connected and the same current, IL, flows
through it as in the circuit of Fig. 3.
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The Thevenin voltage, VTH and the Thevenin resistance, RTH are established
from the circuit of Fig. 3 as follows:
The Thevenin voltage, VTH is established as the output open-circuit voltage
measured at the output terminals of the circuit with the load disconnected.
The Thevenin resistance, RTH is established as the resistance seen looking back into
the output terminals of the circuit with the load disconnected.
For the purposes of establishing the Thevenin resistance all active driving sources
must be made inactive. In this case a non-ideal voltage source is made inactive by
‘shorting out’ the cell voltage (effectively making it zero) and replacing the nonideal source with its internal resistance so that this is accounted for. A non-ideal
current source is made inactive by ‘open-circuiting’ the current source (effectively
making it zero) and replacing the non-ideal source with its internal resistance.
6.3 Case Study 1
Consider a simpler example circuit with only a single voltage source as shown in
Fig. 5. The load resistance, RL is disconnected from the circuit in order to find
the Thevenin equivalent. This means that the output is then under open circuit
conditions.
R2
R4
A
5kΩ
R1
5kΩ
I1
E1
2kΩ
R3
4kΩ
VTH = VO/C
RL
12V
Fig. 5 A Circuit Having a Single Voltage Source and Resistors
With the load disconnected so that the output terminals are open-circuit no
current can flow through the resistor R4. This means that there is no voltage
drop across it and hence the output voltage must be the same as the voltage at
Node A in the circuit. This means that current only flows around the loop
containing the voltage source, E1 and the resistors R1, R2 and R3. Since these
resistors are in series the current can be found as:
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I1 =
E1
R1 + R 2 + R 3
The potential at node A relative to ground is essentially the potential drop across
resistor R3 which gives:
VA = I1R 3
Since this is the open-circuit output voltage we have the Thevenin voltage as:
VTH = VA = I1R 3 =
R3
E1
R1 + R 2 + R 3
For the component values given in the circuit:
VTH
4 × 103
=
× 12 V
1 × 103 + 5 × 103 + 4 × 103
VTH
4 × 103
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=
×
12
V
=
= 4.8V
10 × 103
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In order to evaluate the Thevenin resistance the voltage source must be shorted
out. It can be assumed that R1 is its internal resistance so this replaces the
source. The circuit can then be modified as shown in Fig. 6. The Thevenin
resistance is the equivalent resistance seen looking back into the output
terminals as viewed by the load. This can be seen to be resistor R4 in series with
the combination of R3 in parallel with the pair of R1 and R2 in series.
R1
1kΩ
R2
R4
5kΩ
2kΩ
R3
4kΩ
RTH
Fig. 6 The Circuit Redrawn to Identify the Thevenin Resistance
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This can be written as:
RTH = R4 + R3 //( R1 + R2 )
If the series combination of R1 + R2 is treated as equivalent to a single resistor,
then the product / sum rule can be used to determine the parallel combination so
that:
RTH = R4 +
( R1 + R2 ) R3
R1 + R2 + R3
For the component values given in the circuit:
RTH
R TH
(1 × 103 + 5 × 10 3 ) 4 × 103
= 2 × 10 +
1 × 103 + 5 × 103 + 4 × 103
3
6 × 10 3 × 4 × 10 3
24 × 10 3
3
= 2 × 10 +
= 2 × 10 +
10 × 10 3
10
3
RTH = 2 × 103 + 2.4 × 103 = 4.4 × 103 = 4.4kΩ
So then the Thevenin equivalent circuit for the circuit of Fig. 5 can be drawn as
shown in Fig. 7 below. This circuit will behave in all respects in the same manner
as that of Fig. 5 for all values of load resistance, RL.
IL
RTH
4.4kΩ
VTH
Fig. 7
VL
4.8V
RL
The Thevenin Equivalent Circuit for the Circuit of Fig. 5
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6.4 Norton’s Theorem
A second theorem surrounds an alternative form of equivalent circuit
based on a non-ideal current source. This alternative was proposed by Edward
Lawry Norton (1898-1983) while working as an electrical engineer in Bell Labs.
In modern terms specific to our analysis:
Norton’s Theorem states that: ‘as seen by a resistive load connected to it, any linear
electric circuit consisting of a combination of voltage or current sources and
resistors can be replaced by a single current source with the Norton Current, INR
and a single internal resistance equal to the Norton resistance, RNR.’
Norton equivalent
current source
INR
Fig. 8
RNR
IL
VL
RL
The Norton Equivalent Circuit
It is therefore also possible to find a Norton equivalent circuit for the circuit of
Fig. 5.
The Norton Current, INR, is established as the output short-circuit current
measured at the output terminals of the circuit with the load disconnected and
replaced by a short-circuit.
The Norton resistance, RNR is established as the resistance seen looking back into
the output terminals of the circuit with the load disconnected. It is therefore
identical to the Thevenin resistance and is found in exactly the same way.
It is a little more difficult to establish the Norton parameters for the circuit of
Fig. 5 as it involves other circuit analysis techniques not yet studied. However, a
simpler approach can be taken on the basis that if both the Thevenin and the
Norton equivalent circuits are valid, then they should both behave in exactly the
same manner for all load conditions. This should also include open-circuit and
short-circuit load conditions. This means that the Thevenin equivalent model
should produce the same current into a short-circuit load as the Norton
equivalent model, namely the Norton current. It also means that the open-circuit
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voltage produced by the Norton equivalent model should be identical to the
Thevenin voltage. This is illustrated in Fig. 9 below.
Thevenin
equivalent model
Norton
equivalent model
IS/C
RTH
IO/C = 0
INR
VTH
INR RNR
VS/C = 0
VO/C
Fig. 9 Thevenin Model under s/c and Norton Model under o/c conditions
Then for the Thevenin model:
IS / C =
VTH
= I NR
R TH
VTH
= R TH
I NR
⇒
and for the Norton model:
VO / C = I NR R NR = VTH
⇒
VTH
= R NR
I NR
This is essentially an Ohm’s Law for the equivalent circuits and proves that as
stated RNR = RTH which allows the value of the current source in the Norton
model to be evaluated for the circuit example of Fig. 5 and the Norton model to
be drawn as in Fig. 10:
I NR =
VTH
4.8V
=
= 1.09 mA
R TH 4.4kΩ
Norton equivalent
current source
INR
IL
RNR
4.4kΩ
VL
RL
1.09mA
Fig. 10 The Norton Equivalent Circuit for the Circuit of Fig. 5.
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