SUPERPOSITION, THEVENIN, NORTON AND
MAXIMUM POWER TRANSFER
Introduction
In this experiment, it is aimed to introduce Thevenin’s and Norton’s theorems to
students for simplifying circuit analysis. These theorems will be separately applied to linear
circuits. In addition, concepts of superposition and maximum power transfer will be
introduced.
Theory
The superposition approach is alternative to nodal or mesh analysis to determine the
value of a specific variable (voltage or current).Superposition is analyzing a linear circuit with
more than one independent source by calculating the contribution of each independent source
to the variable (voltage or current) separately and then adding them. When applying
superposition principle, turn off all independent sources except one source. Figure 1 shows
the application of the superposition principle. According to Figure 1, the voltage of the
resistor (R2) is calculated as V=v1+v2.
R1
Vdc
R2
V
Idc
(a)
R1
R1
i2
i3
Vdc
i1
(b)
R 2 v1
R 2 v2
(c)
Figure 1. (a) Circuit, (b) turn off current source, (c) turn off voltage source.
Idc
Thevenin equivalent circuit theorem was developed in 1883 by a French telegraph
engineer M. Leon Thevenin (1857–1926). Thevenin’s theorem provides a technique by which
the fixed part of the linear two-terminal circuit is replaced by an equivalent circuit to avoid
analyzing the entire circuit everything again when the variable load element is changed each
time. Figure 2 shows the replacing a linear two-terminal circuit by its Thevenin equivalent. In
Figure 2, the equivalent circuit is consisting of a voltage source VTh the open-circuit voltage at
the terminals, in series with a resistor RTh the input or equivalent resistance at the terminals,
when the independent sources are turned off.
I
RTh
a
Linear
twoterminal
circuit
V
I
a
VTh
Load
V
Load
b
b
(a)
(b)
Figure 2. (a) Original linear two-terminal circuit, (b) the Thevenin equivalent circuit.
Norton’s Theorem was proposed by E. L. Norton, an American engineer at Bell
Telephone Laboratories in 1926, about 43 years after Thevenin’s Theorem was published.
Like Thevenin’s Theorem, Norton’s theorem states that a linear two-terminal circuit can be
replaced by an equivalent circuit consisting of a current source IN, the short-circuit current
through the terminals, in parallel with a resistor RN, the input or equivalent resistance at the
terminals when the independent sources are turned off. In fact, the Thevenin and Norton
resistances are equal. Figure 3 shows the replacing a linear two-terminal circuit by its Norton
equivalent.
a
Linear
twoterminal
circuit
a
IN
RN
b
(a)
b
(b)
Figure 3. (a) Original linear two-terminal circuit, (b) the Norton equivalent circuit.
A linear circuit is generally designed to provide the maximum power to a load. In the
Thevenin equivalent circuit, Maximum power transfer to the load can be occurred when the load
resistance equals the Thevenin resistance. Figure 4 shows the maximum power transfer to the
load.
P
RTh
a
i
Pmax
RL
VTh
b
RL
RTh
(a)
(b)
Figure 4. (a) The Thevenin equivalent circuit used for maximum power transfer, (b) the
delivered load power as a function of RL.
Equipment
Adjustable DC Power supply
Digital Multimeter
2kΩ
4.7kΩ
1kΩ
10kΩ
Preliminary work (Calculation and simulation section)
1) Superposition.
V1 =5 V
R1 =1 kΩ
R2 =2 kΩ
I1
I2
R3 =4.7 kΩ
Vout
V2 =15 V
Figure 5. Simple resistive circuit
a) Calculate the voltage Vout, the current I1 and I2 by using superposition method.
b) Specify the voltage Vout, the current I1 and I2 by using simulation program.
2) Thevenin’s and Norton’s Theorem
R1 =1 kΩ
R3 =1 kΩ
R5 =1 kΩ
a
IL
V1 =10 V
R2 =2 kΩ
Vab
R4 =2 kΩ
b
Figure 6. Simple resistive circuit
a) Calculate the current IL and the voltage Vab.
b) Find the Thevenin and Norton equivalent circuits at terminals a-b.
c) Specify the current IL and voltage Vab by using simulation program.
3) Maximum power transfer
a
IL
R1 =2 kΩ
R3 =10 kΩ
V1 =15 V
Vab
R2 =10 kΩ
R4 =2 kΩ
b
Figure 7. Simple resistive circuit
a) Calculate the value of R for maximum power
b) Determine the maximum power absorbed by R.
c) Specify the maximum power by using simulation program.
R
RL =4.7 kΩ
Experiment
1) Construct the circuit of Figure 5. Measure the voltage Vout, from the voltmeter, the
current I1 and I2 from the ammeter. Fill Table 1 according to the V1 and V2.
Table 1. Results table
I1 (mA)
I2 (mA)
Supply
Vout (V)
V1 (V)
V2 (V)
Experimental Theory Experimental Theory Experimental Theory
5
15
5
Short
Short
15
15
5
15
Short
Short
5
2) Construct the circuit of Figure 6. Measure the current IL using from the ammeter.
Then remove the load resistor RL by opening the terminals a-b and note down the
corresponding Vab or VTh from the voltmeter. Find out the RTh and draw the Thevenin
equivalent circuits. Fill Table 2 according to the V1 and RL.
Table 2. Results table
Supply Parameter
IL (mA)
Vab (VTh) (V)
RTh (kΩ)
V1 (V)
RL (kΩ)
Experimental Theory Experimental Theory Experimental Theory
10
4,7
15
4,7
10
10
15
10
3) Construct the circuit of Figure 6 again. The current IL was measured in Table 2.
Now, remove the load resistor RL by shorting the terminals a-b and note down the
corresponding Iab or IN from the ammeter. Find out the RN and draw the Norton equivalent
circuits. Fill Table 3 according to the V1 and RL.
Table 3. Results table
IL (mA)
Iab (IN) (mA)
RN (kΩ)
Supply Parameter
V1 (kΩ) RL (kΩ)
Experimental Theory Experimental Theory Experimental Theory
10
4,7
15
4,7
10
10
15
10
4) Construct the Thevenin equivalent of circuit in Figure 7. After construction, vary
the load resistance R from the minimum value to maximum value. Plot the graph between R
and Power (IL2R) where, theoretical IL=[VTh/(RTh+R)], for seven value (one of them is equal to
the RTh, three of them are smaller than RTh, the others are greater than RTh) of R on Graph 1.
Fill Table 4 according to the R.
Table 4. Results table
R (kΩ)
IL (mA)
P = IL 2 R
(mW)
RTh=
P (mW)
IL2R
R (kΩ)
RTh
Graph 1. Model Graph.