This document describes an experiment to verify several network theorems including Thevenin's theorem, Norton's theorem, superposition theorem, and the maximum power transfer theorem. The experiment uses a Wheatstone bridge circuit with resistors R1-R4 and a voltage source. Measurements are taken at various load resistances RL and graphs are plotted to experimentally determine the Thevenin resistance Rth, Thevenin voltage Vth, Norton current In, and maximum power transfer. Direct measurements are also taken and compared to theoretical calculations to verify the network theorems.

1. Experimental Verification of Network Theorems
S. Paul, Scottish Church College, Calcutta-6
1 Network Theorems
Network theorems are analytical method used to convert a complex circuit into a simple equivalent circuit.
1.1 Thevenin’s Theorem
Any linear complex circuit seen from any two terminals may be represented by a single voltage source (Thevenin
Voltage, Vth) in series with an impedance (Thevenin Impedance, Zth) where Thevenin voltage is the open circuit
voltage between that terminals and Thevenin impedance is equal to the impedance seen from that terminals,
after short-circuiting all independent ideal voltage sources and opening the independent ideal current sources.
1.2 Norton’s Theorem:
Any linear complex circuit seen from any two terminals may be represented by a single current source (Nortin
Current, IN ) in parallel with an impedance (Norton Impedance, ZN ), where Norton current is the short circuit
current between that terminals and Norton impedance is equal to the impedance seen from that terminals, after
short-circuiting all independent ideal voltage sources and opening the independent ideal current sources. The
Thvenin’s and Norton’s theorems are dual and applicable to linear circuits only.
1.3 Objective of the Experiment:
To verify Thevenin and Norton theorem by using a given circuit.
1.3.1 Circuit diagram and working formula:
Figure 1 shows the working circuit for verification of Network theorems. It is an unbalanced resistive Wheatstone
bridge network having power supply with ignorable internal resistance connected to load resistance RL.
Thevenin’s Theorem verification:
Let us consider the the terminals B and D between which we apply Thevenin theorem. The Thevenin voltage
measured across B and D, can be written as
Vth = Vin
R2
R1 + R2
∼
R3
R3 + R4
(1)
Thevenin resistance(c.f. figure:2) Rth is given by,
Rth =
R1 · R2
R1 + R2
+
R3 · R4
R3 + R4
(2)
+
_
REGULATEDPOWERSUPPLY
R
1
R
3
R
4
R
2
A
B
C
D
PSfrag replacements
Rth
RL
vth
Figure 1: Schematic Circuit diagram for study of Network Theorems
1

2. B
R1
R2
A C
R3
R4
D
Figure 2: Finding of Rth.
+
_
PSfrag replacements
Vth
Rth
RL
Figure 3: Thevenin equivalent Circuit.
Thevenin equivalent circuit is shown in the figure 3 and VL can be written written as,
VL = Vth − IL · Rth (3)
If a graph is plotted with IL as abscissa and VL as ordinate, must be a straight line having slope −Rth and the
intercept on ordinate is Thevenin voltage, Vth as shown in the figure 4. Note that this is open-circuit voltage,
i.e. RL = ∞ or IL = 0.
∴ From figure 4 ,
vL
IL
Vth
Figure 4: VL -IL graph. Intercept of the extrapolated st. line is denoted by dot on the VL axis and it is the the
open circuit volatge(IL = 0), i.e. Vth.
Rth =
PM
MN
and Vth = OQ (4)
Thus from equation 1 and 2 we get theoretical values of Rth and Vth respectively and comparing these values
with the experimentally obtained values from equation 4, Thevenin’s theorem can be verified.
Norton’s Theorem Verification:
Referring to the circuit 1, we can say according to Norton’s theorem, Norton’s resistance between the terminals
B and D is
Rn = Rth (5)
i.e. same as Thevenin resistance. Short-circuit current between the terminals B and D is In given by
In =
Vth
Rth
(6)
2

3. Therefore Norton’s equivalent circuit of the circuit 1 is depicted in the figure 5
Figure 5: Norton equivalent circuit.
From Norton’s equivalent circuit, figure 5, we can write,
IL = In −
VL
Rn
(7)
If we plot IL as ordinate and VL as abscissa, graph would have been a straight line line as shown in the
fig 6. Intercept of this line on the ordinate gives the value of In, i.e. short-circuit current or Norton’s current.
Magnitude of the slope of the st. line corresponds to 1
Rn
IL
VL
In
Figure 6: IL- VL Graph.
From figure 6
∴
Rn =
MN
PM
; In = OQ (8)
From IL − VL graph, by equation 8, Rn and In can be obtained experimentally.
Thus from equation 1 and 2 we get theoretical values of Rth(=Rn) and Vth respectively and from 6, In may be
computed theoretically.
In and Rn can also be measured directly. Remove RL and insert an ammeter across B and D terminal gives
the short-circuit current,i.e. Norton current. To measure Rn, the power supply is removed and A, C terminals
are short circuited by a conducting wire. Now RL is also removed and place an multimeter (dial in the Ohm)
across B,D terminals gives Rn.
Thus by comparing directly measured, from VL −IL graph and from theoretical values of Rn and In, Norton’s
theorem may be verified.
3

4. Apparatus:
1. A Breadboard
2. A regulated power supply(0-10V)
3. Four carbon resistors(may be R1=100Ω, R2 =220Ω, R3=330Ω ,R4=470Ω) / PO box and a resistance box.
4. Digital Voltmeter/ Analog Voltmeter/ Multimeter (0-10V)
5. Digital Ammeter/ Analog Ammeter/ Multimeter (0-100mA)
6. Resistance box / Few discrete carbon resistance as a RL. It depends on value of Rth, i.e. choices of R1,
R2, R3, R4. (It may be 50Ω, 100Ω, 220Ω, 330Ω, 470Ω, 560Ω, 1kΩ, 1.5kΩ, 2kΩ.)
1.3.2 Experimental Procedure:
1. Circuit is made as shown in the figure 1 on the breadboard or with the help of PO box plus resistance
box. Measure the values of R1, R2, R3, R4 with the multimeter. Record the input voltage(may be 6V) of
the power supply and it should be fixed throughout the experiments.
2. Compute the values of Rth, Vth and In theoretically by using equations 1, 2 and 6
3. Remove RL from the circuit. Now voltmeter gives the open circuit voltage, i.e. Thevenin voltage, Vth.
Hereafter, disconnect voltmeter and ammeter and power supply from the circuit. Short-circuit the ter-
minals by a conducting wire between the terminals A,C. Now measure the resistance by a multimeter
between the terminals B, D. This resistance corresponds to Rth = Rn. These set of values are directly
measured.
4. Re-connect the power supply, ammeter, voltmeter and RL. Now gradually change the values of RL from
highest value to lowest value including RL = Rth. Vi must be constant and checked before each reading.
5. Plot VL − IL graph (like fig 4), calculate Rth and Vth by using equation 4
6. Plot IL − VL graph (like fig 6), calculate Rth and Vth by using equation 8.
1.3.3 Observations
Verification of Thevenin’s and Norton’s theorem
Table 1
Resistances
Color coded value
in Ω
Measured by Multimeter
in Ω
R1
R2
R3
R4
Table 2
Theoretically Calculated Directly measured From VL − IL graph Remarks
Rth ...Ω ...Ω ...Ω
Vth ...V ...Ω ...Ω
In ...mA ...mA ...mA
4

5. Table 3
No. of Obs.
RL
in Ω
VL
in V
IL
in mA
1
2
3
...
...
1.4 Superposition Theorem:
The superposition theorem for electrical circuits states that for a linear system the response (voltage or current)
in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum
of the responses caused by each independent source acting alone, where all the other independent sources are
replaced by their internal impedances.
1.4.1 Objective of the Experiment:
To verify Superposition by using a given circuit.
1.4.2 Circuit diagram and working formula:
Figure 7: (a) Circuit diagram for verification of superposition theorem. (b) Load voltage VL1 due to voltage
source V1 (c) Load voltage VL2 due to voltage source V2
1.4.3 Apparatus:
1. Two regulated power supply(0-10V).
2. Four carbon resistance.
3. A Breadboard.
4. Multimeter/ Digital Voltmeter/ Analog Voltmeter.
5. Connecting wire.
5

6. 1.4.4 Experimental Procedure:
1. Construct the circuit 7(a) on the breadboard. Note the load voltage VL across RL.
2. Replace voltage source V2 by a short circuit, i.e. insert a conducting wire as shown in the figure 7(b).
Measure the load voltage VL1 due to V1.
3. In the similar way, replace voltage source V1 by a short circuit, i.e. insert a conducting wire as shown in
the figure 7(c). Measure the load voltage VL2 due to V2.
4. It will be observed that, VL = VL1 + VL2, Thus superposition theorem is verified.
1.4.5 Observation:
Verification of superposition theorem.
V1
in Volt
V2
in Volt
VL
in Volt
VL1
in Volt
VL2
in Volt
ON(...) ON(...) ... =C X X
ON(...) OFF(Shorted) X ...=A X
OFF(Shorted) ON(...) X X ...=B
1.5 Maximum Power Transfer Theorem:
Maximum external power from a source with a finite internal resistance can be drawn when the resistance of
the load is equal to the resistance of the source as viewed from its output terminals.
Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as
”Jacobi’s law”
The power dissipated in the load resistance
P =
V 2
L
RL
(9)
The maximum power will be supplied by the network to load resistance RL only when RL = Rth.
Therefore for the different values of RL note the corresponding values of VL and IL. Plot P(=VL · IL) as a
ordinate and RL as abscissa as shown in the figure 9. P − RL graph will show, power reaches maximum value
only when RL = Rth. Thus maximum power transfer theorem is verified.
Here the working circuit may be figure 1 or figure 8. For fig 8,
Rth can be written as
Rth = R3 + R2 R1
Thus
Rth = R3 +
R1 · R2
R1 + R2
Figure 8: Circuit diagram for verification of maximum power transfer theorem
6

7. 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 200 400 600 800 1000 1200
’max_power’
PSfrag replacements
RL in Ohm
PinmW
Figure 9: Maximum power transfer theorem
1.5.1 Observation
No. of Obs.
VL
in Volt
IL
in mA
P = VL · IL
in mW
1
2
3
...
...
Precautions and Discussions:
1. Internal resistance of the power supply is very small and usually neglected.
2. Large current should not pass through the circuit because it generates joulian heat(i2
Rt) in the resistance
which in turn makes them nonlinear. Network theorems are not valid in nonlinear regime.
3. Constancy of the input voltage Vin must be checked before each reading.
4. While measuring directly open circuit voltage, voltmeter should have been chosen of high internal resis-
tance. Usually electronic voltmeter should be taken for direct measurement.
5. Circuit is being shorted by the ammeter while taking short-circuit current, i.e. Norton’s current. But
ammeter has finite internal resistance which is usually neglected in student’s laboratory practice. If it is
not ignorable, then its effect should be included in the Norton’s current in the following manner:
Let Ra be the internal resistance of the ammeter used in the circuit.
∴
In,measured =
Vth
Rth + Ra
→ In,measured =
Vth
Rth
1 + Ra
Rth
=
In,true
1 + Ra
Rth
∴ In,true = 1 +
Ra
Rth
In,measured
7

8. 1.5.2 Viva-Voce Questions:
1. What do you mean by linear and non-linear circuit element?
2. What is the use of network theorem?
3. What is ideal voltage source?
4. What is ideal current source?
5. What is bilateral element? Explain with example.
6. What is active and passive elements. Explain with examples.
8