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Department of Electronic and Information Engineering
ENG 237-E02/1
ENG237-E02: Kirchhoff’s Laws, equivalent resistance, and
the maximum power transfer theorem
Introduction
The objectives of this experiment are to (1) verify Kirchhoff’s laws, (2) determine the
equivalent resistance of a network, and (3) study the maximum power transfer theorem.
Sign conventions for current
IXY
X
Y
IXY = -IYX
Where IXY is the current flowing from node X to Y.
Sign conventions for potential difference
VXY
X
Y
VXY = -VYX
Where VXY is the voltage across XY and is positive when point X is at a higher potential than
point Y.
Apparatus
1. Power supply
2. Multi-meter
3. Experimental board
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Kirchhoff’s Laws
Introduction
The interconnection of circuit elements imposes constraints on the relationship between the
terminal voltages and currents. These constraints are referred to as Kirchhoff’s laws. The
two laws that state the constraints in mathematical form are known as Kirchhoff’s Current
law (KCL) and Kirchhoff’s Voltage law (KVL).
Kirchhoff’s Current law (KCL) states that the algebraic sum of the currents at any node
(junction) is zero. In mathematical form, for n branches converging into a node, KCL states
that
I1+I2+…+Ik+…+In = 0
where Ik is the current flowing in the kth branch and its direction is assumed to be pointing
towards the node.
To use Kirchhoff’s current law, an algebraic sign corresponding to a reference direction must
be assigned to every current at a node. Figure 1 shows the conventional notation that is used
to specify the algebraic sign. In general, a positive sign is assigned to a current entering a
node. Conversely, a negative sign is assigned to a current leaving a node. For example, at
node 2 in Figure I, we have
IA - IB - ID = 0
IF
IA
1
IB
2
ID
IC
4
Figure 1
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IE
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ENG 237-E02/3
Kirchhoff'
s Voltage law (KVL) states that the algebraic sum of the voltages between
successive nodes in a closed path in a circuit is equal to zero. In mathematical form, for a
closed path with successive nodes 1,2, ...n, KVL states that
V1,2+V2,3+…+Vn-1,n+Vn,1 = 0
From Figure 2, we have
V1+V2+V3+V4 = 0
+ V2 -
V1 DC
DC
- V4 +
Figure 2
Procedures
1. Connect the DC power supply as shown in Figure 3
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ENG 237-E02/4
2. Kirchhoff'
s Current Law
(a) Adjust the D.C. supply to keep the voltage across Nodes A and C to 5 V.
(b) Calculate the currents at Node B by measuring the voltages across the branches that
connect to Node B. Complete the following table:
Voltage (V)
Current entering Node B (A)
VAB=
IAB =
VMB =
IMB =
VCB =
ICB =
(c) Calculate IAB + IMB + ICB.
(d) Repeat Step (b) for Node M. Complete the following table:
Voltage (V)
Current entering Node B (A)
VAM=
IAM =
VCM =
ICM =
VBM =
IBM =
(e) Calculate IAM + ICM + IBM .
(f) Comment on the results of (c) and (e).
3. Kirchhoff'
s Voltage Law
(a) Adjust the voltage across AC to 5 V.
(b) Measure the voltage around loop AMBCA. Complete the following table:
Loop AMBCA
Voltage (V)
VAM
VMB
VBC
VCA
(c) Calculate VAM + VMB + VBC + VCA.
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(d) Measure the voltage around loop ABMCA. Complete the following table:
Loop ABMCA
Voltage(V)
VAB
VBM
VMC
VCA
(e) Calculate VAB + VBM + VMC +VCA.
(f) Comment on the results of (c) and (e).
Equivalent resistance
Introduction
The circuit shown in Figure 3 is difficult to solve, since series/parallel reduction technique
and voltage/current division principle cannot apply here to reduce the interconnected
resistors to a single equivalent resistance. However, the interconnected resistors can be
reduced to a single equivalent resistance by means of a delta-to-star or star-to-delta
equivalent circuit.
Star-to-Delta Transform
Figure 4 shows how a star-configuration can be converted into a delta-configuration
RX
1'
1
RA
RB
RA
2'
RB
2
RY
RC
RC
3
3'
Figure 4
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RZ
where
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Department of Electronic and Information Engineering
R R + RB RC + RC R A
RX = A B
,
RC
RY =
R A RB + R B RC + RC R A
, and
RB
RZ =
R A RB + RB RC + RC R A
.
RA
ENG 237-E02/6
Delta-to-Star Transform
Figure 5 shows how a delta-configuration can be converted into a star-configuration.
RX
RX
RA
RY
RZ
RY
RB
RC
RZ
Figure 5
where R A =
R X RY
R X RZ
RY RZ
, RB =
, and RC =
.
R X + RY + RZ
R X + RY + RZ
R X + RY + RZ
Procedures
(a) Turn off the power of the supply and use multimeter to measure the equivalent resistance
RAC of the network shown in Figure 3 between terminals A and C. Do not remove the
power supply from the circuit.
(b) Measure again with the removal of power supply. Comment on the result as compared
with (a).
(c) Using the star-to-delta or the delta-to-star transformation, calculate the equivalent
resistance of the network shown in Figure 3.
(d) Compare the results between (b) and (c).
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ENG 237-E02/7
Maximum Power transfer theorem
Introduction
The maximum power transfer theorem states that in DC circuits, the maximum possible
power that can be delivered to a load occurs when the load resistance (RL) equals the internal
resistance (Ri) of the DC generator.
Ri
RL
V
Figure 6
In Figure 6, the power PL dissipated in the load resistor RL can be written as:
2
V
PL = L
RL
where VL is the voltage across the load RL and
VL =
VRL
.
Ri + RL
Procedures
(a) Connect the circuit as shown in Figure 7 with Ri = 5 k Ω and adjust the power supply to
5V DC.
Ri
+
DC
Supply
V
L
Figure 7
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V
R
L
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(b) Vary the load RL from l k Ω to 25 k Ω resistances in "small" but "suitable" steps and
record the corresponding values of VL and IL in each step.
Ri = 5 k
VL (V)
IL (mA)
RL ( )
PL (mW)
(c) Calculate the relative values of RL.
(d) Plot the power delivered to the load against RL.
(e) From the power curve obtained, find the maximum power dissipated and the
corresponding value of RL.
(f) Determine the total resistance of the supply section using the condition for maximum
power transfer.
(g) Repeat the above procedures with Ri = 10 k Ω and Ri = 20 k Ω , so as to obtain families
of P-curves.
Ri = 10 k
(with RL varying from 5 k
to 15 k )
VL (V)
IL (mA)
RL ( )
PL (mW)
Ri = 20 k
(with RL varying from 15 k
to 25 k )
VL (V)
IL (mA)
RL ( )
PL (mW)
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Discussions
(1) A person accidentally grabs conductors connected to each end of a DC voltage source,
one in each hand.
(a) What is the minimum source voltage that can produce electrical shock sufficient to cause
paralysis, preventing the person from letting go of the conductors? You may assume that
the characteristic resistance of a human arm is 400 Ω and the current level for muscle
paralysis is 50 to 70mA.
(b) Is there a significant risk of this type of accident occurring while servicing a personal
computer, which has typically 5V and 12V sources?
(2) State the criterion for maximum power transfer from a source to a load. If you are free to
select Ri without changing the source voltage, how can you maximize the power delivered
to the load?
(3) Prove the maximum power transfer theorem.
References
1. C.K. Tse, Linear Circuit Analysis, 1st Ed., Addison-Wesley, England, 1998.
2. K. Harry, Electronics, a self-teaching guide, 1st Ed., John Wiley & Sons, USA, 1979.
3. J.W. Nilsson and S. A. Riedel, Electric Circuits, 5th Ed., Addison-Wesley, USA, 1996.
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