THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
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
Where I xy
is the current flowing from node X to Y .
Sign conventions for potential difference
Where V
XY
is the voltage across the terminals X and Y and is positive when point X is at a higher potential than point Y .
Apparatus
1.
DC power supply
2.
Multi-meter
3.
Experimental board
Page 1

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
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
Where I k
is the current flowing in the k th 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 1, we have
Figure 1
Page 2

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
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
From Figure 2, we have
Procedures
1.
Set the current limit of the power supply to 0.5 A. Then connect it to the experimental board as shown in Figure 3
Figure 3
Page 3

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
2.
Kirchhoff’s Current Law
(a) Adjust the D.C. supply to keep the voltage across Node 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:
V
V
V
AB
MB
CB
=
=
=
Voltage (V) Current entering Node B (A)
I
AB
=
I
MB
=
I
CB
=
(c) Calculate I
AB
+ I
MB
+ I
CB
(d) Repeat Step (b) for Node M . Complete the following table:
V
V
V
AM
CM
BM
=
=
=
Voltage (V) Current entering Node M (A)
I
AM
=
I
CM
=
I
BM
=
(e) Calculate I
AM
+ I
CM
+ I
BM .
(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
V
AM
V
MB
V
BC
V
CA
Voltage (V)
(c) Calculate V
AM
+ V
MB
+ V
BC
+ V
CA
Page 4

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
(d) Measure the voltage around loop ABMCA . Complete the following table:
Loop ABMCA
V
AM
V
BM
V
MC
V
CA
Voltage (V)
(e) Calculate V
AB
+ V
BM
+ V
MC
+ V
CA
(f) Comment on the results of (c) and (e).
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
Page 5

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
Delta-to-Star Transform
Figure 5 shows how a delta-configuration can be converted into a star-configuration.
Procedures
(a) Turn off the power supply and use multi-meter to measure the equivalent resistance
R
AC
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 results 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).
Page 6

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
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 ( R
L
) equals the internal resistance ( R i
) of the DC power supply.
In Figure 6, the power P
L
dissipated in the load resistor R
L
can be written as:
Where V
L
is the voltage across the load R
L
and
Procedures
(a) Connect the circuit as shown in Figure 7 with R i
= 5 k Ω and adjust the power supply to
5 V DC.
Page 7

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
(b) Vary the load R
L
from 1 k Ω to 25 k Ω resistances in “small” but “suitable” steps and record the values of V
L
and the corresponding I
L
in each step.
R i
= 5 k Ω
R
L
( Ω )
V
L
(V)
I
L
(mA)
P
L
(mW)
(c) Plot the power delivered to the load against R
L
.
(d) From the power curve obtained, find the maximum power dissipated and the corresponding value of R
L
.
(e) Determine the total resistance of the supply section using the condition for the maximum power transfer.
(f) Repeat the above procedures with R i
= 10 k Ω and R i
= 20 k Ω , so as to obtain a family of power-curves.
(g) Estimate the internal resistance R x
of the DC power supply.
R i
= 10 k Ω (with R
L
varying from 5 k Ω to 15 k Ω )
R
L
( Ω )
V
L
(V)
I
L
(mA)
P
L
(mW)
R i
= 20 k Ω (with R
L
varying from 15 k Ω to 25 k Ω )
R
L
( Ω )
V
L
(V)
I
L
(mA)
P
L
(mW)
Page 8

THE HONG KONG POLYTECHNIC UNIVERSITY
ENG237-E02r1 r1
Department of Electronic and Information Engineering
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 70 mA.
(b) Is there a significant risk of this type of accident occurring while servicing a personal computer, which has typically 5 V and 12 V sources?
(2) State the criterion for maximum power transfer from a source to a load. If you are free to select R i
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 , 1 st Ed., Addison-Wesley, England, 1998.
2.
K. Harry, Electronics, a self-teaching guide , 1 st Ed., John Wiley & Sons, USA, 1979.
3.
J.W. Nilsson and S. A. Riedel, Electric Circuits , 5 th Ed., Addison-Wesley, USA, 1996.
---END---
Page 9