TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
3. POWER SUPPLY EFFICIENCY
1. Objective
Determination of the power supply efficiency and effective power depending on current and
the ratio of internal and external resistance.
2. Equipment needed
Stand with voltmeter, ammeter, two elements, two (or three) rheostats and a switch.
3. Theory
Whatever electrical circuits can be observed as consisting of internal and external part:
Internal part consists of power supply and its resistance, external part consists of connecting
wires and consumer (ballast resistor).
According to the Ohm`s law the current is determined by electromotive force of the power
supply and the total resistance of the circuit. While usually the connecting wires are chosen so
that their resistance is inconsiderably small compared to the other resistances of the circuits
elements, then in eventual calculations their resistance can be negligible. Thus the current in
circuit can be found by the formula:
I=
ε
R+r
,
where R is external resistance of the circuit (here consumer resistance) and r is the resistance
of the power supply (internal resistance).
The resistance of the power supply r and its emf ε are characteristic quantities of the given
power supply, which are determined by its construction. ε and r don’t depend on the current
of the power supply but could depend on other operational conditions of the power supply
(temperature, aging of the power supply etc.). The voltage on the external resistance
(consumer) can be expressed:
U = I ⋅R =ε
R
.
R+r
This is always smaller than electromotive force, because R<R+r. If the ballast resistance is
greater than the resistance of the power supply (R>>r) then the voltage on the consumer and
 R

on the terminals of power supply will be equal to the electromotive force 
≈ 1 . For that
R + r

reason the electromotive force of the power supply with small resistance can be measured
quite accurately by the voltmeter with great internal resistance.
According to the definition of the electromotive force, it is known that leading the charge q
through the total circuit work will be done.
A =ε ⋅q
1
TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
Consequently the total power of the power supply is
N=
A
q
ε2
=ε ⋅ =ε ⋅I =
.
t
t
R+r
Meanwhile the power dissociated on consumer or effective power can be expressed
N1 = IU = I 2 R =
ε
2
R
.
(R + r )2
(1)
From the penultimate formula follows that the total power of the power supply N is maximum
in case of short circuit ( R → 0 ). But then total power dissociates on the resistance r of the
power supply and the effective power N1 is equal to zero (formula 1).
If the external resistance R is increasing then the total power N is decreasing and N → 0 ,
when R → ∞ . Effective power on the contrary is equal to zero in two cases: in short circuit
( R → 0 ) and in the case of the open circuit ( R → ∞ ). Consequently in case of the external
resistance, which has increased to a given value, the effective power must obtain the
maximum.
To find this resistance Rm at which the effective power N1 is highest, we first find the
derivation by the resistance using the formula (1).
Let R be marked as Rm and when the result is equal to zero, then we get:
r − Rm
∂N1
=ε2
= 0.
∂R
(r + Rm )
From here Rm = r .
Consequently the effective power is highest when the external resistance R and the
resistance of the power supply are equal. The highest effective power can be expressed:
N1m =
ε2
4r
Using power supplies it is not only the power which is significant but also efficiency.
The efficiency is defined as a ratio of total and effective power and can be found by formula
N1 UI U
=
=
N ε I ε
(2)
IR
R
=
.
I (R + r) R + r
(3)
η=
or
η=
From the formulas (1) and (3) follows that the conditions about obtaining the highest effective
power and efficiency do not coincide. If the effective power is maximal ( Rm = r ), then the
efficiency is 0,5. If the efficiency η is approaching the value “one”, the effective power N1 is
forming a part of its highest value N1m . According to the formulas (1) and (3) the effective
power and efficiency are greater in case of the same external resistance and electromotive
force in this power supply which resistance is smaller.
The scheme of the experimental machine used in this work is given on the figure 3.1.
2
TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
In this figure the power supply is constituted by the elements ε 1 and ε 2 of battery, where the
rheostat r is connected in series to increase artificially the resistance of power supply.
Two rheostats R1 and R2 in external circuit are given for smooth changing of the current in
extensive district.
Figure 3.1
4. Experimental procedure
1.
2.
3.
4.
Record the data of measuring devices.
Set up a circuit as shown in figure 3.1.
Ask the instructor to control the scheme and to give the task.
Put the sliding contacts of the rheostats so that r should be the highest, R1 and R2 should
be minimal.
5. Close the switch K and regulate the short circuit current by the rheostat r to the value
given by instructor. During the further experiment leave the sliding contact of the rheostat
r unchangeable.
6. Decrease the current with the rheostats R1 and R2 by the step given by your instructor (2-5
mA) to almost zero, registering every time readings of the voltmeter and ammeter. Record
the results into data table 3.1.
Table 3.1
The determination of power supply efficiency and effective power
Nr
I, mA
U, V
N1, mW
η%
ε −U , V
r, Ω
R, Ω
R
r
1.
...
25.
ε = ...... V
7. Open the switch K and register the reading of voltmeter, which in this case, with enough
accuracy for given experiment, is equal to total electromotive force ε of battery
consisting of two elements.
8. Present measuring results to your instructor for control and thereafter disconnect the
circuit.
3
TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
9. Find the effective power N1 (formula 1) and efficiency η (formula 2) for each measured
value (I and U). According to the computed values plot the graphs N1 = f ( I ) and
η = f ( I ) in the same axis (in both cases take the x-axis as current I-axis and use two
different scales of y-axis). Smooth the obtained graphs.
10. By the Ohm`s Law calculate for each measured value the external resistance of circuit R,
R
internal resistance r and the ratio of .
r
R
R
R
11. Plot the graphs N 1 = f   , η = f   with common   axis (x-axis) and with two
r
r
r
different y-axis. Smooth the obtained graphs.
12. Find the combined uncertainty U C ( N1 ) = ( I ⋅ ∆U ) 2 + (U ⋅ ∆I ) 2 + 2 ⋅ I ⋅ ∆U ⋅ U ⋅ ∆I
and according to the same principle find the uncertainty U C (η ) for pre-determined cases
by your instructor. (As among the internal quantities I and U there are also dominating a
functional dependence among the internal quantities U and ε , in case of which the
correlation coefficient is equal to “one”.
Look at “Methodical instruction of practical training in Physics I”, Tallinn, TPI, 1987, p.
22, formula (15) and R. Laaneots, O. Mathiesen “Measuring principles”, Tallinn, TTÜ,
2002, p. 85-90)
13. Find r and it’s A-type modifier uncertainty U A (r ) .
5. Questions and tasks
1.
2.
3.
4.
5.
Define the concept of power and efficiency.
Define the concept of electromotive force.
In what lies the disparity between the voltage and the potential difference?
Formulate the Ohm’s law about a part of circuit and about enclosed circuit.
Formulate the maximum conditions of efficiency and effective power and give an
argument.
6. How do the effective power and the efficiency of power supply depend on the external
resistance?
7. An element, which electromotive force is 1,1 V and internal resistance 10 Ω, is loaded
with external resistance 9 Ω. Find the current in circuit, the voltage on external resistance
and the voltage on internal resistance and the efficiency.
6. Literature
1. Halliday, D., Resnick, R., Walker, J. Fundamentals of Physics – 6th ed., USA, John
Wiley&Sons, Inc., 2001. § 28-2, §28-7
4