1E6 Electrical Engineering
DC Circuit Analysis
Lecture 5: DC Electric Sources, Energy and Power
5.1 Introduction
DC sources refer to sources of electrical energy which are associated with
constant voltages and currents. A dc power supply can be constructed as an
electronic circuit operating from the ac mains electricity supply and designed for
purpose. Alternatively it can be obtained from a battery, with the latter being
used in portable equipment and machines where a connection to the mains ac
supply is not convenient or practical. DC circuits essentially contain only dc
power sources and resistive elements and therefore form a suitable basis for
studying the fundamental principles of electrical circuit analysis
5.2 Batteries
The dc battery is commonplace today. Batteries are used in the widest
range of scenarios, from the smallest applications in hearing aids and small
digital watches to the large heavy-duty lead acid batteries used in the automotive
industry.
The voltage cell was invented by Alessandro Volta (1745-1827), an Italian
physicist in 1792 during his work on electrolysis and the first battery as a stack
of such cells in 1800. Today, the term cell and battery are used almost
interchangeably, but many low-voltage batteries are in fact single voltaic cells
while strictly-speaking a battery is a number of cells stacked in series to obtain
higher voltages than a single cell can provide.
A battery is essentially a source of dc electrical energy. It converts stored
chemical energy into electrical energy through an electrochemical process. This
then provides a source of electromotive force or emf to enable currents to flow in
electric and electronic circuits. There are basically two classes of battery,
disposable and rechargeable.
The disposable battery, as the name suggests, is intended for a single use
only, so that once the energy contained within the chemical constituents of
battery is converted into electrical form then the battery is ‘used up’ and is
disposed of. These batteries are sometimes referred to as primary cells and
include the common Zinc-Carbon (ZnC) AAA, AA, C and D cells or their
equivalent alkaline Manganese-Dioxide (MnO2) versions as well as the myriad of
small button cells using Zinc-Oxide (ZnO), Silver-Oxide (AgO) or ChromiumDioxide (CrO2) among other materials. The second class of battery is the wellknown rechargeable type, which has gained widely increased usage in the past
two or three decades. In this type of battery when the chemical energy stored
1
has been used up it can be replaced by a reversal of the chemical process
through the use of electricity to ‘recharge’ it which can be done from a mains
supply. Thus the charge stored by this type of battery can be replenished and the
battery can be used in sequential charge and recharge cycles. Eventually
however, the materials in a rechargeable battery degrade and it reaches the end
of its life. Rechargeable batteries include the equivalent of the standard cells
such as Nickel-Cadmium (NiCd) or Nickel-Metal Hydride (NiMH) or the higher
voltage Lithium-Ion (Li-ion) cells right up to the classic Lead-Acid (PbH2SO4)
car battery.
The construction and use of a typical C or D type cell is illustrated in
Fig.1. The outer metal case in the form of a cylindrical container is made of Zinc
and acts as the negative electrode of the cell. Its base also serves as the negative
terminal of the battery. The cylinder is filled with a chemical compound which
acts as an electrolyte. In modern batteries this is in non-liquid form of a paste or
dry compound. The positive electrode of the cell takes the form of a Carbon or
Graphite rod with a metal cap which is inserted into the electrolyte in the centre
of the cylinder. The metal cap on the rod serves as the positive terminal of the
battery.
Fig. 1 The Construction and Operation of a Battery
When a conducting resistive load is connected between the positive and
negative terminals of the battery a closed electrical circuit is formed. Under this
condition a number of chemical reactions take place in the electrolyte which
results in the generation of positively charged ions and free negatively charged
electrons within it. The positive ions migrate through the electrolyte towards the
carbon rod and become deposited on it. The electrons, on the other hand, cannot
migrate through the electrolyte because its chemical composition forms a
barrier which inhibits the passage of electrons through it. Instead, the electrons
2
accumulate at the negative electrode of the cell. This gives rise to a potential
difference between the two terminals of the battery, which results in an emf or
electric field across the resistive load connected between them. The emf then
causes the electrons to flow in the external electric circuit through the load and
finally to the positive terminal of the battery. This gives rise to a continuous flow
of current in the electric circuit. In the circuit shown in Fig. 1 the electrical load
is the light bulb and the energy drawn from the battery by the bulb is emitted as
visible light. As long as the closed electric circuit exists the current continues to
flow and the electrochemical process in the electrolyte continues with the
constituent chemicals being converted into other chemicals. Eventually the
supply of original chemicals in the electrolyte becomes depleted and the emf
generated between the terminals of the battery drops, ultimately to zero, and the
battery becomes discharged. At this stage a disposable battery is discarded,
while a rechargeable battery will be placed on a charger which reverses the
electrochemical process in the electrolyte and restores charge to the battery by
passing an electric current through it in the reverse direction for a sufficient
period of time.
Thus it can be seen that there is a limit to the length of time for which a
battery can generate electricity and consequently has a limited lifetime or cycle
time. The length of time for which a battery lasts is determined by the amount of
charge it stores in total and the rate at which this charge is used, which in turn
depends on the magnitude of the current drawn from it. A battery will last
longer when a low value of current is drawn from it than it will when a high
value of current is demanded. This is indicated in Fig. 2, where the terminal
voltage of a battery is plotted against time for different values of current drawn
from it with I4 > I3 > I2 > I1.
Terminal
Voltage (V)
VOC
I1
I2
I3
I4
Time (hrs)
Fig. 2 The Discharge Profile of a Battery at Different Currents
3
The operating lifetime (disposable) or cycle time (rechargeable) depends
essentially on the amount of charge it stores in the electrolyte, which can be
converted into free electrons to provide the current in an electric circuit. One
might then expect this battery capacity to be expressed as a quantity of charge in
Coulombs. However, in practice, it proves more useful to express the battery
capacity in terms of the product of current (in Amperes) and time (in hours).
Battery capacity is therefore expressed in units of Ampere-hours (Ahr). This
allows the effective lifetime of the battery to be calculated for different levels of
current drawn from it as indicated in Table 1.
Table 1 Battery Lifetime vs Current Drawn
Battery Capacity
Current Drawn
Lifetime
10 Ahr
10 A
1 hr
10 Ahr
1A
10 hr
10 Ahr
20 A
30 mins
10 Ahr
0.25A
40 hrs
1 Ahr
1A
1hr
1 Ahr
5A
12 mins
1 Ahr
100 mA
10 hr
It is also important, however, to realise that in practice there is a
maximum current which a battery is able to deliver and this must also be taken
into account when choosing a suitable battery for a particular application. For
example, the 1 Ahr battery of Table 1 may not be able to deliver a current as
high as 5A due to the limitations of its chemistry and in this case could not be
used in a scenario where this level of current is demanded, even for the short
period of 12 mins.
5.3 The Ideal Voltage Source
The symbol already used for a dc battery is used for an ideal dc voltage source
as shown in Fig. 3. The emf of an ideal battery is the sum of the cell voltages
which are stacked to obtain a higher voltage than a single cell can provide. The
voltage measured between the terminals of the battery is the output voltage, VO.
The load connected to the battery is shown as a single resistor, RL, which could
of course represent the equivalent resistance of a more complicated resistive
configuration. The current drawn from the voltage source and flowing through
the load resistance is labelled, IL. An ideal voltage source is one which provides a
constant output voltage regardless of the load placed on it.
4
IL
E
VO
RL
VL
Fig. 3 An Ideal Voltage Source Driving a Resistive Load.
The definitive characteristic of the ideal voltage source is therefore:
VO = E
∀ RL
That is, the output or terminal voltage of the battery as measured between its
positive and negative terminals is always the internal collective cell voltage, E.
Since the output voltage of the battery, VO, is in this case identical to the voltage
across the single load resistor VL, then from Ohm’s law we have:
IL =
VL
E
=
RL RL
∀RL
This shows that the current through the load is a function of the resistance, RL,
with the voltage across the load being independent of it. This implies that the
source is capable of providing whatever current is demanded of it. This in turn
suggests that if a ‘short-circuit’ is placed across the source with RL= 0 then the
current would be unlimited with IL → ∞. Clearly, a situation like this cannot
prevail in reality. For example, if a piece of heavy-duty conducting cable were
placed across a 12V lead-acid car battery, the battery would rapidly overheat,
vent Hydrogen gas, melt and possibly explode. Therefore, the concept of a shotcircuit load is primarily a theoretical one to be used only on paper for the
purposes of circuit analysis. However, there are scenarios in practice where
electronic equipment must be protected against damage in the event of a shortcircuit occurring unintentionally or inadvertently.
5
5.4 The Ideal Current Source
It is sometimes necessary to generate a defined and constant value of
current to drive a circuit or load rather than a constant voltage. This is known
as a current source, the most common symbol for which used is the double
overlapping circles shown in Fig. 4. Note that the direction of the current
generated to flow out of the terminals of the source must be indicated in some
manner, usually by a directed arrow. Current sources do not occur naturally in
cell form like batteries and are constructed using electronic circuits which are in
turn powered from a voltage source.
IL
I
VO
RL
VL
Fig. 4 An Ideal Current Source Driving a Resistive Load.
The definitive characteristic of the ideal current source is that:
IL = I
∀RL
That is, the current which flows out of the positive terminal of the current
source, around the circuit through the load resistor, RL, and back into the
negative terminal of the source is always equal to the nominal value of the
current source, I. This value is independent of the value of the load resistance,
RL. The voltage developed across the load, VL, is given by Ohm’s law as:
VL = I L R L = IR L
∀RL
This shows that the voltage across the load, which is also the voltage which is
developed across the current source itself, is a function of the resistance, RL.
A battery charger is a good working example of a current source. The
current source is powered from mains electricity and the user sets the value of
the constant current while the battery to be charged forms the load as illustrated
by the set-up shown in Fig. 5. The voltage developed across the terminal of the
current source will adjust itself to be equal to the battery voltage.
6
L
IL
battery
current
source
N
E
powered
from
mains
supply
VO
I
Fig. 5 A Constant Current Source Used to Charge a Battery
5.5 The Non-Ideal Voltage Source
In practice a voltage source is not ideal and does not provide unlimited current.
When the battery or voltage source is not connected to a load, the voltage
between its terminals is referred to as its open-circuit terminal voltage, VOC, and
is essentially the same as the cell voltage, E. However, when a load is connected
to the source, the terminal voltage drops as current is drawn from it so that:
VO < E
or
VO < VOC
This effect can be observed in the curves shown in Fig. 2 where the voltage
available from the battery is slightly lower than the open circuit voltage, VOC,
and the drop in voltage becomes more pronounced as the current drawn from
the battery is increased. This effect can be modelled by attributing an internal or
source resistance, RS, to the non-ideal voltage source. This can then be
represented as an ideal voltage source generating the cell voltage, E, with an
internal source resistance, RS connected in series with the ideal source and its
output terminals as shown in Fig. 6. In this case the current drawn from the
supply flows through the internal source resistance, RS, giving rise to a potential
drop across it, VS. In this case by Kirchhoff’s Law:
VO = E − VS
But from Ohm’s Law:
VS = I L R S
7
non-ideal voltage
source
RS
IL
VO
E
RL
VL
Fig. 6 An Non-Ideal Voltage Source Driving a Resistive Load.
so that:
VO = E − I L R S
Note also that for the load:
VL = I L R L
From the relation for resistors connected in series we have:
IL =
E
RL + RS
So that finally:
VL =
RL
E
RL + RS
This shows that there is essentially a potential divider action between the
internal resistance of the voltage source, RS, and the resistance of the load, RL,
with the same current flowing through both resistances. This has the effect of
reducing the effective output voltage of the battery.
8
5.6 The Non-Ideal Current Source
In a similar manner in practice, a current source is not ideal. The output
current provided by a non-ideal current source varies slightly with a change in
the load resistance connected to it. This effect can be modelled by attributing an
internal resistance to the current source in a similar manner to the non-ideal
voltage source. However, the internal resistance is connected across the ideal
current source in this case rather than in series with it as shown in Fig. 7.
non-ideal current
source
IL
IS
I
VO
RS
RL
VL
Fig. 7 A Non-Ideal Current Source Driving a Resistive Load
In the case of the non-ideal current source the internal resistance, RS, is
much higher than that in the case of the non-ideal voltage source. The effect of
the internal resistance in the non-ideal current source is to shunt some of the
current generated by the ideal current source, I, so that the current which flows
through the load, IL, is less than the ideal value. In this case:
IL < I
The degree of drop in the output current from the ideal value depends on the
value of the load resistance, RL, by comparison with the internal source
resistance, RS. If Kirchhoff’s Current Law is applied to the positive output
terminal of the current source we have:
I = IS + I L
From previous work on current splitting between resistors in parallel:
IL =
RS
I
RS + RL
9
This shows that there is essentially a current splitting action between the
internal source resistance, RS and the load resistance, RL.
Note also that for the load:
VL = I L R L
so that:
VL =
R SR L
I
RS + RL
5.7 Energy Expenditure and Power Dissipation
In the circuits above the load resistance, RL, represents the electrical
equivalent of some form of load which demands or uses energy. For example,
when the bulb in a torch powered by batteries lights up, electrical energy is
drawn from the batteries and converted into light. This uses up the energy
stored in the batteries and the rate at which the energy is depleted depends on
the brightness of the bulb, often referred as its wattage. The question is simply
what energy or power does the electrical load dissipate. If it is recalled that
power dissipated is the rate at which energy is expended per unit time, then:
Power =
Energy Energy Ch arg e
=
×
= Voltage × Current
Time
Ch arg e
Time
The unit of Energy is the Joule (J), called after the English physicist James
Prescott Joule (1818-89), who discovered the first law of thermodynamics. The
unit of power is the Watt (W), named after James Watt (1736-1819), a Scottish
mechanical engineer and developer of the steam engine. Then for a resistive
element in an electric circuit with a potential drop, V across it and a current, I
flowing through it we have:
P = VI
But from Ohm’s Law we recall:
V = IR
or
I=
so that:
2
V
P = VI = I 2 R =
R
10
V
R