Introduction:

DC sources refer to the sources of electrical energy that are related with constant voltages and currents. The dc power supply can be constructed as an electronic circuit operating from ac mains electricity supply and designed for a purpose. Alternatively it can be obtained from a battery, with the latter being employed in portable machines and equipment where a connection to the mains ac supply is not suitable or practical. DC circuits basically contain just dc power sources and resistive elements and thus form an appropriate basis for studying the basic principles of electrical circuit analysis

Batteries:

The dc battery is ordinary nowadays. Batteries are employed in the broad range of scenarios, from the smallest applications in hearing aids and small digital watches to large heavy-duty lead acid batteries utilized in the automotive industry.

The voltage cell was discovered by Italian physicist Alessandro Volta (1745-1827), in the year 1792 throughout his work on electrolysis and the first battery as a stack of these cells in the year 1800. Nowadays, the term cell and battery are employed almost interchangeably, however many low-voltage batteries are however single voltaic cells while strictly-speaking a battery is a number of cells stacked in sequence to receive higher voltages than a single cell can offer. 
   
A battery is basically a source of dc electrical energy. It transforms stored chemical energy into electrical energy via an electrochemical procedure. This then gives a source of electromotive force or emf to allow currents to flow in electric and electronic circuits. There are fundamentally two groups of battery: disposable and rechargeable.

The disposable battery, as the name proposes, is intended for a single use merely, and hence once the energy contained in the chemical constituents of battery is transformed into electrical form then the battery is ‘used up’ and is disposed of. Such batteries are sometimes termed to as primary cells and comprise the common Zinc-Carbon (ZnC) AAA, AA, C and D cells or their corresponding alkaline Manganese-Dioxide (MnO₂) versions and also the myriad of small button cells employing Silver-Oxide (AgO), Zinc-Oxide (ZnO) or Chromium-Dioxide (CrO₂) among other materials. The second class of battery is famous rechargeable type, which has gained broadly raised usage in the past two or three decades. In this kind of battery whenever the chemical energy stored has been used up it can be substituted by a reversal of the chemical procedure via the use of electricity to ‘recharge’ it that can be done from the mains supply. Therefore the charge stored by this kind of battery can be replenished and the battery can be utilized in sequential charge and recharge cycles. Eventually though, the materials in a rechargeable battery degrade and it reaches the end of its life. Rechargeable batteries comprise the equivalent of the standard cells like Nickel-Metal Hydride (NiMH) or Nickel-Cadmium (NiCd) or the higher voltage Lithium-Ion (Li-ion) cells right up to the classic Lead-Acid (PbH₂SO₄) car battery.

The use and construction of a typical C or D type cell is explained in figure shown below. The outer metal case in the form of cylindrical container is made up of Zinc and acts as the negative electrode of cell. Its base as well serves as the negative terminal of the battery. The cylinder is filled with chemical compound that acts as an electrolyte. In current batteries this is in non-liquid form of a paste or dry compound. Positive electrode of the cell takes the form of a Carbon or Graphite rod with a metal cap that is inserted into the electrolyte in the centre of cylinder. The metal cap on the rod serves as the positive terminal of battery.

Whenever a conducting resistive load is joined between the positive and negative terminals of the battery, a closed electrical circuit is made. Under this situation a number of chemical reactions occur in the electrolyte which outcomes in the generation of positively charged ions and free negatively charged electrons in it. The positive ions migrate via the electrolyte towards the carbon rod and become deposited on it. Electrons, on the other hand, can’t migrate via the electrolyte since its chemical composition forms a barrier that inhibits the passage of electrons via it. Rather, the electrons accumulate at the negative electrode of cell. This gives increase to a potential difference among the two terminals of the battery that outcomes in an emf or electric field across the resistive load joined between them. EMF then causes the electrons to flow in external electric circuit via the load and ultimately to the positive terminal of battery. This gives increase to a continuous flow of current in the electric circuit. In the circuit shown in figure above the electrical load is the light bulb and the energy drawn from the battery by bulb is emitted as visible light. As long as closed electric circuit subsists the current continues to flow and the electrochemical procedure in the electrolyte continues with the constituent chemicals being transformed into other chemicals. Ultimately the supply of original chemicals in an electrolyte becomes depleted and the emf produced between the terminals of the battery drops, finally to zero, and the battery becomes discharged. At this phase a disposable battery is discarded, whereas a rechargeable battery will be put on a charger that reverses the electrochemical process in an electrolyte and restores charge to the battery by passing an electric current via it in the reverse direction for an adequate period of time.

Therefore it can be observe that there is a limit to the length of time for which a battery can produce electricity and consequently has a limited life-time or cycle time. The length of time for which a battery lasts is recognized by the amount of charge it stores in total and the rate at which this charge is utilized, that in turn depends on the magnitude of current drawn from it. A battery will last longer whenever a low value of current is drawn from it than it will if a high value of current is demanded. This is point out in figure shown below, where the terminal voltage of a battery is plotted against time for distinct values of current drawn from it with I₄ > I₃ > I₂ > I₁.

Figure: Discharge Profile of a Battery at Different Currents

The operating life-time (or disposable) or cycle time (or rechargeable) depends basically on the amount of charge it stores in electrolyte that can be transformed into free electrons to give the current in an electric circuit. One might then suppose this battery capacity to be expressed as a quantity of charge in Coulombs. Though, in practice, it proves more helpful to express the battery capacity in terms of product of current (in Amperes) and time (in hrs). Battery capacity is thus expressed in the units of Ampere-hours (Ahr). This permits the efficient lifetime of the battery to be computed for various levels of current drawn from it as indicated in table shown below.

Table: Battery Lifetime vs. Current Drawn:

This is as well significant, though, to realise that in practice there is a maximum current that a battery is capable to deliver and this should also be taken into account when selecting an appropriate battery for a specific application. For illustration, the 1 Ahr battery of table above might not be capable to deliver a current as high as 5A due to the restrictions of its chemistry and in this case could not be employed in a scenario where this level of current is demanded, even for short period of 12 minutes.

The Ideal Voltage Source:

The symbol already employed for a dc battery which is employed for an ideal dc voltage source as shown in figure below. The emf of an ideal battery is the sum of cell voltages that are stacked to obtain a higher voltage than a single cell can offer. The voltage evaluated between the terminals of the battery is the output voltage, VO. The load joined to the battery is shown as a single resistor, RL that could of course symbolize the equivalent resistance of a more complex resistive configuration. The current drawn from the voltage source and flowing via the load resistance is labelled, IL. An ideal voltage source is one that gives a constant output voltage despite of the load placed on it.

Figure: An Ideal Voltage Source Driving a Resistive Load.

The definitive feature of the ideal voltage source is thus:

V_o = E    ∀ R_L

That is, the output or terminal voltage of the battery as evaluated between its positive and negative terminals is always the internal collective cell voltage, E. As the output voltage of battery, V_O, is in this case similar to the voltage across the single load resistor V_L, then from Ohm’s law we encompass:

I_L = V_L/R_L = E/R_L   ∀

This exhibits that the current via the load is a function of resistance, R_L, with the voltage across load being independent of it. This implies that the source is able of providing whatsoever current is demanded of it. This in turn recommends that when a ‘short-circuit’ is positioned across the source with R_L= 0 then the current would be limitless with I_L → ∞. Clearly, a condition like this can’t prevail in reality. For illustration, if a piece of heavy-duty conducting cable were positioned across a 12V Lead-acid car battery, the battery would quickly overheat, vent Hydrogen gas, melt and possibly blow up. Thus, the concept of a short-circuit load is mainly a theoretical one to be employed only on paper for the aims of circuit analysis. Though, there are scenarios in practice where electronic equipment should be protected against damage in the event of a short-circuit happening inadvertently or unintentionally.

The Ideal Current Source:

This is sometimes essential to generate a defined and constant value of current to drive a circuit or load instead of a constant voltage. This is termed as a current source; the most general symbol for which it is used is the double overlapping circles shown in figure below. Note that the direction of the current produced to flow out of the terminals of the source should be indicated in some manner, generally by a directed arrow. Current sources do not take place naturally in cell form such as batteries and are constructed by using electronic circuits that are in turn powered from a voltage source.

Figure: An Ideal Current Source Driving a Resistive Load

The definitive feature of the ideal current source is that:

I_L = I   ∀ R_L

That is, the current that flows out of positive terminal of the current source, around the circuit via the load resistor, R_L, and back into the negative terminal of source is always equivalent to the nominal value of the current source, I. This value is independent of the value of load resistance, R_L. The voltage developed across load, V_L, is given by the Ohm’s law as:

V_L = I_LR_L = IR_L   ∀ R_L

This exhibits that the voltage across the load that is as well the voltage that is developed across the current source itself, is a function of resistance, RL.

A battery charger is a good working illustration of a current source. Current source is powered from mains electricity and the user sets the value of constant current whereas the battery to be charged forms the load as stated by the set-up shown in figure below. The voltage developed across the terminal of current source will adjust itself to be equivalent to the battery voltage.

Figure: A Constant Current Source employed to Charge a Battery

Non-Ideal Voltage Source:

In practice a voltage source is not perfect and does not give unlimited current. Whenever the battery or voltage source is not joined to a load, the voltage between its terminals is termed to as its open-circuit terminal voltage, VOC, and is basically similar as the cell voltage, E. Though, whenever a load is joined to the source, the terminal voltage drops as the current is drawn from it and hence:

V_o < E or V_o <  V_oc

This consequence can be examined in the curves shown in figure above where the voltage available from the battery is little lower than the open circuit voltage, VOC, and the drop in voltage becomes much pronounced as the current drawn from the battery is raised. This consequence can be modelled by attributing an internal or source resistance, RS, to non-ideal voltage source. This can then be symbolized as an ideal voltage source producing the cell voltage, E, with an internal source resistance, RS joined in series with the ideal source and its output terminals as shown in figure below.  In this situation, the current drawn from the supply flows via the internal source resistance, RS, giving mount to a potential drop across it, VS. In this condition by Kirchhoff’s Law:

V_O = E - V_S

However from Ohm’s Law:

V_S = I_LR_S

Figure: A Non-Ideal Voltage Source Driving a Resistive Load.

So that:

V_O = E - I_LR_S

Also note that for the load:

V_L = I_LR_L

From the relation for resistors joined in series we encompass:

I_L = E/(R_L + R_S)

And hence finally:

V_L = [R_L/(R_L + R_S)]E

This exhibits that there is basically a potential divider action among the internal resistance of the voltage source, RS, and the resistance of load, RL, with similar current flowing via both resistances. This has the consequence of decreasing the efficient output voltage of the battery.

The Non-Ideal Current Source:

In a similar way in practice, a current source is not perfect. The output current given by a non-ideal current source differs slightly with a change in the load resistance joined to it. This result can be modelled by attributing an internal resistance to the current source in a similar way to the non-ideal voltage source. Though, the internal resistance is joined across the ideal current source in this case instead of in series with it as shown in figure below:

Figure: Non-Ideal Current Sources Driving a Resistive Load

In the case of non-ideal current source, the internal resistance, RS, is very higher than that in the case of non-ideal voltage source. The consequence of the internal resistance in the non-ideal current source is to shunt some of the current produced by the ideal current source, I, and hence the current that flows via the load, IL, is less than the ideal value. In this situation:

I_L < I

The degree of drop in an output current from ideal value based on the value of load resistance, R_L, by comparison with the internal source resistance, R_S. When Kirchhoff’s Current Law is applied to the positive output terminal of current source we encompass:

I = I_S + I_L

From prior work on current splitting among resistors in parallel:

I_L = [R_S/(R_L + R_S)] I

This exhibits that there is essentially a current splitting action among the internal source resistance, R_S and the load resistance, R_L.

Also note that for the load:

V_L = I_LR_L

And hence,

V_L = [(R_SR_L)/(R_S + R_L)]I

Energy Expenditure and Power Dissipation:

In the circuits above, load resistance, R_L, symbolizes the electrical equivalent of some form of load that demands or employs energy. For illustration, whenever the bulb in a torch powered by batteries lights up, the electrical energy is drawn from batteries and converted into light. This employs up the energy stored in batteries and the rate at which the energy is depleted based on the brightness of bulb, frequently termed as its wattage. The question is simply what power or energy does the electrical load dissipate. When it is recalled, that power dissipated is the rate at which the energy is expended per unit time and then:

Power = Energy/Time = (Energy/Charge) x (Charge/Time) = Voltage x Current

The unit of Energy is Joule (J), termed after the English physicist James Prescott Joule (1818-89), who introduced the First Law of Thermodynamics. The unit of power is Watt (W), named after a Scottish mechanical engineer and developer of the steam engine James Watt (1736-1819). Then for a resistive element in an electric circuit with potential drop, V across it and a current, I flowing via it we have:

P = VI

However from Ohm’s Law we remember:

V = IR or I = V/R

And hence,

P = VI = I²R = V²/R

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