Introduction:

So far electricity has been considered mainly in its own right in terms of atomic charge, electric field, potential and current as the flow of charge. Though, electricity doesn’t appear in daily life in an abstract form however is harnessed and employed to provide a source of energy, generally with the aim of transforming this energy into another form like mechanical energy. In this context it is employed to do work as whenever used in domestic or industrial equipment or machines. Alternatively, on a lower scale of energy, it can be employed in electronic engineering to provide the source of power required to control semiconductor devices and integrated circuits employed in so much of today’s instrumentation, communication and computing the applications.

In this regard, electricity is employed to provide the electromotive force needed to allow current to flow via electrical loads of different forms. This in turn needs an electric or electronic circuit to join the source of emf and the load together in a loop and hence current can circulate between the two. This essentially comprises the formation of an electric circuit. The electricity can be exploited in two forms in electric circuits. The first of such is direct current, generally termed as dc, where the source voltage or emf is constant in polarity and magnitude, as is the current flowing in the circuit beneath steady-state conditions. The second is alternating current, generally termed as ac, where the direction of source emf and the resultant current that flows continually reverses direction and differs instantaneously in magnitude. The simplest of these to understand is dc and the basic circuit laws are more simply assimilated in context of dc circuits. Thus dc circuits will be considered and analysed first and afterwards attention will be turned to the ac circuits.

Ohm’s Law:

Figure below shows the simplest form of dc electric circuit where a dc battery, that will be examined in more detail afterward, is employed as the source of emf or voltage E, to drive current, I, via the load in the form of resistor, R, developing the potential difference across it, measured as the voltage, V.

It was German physicist Georg Ohm (1789 - 1854) who first formally explained the relationship among the voltage developed across a load and the current that flows via the load. The law that bears his name is the most fundamental laws governing electric circuits and was first formally published in the year1827, almost two hundred years ago.

Ohm’s Law defines that: ‘the potential difference that is developed across a conductor at any point in an electric circuit is proportional to current flowing via the conductor at that point and the resistance of conductor is the constant of proportionality’.

Figure: A dc Battery Connected to a Resistive Load

Ohm’s Law is formally written as:

V = IR where, V is in volts or
I = V/R Where I in Amps or
R = V/I Where R in Ohms

Note: The battery voltage E is the source of emf that gives the electric force to drive current around the circuit. The voltage V is the potential difference, frequently termed to as the voltage drop, across the resistor as an outcome of its resistance to current flow. The emf E exists in its own right as a source of energy, for illustration as the terminal voltage of a charged battery. Though, the potential drop, V, exists only as an outcome of the current flowing in the circuit that in turn is caused by the emf, E. When there was no emf present, no current would flow in the circuit and there would be no potential drop across the resistor, R. In this situation, as there is only a single resistor as the load in circuit, the full emf emerges across this resistor as a potential drop and E = V. In a more complicated circuit this would not essentially be the case.

Note also that word ‘point’ employed in the statement of the law above would more precisely be term ‘branch’ or ‘arm’ in a circuit as it doesn’t refer to a single node however rather a conducting element joined at a location in a circuit. 

Figure below shows Ohm’s law as a linear relationship for different values of resistance R, where R₁ > R₂ > R₃ > R₄. Note that the higher the value of resistance, the lower the value of current that flows via it for the same potential drop across it.

Figure: Ohm’s Law Plotted for a Range of Values of Resistance

Kirchhoff’s Circuit Laws:

Subsequent to Ohm’s Law in the basic rules that govern the behaviour of electric circuits are Kirchhoff’s Circuit Laws. These were formulated by German physicist Gustav Kirchhoff (1824 – 1887), published in the year 1845. They are termed to as his circuit laws as he contributed laws in other related fields as well, like Radiation and Electrochemistry. He formulated two circuit laws, one which fundamentally establishes the conservation of charge and the other that establishes the conservation of potential.

Kirchhoff’s Current Law:

Kirchhoff’s first law, termed as Kirchhoff’s Current Law, KCL, or sometimes as Kirchhoff’s Junction Rule, basically expresses the conservation of charge, that can be thought of as the conservation of matter when the charge is considered as a quantity of charged particles. This implies that charge can’t appear from nothing at any point in a circuit, neither can it vanish into oblivion at any point.

Kirchhoff’s Current Law defines that ‘the total sum of currents flowing at a node in an electric circuit is zero’.

_NΣⁿ⁼¹ I_n = 0

In modern terms this is re-stated as:

Kirchhoff’s Current Law defines that ‘the total sum of currents flowing into a node in an electric circuit is equivalent to the sum of currents flowing out of that node’.

_ni=1Σ^Ni I_ni = _no=1Σ^No I_no

In employing this law in circuit analysis it is necessary to adopt a consistent sign convention with regard to the polarity of currents. The normal and most consistent convention is that currents flowing into the node are considered as positive whereas currents flowing out of the node are considered as negative. The diagram in figure below shows some currents at a single node in a circuit. The dot symbolizes the node while the arrows are used both to represent wires connected to the node and the direction of the current carried by each wire into or out of the node. All currents are individually labelled, Ii that is intended to point out the magnitude of the current whereas the arrow head denotes the direction of flow.

Figure: A Circuit Node with Several Associated Currents

Kirchhoff’s Current Law gives:

_NΣⁿ⁼¹ I_n = 0 Or I₁ + I₂ - I₃ - I₄ + I₅ - I₆ = 0

Or in its alternative form provides:

_ni=1Σ^Ni I_ni = _no=1Σ^No I_{no Or} I₁ + I₂ + I₅ = I₃ + I₄ + I₆

Note that both forms are fully mathematically consistent.

Wires are employed in a circuit to join points of the same potential altogether and are considered to be perfect conductors containing no resistivity and thus no potential drop along them whenever carrying current. Thus wires can be employed to enlarge the position of a node for visual improvement in a circuit diagram although without any effect on the exclusive potential at the node. The schematic layout in figure below exhibits this where all the points in this diagram are at same potential as the single nodal point shown in figure above.

Figure: An Equivalent Representation of the Node of figure above

Note that in analysing a circuit, the labels and directions of currents are frequently assigned randomly. Though, Kirchhoff’s Law should be applied to the analysis constantly with the assignment. Then any value of current computed which works out to be negative simply points out that in practice, the current is really flowing in a direction opposite to that assigned in the schematic diagram of circuit.

Kirchhoff’s Voltage Law:

Kirchhoff’s second circuit law, termed as Kirchhoff’s Voltage Law, KVL, or many times as Kirchhoff’s Loop Rule, basically formulates the conservation of energy in form of electric potential around a circuit in which the current is flowing. This signifies that no net voltage can be formed or destroyed around the loop of closed circuit.

Kirchhoff’s Voltage Law defines that ‘the total sum of potentials around a closed electric circuit is zero’

_n=1Σ^N V_n = 0

In modern terms this can be re-stated as:

_nEMF=1Σ^NEMF E_nEMF = _nPD=1Σ^NPD V_nPD

Kirchhoff’s Voltage Law defines that ‘the sum of the emfs around a closed electric circuit is equivalent to the sum of potential drops around similar circuit’

This basically signifies that the total emf in a closed circuit, that might be the sum of number of emfs at various locations and of different polarities and magnitudes, must equivalent the sum of potential differences produced across all conducting elements due to current circulating around the circuit.

Figure below exhibits the application of Kirchhoff’s Voltage Law to a simple circuit. Since with the first law, it is necessary to have agreed conventions with regard to polarities and directions of potentials and to apply such conventions consistently. The simplest way to accomplish this is to assign a direction to the total current considered to be flowing around the circuit loop. This could be for illustration either anticlockwise or clockwise, with the choice being random. This is as well frequently convenient to nominate certain point in the circuit as a reference ground or 0V point and to consider the loop as starting and ending at this reference point.

Following this, the potentials, both potential drops and emfs across conducting elements, should also be assigned around the loop according to convention. The polarities of batteries serving as emfs are intrinsically determined by the orientation of their terminals in connection to the circuit. Long-terminal is the positive side and short terminal is the negative side of battery.

Emfs in red, are summed around the loop of circuit as being positive whenever they support the direction of assigned current flow (that is, tend to produce a current in this direction) and as negative whenever they oppose the direction of assigned current flow (that is, whenever they tend to produce a current flowing in the opposite direction).

Potential drops can be considered as the losses of overall emf in the loop by manner of its distribution among the conducting elements. Convention is to assign all potential drops, shown in blue, across conducting elements in a polarity consistent with assigned direction of the current flowing via them. In this situation they all will be treated as negative potentials around the circuit loop as their polarity emerges as opposite to that of the direction of current flow. Though, when the second form of Kirchhoff’s Voltage Law is employed they will emerge as positive potentials on the right hand side of the summation equation, and hence both forms are mathematically consistent. The labels En and Vn are intended to symbolize the magnitude of the related voltages or potentials.

Figure: Kirchhoff’s Voltage Law applied to Closed Circuit

Kirchhoff’s Voltage Law offers:

_n=1Σ^N V_n = 0  Or E₁ - V₁ - E₂ - V₂ - E₃ + E₄ - V₃ = 0

Or in its different form offers:

_nEMF=1Σ^NEMF E_nEMF = _nPD=1Σ^NPD V_nPD  Or E₁ - E₂ - E₃ + E₄ = V₁ + V₂ + V₃

Note that, merely as with Kirchhoff’s Current Law, consistency should be exercised whenever applying Kirchhoff’s Voltage Law to the closed circuit. Though, in more complicated circuits a particular conducting element might emerge as branch of more than one closed loop and in this situation the potential drop might appear in one loop in an inconsistent direction with that of assigned current flow. Nevertheless, only one direction can be allocated to a given potential and this principle should be adhered to. Whenever this is the case, this potential drop might be computed as a negative value that simply signifies that its real polarity in practice is opposite to that assigned for the aim of analysis.

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