Electric Circuits and Network Theorems

Theorems, when applied to the solutions of electric networks, either simplify the network itself or render their analytical solution very easy.
These theorems can also be applied to an a.c. system, with the only difference that impedances replace the ohmic resistance of d.c. system.
Different electric circuits (according to their properties) are defined below :

Current is the rate of flow of electric charge in a circuit.
The term is often used to describe the flow of electric charge, e.g. a current is flowing in a circuit; this is ambiguous but is so common that we have to accept it.
A source supplies energy to a system.
A load accepts energy from a system.
Electric charge may be either positive or negative. Negative electrons are free to move around a circuit thus transporting energy from source to load.
To maintain a current, the source must provide a driving force called the electromotive force (e.m.f.).
The potential difference across a load indicates in volts the energy lost per coulomb of charge passing through the load.
Since the current is the rate of flow, its product with the voltage gives the rate of energy transmission, i.e. the power.
Resistance is a measure of the opposition to the flow of charge through a load.
Ohm’s law states that the ratio of voltage to current is constant, provided other physical factors such as temperature remain unchanged.
Circuit: A circuit is a closed conducting path through which an electric current either flows or is intended flow.
Parameters: The various elements of an electric circuit are called its parameters like resistance, inductance and capacitance. These parameters may be lumped or distributed.
Linear Circuit: A linear circuit is one whose parameters are constant i.e. they do not change with voltage or current.
Non-linear Circuit: It is that circuit whose parameters change with voltage or current.
Bilateral Circuit: A bilateral circuit is one whose properties or characteristics are the same in either direction. The usual transmission line is bilateral, because it can be made to perform its function equally well in either direction.
Unilateral Circuit: It is that circuit whose properties or characteristics change with the direction of its operation. A diode rectifier is a unilateral circuit, because it cannot perform rectification in both directions.
Electric Network: A combination of various electric elements, connected in any manner whatsoever, is called an electric network.
Passive Network: is one which contains no source of e.m.f. in it.
Active Network: is one which contains one or more than one source of e.m.f.
Node is a junction in a circuit where two or more circuit elements are connected together.
Branch is that part of a network which lies between two junctions.
Loop: It is a close path in a circuit in which no element or node is encountered more than once.
Mesh: It is a loop that contains no other loop within it.

Above figure has seven branches, six nodes, three loops and two meshes in the first figure, whereas the second circuit has four branches, two nodes, six loops and three meshes.
Symbol of basic electric quantities are mentioned below:

Ideal Constant Voltage Source:

It is that voltage source (or generator) whose output voltage remains absolutely constant whatever is the change in load current.
Such a voltage source must possess zero internal resistance so that internal voltage drop in the source is zero.
In that case, output voltage provided by the source would remain constant irrespective of the amount of current drawn from it.
In practice, none such ideal constant voltage source can be obtained.
However, smaller the internal resistance r of a voltage source, closer it comes to the ideal sources described above.

Ideal Constant-Current Source:

It is that voltage source whose internal resistance is infinity.
In practice, it is approached by a source which posses very high resistance as compared to that of the external load resistance.

Kirchhoff Laws

These laws are more comprehensive than Ohm’s law and are used for solving electrical networks which may not be readily solved by the latter.
Kirchhoff laws, two in number, are particularly useful (a) in determining the equivalent resistance of a complicated network of conductors and (b) for calculating the currents flowing in the various conductors

Kirchhoff Point Law or Current Law (KCL):

It states that: In any electrical network, the algebraic sum of the currents meeting at a point (or junction) is zero.
It simply means that the total current leaving a junction is equal to the total current entering that junction.
It is obviously true because there is no accumulation of charge at the junction of the network.

Kirchhoff Mesh Law or Voltage Law (KVL):

It states that: The algebraic sum of the products of currents and resistances in each of the conductors in any closed path (or mesh) in a network plus the algebraic sum of the e.m.f. in that path is zero.

Determination of Voltage Sign:

Sign of Battery E.M.F.: A rise in voltage should be given a positive sign and a fall in voltage a negative sign.
As we go from the negative terminal of a battery to its positive terminal, there is a rise in potential, hence this voltage should be given a positive sign.
If, on the other hand, we go from positive terminal to negative terminal, then there is a fall in potential, hence this voltage should be preceded by a negative sign.
Sign of the battery e.m.f. is independent of the direction of the current through that branch.

Sign of IR Drop

If we go through a resistor in the same direction as the current, then there is a fall in potential because current flows from a higher to a lower potential.
Hence, this voltage fall should be taken negative.
However, if we go in a direction opposite to that of the current, then there is a rise in voltage.
Hence, this voltage rise should be given a positive sign.
Sign of voltage drop across a resistor depends on the direction of current through that resistor but it is independent of the polarity of any other source of e.m.f. in the circuit under consideration.

The given circuit of a mesh and the equation gives an idea of writing KVL equation.