Monday, 25 July 2016

networks


Introduction of Network


Electric Network: A physical electric network or electrical circuit, is a system of interconnected components. The word components include source energy such as voltage source or current source, electrical elements such as resistors, inductors and capacitors.
  • Circuit elements can be classified in two categories, passive elements and active elements.
  • Passive Element: The element which receives energy (or absorbs energy) and then either converts it into heat (R) or stored it in an electric (C) or magnetic (L ) field is called passive element.
    • Transformer is an example of passive element.
  • Active Element: The elements that supply energy to the circuit is called active element.
    • Examples of active elements include voltage and current sources, generators, and electronic devices that require power supplies.  A transistor is an active circuit element, meaning that it can amplify power of a signal.

A circuit which contains at least one source of energy is called active. An energy source may be a voltage or current source.

A circuit which does not contain any energy source is called passive circuit.

Charge: There are two types of charge: positive (corresponding to a proton), and negative (corresponding to an electron). Electric charge is conservative. It can not be created or destroyed.

Current: The idea of “transfer of charge” or “charge in motion” is vital importance to us in studying electric circuits because, in moving a charge from place to place, we may also transfer energy from one point to another.
  • Electric current (i) = dq/dt, where i = current in amperes, q = charge in coulombs, t = time in seconds.
  • The unit of ampere can be derived as 1 A = 1C/s.
  • A direct current (dc) is a current that remains constant with time.
  • An alternating current (ac) is a current that varies sinusoidally with time.

Voltage: Voltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V).
  • Electric potential difference (voltage) between two points can be defined as the work dw done by the electric field in moving a small amount of positive charge dq from one point to other.
  • v= dw/dq, where v is voltage in volts, energy in joules, q= charge in coulombs
Power & Energy: If one joule of energy is expended in transferring one coulomb of charge through the device in one second, then the rate of energy transfer is one wait. The absorbed power must be proportion both to the number of coulomb transferred per second (current), and to the energy needed to transfer one coulomb through the element (voltage).
  • Power is the time rate of expending or absorbing energy, measured in watts (W)
  • p=dw/dt=v.i
  • The law of conservation of energy ∑ p = 0
  • Energy is the capacity to do work, measured in joules (J)

Symbol and Units of Electrical quantities:


The Classification of Network:
  • A circuit or network whose parameters i.e., elements like resistances, inductances and capacitances are always constant irrespective of the change in time, voltage, temperature etc., is known as linear network. The Ohm’s law can be applied to such network.
  • A circuit whose parameters change their values with change in time, temperature, voltage etc., is known as non-linear network. The Ohm’s law may not be applied to such network.
  • A circuit whose characteristics, behaviour is same irrespective of the direction of current through various elements of it, is called bilateral network.
  • A circuit whose operation, behaviour is dependent on the direction of the current through various elements is called unilateral network.
  • A network in which all the network elements are physically separable is known as lumped network.
  • Most of the electric networks are lumped in nature, which consists elements like RLC voltage source etc.
  • A network in which the circuit elements like resistance, inductance etc., are not physically separable for analysis purposes, is called distributed network.

Circuit Elements
Resistor: Resistance is the property of the material by which it opposes the flow of current through it.
R = ρ L / A
Where, l = Length in metre, A = Cross-sectional area in square-metre, ρ = Resistivity in ohm-metre, R = Resistance in ohm.

  • 4.186 joule = 1cal
  • 1 joule = 0.24 cal

Thus, unit 1 ohm can be defined as that resistance of the circuit if it develops 0.24 cal of heat, when one ampere current flow through the circuit for one second.

Inductor: An inductance is the passive element in which energy is stored in the form of electromagnetic field.
  • An inductor consists of a coil of conducting wire.
  • Inductance (L) is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H).
  • The Inductance L is a measure of the voltage drop across an inductor (e.g. a solenoid) per the rate change of current.
V= – L ΔI/Δt
  • Since the magnetic field inside a solenoid is proportional to a current, the rate-change of the magnetic flux is proportional to the rate-change of the current.
  • The magnetic field inside a solenoid is: B = μ0 (N/ l) I
ΔB/Δt =  μ0 (N/ l) ΔI /Δt
  • The voltage across the solenoid is given by the rate-change of the flux times the number of loops:
V=-NA ΔB/Δt = -μ0 l (N/ l)2 A ΔI/Δt
  • Therefore, the inductance L can be given in terms of the geometry of the solenoid.
L= μ0 l (N/ l)2 A = μ0 N2A/l

where, μ0 = Permeability of the core, N = Total number of turns in coil, A = Area of cross-section of coil, and l = Length of the coil in metre.

Capacitor: It is a passive element in which energy is stored in the form of an electrostatic field.
  • A capacitor consists of two conducting plates separated by an insulator (or dielectric).
  • Capacitance (C) of the capacitor is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F).
C = ε A/d
where A is the surface area of each plate, d is the distance between the plates, and is the permittivity of the dielectric material between the plates.
  • Capacitors are used extensively in electronics, communications, computers, and power systems.

Gyrator:
An ideal gyrator is a two port network whose input (respectively output) voltage (respectively output) is directly proportional to the output (respectively input) current. The ratio α is usually called the “gyration resistance”.
In the case of an asymmetrical representation, a gyrator is illustrated by:

The relations between current and voltage are : V1= α i2, and V2= -α i1

Transformer: A transformer is a static electrical device that transfers energy by inductive coupling between its winding circuits. A varying current in the primary winding creates a varying magnetic flux in the transformer’s core and thus a varying magnetic flux through the secondary winding. This varying magnetic flux induces a varying Electromotive Force (EMF), or voltage, in the secondary winding.

Current-voltage relationship for a capacitor: i = C dv/dt

Energy Sources: Energy source is defined as the device that generates electrical energy. The classification of energy sources is given below.
  • Independent Sources: Independent sources are those in which source voltage or current are not dependent on a voltage or current.
  • Ideal Voltage Source: It is a voltage generator whose output voltage remains absolutely constant whatever be the value of the output current.
  • Practical Voltage Source: The voltage does not remain constant but falls slightly; this is taken care of by connecting a small resistance (Rs) in series with ideal source.
  • Ideal Current Source: It produces a constant current value of the irrespective of the voltage across it.
  • Practical Current Source: The output current does not remain constant but decreases with increase in voltage.

Ohm’s Law: The current flowing through the electric circuit is directly proportional to the potential difference across the circuit and inversely proportional to the resistance of the circuit, provided the temperature remains constant.
V ∝ I
V/I= constant
V/I = R
The ratio of potential difference (V) between any two points of a conductor to the current (I) flowing between them is constant, provided that the temperature of the conductor remains constant.
  • When R=0 (Short Circuit), V=0 Volts
  • When R= ∝ (Open Circuit), I=0 Amps

Note: Ohm’s law can be applied either to the entire circuit or to the part of a circuit.

Limitations of Ohm’s Law: It is not applicable to the non-linear devices such as diodes, zener diodes, voltage regulators etc.
  • It does not hold good for non-metallic conductors such as silicon carbide.
  • The law for such conductors is given by V = Kim where K and m are constants.

Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents meeting at a junction point is always zero.
  • Mathematically, 
where N is the number of branches connected to the node and in is the nth current entering (or leaving) the node.

Note: Current flowing towards a junction point are assumed to be positive while current flowing away from a junction point assumed to be negative.

Example-1: Consider the following Node analysis of currents entering /leaving from node point p.

Using KCL, entering a node p may be regarded as positive, while currents leaving the node may be taken as negative or vice versa.
i1 + (-i2) + i3 + i4 + (-i5) = 0
i1 +  i3 + i4 = i2 + i5

Note: The sum of the currents entering a node is equal to the sum of the currents leaving the node.

Kirchhoff’s Voltage Law (KVL): In any network, the algebraic sum of the voltage drops across the circuit elements of any closed path (or loop or mesh) is equal to the algebraic sum of the emfs in the path.
  • Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero

  • KVL can be interpreted as: Sum of voltage drops =  Sum of voltage rises

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