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Circuit Elements And Signal WaveForm
Introduction to Circuit Elements : Basic elements of networks are resistance, inductance and capacitance
Resistance : In network, the resistance is defined as per ohm's law.
V = R I
Constant of proportionality 'R' is termed as resistance.
R, depends upon the nature of material and also on its geometry
where ρ → resistivity, unit ohmmetre
I → length of wire, unit : metre
A → Area of cross section unit : m^{2}
1/ρ = σ (conductivity)
unit of conductivity  slemens m^{–1} or mho m^{–1 }
Capacitance : The physical device that permits storage of charge is a capacitor and its ability to store charge is its capacitance. The reciprocal of capacitance is defined as elastance The charge Q is proportional to the potential difference V, i.e.
Q ∝ V
or Q = CV or C = Q/V
unit of capacitance is farad or coulombs/voit
The total energy required to transfer Q coulombs of charge, resulting in a final potential of V volts between the plates is
Example : A parallel plate capacitor of 5pF capacitance has a charge of 0.1 μC on its plates. What is the energy stored in the capacitor?
(a) 1 mJ
(b) 1 μJ
(c) 1 nJ
(d) 1 pJ
Solution : (a)
Energy store = 1/2 VQ
= Q^{2}/2C = 1mJ
C ∝ A
The above equation is for vacuum, for any dielectric medium of dielectric constant ' ∈ ' in SI units.
∈_{t} → relative permittivity or dielectric constant of the medium between the plates
In each case C will be function of the geometry of the conductors and ∈ .
Inductance :
N → total no. of turns.
S → area of cross–section of coil
Then
where I → length of the coil
'L' is called the self inductance of the coil.
Relationship of parameters:
Definitions: Before going into details of discussion of network analysis, it is desirable to introduce certain definitions.
Lumped and distributed Network
Lumped Network :A network in which we can separate resistors, capacitors and inductors physically.
Distributed Network : A circuit in which the voltage and current are functions of time and position is called, a distributed parameter circuit. While a circuit in which the voltage and current are functions of time only is called a lumped parameter circuit.
Active and Passive network
Passive Network : A network containing circuit elements without any energy sources.
Active Network : A network containing energy sources together with other circuit elements.
Linear Element :
A circuit element is linear if the relation between current and voltage involves a constant coefficient e.g.
Average value : The general periodic function y (t) with period T has an averange value Y_{av} given by
RMS of effective value : The general function y(t), with period T has an effective value Y_{rms} given by
RMS value for several sine and cosine terms :
The function
y(t) = a_{0} + (a_{1} cos ωt + b_{2} sin 2 ωl) has an effective value given by
also if A, is the effective value of a cos ωt then
Example : A series R–C circuit with R = 3 Ω and X_{C} = 4 Ω at 50 Hz is supplied with a voltage V = 50 + 141 4sin 314 t. What isthe RMS value of the current flowing through the circuit?
(A) 5 A (B) 10 A
(C) 20 A (D) 22.36 A
Solution : (D) Impedance,
Z = R – jX_{C} = 3 – j4 = 5 ∠ – 53.13°
= 22.36A
Definitions of Certain Terms
Node Any point in a circuit where the terminals of two or more elements are connected together.
Branch A branch is a part of the circuit which extends from one node to another A branch may contain one element or several elements in series. It has two terminals.
Essential Node If three or more elements are connected together at a node, then that node is sometimes called essential node.
Essential Branch A path which connects two essential nodes without passing through an essential node is called an essential branch.
Mesh Any path which contains no other paths within it is called a mesh.
Loop A path which contains more than two meshes is called a loop. Thus a loop contains meshes but a mesh does not contain a loop.
Kirchoff's Voltage Law (KVL) The algebraic sum of all branch voltages around any closed loop of a network is zero at all instants of time.
Mathematically ∑v(t ) = 0
Illustration :
Kirchoff's Current Law (KCL) At any instant of time, the algebraic sum of currents at a node is zero
Mathematically ∑i(t ) = 0
At nodes : Current entering → node are assigned positive sign
Current leaving → node, will assigned negative sign or vice versa
Applying KCL at node
Resistor in series :
Resistance in series and its equivalent
R_{eq} = R_{1} + R_{2 }+ .....+ R_{n}
Resistances in parallel :
Resistance in parallel and its equivalent
Voltage Division Equations : Between two resistors in series
Concept can be extended to 'n' series resistances
Current divider equations :
Star ⇔ DELTA (OR T ⇔ π) Transformation :
(a) Delta to star (or π to T) Transoformation
(b) Starto delta (or T to π ) Transformation
Superimposed delta and star networks
Inductors in series
L_{s} = L_{1} + L_{2} + L_{3 }+ .............. + Ln
Inductors in parallel :
Inductors in parallel
Capacitors in series :
Capacitors in parallel :
i = C_{p}
C_{P} = C_{1} + C_{2} + C_{3} + .... + C_{n} .
Source  Transformation :
a. voltage to current transformation
b. current to voltage transformation
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