Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) PDF Download

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.

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

V = R I
Constant of proportionality 'R' is termed as resistance.
R, depends upon the nature of material and also on its geometry

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

where ρ →  resistivity, unit ohm-metre

I → length of wire, unit : metre

A → Area of cross section unit : m2

1/ρ = σ (conductivity)

unit of conductivity - slemens m–1 or mho m–1 

  • If length of wire is doubled and radius is half or halved, then resistivity remains same but resistance becomes 8 times. 
  • Extension of wire result into increase in length and decrease in area of cross-section therefore resistance of wire increases 
  • For compression resistance decreases.

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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

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

The total energy required to transfer Q coulombs of charge, resulting in a final potential of V volts between the plates is

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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

 = Q2/2C = 1mJ

  • For the parallel plate capacitor shows above

C ∝ A

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

The above equation is for vacuum, for any dielectric medium of dielectric constant ' ∈ ' in SI units.

  Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

t → relative permittivity or dielectric constant of the medium between the plates 

  • For other geometric shapes such as two spheres two conductors, or one conductor and and ground, the formula would be different.

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  Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
where I → length of the coil

'L' is called the self inductance of the coil.

Relationship of parameters: 

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Definitions: Before going into details of discussion of network analysis, it is desirable to introduce certain definitions.

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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. 

  • In distributed network, resistors, capacitors and inductors cannot be electrically separated and individually isolated as separate elements. Transmission line is such a network.

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.

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  • A linear network is one in which the principle of superposition (linearity test) holds, A nonlinear network is one which is not linear.

Average value : The general periodic function y (t) with period T has an averange value Yav given by

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

RMS of effective value : The general function y(t), with period T has an effective value Yrms given by

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  • R.M.S. value of the functions asin ωt and acos ωt is a / √2

RMS value for several sine and cosine terms : 

The function
y(t) = a0 + (a1 cos ωt + b2 sin 2 ωl) has an effective value given by


Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

also if A, is the effective value of a cos ωt then
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Example : A series R–C circuit with R = 3 Ω and XC = 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 – jXC = 3 – j4 = 5 ∠ – 53.13°

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

= 22.36A

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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

  • In most of the cases wer shall choose clockwise direction for traversing the closed path. 
  • Also the voltage across elements that are traversed from – to + are taken positive and the voltages across elements that are traversed from + to – are taken negative (or you can choose vice-versa).

Illustration :

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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 

  • Sign convention is arbitrary

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Applying KCL at node
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resistor in series :

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resistance in series and its equivalent
Req = R1 + R+ .....+ Rn

Resistances in parallel :

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resistance in parallel and its equivalent

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Voltage Division Equations : Between two resistors in series

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Concept can be extended to 'n' series resistances

Current divider equations :

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  • This concept can also be extended to 'n' parallel resistances.

[Question: 0]

Star ⇔ DELTA (OR T ⇔ π) Transformation :

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)  

(a) Delta to star (or π to T) Transoformation

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

(b) Star-to delta (or T to π ) Transformation

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
 

Superimposed delta and star networks

  Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Inductors in series

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Ls = L1 + L2 + L+ .............. + Ln

Inductors in parallel :
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Inductors in parallel

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Capacitors in series :

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) 

Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

 Capacitors in parallel :

i = Cp
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

CP = C1 + C2 + C3 + .... + Cn .

Source - Transformation :
a. voltage to current transformation
Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

b. current to voltage transformation

 Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

The document Circuit Elements & Signal Waveform | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electrical Engineering SSC JE (Technical).
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FAQs on Circuit Elements & Signal Waveform - Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

1. What is the difference between capacitance and inductance in circuit elements?
Ans. Capacitance is the ability of a component to store electrical energy in an electric field, while inductance is the ability of a component to store electrical energy in a magnetic field.
2. What is the difference between lumped and distributed networks in electrical engineering?
Ans. Lumped networks are those where the components are concentrated at specific points, while distributed networks have components spread out over a larger area.
3. What is the distinction between active and passive networks in circuit elements?
Ans. Active networks contain a source of energy, such as a battery or generator, while passive networks do not contain any source of energy and rely on stored energy in components.
4. How do linear elements differ from non-linear elements in circuit analysis?
Ans. Linear elements follow the principle of superposition and have a linear relationship between voltage and current, while non-linear elements do not follow this principle and have a non-linear relationship between voltage and current.
5. What is the significance of Star-Delta (or T-π) transformation in electrical engineering?
Ans. The Star-Delta transformation is used to simplify complex circuits by converting between different circuit configurations, making it easier to analyze and solve circuit problems.
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