Network Theory EE K-Notes

Contents Manual for K-Notes ................................................................................. 2 Network Elements .................................................................................. 3 Graph Theory .......................................................................................... 9 Circuit Theorems ................................................................................... 11 Transient Analysis ................................................................................. 15 Sinusoidal steady state analysis ............................................................ 19 Resonance............................................................................................. 23 Circuits analysis in Laplace domain ....................................................... 25 Two Port Network ................................................................................. 26 Magnetically coupled circuits................................................................ 29 Three Phase Circuits.............................................................................. 31 Electrical & magnetic fields ................................................................... 33

© 2014 Kreatryx. All Rights Reserved. 1

Manual for K-Notes Why K-Notes? Towards the end of preparation, a student has lost the time to revise all the chapters from his / her class notes / standard text books. This is the reason why K-Notes is specifically intended for Quick Revision and should not be considered as comprehensive study material. What are K-Notes? A 40 page or less notebook for each subject which contains all concepts covered in GATE Curriculum in a concise manner to aid a student in final stages of his/her preparation. It is highly useful for both the students as well as working professionals who are preparing for GATE as it comes handy while traveling long distances. When do I start using K-Notes? It is highly recommended to use K-Notes in the last 2 months before GATE Exam (November end onwards). How do I use K-Notes? Once you finish the entire K-Notes for a particular subject, you should practice the respective Subject Test / Mixed Question Bag containing questions from all the Chapters to make best use of it.

© 2014 Kreatryx. All Rights Reserved. 2

Network Elements Active & Passive Elements If any elements absorb, dissipate, waste, convert electrical energy it is called as passive element. Eg. Resistor, Inductor, Capacitor. If any elements energize, deliver, give out, drive the electrical energy it is called as active element. Eg. BJT, MOSFET. Network Technologies Node : It is a point of interconnection or junction between two or more components. Branch : It is an elemental connection between two nodes. Mesh: A mesh is a close path which should not have any further closed path in it. Loop : All possible close path. Ohm’s law At constant temperature and for uniform cross section of conductor. J  E

σ= conductivity,



1  resistivity . 

V  IR l R A l  lenght of conductor

Circuit Symbol:

A = Area of conductor. Conductance of circuit elements is

G

1 R

Sign Convention To apply ohm’s law, we must apply following sign convention.

3

Short circuit & open circuit Voltage across terminals of a short circuit is always zero, regardless of the value of current which could be any value. (R = 0) The current through an open circuit is always zero, regardless of voltage across the terminals which could be any value. R    Power of resistor

V2  I2R R Resistance always absorbs or dissipates power. P  VI 

Kirchoff’s laws 

Kirchoff’s current law(KCL) It states that any instant the algebraic sum of current leaving any junction (or node) in a network is zero. In other words, current entering a node is equal to current leaving the node.

n in  t   0

 ientering   ileaving i1  i3  i5  i2  i4



Kirchoff’s voltage law (KVL) It states that any instant the algebraic sum of the voltage around any closed path (or loop) within a network is zero. In other words, the sum of voltage drops is equal to sum of voltage rises.

 V t  0 n

n

 Vdrop   Vrise V1  V2  V3  V4  V5  0

4

Series resistance or voltage division Two or more circuit elements are connected in series means that current through all elements in same. If ‘N’ resistors, with resistance R1 ,R2 ,........Rn are connected in series R eq  R1  R 2  ...................RN

V1  V2 

V R1 

R1  R 2 V R 2 

R1  R 2

Parallel resistance or current division Two or more circuit elements are connected in parallel means that voltage across all elements is same. If ‘N’ resistors are connected in parallel R1 ,R 2 ,............RN 1 1 1 1    ...........  R eq R1 R 2 RN

I1  I2 

I R 2 

R1  R 2 I R1 

R1  R 2

Star Delta Conversion Start to Delta

Ra 

R1R 2  R1R 3  R 2R 3 R1

Rb 

R1R 2  R1R 3  R 2R 3 R2

Rc 

R1R 2  R1R 3  R 2R 3 R3 5

Delta to start conversion

R1 

RbR c R a  Rb  R c

R2 

R aR c R a  Rb  R c

R3 

R aR b R a  Rb  R c

Sources Independent Voltage Source An ideal independent voltage source maintains a specified voltage across its terminals. The voltage is independent of current flowing through it. Independent current source An ideal independent current source maintains a specified current to flow through it. The current through this is independent of voltage across it. Dependent Source Voltage controlled voltage source (VCVS) ; V  AVX Current controlled voltage source (CCVS) ; V  AiX Voltage controlled current source (VCCS) ; i  AVx Current controlled current source (CCCS) ; i  Aix

Capacitor A capacitor is a combination of a two conducting plates separated by a non-conducting material. Capacitance is donated by ‘C’ A C d ϵ= Permittivity of medium A = Area of Plates D = distance between the plates. Charge on Plates,

Q = CV

V = Potential difference between the plates.

6

Voltage Current relationship

i t 

dq  t 

i t  C

dt dv  t  dt

Sign Convention Energy Stored in a capacitor

1 E  cv 2 2

Q  t  t  2C

2

1  Q t v t  2

Properties of ideal capacitor 

If voltage across capacitor is constant (dc) then current through capacitor is zero & it acts as open circuit.



The voltage across capacitor must be continuous, if it as discontinuous, then i  C

dv  t  dt

so voltage across capacitor cannot change instantaneously. 

A capacitor never dissipates energy, it only stores energy.

Capacitor in series & parallel 

In ‘n’ capacitors are connected in series C1 ,C2 ,.............Cn

Ceq 



1 1 1 1   ................. C1 C2 Cn

If ‘n’ capacitors are connected in parallel, C1 ,C2 ,..........................Cn Ceq  C1  C2  .......................  Cn

In series connection, charge is same whereas in parallel connection voltage is same.

7

is infinite,

Inductor It is a two terminals element consisting of winding of ‘N’ turns.   N2 A L 0 r l  0 = Permeability of free space r  relative Permeability

N = number of turns A = area of cross section of coil l = length of inductor Current voltage relationship

v t  L

di  t 

dt L is constant, called as inductance Energy Stored

1 E  L i t 2

 

2

Like, Capacitor, inductor also stores energy but in electro-magnetic terms. Series & parallel Connection If ‘n’ inductors L1 ,L2 ,...............,Ln are connected in series then Leq  L1  L2  .............  Ln

In ‘n’ inductors are connected in parallel L1 ,L2 ,.............Ln 1 1 1 1    .............  L eq L1 L 2 Ln

In series connection current in same, through all elements & in parallel connection voltage is same across all elements. Duality Two circuits N1 & N2 are called dual circuit if the branches KCL, KVL & branch v - i relationship becomes respectively KVL, KCL.

8

Resistance Capacitance Inductance Open Circuit Short Circuit Voltage Source Current Source

Dual Elements Conductance R CF LH

GR

L=CH C=LF

VS

Inductance Capacitance Short Circuit Open Circuit Current Source

IS

Voltage Source

V  IS

Series Connection Parallel Connection

I  VS

Parallel Connection Series Connection

Eg.

Graph Theory Network Graph: If all elements of a circuits are replaced a line segment between 2 end points called as nodes.

Directed Graphs: If the branches of a graph has directions then it is called as a directed graph.

9

Sub graph It consists of less or equal number of verticals (nodes) & edges, as in its complete graph.

True & Co-tree A connected sub-graph of a network which has its nodes same as original graph but does not contain any closed path is called tree of network. A tree always has (n - 1) branches. Eg. The following trees can be made from graph shown before.

The set of branches of a network which are remove to form a tree is called co-tree of graph. Twigs & Links The branches of a tree are called as its twigs & branches of a co-tree are called as chords or links.

10

Incidence Matrix The dimension of incidence matrix is (nxb) N = no. of nodes B = no. of branches It is represented by A aij = + 1 , If jth branch is oriented away from ith node aij = -1 ,

If jth branch is oriented into ith node.

aij = 0 ,

If jth branch is not connected to ith node

 a b c d e f   0 0 1 1 1 0 1 A  2  1 1 0 1 0 0   1 0 0 1 1  3 0 4  0 0 1 1 1 0 

If one of nodes is considered as ground & that particulars row is neglected while writing the incidence matrix, then it is reduced incidence matrix. Order  n  1   b T Number of trees of any graph  det  Ar  Ar     Ar = reduced incidence matrix

Circuit Theorems Linearity A system is linear if it satisfies the following two properties. 1. Homogeneity Property It requires that if input is multiplied by constant hen output is multiplied by same constant. eg. V = IR is I becomes KI V’ = KIR = KV So, resistance is a linear element & so are inductor & capacitor. 2. Additivity Property It requires that response to sum of inputs is sum of response to each input applied separately. V1  I1R

V2  I2R If we apply

I

We get

V3   I1  I2  R  V1  V2

1

 I2 

11

Superposition It states that, in any linear circuit containing multiple independent sources, the total current through or voltage across an element can be determined by algebraically adding the voltage or current due to each independent source acting alone with all other independent source set to zero. Source Transformation It states that as independent voltage source VS in series with a resistance R is equivalent to independent current source IS  Vs / R in parallel with a resistance R. Or An independent current source IS in parallel with a resistance R is equivalent to a dependent source VS  ISR in series with a resistance R.

Thevenin’s Theorem It states that any network composed of ideal voltage and current source, and of linear resistor, may be represented by an equivalent circuit consisting of an ideal voltage source VTH in series with an equivalent resistance R TH . Methods to calculate thevein equivalent The therein voltage  VTH  is equal to open circuit voltage across load terminals. Therein resistance is input or equivalent resistance at open circuit terminals (load terminals) when all independent source are set to zero (voltage sources replaced by short circuit & current source by OC) Case – 1 : Circuit with independent sources only

12

To calculate VTH , open circuit of RL

Using sources transformation

VTH 

2  24  12V 22

To calculate Rth Short I & V sources & open 6mA source

R th  1  2

1  1  2k 13

Case – 1 : Circuit with both dependent & independent sources Methodology 1: 

VTh can be found in same way.



For R TH set all independent sources to zero.



Remove load & put a test source Vtest across its terminals, let current through test source is Itest .



Thevenin resistance ,



This method is must if independent sources are absent.

R TH 

Vtest Itest

Methodology 2:    

VTH is calculated in same way. For R TH short circuit load terminals & leave independent sources as it is Obtain ISC through load terminals. R TH  VOC ISC Norton’s Theorem Any network composed ideal voltage & current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal current source IN in parallel with an equivalent resistance R N .



RN  R TH



To calculate IN we short circuit load terminals & calculate short circuit current.



Therein equivalent & Norton equivalent are dual of each other.

14

Maximum Power Transfer Theorem: A load resistance RL will receive maximum power from a circuit when load resistance is equal to Thevenin’s/ Norton’s resistance seen at load terminals. RL  R Th

In case of AC circuit, this condition translates to

ZL  Z*th 

But if load is resistive in AC circuit then

RL  Z Th  For maximum power transfer 2

RL  R2Th   XL  X Th 

 For maximum power transfer

XL  XTh  0

Transient Analysis Time Constant : It is the time required for the response to delay by a factor of 1 e or 36.8 % of its initial value. It is represented by τ. For a RC circuit   RC For a RL circuit   LR R is the therein resistance across inductor or capacitor terminals. 15

General method of analysis  t t  x t  x    x t0  x   e  o  , t  0  



   

 

If switching is done at t=t0

 

x t0  initial value of x  t  at t  t0 x     final value of x  t  at t  

Algorithm 1. Choose any voltage & current in the circuit which has to be determined. 2. Assume circuit had reached steady state before switch was thrown at t  t0 . Draw the circuit at t  t 0 with capacitor replaced by open circuit and inductor replaced by short circuit. Solve for

 

 

3. Voltage

across

v C t0 & iL t0 . capacitor

   V t   V t  i t   i t   i t  VC t L

 0

 0

C

L

 0

 0

C

L

and

inductor

current

cannot

change

instantaneously.

0

0

4. Draw the circuit for t  t  with switches in new position. Replace a capacitor with a voltage source

   

 

  of variable x  t  .

VC t0  VC t0 and inductor with a current source of value iL t0  iL t0 . Solve for initial value  0

5. Draw the circuit for t   , in a similar manner as step-2 and calculate x    . Calculate time constant of circuit 6. τ=Rth C or τ=L/Rth 7. Substitute all value to calculate x(t). Example In the circuit shown below, V1  t  for t > 0 will be given as

16

Solution Step 1 : For t < 0 30u  t   0 & 3u  t   0

 

V1 0  0V For t   V1     3mA  10k

= -30 V Step 2 : At t  0

V1  0   30 20k

 3mA 

V1  0 

3 V1  0   1.5mA 20k V1  0   10V

10k

t

V1  t   30   10  30  e

0



R TH  30k ;   R THC  0.3s V1  t    30  20e

t

0.3



 u t V 



Series RLC circuit Without Source

V 0 

0

1  i  t  dt  V0 C 

i  o   I0 By KVL

Ri  t   L

di  t  dt



t

1  i  t  dt  0 C 

Difference both sides d2 i  t  dt

2



R di  t  1  i t  0 L dt LC

17

Substitute i  t   Aest



Aest S2  R s  1

L

S1   R

LC

0



S2  R s  1  0 L LC 2

2

R  R S2     1 LC 2L  2L 

R     1 , 2L LC  2L 

S1 ,S2     2  w 02 ;   R 2L ; w 0 

1 LC

1. If   w0 roots are real & unequal (over-damped)

i  t   Aes1t  Bes2t

2. If   w0 , roots are real & equal (critically damped)

i  t    A  Bt  et

3. If   w0 , roots are complex conjugate (under-damped)

i  t   et  A cos wdt  Bsinw dt  wd  w20  2 Calculate A & B using initial conditions. With a Source

v  t   VS  Ae 1  Be 2 (Over-damped) st

s t

v  t   VS  A  Bt  et (Critically damped)





v  t   VS A cos wdt  Bsinwdt et under  damped

Parallel RCL Circuit Without Source

i 0 

0

1  v  t  dt L 

v  0   V0

By KCL

dv  t  1 1 v  t    v    d  C 0 R L  dt t

18

Characteristics equation s2 

1 1 s 0 RC LC

;



1 , w0  2RC

1 LC

S1 ,S2    2  w02 v  t   Ae 1  Be st

S2t

 over  damped

v  t    A  Bt  et  critically damped v  t   et  A cos wdt  Bsinwdt  under  damped With a step input



i  t   Is  Ae 1  Be st

S2 t

 Over  damped

i  t   Is   A  Bt  et Critically damped i  t   Is   A cos wdt  Bsinwdt  et Under  damped Steps: 1. Write differential equation that describe the circuit. 2. From differential equation model, construct characteristics equation & find roots. 3. Roots of characteristics equation determine the type of response over-damped, critically damped & under-damped. 4. Obtain the constant using initial conditions.

Sinusoidal steady state analysis Lagging & Leading We can compare the phases of two sinusoids provided that 

Both V1  t  & V2  t  are expressed in form of either sine function or cosine function.



Both V1  t  & V2  t  are written with positive amplitude though they may not have same amplitude. Both V1  t  & V2  t  have same frequency.

19

If V1  t   A sinwt

V2  t   Bsin  wt    ;   00

V2  t  leads v1  t  by an angle 

V1  t  lags v 2  t  by an angle  PHASORS

A phasor is complex number that represents the amplitude & phase angle information of a sinusoidal function.

v  t   Vm sin  wt   

Phasor representation, V  Vrms magnitude  Vrms phase  

Networks Elements 1. Resistor

V  RI 2. Inductor

V   jl I

  2f ; f  frequency of source

3. Capacitor

I   j c  V

Impedance & Admittance

Impedance , Z 

V Vrms     v  i  I Irms

Unit of impedance  ohm    Z  R  jX  Z 

R = resistive component X = relative component 20

 R

Phase angle,   tan1 X

Z  R2  X2

For Resistor ZR  R For Inductor ZL  jL For Capacitor Z C 

1 j  c j L

Inductive reactance, XL  L Capacitive reactance, XL 

1 c

If X = 0, impedance is resistive; current & voltage are in same phase. If X > 0, impedance is inductive; current lags voltage. If X < 0, impedance is capacitive; current leads voltage. Admittance, Y 



1  G  jB Z

G = Capacitance B = Susceptance Impedance in AC circuits behave like resistance in DC circuits and all the laws remain same like Series combination

Zeq  Z1  Z2  ................  Zn 

Parallel combination

1 1 1 1    ......................  Z eq Z1 Z 2 Zn

Yeq  Y1  Y2  .........  Yn 

Star-delta conversion also remain same here. Circuits analysis in AC domain

1. Identify the sinusoidal source & note the excitation frequency. 2. Convert source to phasor form. 3. Represent each circuit element by its impedance. 4. Solve circuits using circuit techniques (nodal analysis mesh analysis etc.) 5. All circuits’ theorems are applicable here as well

21

Power analysis Real Power

P  VrmsIrms cos  v  i  In a resistance P

2 Vrms 2  Irms R R

Complex Power * S  VrmsIrms

 Vrms Irms   v  i 

  Real part of S  Q  Vrms Irms sin  v  i  Real part of S  P  Vrms Irms cos v  i

Reactive Power = Q Q = 0 for resistive loads. Q < 0 for capacitive loads  v  i  . Q > 0 for inductive loads  v  i    

If the current goes into an element, then it absorbs power and if current comes out it delivers power. Hence, a capacitor absorbs leading reactive power. We can also say it delivers lagging reactive power. Same way, inductor absorbs lagging reactive power & delivers leading reactive power. Power Factor

pf 

P  cos  v  i  S

  v  i  power factor angle If   0 , power factor is lagging If   0 , power factor is leading If   0 , power factor is unity

22

Resonance Series resonance For resonance

Im  Z   0



Z  R  j L  1 L  1

c

c

 1

 

LC

rad s

The frequency at which impedance of circuits is purely resistive is called resonant frequency.

1

0 

LC

rad s

At resonance I 

VS R

VR  IR  VS  VS R 

   

j  j  VS I  0c oC  R

   

VL  joL I  joL  Vc 

At   0 ; XL  XC , net reactance is capacitive so circuits operates at leading pf. At   0 ; XL  XC , net reactance is zero, so circuits operates at unity pf. At   0 ; XL  XC , net reactance is inductive, so circuits operates at lagging pf. Bandwidth: range of frequency for which power delivered to R is half of power at resonance. Bw  R L Quality factor 1 2 1 2 I XL I XC Re active power 2 2 Q   1 2 1 2 Average power I R I R 2 2

Q

XL R



XC R



1 L R C 23

Parallel Resonance Y1

R



1  j C j L

At resonance

Im  Y   0

1

o 

LC

rad s

At resonance V  IS R

IR  IS IL 

IR V  S j0L j0L

IC  V  j0C   ISR  j0C  Bandwidth 

1 RC

Quality factor, Q 

V Q

2

2 XC V

2

2R

V 

2

2 XL V

Re active Power Average Power

2



R R C  R XL XC L

2R

24

Circuits analysis in Laplace domain For basic of laplace transform, refer to signal & system k-notes. Laplace transform. Resistor V(s)  RI(s)

Inductor

 V(s)  sL  I(s)  Li(0 ) Or I(s)  V(s)  i(0 )

sL

s

Capacitor

I(s)  C sV(s)  V(0  )

Or I(s) 

1 V(0  ) I(s)  sC s

Methodology 1. Draw circuit in s-domain by substituting s-domain equivalent for each circuit element. 2. Apply circuit analysis to obtain desired voltage or current in s-domain. 3. Take inverse Laplace transform to convert voltage and current back in time-domain. 25

Two Port Network Impedance Parameters V1  Z11 I1  Z12 I2 V2  Z21 I1  Z22 I2

In matrix form

Z11  Z12  Z 21  Z 22 

V1 I1 V1 I2 V2 I1 V2 I2

 V1   Z11    V2   Z 21

Z12   I1    Z 22  I2 

 open circuit input impedance I2 0

 open circuit transfer impedance form part 1 to part 2 I1 0

 open circuit transfer impedance form part 2 to part 1. I2 0

 open circuit output impedance I1 0

Admittance parameters I1  y11 V1  y12 V2 I2  y 21 V1  y 22 V2

In matrix form,  I1   y11   I2   y 21

y11  y12 

I1 V1 I1 V2

y12   V1    y 22   V2  = short circuit input admittance.

V2 0

= short circuit transfer admittance from part 1 to part 2. V1 0

26

y 21  y 22   y11 y  21

I2 V1

= short circuit transfer admittance from part 2 to part 1. V2 0

I2 V2

= short circuit output admittance. V1 0

y12   Z11  y 22   Z 21

Z12  Z 22 

1

Hybrid parameters V1  h11 I1  h12 V2 I2  h21 I1  h22 V2

In matrix form,  V1  h11 h12   I1       I2  h21 h22   V2 

h11 

h12  h21 

h22 

V1 I1

V1 V2 I2 I1

= short circuit input impedance. V2 0

= open circuit reverse voltage gain. I1 0

= short circuit forward current gain. V2 0

I2 V2 I

= open circuit output admittance.

1 0

Inverse hybrid parameters  I1  g11    V2  g21

g11 g  21

g12   V1    g22   I2 

g12  h11 g22  h21

h12  h22 

1

27

Transmission parameters V1  AV2  BI2 I1  CV2  DI2

 V1   A B   V2   I    C D   I   1   2

A

B C

V1 V2

V1 I2 I1 V2

D

= open circuits voltage ratio I2 0

= negative short circuit transfer impedance. V2 0

= open circuit transfer admittance. I2 0

I1 I2

= negative short circuit current ratio. V2 0

Symmetrical & Reciprocal N/w For a reciprocal 2-part Network: Z12  Z21 Y12  Y21 h12  h21

AD  BC  1 g12  g21

For a symmetric 2-part network: Z11  Z22 Y11  Y22 h11h22  h21h12  1

A=D g11g22  g21g12  1

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Interconnection of 2-part networks 

For a series connection of two networks Na & Nb having z-parameters metric

 Z a  &  Zb 

Z   Z   Z   eq   a   b 

Z – Parameter matrices are added.



For a parallel connection of two networks Na & Nb having y-parameter matrices

 ya  &  yo 

 yeq    ya    yb  Y – Parameter matrices are added.



For a cascade connection of two networks Na & Nb having transmission parameters matrices

 Ta  & Tb  Teq    Ta  Tb  Transmission parameter matrices are multiplied.

Magnetically coupled circuits If change in flux of one coil induces a voltage in second coil then both coils are said to be magnetically coupled.

v 2  t   M21

v1  t   M12

di1  t  dt

di2  t  dt

M12  M21  M

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Dot convention If a current enters the dotted terminals of one coil, then induced voltage in second coil has a positive voltage reference at dotted terminal of second coil. If a current enters undotted terminals of one coil, then induced voltage n second coil as a positive voltage reference at undotted terminals of second coil.

Series connection of coupled coils

Leq  L1  L2  2M

Leq  L1  L2  2M

Parallel connection of coupled coils

L eq 

L1L 2  M2 L1  L 2  2M

Coefficient of coupling k 

L eq  M L1L2

30

L1L 2  M2 L1  L 2  2M

Equivalent circuits of linear transformers T-equivalent circuit

La  L1  M Lb  L 2  M LC  M

π- equivalent circuit

LA 

L1L 2  M2 L2  M

L1L 2  M2 LB  L1  M LC 

L1L2  M2 M

Three Phase Circuits Balanced three phase system A system in which all three voltage have equal voltage magnitude and are phase displaced by 1200 with respect to each other. Positive Sequence (abc)

Van  VP00

Vbn  VP  1200 Vcn  VP  240  VP1200 31

Negative Sequence (acb)

Van  VP00

Vcn  VP  1200 Vbn  VP  2400  VP1200 For both phase sequence, Van  Vbn  Vcn

Connections Star Connection

VL  3VP 300 IL  Ip Vab = line to line voltage or line voltage

Vab  Van  Vbn  3VP 300 So, line voltage Vab , Vbc , Vca are 3 times the phase voltage & lead the respective phase voltage by 30 0 . Line currents Ia , Ib , Ic in this connection are equal to phase currents. Phase current are the currents that flow in individual phases.

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Delta Connection

VL  VP

IL  3IP   300

Line current is 3 times phase current & lags respective phase current by 30 0 .

Power in a balanced 3   system

P  3VL IL cos   3VP IP cos 

Q  3VL IL sin   3VP IP sin  S  3VL IL *  3VP IP* For power measurement, refer to electrical measurements k-notes, two wattmeter method.

Electrical & magnetic fields Coulomb’s law Coulombs law states that magnitude of force between two point charges is directly proportional to square of distance between them & direction of force is along the line joining the charges.

F

Q1 Q2

4  R

2

aˆR

or ; o  8.854  1012 F m = permittivity of free space

r = relative permittivity or dielectric constant.

33

Electric field intensity

E

F Q  aˆR q 4  R 2

Electric field direction is away from a positive charge & towards negative charge. Charge densities 1) Linear charge density It is denoted by '  ' . It is equal to charge per unit length. q    c m l 2) Surface charge density It is denoted by '  ' . It is equal to charge per unit area. q  c m2 A





3) Volume charge density It is denoted by '  ' . It is equal to charge per unit volume.



q c m3 V





Electric field due to continuous charge distribution 1) Infinite line charge Electric field intensity at a distance ‘r’ from a line charge of linear charge density 

E

 ˆ a 2o r r

2) Infinite sheet charge Electric field at a distance ‘h’ from an infinite charged sheet with charge density



 E aˆ ; aˆ n  Normal unit vector 2 n 3) Conducting sphere If a conducting sphere of radius ‘R’ is charged with a charge ‘Q’ then electric field.

0  E Q   4  r2

r R r R

Electric field inside conducting sphere is zero.

34

is

Electrical potential The amount of work done in bringing a unit positive charge from infinity to a certain point in an electric field is called electric potential. A

VA   E.dL 

E  V

 = represent gradiant For vector operations, refer engineering mathematics k-notes. Electric Flux Density D  E

Electrical flux  

 D.dS S

SI unit of electric flux is coulomb. Gauss’s law It states that total electric flux through any closed surface is equal to charge enclosed by that surface.

S D.dS  b dV By Gauss’s Divergence theorem

.D   Magnetic flux Density Magnetic flux per unit area is called magnetic flux density. It is a vector quantity and denoted by B & its unit is tesla (T).

Flux    B. dS

35

Magnetic field intensity Represented by H .

B  H

 = permeability.   or r = relative permeability o = permeability of free space o  4  107 H m

Biot – Savart’s law

d H

I 4 R

2

dL  aˆ  R

Magnetic field due to infinite line current H

I aˆ  2

 = perpendicular distance of point from line current.

aˆ  = Unit vector in cylindrical co-ordinates. Ampere’s Circuital law It states that line integral of magnetic field intensity H around any closed path is exactly equal to net current enclosed by that path.

 H . dL  I

enclosed

 H . dL   J . ds By stokes theorem

 H  J

36

Maxwell equations d B . dS dt 

B t

1)

 E . dL 

2)

 E . dS    dv

3)

 B . dS  0

4)

 B . dL  0  J . ds  o o dt  E . ds

1

or

or

or

 E 

 . E  

.B 0

d

or 

  B  o  J  o  

E   t 

37