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Engineering Electromagnetics Notes

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Engineering Electromagnetics Notes 10EC36 ENGINEERING ELECTROMAGNETICS NOTES 10EC36 Introduction to vectors The behavior of a physical device subjected to electric field can be studied either by Field approach or by Circuit approach. The Circuit approach uses discrete circuit parameters like RLCM, voltage and current sources. At higher frequencies (MHz or GHz) parameters would no longer be discrete. They may become non linear also depending on material property and strength of v and i associated. This makes circuit approach to be difficult and may not give very accurate results. Thus at high frequencies, Field approach is necessary to get a better understanding of performance of the device. The ‗Vector approach‘ provides better insight into the various as ects of Electromagnetic phenomenon. Vector analysis is therefore an essential tool for the study of . The ‗Vector Analysis‘ comprises of ‗Vector Algebra‘ and ‗Vect r Calculus‘. Any physical quantity may be ‗Scalar quantity‘ or ‗Vector quantity‘. A ‗Scalar quantity‘ is specified by magnitude only while for a ‗Vector quantity‘ requires both magnitude and direction to be specified. Examples : Scalar quantity : Mass, Time, Charge, D nsity, Pote tial, Energy etc., Represented by alphab ts – A, B, q, t etc Vector quantity : Electric field, force, velocity, acceleration, weight etc., represented by alphabets with arrow on top. A, B, E, B etc., Vector algebra : If A, B, C are vectors and m, n are scalars then (1) Addition A B B A Commutativ law citystudentsgroup A ( B C) (A B) C Associative law (2) Subtraction A - B A (- B) (3) Multiplication by a scalar m A A m Commutative law m (n A) n (m A) Associative law (m n) A m A n A Distributive law m (A B) m A m B Distributive law Department of ECE Page 4 10EC36 A ‗vector‘ is represented graphically by a directed line segment. A ‗Unit vector‘ is a vector of unit magnitude and directed along ‗that vector‘. aˆ A is a Unit vector along the direction of A . Thus, the graphical representation of A and aˆ A are A Vector A Unit vector aˆ A Also aˆ A A / A or A aˆ A A citystudentsgroup Product of two or more vectors : (1) Dot Product ( . ) A . B A ( B COS θ OR { A COS θ } B , 0 θ π B B A Cos θ A B Cos θ A A . B = B . A (A Scalar quantity) (2) CROSS PRODUCT (X) C = A x B = A B SIN θ ˆ Ex , where ' θ ' is angle between A and B ( 0 θ π ) and nˆ is unit vector perpendicular to plane of A and B directed such that A B C form a right handed system of vectors A x B - B x A A x ( B C) A x B A x C Department of ECE Page 5 10EC36 CO-ORDINATE SYSTEMS : For an explicit representation of a vector quantity, a ‗co-ordinate system‘ is essential. Different systems used : Sl.No. System Co-ordinate variables Unit vectors 1. Rectangular x, y, z ax , ay , az 2. Cylindrical ρ, , z aρ , a , az 3. Spherical r, , ar , a , a These are ‗ORTHOGONAL‗ i.e., unit vectors in such system of co-ordinates are mutually perpendicular in the right circular way. citystudentsgroup i.e., x y z , z , r RECTANGULAR CO-ORDINATE SYSTEM : Z x=0 plane az p y=0 Y plane ay ax z=0 plane X a x . a y a y . a z a z . a x 0 a x x a y a z a y x a z a x a z x a x a y az is in direction of ‗advance‘ of a right circular screw as is turned from ax to ay Co-ordinate variable ‗x‘ is intersection of planes OYX and OXZ .e, z = 0 & y = 0 Location of point P : If the point P is at a distance of r from O, then If the components of r along X, Y, Z are x, y, z then r x a x y a y z az r ar Department of ECE Page 6 10EC36 Equation of Vector AB : If OA A Ax a x Ay a y Az a z B and OB B Bx a x By a y Bz a z then B AB A AB B or AB B - A 0 A A where As , Ay & Az are components of A along X, Y and Z and Bs , By & Bz are components of B along X, Y and Z Dot and Cross Products : A . B (Ax a x Ay a y Az a z ) . (Bx a x By a y Bz a z ) Ax Bx Ay By Az Cz citystudentsgroup A a A A x B (Ax a x y y z az ) x (Bx a x By a y Bz a z ) Taking 'Cross products' term by term and grouping, we get a x a y a z A x B Ax Ay Az Bx By Bz Ax Ay Az B B B A . (B x C ) x y z Cx Cy Cz If A, B and C are non zero vectors, 0 (i) A . B 0 then Cos θ 0 .e., θ 90 A and B are perpendicular A x B 0 then Sin θ 0 θ 0 A and B are parallel (ii) A . ( B x C) represents the volume of a parallelopoid of sides A , B and C Unit Vector along AB AB a AB AB where Vector length AB AB (AB . AB) Department of ECE Page 7 10EC36 Differential length, surface and volume elements in rectangular co-ordinate systems r x aˆ x y aˆ y z aˆ z r r r dr x dx y dy z dz dr dx aˆ x dy aˆ y dz aˆ z 2 2 2 1/2 Differential length dr [ dx dy dz ] - - - - - 1 Differential surface element, ds r 1. to z : dxdy aˆ z r 2. to z : dxdy aˆ z ------ 2 r citystudentsgroup 3. to z : dxdy aˆ z Differential Volume element dv = dx dy dz ------ 3 z dx p‘ p dz dy r r d r 0 y x Other Co-ordinate sy tems :- Depending on the geometry of problem it is easier if we use the appropriate co-ordinate system than to use the Car esian co-ordinate system always. For problems having cylindrical symmetry cylindrical -ordinate system is to be used while for applications having spherical symmetry spherical o-ordinate system is preferred. Cylindrical Co-ordiante systems :- z P(ρ, , z) x = ρ Cos y = ρ Sin a r ρ z = z z 2 2 0 ρ x y -1 ap r y φ tan y / x z z ρ x Department of ECE Page 8 10EC36 r x aˆ x y aˆ y z aˆ z r ρ Cos aˆ x ρ Sin aˆ y z aˆ z r r r dr ρ d ρ d z dz - - - - - - 1 r r h r ρ Cos aˆ x Sin aˆ y ρ aˆ hρ aˆρ ; ρ ρ 1 r r r - ρ Sin aˆ ρ Cos aˆ aˆ ρ aˆ ; h ρ x y r aˆ h r 1 z z z z citystudentsgroup Thus unit vectors in (ρ, , z) systems can be expressed in (x,y,z) system as aρ Cos a x Sin a y a x Cos a Sin a a - Sin a x Cos a y a y Sin a Cos a a z a z ; a , a and a z are orthogonal Further , dr d ρ aˆ ρ d aˆ dz aˆ z - - - - - - 2 2 2 2 2 and dr d ρ (ρ d) (dz) Differential areas : ds aˆz (d ρ) (ρ d) . aˆz ds aˆ (dz) (ρ d) . aˆ - - - - - - - 3 ds aˆ (d ρ dz) aˆ Differential volume : d (d ρ) (ρ d) (dz) or d ρ d ρ dz - - - - - 4 Spherical Co-ordinate Systems :- Z X = r Sin Cos Y = r Sin Sin z p Z = r Cos R r 0 y Y x r Sin X Department of ECE Page 9 10EC36 R r Sin Cos aˆ x r Sin Sin aˆ y r Cos aˆ z aˆ R / R Sin Cos aˆ Sin Sin aˆ Cos aˆ r x y z r r R R aˆ / Cos Cos aˆ x Cos Sin aˆ y Sin aˆ z R R aˆ / - Sin aˆ x Cos aˆ y R R R dR dr d d r dR dr aˆ r r d aˆ r Sin d aˆ 2 d S r r Sin d d d S r2 Sin dr d d S r dr d 2 d v r Sin dr d d General Orthogonal Curvilinear Co-ordinates :- z u1 a3 u3 a1 u2 a2 y x Co-ordinate Variables : (u1 , u2, u3) ; Here u1 is Intersection of surfaces u2 = C & u3 = C u2 is Intersection of surfaces u1 = C & u3 = C u3 is Intersection of surfaces u1 = C & u2 = C aˆ1 , aˆ 2 , aˆ3 are ubnit vectors tangential to u1 , u2 & u3 System is Orthogonal if aˆ1 . aˆ 2 0 , aˆ 2 . aˆ3 0 & aˆ3 . aˆ1 0 If R x aˆ x y aˆ y z aˆ z & x, y, z are functions of u1 , u2 & u3 then d R R du R du R du u 1 1 u 2 2 u 3 3 h1 du1 aˆ1 h2 du2 aˆ 2 h3 du3 aˆ3 where h1 , h2 , h3 are scale factors ; R R R h , h , h 3 1 2 u1 u2 u3 Department of ECE Page 10 10EC36 Co-ordinate Variables, unit Vectors and Scale factors in different systems Systems Co-ordinate Variables Unit Vector Scale factors General u1 u2 u3 a1 a2 a3 h1 h2 h3 Rectangular x y z ax ay az 1 1 1 Cylindrical ρ z a ρ a az 1 ρ 1 Spherical r ar a a 1 r r sin Transformation equations (x,y,z interms of cylindrical and spherical co-ordinate system variables) Cylindrical : x = ρ Cos , y = ρ Sin , z = z ; ρ 0, 0 2 - < z < Spherical x = r Sin Cos , y = r Sin Cos , z = r Sin r 0 , 0 , 0 2 V 1 v aˆ 1 v aˆ 1 v aˆ 1 h 2 3 h 1 u 1 2 u 2 h 3 u 3 1 h .
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