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15A02501 ELECTRICAL MEASUREMENTS 10.2 General Wave Equations In general the wave equations can be obtained by relating the space and time variations of the electric and magnetic fields, using the Maxwell's equations. To obtain general wave equations, let us assume that the electric and magnetic fields exist in a linear, homogeneous and isotropic medium with the parameters 1, € and o. Also assume that the medium is source free which clearly gives the idea about the charge free medium. Assume that the medium obeys the ohm’s law i.e. J = o E. Then the Maxwell's equations are given by, VxE = AS ---(1) Vx = oE+e5- ++(2) V.B = 0 ie. V.H=0 ---8) ‘ V.D = 0 ie. V.E=0 (4) To eliminate H from equation (1), taking curl on both the sides of equation (1), we get, Vx(VxE) = -o( v3 | (5) V_ operates indicates differentiation with respect to space while a operates differentiation with respect to time. Both are independent of each other, the operators can be interchanged. So we get, VxVxE = -udoxm -(6) Substituting value of Vx H from equation (2), we get, VxVxE = 13 [ob+e5F| z aE 2E VxVXxE = “noes +7) Now according to the vector identity, VxVxE = V(V.E)-V7E «.(8) Substituting V.E = 0 from equation (4), we can modify equation (8) as, VxVxE = -V7E +(9) Substituting value of VxV x E from equation (9) in equation (2) we get, 117 15A02501 ELECTRICAL MEASUREMENTS aE 2E 78 = ott net os aE 2E V’E = wos tnels -..(10) This is the wave equation for the electric field E Now multiplying both the sides of equation (10) by ¢, = deE OE 2 = palate V7(eE) = po H THE 2 ; 5 aD 2D 2 = — ie v*D HO- tHE 7 «-(11) This is the wave equation for D in uniform medium. Exactly on the similar lines, the wave equation for H can be obtained by taking curl on both the sides of equation (2), we get, _ vx(VxF) = Vx(oB)+eVx «..(12) As V operator and 2 represent independent relationship between the two, we can interchange them as follows. VxVxH = OV xB +e 2 (VE) «+.(13) Substituting Vx E = “1 in equation (12), we get, {ot}(o2) VxVxH = nope «= (14) VxVxH From the vector indentity, VxVxH = V(V.H)-V°H ++(15) Substituting V.H = 0 from equation (4) in equation (15), we get VxVxH = -V?H (16) Substituting value of VxVxH _ in equation (14) we get, -V7H = “ieee = = ie. V7H = pore eH ea Wj This is the wave equation for the magnetic field H. Now multiplying both the sides by LL, we get, rt onH a?(uH) Vv2q.H) = ho + he 118 15A02501 ELECTRICAL MEASUREMENTS Expressing D in rectangular co-ordinate system, Vx = £ [D. a, +D, a, +D.a,] Writing curl of H on left of equation, OH, @Hy]_ [AH, O@H,]_ OHy dH, ]_ [he az 7x |?” * Ox oy = é = & = i [Dx ax + Dy ay +D, az] As His in y-direction, H, =H, = 0, aH). SH Bn = = = ap tas 2 =5y [Dx dx +Dy ay +Dz az] oH Also Hy is not changing with x, as it is uniform in x-y plane, so =0 a a sec ap By LDs ax + Dy ay +D2 a2] Equating L.H.S. and R.H.S. of above equation directionwise, we can write, _OHy _ dDy oz ~~ at éHy ek, a Weil “an ft prey D=cE eHy oEX = = Pay 8 Oz ° ot a) Now consider Maxwell's equation derived from Faraday's law, = aB VxE = = ae Using rectangular co-ordinate system, we can write, OEy OE], [OE OF], [0B Ey], dz dy|* |dz Ox|** |éy dx |°* =-£ [B.a, +B, ay +B, az] 120 15A02501 ELECTRICAL MEASUREMENTS As E is in x-direction, Ey =E, =0, . OE, _ | OE. Om x < ~ . Bz BY + By Be =~ Ze [Bx Bs + By ay +B xa] Also component E, is not varying with y, as it is uniform in x-y plane, so 2E =0, ey OE, _ é a oe = tis A [Bx ax + By ay +B,a,] Equating L.H.S. and R.H.S. of above equation directionwise, we can write, OE, _ _6By oz ~—stétk dE, _ _ oHy a an ep B=npH éHy 1 Ex ot pez ~@) Differentiating equation (1) with respect to t, a [eHy a2 Ey 3(52 * oe “er Now differentiating equation (2) with respect to z, 6f@Hy]_ 1 aE, Bat * “1 oe ollie Now observe L.H.S. of equations (3) and (4). As we can change order of differentiation, L.H.S. of equations (3) and (4) is same. So equating R.H.S. of both the equations, 22 Es 107 Es et? uM éz? 6? Ey 1 0? Ex =— «i ® ot? He dz? ® According to the results in physics, 1 vs —== jue where v is the velocity of propagation also called wave velocity. For the free space it is denoted by c and its value is 3x 10° m/s. 121 15A02501 ELECTRICAL MEASUREMENTS This equation is similar to equation (7) representing two components of a magnetic field one in forward direction while other in backward direction. From equations (7) and (11) it is clear that when we assume x component for E, it results in y component for H. Both EandHi are in time phase and both are perpendicular to each other. Both these fields lie in a plane which is perpendicular to the direction of wave propagation. Thus E and H together form transverse electromagnetic (TEM) wave ; with one forward travelling wave in the positive z-direction with velocity i and another backward travelling wave in negative z-direction with the same velocity. Thus E and H are only the functions of direction of travel and time. In general, when any wave propagates in the medium, it gets attenuated. The amplitude of the signal reduces. This is represented by an attenuation constant a. It is measured in neper per meter (Np/m). But practically it is expressed in decibel (dB). The conversion between a basic unit neper (Np) and decibel (dB) is given by 1 Np = 8.686 dB It is also observed that when a wave propagates, phase change also takes place. Such a phase change is expressed by a phase constant f. It is measured in radian per meter (rad/m). So attenuation constant (a) and phase constant (B) together constitutes a propagation constant of medium for uniform plane wave. It is represent by y. It is expressed per unit length as y = a+jp w+ (12) The ratio of amplitudes of E to H of the waves in either direction is called intrinsic impedance of the material in which wave is travelling. It is denoted by n and given by, +» (13) ne ti-Jf4 a .. (14) 123 15A02501 ELECTRICAL MEASUREMENTS For free space, intrinsic impedance is denoted by no, no = [us =12022=3772 ww. (15) 0 and g eeu! =3x108 m/s w. (16) Moto In general, wave repeats itself after 2x radians. In otherwords, if 4 is the length of one cycle of sinusoidal signal, then signal changes phase by 360° or 2x radians. So we can write relation between ’ and f as, 2n em oy m a. (17) Multiplying both sides of equation (17) by frequency f, Thus velocity of propagation or wave velocity is given by, v= fk m/s ww (18) 9.3 Wave Equations in Phasor Form An electromagnetic wave in a medium can be completely defined if intrinsic impedance (n) and propagation constant (y) of a medium is known. Thus it is necessary to derive the expressions for ny and y in terms of the properties of a medium such as permeability (4), permittivity (¢), conductivity (o) etc. Consider Maxwell's equation derived from Faraday's law, VxE = Fa at) Taking curl on both the sides of the equation, VxVxE = -1 fox] ™ a wv VxVxE = «| 5 (exH)| wat 124 15A02501 ELECTRICAL MEASUREMENTS In similar way, we can write another phasor equation as, V?H = [jou(o+jos)JH w (7) The terms inside the bracket in equations (6) and (7) are exactly similar and represent the properties of the medium in which wave is propagating. The total bracket is the square of a propagation constant y, hence we can rewrite equations (6) and (7) as, V2E = y2E and V?H = 7? So the propagation constant y can be expressed in terms of properties of the medium as, 1 = a 4)P = \jou(e + joe) 8) The real and imaginary parts of y are attenuation constant (a) and phase constant (B) and both can be expressed in terms of the properties of the medium, 2 ee Sey me 1+(&) =| (9) \ 2 and p= 0 Al h (3) «| vw» (10) The intrinsic impedance of a medium can be expressed interms of the properties of a medium and is given by, = {Jeu n o+joe old) It can also be expressed in polar form as |n| 28 where Oe and n+(Z) In| = tan20 = 2 0° 9< 45° Ee Let us summarize the equations which are helpful in describing the electromagnetic waves (uniform plane waves). Table 9.1 lists equations describing the propagation of EM waves in a medium. 126 15A02501 ELECTRICAL MEASUREMENTS 9.4 Uniform Plane Wave in Perfect Dielectric If a medium, through which the uniform plane wave is propagating, is perfect dielectric (which is also called lossless dielectric), then the conductivity is zero i.e. o=0. Let the permittivity permeability of the medium be e=e9e, and p=popr respectively. The propagation constant y is given by, Y= Vjou (0+ jae) = tjoJue y =a+jp=tjojpe we) From equation (1) it is clear that, propagation constant is purely imaginary. It indicates in a perfect dielectric medium, attenuation constant a is zero. Let us select value of B which gives propagation of wave in positive z-direction. .a=0, B = jue s+» (2) Similarly an intrinsic impedance for a perfect dielectric medium is given by, jou in “ pee = iE 2 + @) Thus intrinsic impedance n is real resistive. That means phase angle of intrinsic impedance is zero. But the phase angle of intrinsic impedance is zero means phase difference between E and H is zero. In other words, for a perfect dielectric, both the fields E and H, are in phase. As in perfect dielectric, o =0, attenuation constant (a) is also zero. As wave propagates, only the phase ($8) changes. Thus no attenuation i.e. a = 0 means no loss. Key Point: So perfect diectric medium is also called lossless dielectric. The velocity of propagation in the perfect dielectric is given by, 1 @ =< » @) - Jue B If 4 is the wavelength of one cycle of the propagating wave then velocity is given by v = Af m/s >we, the loss tangent is very high, thus a medium is said to be good conductor. When o<>1 @e The propagation constant y is given by, y = yjou(o +joe) ~» (1) As o>>we, we can neglect imaginary part (jwe), Y= Jjous Jou Jj But j = 1290° you o yi 290° ona 245° ous [cos45°+jsin 45°] Dh ask rae = \(2nf)us cB (1+j »| . ¥ a +jp= Vxfus +j Vrfuo a» (2) Thus for good conductor, a= xfs Np/m and B= Jxfuo rad/m ~ " ~~ i] " ~ i ~ i 132 15A02501 ELECTRICAL MEASUREMENTS For good conductor, a and f are equal and both aré directly proportional to the square root of frequency (f) and conductivity (o). The intrinsic impedance of a good conductor is given by - [ie 4 oO +jwe + @) But for good conductor, 5 >>jwe, « tn ag VE Vo o Wi But jj = {1 290° = 1 245°= cos 45°+jsin 45° ‘= (id) Substituting value of Jj , Po pied Jy i i ae i Joe j ig (+i 2nf; Z 1 x (1+j 1) . 4) The angle of intrinsic impedance is 45°. As we have already studied that for perfect dielectric ie. zero conductivity, the intrinsic impedance angle is zero and for the good conductor angle is 45°. Moreover the intrinsic impedance has only a positive angle. This clearly indicates that the field H may lag the field E by at the most 45°. Consider only the component of the electric field E, travelling in positive z-direction. When it travels in good conductor, the conductivity is very high and attenuation constant a is also high. Thus we can write such a component in phasor form as E, = Ej, e-*%e-ife w. (5) 133 15A02501 ELECTRICAL MEASUREMENTS (0.37) E, Fig. 9.3 Exponential decay in the amplitude of field component of E with the distance along +z direction From above plot, it is clear that in 15 distance, amplitude reduces to 37% of its original value. For a good conductor, amplitude reduces to almost zero within 26 or35 distance. Thus uniform plane wave can not travel large distance through good conductor. This concept is used in a shielding of a conductor. In a co-axial cable, the inner conductor carries the signal while the outer is shield which is made up of a material having properties of good conductor. So even if there is an external interference, its amplitude reduces to zero within a very short distance due to the outer conductor. Hence the signal carried by inner conductor is not interfered by an external interference. 135 15A02501 ELECTRICAL MEASUREMENTS Extemal interference Inner conductor Fig. 9.4 Shielding of a co-axial cable The intrinsic impedance of a good conductor interms of skin depth 6 is given by, £3 \ at n= ( + ay) a0 w (7) os Gd The velocity of propagation is given by, y 8 7 Jxfao 1 V2Jnfuo " ~ fous v= 2% = 0S m/s » (8) Ho Similarly the wavelength A is given by, A= oF 2x5 m #'(9) B Equations (8) and (9) gives the velocity of propagation and wavelength expressed interms of skin depth 5 (ss p-3) P 8.9 Poynting Vector and Poynting Theorem By the means of electromagnetic (EM) waves, an energy can be transported from transmitter to receiver. The energy stored in an electric field and magnetic field is transmitted at a certain rate of energy flow which can be calculated with the help of Poynting theorem. As we know E and H are basic fields. E is electric field expressed in V/m; while H is magnetic field measured in A/m. So if we take product of two 136 
