ENE 325 Electromagnetic Fields and Waves Lecture 12 Uniform plane waves RS

Introduction (2) From Maxwell’s equations, if the electric field is changing with time, then the magnetic field varies spatially in a direction normal to its orientation direction Knowledge of fields in media and boundary conditions allows useful applications of material properties to microwave components A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation RS

Uniform plane wave propagation direction Wavefront 12/3/2018 RS http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/images/07-EB_Light_320.jpg http://astro1.panet.utoledo.edu/~lsa/_color/10_wavefront.gif RS 12/3/2018

Maxwell’s equations (1) (2) (3) (4) RS

Integral forms of Maxwell’s equations RS

Divergence theorem Stokes’ theorem RS

Maxwell’s equations in phasor form Assume and ejt time dependence, (1) (2) (3) (4) RS

Fields are assumed to be sinusoidal or harmonic, and time dependence with steady-state conditions Time dependence form: Phasor form: RS

Fields in dielectric media (1)   RS

Fields in dielectric media (2) may be complex then  can be complex and can be expressed as Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. The loss of dielectric material may be considered as an equivalent conductor loss if the material has a conductivity  . Loss tangent is defined as RS

Anisotropic dielectrics The most general linear relation of anisotropic dielectrics can be expressed in the form of a tensor which can be written in matrix form as RS

Analogous situations for magnetic media (1)   RS

Analogous situations for magnetic media (2) may be complex then  can be complex and can be expressed as Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. RS

Anisotropic magnetic material The most general linear relation of anisotropic material can be expressed in the form of a tensor which can be written in matrix form as RS

General plane wave equations (1) Consider medium free of charge For linear, isotropic, homogeneous, and time-invariant medium, assuming no free magnetic current, (1) (2) RS

Maxwell’s equations in free space  = 0, r = 1, r = 1 0 = 4x10-7 Henrys/m 0 = 8.854x10-12 Farad/m Ampère’s law Faraday’s law RS

General plane wave equations (2) Take curl of (2), we yield From then For charge free medium RS

Helmholtz wave equations For electric field For magnetic field RS

Plane waves in a general lossy medium Transformation from time to frequency domain Therefore RS

Plane waves in a general lossy medium or where This  term is called propagation constant or we can write  = +j where  = attenuation constant (Np/m)  = phase constant (rad/m) RS

Solutions of Helmholtz equations Assuming the electric field is in x-direction and the wave is propagating in z- direction The instantaneous form of the solutions Consider only the forward-propagating wave, we have Use Maxwell’s equation, we get RS

Solutions of Helmholtz equations in phasor forms Showing the forward-propagating fields without time-harmonic terms. Conversion between instantaneous and phasor form Instantaneous field = Re(ejt  phasor field) RS

Wave impedance For any medium, For free space RS

Propagating fields relations   RS

Propagation in lossless-charge free media Attenuation constant  = 0, conductivity  = 0 Propagation constant Propagation velocity for free space up = 3108 m/s (speed of light) for non-magnetic lossless dielectric (r = 1), RS

Propagation in lossless-charge free media intrinsic impedance wavelength RS

The Poynting theorem and power transmission Total power leaving the surface Joule’s law for instantaneous power dissipated per volume (dissi- pated by heat) Rate of change of energy stored In the fields Instantaneous Poynting vector RS

Time averaged power density Time-averaged power density is easily calculated in the phasor form. For lossless medium W/m2 RS

Uniform plane wave (UPW) power transmission for lossless medium W/m2 amount of power W RS

Reflection and transmission of UPW at normal incidence RS

Reflection coefficient From and We can find the reflection coefficient as RS 12/3/2018

Transmission coefficient and we can find the transmission coefficient as RS

Power transmission for 2 perfect dielectrics (1) 1 and 2 are both real positive quantities and 1 = 2 = 0.  Average incident power densities W/m2 RS

Power transmission for 2 perfect dielectrics (2) Average reflected power densities W/m2 RS

Power transmission for 2 perfect dielectrics (3) Average transmitted power densities W/m2 or RS