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4/26/2005 Wave Propagation present 1/1 Jim Stiles The Univ. of Kansas Dept. of EECS III Antenna Fundamentals Now we will discuss what occurs between the transmitter and receiver. Recall this region is called the channel, and we couple an electromagnetic wave to/from the channel using an antenna. A. Wave Propagation We must first review the basics of electromagnetic propagation in free-space. HO: EM Wave Propagation in Free-Space HO: The Poynting Vector11/8/2006 Electromagnetic Wave Propagation 1/9 Jim Stiles The Univ. of Kansas Dept. of EECS Electromagnetic Wave Propagation Maxwell’s equations were cobbled together from a variety of results from different scientists (e.g. Ampere, Faraday), whose work mainly was done using either static or slowly time-varying sources and fields. Maxwell brought these results together to form a complete theory of electromagnetics—a theory that then predicted a most startling result! To see this result, consider first the free-space Maxwell’s Equations in a source-free region (e.g., a vacuum). In other words, the fields in a region far away from the current and charges that created them: ()()()()()()00r,xr,r,xr,r, 0r, 0ttttttttµε∂∇=∂∂∇=−∂∇⋅ =∇⋅ =EBBEEB11/8/2006 Electromagnetic Wave Propagation 2/9 Jim Stiles The Univ. of Kansas Dept. of EECS Say we take the curl of Faraday’s Law: ()()xr,xx r,ttt∂∇∇∇ =−∂BE Inserting Ampere’s Law into this, we get: ()()()002002r,xx r,r,ttttttµεµε∂⎛⎞∂∇∇ =−⎜⎟∂∂⎝⎠∂=−∂EEE Recalling that if ()0r∇⋅ =E then () ()2xxrr∇∇ ∇EE , we can write the following differential equation, one which describes the behavior on an electric field in a vacuum: ()()22002r,0tr,ttµε∂∇+ =∂EE This result is none as the vector wave equation, and is very similar to the transmission line wave equations we studied at the beginning of this class. This result means that electric field ()r,tE cannot be any arbitrary function of position r and time t. Instead, an electric field ()r,tE is physically possible only if it satisfies the differential equation above!11/8/2006 Electromagnetic Wave Propagation 3/9 Jim Stiles The Univ. of Kansas Dept. of EECS Q: So, what are some solutions to this equation? A: The simplest solution is the plane-wave solution. It is: ()()()00jtzxyˆˆr,t E E eωµε−=+Exy For this solution, the electric field is varying with time in a sinusoidal manner (that eigen function thing!), with an angular frequency of radians/secω. Note this field is a function of spatial coordinate z only, but the direction of the electric field is orthogonal to the z-axis. Q: What does this equation tell us about ()r,tE? What is this electric field doing?? A: Lets plot (){}Re r ,tE as a function of position z, for different times t, and find out!11/8/2006 Electromagnetic Wave Propagation 4/9 Jim Stiles The Univ. of Kansas Dept. of EECS Here the red dot indicates plane of constant phase, for this case a phase of zero radians, i.e., ()000tzφω µε=−=. Note that this dot appears to be moving forward along the z- axis as a function of time. The electric field is moving ! (){}Re , 1zt=E z z z t =1 t =2 t =3 (){}Re , 2zt=E (){}Re , 3zt=E11/8/2006 Electromagnetic Wave Propagation 5/9 Jim Stiles The Univ. of Kansas Dept. of EECS Q: How fast is it moving? A: Lets see how fast the red dot (i.e., the plane of constant phase) is moving! Rearranging ()000tzωµε−= , we get the position z of the dot as a function of time t : 00tzµε= Its velocity is just the time derivative of its position: 001pdzvdtµε== Hey we can calculate this! The electric field is moving at a velocity of: ()( )00-7 -12114x10 8854x10meters secondpv.µεπ==⎡⎤=⎢⎥⎣⎦83x10 A: True! We find that the magnetic field will likewise move in the same direction and with the same velocity as the electric field. Q: Hey wait a minute! 3 x 108 meters/second—that’s the speed of light!?!11/8/2006 Electromagnetic Wave Propagation 6/9 Jim Stiles The Univ. of Kansas Dept. of EECS We call the combination of the two fields a propagating (i.e., moving) electromagnetic wave. Light is a propagating electromagnetic wave! This was a stunning result in Maxwell’s time. No one had linked light with the phenomena of electricity and magnetism. Among other things, it meant that “light” could be made with much greater wavelengths (i.e., lower frequencies) than the light visible to us humans. Henrich Hertz first succeeded in creating and measuring this low frequency “light”. Since then, humans have put this low-frequency light to great use. We often refer to it as a “radio waves”—a propagating electromagnetic wave with a frequency in the range of 1 MHz to 20 GHz. We use it for all “wireless” technologies !11/8/2006 Electromagnetic Wave Propagation 7/9 Jim Stiles The Univ. of Kansas Dept. of EECS Given the results above, we can rewrite our plane-wave solution as: ()()()()()()()00jtzxyzjtcxyjzjtcxyˆˆr,t E E eˆˆEEeˆˆEEeeωµεωωω−−−=+=+=+Exyxyxy Now, making the definition: 02/kradiansmetercωπλ==⎡⎤⎣⎦ We get: ()()0jkz j txyˆˆr,t E E e eω−=+Exy Q: This plane-wave solution reminds me somewhat of the solution to the telegrapher’s equations, with 0kanalogous to β. Is this just a coincidence? A: Nope! Since we have voltages and currents along our transmission line, we must also have electric fields and magnetic fields. In fact, the voltage and current wave solutions for a transmission line can likewise be expressed as propagating electric and magnetic (i.e., electromagnetic) fields. But, there is one super-huge difference between the transmission line solutions and the plane wave solution presented here!11/8/2006 Electromagnetic Wave Propagation 8/9 Jim Stiles The Univ. of Kansas Dept. of EECS The propagating wave along at transmission line is constrained to one of two directions—the plus z direction or the minus z direction. In contrast, nothing constrains a plane wave in free space—it can propagate in any and all directions! Although the plane-wave solution shown above propagates in the ˆz direction, the solution would be equally valid in the ˆ−y direction or ˆx direction, or any arbitrary direction ˆk . The only constraint is that the direction of the electric field vector be orthogonal to the direction of wave


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KU EECS 622 - III Antenna Fundamentals

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