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ELEC 401 – Microwave Electronics ELEC 401 MICROWAVE ELECTRONICS Lecture 3 Instructor: M. İrşadi Aksun Acknowledgements: 1. Animation on the visualization of EM waves was taken from the following web page: http://web.mit.edu/~sdavies/MacData/afs.course.lockers/8/8.901/2007/Tues dayFeb20/graphics/ M. I. Aksun Koç University 1/18 ELEC 401 – Microwave Electronics Outline Chapter 1: Motivation & Introduction Chapter 2: Review of EM Wave Theory Chapter 3: Plane Electromagnetic Waves Chapter 4: Transmission Lines (TL) Chapter 5: Microwave Network Characterization Chapter 6: Smith Chart & Impedance Matching Chapter 7: Passive Microwave Components M. I. Aksun Koç University 2/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Wave equations are the governing differential equations of EM waves ~ E 2 E 0 t 2~ 2E 2E 0 ~ 2~ H 2 H 0 t 2 2 Wave equation Helmholtz equation 2H 2H 0 Facts: Wave equations - have solutions representing wave nature of the fields; - are second order partial differential equations; - have unique solutions if two boundary conditions are introduced. So, we need BOUNDRY CONDITIONS M. I. Aksun Koç University 3/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Boundary Conditions on E and H: E dl j B ds C D1 ,B1 ̂ n̂ n̂1 Pill-box n̂ ̂ ds S D 2 ,B 2 2 , 2 H dl j D ds J ds C l S S H1 τˆl H2 τˆl jωD nˆ1l h J nˆ1l h lim J nˆ1h nˆ1J s Contour C E2 ,H 2 Δh 0 as h0 M. I. Aksun Koç University E1 E2 as h0 Medium 2: h E1 τˆl E2 τˆl j B nˆ1l h Medium 1: 1 , 1 E1 , H1 S A/m H1 H 2 J s 4/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Comments on the surface current density Js : lim J nˆ1h nˆ1J s Δh 0 Medium 1: 1 , 1 Volume current density must go to as h goes zero. Conducting Medium 2: Js J s (0)e z M. I. Aksun Koç University A/m z 2 , 2 , Current Sheet: Current is distributed over a very narrow sheet of conductor. Facts: 1. Current flow in a good conductor is practically confined to the layer next to the surface, whose thickness depends inversely on both the conductivity of the material and the frequency of the field. 2. Conductor is modeled as a sheet of current with a finite surface current density. 3. This layer of finite thickness is called the skin depth of that specific conducting material. 5/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Boundary Conditions on D and B: D ds dv S D1 ,B1 ̂ n̂ n̂1 Pill-box n̂ ̂ Medium 1: 1 , 1 Medium 2: 2 , 2 h D 2 ,B 2 l D1 nˆ S D2 nˆ S sw ρ S Δh lim h s S E1 , H1 V h 0 C/m 2 as h0 D1n D2n s Similarly, from B ds 0 S Contour C E2 ,H 2 M. I. Aksun Koç University B1n B2 n 6/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Let us review the boundary conditions with words: 1. Tangential E fields are continuous D1 ,B1 ̂ n̂ Pill-box n̂1 n̂ ̂ Medium 1: 1 , 1 S Medium 2: E1 , H1 h D 2 ,B 2 l 2 , 2 E1 E2 2. Tangential H fields are discontinuous by the amount of the surface current H1 H 2 J s 3. Normal components of D are discontinuous by the amount of the surface charge D1n D2 n s Contour C E2 ,H 2 4. Normal components of B are continuous B1n B2 n M. I. Aksun Koç University 7/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Special case: Dielectric-PEC interface ( ) 1. Tangential E field is zero E1 0 Medium 1: 1 , 1 Conducting Medium 2: Js J s (0)e z z 2 , 2 , 2. Tangential H field is equal to the surface current density H1 J s 3. Normal component of D is equal to the surface charge density D1n s 4. Normal component of B is zero B1n 0 M. I. Aksun Koç University 8/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory So far, we have derived the governing equation of the fields involved in waves, and the necessary boundary conditions; Now, it is time to see them in action to determine the fields in a wave generated by a time-varying current source !!!! M. I. Aksun Koç University 9/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Example: Assume an infinite sheet of electric surface current density J s xˆJ 0 A/m is placed on a z 0 plane between free space for z < 0 and a dielectric medium with for z > 0. Find the resulting electric and magnetic fields in both regions. x About the problem: 0 0 r x̂J 0 A/m 1. The surface current density is uniform on the z = 0 plane; z 2. Magnitude of the current density is j t J0 with e time dependence; 3. We assume that layers are lossless, isotropic, homogenous, and semiinfinite in extent. M. I. Aksun Koç University 10/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory About the solution: The best way to solve the wave equation in a piecewise homogeneous geometry is 1. to find the source-free solutions with unknown coefficients in each homogeneous sub-region; 2. to apply the necessary boundary conditions at the interfaces to account for the boundaries between different media; and then 3. to apply the boundary conditions at the sources to incorporate the influence of the sources into the solution. M. I. Aksun Koç University 11/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory So, let us implement these steps one-by-one: 1. Write the frequency-domain wave equation in a source-free medium as 2 E ω2 με E 0 From the source distribution and geometry, the functional form of the solutions can be predicted. E xˆE x z yˆ E y z zˆE z z H xˆH x z yˆ H y z zˆH z z Source and geometry are independent of x and y, and so are the solutions!!! WHY ? M. I. Aksun Koç University 12/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory As we have noted earlier, the field components must satisfy Maxwell’s equations, so using 0, 0, 0 x y z Maxwell’s equations can be written as follows: E xˆ E y yˆ E x jH x xˆ jH y yˆ z z H xˆ H y yˆ H x jωE x xˆ jωE y yˆ z z As a result, two decoupled set of solutions are obtained: Ex jH y ; 0 H z z 2Ey H y H x jE y ; jEx ; 0 Ez z z 2 Ex E y z jH x ; M. I. Aksun Koç University z z 2 2 2E y 0 E y , H x 2E x 0 Ex , H y 13/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory - Note that these solutions are possible solutions; either only one of them or both with different weights can exist depending upon the boundary conditions. x - Since the surface current density is in xdirection, it gives rise to a discontinuity only on Hy . 0 0 r z x̂J 0 A/m - Therefore, it can be interpreted that the source excites only one of the two possible solutions, Ex , H y with the following governing equation: 2 Ex z 2 2E x 0 M. I. Aksun Koç University 0 z < 0 k0 200 z < 0 0 r z > 0 k 20r 0 z > 0 14/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory - Then, the solutions can be written as E x z Ae jk0 z Be jk0 z , for z < 0 E x z Ce jkz De jkz , for z > 0 where A, B, C, D are unknown coefficients, and k0 and k are the wavenumbers in free-space and materials, respectively. - The corresponding magnetic field Hy can be obtained from FaradayMaxwell equations as follows: E x z jH y z z H y z H y z M. I. Aksun Koç University k0 Ae jk0 z k Ce jkz k0 Be jk0 z , for z < 0 k De jkz , for z > 0 15/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory - There are four unknowns (A, B, C and D) to be determined by using the necessary boundary conditions: 1. Source is at z=0, and no wave can travel from infinity towards source, implying that A and D must be zero. Be , for z < 0 Ex z jkz Ce , for z > 0 jk0 z k0 jk0 z Be , for z < 0 H y z k Ce jkz , for z > 0 2. The remaining boundary conditions are at the interface: E x z 0 E x z 0 H y z 0 H y z 0 J 0 M. I. Aksun Koç University BC B J 0 k0 k 16/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory Energy Flow and Poynting’s Vector -The general law of the conservation of energy states that if an object radiates electromagnetic waves (for example, light), it loses energy. - From Maxwell’s equations, one can get the following relation: ~ ~ ~ ~ 1 ~ ~ 1 ~ ~ E H ds E J E E H H dV t V 2 2 V S Inflow of energy due to source outsideV Energy flux density (W/m2) M. I. Aksun Koç University Energy dissipated in volume V Rate of increase in stored energy ~ ~ ~ P r, t E r, t Hr, t Instantaneous Poynting’s vector 17/18 ELEC 401 – Microwave Electronics Review of EM Wave Theory For most practical applications, the time-averaged energy and power quantities are required because most systems usually respond to the average power, rather than its instantaneous values. T 1 ~ Pr Pr, t dt : time-averaged power density T 0 For time-harmonic fields, the instantaneous fields are written as 1 ~ Er, t ReEr e j t Er e j t E r e j t 2 1 ~ Hr, t ReHr e j t Hr e j t H r e j t 2 2 2 1 1 j 2 t j 2 t Pr dt E H E H E H e dt E H e 2π 4 0 0 2 Re E H 0 0 M. I. Aksun Koç University 0 dt 2 1 Re E H* 2 18/18