Transmission Lines Embedded in Arbitrary Anisotropic Media

Transcription

1 ECE 53 1 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) rcrumpf@utep.edu Lecture #4 Transmission Lines Embedded in Arbitrary Anisotropic Media Lecture 4 1 Lecture Outline RF transmission lines Transmission lines embedded in anisotropic media Finite-difference analysis of transmission lines embedded in arbitrary anisotropic media Formulation Implementation Eample Generalization of the method Lecture 4 Slide 1

2 RF Transmission Lines What are Transmission Lines? Transmission lines are metallic structures that guide electromagnetic waes from DC to ery high frequencies. Microstrip Coaial Cable Stripline Slotline Lecture 4 Slide 4

3 Characteristic Impedance The ratio of the oltage to current as well as the electric to magnetic field is a constant along the length of the transmission line. This ratio is the characteristic impedance. L Zc C The alue of the characteristic impedance alone has little meaning. Reflections are caused wheneer the impedance changes. For small H/W, 377H H H H Zc 1 1 rln r r rw rw W W For large H/W, H 1 W ln 8 377H W 3 H Zc 1 r H W 1 r 1 W ln W H 161 r r H S. Y. Poh, W. C. Chew, J. A. Kong, Approimate Formulas for Line Capacitance and Characteristic Impedance Lecture 4 of Microstrip Line, IEEE Trans. Microwae Theory Tech., ol. MTT 9, no., pp , May Slide 5 1 Approimate Equations Coaial 60 b Z0 ln a r Twin Lead cosh d Z ab r Parallel Plate 10 d Z0 w r Microstrip 60 8d w ln 1 D 4d w d Z0 10 w 1 D 3ln w d1.393 w d d r 1 r 1 D 11d 1st Century Electromagnetics 6 3

4 Analysis: Electrostatic Approimation Transmission lines are waeguides. To be rigorous, they should be modeled as such. This can be rather computationally intensie. An alternatie is to analyze transmission lines using Laplace s equation. The dimensions of transmission lines are typically much smaller than the operating waelength so the wae nature of electromagnetics is less important to consider. Therefore, we are essentially soling Mawell s equations as 0. Setting =0 is called the electrostatic approimation and the solution approimates the TEM solution of a waeguide. B 0 B 0 All of the field components are separable. The transmission line D 0 D 0 is TEM. H J D t H J Rigorously, the transmission line E B t E 0 is quasi TEM. Lecture 4 Slide 7 Analysis: Inhomogeneous Isotropic Laplace s Equation The diergence condition for the electric field is D 0 Eq. (1) But, we know that D= r E, so this equation becomes E Eq. () r 0 The electric field E is related to the scalar potential V as follows. E V Eq. (3) The inhomogeneous Laplace s equation is deried by substituting Eq. (3) into Eq. (). V r r V Lecture 4 Slide 8 0 4

5 Analysis: Calculating Distributed Capacitance In the electrostatic approimation, the transmission line is a capacitor. The total energy stored in a capacitor is U DE da 1 A The integral is performed oer the entire cross section of deice. For open deices like microstrips, this is infinite area. In practice, we integrate oer a large enough area to incorporate as much of of the electric field as possible. This equation is alid in anisotropic media. The capacitance is related to the total stored energy through CV U 0 V 0 is the oltage across the capacitor. If we set the aboe equations equal and substitute E V into the epression, we derie the equation for the distributed capacitance. 1 C DEdA V 0 A Lecture 4 Slide 9 Analysis: Calculating Distributed Inductance The oltage along the transmission line traels at the same elocity as the electric field so we can write 1 1 r r V E LC LC c Assuming the surrounding medium is homogeneous, we can calculate the distributed inductance from the aboe equation as L This means that for the dielectric only case, we can calculate the cc distributed inductance directly from the distributed capacitance. r r 0 Dielectric materials should not alter the inductance. Howeer if we use the alue of C calculated preiously, it will. This is incorrect. The solution is to calculate capacitance with a homogeneous dielectric C h and then calculate inductance from this. It is simplest to use just air. 1 0 Lecture 4 Slide 10 0 L cch 5

6 Analysis: Calculating TL Parameters The characteristic impedance is calculated from the distributed inductance and capacitance through Z c L C The elocity of a signal traelling along the transmission line is 1 LC The effectie refractie inde n eff is therefore c0 neff c0 LC Both Z c and n eff are needed to analyze transmission line circuits using network theory. Lecture 4 Slide 11 Two Step Modeling Approach Homogeneous Case air outer conductor Step 1 inner conductor distributed inductance L air Inhomogeneous Case Step 1 outer conductor inner conductor distributed capacitance C air Lecture 4 Slide 1 6

7 How We Will Analyze Transmission Lines Step 1 Construct homogeneous TL Step 4 Construct inhomogeneous TL 1 air outer conductor inner conductor air outer conductor inner conductor Step Construct and sole matri equation Vh 0 L h h L h air Zc n c0 LC 1 L C src Lecture 4 Slide 13 src 1 h src Step 5 Construct and sole matri equation r V V0 r L src Step 3 Calculate distributed inductance C D E da 0 h h h V0 A 1 L cc 0 h Step 6 Calculate distributed capacitance C DEdA V 0 0 A Step 7 Calculate TL parameters L Transmission Lines Embedded in Anisotropic Media 7

8 Effect of Strength of Anisotropy Dielectric anisotropy has no effect on the distributed inductance. Capacitance increases and impedance lowers with increasing yy. Field deelops most strongly in the direction with highest permittiity. The degree to which field follows highest is proportional to the strength of the anisotropy. Lecture 4 15 Effect of Spatially Varying the Tensor Orientation Impedance changes in the presences of the anisotropic medium. Field actually follows the anisotropy. Impedance changes slightly with tilt. Impedance relatiely constant regardless of spatial ariance. Lecture

9 Isolation of a Microstrip Ordinary microstrip with metal ball in proimity. Near field is perturbed affecting the signal in the line. Same microstrip embedded in an anisotropic material with dielectric tensor tilted away from ball. Nearfield is less perturbed, affecting the line much less. Lecture 4 17 Eperimental Results Ordinary Microstrip Embedded Microstrip Lecture

10 Finite-Difference Analysis of Transmission Lines Embedded in Arbitrary Anisotropic Media Formulation The Grid We hae two unknowns, V and E, so we adopt a staggered grid. We can use our yeeder() function to construct the deriatie operators. V E E y Lecture

11 Goerning Equations D D 0 D y 0 y z Dz D y z E D E D E y y yy yz y D z z zy zz E z E E V E y V y E z z Lecture 4 1 Reduction to Two Dimensions D 0 Dz D y 0 D y z 0 D D z y y D z 0 y z E E Dy y yy D yz E y y E D D z z zy zz Ez y y yy E y E Ey y V E z z E V E y y Transmission line properties are independent of z, yz, z, zy, and zz. Lecture 4 11

12 Finite-Difference Approimation D e e d 0 y y D D D 0 y d y D ye d ε εy e D y y yy E y d y εy εyyey E e D V E y y e y D y??? Lecture 4 3 The Problem with the Tensor D D E E ye D E D E E y y y y yy y y y yy y D D??? i, j i, j i, j i, j E y E i, j y??? i, j i, j i, j i, j i, j y y E yy Ey In the first finite difference equation, E y is in a physically different position than E and D. Similarly, in the second finite difference equation, E is in a physically different position than E y and D y. The finite difference equations aboe are incorrect because not all of the terms eist at the same position in space. Lecture 4 4 1

13 The Correction Recall the interpolation matrices R, R, R, R, R, and R Computational Electromagnetics. y z y z from Lecture 10 in D D E E E E E 4 E E E E i, j i, j i, j i, j i, j i1, j i1, j i, j1 i, j1 i1, j1 i1, j1 y y y y y y y y i, j i, j i1, j i1, j i, j1 i, j1 i1, j1 i1, j1 i, j y y y y i, j i, j y yy Ey 4 d ε e R R ε e y y y d R R ε e εyey y y y R a interpolation matri a ais along which aerage is taken direction of adjacent cell Lecture 4 5 Reised Finite-Difference Approimation We modify our tensor by incorporating the interpolation matrices. D e e d 0 y y D D D 0 y d y D y E d ε RRyε y e D y y yy E y dy RRε y y εyy ey E e D V E y y e y D y Lecture

14 Matri Form of the Inhomogeneous Laplace s Equation We now derie the matri form of the inhomogeneous Laplaces equation. Eq. 1 Eq. Eq. 3 e e d D D y 0 d y d ε R R ε e ey y y dy RRε y y εyy e D e y D y Step 1: Substitute Eq. () into Eq. (1) ε e e RRyεy e D D y 0 RRε y y εyy e y Step : Substitute Eq. (3) into the aboe ε e e RRyε y D D D y 0 RRε y y εyy Dy Step 3: Drop the negatie sign ε e e RRyε y D D D y 0 RRε y y εyy Dy Lecture 4 7 Matri Form of the Homogeneous Laplace s Equation For the homogeneous case, our tensor reduces to ε RRyε y I 0 RRε y y εyy 0 I Substituting this into the inhomogeneous Laplace s equation gies us e e I 0D D Dy 0 0 ID y D e e D Dy 0 D y Lecture

15 The Ecitation Vector b We hae L = 0. This is not solable because we hae not stated the applied potentials. When we do this, our equation becomes L = b.? V 0 + V z L 0 L b Lecture 4 9 Enforcing the Known Potentials It turns out we must also modify L in addition to constructing b. Each point on the grid containing a metal must be forced to some known potential. We do this by modifying the corresponding row in the matri as follows: 1. Replace entire row in L with all zeroes.. Place a 1 in the position along the center diagonal. 3. Place the alue of the applied potential in the same row in b. # # # # # # V1 0 # # # # # # V 0 metal applied metal applied V Vm Vm m Vm # # # # # # V NNy1 0 # # # # # # V N 0 Ny L' b Lecture

16 Fancy Way of Forcing Potentials We construct two terms: F Force matri f forced potentials Diagonal matri containing 1 s in the positions corresponding to alues we wish to force. Column ector containing the forced potentials. Numbers in positions not being forced are ignored. 1. Replace forced rows in L with all zeros. L IF L # # # # # # V1 0. Place a 1 in the diagonal element. # # # # # # V 0 metal applied metal applied L FIFL V Vm Vm m Vm. Place the forced potentials in b. # # # # # # V NNy1 0 # # # # # # V N 0 Ny b F Use same b for both f L' b homogeneous and inhomogeneous cases. Lecture 4 31 Calculating Transmission Line Parameters Calculate the field quantities e D e, h D e y Dy ey, h D y d ε RRyε y e dh, eh, dy RRε y y εyy ey dy, h ey, h C T d e 0y d y ey Calculate the transmission line parameters Z L C n c LC c eff 0 r,eff n eff Lecture 4 3 h T dh, eh, 1 Ch 0y L d yh, e yh, cc 0 h Note: we hae moed the constant 0 to the capacitance equation for conenience. Calculate the inductance and capacitance using numerical integration CAUTION: Be careful that you are consistent with your units. For eample, it is incorrect to set centimeters=1 and then set e0=8.854e-1 because 0 has units of F/m. 16

17 Finite-Difference Analysis of Transmission Lines Embedded in Arbitrary Anisotropic Media Implementation Step 1: Choose a Transmission Line We will choose to model a microstrip transmission line with the following parameters: r,sup r,sub w h r,sup r,sub air superstrate dielectric substrate w = 4.0 mm h = 3.0 mm Lecture

18 Step : Build Deice on Grid (1 of ) N unit cells N y unit cells We will use Dirichlet boundary conditions so there must be sufficient space between the microstrip and boundaries. Rule of Thumb: Spacer regions should be around three times the width of the line. Lecture 4 35 Step : Build Deice on Grid ( of ) We use si arrays to describe the transmission line. 1 grid ER ERy ERy ERyy grid 0 s 0 s 9 s 0 s 0 s 9 s We do not need to build ERz, ERyz, ERz, ERzy, or ERzz. Lecture

19 Step 3: Construct Matri Operators We use yeeder() to calculate the deriatie operators. % CALL FUNCTION TO CONSTRUCT DERIVATIVE OPERATORS NS = [N Ny]; %grid size RES = [d dy]; %grid resolution BC = [0 0]; %Dirichlet BCs [DVX,DVY,DEX,DEY] = yeeder(ns,res,bc); %Build matrices Since we are using Dirichlet boundary conditions, we can construct our interpolation matrices directly from the deriatie operators. y e R D R D R R R y y y f f f f i 1 i i1 i % FORM INTERPOLATION MATRIX FROM DERIVATIVE OPERATORS RXP = (d/)*abs(dvx); RYM = (dy/)*abs(dey); R = RXP*RYM; Lecture 4 37 Step 4: Construct Diagonalized Tensor Step 1: Parse materials from grid to 1 grid. ER = ER(::N,1::Ny); ERy = ERy(1::N,::Ny); ERy = ERy(::N,1::Ny); ERyy = ERyy(1::N,::Ny); Step : Diagonalize all four tensor elements. ER = diag(sparse(er(:))); ERy = diag(sparse(ery(:))); ERy = diag(sparse(ery(:))); ERyy = diag(sparse(eryy(:))); Step 3: Construct composite tensor ER = [ ER, R*ERy ; R *ERy, ERyy ]; Note the comple transpose here. Lecture

20 Step 5: Build L and L h Step 1. Build the inhomogeneous matri L L = [ DEX DEY ] * ER * [ DVX ; DVY ]; Step. Build the homogeneous matri L h Lh = [ DEX DEY ] * [ DVX ; DVY ]; L Lecture 4 39 Step 6: Force Known Potentials Step 1. Build the force matri F that identifies the locations of the forced potentials. F = SIG GND; F = diag(sparse(f(:))); Step : Build the column ector containing the alues of the forced potentials. f = 1*SIG + 0*GND; For two conductors 1 olt applied to SIG f = 0.5*SIG1 0.5*SIG + 0*GND; Step 3: Force the known potentials. L = (I - F)*L + F; Lh = (I - F)*Lh + F; b = F*f(:); For multiple conductors F = SIG1 SIG GND; Lecture

21 Step 7: Compute the Fields Step 1. Compute the potentials. BIG COMPUTATIONAL STEP!!!! 1 L b h L b 1 h Step : Calculate the E fields. e D e e y D y Step 3: Compute the D fields. d ε Rεy e d H dy R εy εyy ey e h d h e D h, e yh, D y d, h e, h d e yh, yh, h Lecture 4 41 Step 8: Compute TL Parameters Step 1. Compute the distributed capacitance C. 1 C DEdA V 0 A Step : Calculate the distributed inductance L. C = (e0*d*dy)*dy'*ey; Ch = (e0*d*dy)*dyh'*eyh; 1 C D E da h h h V0 A 1 c0ch L Step 3: Calculate characteristic impedance impedance L = 1/(c0^*Ch); Recall that we moed the 0 term to this equation for conenience. Z 0 L C Be sure you are consistent with your units! Lecture 4 4 1

22 Step 9: Visualize the Fields Step 1. Etract the and y components of E. e E = ey(1:m); e Ey = ey(m+1:*m); e y Step : Reshape from column ectors to D arrays. = reshape(,n,ny); E = reshape(e,n,ny); Ey = reshape(ey,n,ny); Step 3: Visualize in MATLAB using imagesc(), pcolor(), and/or quier(). For day to day data analysis, I use imagesc(). For generating pictures for publications and presentations, I use pcolor(). Lecture 4 43 Finite-Difference Analysis of Transmission Lines Embedded in Arbitrary Anisotropic Media Eamples

23 Eample #1: Ordinary Microstrip (1 of ) r,sup r,sub w h w = 4.0 mm h = 3.0 mm r,sup r,sub air dielectric Lecture 4 45 Eample #1: Ordinary Microstrip ( of ) N = 00 Ny = 150 d = dy = Z0 = ohms Lecture

24 Eample #: Anisotropic Microstrip (1 of 3) z y r,sup r,sub w h w = 4.0 mm h = 3.0 mm r,sup r,sub Lecture 4 47 Eample #: Anisotropic Microstrip ( of 3) To generate correct tensors, we start with the diagonalized tensor and the angle that it should be rotated around the z ais r z The rotation matri is cos 30 sin R R 30 sin 30 cos 30 0 z The rotated tensor is then R R r 1 z y z Lecture

25 Eample #: Anisotropic Microstrip (3 of 3) N = 00 Ny = 150 d = dy = Z0 = ohms Lecture 4 49 Generalization of the Method 5

26 Method Can Do More than Transmission Lines! The model we just deeloped simply calculates the potential in the icinity of multiple forced potentials. We used it to model a transmission line, but the method has other applications. V1 5 V V 7 V 5 V 7 V Potential V 0 V 0 V Lecture 4 51 Think of Laplace s Equation as a Linear Number Filler Inner Gien known alues at certain points (boundary conditions), Laplace s equation calculates the numbers eerywhere else so they ary linearly. Map of known alues (boundary alues) Laplace s equation is soled to fill in the alues away from the boundaries. Lecture 4 5 6