ࡱ> { ibjbjzz  5a X b nnn8t.4'b^WWW&&&&&&&$(+h'nW@WWW'nnW']#]#]#Wnn&]#W&]#]#r$T3%睃O%&'0'%%, "T,3%,n3%WW]#WWWWW'']#WWW'WWWW,WWWWWWWWW : ECE 222L Lab 6 Further Explorations with MATLAB In this lab we will continue to explore the capabilities of MATLAB. We will be looking at three areas: 1) symbolic processing of algebraic expressions, differential equations and Laplace transforms, 2) creating an LTI object to represent a system and 3) an introduction to Simulink modeling. In each section, there is first a description of the commands, followed by some examples that you should try on your own but dont turn in, then finally a problem which you will turn in. Dont turn in this whole handout; put the problems in a separate file. Part I. Symbolic Processing. Previously we have performed numerical calculations with MATLAB. We will now explore how to represent and manipulate functions in symbolic, or variable form. The first step is to create symbolic objects, i.e. to define the variable (t, s, x, etc.) as a symbol. There is a long way and a short way: x = sym(x) or syms x The short way is not only shorter but can create multiple symbols at once: syms x, y, z There are many operations in the Symbolic Toolbox that are used to manipulate symbolic functions. Here are some useful ones; you can explore others on your own by typing help symbolic. For more information on any of the commands - syntax, options, examples - type help command. For all of these, the expression F can either be defined first or within the command itself. For example, F = (x 2)^2 collect(F) or collect((x 2)^2) will do the same thing. collect(F), expand(F) reduces or expands an expression F by collecting together or multiplying out terms factor(F) finds factors of a polynomial pretty(F), simple(F), simplify(F) used to display an expression in its clearest form. pretty displays the expression in a form that resembles typing, simple finds the form with the least characters, and shows the steps to get there, and simplify uses Maples simplification rules. poly2sym(p), sym2poly(F) converts from a coefficient vector p to a polynomial, and vice-versa Probably the three most common operations we perform are to plot a function, solve an equation (or set of equations) and evaluate a function. The commands for these are: ezplot(F) plots the one-variable expression F. Format commands that we use in numerical plotting such as axis, xlabel, ylabel, etc. can be used in exactly the same way solve(eqn) or solve(eqn1, eqn2) - solves a single algebraic or transcendental equation eqn or simultaneous equations eqn1, eqn2. For this command, the equations are defined as a string, e.g. x^2 2 = 0. For simultaneous equations, you can get the output in one step by defining a vector output: [x,y] = solve(eqn1,eqn2) Otherwise, you need to type ans.x and ans.y to see the output. subs(F,symbol,number) replaces the symbol in the expression F with the number, and evaluates the expression at that number Finding derivatives and integrals is also straightforward: diff(F), int(F) Some useful options: diff(F,n) - finds the nth derivative diff(F,x) partial derivative with respect to variable x int(F,a,b) integral with limits a, b, where a, b can be numbers or symbols Some other functions you may want to explore: taylor, symsum, limit Symbolic MATLAB is useful for setting up and solving differential equations. The letter D can also be used (as well as diff) to represent the first derivative, D2 the second, and so on. Dt represents the first derivative with respect to t (x is the default.) The command to solve the differential equation is dsolve(eqn). Arbitrary constants in the solution are given by C1, C2, etc., or initial conditions can be included in the dsolve command. You can also use dsolve for simultaneous differential equations. See help dsolve for more information the syntax on this one is a bit tricky! Finally, we can find Laplace transforms and inverse Laplace transforms with the commands laplace(F) and ilaplace(F). Here are some examples to try. You do not need to turn these in. 1. Let F1 = x3 15x2 + 75x 125 and F2 = (x + 5)2 20x. Explore the various commands, e.g.: find F1 * F2 and F1/F2 and express in simplest form evaluate F1 + F2 at x = 7.1 solve F1 = 0 form the coefficient vector p1 plot F1 and F2 on one plot (you can use hold on to keep the plot open) 2. Let F = sinh(3x)cosh(5x), find the derivative dF/dx and evaluate at x = 0.2. Let G = 5cos(2x)ln(4y), find the second derivative of G with respect to y. (Remember log is ln!) Let H = xsin(3x), find the integral of H 3. Solve the simultaneous equations E1 = 6x +2y = 14 and E2 = (x 1)2 + 5y = 6 4. Find the Laplace transforms of 1 e-at and cos(bt). Take the inverse transforms of your answers. Lab problem 1. This problem you turn in. Include your commands (either print the screen or save in an m.file) and your results (numbers or plots.) Add comments to your commands to explain what you are doing.  The figure above shows a robot arm with two motor joints. The lengths are L1 = 4 and L2 = 3 (keep the units in feet.) The variables are the joint angles q1 and q2. The equations for the x, y coordinates of the hand are: x = L1cos q1 + L2cos(q1 + q2) y = L1sin q1 + L2sin(q1 + q2) a) Find q1 and q2 to position the hand at x = 6 and y = 2 . Give the answer in degrees. You should get two solutions (elbow up or elbow down.) b) We now want to keep x fixed at 6 and move the hand in a horizontal line. Obtain symbolic solutions for q1 and q2 as functions of y, then evaluate and plot them for y = 0.1 to 3.6 . Use subplot to put the two plots on one page. Remember the angles should be in degrees. Check the point y = 2 to verify your plot agrees with part a). Part II. System Representation. A circuit with no initial conditions is an example of an LTI (linear time-invariant) system. Weve talked about what linearity means in class. Time-invariance means that if the system input has a time delay, the system output will have the same time delay but is otherwise not affected. I.e., if input x(t) has output y(t), then input x(t t0) has output y(t t0). A transfer function is a way to represent an LTI system. We will discuss transfer functions (section 13.5) in class, but it is possible when you read this that we havent gotten there yet. Simply, a transfer function is the output/input, all in Laplace form. The usual notation is H(s) = Y(s)/X(s), where output Y and input X could be voltages or currents. This applies to any circuit with a single independent source as the input and no initial conditions. An example is an op amp circuit, where we find Vo(s)/Vi(s). Its a way to look at what the circuit is doing, independent of any particular input. When we find the transfer function in MATLAB, we create an LTI object. We can then easily analyze properties of the system such as frequency response and time behavior. The command to create an LTI object in transfer function form is tf. The syntax is: sys = tf([num],[den]) where [num] and [den] are the numerator and denominator polynomials represented as coefficient vectors. There is also a state-space form, but we will not go into that in this lab. Some of the commands we will use to analyze the system are: impulse(sys) response of system to a unit impulse input (natural response) step(sys) response of system to a unit step input bode(sys) plots the magnitude (in dB) and phase of the transfer function versus log frequency, called a Bode plot. The command bodemag(sys) plots magnitude only. pzmap(sys) plots the poles and zeros of the transfer function in the complex plane Example (try this but dont turn it in) Given a transfer function is H(s) =  EMBED Equation.3 , create an LTI object using sys = tf([1 -1],[1 3 4]). Plot the frequency response (bode), impulse response, step response and pole-zero map. Lab problem 2. This problem you turn in. Turn in your calculations, commands and plots. a) A series RLC circuit has an input voltage source vi, and the output is taken as the voltage across the resistor, vR. Find the transfer function H(s) = VR(s)/Vi(s). b) If the input is ac, s =jw only. Show that |H(jw)| =  EMBED Equation.3  Hint: remember  EMBED Equation.3  Let C = 10 mF and L = 5 mH. For R = 10, 100 and 1000 W, use ezplot to plot |H| vs w for w from 100 to 100 krad/s. Use hold on to put the plots for the three resistor values on one graph. c) Now create the same plot from H(s) using tf and bodemag. You should find this a much shorter way to do the same thing! (If you want, change the units of H in b) to dB, and change the x-axis of the ezplot to log scale in the graphics window.) ( problem continues ! ) d) Make plots of the impulse and step response. For the step response you can put the three R values on one plot, but for the impulse you will probably want to look at them separately. Use your judgment as to what best displays the information. Describe what you see in the frequency domain and time domain as R increases. e) In the time domain, the differential equation for vR is:  EMBED Equation.3 . You dont have to derive this, but you should be able to satisfy yourself that it is true. Use dsolve to find the natural response of vR(t) if vi(t) = 0. Use initial conditions v(0) = 1 and dv/dt(0) = 0. Compare to the result using the second-order circuit formulas from chapter 8. Part III. Simulink. Simulink is a graphical interface that allows you to model dynamic systems using block diagrams. If youve never used it before, I recommend you go to  HYPERLINK "http://www.mathworks.com" www.mathworks.com, Academia, Interactive Tutorials, and watch the introductory tutorial for a while until you feel ready to get started. Refer back to the tutorial or the documentation as needed. Type simulink in the MATLAB command window to start. Open a new model (either the blank paper icon or use the File - New menu.) Take a few minutes to explore through the libraries of blocks to get an idea of what is available. We will not attempt to discuss all the many options in this lab; we will just work through a couple of simple examples to give you a feel for the program. Example 1. (Dont turn these examples in.) Lets model the RLC circuit from the last problem with R = 10 W. In the Continuous library, find the Transfer function block and drag it into your model. Double-click to open it and set the transfer function by entering the numerator and denominator coefficient vectors. For an input, go to the Sources library and select the step input. To observe the output, go to the Sinks library and put the scope in your model so we can visually see the output. Connect the blocks together. This is a fast system, so we want to adjust the step and simulation times. Double-click on the step and change the time it turns on from 1 to 0.01, and set the simulation time to 0.02 (in the simulation menu or there is a box on the toolbar.) Theres nothing critical about these numbers just a better scale for this example. Now run the simulation by clicking the arrow on the toolbar and double-click on the scope to view the results. You can zoom in to observe the response better. Its nice to view the input and output simultaneously. From the Signal Routing library, put a MUX in your model in front of the scope. Change the output of the transfer function block to go to the MUX first and then to the scope. Connect the other MUX input to the line from the step to the transfer function. Run it again to observe. Another useful output option to try is simout, which sends the output results to the MATLAB workspace for further analysis or plotting. Replace the scope with the simout block, and set the output format to array. After running the simulation, go back to the MATLAB command window and type simout. Try the simulation with the other R values if you like, other input sources, or any other variations you want to try. Example 2. The transfer function of the integrator shown in (i) below is H(s) = 1/sRC. We could put this in the transfer function block or we can model it as an integrator block (1/s) combined with a gain block. Both of these are in the Commonly Used Blocks library. Connect the integrator with a gain of 10 (i.e., 1/RC = 10) with a sine wave input, and connect both the input and output through a MUX to the scope as before. Run the simulation, adjusting the input frequency, amplitude and the simulation time as needed to create a nice signal on the scope, and observe that it is performing integration.  (i)   123Y  \ ] z    + 0 N R  ƺ}u}qhcThHhV:C5hV:Ch0/hV:Ch>5h>hIO\hHvt5>*hhAQh hHvthOhBhh CJaJhh hh 5CJaJhO5CJaJhh h5CJaJhh hVb5CJaJhVbhVb5CJaJhVbh5CJaJ, 23Y ] z   4 T~gdIO\xxgdIO\ d`gdB x`gdIO\hgdBxgd`Y$a$gdIO\ $<a$gdB ! & 2 3 4 8 [ b r T]~FinvŽݲhYrhYr5hh hIO\hcT6hIO\hH6hIO\hIO\6hH hcThHhcThH5 h h= h=5h=h=5h=hAQh h h h>hcThcT5hcTh 6P&\ixPm{7xhgdIO\ x`gdIO\xgd`YxxgdIO\%PVWYZ[\`fgijklnoqrstu&̸̸ָָh h 5h hYr5hAQhh0/ h5 h0/5 h575 hYrhYrh hYr h 5hYrhYr5hYrhcT5C[\&i.Nk}r߭h$h$5h2w_ h575h57h575hAQh$ h$h57h57 hh0/hh0/5h0/h= h0/h h0/h=5h0/h 5h hcThq8h hq8hcT5hq8h 54 \fkvwx"#'(./123[\`auv67=žͶͶͶͭhAQhh]h{Gh{Gh&5h& h8H*h8h8H* h8h8hIO\h8H*h8hIO\hIO\H*hIO\hIO\H* h57h57h57h575h57hIO\h h2w_77 []$g h !f$f%|%1&m&&xgdS gdS gdAQgdAQxgdAQxgd`Y P^P`gdAQgdAQ  :;HKZ[\]hmv󿷳vq h H*h h OJQJh hIOJQJh hIH*hIhojhohoUhhXbh hh h5jh0I0J5UhhH*h$huhuH*huhuH*huhH*huhhh5, "  6e f g h n o q !!!!!!!!!Y"h"##ԿԿԿԿԷԷԿha`h] h6hS H*h6 hS 5>*hIO\hS 5>*hS hhXbh  h H*h h OJQJh h H*hIE#$$$$ $ $c$e$$R%T%f%|%1&m&y&&&&&&&&&&&&&&''@'D'M'p'w'|''''''(((4(5(H(I(J(K(h(i(ÿÿÿ෿âÕjtkhdh5cGEHUj"T h5cGCJUVjha4Uh5cGha4>* h5cG5h5cGh5cG5h5cGha4 ha45 h=h= h=5h=h=5h=h] h] 5h] ha`h6h6H*h65&&''(i(((6))n**, .M00111]34gdxgdjGgdlgdjGxgdexgdS xgda`xgda`gda`hgd5cGxgd5cGxgd=i(((((((() )")))*)5)6)8)l)m)))))))))**.*0*:*<*b*d*f*h********ε{njph|h|EHUj'T h|CJUVjmh|h|EHUj&T h|CJUVjh|Uh|h|OJQJh|h6h5cGH* h5cGH*h|h5cGH*hjGh)h&hw h5cG5h h5cG5ha4ha`h5cGh5cGh5cG5*****+,+8+D+d+f+p+r++++++,,,,, ,!,:,D,J,L,Q,X,,,,,,,,,----. ..n... ///K0L0M0O00000000־־¶ֲֲֶֶֶ֦֢֚jh0Uh0h0hwH*hlheh)h8h1hjGhh&hwhw5hwhwh#XOJQJhwh|5h|h#Xhwh|OJQJ<00000001 11*1+1,14151;1<1?1^1c1p1r1111111122Y2i2o2p222222223 3b3j33C4V4սչչյ{h 2h 2hR5hRh6_hU0JjhUUh#yhU hjG5>*hIO\hjG5>*hjGhpAh&h0h0H*h0h05hwhj>h0jh0Ujsh0h0EHUj*T h0CJUV0V4W4v44444444444 55>566P6r666(77 888*8L8S8X8888889999r:::;; <<K<l<m<<<<<ܽŵܱܱܱܱܠ h"eaJhLNIhLNIaJh"eh5aJhLNIhaJhmhrX5hrXhmh5hmh$c5hhmh 25h$ch$cOJQJh$chhPhPhP>*h'h0^h#yh 2548:;==???????ddee7>T>X>a>? ???????????dddd1dddddddddde9eHeMeSeTeheieteмȼЪȨȤ̸̜̜ h'5h hU5 hUhUh:hOhn(UjhUUmHnHuh'h Ch"ehP5hG\hUhPhn(hP>* hLNIh"ehBh"e hrXaJ h"eaJhLNIhLNIaJ4 (ii) If you use the integrator block, you can model the effect of the power supply limits in an op amp. Try the simulation with the integrator limits set to 12. Try this with a step or ramp input to really see the difference! Lab problem 3. This problem you turn in. Turn in your calculations, model schematic and results (where we say plot, it means paste the scope screen in the report.) a) Write the transfer function H(s) = Vo(s)/Vi(s) for the op amp circuit in (ii) above. Use R1 = 1 kW, R2 = 10 kW and C = 50 mF. b) Model the transfer function in Simulink with a sine wave input and scope output, displaying both the input and output on the scope. Plot with input frequencies of 1, 10 and 100 rad/sec, adjusting the simulation time so that you see 5 10 cycles in each case. Why does the amplitude change with frequency? c) Replace the input with a unit step. What is the output function in the time domain? This is a very quick introduction to Simulink. I hope you will see that it is a useful tool, and that you have the opportunity to explore it more and to use it again in future classes.  This problem was shamelessly stolen from Introduction to MATLAB 6 for Engineers by W. Palm     teeeeeeeeeeeeeff f fff4f6f:f p#I@!o⷗bxߡ @; @_Y @ P\@/` @ @ 6 @ @??{; @_Y @n^&6'W @l4t*H%֚'@@'2X.?~7>F'@ ZRAX\Xﯞŷ} s| ]  N%ͷ\n0  @`}oY& @@>w#67ZQ  @=nj @mT @==nj @mT @==nj @mT @==nj @mT @==njxWx=HtP @B @  @(. _ @ @B @  @(. _ @ @B @  @(. _ @ @B @  @(. _ @ @B @  @(.rxXVkޟRi喢ax  @ )9)Ѷਜ N@O  @+=}=U#@@7Kmo/ףZX  x_|#@ ; @ l< @B3@ @_|#@ ; @ l< @B3@ @_|#@ ; @ l< @B3@ @_|#@ ; @ l< @B3@ @_|#@ ; @ l< @B3@.wPx7x{A>T!@ߡ @ P\@/` @ @ 6 @ @ P\@/` @ @ 6 @ @ P\@/` @ @\.#鏠P?SwT g؂+ڿBؗ @ & rE5 @S @ P\{/x>~޹~,o+rz7?߭ !IU5 h\DBP @ @&T>稜 @ g^&@E@GX?Kr] @8I@? mx&c~ǃ *N+q @8 }|: c?$t98" Q"w#^KZ% YOvU~]  @`U@n?~tGk kO&קyXH@_hYZ%P\ 6ٻ3| p^?A- p@[>|  Ml @BgP @dmD? @: A#@ M@϶ @, wU@=?uwMGoJ((p]Dus{ICm֟7Pim  /Ok@utiF¨ڗn !q 7p{78U@?%p ݊м+ +nMZG<@ʖ-nx5K72 m|yok<ꞸyOD))qvZd:0s{+KDov kQTk6u˳Mҟ~pϾpw E7 73@v/3?/TO>_n=O^kw{/x'* @']!PA@<ׁ|l!:Ov/ *Zxw}O[X@{^";XZ@_z} ,5&{ϖE5CAEEmte]Fr= [?*./mk+U3~"+Zt;K ܆`ѐ ~t/α:^Na/u4OSm\3xx HX}FDl g/{1az @8 <~[gv<\p#vw+<À'ވ& xҿ ~/p;@B#pXm~}\ z^;SovR ֡ f$Nc-Yjk~z7ۙ (' [L? ZE551a<%~a/Li.' F64%v"^B@w< p2(+'v7l` ? ΍mw1u<W(3VnƦj3ۇ{&;pz$թ J <}x$7Ь8lS =N]<L@oIm&=6#]3f~rGw,, /<H(psۯ:N4ٰҍLho->5✷L̽{}=zskщSj|Neק>y<ϳ ((p<Dܻ򫀧N]!@^?t꧿}^\UݯS$BEh@Y_A޽R' &?m5?p%@SS->91gw9zFѿe-_o$.EK C{\|Y_v2>b]o.Wn\ <3Ͻ|=:97|{ߞU @S @/_:j|dgBӯS{x\ӾW G< s^pGte]z˚'pTOWwtƮ3~LZ 0V{NY'퉹ۋ_/;- Jq x?VY.zyߣxK~gwIީ1B+}e簭/ ߑ  @oq98 w/9P$@`!?ȻвJ* oGu_?G2o?8.%@เ'NtntoIANd @՟}?anFa[R|t@~qyPg`|$fD Z4 ܛ XF@_fU%@@IqZ E@6?FC]]6mN Lh#T$@  @r7=m&@`?{O==lJ I-ˬ + ; @ }C#@`"66( tyrR⠖U7+ TZ2zͮ9Z0-Kq Nݎs|eV3uNb[qH!pFOq4A4BUhZs_(J @W)_w  @Q@w* 0P/ܥXY@_y{z'@`rĿq#sݕNV'ʰT@T8"04KGV5I@if%@`/Oݪ @` =s_` @:~"@Fe&%K67X '~0 `%@{_ x (K3WXY@_y{z'@@cKl# qwW C?&"Kݔ"@B77~E ;fuLr)v8\[ p'YZ~  k @l @ ]ėmHCXM@_mc%@`/pS @`U @tO@ @ c}U'@ 0]@蟾  @+ U @t 4@ @`?Wu @+ @BX_  @L@ @ c}U'@ 0]@蟾  @+ U @t 4@ @`?Wu @+ @BX_  @L@ @ c}U'@ 0]@蟾  @+ U @t 4@ @`?Wu @+ @BX_  @Ls\7l%3?G\a  @*I@ @_|#@`o0  @SS1 @ ğk!@@J?Z4E6U]@EĿMݑQq?*L@l%@_U3 @SS-,%ݎ+ +nMKw @]@or%xe @"B"& @BQ9#@ пȢI @਀T @," /(m @8* :xogw{A _g/ PKHj-4!SV|{F$@~T.nn пѲJ@*(xY4Cl |Q  @ ,=aB @?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @&T @@n?~tG @ ,  @ [@Ͻ @ aB @sGw @BP @ @EYIENDB`n:-')4ZPNG  IHDRsRGB:{IDATx^m6[Y/6T_DI'? h. @+ufe @%  @ ly @~{ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @ @ P\@/>`#@  @ ly @~{ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @ @ P\@/>`#@  @ ly @~{ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @ @ P\@/>`#@  @ ly @~{ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @ @ P\@/>`#@  @ ly @~{ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @ @ P\@/>`#@  @ ly @~{ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @ @ P\@/>`#@  @ ly @~{ @@q- @o @(.  @ @< @B=@ @K<j,R4>e R@1 @I=ʽ񟋈Mp xg; @A9o~lؼ.!@BmA @{ %p =+>!@NO`i%K п- @`=[1F3Io <  @ༀ=o8_o%-}qU|h+*n @xu}Y&  " 0*B4%~ x2j"@޼~?ܓ=W5 @B_8"@_B}@rv ;\Nock @ Пv4#@|% ) oʱh ol~ -ڇ| 8~@T"{qh7N @[Q    G|<_UxN=A @< @B=@ @_|G @@ @ @cnO @_lC[>=FPd3MC/!pcc @@  @lB<OnB7Z  0}{4z=T$@9Ygo4 H.!@zQc޴B[!>>F" @`Ri?J @`?ޓ  I ckjrͱ#@x>ڿIJH 㸽f @DBah65rG{../@L#7?!Zj{Ռ  PU@:Y"@@@cn3Xp_{VGi&sc&v  J@/ @ @%O\s8֧ 0J@% p,=kW * ?^ @ϩwnB *- 􏞀 @`rzy$|'@k @0w a"  @ Пc @IvW?ݕB@bL$@4߀>}!ifn rrͱZ*\s"@@~?tHۯ<>铍J,# /3j %@@gG>38+ t?< ܎G}@B¡h ;n}YgoטU @rQ}H !@Hk_HG * r*F @ Пo&:"@}>e0 %mc>ݣ  N@7 @,LQL( O84- @# r- @  e @{=Z%@ 0?дL @`пG˵ @&'  @h @BC2 |=Y @`I['z SOB@gZb}5gp?m 8%t]]j p'  )"KEGr#@'?>K67DxN5iUP@phZ&@@3ω>X. P\@/>`#@@q? ABsj @m @k kݪ  @ @) 9w&@?_F<O@ @ @1ЋJ @`п @+t;*K%Qa9A &@@7X@!?~g K$@`!i sRT+MZ 0@=Ƿ_9`IBZ.hI-\Ҭ @`Eũ[3~2}x  @;?^lXQЇni@y'Gl @ '@(/  @.  @_~H п~ @B[  @B;  % n^Չ:   p bM(Ao~[ @@q- @o @(.  @ @< @B=@ @_|G @@ @ @ @B[ @? @`wm^W fDRtpߕWq @3 @BW^  @@ @ ]y'@ 0^@? @*+;*N&_TwE_9\uttC@'}\U%@ F@O3  @#> @`{"n^,ѵfHS+tpߍVa @9s @nB7Z  @sA @ h&@ C@1] @&+;*Lss}fuRE85 @ @ @B[ @?k pPG7޼v}Ow\?  CK-ଅ@a'kiOY($ -7/Ƚ> @BRRXX@y? ~%ª @ Пf!PTzݿ-r K 8_jKM|G A: `%@ Z6sܼ`k\=yZCSݿՙm \-jq#@@ @BG`Q/,:x&@B9@]qlF d<( 85=H-|/ @`!["gD;I*;7*Nv+W [>{g7 @@  rK @@o"^ @p4@`-kj  @ Пc PZ7 @S"K ߨM72 (/ि- @`u` @#@ @w @ˏ  @VWO`S`7o`(?tNGcR@R۳'p:?uN(?JR?JC+MZ W@+zL, O<< @ϭJ <_|?2} }E2õ4 P.#sdbx~ Ej @`I/;W=LJs N@O7 H%p;x>TiFaTs sK|aP < r{o`yV?( 85=}iɮ.y@_~gA~5o)rQ_pLjBj_}'-!Gދ70ȣE!Jkk=@5L^m|BZZ/[*zz=򪺛Iznݟ*\ F ԞXP*. iol^q빽߽ͧ( e8+p wρRl1 Ʃ R% ܑ{Sg0>:О[D{S SORjiznx-(Z J< 6&' hӍ5vyÙ{sكoK @`:i8FДQp'sk,~e̎Qũ=@w}#Z~=gl%hG};ҽ9|{o2{~ڲؽ{1_q~g{  ,߮ Z.~w~p9vW;i+x˥9O5q @^@t85= ћɆwF>H0n= @A= !Tr{Ecv>l \ =6+PXC{?tL@9H-P!{op?P" 2S@sn# vw qoۇ}+ " a\O<3!}d;A=@Y<1ַ7\vf& \hؖA!&~#e}fkLjS#2}\vf$ wUYk,^͸2?/=Jp@UTu,Mkz>_*.@AI$"FK戂x~֕OZŚuD5ntzOW^ ,*݆ky<#Fzs$~Ž>N1"@Q>@F'A>dn 8?k@ Пv4#0ǜc>.'C@cN$0cا;~M=s+7 pgXH]h>}( 7B'O淬\ݿ?٧ c=U#@f6]rqo:ڼ=ߖ @q5}z/Su- }or8\Iz_y'XH9+\Or/+?k \9?sM~9Y~%eOA-w}T~3G`R ơn%mXاX8I &&n"=/_s׊.y|Q;+K: pҟ{>#@/Oa|1ӈmN@o / 7V= qcn'\@O>  @?<^2%W@+zZ@$~ݞ[Te ,{=cegH^W׵,3úZFk kݪ /0{C{Cb?~޳”3 x'}H9/]xpoK @'NkE"@ Cp@@_ Y@ܷC pG @ q߶ @eBeDt-أiq XT/Ͳ - 􏞀 @` q9[%IeFi!+ ;; @ }B#@`92P&y/L-'5['NX?S5 dr|{_ok p{.8h򍗽\yƌS  ि%!{d~,o޾yA[CaPp_dA@u}> @߅UQ * Wu @`C~`@ܷ 0?M/p+AqB8{O&@  S#6~[^ veFi!("@@?tA~9B0rT@ܟtp&@@ߢT+O o# @`]q[9  r  𷀸o# @`)q[,}{+ +Nݚ XSsj gl( (.V?ŗmy p' XH@_hؖJ @jj^ 8 PSq怭{=Z%@TT,: q&@ A@0= @( wU @@?@ @Wi @ S @BG\  @d3LA @: q&@ A@0= @( wU @@?@ @Wi @ S @BG\  @d3LA @: q&@ Aw>@J___ٖwlpg @㞳%6E @ П`Z @@3]՚ @ {Fg+p>*mga  B>yFO4A4BQhZs_k+Y D% R`HFx t8Kg4fg A/ϰH1BwO%@`) rĿԎXG~HCm %Sp! 1'] PC   @+ U @p4@ @Wu @# @B__  @ @ @ }}U'@ 0\@>  @+ U @p4@ @Wu @# @B__  @ @ @ }}U'@ 0\@>  @+ U @p4@ @Wu @# @>AuPi~or 8/0DK @<= F@O3  @`?B3  p{<r| ӍLL/p,ʷ<p?|z B#O?u @k  O@Ϸ @@_* wUe28>-] [ A˷ @j'W{pR  PL@/6P!@`-&}-k%@B; 'o_  0?4Md2T|HxrD@dP$@@j @`!i>Q).  p}w @`?B3  I@4  @߅UQ4 |]|2k*IiL+ ?nES{C/ kSkv3{2k ,~_ xg[0IyeI ) 看XBoɽ_E @@{ !@ UI @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@  @B @@i9B]IENDB`tkDd J  C A@?"`"j9Rֈ3 }jD&@=j9Rֈ3 }j?sxjxڥWukUs/Oչ\*9RRA2H$ YTLPY*NݿsmU_kwx~\\].sg/rޱGryow]kۏܱ=ܱ?_}_.؏˿__[o|gy|s\yw1f65'!}fLrss<1ךg86^k39w.i;:sy|?cg?k?cw93=xp=7Ν7b{11w27w~{}9y=3 sfƜ2μ}<ُ424|+w^5sn~y3~N^ſ9\{w̵c|qM|uk$s}9.2>kF?.s`6ٟy6\'uzښ?\g}e9x]!Ǯ/Y1i߹pMzlֵ\?y1מ͸>^G~snq[mP9+nɃǑ׳࿽NCC{ދa/=qq{qK7% vm|Fu{}yy#93N\>wO|?g7>a<2س_>#o͘dž Wf,s(qO6:c}7B=k1c޼Nkk5U5+͹'v6cٜ3f}1cm98Ωُ<Οa3w`<1ish13''n88˜ochh=e{3s`Ɛs윟߾W.`'3z̸r||sy+w圗k|?w97sSp>~w>aweɽx9ރ7Y3^3~s>׬A!wmCǯsr5`zclь ueűy]p,^gv:wJn6\߹Kk|w38/}>\ 9O?zuX ޟ+cdkys:;D}oq0N`pfPˤ<8x3oŻZϤ;97c:8z5lŬǘkqe8ZG뿾.]83lr\Zgk60߃󡴳|3f\.gfnQ {Fq-fm 8fq.j1lKe)/y?F7ތ?s,jelg ~d'|/~/}k/??|Ӟ d}~.x+~ٗ}U7_^җ^s/| ?S>Ͽ/o| /??}y ^py;q~%Wٍ򕯼3p0({ qw$G#lCMߴVjI>a#n6mƱa-6jk1g>sg~*3o~.~W~W^]LsOO^'_|\雾;;w>WO99ٟ|'菮Gg}w~U_\~~w=輧>wʗmn6ypͮjؾ#9'c0666fa.nᫌ` XΙۯַIIW-~ulH-.-:'o|?.=zn_5_s~//ih.ۿ}|E/b6,bÅguf3l1E{ݝa2fWxӛf4}ݎm28=\7^d&rW<Լgޗ95^< s>r 6(>qʳϜ-9φo>bxc/`q +)S:EW)P_&.FCLPι(jX;.QyY;XS8r]4ru\EĞ'B=E#Vs=ZYpިZ^Sy4:kȒGA1;E3rfȇ#z8]H!21\[yM ˗ģdžXƅ6|䁲K|q`u&@M؞:o뵃P֎&^: -m*>3?Q&o<#G﹆qwOɼ12?i;ϻw 8[}hYGָ!d9'#k-;W|_O/؍ 5 :ײ|%$YFF)g]uJͷkYvM u ض]~kf6k|hc:9[6eWwK5%ֶZºh-1>>/K.8ooEà瘿YshPXs6k-ut"1w3];A=5קױ]qZ{.uݝxvr3Ϙ¬G5Ǵ#&˿}N12,k6߷ΜȮrmc|sm4gZa6Gx>9~9w}ホex{=Xy1W^)o>rl.[0JN~~WpUP? Ûwl}>f,tww͎sJKm\y$sGUVxK-h60>[s;b g%X{Pƹlzrӭ{[-q|Omu1eOb 3g(}@rn'`0yC.~સ~ϹS?fs~g۫#HS2#^W릦Gh61eѻG6Gm6נoY^D'4ј1=uSǓs-J+++֙W>rr{r9&WH/;[yu*RO81wlRgO Ƒ`\<Ǭ|6BI^%&ZL1:Zlϸa?wܺOܗ;Wu bH͜?km r=q=y?>_|&FԊ}G$rew}^g|o7zO9>9g}g]jR;Ϝ3~r{x;eϜ?ε}S˖gel{g]Γ̱h~-C|uggs3[~67w?~Y>މos#g8&}{_mԏ]y~Z[4yJs#}Kd;S|X|~b%{Ndo#Of||?9&9_1[~߳ Y~ڔNE{>s-3x|cFtgb{[:3eߩ)fl9 N3&'9rlw{]>s 18f-;w2w6ysΘ }{o=`q}On"?X4vFzԣ>N8*Ur |!kK^"G3ql!rmq&V#kmS s;gm#tmR_AƟd/3k/rFn_0t<.͜3p]'|k61GQbt56Bǻbl\l=I2:Yf K-ͯ^3\;fS^]W$Tq}$˘Kp 3,ojMβ>w\t8ňc j39mcw1NN0q%7ߧVיG57D#"~(_=rpԼ\k={b}7m?9\Ҏe~+J@3O`~cU✉%po'Ӹp.psޑPm}>;xe7\!b];♼G6ʑ#zZh|uc']wù0i`mĜ2LxnϞ1K1XbCbG?Y9@FkYK$O{wоm]ք5\w9bc{yktDїr۝bDZ?l9o|FcR7M&u7[E!tM?Rs b Gc1 i_{z5'}_u$w:oosiS-K8s3++:# 5,=+~VcM,ktx7}ro$kD&.zpju͞.w{h<ߓ_/˗1֋7sr^n|G|MVnyǀ#Ȝpd^ >X%Kƿ9/~dclL19谏œoX\ܿm~g=gko[j}ݏ0GޭfݶV#j>N_~yDɅi2H2`X_[Ϻ'-,뭎HFjÌwܓ`ouQ|<6s<9ݒx{0{6y\zRgI+ثȜ^`)b3 %XO/o3 7~ֻ䈛qniq2zYmm[.x3{Tݸ=g-v̛0'19Sk[%(HK;"̒6Sq]?IKN/rL!mH~cp\$3x\԰%?x}sy cY~Ng8H3*p~ cvl5bu@^Cy7?.x#9ɏ5]W[=8gP|7|4>V2r1Fʌ0[g w('ku}VWZ|D(6HlsK[80 AZƁ<ߛ|ؾ;>lv|sCǐm?GҼ{īQac,_\ӛ->V]zΡcbqq+KεtGf<1.!J 3]|Ԟ~ 5GŹ=糾1׸2uNo߁G5yͻH{[Vg?ه@ּ72v䗌}xm65<̏ '9-s=O.:XCs5fkd84b?/jrc1̅}|}ߋc=A16x<">=,}ŷ~䵟oySZ /NsL!~%uu%y޴F{a+nGoDy߱Ӌ65k|s5v &9i[<Vo;ِSnfq-?/i|?=l#c̽=6s̳f:6[{Ur;em[WsqS}%Q7xԚ7oƱ Ϻnl<^%f-,w]'Fq6'e\b9uueuS|8~fϼ y⧭'c[}8F<iC^Sl7:nizرNg n \,#-glaN9.=e 0~l;2VAվ2R2V0c;f_nny@[ V{|`e1vC/x#lm#ݦjI>q`={h=47q1?zjz>lq3wO َSfscM4 *h-]jWË6|j-?>g¾[36!sܼceʵm>'/q^kms~6.bᖷ28\K bWO _GܵmUyEH=u ֹ /lQ2My kD͕)'ԽYW=/lϛeN5tL#|'>ze}gpYr`Oj"SG=Z4\3_yt[8#_}sc#yGLL5z0/:IB<k0\a#x:Qnya}M.i:79e*se>'b36ν$ԑ们ѱkg߉ԁo;kbdlk:ޒi0=c)FDEfeLRWf1yrϾ9G96`\wM\#oSs`ƒw1G< m;Mc򻱏ƑEMnwok":˽l5VӮcw"6]sgƫCVkΉj޻|R6mbW0qlwN2'ؒw?i8{#y'GqSn5i}U>is|\tI92F߰%i\ڣLVb?>X?|eƇqyrض{T%58-o܇kBbsfēl|GgtbrP 1#kZU9ܼq05O6'8 \XOǝYCm/;\;fû;,wә흹ƀz_ktn=?䃌{m絼zk/8}Ѓt͍y.g9s<7lX fǛ52[ڸ7lo hT9~lx'ㅏrܰmMa^:*=5mhsTtd>gފ{pI~=;8f'׎:cՠl=Hz4#N&'a76GAKqg҆l?7v79]ñ{˺zΛ C|䢂xCzy~WSo홽Ck!vwoO6ݲBzWr/5ȌӸ;Fdǚ E2Vl@Q"]gq_Vmz-j.-[ qoL5i6&f~fWx[°As[_1ʲ̿'YUΔAZ}-GGs1<|iq!Եo7Ɣ:\\3 nx`s|ԗ?j?xCs\Ι 99S4.!Gk s6CV0N+OpRrִ8>,s_";hEqmŧrO2垶m ^ǜT?gaM܏gxȝgӵgynp0o>٦lۙKй;q^K挚gfkf4Jn_Rq-2?})cm6nh2!g̈qFS8UJHldRS3wڒs"xE;'2)ػ[#1]ۓ[Y[ `1]cin=yb~|dl;g[B&!umr/Ca䥞Ż k Nc[K<喓m1;gtT`?N8 Nd>7%5ΰn6a-8゙|5}bNm5Gyf73O͆w1%$N?7QMl}[mG}[oӆ`=ا`H㞛cy ]^-_/~,{99G8ėG9bB1?c$a\8;e5֙m,X{xް7{ƺ8­613(׼ΛXT;c"_ׇ+˜+17u̜8_q4Ϲ4KO˄&Ø06q'֜s?`\Ƴ>q|諒2m::Nvλ46,[1Ocl8*1SK-m\?!Ym=Q>}ku+ Ac,nzf69>3a☭ϳd,+5 hv<';i|ϖ#ʹ\[.,Xڒs;|ʗyL,vu⅌,6c8Jp1M~4 ~ŴM,!lĠ/ E0y^3qֵ`c;Ҟ.`}R =5?sSGcyb}[{^'ZN#81#.qV]hQIwuSE:XLn/ x14c7/inߤ-5:\\3@|[d!+}t|ݹ4WmG3ǼwOX1/{}0=elOkXr=ؑ.}Zomʞ>gz诩Oq,_-No[׃׵o|XEs ٢?7mL'ƌsp<{w61 F2gn{-1ϩ\/N>p;?:mdu۲ȸ8-x6@]ܜe:ҧ'5/¼K,{wOp,ѽqk f_'czka&xUkJӷ +~alh6=q3u!s[|m\O;8@2<߆cΊk˨s2c'l90s־?r 1K~s9IG>ڊֳ0f']Fks9T?:5ԑpFm8V_ZΌ,ub7w;ڸ6Wi9,c_sfW?5ԫh2~7 NJmq[Ǩ&c6.Mz<:_W4nP$G}OxsUxzɶkoyߜ6߭-~GQu|5֭K m`c~kz;[׃auN n}Gh[fՍۆ} Y:θ$gNMCky>Y/ ɍ)v}&fS~zq^>dX0^g=l3_cz7Jx[Cy2{/9=l(2IڸflsGvm%y"ݓK^yVk̡ߍ08xgoqVqsX[nr}Fp_w>HF 3n c?yOd꼚oqeGwŶfLjw`DhSG^W_fr埜_d{X#ڼޱ];ng9tQ sDqM|~⭶oo=nCQ=c"ֵIÂKl>o}[W#.?=6%|0,\]vgxEf86QMmҮ>_Mn}Qy-a#[l~Oj۳nFuƵqt?ܞaxfK[G䆩7&^I=n&z96 x??6oĖ[zkۮ=ۑad}z/ƩjaY5m%GhK*qp鵙:/c8β_+6zk&&xN.xi+:.[Bێlmv03ss5$=y]j =f%{g~6NU#'w9:T/[nra<;yZԹ?ʛ3 m=m>;񎻌w^9JYu(%klk,g͈>1OF?=n~;h^?,;]>a\;wd3o3p/sZk\K{gs0u3'N{_lVePrm}.婧_=DWyk^sy9|Cs?;vɼB ʚz3wxEpc]60W؏41"NsX^<#֕cۗ{<~횹_;Kl=3?ƟXy8~~L?x!yʗ{+ƎxZ(wĹ cF8_Q4vq؇1OoSk&wYcˎz3ĺFWQj\e.G&s}0r/Kc95gbU[H,^4^_]e̵?l}ۥk>En}4ݛ#> sȱӟqo]cշ^Ê @>G/zыvV[X'OkD8wy}D=ODEu{L/}}S3kxq|t{ONgb>Ȗgv~uplx؇k[w^^ƁѰy޹];1}27LVL]WOx.xaK_ҫM ؒs.ꝗl5Wr7ӆqot|06Fxry-bGMM^8p1M4ѱ[#Xlwya`&GN9r+g]\_\mJnȖߞ]U[0_G}nŘqo)ɲ.~[*jWLaӱ=fwmuçe}zjR˶Mu[rTG,3kE6 aϊ|$g?#EZH|O}ozŶFzg}M^ZGY|M{tNtӶnc¸׆]lc0|[m9g>>Xѫ1o>ưm}YmR?to>ʴwli[O3 ǟN$MtL{0@ZOo6}{6k7pKdO5fZYf96~stnƹx j1bxY;wcix_{k.&ؘ0BoN55\{p<[՟|k1b3&̞067i&8K݆6e^&c'S"y!{s2g~a1`+X&yNS >sO{x}?9[> s|ƨ 2&#g^%_Fq.q:~W!z{<=ZރXÝq}ڜ5Ē3bb0*Ж'|]Yq dMjq%H}97g<7 w;5n5̽M%64傲DU[S\k~ؚ硾YW2|~o[3 0_33פsOqFތ;mALq_9Qg񣟈)"N}`3AƮQWg0^l#x&Nipݻ.|xxo={G#83{v> ~}WLvC+B+5;}r||^5n>:>~sdž'i99ƶkD[錟cϛB^D~4~nl@r~b#C.и>甹32j(k!7#5yLY#V8Pr>klx]7n' yX_J #_[`l/`܌;'Y Ӗ0fѻLcDvy>dܐvM MmK׺OH>פs{˭L =+8s7G|%8WlJνcֶ]?k;JrXl@~I~:yjkx ےfRƖ4m߆og vy6,yvs='{Pp iy7ѵrhP7w-o~{w<xlyцZi-$ƾ =%sx[gpV)rXfNƦɻu pW4͆sNz:lIgm8>|q>׵yE9iumnGyoGqGhhrÒ{ 7~1N0 up>S 61)9&*~n=6blsM*7{k7=+[}XPg̜n-Kf{z[b~-yc6>7_FYs>_rkn"~ukrظ8YMfmgωAگjr^x쓚,r|p/hs2%σh_糏0 S^̡dsb.Nc<.W8ݎy'O\i=M4R?@=ay}~k1XvطikwoskSZ$uIl=v`b!G9h/o|'k|(o~~g1u{zNkwc81ٗM5ٜJ-6K[G&:tlO$h\S果.po: k1F)vSj^\& x|o޴pkt0X% /oh޶XUAzs"^9{36=x\:m|= pҧbq[ll@3ر_W'?d&y$Wjsf i>i\ \|9,_ s>xp׵]^[#7-;V`2+wgo[-ZYGyy0[\G<<+]ccy1!mȭqG1؂+NL#XHm,cK}pT17y˽Kye‰^r#:M,s,wz<1I1ֽW?|y~u9m7̌1XwLR ku{M:D?8o}p?| O|t"y#kѓMOmpqK;W**s\wrpia { YX#Hc/?OyG5cXtgb9W.N<&_bb=zos292 l<=P;a8&6 $sֱKx's,+=ۯɘ-|sG2ON9su">!^" OpTw=^ x5^:GpmYz{.P\7uĢxp:k_ ycXqʹ/ q3O8|˝xc n%^ kCZb61y5>Zd̘W8#'nd1y̕*[8W~O'D=sؖgAoumqsy͗6F7 qN?6q-+E|/p[3{r7^8cktũ+KT"#s#=A>Wч=aWG?$P]6_ϾJ\8{{]}3y=(]pn;-x^|äy /)Gƛƅי dg7-\yG<5θg<+7y!f vp}Y#,>wվKl=1r-6&ىμ*[ gz8?ʹ؎3/>E s1E3sA5!bgC8oq⼲}󱻌oFdQ_ߝ-vb<ϸjWg۷qH68&G68֛wⓖ9c3gc :[Jrkx6~6mkę,ޚk/Q]G_5*2E0z|]l6h['7~ha\צ]۷ouG>fG6ֱ_e!~ٲDAbw'eyFާoF̾G-9Oh$;zKd^a8[H?#?.mlDk>آmֻ=m89긘a#̿x/+8tb'G.٘ c59k/"ƺ\ WVn#ZwHmg=i%nse1gpH6Qf\GKl#3 ?^܅ߍ8w>4.G~O絹Z/4u2ϛ1H;jrƢqzȜ c"_ycO|1V89">o/9ާj;|1AMN|+X'>.؄cw:` Ǖm\`9O>#6 AiG䱾zf? _3Z#wK"w]㆟w1R96p t=+7o| f'Wc|ǐɜUS&؊'Ʀ= +65y->}V@Y&ތ mr;dvݰN|oo< o{+:v0IƵ}_w}ό5ٯ4־]Mܟݺ6k|yx|t'`3CzCrOFFG\mtl5|6wrB|'gs[3蘚jS;2c6}eXk[r]ٰ-BY&X?;0\g59GKlڄ^Wuu$͆z3vlGيn >.^Q"gۃ9~,|6s<%uvs㾭X?nO/)8G]s.\r?i9{I^kgxM$͍jm^ɼs>o0qࠂOGp9~p' 5aɡ=fUf3wO`5l euG[,al_cFO3>SOlFm9-F0Սߺ%qwײzg|ݒ1F9.俘g\qbG-us5G`=qZ9cU:1;G {=:7d'96Gjg8LF~Gζe87j#Ɯͷvmvk~2G.&Q#q=7s[?"ʝp mfmԸ5ZƛB{.׆V;m:M[\暫|a1Ӱ7Fqm5 Ʃt}o0_fiﺈvoHb׮͇eYi}=&sM7۸%Ѻk}xlI{?< 69גR&ejHy軙c^s^?.G0>=㞠~u0v[njmqƽfַuyGp2~ɚ]x`F&G>'=-G3+g5cg=j6;[XnpUvq5^rm.mY_toV;fm}~-87Ֆk GjsN 7 .'\`^o3[(ׁnNkYja no46϶)<:fqb4kmcyVs\GVG{TwK,2ߜw چ>l6BBv/|f9#[ݸgodn>9~Qo˷X?7ݽxZpڛl6~[:mMb[a,3)o`v&gUCzG2i^Y\qƏ1e{澺èM~0z1<&WSz~J ?Gۣ8Ao(FYaIZDn2Ys<{ӭ\5Q&O硉հ?jxS2={L0W&q܇y\qޙ!#v#o{:8V\@<NCgkq*ڎjx(u伄qe+2-_no- |p=0c*7g?>荟v=qaPA a8y:zmclp?G81|EkhƢwD9r/5kstm'rM(I4w/s}>YÇkL/B}185im-l5eN~Č7qk<odsr}r>l^tλ85䨷f_n46Ҿ)+1ܫh-~36r-~!O `785>^˦1JyϓG7n$km7e>S_Ǎ\eYc4uc-L{s{)%~-y4?1kz姚>3/ɖmۜ؜g f#[$J[l+G<;4X㔳M~Q]5պ {c7Z<cmy_}_uL_M{p l[K4ܗ1\q1%ƕT>ܜ5nkٖܸMOq |w6Nu S3|63[oHOIO:;8vmmtk{&[96lP?5޹w]^e6}e厹97o6nbb'1M` M_ &ۺC_ڶ$߇ytY;g w\|˽ɴa6m6aPv%e8|Kx/eΫlGַ؏ ϲqOn܀[v ޞ6Zwbh:qxm,ݩS-5|?Z\>gIQWgk{=%89crKlY-i.h_tnww>4Yݸژ٧ޏ)Z|aq۸ň~V?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnoqrstuvwxyz{|}~Root Entry Fp Data pnvWordDocument ObjectPool >坃p_1422795484F>坃e坃Ole CompObjfObjInfo  !"#%&'()*+- FMicrosoft Equation 3.0 DS Equation Equation.39qM! s"1s 2  +3s+4 FMicrosoft Equation 3.0 DS EqEquation Native i_1422796311 Fe坃坃Ole CompObj fuation Equation.39q$$  RL  ( 1LC " 2 ) 2 +( RL ) 2 . FMicrosoft Equation 3.0 DS EqObjInfo Equation Native  _1422796558F坃坃Ole CompObjfObjInfoEquation Native _1422797448F坃坃uation Equation.39qh\ |z|= Re{z} 2 +Im{z} 2 FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native h8 d 2 v R dt  2 +RLdv R dt+1LCv R =RLdv i dtOh+'0Dx   `?2˦,&k&`!˦,&`@hn TxڝRJA};w%-DRH ˀ0WXDKmD[+;|E/ARIE4EjL^0TgP}~͗Q "{a#GdRW|Wqs:uGNyG ̞-97֩  hs H/쾖m%6kmW/•M|Q/-I4)\)c$`)j[㙻yFh->KI<x t7W7Ex DcKQsހm9gv8*BDd ` b  c $A? ?3"`?2Srț q&`!Srț @xڝR-OA~ܵ !mmP @ @`pH4@±GJIޛ}of3tzB4?:*͑ uI CS4L`21gL4IX6* q}GGc1Kd1|&E(ޑ#֐:=VaF9x*OYKD} hRͩZЪ o.ݬ>(q[⃘~%Sv$9weV'mGp8ʱqGvxbC~_fCe٨\D̟o'gl)đE6[&dob]$hy`I8> ?ߕ [Dd p b  c $A? ?3"`?26L۔q8s&`! L۔q8S!`\xcdd``$d@9`,&FF(`TK A?dbQZĒʂT @_L TȠ1[XY`bݬ0l01QvNX7L&-[+*oL`A $37X/\!(?71!bD VpZ-݌`q ( g 1AP ~\p8+ a/QdGo+}KYcd`$K 28?, d8?ͅiȈL=,ɟ@dB[0ZiT%P 27)? B8.&X| -@|m8?炦r.pN- `padbR ,.IexC `g!0bna1Table,SummaryInformation(tDocumentSummaryInformation8$CompObj,r $,4<ECE 202Melinda Holtzman 231372C3 holtzman6Microsoft Office Word@J@<~@| -՜.+,D՜.+,L hp  Portland State Universitya5 ECE 202 Title 8@ _PID_HLINKSAlJ http://www.mathworks.com/  F Microsoft Word 97-2003 Document MSWordDocWord.Document.89q^ 2 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~_HmH nH sH tH 8`8 Normal_HmH sH tH T@T  Heading 2$$x<@&a$5:CJOJQJDA D Default Paragraph FontViV  Table Normal :V 44 la (k (No List 2B@2 Body TextCJJ"@JCaption$xxa$5CJmHnHu6@6 0I Footnote Text@&`!@ 0IFootnote ReferenceH*6U`16 U Hyperlink >*B*phPK![Content_Types].xmlN0EH-J@%ǎǢ|ș$زULTB l,3;rØJB+$G]7O٭V$ !)O^rC$y@/yH*񄴽)޵߻UDb`}"qۋJחX^)I`nEp)liV[]1M<OP6r=zgbIguSebORD۫qu gZo~ٺlAplxpT0+[}`jzAV2Fi@qv֬5\|ʜ̭NleXdsjcs7f W+Ն7`g ȘJj|h(KD- dXiJ؇(x$( :;˹! I_TS 1?E??ZBΪmU/?~xY'y5g&΋/ɋ>GMGeD3Vq%'#q$8K)fw9:ĵ x}rxwr:\TZaG*y8IjbRc|XŻǿI u3KGnD1NIBs RuK>V.EL+M2#'fi ~V vl{u8zH *:(W☕ ~JTe\O*tHGHY}KNP*ݾ˦TѼ9/#A7qZ$*c?qUnwN%Oi4 =3N)cbJ uV4(Tn 7_?m-ٛ{UBwznʜ"Z xJZp; {/<P;,)''KQk5qpN8KGbe Sd̛\17 pa>SR! 3K4'+rzQ TTIIvt]Kc⫲K#v5+|D~O@%\w_nN[L9KqgVhn R!y+Un;*&/HrT >>\ t=.Tġ S; Z~!P9giCڧ!# B,;X=ۻ,I2UWV9$lk=Aj;{AP79|s*Y;̠[MCۿhf]o{oY=1kyVV5E8Vk+֜\80X4D)!!?*|fv u"xA@T_q64)kڬuV7 t '%;i9s9x,ڎ-45xd8?ǘd/Y|t &LILJ`& -Gt/PK! ѐ'theme/theme/_rels/themeManager.xml.relsM 0wooӺ&݈Э5 6?$Q ,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧6 0_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!0C)theme/theme/theme1.xmlPK-! ѐ' theme/theme/_rels/themeManager.xml.relsPK] @6`c@6n  #i(*0V4<tei "$%&()*,-./157&4ii!#'+06/CE ' ) < P R ###%%%@6::::X/Xb$^Yi60nb$-')4Z:3 0e0e     @  5% 8c8c     ?A)BCD|E||@((  t  0_6Z(k7A? #" `?t  0 ?^002A? #" `?B S  ?W1X1@654}taAt   % W Z   % i k ag\cksMOPSTVMOgikrjlm o @!B!G!N!!!##"$($I$K$$$&&w-}---s.y.2/3/f1g152646567686:6;6=6A6;>R[lT[~, 7 P V   \ d  i k "\d QYns|EJPTNTlrj*  adw|hpks-/dgfh < S k l !!!! ""c"k"#####I$L$u$w$$$[%a%%%|&&' ( ((,(1(**+++,....v1x1<2I2[2i2f3g34452646567686:6;6=6A6333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333\/F * < S #"#"##1%9%A%A%x%%%%b'u''''''''''' (,(P-.222333Z334445555506A6i  n  n B*e hh^h`o(.hh^h`o(.hh^h`o(.hh^h`o(.n i B*en`_] S  q8u8]dh $"J$&n(UD+0/M1 2=j>B CV:C5cG{G0ILNIAQrX`YG\IO\]2w_a`VbXb$c"eYrHvt2v3w3x#ypAo#XH& a4K0wUORZuP|Ol=ub+1[7)0^8cT'mm I:e>jG ISvU65755@.... "#j(j)h1h3h4@6@@@*X@.`@6p@@f@UnknownG*Ax Times New Roman5Symbol3. *Cx ArialA$BCambria Math"Ahx>-a-a!24553HP?Vb2!xx ECE 202Melinda Holtzmanholtzman