ECE-202 Homework Problems (Se 1) Spring 18 TO THE STUDENT: ALWAYS CHECK THE ERRATA on he web. ANCIENT ASIAN/AFRICAN/NATIVE AMERICAN/SOUTH AMERICAN ETC. PROVERB: If you give someone a fish, you give hem food for a day; each a person o fish and you give hem food for a lifeime. The Moral of he Proverb is: (i) solve problems so ha you learn o fish he waers of circui heory; (ii) if he TA gives you he soluion, hen you have fish for he day, bu will sarve on he day of he exam. RAY S VIEW OF EDUCATION: The success of your educaion depends on your invesmen in ha educaion. All invesmen is risky in regards o success. Invesmen planners always alk abou he long erm reurns. Learn he maerial of his course for he long erm, no he shor-erm grade. Informaion, KVL, KCL, Ohm s law, is only par of he invesmen porfolio. The developmen of your hinking paerns consiues he long-erm goal. Successful professors have hinking paerns you would do well o emulae, as a musician migh learn he rhyhm and blues progressions of he bes guiariss. Learn how o hink abou a problem, break i ino small pieces, and solve i, o become he bes you can be. Mediocriy is mosly a choice move o a higher level? ADVICE: (i) Carefully read he lecure noes and do he examples herein before aemping HW. (ii) Do as many problems in each chaper of he ex as you can in addiion o he HW. Try he easy ones firs. (iii) Work back exams wihou noes or looking a soluions. (iv) When sudying for an exam, work backwards hrough he maerial saring wih he mos recen lecure. Playing cach-up seldom works. DUE FRIDAY WEEK 1 (JANUARY 12) HW1 1. (Signal represenaion and Laplace ransforms) Recall ha Kδ () denoes he so-called dela funcion wih weigh or area K, u() = r() = u() = δ (τ )dτ denoes he uni sep funcion, and u(τ )dτ he ramp funcion. (The dela-funcion is NOT really a funcion, bu raher a so-called disribuion; neverheless we rea i as a funcion.) The above are simple signals and can be hough of as a basis for he signal space. (a) Below are hree signals, f 1 (), f 2 (), f 3 (). Using graph paper, skech (i) f 1 ( +1) and represen i as a sum of simple signals; (ii) f 2 ( 2) and represen i as a sum of simple signals; (iii) f 3 ( 1) and represen i as a sum of simple signals.
202 HW Probs page 2 Spring 18 (iii) (b) Compue he Laplace ransforms H 1 (s) = L f 1 ( +1), H 2 (s) = L f 2 ( 2), and H 3 (s) = L f 3 ( 1). (c) Repea pars (a) and (b) for he derivaives of he signals, i.e., (i) g 1 () = d d f 1 ( +1) ; hin for (b): consider ĝ 1 () = g 1 () 0 0 oherwise (ii) g 2 () = d d f 2 ( 2) ; (iii) g 3 () = d d f 3 ( 1).. 2. (a) Below is code for generaing an mfile in MATLAB for generaing he sep funcion. Use his code o produce an mfile in MATLAB for your use when ploing funcions in his HW and hroughou he course. Now o be urned in as par (a) of his problem is your code for generaing an mfile o produce he ramp funcion, r(). A direc approach migh be o modify he code given below for he sep funcion. funcion f = usep() = +1e-12; f = (sign()+1)*0.5; (b) Do no evaluae he Laplace ransform inegral for any par below. Use properies and he ables/class noes. Skech he indicaed signals on graph paper or in MATLAB using he code from par (a). Typically, I do somehing like: = 0:0.02:4; y = usep(); yy =.*usep(); plo(,y,,yy) 2
202 HW Probs page 3 Spring 18 Yours will of course be more complicaed. Wha do you observe in case (i)? (i) Wrie f 1 () = 2u( +1)r(1 ) as a SUM of seps, ramps ec bu NO producs of funcions. THINK? (ii) Repea par (i) for f 2 () = u()r( +1)u(2 ). (iii) Compue he Laplace ransforms of he signals in b-(i) and b-(ii) above. Be careful of (i). Again, do no evaluae he Laplace ransform inegral. 3. (Laplace ransforms wih Circuis--A 201 Review) (a) For he circui below, find he Laplace ransforms of i in (), v in (), i 2 () and i 3 () when i in ()= 100e 2 u() u( 1) ma. This is ricky apply he heorems carefully; you need o make a modificaion o he inpu in order o find he Laplace ransforms. In his course capial leers denoe Laplace ransforms of he ime funcions. (b) (Laplace Transforms wih Volage Division Review) For he circui below, R s = 40Ω, R 1 = 200 Ω, R 2 = 200 Ω, R 3 = 400 Ω, and R 4 = 200Ω. Use class examples and noes o compue V ou (s) = L v ou () [ ] for he inpu v in ()= 50r() 50r( 1) 50u( 1) V. (c) For he circui below, suppose R 1 = 40 Ω, R 2 = 10 Ω, R 3 = 5 Ω, R 4 = 20 Ω, and K = 2. Compue he volage v d when i in ()= 10cos(2)u() 10sin(4)u() A. Wrie down he Laplace ransform of he volage v d using he ransform ables or class noes. 3
202 HW Probs page 4 Spring 18 4. Using properies and previously (in class) compued ransforms, find he Laplace ransform of each of he signals below. When dela-funcions are presen you may use he Laplace ransform inegral o obain he answer. When dela-funcions are no presen in a erm, you should NOT use he Laplace ransform inegral, bu raher rewrie he expression in a form where properies and class resuls (or ables) can be applied easily. (a) f 1 ()= K 5 cos(2 T)e δ( T)u(),T > 0. (You may use Laplace ransform inegral.) (b) f 2 () = K 1 δ ( T ) + K 2 δ ( + T ) K 3 e ( T ) sin(2 T )δ (), T > 0. (You may use he Laplace ransform inegral.) (c) f 3 ()= 2 +1 u( +2)u(2 ). Firs represen his signal as a sum ha allows you o apply he previously compued ransforms and he ransform properies, specifically lineariy and ime shif. Hin 1: Wha is he lower limi of he Laplace ransform inegral and would ha affec he resul? (d) f 4 ()= e ( 2) u( 1) 2e u( 2) Use he properies and ransform ables wherever possible. Noe: e a e a = 1.) (e) Find a simple expression for he waveform, f (), below (ha shows a parial sinusoid) in erms of a sum of a sinusoid and a shifed sinusoid. Then deermine he Laplace ransform of he signal using he ransform properies. The signal f () = 0 for [0,2]. 2 1.8 1.6 1.4 f=2*cos(0.25*pi*) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime in sec 4
202 HW Probs page 5 Spring 18 DUE FRIDAY WEEK 2 (JANUARY 19) HW2 5. For each of he expressions below, compue by hand: (i) he SIMPLIFIED parial fracion expansions, and (ii) he inverse Laplace ransforms. Here a, b > 0. Show ALL work. No deails, no credi: (a) F 1 (s)= 2as + 4a2 8a s 2 + 4as + 4a 2 =???? s +2a +???? (s +2a) 2 ( ) (b) F 2 (s)= (2a+5)s2 (10b+ 4ab)s 5a 2 +5b 2 ; Check: one erm is 5u(). s s 2 2bs + b 2 a 2 (c) F 3 (s)= (2a 2)s2 +(4a 2 16a)s 8a 2 s(s +2a) 2 ; (d) F 4 (s)= s3 +(a 1)s 2 (4a+ 4)s + 4a s 2 (s 2) 2 ; Check: one erm is 1/s. (e) The Laplace ransform of f 5 () = K 1 e a cos(ω) + K 2 e a sin(ω) + K 3 e b u() is 9s 2 + s +32 F 5 (s)= s 3 +5s 2. Find a, b, K 1, K 2, K 3, and ω. Hin: roos(coef) where coef +17s +13 = [coefficiens of he polynomial] produces he roos or zeros of he polynomial in MATLAB. Remark: check answers o (a) and (d) using MATLAB s residue command where possible. Type help residue. 6. This problem requires ha you find parial fracion expansions and he inverse Laplace ransforms of each of he indicaed quaniies volages. All answers mus be in erms of real funcions wih real coefficiens or symbols. Show ALL work. No deails, no credi. (a) For he circui of figure 6a, find V ou (s), he parial fracion expansion of V ou (s), and hen find v ou () when V s1 (s)= 3s2 +24s + 40 s (s +2) 2 + 4 and V s2 (s)= 10 s. (b) For he LINEAR circui of figure 6b R 1 = R 3 = R 4 = 2Ω, R 2 = 4Ω. 5
202 HW Probs page 6 Spring 18 (i) Because he circui is linear, V 2 (s) = K 1 I s1 (s) + K 2 V s2 (s) or equivalenly v 2 () = K 1 i s1 () + K 2 v s2 (). Compue K 1 and K 2. (ii) Suppose I s1 (s) = expansion of V 2 (s). 10 (s +1) and V 2 s2 = s2 + s + 2 s(s 2 2s + 2). Find V 2 (s) and he parial fracion (iii) Find v 2 (). Figure 6.a Figure 6.b 7. Use Laplace ransform properies o find he indicaed ransform for each quesion below. For each par indicae he specific propery or properies you used for each sep. (a) The Laplace ransform of a Speabok calorie inverer circui is H(s)= 4s s 2. Define + 4 G(s) = L e 2 h() where h() = L 1 [ H (s)]. Then compue G(s). (b) Reconsider problem (a) and compue a parial fracion expansion of G(s) = L d d h().???? (c) Reconsider problem (a) excep suppose now ha G(s) = L[ h() ]. Then compue G(s) = (s 2 + 4) 2. (d) Reconsider problem (a). Find G(s) = L d e 2 ( h() ) d by inspecion. (e) Suppose F(s)= 2 s + 4a (s +2a) 2, a > 0. If g() = e2a f ( T )u( T ), T > 0, hen compue G(s). 8. (a) Consider f () in he figure below. 6
202 HW Probs page 7 Spring 18 F(s). (i) Express f () as a sum of appropriae (shifed) sep funcions and ramp funcions. Compue (ii) Compue L g 1 () (iii) Compue L g 2 () [ ] = L d d f () [ ] = L f (τ )dτ using he derivaive propery. using he inegral propery. (b) Repea par (a)-(iii) for f new ()= f ( +3). DUE WEDNESDAY WEEK 3 (JANUARY 24) HW3 9. Consider he inegro differenial equaion below, dv ou () + 4v d ou ()+8 v ou (q)dq = v in () 4 v in (q)dq 0 (a) Conver he equaion o a purely differenial equaion. How do you ge rid of he inegrals? (b) Deermine he relaionship beween V ou (s) and V in (s), v in (0 ), v ou (0 ), and dv ou d (c) When he iniial condiions are all zero and v in (0 ) = 0, one can define he relaionship 0 (0 ). H(s) = V ou (s) associaed wih he inegro-differenial equaion. Laer we will call H (s) he ransfer V in (s) funcion meaning ha V ou (s) = H (s)v in (s) when all IC s are zero. Compue H (s). (d) Given your answers o (a) and (b), show ha in he s-world: V ou (s) = H (s)v in (s) + Somehing ( )v ou (0 ) + (Somehing else) dv ou (0 ) d 7
202 HW Probs page 8 Spring 18 Noe: We noe here ha he oupu is linear in he inpu V in (s) when he IC s are zero. Look up wha linear means before you answer!!!!! Quesion: Wha can you say abou he lineariy of he soluion when he inpu is zero? (e) Find v ou () assuming v in () = 12e 2 u() V for 0 and all IC s are zero, e.g. by virue of he sep funcion v in (0 ) =?. Proceed as follows: (i) Compue he parial fracion expansion of V ou (s) = H (s)v in (s) showing ALL work. DO NOT USE COMPLEX ARITHMETIC. Any erms wih complex poles need o be of he form Bs + C (s + a) 2. For he above equaion wha are a and b. Noe ha a pair of complex poles has he form 2 + b (s + a + jb)(s + a jb) = (s + a) 2 + b 2. Bs + C (ii) From class noes, recall how o break up (s + a) 2 so ha one can inver he s-domain 2 + b funcion o ge a decaying cosine and a decaying sine. (iii) Inver your parial fracion expansion o obain v ou () when all IC s are zero. This is called he zero-sae response where zero-sae means he equaion has zero iniial condiions. In he fuure we designae his par of he response as v ou,zs (). (f) Now suppose v in () = 0 in which case v in (0 ) = 0, and v ou (0 ) = 3 V, dv ou (0 ) = 3 V/s. Find d v ou (). Since he inpu is zero and v ou () depends only on he iniial condiions, his response is called he zero-inpu response (so creaive a name). In he fuure we designae his par of he response as v ou,zi (). 10. (a) In he circui below, R 1 = 2 Ω, R 2 = 2 Ω, and C 1 = C 2 = 0.2 F. The iniial capacior volage is zero. Define V ou (s) = V C1 (s) +V C2 (s). Compue he raio V ou (s) V in (s) v in () = 10e 10 u() V, compue V ou (s), V C1 (s), and v ou (). using volage division. If 8
202 HW Probs page 9 Spring 18 (b) In he circui below, L = 1 H, R = 2 Ω, C = 0.1 F, and I in (s) = 1. When all iniial s +1 condiions are zero, compue V ou (s), a parial fracion expansion of V ou (s), and he zero-sae (zero iniial condiions) response, v ou (). (c) Consider he circui below in which R = 10 Ω and L = 0.5 H. Suppose v in () = 50 0 e 5q u(q)dq V and all iniial condiions are ZERO. Compue I L (s), he parial fracion expansion of I L (s), and i L (). 11. (a) For he circui below, find V ou (s) and he response v ou () when R 1 = 1 Ω, R 2 = 4 Ω, L = 2 H and he inpu v in () is given in he graph below he figure. All iniial condiions are zero. Plo v ou () in MATLAB using he plo command for 0 5 s: >> = 0:0.02:5; >> vou =????? >> plo(,vou) >> grid >> xlabel('???') >> ylabel('???') 9
202 HW Probs page 10 Spring 18 (b) In he circui below, all iniial condiions are zero, R 1 = R 2 = 10 Ω, and C 1 = C 2 = 1 F. Le Z in (s) be he inpu impedance seen by he curren source. (i) Find he impedance, say Z 1 (s) and Y 1 (s), of he R 1 C 1 pair; (ii) Repea for he R 2 C 2 pair; (iii) Find Z in (s) and Y in (s) ; (iv) Find an expression for V C1 (s) in erms of I in (s) (hin: Ohm s law for impedances); (v) Find an expression for I C2 (s) (hin: ry curren division on he final RC pair); (vi) find v c1 () and i c2 () when I in (s) = 10 s + 0.1. Hin: V(s) = Z(s)I(s). (c) For his par, simply find Y h (s) and Z h (s) for he circui below. Recall ha if you connec an imaginary curren source o he fron end of he circui labeled I in (s) and proceed o compue an expression for V in (s) (a volage across he erminals), hen by he famous mehod of paern recogniion, Y in (s)v in (s) = I in (s) or equivalenly, V in (s) = Z in (s)i in (s). Le R 1 = 4 Ω, R 2 = 2 Ω, L = 2 H, and β = 0.25 mho. 1 0
202 HW Probs page 11 Spring 18 12. Consider he circui below in which R 1 = 2 Ω, R 2 = 2 Ω, R 3 = 1 Ω, and C = 0.125 F. ALL iniial condiions are zero. (a) Wrie a node equaion in V opnode (s), he s-domain volage a he op node of he circui: (b) Solve for V opnode (s) in erms of V s1 (s) and V s2 (s). (c) Find V opnode (s) and v opnode () when v s1 ()= 10e 4 u() V and v s2 ()= 10e 4 u() V. (d) Use Ohm s law in he s-world o find I C (s) in erms of V opnode (s). Then plug in using your answer in (c) and find i C (). ----------------- NON REQUIRED PROBLEM THAT IS PERTINENT TO EXAM Maerial. 4s + 36 NR-Problem. (a) Find an RC circui realizaion for Z in ( s) = s + 8. 2s +12 2s +12 (b) Find an RL circui realizaion for Y in ( s) = + s + 2 s + 4. (c) Find an LC realizaion of Z in ( s) = s2 + 4 + 2s 2s s 2 + 4 (d) Find an LC realizaion of Y in s ( ) = s2 +1 2s + 0.25s s 2 + 4. 1 1