EE40 Summer 2005: Lecture 2 Instructor: Octavian Florescu 1. Measuring Voltages and Currents

Transcription

1 Announemens HW # Due oday a 6pm. HW # posed online oday and due nex Tuesday a 6pm. Due o sheduling onflis wih some sudens, lasses will resume normally his week and nex. Miderm enaively 7/. EE4 Summer 5: eure Insruor: Oavian Floresu eview Mesh and Nodal Analysis Superposiion Equivalen iruis Thevenin Noron Measuring Volages and urrens EE4 Summer 5: eure Insruor: Oavian Floresu

2 eview: Thevenin Equivalen Example Find he Thevenin equivalen wih respe o he erminals a,b: EE4 Summer 5: eure Insruor: Oavian Floresu 3 eure #4 OUTINE The apaior The induor s Order iruis Transien and Seady-Sae response eading haper 3, hap EE4 Summer 5: eure Insruor: Oavian Floresu 4

3 The apaior Two onduors (a,b) separaed by an insulaor: differene in poenial V ab > equal & opposie harge Q on onduors Q V ab (sored harge in erms of volage) where is he apaiane of he sruure, posiive () harge is on he onduor a higher poenial Parallel-plae apaior: area of he plaes A (m ) separaion beween plaes d (m) dieleri permiiviy of insulaor ε (F/m) Aε > apaiane F(F) d EE4 Summer 5: eure Insruor: Oavian Floresu 5 apaior Symbol: Unis: Farads (oulombs/vol) or (ypial range of values: pf o μf; for superapaiors up o a few F!) urren-volage relaionship: i dq d dv d v d d If (geomery) is unhanging, i dv /d i v Elerolyi (polarized) apaior Noe: Q (v ) mus be a oninuous funion of ime EE4 Summer 5: eure Insruor: Oavian Floresu 6 3

4 Volage in Terms of urren Q( ) v ( ) i ( ) d Q() Q() i ( ) d i ( ) d v () Uses: apaiors are used o sore energy for amera flashbulbs, in filers ha separae various frequeny signals, and hey appear as undesired parasii elemens in iruis where hey usually degrade irui performane EE4 Summer 5: eure Insruor: Oavian Floresu 7 Sored Energy APAITOS STOE EETI ENEGY You migh hink he energy sored on a apaior is QV V, whih has he dimension of Joules. Bu during harging, he average volage aross he apaior was only half he final value of V for a linear apaior. QV V Thus, energy is. Example: A pf apaiane harged o 5 Vols has ½(5V) (pf).5 pj (A 5F superapaior harged o 5 vols sores 63 J; if i disharged a a onsan rae in ms energy is disharged a a 63 kw rae!) EE4 Summer 5: eure Insruor: Oavian Floresu 8 4

5 A more rigorous derivaion i v w Final Iniial v v V w v v V Final Iniial i dv v V d v v V V Final Final Iniial dq v V d v d v V V Iniial Final Iniial dq EE4 Summer 5: eure Insruor: Oavian Floresu 9 Example: urren, Power & Energy for a apaior i() v( ) i( ) d v() v (V) τ τ v() μf i (μa) i dv d (μs) (μs) EE4 Summer 5: eure Insruor: Oavian Floresu v and q mus be oninuous funions of ime; however, i an be disoninuous. Noe: In seady sae (d operaion), ime derivaives are zero is an open irui 5

6 Example: urren, Power & Energy for a apaior p (W) i() v() (μs) μf p vi w (J) (μs) w pd τ v EE4 Summer 5: eure Insruor: Oavian Floresu apaiors in Series v () v () i() i() eq v()v ()v () Proof: eq EE4 Summer 5: eure Insruor: Oavian Floresu 6

7 apaiors in Parallel i()i ()i () v() _ i () i () v() _ eq eq Proof: EE4 Summer 5: eure Insruor: Oavian Floresu 3 Praial apaiors A apaior an be onsrued by inerleaving he plaes wih wo dieleri layers and rolling hem up, o ahieve a ompa size. To ahieve a small volume, a very hin dieleri wih a high dieleri onsan is desirable. However, dieleri maerials break down and beome onduors when he eleri field (unis: V/m) is oo high. eal apaiors have maximum volage raings An engineering rade-off exiss beween ompa size and high volage raing EE4 Summer 5: eure Insruor: Oavian Floresu 4 7

8 Induor Symbol: Unis: Henrys (Vols seond / Ampere) (ypial range of values: μh o H) urren in erms of volage: di v ( ) d i ( ) v ( τ ) dτ i( v Noe: i mus be a oninuous funion of ime ) i EE4 Summer 5: eure Insruor: Oavian Floresu 5 Sored Energy INDUTOS STOE MAGNETI ENEGY onsider an induor having an iniial urren i( ) i p( ) v( ) i( ) w( ) w( ) p( τ ) dτ i i EE4 Summer 5: eure Insruor: Oavian Floresu 6 8

9 Induors in Series v () v () i() i() eq v()v ()v () eq EE4 Summer 5: eure Insruor: Oavian Floresu 7 Induors in Parallel i()i ()i () v() _ i () i () v() _ eq eq EE4 Summer 5: eure Insruor: Oavian Floresu 8 9

10 Firs-Order iruis A irui ha onains only soures, resisors and an induor is alled an irui. A irui ha onains only soures, resisors and a apaior is alled an irui. and iruis are alled firs-order iruis beause heir volages and urrens are desribed by firs-order differenial equaions. v s i v s i EE4 Summer 5: eure Insruor: Oavian Floresu 9 Transien vs. Seady-Sae esponse The momenary behavior of a irui (in response o a hange in simulaion) is referred o as is ransien response. The behavior of a irui a long ime (many ime onsans) afer he hange in volage or urren is alled he seady-sae response. EE4 Summer 5: eure Insruor: Oavian Floresu

11 eview (onepual) Any* firs-order irui an be redued o a Thévenin (or Noron) equivalen onneed o eiher a single equivalen induor or apaior. Th I Th Th V Th In seady sae, an induor behaves like a shor irui In seady sae, a apaior behaves like an open irui EE4 Summer 5: eure Insruor: Oavian Floresu esponse The naural response of an or irui is is behavior (i.e., urren and volage) when sored energy in he induor or apaior is released o he resisive par of he nework (onaining no independen soures). The sep response of an or irui is is behavior when a volage or urren soure sep is applied o he irui, or immediaely afer a swih sae is hanged. EE4 Summer 5: eure Insruor: Oavian Floresu

12 Naural esponse of an irui onsider he following irui, for whih he swih is losed for <, and hen opened a : I o o i Noaion: is used o denoe he ime jus prior o swihing is used o denoe he ime immediaely afer swihing < he enire sysem is a seady-sae; and he induor is like shor irui The urren flowing in he induor a is I o and V aross is. EE4 Summer 5: eure Insruor: Oavian Floresu 3 v Solving for he urren ( ) For >, he irui redues o I o o Applying KV o he irui: v()i() i v A, ii, A arbirary >, ii() and di() v () d Soluion: i( ) i() e ( / ) I e -(/) EE4 Summer 5: eure Insruor: Oavian Floresu 4

13 Solving for he Volage ( > ) I o o i I e ( / ) ( ) o v Noe ha he volage hanges abruply: v( ) for >, v( ) i v( I ) I o e ( / ) EE4 Summer 5: eure Insruor: Oavian Floresu 5 Solving for Power and Energy Delivered ( > ) i I e ( / ) ( ) o I o o v p i w o I o e p( x) dx I ( / ) I o e ( / ) ( e ) ( / ) x dx EE4 Summer 5: eure Insruor: Oavian Floresu 6 3

14 Naural esponse of an irui onsider he following irui, for whih he swih is losed for <, and hen opened a : o V o v Noaion: is used o denoe he ime jus prior o swihing is used o denoe he ime immediaely afer swihing The volage on he apaior a is V o EE4 Summer 5: eure Insruor: Oavian Floresu 7 Solving for he Volage ( ) For >, he irui redues o o V o v Applying K o he irui: Soluion: v( ) v() e / EE4 Summer 5: eure Insruor: Oavian Floresu 8 4

15 Solving for he urren ( > ) o V o v v( ) V o e / Noe ha he urren hanges abruply: i( ) v Vo / for >, i( ) e Vo i( ) EE4 Summer 5: eure Insruor: Oavian Floresu 9 Solving for Power and Energy Delivered ( > ) i V o o v Vo p e w p( x) dx Vo / v V o e / ( e ) x / dx v V e / ( ) o EE4 Summer 5: eure Insruor: Oavian Floresu 3 5

16 Naural esponse Summary irui i Induor urren anno hange insananeously i( ) i( i( ) i() e ime onsan ) /τ τ irui v apaior volage anno hange insananeously v( ime onsan ) v( v( ) v() e ) /τ τ EE4 Summer 5: eure Insruor: Oavian Floresu 3 Proedure for Finding Transien esponse. Idenify he variable of ineres For iruis, i is usually he induor urren i () For iruis, i is usually he apaior volage v (). Deermine he iniial value (a ) of he variable eall ha i () and v () are oninuous variables: i ( ) i ( ) and v ( ) v ( ) Assuming ha he irui reahed seady sae before, use he fa ha an induor behaves like a shor irui in seady sae or ha a apaior behaves like an open irui in seady sae EE4 Summer 5: eure Insruor: Oavian Floresu 3 6

17 Proedure (on d) 3. alulae he final value of he variable (is value as ) Again, make use of he fa ha an induor behaves like a shor irui in seady sae ( ) or ha a apaior behaves like an open irui in seady sae ( ) 4. alulae he ime onsan for he irui τ / for an irui, where is he Thévenin equivalen resisane seen by he induor τ for an irui where is he Thévenin equivalen resisane seen by he apaior EE4 Summer 5: eure Insruor: Oavian Floresu 33 Summary apaior Induor dv i d w v v anno hange insananeously i an hange insananeously Do no shor-irui a harged apaior (-> infinie urren!) n n ap. s in series: n ap. s in parallel: eq eq i n i i i v w i i anno hange insananeously v an hange insananeously Do no open-irui an induor wih urren (-> infinie volage!) n ind. s in series: n ind. s in parallel: di d EE4 Summer 5: eure Insruor: Oavian Floresu 34 eq eq n i n i i i 7

18 Summary on d Seady-sae nohing is ime varying. In seady sae, an induor behaves like a shor irui In seady sae, a apaior behaves like an open irui EE4 Summer 5: eure Insruor: Oavian Floresu 35 8