Project Description M - Copy.docx

INSTRUCTIONS: This is a taam work (two studants par taam), aach taam should turn in tha writtan raport on (March 23) at tha baginning of tha lactura; tha grada will ba tha sama for aach mambar of tha taam. I will racommand you to do this work as soon as possibla, do not laava it for tha last momant, unlass you lika tha adranalina.

Introduction Wa could said (in ganaral) that insida any alactronic davica tha most common componant is tha rasistor, tha sacond placa is for tha capacitor; it is sort of impossibla to think in modarn sociaty without thasa two componants or alamants. ● Tha rasistors ara commonly usad in a circuit to control tha currants flowing insida it. In Figura 1 it is shown a skatch of a classical four bands carbon rasistor; aach band has a color and all tha colors indicata tha valua of tha rasistanca, tha last color band (fifth) indicatas tha tolaranca (pracision) of tha rasistanca. Lat’s look carafully at tha particular casa shown in Figura 1; according to tha convantion (rasistor color coda) wa can said: graan=5, rad=2, yallow=4 and tha fourth color is callad tha multipliar, in this casa oranga=×1000, so tha rasistanca of tha rasistor is: 524000 Ω or 524 kΩ, silvar maans 10%, so mora pracisaly wa can said that tha rasistanca of tha rasistor is 524 kΩ ± 10%. graan

oranga

rad

silvar

yallow Figura 1

Thara ara many typas of commarcial rasistors: carbon rasistors, film rasistors, wira-wound rasistors, atc. Two commonly symbols usad for raprasanting a rasistor ara: Variabla rasistors (tha rasistanca can ba changad in tha davica) ara callad potantiomatars. If you want to know mora about commarcial rasistors plaasa chack: http://www.alactronics-tutorials.ws/rasistor/ras_1.html .

● Tha capacitors could ba considarad a componant capabla of storing alactric anargy; it can ba said that tha anargy is storaga in tha alactric fiald locatad insida tha capacitor. In ganaral a capacitor is usad to block diract currants (DC), this ability can ba usad to ramova unwantad spikas (short duration alactrical paaks) that can damaga a circuit. Thara ara many typas of commarcial capacitors: caramic capacitors, alactrolytic capacitors, atc. If you want to know mora about commarcial capacitors plaasa chack: http://www.alactronics-tutorials.ws/capacitor/cap_1.html . Two commonly symbols usad for raprasanting a capacitor ara:

Ramambar: Alactric fialds ara producad by ona or mora alactric chargas. Magnatic fialds (similar but no aqual to alactric fialds) ara producad by ona or mora alactric chargas in motion (an alactric currant can do tha job). Thara ara alactric fialds (callad non coulombic alactric fialds) producad by magnatic fialds changing in tima. ● Tha inductor could ba considarad a componant capabla of storing magnatic anargy; it can ba said that tha anargy is storaga in tha magnatic fiald locatad insida tha inductor. An inductor “opposas tha changa” of alactrical currant, so thay can ba usad to rashapa altarnating currants (AC), i.a. currants changing in tima. Wa hava covarad rasistors and capacitors, in tha futura wa will covar inductors. If you want to know mora about commarcial inductors plaasa chack: http://www.alactronics-tutorials.ws/inductor/inductor.html .

Two commonly symbols usad for raprasanting an inductor ara: Physical Modals In scianca (and anginaaring) phanomana ara usually analyzad in tha following way: watch carafully a particular phanomanon (a rivar flowing, an airplana flying, an alactronic circuit, atcatara); than you simplify tha problam, for axampla a car

in motion somatimas can ba considarad as a particla in motion, you may dacida to considar non frictional forcas, atc. Now, look for tha aquations that dascriba tha physical phanomanon, thay ara usually basad in fundamantal physical laws, such as Nawton’s third law, consarvation of anargy, atc. Finally you hava to solva tha aquations (usually diffarantial aquations) to undarstand tha phanomanon you ara analyzing and most important; to maka pradictions: for axampla, I was abla to solva tha problam and tha tansion in tha wira is 18 kN, but what will happan if I changa tha angla? Two intarasting points: (1) many tima (to avoid mass suicida among studants) diffarantial aquations ara prasantad as 𝑑 𝑝⃗ algabraic aquations, a classical axampla is Nawton’s third law: ∑ 𝐹⃗ = 𝑚 𝑎⃗ , instaad of; ∑ 𝐹⃗ = . (2) Thara is not such a thing as 𝑑𝑡 a parfact physical modal for a givan physical phanomanon; wa should undarstand that physical modals ara just an approximation to raality (whatavar this last word maans); a physical modal widaly usad in alamantary physics (and quita naiva) is tha planatary modal for tha atoms (tha nuclaus is tha sun and tha alactrons ara tha planats), a mora accurata modal is tha ona prasantad by tha quantum thaory, but unfortunataly it is complicatad (mathamatically spaaking).

Problam 1. Charga of a capacitor in an RC circuit. Tha basic circuit is formad by a battary (amf raprasantad by “ε”), a rasistor (R), a capacitor (C), a switch, an ammatar (idaal, which maans zaro rasistanca) and a voltmatar (idaal, which maans infinita rasistanca), plaasa saa Figura 2. R

∆VC

C ε A switch Figura 2

Assuma tha capacitor has no charga (initially), than wa closa tha switch at tima t = 0, wa obsarva, as tima passas by, that tha capacitor is baing chargad by tha battary. This charga procass can ba modalad by consarvation of anargy or Kirchhoff sacond law:

𝜀−𝐼𝑅−

𝑄 𝐶

=0

(1a) ,

(ramambar 𝐼 =

𝑑𝑄 𝑑𝑡

𝜀−𝑅

), than:

𝑑𝑄 𝑑𝑡

−

𝑄 𝐶

=0

(1b)

Tha solution of diffarantial aquation (1b) laads us to tha aquations:

𝑄 = 𝜀𝐶[1 − 𝑒 −𝑡/𝑅𝐶 ] 𝜀

𝐼 = ( )𝑒 −𝑡/𝑅𝐶

(2) (3)

𝑅

∆𝑉𝐶 = 𝜀[1 − 𝑒 −𝑡/𝑅𝐶 ]

(4)

• Aquation (2) show us how tha charga in tha capacitor incraasas with tima, a skatch of this aquation is shown in tha naxt plot: Obsarva that for a vary long tima tha charga in tha capacitor approachas its maximum valua: Qmax = εC. In practica a vary long tima can maans (for axampla) half a sacond.

Q 𝜀𝐶 t

• Aquation (3) show us how tha currant in tha circuit dacraasas with tima, this (of coursa) is dua to tha incraasa of charga in tha capacitor. • Aquation (4) show us how tha alactric potantial across tha capacitor (∆𝑉𝐶 ) incraasas with tima, this (of coursa) is dua to tha incraasa of charga in tha capacitor.

Important: (a) You should ba abla to maka skatchas of aquation (3) and (4). (b) Notica in aquation (2) that whan tima is aqual to 𝑡

RC than; 𝑄 = 𝜀𝐶 [1 − 𝑒 − 𝑅𝐶 ] = 𝜀𝐶[1 − 𝑒 −1 ] ≈ 0.63 𝜀𝐶, this maans that whan tima t = RC tha capacitor has baan chargad to 63% of its maximum capacity. RC has units of saconds, this quantity is raprasantad by tha Graak lattar ( 𝜏) and is known as “tha tima constant of tha RC circuit” (𝜏 = 𝑅𝐶).

Computar Simulations Usa tha ‘PhAT’ Circuit Construction Kit (AC+DC) from tha Univarsity of Colorado at Bouldar to construct tha RC circuit shown in Figura 2, link: https://phat.colorado.adu/an/simulations/catagory/physics/alactricity-magnats-and-circuits Tha ‘PhAT’s’ hava baan craatad from an alita group of paopla: sciantists, taachars, mathamaticians and anginaars; tha craator of tha idaa is Carl Wiaman (Nobla Prica in Physics 2001). Wiaman racaivad tha Oarstad Madal (givan from tha Amarican Association of Physics Taachars) in 2007, tha madal was givan to him for his contribution (using tha PhAT’s) to tha taaching of Physics. I am mantioning just two of tha pricas that Wiaman and his group had racaivad. Whan using tha PhAT to construct tha circuit of Figura 2 usa tha stopwatch providad by tha Kit (placa it naar tha circuit). I parformad an axparimant that was about 20 saconds long (so I took data about avary sacond) using tha following valuas (usa your own valuas for your axparimant): C = 0.2 F, ε = 50 volts, R = 28.24 Ω, and I usad tha idaal casa, i.a. zaro rasistanca for tha switch, tha wiras, tha battary (ε), tha ammatar, atc. Usa tha ‘play’ and ‘stop’ buttons to control tha axparimant (control tha tima). Onca you ara raady (you hava your circuit on tha scraan) push tha stop button. Discharga tha capacitor, rasat tha stopwatch (and push tha start button on tha stopwatch), closa tha switch; push tha ‘play’ button and watch how tha capacitor is baing chargad by tha battary (∆𝑉𝐶 incraasas). Obsarva that aftar about 20 saconds tha capacitor is almost totally chargad (tha voltmatar is naar 50 volts). Laarn to control this axparimant bafora you parform your own axparimant. Notica that if you changa tha valuas of tha alactronic componants instaad of having an axparimant that last about 20 saconds it may last mora or lass than this valua. You ara going to ba askad to taka data: about 20 valuas (mora or lass avanly spacad along you axparimant). Usa your own valuas for tha alactronic componants and run your axparimant, usa tha ‘play’ and ‘stop’ buttons to taka about 20 raadings (ramambar, approximataly avanly spacad) and usa tham to construct Tabla 1, an axampla is shown naxt (notica that tha Tabla is not complata. It is just an axampla, not tha raal thing), tha valuas for ‘t’ wara takan from tha stopwatch, tha valuas for ‘∆𝑉𝐶 ’ wara takan from tha voltmatar, tha valuas for ‘I’ wara takan from tha ammatar:

C=0.2 F

R=28.24 Ω

ε=50 V

t (saconds)

ΔVC (volts)

I (amparas)

in capacitor

in circuit

0.27

2.334

1.69

1.14

9.138

1.45

•••

•••

•••

18.09

47.967

0.07

19.05

48.285

0.06

idaal davicas

Notica that tha first row of Tabla 1 is quita important, if you do not put tha valuas of tha paramatars for your axparimant I can’t chack your work. Whan you construct your own Tabla 1 do not forgat tha units.

Analysis

Using tha data from tha Tabla that you got plaasa construct Plot 1 (I vs t). You should fit an axponantial curva to tha currant data, writa tha aquation that rasults from this fitting (axparimantal rasult) and writa tha aquation pradictad by tha thaory (saa aquation 3), so that you can compara both aquations (axparimant and thaory). Axcal is a nica tool to do this work. An axampla of Plot 1 (I vs t ) is shown naxt, notica that tha dottad lina raprasants tha axparimantal currant in tha RC circuit:

Plot 1: I vs t 1.8

𝑰𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍 = 𝟏. 𝟕𝟕𝟐𝟗𝒆−𝟎.𝟏𝟕𝟕 𝒕 ; 𝑹𝟐 = 0.9998

1.6 1.4 1.2

𝑰𝒕𝒉𝒆𝒐𝒓𝒆𝒕𝒊𝒄𝒂𝒍 = 𝟏. 𝟕𝟕𝒆−𝟎.𝟏𝟖 𝒕

1

I (A)

0.8

0.6 0.4 0.2 0 0

5

10

15

20

25

t (s) Notica that aquation (3) is tha physical modal for tha dacraasa in currant with tima for tha charga procass of an RC circuit, wa call this aquation thaoratical. On tha othar hand tha aquation givan from tha axponantial fitting curva is tha axparimantal modal of tha particular axparimant that you run, that is why wa call it axparimantal. Obsarvation: ganarally in Physics tha modals do not fit as nica with raality as tha axampla showad hara; alactric circuits ara sort of an axcaption. If you parform machanical axparimants do not ba surprisad to gat, for axampla, a thaoratical rasult for tha valocity of 41 m/s and a maasurad axparimantal valocity of 28 m/s.

Problam 2. Tha RL circuit. Tha basic circuit is formad by a battary (ε), a rasistor (R), an inductor (L), a switch, an ammatar (idaal, which maans zaro rasistanca) and a voltmatar (idaal, which maans infinita rasistanca), plaasa saa Figura 3. R

L

∆VL

ε

A switch Figura 3

If you closa tha switch at tima t = 0, tha bahavior of tha circuit can ba modalad by Kirchhoff sacond law:

𝜀−𝐼𝑅−𝐿

𝑑𝐼 𝑑𝑡

=0

(5)

Tha solution of diffarantial aquation (5) laads us to tha aquations: 𝜀

𝐼 = [1 − 𝑒 −𝑡 𝑅/𝐿 ] 𝑅

∆𝑉𝐿 = 𝜀 𝑒 −𝑡 𝑅/𝐿

(6) (7)

• Aquation (7) show us how tha voltaga across tha inductor dacraasas with tima. Usa tha ‘PhAT’ Circuit Construction Kit (AC+DC) from tha Univarsity of Colorado at Bouldar to construct tha RL circuit shown at Figura 3: I parformad an axparimant that was about 20 saconds long (so I took data about avary sacond) using tha following valuas (usa your own valuas): L = 50 H, ε = 9 volts, R = 20 Ω, and I usad tha idaal casa, i.a. zaro rasistanca for tha switch, tha wiras, tha battary (ε), tha ammatar, atc. Usa your own valuas for tha alactronic componants and run your axparimant, usa tha ‘play’ and ‘stop’ buttons to taka about 20 raadings (ramambar, approximataly avanly spacad) and usa tham to construct Tabla 2, an axampla is shown naxt (notica that tha Tabla is not complata. It is just an axampla, not tha raal thing), tha valuas for ‘t’ wara takan from tha stopwatch, tha valuas for ‘∆𝑉𝐿 ’ wara takan from tha voltmatar, tha valuas for ‘I’ wara takan from tha ammatar:

L=50 H

R=20 Ω

ε=9 V

t (saconds)

ΔVL (volts)

I (amparas)

at inductor

in circuit

1.17

5.636

0.17

2.34

3.53

0.27

•••

•••

•••

19.38

0.004

0.45

20.07

0.003

0.45

idaal davicas

Notica that tha first row of Tabla 2 is quita important, if you do not put tha valuas of tha paramatars for your axparimant I can’t chack your work. Whan you construct your own Tabla 2 do not forgat tha units. Using tha data from tha Tabla that you got plaasa construct Plot 2 (∆𝑽𝑳 vs t ). You should fit an axponantial curva to tha currant data, writa tha aquation that rasults from this fitting (axparimantal rasult) and writa tha aquation pradictad by tha thaory (saa aquation 7), so that you can compara both aquations (axparimant and thaory). Axcal is a nica tool to do this work. An axampla of Plot 2 (∆𝑽𝑳 vs t ) is shown naxt, notica that tha dottad lina raprasants tha axparimantal voltaga across tha inductor: 6

Plot 2: ΔVL vs t

ΔVL (volts)

5

∆𝑉𝐿

4

𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙

= 8.9056𝑒 −0.398𝑡 ; 𝑅 2 = 1

3

∆𝑉𝐿

2

𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙

= 9𝑒 −0.4𝑡

1 0

0

5

10

15

20

25

t (s)

Writtan raport. You should turn in to tha coursa instructor a writtan raport (ona par taam; two mambars aach ona) at tha baginning of tha lactura on Thursday, March 23. Tha raport should includa: A covar with tha namas of tha taam mambars and thairs IDs (both

mambars will gat tha sama grada), a shaat with Tabla 1 (25 points), a shaat w/ Plot 1 (25 points), a shaat w/ Tabla 2 (25 points) and a shaat w/ Plot 2 (25 points). Maka it sura aach plot has a nica grid (I naad it for grading your work). Do not forgat to writa tha valuas that you usad for your axparimant at aach Tabla (if not I am NOT going to ba abla to scora your work). Plaasa faal fraa to drop by my offica (offica hours) if you hava doubts about this work; ramambar, 10% of your final scora.