Several schematic symbols for a capacitor are shown below. The symbol resembles the two conducting surfaces separated with a dielectric.

Transcription

1 Capacitor Capacitor are two terminal, paive energy torage device. They tore electrical potential energy in the form of an electric field or charge between two conducting urface eparated by an inulator called a dielectric. Becaue an electrical inulator eparate the plate, a cannot not conduct direct current. It doe however, conduct alternating current. Several chematic ymbol for a are hown below. The ymbol reemble the two conducting urface eparated with a dielectric. C C3. unpecified type uf polarized pf variable S Figure : Capacitor Schematic Symbol t= One of the main ditinguihing characteritic of a i it dielectric. I The dielectric material may be air, Mylar, polyeter, V mica or a variety of other material. The dielectric affect many of the G parameter of the uch a it temperature tability, breakdown ma voltage and ize. Some are polarized, uch a electrolytic or tantalum. Thee require connection be made to it oberving a particular polarity. Capacitor are alo differentiated a between being fixed or variable in value. In the air variable, there are conducting plate on a rotatable haft that are andwiched between plate on a fixed frame. Rotating the haft change the degree of overlap between the plate, effecting the capacitance value. The are alo variable C C3 that ue Ceq a thin = platic + C + dielectric. C3 Below we ee everal different type of. From top left are two variable called trimmer. Thee are generally mall in value, -5pF, and are ued at radio frequencie. Moving right, the next i a mica, o called becaue of it mica dielectric. It i a very lowlo C C3 alo ueful at radio Ceq frequencie. The two twited wire (/)+(/C)+(/C3) form what i known a a gimmick. It provide a one-time trimming that i et by clipping off a little of the two wire until the deired capacitance i achieved. initially ch to volt

2 Figure : Variou On the bottom row tarting from left, we have a Mylar, followed by a ceramic and an electrolytic. The Mylar cap i ueful at lower frequencie and alo ha very low DC leakage. No dielectric i perfect o there i alway ome DC leakage in a. The ceramic cap i a very general purpoe and i ueful from audio to high frequencie. The electrolytic on the far right i an example of a polarized. The black tripe indicate the negative terminal. Connecting the negative lead of an electrolytic to a higher potential than the poitive lead, can caue permanent damage and poibly a rupture of the cae. The amount of capacitance available in a depend on it phyical dimenion. For a parallel plate ; C = ɛa/d where C i capacitance in Farad, A i urface area between the plate, ɛ i the permitivity of the dielectric and d i the ditance between the plate. We often think of a parallel plate but they may be found in many phyical configuration. However, the general meaning of thi equation hold for other phyical configuration. In other word, the cloer the two urface and the larger their urface area i, the greater the capacitance. In term of charge and voltage, the capacitance i defined a; C = Q/V where Q i charge in Coulomb and V i the voltage exiting between the plate; thu the unit of capacitance are coulomb per volt.

3 C C3. unpecified type uf polarized pf variable S t= Capacitor in Serie and Parallel I From our work on reducing reitor network to a implified form, we are often preented with V G parallel and erie combination of. The ame technique mamay be applied to to find an equivalent for a network of. However, unlike parallel reitor, parallel have an equivalent capacitance that i obtained by umming their value. Converely, the equivalent to a et of erie i the reciprocal of the um of the. initially charged to volt C C3 Ceq = + C + C3 C C3 Ceq (/)+(/C)+(/C3) Figure 3: Capacitor in Serie and Parallel Capacitor Current-Voltage Relationhip The current-voltage relationhip for the i; i C = C dv dt where i i the current through the and dv i the change in voltage per unit time. We dt could alo ay; C i C = dv dt Note that the uual capital I and V ha been exchanged for a lower cae i and v. Thi i to emphaize the time varying nature of the current and voltage. Thee equation above tell u everal thing very clearly. The firt one make clear that if the voltage i not changing, i.e.; dv dt = (a DC ource) then no current i flowing. Therefore, a i an open circuit to DC current. Secondly, if a contant current i applied to the, the change in voltage acro it terminal with repect to time i contant, i.e., a traight line. A Spice imulation illutrate thi well. 3

4 A charging from a contant current ource Iin gnd cap_in PULSE( ma m n n 5m 5m) ;am current ource c cap_in gnd r cap_in cap_in G ;reitor required for convergence.control tran u m plot v(cap_in) gnuplot cap_irc_charging V(cap_in) xl u m ;make.ep for latex et noakquit.endc.end In thi circuit, the G reitor i only required o that pice ha a ground reference, otherwie it will have convergence error. The reitor i o big, it may be ignored. Running thi imulation C C3 yield: 9. unpecified type a charging from a contant current ource uf polarized v(cap_in) pf variable V V S t= I ma G (a) Spice Output (b) Schematic Figure : ma Contant Current Source Charging a Capacitor Working the equation out yield: C C3 dv Ceq = + C + C3 dt = C i C dv dt = ( 3 e A ) F dv dt = V m C C3 Ceq Thi make ene a the graph how a voltage ramp (/)+(/C)+(/C3) of volt per milliecond. initially charg to volt Finally, we can ee that the voltage acro a cannot change intantaneouly. For the voltage to change in zero time, then i(c) would have to be infinite.

5 RC Circuit Behavior C C3 When a i charged from a voltage ource in erie with a reitor, we can ee that the voltage acro the terminal. charge differently. uf See pf the chematic below. unpecified type polarized variable S V t= I ma G Figure 5: Circuit to Determine Capacitor Tranient Repone Auming that initially, there i no tored charge on the at (t = ), when the witch i cloed, the charge on the battery begin Cto charge C3 the Ceq = through + C the + C3reitor. A the voltage acro the terminal charge to the ource potential, the current flow aymptotically approache zero. We ay at thi point the i charged. If we diconnect the from the circuit, we would find the voltage acro it remain equal to the battery voltage. initially to v For thi particular circuit, it ha been Cdetermined C3 that the voltage Ceq acro the i expreed (/)+(/C)+(/C3) by the equation: V c (t) = V rc ( e t/rc ) Thi relationhip can alo be een from a ngpice imulation a hown below. RC network - charging Vin vin. PULSE(. m n n 5m 5m); V ource r vin cap_in k c cap_in gnd.control tran n m plot rc_im V(vin) V(cap_in) xl u m * gnuplot rc_im V(vin) V(cap_in) xl u m ;make imout.ep for latex et noakquit.endc *meaure the time difference between input reaching.v to *cap_in reaching.3...i.e, the RC time contant.mea tran tdiff trig v(vin) val=. rie= td=5p + targ v(cap_in) val=.3 rie= td=5n.end 5

6 rc network - charging v(vin) v(cap_in) V Figure : Capacitor charging through a reitor Although there i no witch in thi circuit, the voltage ource i not turned on until. ec. We can clearly ee the exponential charging curve. If we chooe t equal to the product of R and C, the voltage expreion become: V c (t) = V rc ( e ) V c (t) = V rc (.3) V c (t) =.3V rc The product RC i called the RC time contant or τ and ha unit of econd. When time t i equal to RC, the voltage acro the will have reached 3% of final value of V rc. Our pice imulation found the time for the target voltage to reach.3 volt with the.meaure tatement. The value it found wa: Meaurement for Tranient Analyi tdiff = 9.997e- Thi i almot exactly m, our time contant τ, a () = 3 econd, or milliecond

7 If we tart with a charged to V rc and dicharge it, we alo ee a different exponential behavior. For that circuit, without a ource but with only a charged and reitor, the equation for the voltage acro the i given by: ( V c (t) = V rc e t/rc) If we let t equal RC we get: V c (t) =.3V rc So, the voltage acro the cap after one time contant, τ, will be.3 time the initial voltage the wa charged to. Again, we can run an Spice imulation. RC network - Dicharging charged through a reitor.ic v(cap_in)=v ; cap initally charged to volt r cap_in gnd k c cap_in gnd.control tran n m * gnuplot rc_im_dich V(cap_in) xl u m ;make.ep file for latex plot rc_im_dich V(cap_in) xl u m et noakquit.endc *meaure the time difference between cap_in tarting from 9.99v to *reaching 3...i.e, the RC time contant.mea tran tdiff trig v(cap_in) val=9.99 fall= td=p + targ v(cap_in) val=3. fall= td=p.end Meaurement for Tranient Analyi tdiff=.9e-3 Again, almot exactly m, our time contant, a = 3 7

8 type S V t= rc network - dicharging v(cap_in) I ma G V initially charged to volt C C3 Ceq = + C + C (a) Simulation of Capacitor dicharging (b) Schematic, Capacitor dicharging C C3 Ceq (/)+(/C)+(/C3) Figure 7: Capacitor dicharging through a reitor Capacitor Application Capacitor are ued in many if not mot electronic circuit. They primarily erve in to broad categorie; a timing element and to provide election or rejection of certain frequencie. We have een already how a can provide a time delay to an input voltage when ued in conjunction with a reitor. The RC time contant τ allow calculation of a time period before a voltage waveform reache a particular value. If we have an element that can make a deciion or caue a witch to be actuated at thi voltage then we can alo aume that the delay from the onet of the applied voltage to the witch actuation will be given by τ. Capacitor are frequently ued in AC to DC power upplie. The input AC waveform i converted to a pulating DC waveform that although i DC, i unuable to a device attached to it. The problem i that the DC waveform while alway flowing in one direction and poitive, pulate between zero and a little over eight volt. However, we can ue the energy torage propertie of the to upply the current between the hump in the waveform. When the i ued in thi way, it alo known a a filter. Eentially, it filtering out the big bump in the DC waveform, providing a ueful output voltage. In the econd waveform you can ee the output voltage a the green trace. When the input voltage begin it downward path, the upplie current to the load reitor. A it doe, we ee the beginning of the exponential dicharge curve in the output voltage. But before the voltage on the can drop too low, the next hump come and both upplie current to the load reitor and charge the.

9 fullwave bridge rectifier without v() v(,3) fullwave bridge rectifier with filter v() v(,3) (a) Input and Output from Fullwave Rectifier (b) Fullwave Rectifier with Filter Capacitor 9