Capacitors

1

• Chapter 10
• Capacitors Capacitance

2

• 10.1 Capacitance (p. 386)
• Capacitor
• Consists of 2 conductors separated by an
  insulator.
• Basic form parallel-plate capacitor Fig. 10-1,
  page 386 which consists of two metal plates
  separated by dielectric (e.g. air).
• Charging capcitor
• When DC source is connected to two metal plates
  Fig. 10-2, page 386, electrons are removed from
  ve plate and an equal number of electrons will
  be deposited on -ve plate.
• This leaves top plate positively charges and
  bottom plate negatively charged.
• Capacitor can store charge, i.e. remains charged
  after DC source is removed.

3

• Discharging Capacitor
• By shorting a wire/resistor across the two leads.
• Definition of Capacitance
• Charge stored depends on applied voltage
• Q CV or C Q/V unit Farads, F
• C is defined as the capacitance of the capacitor
• Example 10-1, page 387
• a. How much charge is stored on a 10-?F
  capacitor when it is connected to
• a 24-volt source?
• b. The charge on a 20-nF capacitor is 1.7 ?C.
  What is its voltage?
• Solution
• a. From Equation 10-1, Q CV. Thus, Q (10 x
  10-6 F)(24 V) 240 ?C.
• b. Rearranging Equation 10-1, V Q/C (1.7 x
  10-6 C)/(20 x 10-9 F) 85 V.

4

• 10.2 Factors Affecting Capacitance (p. 387)
• Effect of area
• The more charge put on a capacitors plate for a
  given voltage, the greater will be its
  capacitance.
• Capacitance is directly proportional to plate
  area. Fig. 10-4, page 387
• Effect of Spacing
• As space between plated decreases, the force of
  attraction increases and pulls more electrons
  from one plate to other.
• Capacitance is inversely proportional to plate
  spacing.
• Effect of Dielectric
• Capacitance varies for different materials Table
  10-1, page 388.
• This factor is called relative dielectric onstant
  or relative perrmittivity of a material Fig.
  10-6, page 388

5

• Capacitance of a Parallel Capacitor (p. 389)
• C ? A / d (F)
• A plate area
• d spacing
• ? absolute dielectric constant (F/m)
• For air or vacuum, ? ? o 8.85 x 10-12 F/m
• Other materials, ? expressed as the product of
  the relative dielectric constant,
• ? r times ? o , i.e. ? ? o ? r

6

• Example 10-3, page 389
• A parallel-plate capacitor with air dielectric
  has a value of C 12 pF. What is the
  capacitance of a capacitor that has the
  following
• a. The same separation and dielectrric but
  five times the plate area?
• b. The same dielectric but four times the
  area and one fifth the plate spacing?
• c. A dry paper dielectric, six times the
  plate area, and twice the plate spacing?
• Solution
• a. Since the plate area has increased by a
  factor of five and everything else remains
• the same, C increases by a factor of five.
  Thus, C 5(12 pF) 60 pF.
• b. With four times the plate area, C increases
  by a factor of four. With one fifth the
• plate spacing, C increases by a factor
  of five. Thus, C (4)(5)(12pF) 240 pF.
• c. Dry paper increases C by a factor of 2.2.
  The increases in plate area increases
• C by a factor of six. Doubling the plate
  spacing reduces C by one half. Thus,
• C (2.2)(6)(½)(12 pF) 79.2 pF.

7

• Quiz
• For a given capacitor, it stores Q1 charges
  when voltage V1 applied across its two ends. It
  will store Q2 charges when voltage V2 applied
  across its two ends.
• If V1 10V and V2 5V, Q1 is greater than
  Q2 by 2?C. Find the a capacitance of the
  capacitor.

8

• 10.3 Electric Fields (p. 390)
• Electric flux
• Electric fields are force fields that exist in
  the region surrounding charged bodies. Fig.
  10-8, page 391
• The direction of the field is defined as the
  direction of force on a positive charge. It is
  directed outward from the ve charge and inward
  toward the -ve charge.
• Field lines never cross and density of lines
  indicates the strength of the field.
• Fig. 10-8 (a) Field about a pair of
  positive and negative charges
• (b) Field of parallel plate capacitor

9

• Electric field of parallel-plate capacitor is
  uniform across the gap with some fringing near
  edges.
• Electric flux lines are represented by
• Electric Field Intensity, ?
• Strength of an electric field, i.e. force that
  the field exerts on a small, ve test charge, Qt
• ? F / Qt unit newtons / coulomb (N/C)
• F Q Qt / (4 ? ? r2) for a point charge, Q
• Hence ? Q / (4 ? ? r2)

10

• Electric Flux Density, D (p. 391)
• represent the density fo flux lnes in space
• independent of the medium
• D ? ?
• or D Total flux / area
• ? / A
• The number of flux lines emanating from a charge
  is equal to the charge itself, i.e. ? Q
• Figure 10-9 - In the SI system, total flux ?
  equals charge Q. page 392

11

• Figure 10-10 - Work moving test charge Qt is
  force times distance. page 392
• Work required to move the charge against the
  force (F) through distance (d)
• W F d
• Define voltage, V W / Qt for a test charge Qt
• F d / Qt
• ? V / d as ? F / Qt
• i.e. electric field strength between capacitor
  plates
• voltage across the plates divided by the
  distance between them.

12

• Capacitance, C Q / V
• ? / V
• A D / ?d
• (D / ?) (A / d)
• ? A / d
• EXAMPLE 10-4, page 392 Suppose that the
  electric field intensity between the plates of a
  capacitor is 50 000 V/m when 80 V is applied
• a. What is the plate spacing if the dielectric
  is air? If the dielectric is
• ceramic?
• b. What is ? if the plate spacing is halved?
• Solution
• a. ? V/ d, independent of dielectric. Thus, d
  V/? 80V/50x103V/m
• 1.6x10-3 m
• b. Since ? V/d, will double to 100 000 V/m.

13

• 10.4 Dielectrics (p. 393)
• Voltage breakdown
• If voltage applied across capacitor is increased
  beyond a critical value, force on electronics is
  so great that they are torn from orbit.
• This is called dielectric breakdown.
• Electric field intensity at breakdown is called
  dielectric strength of a material. E.g. air
  3kV / mm
• Table 10-2, page 393
• MATERIAL kV/mm
• Air 3
• Ceramic (high ?r) 3
• Mica 40
• Mylar 16
• Oil 15
• Polystyrene 24
• Rubber 18
• Teflon 60

14

• Voltage rating
• because of dielectric breakdown, capacitors are
  rated for max operating voltage (called working
  voltage).
• EXAMPLE 10-5, page 394 A capacitor with plate
  dimensions of 2.5 cm by 2.5cm and a ceramic
  dielectric with ? r 7500 experiences breakdown
  at 2400 V. What is C?
• Solution From Table 10-2 dielectric strength
  3 kV/mm. Thus, d 2400 V/3000 V/mm 0.8 mm 8
  x 10-4 m. So
• C ?r ?o A/d
• (7500) (8.85 x 10-12) (0.025m)2/(8x10-4m)
  
• 51.9 nF

15

• 10.5 Nonideal Effect (p. 394)
• Leakage current
• charged capacitor will discharge after
  disconnected from source
• small leakage current will pass through
  dielectric when capacitor connected to a source.
• Charge leaks through the dielectric Fig. 10-12,
  page 394
• R hundreds of Mohm
• Figure 10-12 - Leakage current. page 394

16

• Equivalent series resistance
• resistance (RS) may develop in capacitors leads
  as its internal connections begins to fail
• cause problems in high-frequency circuits
• dissipation factor
• D RS / XC where XC 1/2?fC
• Dielectric absorption
• after shorting two leads of a capacitor, a
  residual voltage remains
• Disadv. upset circuit voltage levels
• Temperature coefficient
• ve / zero / -ve temperature coefficient imply
  capacitance increases / no change / decreases
  with increasing temp.
• in parts per million (ppm)

17

• 10.6 Types of Capacitors (p. 395)
• Fixed capacitor Fig. 10-13, 10-14, page 396
• Variable capacitor Fig. 10-19, page 399
• Figure 10-13 Stacked capacitor construction.
  The stack is compressed, leads attached, and
  the unit coated with epoxy resin or other
  insulting material.

18

• Fixed capacitor
• Ceramic
• relative permittivity 30 to 7500
• extremely high permittivity permit small
  packaging but characteristics vary wifely with
  temp. and operating voltage. Use in limited temp
  applications where small size and cost are
  important
• many surface mount capacitors use ceramic
  dielectrics
• Plastic film
• film/foil Fig. 10-14, page 396 use metal foil
  use metal foil separated by plastic film
• metallized-film have foil material vaccum
  deposited directly onto plastic film
• Mica
• low in cost with low leakage and good stability

19

• Electrolytic
• provide large apacitance up to hundred thousand
  microfarads
• relatively low cost
• leakage is relatively high and breakdown voltage
  is relatively low
• Surface mount extremely small and provide high
  packaging density
• Variable capacitor
• used in radio tuning circuits
• stationary plates and set of movable plates which
  are ganged together and mounted on a shaft
• as shaft is rotated, the effective area change

20

• 10.7 Capacitors in parallel and series
• capacitors in parallel fig. 10-20, page 400
• . Voltage is the same across each.

21

• Example 10-6, page 401
• Example 10-6 A 10-?F, a 15-?F, and a 100-?F
  capacitor are connected in parallel across a 50-V
  source. Determine the following
• a. Total capacitance.
• b. Total charge stored.
• c. Charge on each capacitor.
• Solution
• a. Cr C1 C2 c3 10?F 15 ?F 100 ?F
  125 ?F
• b. QT CTV (125 ?F)(50 V) 6.25 mC
• c. Q1 C1V (10 . Q1 C1V (10 ?F)(50 V)
  0.5 mC)(50 V) 0.5 mC
• Q2 C2V (15 ?F)(50 V) 0.75 mC
• Q3 C3V (100 ?F)(50 V) 5.0 mC
• Note Q1 Q2 Q3 (0.5 0.75 5.0) mC 6.25
  mC, which checks with (b).

22

• Capacitors in series Fig. 10-21, page 401
• Charge is the same on each.

23

• EXAMPLE 10-7 (p. 402)
• Refer to Figure 10-22(a)
• a. Determine Cr .
• b. If 50 V is applied across the capacitors,
  determine Q.
• d. Determine the voltage on each capacitor.
• Solution
• a. 1/Cr 1/C1 1/C2 1/C3 1/30µF 1/60µF
  1/20µF
• 0.0333 x 106 0.0167 x 106 106 0.1 x 106
• Therefore,
• Cr 1/(0.1 x 10-6) 10 ?F
• b. Q CTV (10 x10-6F)(50 V) 0.5 mC
• c. V1 Q/C1 (0.5 x 10-3 C) / (30 x 10-6F)
  16.7V
• V2 Q/C2 (0.5x10-3C)/(60x10-6F) 8.3V
• V3 Q/C3 (0.5x10-3C)/(20x10-6F) 25.0V

24

• EXAMPLE 10-8 page 403
• For the circuit of Figure 10-23(a), determine CT
• Refer to Figure 10-23 Systematic reduction.

25

• Quiz Consider capacitors of 1?F, 1.5?F, and
  10?F. If equivalent capacitance is equal to
  10.6?F. How are the capacitors connected?
• Solution
• To yield 10.6 ?F, the 10 ?F capacitor must be
  connected in parallel with some combination of
  the 1 ?F and 1.5?F capacitors. Only the
  connection shown below works.
• 10.8 Capacitor current and voltage page 404
• (Refer to Fig. 10-25, page 405)
• During charging
• movement of electrons constitutes current
• current lasts for capacitor to be charged
• no current pass through dielectric
• capacitor voltage builds as charge deposited on
  plates
• as capacitor voltage increases, charging current
  decreases

26

• Capacitor V-1 relationship, page 405
• q C VC
• ic dq / dt d (C VC) / dt
• ic C d Vc / dt
• Current through a capacitor is equal to C times
  the rate of change of voltage across it.
• Example 10-9 page 406 A signal generator
  applies voltage to a 5-µF capacitor with a
  wavefrom as in Figure 10-27(a). The voltage
  rises linearly from 0 to 10V in 1ms, falls
  linearly to -10V at t3ms, remains constant until
  t4ms, rises to 10V at t5ms, and remains
  constant thereafter.
• a. Determine the slope of vc.
• b. Determine the current and sketch its graph.

27

• Solution
• a. We need the slope of vc during each time
  interval where slope rise/run ? v/?t.
• 0 ms to 1 ms ?v 10 V ?t 1 ms Therefore,
  slope 10 V/1 ms 10 000 V/s.
• 1 ms to 3 ms Slope -20 V/2 ms -10 000 V/s.
• 3 ms to 4 ms Slope 0 V/s.
• 4 ms to 5 ms Slope 20 V/1 ms 20 000 V/s.
• b. ic Cdvc/dt C times slope. Thus,
• 0 ms to 1 ms i (5 x 10-6F) (10 000 V/s) 50
  mA.
• 1 ms to 3 ms i -(5 x 10-6F) (10 000 V/s)
  -50 mA.
• 3 ms to 4 ms i (5 x 10-6F) (0 V/s) 0 mA.
• 4 ms to 5 ms i (5 x 10-6F) (20 000 V/s) 100
  mA.
• Refer to Figure 10-27, page 406

28

• 10.9 Energy stored by a capacitor page 407
• an ideal capacitor does not dissipate power
• when power is transferred to a capacitor, all of
  it is stored as energy in the capacitors
  electric field
• Stored energy, W 1/2 C V2
• 10.10 Capacitor failures and troubleshooting
  page 408
• Failures
• short internally
• leads open
• dielectric leaky
• (Noted if electrolytic capacitor is connected
  with its polarity reversed, it may explode.)

29

• Capacitors fail because of
• misapplication
• excessive voltage, current, or temperature
• aging
• Basic testing with an ohmmeter
• out-of-circuit tests with analog ohmmeter
• - detect opens and shorts
• - leaky dielectrics
• (Noted discharge capacitor first before
  measurement)
• for normal capacitor, the ohmmeter reading should
  be low initially and gradually increase to
  infinity.

30

• Capacitor testers page 408
• some digital multimeter can measure capacitance
• LCR (inductance, capacitance, resistance)
  analyzer can determine capacitance as well as
  detect opens and short