Chapter 26B Chapter 26B -- Capacitor Capacitor CircuitsCircuits

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

Objectives: Objectives: After completing this After completing this module, you should be able to:module, you should be able to:

• Calculate the equivalent capacitance of a number of capacitors connected in series or in parallel.

• Determine the charge and voltage across any chosen capacitor in a network when given capacitances and the externally applied potential difference.

Electrical Circuit SymbolsElectrical Circuit Symbols

Electrical circuitsElectrical circuits often contain two or more often contain two or more capacitors grouped together and attached capacitors grouped together and attached to an energy source, such as a battery.to an energy source, such as a battery.

The following symbols are often used:The following symbols are often used:

+

Capacitor

+--+ - + -

- + - + -

Ground Battery-+

Series CircuitsSeries Circuits

Capacitors or other devices connected Capacitors or other devices connected along a single path are said to be along a single path are said to be connected in connected in seriesseries. See circuit below:. See circuit below:

Series connection of capacitors.

“+ to – to + …”

Charge inside dots is induced.

Battery

C1 C2C3

++

--

++

++

--

--

Charge on Capacitors in SeriesCharge on Capacitors in Series

Since inside charge is only Since inside charge is only inducedinduced, the , the chargecharge on each capacitor is the on each capacitor is the samesame..

Charge is same: series connection

of capacitors.

Q = Q1 = Q2 =Q3Battery

C1 C2C3

++

--

++

++

--

--

Q1 Q2 Q3

Voltage on Capacitors in SeriesVoltage on Capacitors in SeriesSince the Since the potential differencepotential difference between between points points AA and and BB is independent of path, the is independent of path, the battery voltage battery voltage V V must equal the sum of must equal the sum of the voltages across each capacitor.the voltages across each capacitor.

Total voltage V Series connection Sum of voltages

V = V1 + V2 + V3Battery

C1 C2C3

++

--

++

++

--

--

V1 V2 V3

• •A B

Equivalent Capacitance: SeriesEquivalent Capacitance: Series

V = V1 + V2 + V3

Q1= Q2 = Q3

++

--

++

++

--

--

C1 C2 C3

V1 V2 V3 ; Q QC VV C

31 2

1 2 3

QQ Q QC C C C

1 2 3

1 1 1 1

eC C C C

Equivalent Equivalent CCee for capacitors for capacitors in series:in series: 1

1 1n

ie iC C

Example 1.Example 1. Find the equivalent capacitance Find the equivalent capacitance of the three capacitors connected in series of the three capacitors connected in series with a 24with a 24--V battery.V battery.

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F1

1 1n

ie iC C

CCee for for series:series:

1 1 1 12 4 6eC F F F

1 0.500 0.250 0.167eC

1 10.917 or 0.917e

e

CC

Ce = 1.09 FCe = 1.09 F

Example 1 (Cont.):Example 1 (Cont.): The equivalent circuit can The equivalent circuit can be shown as follows with single be shown as follows with single CCee..

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F 1.09 F

Ce

24 V

1

1 1n

ie iC C

Ce = 1.09 FCe = 1.09 F

Note that the equivalent capacitance Note that the equivalent capacitance CCee for capacitors in for capacitors in seriesseries is always is always less than less than the leastthe least in the circuit. (1.09 in the circuit. (1.09 F < 2 < 2 F)

1.09 F

Ce

24 V

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F QCV

Q CV

Ce = 1.09 FCe = 1.09 F

QQTT = = CCee VV = = (1.09 (1.09 F)(24 V);F)(24 V); QT = 26.2 CQT = 26.2 C

For series circuits: For series circuits: QQTT = Q= Q11 = Q= Q22 = Q= Q33

Q1 = Q2 = Q3 = 26.2 CQ1 = Q2 = Q3 = 26.2 C

Example 1 (Cont.):Example 1 (Cont.): What is the total charge What is the total charge and the charge on each capacitor?and the charge on each capacitor?

++

--

++

++

--

--

2 F

C1 C2 C3

24 V

4 F 6 F

; Q QC VV C

VT = 24 VVT = 24 V

11

1

26.2 1C 3.1 V2 F

QVC

22

2

26.2 6C .55 V4 F

QVC

33

3

26.2 4C .37 V6 F

QVC

Note: VT = 13.1 V + 6.55 V + 4.37 V = 24.0 VNote: VT = 13.1 V + 6.55 V + 4.37 V = 24.0 V

Example 1 (Cont.):Example 1 (Cont.): What is the voltage across What is the voltage across each capacitor?each capacitor?

Short Cut: Two Series CapacitorsShort Cut: Two Series CapacitorsThe equivalent capacitance The equivalent capacitance CCee for for twotwo series series capacitors is the capacitors is the product divided by the sumproduct divided by the sum..

1 2

1 1 1 ;eC C C 1 2

1 2e

C CCC C

3 F 6 F

++

--

++

--

C1 C2

Example:Example: (3 F)(6 F)3 F 6 FeC

Ce = 2 FCe = 2 F

Parallel CircuitsParallel CircuitsCapacitors which are all connected to the Capacitors which are all connected to the same source of potential are said to be same source of potential are said to be connected in connected in parallelparallel. See below:. See below:

Parallel capacitors: “+ to +; - to -”

C2 C3C1 ++

--

++

--++

--Charges:

QT = Q1 + Q2 + Q3

Voltages: VT = V1 = V2 = V3

Equivalent Capacitance: ParallelEquivalent Capacitance: Parallel

Q = Q1 + Q2 + Q3

; QC Q CVV

Equivalent Equivalent CCee for capacitors for capacitors in parallel:in parallel: 1

n

e ii

C C

Equal Voltages: Equal Voltages: CVCV = C= C11 VV11 + C+ C22 VV22 + C+ C33 VV33

Parallel capacitors in Parallel:

C2 C3C1 ++

--

++

--

++

--

CCee = C= C11 + C+ C22 + C+ C33

Example 2.Example 2. Find the Find the equivalent capacitanceequivalent capacitance of the three capacitors connected in of the three capacitors connected in parallelparallel with a 24with a 24--V battery.V battery.

CCee for for parallel:parallel:

Ce = 12 FCe = 12 F

C2 C3C1

2 F 4 F 6 F

24 V

Q = Q1 + Q2 + Q3

VT = V1 = V2 = V3

1

n

e ii

C C

CCee = (2 + 4 + 6) = (2 + 4 + 6) FF

Note that the equivalent capacitance Note that the equivalent capacitance CCee for for capacitors in capacitors in parallelparallel is always is always greater than greater than the largestthe largest in the circuit. (12 in the circuit. (12 F > 6 > 6 F)

Example 2 (Cont.)Example 2 (Cont.) Find the Find the totaltotal charge Qcharge QTT and and chargecharge across each capacitor.across each capacitor.

Ce = 12 FCe = 12 F

C2 C3C1

2 F 4 F 6 F

24 V

Q = Q1 + Q2 + Q3

V1 = V2 = V3 = 24 V

; QC Q CVV

QQ11 = = (2 (2 F)(24 V) = F)(24 V) = 48 48 CC

QQ11 = = (4 (4 F)(24 V) = F)(24 V) = 96 96 CC

QQ11 = = (6 (6 F)(24 V) = F)(24 V) = 144 144 CC

QQTT = = CCee VV

QQTT = (12 = (12 F)(24 V) F)(24 V)

QT = 288 CQT = 288 C

Example 3. Example 3. Find the equivalent capacitance Find the equivalent capacitance of the circuit drawn below.of the circuit drawn below.

C1

4 F

3 F

6 F

24 V C2

C3

C1

4 F2 F

24 VC3,6 Ce 6 F

24 V

3,6(3 F)(6 F) 2 F3 F 6 F

C

CCee = 4 = 4 F + 2 F + 2 FF

Ce = 6 FCe = 6 F

Example 3 (Cont.)Example 3 (Cont.) Find the total charge Find the total charge QQTT . .

C1

4 F

3 F

6 F

24 V C2

C3

Ce = 6 FCe = 6 F

Q = CVQ = CV = (6 = (6 F)(24 V)F)(24 V)

QT = 144 CQT = 144 C

C1

4 F2 F

24 VC3,6 Ce 6 F

24 V

Example 3 (Cont.)Example 3 (Cont.) Find the charge Find the charge QQ44 and and voltage voltage VV44 across the the 4across the the 4F capacitorF capacitor

C1

4 F

3 F

6 F

24 V C2

C3

V4 = VT = 24 VV4 = VT = 24 V

QQ44 = = (4 (4 F)(24 V)F)(24 V)

Q4 = 96 CQ4 = 96 C

The remainder of the charge: (144 The remainder of the charge: (144 CC –– 96 96 CC) ) is on is on EACH EACH of the other capacitors. (Series)of the other capacitors. (Series)

Q3 = Q6 = 48 CQ3 = Q6 = 48 C This can also be found from Q = C3,6 V3,6 = (2 F)(24 V) This can also be found from Q = C3,6 V3,6 = (2 F)(24 V)

Example 3 (Cont.)Example 3 (Cont.) Find the Find the voltagesvoltages across across the the 33 and and 66--FF capacitorscapacitors

C1

4 F

3 F

6 F

24 V C2

C3

Note: V3 + V6 = 16.0 V + 8.00 V = 24 VNote: V3 + V6 = 16.0 V + 8.00 V = 24 V

Q3 = Q6 = 48 CQ3 = Q6 = 48 C

348 C 16.3

VF

0V

648 C 8.06

VF

0V

Use these techniques to find voltage and capacitance across each capacitor in a circuit.

Use these techniques to find voltage and capacitance across each capacitor in a circuit.

Summary: Series CircuitsSummary: Series Circuits

1

1 1n

ie iC C

Q = Q1 = Q2 = Q3Q = Q1 = Q2 = Q3

V = V1 + V2 + V3V = V1 + V2 + V3

1 2

1 2e

C CCC C

For two capacitors at a time:For two capacitors at a time:

Summary: Parallel CircuitsSummary: Parallel Circuits

Q = Q1 + Q2 + Q3Q = Q1 + Q2 + Q3

V = V1 = V2 =V3V = V1 = V2 =V3

1

n

e ii

C C

For complex circuits, reduce the circuit in steps using the rules for both series and parallel connections until you are able to solve problem.

For complex circuits, reduce the circuit in steps using the rules for both series and parallel connections until you are able to solve problem.

CONCLUSION: Chapter 26BCONCLUSION: Chapter 26B Capacitor CircuitsCapacitor Circuits