Chapter 2.3 Capacitor Charging & Discharging Page 1 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a capacitor

Chapter 2.3 Capacitor Charging & Discharging Page 1 of 23Last Updated: 1/9/2005

Electrical Theory I (ENG3322)Engineering Course Board

Charging of a capacitor

• One of the functions of a capacitor is its ability to store up charge when a potential difference is applied across the positive and negative plates.

• Energy is stored in the electric field. When a voltage is applied across a capacitor, current rushes into the plates of the capacitor, developing a potential difference across the capacitor, therefore the potential difference between the battery and the capacitor become smaller and the flow rate of electrons become smaller.

• The charging process continues until the capacitor becomes fully charged. The charging current follows an exponential curve.



Charging of a capacitor…/2



Charging of a capacitor…/3

C

qV

V

qCVVE C

CCR ,,

qdt

dqRCEC

C

qiRE ,

dt

dqRCqEC

qEC

qECd

RC

dt

qEC

dq

RC

dt,

tConsnIntegratioCqECRC

tC tan,ln 11

RCtCeqEC /1 1................../

2 equationeCqEC RCt



Charging of a capacitor…/4Insert initial (boundary) conditions, if the capacitor is initially uncharged, then at t = 0, q = 0

C2 = EC, substitute into equation 1 RCtECeqEC /

RCteECq /1

1 /RCtC eE

C

qV

RCtedt

dEC

dt

dqi /1

10 /

RCeECi RCt

RCtC eEV /1

RCteR

Ei /



Charging Curve

RCtC eE

C

qV /1



Charging Curve

• From the charging equation, we noticed that the rate of charging is determined by the exponential curve that is in term determined by the RC constant.

• The term RC is termed the time constant since it affects the rate of charge. Mathematically, this is the time taken for the capacitor to reach 0.632 of the fully charged value.

RCtC eE

C

qV /1



Duration of Transient

• Theoretically, capacitors will never be fully charged according to the charging equation.

• Thus, for all practical purposes, transients can be considered to last for only five times of the time constant. I.e. the capacitor is said fully charged after

5× RC.

• After 5 time constant, q, Vc and I will be over 99% to their final values.



Charging of Capacitor-Example one



Charging of Capacitor-Example one

• An uncharged capacitor of 2000 micro-Farad is connected to a 100 volt D.C. supply in series with a current limiting resistor of 5000 Ohms, calculate

• i) The voltage of the capacitor at the end of 8 seconds charging

• ii) The charging current at the end of 8 seconds• iii) The time taken for the capacitor to be

charged to 80 volts.



Charging of Capacitor-Solution to Example one

1) Using the charge formula,

2)

RCtC eE

C

qV /1

61020005000/81100 eVC

VoltsVC 07.55RCte

R

Ei /

61020005000/8

5000

100 ei

Amperemillii 9866.8



Charging of Capacitor-Solution to Example one…/2

3) For the capacitor to be charged to 80 Volts, using the same formula, RCt

C eEC

qV /1

61020005000/110080 te

10/18.0 te2.010/ te

6094.12.0ln10/ tSecondst 094.16



Charging of a initially charged capacitor

• If at the start of charging, the capacitor is charged to a voltage of E1 Volts,

• Then at t = 0, q = E1, substituting this initial condition

into equation 1, we haveRCteCCEEC /

21

12 EECC RCteEECqEC /

1

RCtRCt CeEeECq /1

/ )1( RCtRCt

C eEEeEC

qV /

1/

RCtRCtC eEeEV /

1/ )1(



Charging of a initially charged capacitor

RCtRCt edt

dCEe

dt

dEC

dt

dqi /

1/1

/1/ RCtRCt eR

Ee

R

Ei

RCteR

EEi /1



Example to charging of a capacitor with residual charge and initial voltage

A 3,000μF capacitor has an initial voltage of 50 Volts is further charged by a 200 volts D.C. supply in series with a 2 k-Ohm resistor. Calculate the voltage across the terminals of the capacitor after 10 seconds.

Solution: Using the formula for capacitor with an initial voltage, RCt

C eEEEV /1

61030002000/1020050200 eVC

1889.0150200 CV

VoltsVC 67.171



Alternative solution to previous problem

• Alternatively, if you do not wish to memorize the formula for charging a capacitor with an initial charge, you can first find the time required to charge an uncharged capacitor to the initial voltage, then add the extra time for charging to find the final voltage.

Solution:

Time required to charge an uncharged 3,000μF capacitor to 50 volts can be found by applying the formula RCt

C eEV /1



Alternative solution to previous problem…/2

Total time equivalent the capacitor is to be charged from zero volt = 1.7261+ 10 = 11.7261 Seconds

61030002000/120050 te

6/1200

50 te

6/75.0 te

6/2877.0 tSecondst 7261.1



Alternative solution to previous problem…/3

Final voltage of the capacitor =

61030002000/7261.111200 eVC

9544.11200 eVC

8583.0200CV

VoltsVC 67.171



Discharge of a capacitor



Discharge of a capacitor…/20 = VR +VC

C

qRi 0

RiC

q

q

dq

RC

dt

qCRC

tln1 tConsC tan1

1/ CRCteq

RCteCq /2

tConsC tan2



Discharge of a capacitor…/3

Substitute boundary condition, at t = 0,

Voltage across C = E, q = EC

C2 = EC

(Note that the negative sign indicate that the current is opposite to the charging current’s direction)

RCtECeq /

RCtC Ee

C

qV /

RCteRC

ECdt

dqi /1 RCte

R

Ei /



Discharge curve

RCteR

Ei /



Discharge of capacitor example

A 1000μF capacitor previously charged to 80 Volts is to be discharged through a resistance of 20 k-Ohms. Find the voltage across the terminals of the capacitor at the end of 15 seconds.



Discharge of capacitor-solution to example

Using the discharge formula,

RCtC Ee

C

qV /

610100020000/1580 eVC

VoltsVC 79.37

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Chapter 2.3 Capacitor Charging & Discharging Page 1 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a capacitor