Electrical Theory I (ENG3322)

1
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.

2
Charging of a capacitor/2
3
Charging of a capacitor/3
4
Charging of a capacitor/4

• Insert initial (boundary) conditions, if the
  capacitor is initially uncharged, then at t 0,
  q 0
• C2 EC, substitute into equation 1

5
Charging Curve
6
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.

7
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.

8
Charging of Capacitor-Example one
9
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.

10
Charging of Capacitor-Solution to Example one

• 1) Using the charge formula,
• 2)

11
Charging of Capacitor-Solution to Example one/2

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

12
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 have

13
Charging of a initially charged capacitor
14
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,

15
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

16
Alternative solution to previous problem/2

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

17
Alternative solution to previous problem/3

• Final voltage of the capacitor

18
Discharge of a capacitor

19
Discharge of a capacitor/2

• 0 VR VC

20
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 currents
  direction)

21
Discharge curve
22
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.

23
Discharge of capacitor-solution to example

• Using the discharge formula,