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 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.

Chapter 2.3 Capacitor Charging & Discharging Page 2 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a capacitor…/2

Chapter 2.3 Capacitor Charging & Discharging Page 3 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a capacitor…/3

Chapter 2.3 Capacitor Charging & Discharging Page 4 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a capacitor…/4 Insert initial (boundary) conditions, if the capacitor is initially uncharged, then at t = 0, q = 0 C 2 = EC, substitute into equation 1

Chapter 2.3 Capacitor Charging & Discharging Page 5 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging Curve

Chapter 2.3 Capacitor Charging & Discharging Page 6 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board 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 of the fully charged value.

Chapter 2.3 Capacitor Charging & Discharging Page 7 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board 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, V c and I will be over 99% to their final values.

Chapter 2.3 Capacitor Charging & Discharging Page 8 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of Capacitor-Example one

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

Chapter 2.3 Capacitor Charging & Discharging Page 10 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of Capacitor-Solution to Example one 1)Using the charge formula, 2)

Chapter 2.3 Capacitor Charging & Discharging Page 11 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of Capacitor-Solution to Example one…/2 3)For the capacitor to be charged to 80 Volts, using the same formula,

Chapter 2.3 Capacitor Charging & Discharging Page 12 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a initially charged capacitor If at the start of charging, the capacitor is charged to a voltage of E 1 Volts, Then at t = 0, q = E 1, substituting this initial condition into equation 1, we have

Chapter 2.3 Capacitor Charging & Discharging Page 13 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Charging of a initially charged capacitor

Chapter 2.3 Capacitor Charging & Discharging Page 14 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Example to charging of a capacitor with residual charge and initial voltage A 3,000 μ F 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,

Chapter 2.3 Capacitor Charging & Discharging Page 15 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board 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 3,000 μ F capacitor to 50 volts can be found by applying the formula

Chapter 2.3 Capacitor Charging & Discharging Page 16 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Alternative solution to previous problem…/2 Total time equivalent the capacitor is to be charged from zero volt = = Seconds

Chapter 2.3 Capacitor Charging & Discharging Page 17 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Alternative solution to previous problem…/3 Final voltage of the capacitor =

Chapter 2.3 Capacitor Charging & Discharging Page 18 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Discharge of a capacitor

Chapter 2.3 Capacitor Charging & Discharging Page 19 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Discharge of a capacitor…/2 0 = V R +V C

Chapter 2.3 Capacitor Charging & Discharging Page 20 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Discharge of a capacitor…/3 Substitute boundary condition, at t = 0, Voltage across C = E, q = EC C 2 = EC (Note that the negative sign indicate that the current is opposite to the charging current’s direction)

Chapter 2.3 Capacitor Charging & Discharging Page 21 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Discharge curve

Chapter 2.3 Capacitor Charging & Discharging Page 22 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Discharge of capacitor example A 1000 μ F 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.

Chapter 2.3 Capacitor Charging & Discharging Page 23 of 23 Last Updated: 1/9/2005 Electrical Theory I (ENG3322) Engineering Course Board Discharge of capacitor-solution to example Using the discharge formula,