1. ELE1110C – Tutorial 2 20-9-2006 Luk Chun Pong Outline -Basic concepts of Capacitors -RC circuits (DC) -Examples

2. Capacitor -Store energy in the electric field dielectric conducting plates Q = CV Q: Total Charge C: Capacitance V: Voltage across the capacitor C: actually depends on the permittivity of the dielectric, the area of the plates and the spacing between them.

3. Important Laws Capacitor in parallel C = C 1 + C 2 Capacitor in series Proof? In series, current passes through the capacitors is the same In parallel, voltage across the capacitor is the same

4. Important Laws (3) Rate of change of voltage across the capacitor the current pass through the capacitor

5. Time domain description (case 1) dV/dt = I/C = Constant Current Source (I is constant) dV dt Slope = dV/dt I and C are constants Not common!

6. Time domain description (case 2) Constant voltage source R V0V0 VCVC I Current at time t Voltage across the capacitor at time t Remember this is a DC circuit Initial Condition, t = 0: Vc = 0

7. Solving the differential equation First order differential equation Separable Differential Equations, integrate both sides Don’t forget the Constant k Substitute using the initial conditions

8. General formula for charging and discharging a capacitor Charging Discharging R C V0V0 VcVc

9. Charging and discharging Charging Discharging Vc(t) t

10. RC circuit: Time constant RC = Resistance of the resistor x Capacitance of the capacitor –How long does it take to charge up/discharge the capacitor –The unit of RC is second t = RC, 63% t = 5RC, 99%

11. Question Find out the time needed for 1)Vc = 0.5 V 0 ? 2)Vc = V 0 ? Transient state = Solving differential equations Steady state = Open circuit DC RC circuit

12. Problem sheet 2 Q2 In the following circuit, prove that the energy required to charge the capacitor from 0 to V 0 volt is 0.5CV 0 2. What is the energy dissipated in discharging the capacitor from V to 0 volt? Remember this result - the energy dissipated DOES NOT depend on the resistor R!

13. Charging

14. Discharging

15. Another Example Consider the following RC circuit 4kΩ 30V +-+- +-+- 24V 5kΩ 3kΩ 0.5mF v + - t = 0 t > 0 The circuit has been operated at the above state for a long time. At t = 0, the switch move to X, find v at t = 1s X

16. Solution For t < 0, v = 15V So, v(0) = 15V Using the general formula, V 0 = 30, Vc(0) = 15, t = 1, CR = 2s Vc(1) = 20.9V