First Order Circuit

1. First Order Circuit • Capacitors and inductors • RC and RL circuits

2. What is First Order Circuit? :: A circuit which contain at least a resistor (R) and a capacitor (C)  simply called RC circuit :: A circuit which contain at least a resistor (R) and an inductor (L)  simply called RL circuit :: Analysis on these RC or RL circuits gives the first order differential equation :: RC and RL circuits can be found in many applications: electronics, commmunications, control system, power electronics, power system, etc. We will study on capacitors and inductors first before we do the analysis on the first order circuit

3. R Capacitors Upper and lower plates made of conducting material These plates are separated by a distance d d The basic structure of a capacitor

4. e e e e + + + + + + + + + + + + +                   e e e e e Capacitors When the switch is closed, electrons start to accumulate on the bottom plate whereas the upper plate looses electrons R Current (or electrons) ceased to conduct when the voltage across the plate equals source voltage

5. + + + + + + + + + + + + + + + +                  Capacitors The amount of charge (q) deposited on the plate is proportional to the voltage between the plate (v) C – is known as the capacitance of the capacitor, measured in farads (F) The capacitance of a capacitor depends on the physical dimension of the capacitor and the material (dielectric) between the plate: where A the surface area of each plate, d is the separation between plate and  is the permittivity of the dielectric

6. r dielectric Air 1.0006 Teflon 2.0 Rubber 3.0 Mica 5.0 Ceramic 7500 Capacitors • With the same A and d, dielectric with high permittivity will have higher capacitance • For vacuum permittivity is 8.85 x 10-12 F/m • The ratio of any dielectric permittivity to the vacuum permittivity is called the relative permittivity , r Some example:

7. From , take the time derivative on both sides : i or Capacitors In circuit theory, we are more interested in the voltage-current relation of a capacitor +  v The voltage in terms of current:

8. Capacitors Energy in capacitor is given by: • Energy is stored in the electric field between the plates • The stored energy can be retrieved • Ideally capacitor DOES NOT dissipate energy Important properties of a capacitor (for SEE1003): 1. It behaves as an open circuit to DC quantities 2. The voltage cannot change abruptly – it has to be continuous

9. Capacitors Types of capacitors

10. Capacitors Construction

11. Capacitors Example An initially uncharged 1-mF capacitor has the current shown below across it. Calculate the voltage across it at t = 2 ms and t = 5 ms.

12. Capacitors Example Under DC conditions, find the energy stored in each capacitor

13. C1 C2 C3 C4 Cs C6 C5 j=1 ,2, ..6 Capacitors Series connection

14. Cp C2 C3 C4 C1 , j=1 ,2, ..6 Capacitors Parallel connection

15. Capacitors Parallel and series connection Find the equivalent capacitance, Ceq

16. is the instantaneous change in flux Inductors Faraday’s law When a coil of N turns is placed in region of changing flux, an emf (voltage) will be induced across a coil determined by Faraday’s law:

17. - + i +  Inductors Lenz’s law When switch is closed, current starts to flow flux starts to build and link the coil flux linking the coil will change with time According to Faraday’s law, emf will be induced: According to Lenz’s law: an induced effect is always such as to oppose the cause that produced it

18. Inductors Inductance Inductance is a measure of the change in flux linking a coil due to a change in current through a coil :  For the same change in current and N, a coil with a larger change in flux has a larger inductance The inductance of a coil depends on the construction of the coil and the magnetic properties of the core:

19. l A Inductors Inductance Example for a toroid core N = number of turns of coil = permeability of the core

20. Inductors Inductance • A core made of ferromagnetic material with high permeability is normally used to increase the inductance of a coil •  is typically written as  = ro where o is the permeability of the free space and r is the relative permeability of a material. o = 4 x 10-7 Wb A-1m-1

21. From , this can be written as: i or Inductors Inductance In circuit theory we normally interested in the voltage-current relation of a inductor. + v  The current in terms of voltage:

22. Inductors Energy in an inductor is given by: • Energy is stored in the magnetic field produced by the coil • The stored energy can be retrieved • Ideally inductor DOES NOT dissipate energy Important properties of an inductor (for SEE1003): 1. It behaves as short circuit to DC quantities 2. The currents cannot change abruptly – it has to be continuous

23. Inductors

24. Inductors Series connection • The equivalent inductance of series-connected inductors is the sum of the individual inductances.

25. Inductors Parallel connection • The equivalent capacitance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

26. Examples Under steady state find i, iL, vc and energy stored in C and L

27. Examples Determine vc,iL, and the energy stored in the capacitor and inductor in the circuit of circuit shown below under dc conditions.