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First Order Circuit
• Capacitors and inductors
• RC and RL circuits
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
Capacitors
Upper and lower plates made
of conducting material
R
These plates are
separated by a distance d
d
The basic structure of a
capacitor
Capacitors
When the switch is closed,
electrons start to accumulate on
the bottom plate whereas the
upper plate looses electrons
R
e
e
e
e
++ +
+ +
+
+ ++
+
+
+
+
  
   


 

e
e
e
e
e
Current (or electrons) ceased to
conduct when the voltage
across the plate equals source
voltage
Capacitors
The amount of charge (q) deposited on the plate is
proportional to the voltage between the plate (v)
q  Cv
++++++++++++++++
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:
A
C
d
where A the surface area of each plate, d is the
separation between plate and  is the permittivity
of the dielectric
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:
C
dielectric
Air
Teflon
A Rubber
Mica
d Ceramic
r
1.0006
2.0
3.0
5.0
7500
Capacitors
In circuit theory, we are more interested in the voltage-current relation of a
capacitor
From
q  Cv , take the time derivative on both sides :
i
iC
dv
dt
+
v

The voltage in terms of current:

1 t
v
i dt
C 
or

1 t
v
i dt  v( t o )
C to
Capacitors
Energy in capacitor is given by:
1 2
w  Cv
2
•
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
Capacitors
Types of capacitors
Capacitors
Construction
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.
Capacitors
Example
Under DC conditions, find the energy stored in each capacitor
Capacitors
Series connection
C1
C2
C3
C4
C6
C5
j=1 ,2, ..6
Cs
Capacitors
Parallel connection
C1
C2
, j=1 ,2, ..6
C3
C4
Cp
Capacitors
Parallel and series connection
Find the equivalent capacitance, Ceq
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:
d
e N
dt
d
dt
is the instantaneous change in flux
Inductors
Lenz’s law
When switch is closed,
current starts to flow
-
+
i
+

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:
e N
d
dt
According to Lenz’s law:
an induced effect is always such as to oppose the cause that produced it
Inductors
Inductance
Inductance is a measure of the change in flux linking a coil due to a change in
current through a coil :
d
L N
di
 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:
N2A
L
l
Inductors
Inductance
Example for a toroid core
N = number of turns of coil
 = permeability of the core
l
A
N2A
L
l
Inductors
Inductance
N2A
L
 henrys (H)
l
•
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
Inductors
Inductance
In circuit theory we normally interested in the voltage-current relation of a
inductor.
From
v N
d
dt
, this can be written as: v  N
i
 v L
d di
.
di dt
+
v

di
dt
The current in terms of voltage:
i
1 t
v dt
L  
or
i
1 t
v dt  i( t o )
L t
o
Inductors
Energy in an inductor is given by:
1 2
w  Li
2
•
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
Inductors
Inductors
Series connection
•
The equivalent inductance of series-connected inductors is the sum of the
individual inductances.
v  v1  v 2  v 3  ....  v n
v  L1
di
di
di
di
 L2
 L3
 ....  L n
dt
dt
dt
dt
v  L1  L 2  L 3  ....  L n 
v  L eq
di
dt
di
dt
L eq  L1  L 2  ...  L N
Inductors
Parallel connection
•
The equivalent capacitance of parallel inductors is the reciprocal of the sum
of the reciprocals of the individual inductances.
i  i1  i2  i3  ....  in
i
1 t
1
vdt 

t
L1 o
L2
t
1
1
t
t
t vdt  L t vdt  ....  L t vdt  i1(t o )  i2 (t o )  ...  in (t o )
o
3
o
n
o
1
1
1
1 t
i   

 ....   t vdt  i1( t o )  i2 ( t o )  ...  in ( t o )
Ln 
 L1 L 2 L 3
o
i
1
L eq
t
t vdt  io,eq (t o )
o
1
1
1
1


 ... 
L eq L1 L 2
LN
Examples
Under steady state find i, iL, vc
and energy stored in C and L
Examples
Determine vc, iL, and the energy stored in the capacitor and inductor in the circuit
of circuit shown below under dc conditions.