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Brief Discussion of inductors and capacitors
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Inductors and Capacitors
Passive Elements
Capacitors
Capacitor
Is consists of two conducting plates separated by an insulator (dielectric).
CAPACITANCE – It is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates and measured in Farads
1 Farad = 1 Coulomb/Volt
d
AC
Where: - permittivity of the dielectric material A - surface area of each plates d - distance between the plate
Where: q- charge stored C- Capacitance v- applied voltage
A capacitor is a passive element designed to store energy in its electric field. (Electro static energy)
q = Cv
Voltage and Current Relation
CAPACITORS oppose changes in voltage by drawing or supplying current as they charge or discharge to the new voltage level.
The flow of electrons “through” a capacitor is directly proportional to the rate of change of voltage across the capacitor.
)( 0tVC
dtiV
c
C
dt
dVCi C
C
Energy
The energy stored in the capacitor is:
Where: C - Capacitance
v - Voltage
2
2
1CvW
Important Properties of Capacitor
The capacitor is an open circuit to DC.
Important Properties of Capacitor
The voltage on capacitor cannot change abruptly.
(a) is ALLOWED (b) NOT ALLOWED ( an abrupt change is not possible)
Waveform of the Voltage across the capacitor:
Capacitors in Series
The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocal of the individual capacitance.
Capacitors in Parallel
The equivalent capacitance of N-parallel connected capacitors is the sum of the individual capacitance.
Problems:
1.
Problems:
2.
Problems:
3.
4.
Problems: 5.
6.
Change in voltage as shown in the figure :
0V – 50V between 0 sec. to 1 sec ( 0 < t < 1 )
50V – (-50V) between 1 sec. to 3 sec ( 1 < t < 3 )
(-50V) – 0V between 3 sec. to 4 sec ( 3 < t < 4 )
Note: The line equation is Y = mX + b; Therefore: V(t) = mt + b
7.
Recall that:
(a) is ALLOWED (b) NOT ALLOWED ( an abrupt change is not possible)
Change of voltage as shown in the figure :
0V – 50V between 0 sec. to 1 sec ( 0 < t < 1 )
50V – (-50V) between 1 sec. to 3 sec ( 1 < t < 3 )
(-50V) – 0V between 3 sec. to 4 sec ( 3 < t < 4 )
0 < t < 1
V(t) = mt + b
When t = 0, V(t) = 0V
0 = m*0 + b
b = 0
t = 1, V(t) = 50V
50 = m*1 + 0
m = 50
Voltage equation
w/ respect to time:
V(t) = 50t
1 < t < 3
t = 1;
50 = m*1 + b
t = 3;
-50 = m*3 + b
m = -50
b = 100
V(t) = -50t + 100
Change of voltage as shown in the figure :
0V – 50V between 0 sec. to 1 sec ( 0 < t < 1 )
50V – (-50V) between 1 sec. to 3 sec ( 1 < t < 3 )
(-50V) – 0V between 3 sec. to 4 sec ( 3 < t < 4 )
3 < t < 4
t = 3;
-50 = m*3 + b
t = 4;
0 = m*4 + b
m = 50
b = -200
V(t) = 50t - 200
0 < t < 1
i = 200 (1x10-6) d 50t
dt i = 10 mA
Change of voltage as shown in the figure :
0V – 50V bet. 0 sec. to 1 sec ( 0 < t < 1 )
50V – (-50V) bet. 1 sec. to 3 sec ( 1 < t < 3 )
(-50V) – 0V bet. 3 sec. to 4 sec ( 3 < t < 4 )
Inductors
Inductor
It is a passive element designed to store energy in
its magnetic field. (Electro magnetic energy)
- It is consists of coil of conducting wires.
INDUCTANCE is the property whereby an inductor exhibits opposition to the change of current flowing through it and it is measured in Henrys (H).
l
ANL
2
where: N – number of turns - permeability of the core A – cross-sectional area l - length
INDUCTOR oppose changes in current through them, by dropping a voltage directly proportional to the rate of change of current.
dt
diLVL
)(1
0tidtVL
i tLL
Voltage and Current Relationship
Energy
The energy stored in the inductor is:
Where: L - Inductance
i - current
2
2
1LiW
Important Properties of Inductor
The inductor is a short circuit to dc.
Important Properties of Inductor
The current through an inductor cannot change instantaneously.
Current though an inductor:
a) Allowed b) Not allowable (an abrupt change is not possible)
Inductors in Series
The equivalent inductance of series-connected inductors is the sum of the of the individual inductance.
Inductors in Parallel
The equivalent capacitance of N-parallel connected inductors is the reciprocal of the sum of the reciprocal individual inductance.
Problems:
1.
Problems:
2.
3.
Problems:
4.