Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Electrical Engineering
Principles & Applications

Chapter 3- Inductance and
Capacitance

Slide 1

Inductance and Capacitance
1
...
Parallel and series connections of capacitors or
inductors
3
...

Plot i(t) and q(t)

q = Cv

Slide 8

dv
i = C
dt

Determining Voltage for a Capacitance Given
Current
Example:

The current through a 0
...
5 sin(104t)
...
5 ×10−4[1 − cos104 t ]
Slide 9

v (t ) = q (t ) / C = 500[1 − cos 10 4 t ]

Capacitor as Energy Storage Device
+

iC

Instantaneous power

pC (t) = vC (t)iC (t)

vC
−

dv c
(t )
dt
dv
pC (t ) = CvC (t ) c
dt

iC (t ) = C

W

Energy is the integral of power
t2

wC (t 2 , t1 ) = ∫ pC ( x )dx

1 t
q (t )
vC (t ) = ∫ iC ( x ) dx = C
C −∞
C
pC (t ) =

t1

d 1 2 
pC ( t ) = C  vC (t ) 
dt  2


pC ( t ) =

1
dq
qC (t ) C (t )
C
dt

1 d 1 2 
 qc (t ) 
C dt  2


If t1 is minus infinity we talk about “energy stored at time t 2”

1 2
1 2
wC (t 2 , t1 ) = Cv C ( t2 ) − Cv C (t1 )
2
2

1 2
1 2
wC ( t 2 , t1 ) = qC ( t2 ) − qC ( t1 )
C
C

If both limits are infinity then we talk about the “total energy stored”
1
q 2 (t )
2
Slide 10

w (t ) =

2

Cv

(t ) =

2C

Example
The voltage v(t) is applied to a 10 µF capacitance
...

 1000 t 0 < t < 1

v (t ) =  1000 1 < t < 3
 500 ( 5 − t ) 3 < t < 5


p (t ) = v(t )i (t )

Slide 11

 10×10−3 A 0 < t <1
dv(t) 
i(t) = C
=  0A 1 < t < 3
dt
− 5×10−3 A 3 < 3 < 5


1
w(t ) = Cv2 (t)
2

Connections of Capacitors
Parallel Connection
i = C eq

Series Connection

dv
dt
v =

1
C

eq

t

∫ i (t )dt

t

0

To get a high voltage than the source, charge capacitors in
parallel then connect them in series
...
85×10

Fm

Construction of Practical Capacitors
• Parallel- plate capacitors are too large
– Rolled-type capacitors are used

• A dielectric with high dielectric constant is used
• Dielectric becomes a conductor if the voltage is high è
capacitors have a maximum voltage rating

Slide 14

Inductance

A time varying flux
causes a voltage to
appear at the
terminals of the device
Inductors store energy in the
form of magnetic field
Slide 15

Inductance
A time varying magnetic flux
induces a voltage

dφ
vL =
dt

Induction law

For a linear inductor the flux is
proportional to the current

φ = Li
1
i (t ) =
L

t

di
v (t ) = L
dt

∫ v (t )dt

Current through an inductor has
to be continuous

+ i (t 0 )

t0

1
Energy: w (t ) =
Li
2

2

(t )

L is the inductance with units of
henries (H) [volt seconds per ampere]
Slide 16

If i is constant
èv=0
DC or steady state behavior
An inductor in steady state acts as a
SHORT CIRCUIT

Example
The current through a 5-H inductance is given

di
v (t ) = L
dt

p(t) =v(t)i(t)

Slide 17

w (t ) =

1
Li 2 (t )
2

Example

1
i (t ) =
L

Slide 18

t

∫ v (t )dt

t0

+ i (t 0

)

Connections of Inductors
di
v=L
dt

=

Slide 19

1
L

∫ v (t )dt

Connections of Inductors

Slide 20



---