Electrical Circuits- L55555555555555555555555555555555

lecture notes

1. ECO111 Electrical Circuits
Chapter 5
Energy Storage Devices
Dr. Wael Mamdouh
Electrical Engineering Department
Egyptian Academy for Engineering and Advanced Technology
Dr. Wael Mamdouh -ECO111-Electrical Circuits 1

2. Energy Storage Devices
1-Inductors
2-Capacitors
Dr. Wael Mamdouh -ECO111-Electrical Circuits 2

3. Inductors
•Generally -coil of conducting wire
•Usually wrapped around a solid core.
•If no core is used, then the inductor is said to have an ‘air core’.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 3

4. Symbols
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5. Alternative Names for Inductors
Reactor:inductorinapowergrid
Choke:designedtoblockaparticularfrequencywhileallowing
currentsatlowerfrequenciesord.c.currentsthrough
Commonly used in RF (radio frequency) circuitry
Coil:oftencoatedwithvarnishand/orwrappedwithinsulating
tapetoprovideadditionalinsulation.
A winding is a coil with taps (terminals).
Solenoid:athreedimensionalcoil.
Also used to denote an electromagnet where the magnetic field is
generated by current flowing through a toroidalinductor.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 5

6. Energy Storage
The flow of current through an inductor creates a magnetic field
(right hand rule).
Ifthecurrentflowingthroughtheinductordrops,themagneticfield
willalsodecreaseandenergyisreleasedthroughthegenerationofa
current.
B field
Dr. Wael Mamdouh -ECO111-Electrical Circuits 6

7. Sign Convention
The sign convention used with an inductor is the same as
for a power dissipating device.
•When current flows into the positive side
of the voltage across the inductor, it is
positive and the inductor is dissipating
power.
•When the inductor releases energy back
into the circuit, the sign of the current will
be negative.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 7

8. Current and Voltage Relationships
•L , inductance, has the units of Henries(H)
=
=
1
1
t
t
LL
L
o
dtv
L
i
dt
di
Lv
Dr. Wael Mamdouh -ECO111-Electrical Circuits 8

9. Power and Energy

==




=
=
=
11
1
0
1
t
t
LL
t
t
L
L
t
t
LLL
L
LL
LLL
oo
diiLdti
dt
di
Lw
dtvv
L
p
dt
di
Lip
ivp
Dr. Wael Mamdouh -ECO111-Electrical Circuits 9

10. Inductors
Stores energy in a magnetic field created by the electric current flowing
through it.
•Inductor opposes change in current flowing through it.
•Current through an inductor is continuous; voltage can be discontinuous.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 10

11. Calculations of L
For a solenoid (toroidalinductor)
Nis the number of turns of wire
Ais the cross-sectional area of the toroidin m
2
.

ris the relative permeability of the core material

ois the vacuum permeability (4π×10
-7
H/m)
lis the length of the wire used to wrap the toroidin meters
ANAN
L
or
22
==
Dr. Wael Mamdouh -ECO111-Electrical Circuits 11

12. Wire
Unfortunately, even bare wire
has inductance.
d is the diameter of the wire in
meters.( )Hx
d
L
7
10214ln
−






−





=


Dr. Wael Mamdouh -ECO111-Electrical Circuits 12

13. Properties of an Inductor
•Acts like a short circuit at steady state when connected to a d.c.voltage or
current source.
•Currentthrough an inductor must be continuous
There are no changes to the current, but there can be abrupt
changes in the voltage across an inductor.
•An ideal inductor does not dissipate energy, it takes power from the circuit
when storing energy and returns it when discharging.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 13

14. Properties of a Real Inductor
Realinductorsdodissipateenergydueresistivelossesinthelengthofwire
andcapacitivecouplingbetweenturnsofthewire.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 14

15. Inductors in Series
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16. Leqfor Inductors in Seriesi 4321eq
4321
4433
2211
4321
L
dt
di
dt
di
dt
di
dt
di
dt
di

dt
di

dt
di

dt
di

dt
di
LLLL
Lv
LLLLv
LvLv
LvLv
vvvvv
eqin
in
in
+++=
=
+++=
==
==
+++=
Dr. Wael Mamdouh -ECO111-Electrical Circuits 16

17. Inductors in Parallel
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18. Leqfor Inductors in Paralleli ()()()() 
1
4321eq
t
t
t
t4
t
t3
t
t2
t
t1
t
t4
4
t
t3
3
t
t2
2
t
t1
1
4321
1111L
vdt
1
vdt
1
vdt
1
vdt
1
vdt
1
vdt
1
vdt
1
vdt
1
vdt
1
1
o
1
o
1
o
1
o
1
o
1
o
1
o
1
o
1
o
−
+++=
=
+++=
==
==
+++=




LLLL
L
i
LLLL
i
L
i
L
i
L
i
L
i
iiiii
eq
in
in
in
Dr. Wael Mamdouh -ECO111-Electrical Circuits 18

19. General Equations for Leq
Series Combination
•If S inductors are in series, then
Parallel Combination
•If P inductors are in parallel, then:1
1
1
−
=







=
P
p p
eq
L
L 
=
=
S
s
seq LL
1
Dr. Wael Mamdouh -ECO111-Electrical Circuits 19

20. Capacitors
❑Deviceforstoringelectricalenergywhichcanthenbe
releasedinacontrolledmanner
❑Consistsoftwoconductors,carryingchargesofqand–q,
thatareseparated,usuallybyanonconductingmaterial-an
insulator
Symbolincircuitsis
❑Ittakeswork,whichisthenstoredaspotentialenergyinthe
electricfieldthatissetupbetweenthetwoplates,toplace
chargesontheconductingplatesofthecapacitor
❑Sincethereisanelectricfieldbetweentheplatesthereis
alsoapotentialdifferencebetweentheplates
Dr. Wael Mamdouh -ECO111-Electrical Circuits 20

21. Capacitors
•We usually talk about
capacitors in terms of
parallel conducting
plates
•They in fact can be any
two conducting objects
Dr. Wael Mamdouh -ECO111-Electrical Circuits 21

22. Capacitanceab
V
Q
C=
Units areVolt
Coulomb
farad
1
1
1 =
The capacitance is defined to be the ratio of the amount
of charge that is on the capacitor to the potential
difference between the plates at this point
Dr. Wael Mamdouh -ECO111-Electrical Circuits 22

23. Calculating the Capacitance
•We start with the simplest form –two parallel
conducting plates separated by vacuum
•Let the conducting plates have area A
and be separated by a distance d
•The magnitude of the electric fieldE
between the two plates is given by
Where is charge densityA
Q
E
00


==
Dr. Wael Mamdouh -ECO111-Electrical Circuits 23

24. Permittivityisamaterialpropertythataffectsthe
Coulombforcebetweentwopointchargesinthe
material.
•We treat the field as being uniform
allowing us to write
Where 
0: is a permittivity vacuumA
dQ
EdV
ab
0

==
Calculating the Capacitance
Relativepermittivityistheratioofthecapacitanceofa
capacitorusingthatmaterialasadielectric,comparedto
asimilarcapacitorthathasvacuumasitsdielectric
Dr. Wael Mamdouh -ECO111-Electrical Circuits 24

25. Calculating the Capacitance
Puttingthisalltogether,wehaveforthe
capacitanced
A
V
Q
C
ab
0==
Thecapacitanceisonlydependentupon
thegeometryofthecapacitord
A
d
A
C
r0==
Dr. Wael Mamdouh -ECO111-Electrical Circuits 25

26. 1 farad Capacitor
Given a 1 farad parallel plate capacitor having a plate
separation of 1mm. What is the area of the plates?
We start withd
A
C
0=
And rearrange to solve
for A, giving()( )
28
12
3
0
101.1
/1085.8
100.10.1
m
mF
mFdC
A
=


==
−
−

This corresponds to a square about 10km on a side!
Dr. Wael Mamdouh -ECO111-Electrical Circuits 26

27. The Current i
c
•Current i
cassociated with the capacitance C is
related to the voltage across the capacitor by

•Where dv
c/ dtis a measure of the change in v
cin a
vanishingly small period of time.
•The function dv
c/ dtis called the derivativeof the voltage
v
cwith respect to time t.
Dr. Wael Mamdouh -ECO111-Electrical Circuits 27

28. Series or Parallel Capacitors
•Sometimes in order to obtain needed values of capacitance,
capacitors are combined in either
or
Series
Parallel
Dr. Wael Mamdouh -ECO111-Electrical Circuits 28

29. Capacitors in Series
Capacitors are often combined in series and the question
then becomes what is the equivalent capacitance?
Given what is
Westartbyputtingavoltage,
Vab,acrossthecapacitors
Dr. Wael Mamdouh -ECO111-Electrical Circuits 29

30. Capacitors in Series
Capacitors become charged because of V
abis applied.
If upper plate of C
1gets a charge of +Q,
Then the lower plate of C
1gets a
charge of -Q
What happens with C
2?
Since there is no source of charge at point c, and we have
effectively put a charge of –Q on the lower plate of C
1, the
upper plate of C
2gets a charge of +Q
This then means that lower plate of C
2has a charge of -Q
Charge Conservation
Dr. Wael Mamdouh -ECO111-Electrical Circuits 30

31. We also have to have that the potential across C
1plus the
potential across C
2should equal the potential drop across
the two capacitors21VVVVV
cbacab +=+=
We have2
2
1
1
C
Q
V
C
Q
V == and
Then21
C
Q
C
Q
V
ab +=
Dividing through by Q, we have21
11
CCQ
V
ab
+=
Capacitors in Series
Dr. Wael Mamdouh -ECO111-Electrical Circuits 31

32. 21
11
CCQ
V
ab
+= The left hand side is the inverse of the
definition of capacitanceQ
V
C
=
1
So we then have for the equivalent capacitance21
111
CCC
eq
+=
If there are more than two capacitors in series, the resultant
capacitance is given by=
iieq CC
11
Theequivalentcapacitorwillalsohavethesamevoltage
acrossit
Capacitors in Series
Dr. Wael Mamdouh -ECO111-Electrical Circuits 32

33. Capacitors in Parallel
Capacitors can also be connected in parallel
Given what is
Again we start by putting a
voltage across a and b
Dr. Wael Mamdouh -ECO111-Electrical Circuits 33

34. The upper plates of both capacitors
are at the same potential Likewise
for the bottom plates
We have thatabVVV ==
21
NowVCQVCQ
or
C
Q
V
C
Q
V
2211
2
2
2
1
1
1
==
==
and
and
Capacitors in Parallel
Dr. Wael Mamdouh -ECO111-Electrical Circuits 34

35. The equivalent capacitor will have the same voltage
across it, as do the capacitors in parallel
But what about the charge on the equivalent capacitor?
The equivalent capacitor will have
the same total charge21QQQ +=
Using this we then have21
21
21
CCC
or
VCVCVC
QQQ
eq
eq
+=
+=
+=
Capacitors in Parallel
Dr. Wael Mamdouh -ECO111-Electrical Circuits 35

36. The equivalent capacitance is just the sum of
the two capacitors
If we have more than two, the
resultant capacitance is just
the sum of the individual
capacitances=
i
ieq
CC
Capacitors in Parallel
Dr. Wael Mamdouh -ECO111-Electrical Circuits 36

37. Example 1
C213
111
CCCC +
+=
321
213 )(
CCC
CCC
C
++
+
=
Where do we start?
Recognize that C
1and C
2are parallel with each other
and combine these to get C
12
This C
12is then in series with withC
3
The resultant capacitance is then given by
Dr. Wael Mamdouh -ECO111-Electrical Circuits 37

38. Energy Stored in a Capacitor
Electrical Potential energy is stored in a capacitor
The energy comes from the work that is done in
charging the capacitor
Let q and v be the intermediate charge and potential
on the capacitor
The incremental work done in bringing an incremental
charge, dq, to the capacitor is then given byC
dqq
dqvdW ==
Dr. Wael Mamdouh -ECO111-Electrical Circuits 38

39. Energy Stored in a Capacitor
The total work done is just the integral of this equation
from 0 to QC
Q
dqq
C
W
Q
2
1
2
0
==
Using the relationship between capacitance, voltage
and charge we also obtainVQVC
C
Q
U
2
1
2
1
2
2
2
===
where U is the stored potential energy
Dr. Wael Mamdouh -ECO111-Electrical Circuits 39

40. Dr. Wael Mamdouh -ECO111-Electrical Circuits 40

41. Energy Storage Elements
Mechanical components VsElectrical components
Dr. Wael Mamdouh -ECO111-Electrical Circuits 41