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ECEN 301 Discussion #10 – Energy Storage 1
Date Day Class
No.
Title Chapters HW
Due date
Lab
Due date
Exam
6 Oct Mon 10 Energy Storage 3.7, 4.1 NO LAB
7 Oct Tue NO LAB
8 Oct Wed 11 Dynamic Circuits 4.2 – 4.4
9 Oct Thu
10 Oct Fri Recitation HW 4
11 Oct Sat
12 Oct Sun
13 Oct Mon 12 Exam 1 Review
LAB 4 EXAM 1
14 Oct Tue
Schedule…
ECEN 301 Discussion #10 – Energy Storage 2
Energy
Moro. 7: 48
48 Wherefore, my beloved brethren, pray unto the Father with all the energy of heart, that ye may be filled with this love, which he hath bestowed upon all who are true followers of his Son, Jesus Christ; that ye may become the sons of God; that when he shall appear we shall be like him, for we shall see him as he is; that we may have this hope; that we may be purified even as he is pure. Amen.
ECEN 301 Discussion #10 – Energy Storage 3
Lecture 10 – Energy Storage
Maximum Power Transfer
Capacitors
Inductors
ECEN 301 Discussion #10 – Energy Storage 4
Maximum Power Transfer
Equivalent circuit representations are important
in power transfer analysis
vT+
–
RT
Load
+
v
–
i
Ideally all power from source is absorbed by the load
But some power will be absorbed by internal circuits
(represented by RT)
ECEN 301 Discussion #10 – Energy Storage 5
Maximum Power Transfer
Efficiently transferring power from source to load means that internal resistances (RT) must be minimized
i.e. for a given RL we want RT as small as possible!
For a given RT there is a specific RL that will maximize power transfer to the load
vT+
–
RT
RLiL
ECEN 301 Discussion #10 – Energy Storage 6
Maximum Power Transfer
Consider the power (PL) absorbed by the load
vT+
–
RT
RLiL
ECEN 301 Discussion #10 – Energy Storage 7
Maximum Power Transfer
Consider the power (PL) absorbed by the load
TL
TL
RR
vivT
+
–
RT
RLiL
LLL RiP2
BUT
L
TL
TL R
RR
vP
2
2
Therefore
ECEN 301 Discussion #10 – Energy Storage 8
Maximum Power Transfer
The maximum power transfer can be found by
calculating dPL/dRL = 0
vT+
–
RT
RLiL
OR
02
4
222
TL
TLLTTLT
L
L
RR
RRRvRRv
dR
dP
022
TLLTL RRRRR
OR
TL RRNB: assumes RT is fixed
and RL is variable
ECEN 301 Discussion #10 – Energy Storage 9
Maximum Power Transfer
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Normalized Resistance (RL/RT)
No
rma
lize
d P
ow
er
(PL/V
T2)
To transfer maximum power to the load, the source and load
resistors must be matched (i.e. RT = RL). Power is attenuated
as RL departs from RT
When RL < RT:PL is lowered since vL goes
down
When RL > RT:PL is lowered since less
current (iL) flows
ECEN 301 Discussion #10 – Energy Storage 10
Maximum Power Transfer
Maximum power theorem: maximum power is delivered by a source (represented by its Thévenin equivalent circuit) is attained when the load (RL) is equal to the Thévenin resistance RT
vT+
–
RT
RLiLL
TL
TL R
RR
vP
2
2
4
4
2
2
TiR
R
vP
L
L
TLMax
NB: Substitute (RT = RL) to get PLMax
ECEN 301 Discussion #10 – Energy Storage 11
Maximum Power Transfer
Efficiency of power transfer (η): the ratio of the power
delivered to the load (Pout), to the power supplied by
the source (Pin)
in
out
p
p
NB: Pout = PL
ECEN 301 Discussion #10 – Energy Storage 12
Maximum Power Transfer
Efficiency of power transfer (η): the ratio of the power
delivered to the load (Pout), to the power supplied by
the source (Pin)
vT+
–
RT
RLiL
ECEN 301 Discussion #10 – Energy Storage 13
Maximum Power Transfer
Efficiency of power transfer (η): the ratio of the power
delivered to the load (Pout), to the power supplied by
the source (Pin)
vT+
–
RT
RLiLLT
TT
TLin
RR
vv
viP
2
2
2
2
TL
L
TinMax
iR
R
vP
NB: Substitute (RT = RL) to get PinMax
ECEN 301 Discussion #10 – Energy Storage 14
Maximum Power Transfer
Efficiency of power transfer (η): the ratio of the power
delivered to the load (Pout), to the power supplied by
the source (Pin)
vT+
–
RT
RLiL
2
1
2
42
2
max
T
L
L
T
inMax
outMax
v
R
R
v
P
P
In other words, at
best, only 50%
efficiency can be
achieved
ECEN 301 Discussion #10 – Energy Storage 15
Maximum Power Transfer
Example1: Find the maximum power delivered to RL
vs = 18V, R1 = 3Ω, R2 = 6Ω, R3 = 2Ω
R2
R1
RL
R3
vs+
–
ECEN 301 Discussion #10 – Energy Storage 16
Maximum Power Transfer
Example1: Find the maximum power delivered to RL
vs = 18V, R1 = 3Ω, R2 = 6Ω, R3 = 2Ω
1. Find Thévenin equivalent circuit
V
vRR
Rv sT
12
21
2
vT+
–
RT
RL
4
||213 RRRRT
ECEN 301 Discussion #10 – Energy Storage 17
Maximum Power Transfer
Example1: Find the maximum power delivered to RL
vs = 18V, R1 = 3Ω, R2 = 6Ω, R3 = 2Ω
1. Find Thévenin equivalent circuit
2. Set RL = RT
3. Calculate PLMax
W
R
v
PP
L
T
inL
9
8
)12(5.0
25.0
5.0
2
2
maxmaxvT
+
–
RT
RL
ECEN 301 Discussion #10 – Energy Storage 18
Dynamic Energy Storage Elements
Capacitor and Inductor
ECEN 301 Discussion #10 – Energy Storage 19
Storage Elements
To this point, circuits have consisted of either:
DC Sources (active element, provide energy)
Resistors (passive element, absorb energy)
Some passive (non-source) elements can store energy
Capacitors – store voltage
Inductors – store current
+
C
–
+
C
–
+
L
–
+ L –+ C –
Capacitor symbols Inductor symbols
ECEN 301 Discussion #10 – Energy Storage 20
Ideal Capacitor
A capacitor (C) is composed of:
2 parallel conducting plates
• cross-sectional area A
Separated distance d by either:
• Air
• Dielectric (insulating) material
d
A
Dielectric (or air)
d
AC
ε – relative permittivity
(dielectric constant)
NB: the parallel plates do not
touch – thus under DC current a
capacitor acts like an open circuit
+
–
C – Capacitance
ECEN 301 Discussion #10 – Energy Storage 21
Ideal Capacitor
Capacitance: a measure of the ability to store energy in
the form of separated charge (or an electric field)
+ ++ ++
+ +++ +
+
– –– –––
–––– ––– – –
The charge q+ on one plate
is equal to the charge q– on
the other plate
Cvq
Farad (F): unit of capacitance.
1 farad = 1 coulomb/volt (C/V)
ECEN 301 Discussion #10 – Energy Storage 22
Ideal Capacitor
No current can flow through a capacitor if the voltage across it is constant (i.e. connected to a DC source)
BUT when a DC source is applied to a capacitor, there is an initial circuit current as the capacitor plates are charged (‘charging’ the capacitor)
dt
tdqti
)()(
+ + + +
– – –– ––vs
+
–
Charging the capacitor: a current
flows as electrons leave one plate to
accumulate on the other.
This occurs until the voltage across
the capacitor = source voltage
i
ECEN 301 Discussion #10 – Energy Storage 23
Ideal Capacitor
Capacitor with an AC source: since AC sources periodically reverse voltage directions, the capacitor plates will change charge polarity accordingly With each polarity change, the capacitor goes through a ‘charging’
state, thus current continuously flows
Just as with a DC source, no electrons cross between the plates
Just as with a DC source, voltage across a capacitor cannot change instantaneously
+ + + +
– – –– ––vs(t)
+
–
i
– – – –
+ + +++vs(t)
–
+
i
ECEN 301 Discussion #10 – Energy Storage 24
Ideal Capacitor
i – v characteristic for an ideal capacitor
)()( tCvtqdt
tdCv
dt
tdq )()(
Differentiate
both sides
dt
tdqti
)()(BUT
dt
tdvCti
)()( OR
t
C diC
tv )(1
)(
ECEN 301 Discussion #10 – Energy Storage 25
Ideal Capacitor
i – v characteristic for an ideal capacitor
t
C diC
tv )(1
)( NB: Assumes we know the value of the
capacitor from time (τ = -∞) until time (τ = t)
)()(1
)( 00
tvdiC
tvt
tC
Instead
Initial time (often t0 = 0)
Initial condition
ECEN 301 Discussion #10 – Energy Storage 26
Ideal Capacitor
Example2: calculate the current through the capacitor C = 0.1μF
with the voltage as shown: v(t) = 5(1-e-t/10-6)
0.0
1.0
2.0
3.0
4.0
5.0
0.00 2.00 4.00 6.00
time (us)
vo
ltag
e (
V)
ECEN 301 Discussion #10 – Energy Storage 27
Ideal Capacitor
Example2: calculate the current through the capacitor C = 0.1μF
with the voltage as shown: v(t) = 5(1-e-t/10-6)
0.0
1.0
2.0
3.0
4.0
5.0
0.00 2.00 4.00 6.00
time (us)
vo
ltag
e (
V)
Ae
e
dt
tdvCti
t
t
6
6
10/
10/
6
7
5.0
10
5)10(
)()(
ECEN 301 Discussion #10 – Energy Storage 28
Ideal Capacitor
Example2: calculate the current through the capacitor C = 0.1μF
with the voltage as shown: v(t) = 5(1-e-t/10-6)
0.0
1.0
2.0
3.0
4.0
5.0
0.00 2.00 4.00 6.00
time (us)
vo
ltag
e (
V)
Aeti t 610/5.0)(
0.0
0.1
0.2
0.3
0.4
0.5
0.00 2.00 4.00 6.00
time (us)cu
rren
t (A
)
NB: the capacitor’s current jumps ‘instantaneously’ to 0.5A.
The ability of a capacitor’s current to change instantaneously is
an important property of capacitors
ECEN 301 Discussion #10 – Energy Storage 29
Ideal Capacitor
Example3: find the voltage v(t) for a capacitor C =
0.5F with the current as shown and v(0) = 0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0
time (s)
cu
rren
t (A
)
ECEN 301 Discussion #10 – Energy Storage 30
Ideal Capacitor
Example3: find the voltage v(t) for a capacitor C =
0.5F with the current as shown and v(0) = 0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0
time (s)
cu
rren
t (A
)
)0(1
)(0
vidC
tvt
02
121
10
00
current Interval
t
t
tt
t
i(t)
ECEN 301 Discussion #10 – Energy Storage 31
Ideal Capacitor
Example3: find the voltage v(t) for a capacitor C =
0.5F with the current as shown and v(0) = 0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0
time (s)
cu
rren
t (A
)
)0(1
)(0
vidC
tvt
)2(02
)1()1(221
0210
00
voltageInterval
0
0
vt
vdt
dt
t
v(t)
t
t
ECEN 301 Discussion #10 – Energy Storage 32
Ideal Capacitor
Example3: find the voltage v(t) for a capacitor C =
0.5F with the current as shown and v(0) = 0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0
time (s)
cu
rren
t (A
)
32
1221
10
00
voltageInterval
2
t
tt
tt
t
v(t)
ECEN 301 Discussion #10 – Energy Storage 33
Ideal Capacitor
Example3: find the voltage v(t) for a capacitor C =
0.5F with the current as shown and v(0) = 0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0 1.0 2.0 3.0 4.0
time (s)
vo
lta
ge
(V
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0
time (s)
cu
rren
t (A
)
NB: The final value of the capacitor voltage after
the current source has stopped charging the
capacitor depends on two things:
1. The initial capacitor voltage
2. The history of the capacitor current 32
1221
10
00
voltageInterval
2
t
tt
tt
t
v(t)
ECEN 301 Discussion #10 – Energy Storage 34
Series Capacitors
Capacitor Series Rule: two or more circuit elements are said to be in series if the current from one element exclusively flows into the next element. Capacitors in series add the same way resistors in parallel add
N
EQ
NEQ
CCCC
CCCCCC 1111
111111
321
321
CEQ
C1
∙ ∙ ∙ ∙ ∙ ∙
C2 C3 Cn CN
ECEN 301 Discussion #10 – Energy Storage 35
Parallel Capacitors
Parallel Rule: two or more circuit elements are said to
be in parallel if the elements share the same terminals
Capacitors in parallel add the same way resistors in series add
C1 C2 C3Cn CN CEQ
N
n
nEQ CC1
ECEN 301 Discussion #10 – Energy Storage 36
Parallel and Series Capacitors
Example4: determine the equivalent capacitance CEQ
C1 = 2mF, C2 = 2mF, C3 = 1mF, C4 = 1/3mF, C5 = 1/3mF,
C6 = 1/3mF
C1
C2
C3
C4
C5
C6
ECEN 301 Discussion #10 – Energy Storage 37
Parallel and Series Capacitors
Example4: determine the equivalent capacitance CEQ
C1 = 2mF, C2 = 2mF, C3 = 1mF, C4 = 1/3mF, C5 = 1/3mF,
C6 = 1/3mF
C1
C2
C3
C4
C5
C6
mF
CCCCCC
CCCCEQ
9
1
3/1
27/1
)3/1)(3/1()3/1)(3/1()3/1)(3/1(
)3/1)(3/1)(3/1(
656454
6541
ECEN 301 Discussion #10 – Energy Storage 38
Parallel and Series Capacitors
Example4: determine the equivalent capacitance CEQ
C1 = 2mF, C2 = 2mF, C3 = 1mF, C4 = 1/3mF, C5 = 1/3mF,
C6 = 1/3mF
mF
CCC EQEQ
9
10
9
1)1(
132
C1
C2
C3 CEQ1
mFCEQ9
11
ECEN 301 Discussion #10 – Energy Storage 39
Parallel and Series Capacitors
Example4: determine the equivalent capacitance CEQ
C1 = 2mF, C2 = 2mF, C3 = 1mF, C4 = 1/3mF, C5 = 1/3mF,
C6 = 1/3mF
C1
C2
CEQ2
mFCEQ9
102 mF
CCCCCC
CCCC
EQEQ
EQ
EQ
19
10
9/209/204
9/40
)9/10)(2()9/10)(2()2)(2(
)9/10)(2)(2(
222121
221
ECEN 301 Discussion #10 – Energy Storage 40
Parallel and Series Capacitors
Example4: determine the equivalent capacitance CEQ
C1 = 2mF, C2 = 2mF, C3 = 1mF, C4 = 1/3mF, C5 = 1/3mF,
C6 = 1/3mF
CEQ mFCEQ19
10
ECEN 301 Discussion #10 – Energy Storage 41
Energy Storage in Capacitors
Capacitor energy WC(t): can be found by taking the
integral of power
Instantaneous power PC = iv
)(
)(
2
)(
)(
2
1
)()(
)()(
)()(
tv
v
tv
v
tC
C
t
CC
t
CC
Cv
vdvC
dd
dvCv
div
dPtW
Since the capacitor is uncharged at t = -∞,
v(- ∞) = 0, thus:
JtCvtWC
2)(2
1)(
ECEN 301 Discussion #10 – Energy Storage 42
Ideal Inductor
Inductors are made by winding a coil of wire around a
core
The core can be an insulator or ferromagnetic material
lN turns
• Cross-sectional area A
• Diameter d
dl
ANL
45.0
2
μ – relative permeability
+
–
L – Inductance
ECEN 301 Discussion #10 – Energy Storage 43
Ideal Inductor
Inductance: a measure of the ability of a device
to store energy in the form of a magnetic field
Henry (H): unit of inductance.
1 henry = 1 volt-second/ampere (V-s/A)
Ideally the resistance through an inductor is zero
(i.e. no voltage drop), thus an inductor acts like
a short circuit in the presence of a DC source.
BUT there is an initial voltage across the
inductor as the current builds up (much like
‘charging’ with capacitors)
i
ECEN 301 Discussion #10 – Energy Storage 44
Ideal Inductor
Inductor with an AC source: since AC sources periodically reverse current directions, the current flow through the inductor also changes
With each current direction change, the current through the inductor must ‘build up’, thus there is a continual voltage drop across the inductor
Just as with a DC source, current across an inductor cannot change instantaneously
vs(t)+
–
i
vs(t)–
+
i
ECEN 301 Discussion #10 – Energy Storage 45
Ideal Inductor
I – v characteristic for an ideal inductor
dt
tdiLtv
)()(
t
L dvL
ti )(1
)(
NB: note the duality between inductors and capacitors
ECEN 301 Discussion #10 – Energy Storage 46
Ideal Inductor
i – v characteristic for an ideal inductor
t
L dvL
ti )(1
)( NB: Assumes we know the value of the
inductor from time (τ = -∞) until time (τ = t)
)()(1
)( 00
tidvL
tit
tL
Instead
Initial time (often t0 = 0)
Initial condition
ECEN 301 Discussion #10 – Energy Storage 47
Ideal Inductor
Example5: find the voltage across an inductor L =
0.1H when the current is: i(t) = 20 t e-2t
0.0
2.0
4.0
0.0 0.5 1.0 1.5
time (s)
cu
rre
nt
(A)
ECEN 301 Discussion #10 – Energy Storage 48
Ideal Inductor
Example5: find the voltage across an inductor L =
0.1H when the current is: i(t) = 20 t e-2t
0.0
2.0
4.0
0.0 0.5 1.0 1.5
time (s)
cu
rre
nt
(A)
Vte
ete
tedt
d
dt
tdiLtv
t
tt
t
)21(2
)2(2
)20(1.0
)()(
2
22
2
ECEN 301 Discussion #10 – Energy Storage 49
Ideal Inductor
Example5: find the voltage across an inductor L =
0.1H when the current is: i(t) = 20 t e-2t
0.0
2.0
4.0
0.0 0.5 1.0 1.5
time (s)
cu
rre
nt
(A)
Vtetv t )21(2)( 2
-0.5
1.5
0.0 0.5 1.0 1.5
time (s)vo
ltag
e (
V)
NB: the inductor’s voltage jumps ‘instantaneously’ to
2V. The ability of an inductor’s voltage to change
instantaneously is an important property of inductors
ECEN 301 Discussion #10 – Energy Storage 50
Series Inductors
Series Rule: two or more circuit elements are said to be in series if the current from one element exclusively flows into the next element. Inductors in series add the same way resistors in series add
N
n
nEQ LL1
LEQ
L1
∙ ∙ ∙ ∙ ∙ ∙
L2 L3 Ln LN
ECEN 301 Discussion #10 – Energy Storage 51
Parallel Inductors
Parallel Rule: two or more circuit elements are said to
be in parallel if the elements share the same terminals
Inductors in parallel add the same way resistors in parallel add
N
EQ
NEQ
LLLL
LLLLLL 1111
111111
321
321
L1 L2 L3 Ln LN LEQ
ECEN 301 Discussion #10 – Energy Storage 52
Energy Storage in Capacitors
Capacitor energy WL(t): can be found by taking the
integral of power
Instantaneous power PL = iv
)(
)(
2
)(
)(
2
1
)()(
)()(
)()(
ti
i
ti
i
t
LL
t
LL
t
LL
Li
idiL
did
diL
div
dPtW
Since the inductor is uncharged at t = -∞,
i(- ∞) = 0, thus:
JtLitWL
2)(2
1)(
ECEN 301 Discussion #10 – Energy Storage 53
Energy Storage in Capacitors
Example6: calculate the power and energy stored in a 0.1-H inductor when: i = 20 t e-2t A (i = 0 for t < 0)
V = 2 e-2t (1- 2t)V (for t >= 0)
ECEN 301 Discussion #10 – Energy Storage 54
Energy Storage in Capacitors
Example6: calculate the power and energy stored in a 0.1-H inductor when: i = 20 t e-2t A (i = 0 for t < 0)
V = 2 e-2t (1- 2t)V (for t >= 0)
Wtte
tete
titvtP
t
tt
LLL
)21(40
20)21(2
)()()(
4
22
ECEN 301 Discussion #10 – Energy Storage 55
Energy Storage in Capacitors
Example6: calculate the power and energy stored in a 0.1-H inductor when: i = 20 t e-2t A (i = 0 for t < 0)
V = 2 e-2t (1- 2t)V (for t >= 0)
Jet
te
tLitW
t
t
LL
42
22
2
20
20)1.0(2
1
)(2
1)(
Wtte
tete
titvtP
t
tt
LLL
)21(40
20)21(2
)()()(
4
22
ECEN 301 Discussion #10 – Energy Storage 56
Ideal Capacitors and Inductors
Inductors Capacitors
Passive sign
convention
Voltage
Current
Power
)()(1
)( 00
tidvL
tit
tL
dt
tdiLtv
)()(
+ L –
i
+ C –
i
)()(1
)( 00
tvdiC
tvt
tC
dt
tdvCti
)()(
dt
tditLitPL
)()()(
dt
tdvtCvtPC
)()()(
ECEN 301 Discussion #10 – Energy Storage 57
Ideal Capacitors and Inductors
Inductors Capacitors
Energy
An instantaneous change
is not permitted in:
Current Voltage
Will permit an
instantaneous change in:
Voltage Current
With DC source element
acts as a:
Short Circuit Open Circuit
2)(2
1)( tLitWL
2)(2
1)( tCvtWC
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