capacitors and inductors(new) - faraday
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EENG223: CIRCUIT THEORY I
DC Circuits:
Capacitors and Inductors Hasan Demirel
EENG223: CIRCUIT THEORY I
• Introduction
• Capacitors
• Series and Parallel Capacitors
• Inductors
• Series and Parallel Inductors
Capacitors and Inductors: Introduction
EENG223: CIRCUIT THEORY I
• Resistors are passive elements which dissipate energy only.
• Two important passive linear circuit elements: 1. Capacitor 2. Inductor
• Capacitors and inductors do not dissipate but store energy, which can be retrieved at a later time.
• Capacitors and inductors are called storage elements.
Capacitors and Inductors: Introduction
EENG223: CIRCUIT THEORY I
Capacitors and Inductors MICHAEL FARADAY (1791–1867)
English chemist and physicist, discovered electromagnetic induction in 1831 which was a breakthrough in engineering because it provided a way of generating electricity. The unit of capacitance, the farad, was named in his honor.
EENG223: CIRCUIT THEORY I
• A capacitor consists of two conducting plates separated by an insulator (or dielectric).
Capacitors and Inductors
Plates may be aluminum foil while the dielectric may be air, ceramic, paper, or mica.
A capacitor with applied voltage v.
EENG223: CIRCUIT THEORY I
• Three factors affecting the value of capacitance:
Capacitors and Inductors
1. Area(A): the larger the area, the greater the capacitance.
2. Spacing between the plates (d) : the smaller the spacing, the greater the capacitance.
3. Material permittivity (e) : the higher the permittivity, the greater the capacitance.
d
εAC
EENG223: CIRCUIT THEORY I
• The relation between the charge in plates and the voltage across a capacitor is given below.
Capacitors and Inductors
Cvq
C/V1F1
v
q Linear
Nonlinear
Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F).
(1 farad = 1 coulomb/volt)
EENG223: CIRCUIT THEORY I
Capacitors and Inductors
Circuit symbols for capacitors: (a) fixed capacitors, (b) variable capacitors.
(a-b)variable capacitors
(a) Polyester capacitor, (b) Ceramic capacitor, (c) Electrolytic capacitor
EENG223: CIRCUIT THEORY I
• Voltage Limit on a Capacitor:
Capacitors and Inductors
The plate charge increases as the voltage increases. Also, the electric field intensity between two plates increases.
If the voltage across the capacitor is so large that the field intensity is large enough to break down the insulation of the dielectric, the capacitor is out of work. Hence, every practical capacitor has a maximum limit on its operating voltage.
Cvq According to:
EENG223: CIRCUIT THEORY I
• Current(i)-voltage(v) relationship of a Capacitor:
Capacitors and Inductors
Taking the derivative of both sides
where
+
-
v
i
C
EENG223: CIRCUIT THEORY I
• Physical Meaning:
Capacitors and Inductors
+
-
v
i
C
• when v is a constant voltage, then i=0; a constant voltage across a capacitor creates no current through the capacitor, the capacitor in this case is the same as an open circuit.
• If v is abruptly changed, then the current will have an infinite value that is practically impossible. Hence, a capacitor is impossible to have an abrupt change in its voltage except an infinite current is applied.
EENG223: CIRCUIT THEORY I
• Important Properties of capacitors:
Capacitors and Inductors
1) A capacitor is an open circuit to dc.
2) The voltage on a capacitor cannot change abruptly.
Voltage across a capacitor: (a) allowed, (b) not allowable; an abrupt change is not possible.
4) A real, nonideal capacitor has a parallel-model leakage resistance. The leakage resistance may be as high as 100 MW and can be neglected for most practical applications. We will assume ideal capacitors.
3) Ideal capacitor does not dissipate energy. It takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit.
EENG223: CIRCUIT THEORY I
• Important Properties of capacitors:
Capacitors and Inductors
• A capacitor has memory.
• Capacitor voltage depends on the past history (-∞ →to) of the capacitor current. Hence, the capacitor has memory.
+
-
v C
EENG223: CIRCUIT THEORY I
• Important Properties of capacitors:
Capacitors and Inductors
• Energy Storing in Capacitor
+
-
v C
instantaneous power delivered to the capacitor is
energy stored in the capacitor:
note that
recall that
EENG223: CIRCUIT THEORY I
Ex. 6.1:(a) Calculate the charge stored on a 3-pF capacitor with 20 V across it. (b) Find the energy stored in the capacitor.
Capacitors and Inductors
(a) since: ,Cvq
pC6020103 12 q
(a) The energy stored: pJ6004001032
1
2
1 122 Cvw
Solution
EENG223: CIRCUIT THEORY I
Ex. 6.2: The voltage across a 5- F capacitor is: Calculate the current through it.
Capacitors and Inductors
(a) By definition current is:
V 6000cos10)( ttv
6105 dt
dvCi
6000105 6
)6000cos10( tdt
d
Att 6000sin3.06000sin10
dt
dvCi
Solution
EENG223: CIRCUIT THEORY I
Ex. 6.3: Determine the voltage across a 2-F capacitor if the current through it is :
Capacitors and Inductors
Since:
mA6)( 3000teti
Assume that the initial capacitor voltage is zero.
Solution
3
0
3000
6106
102
1
dtev
tt
0
30003
3000
103tte
t
vidtC
v0
)0(1
,0)0(and v
V)1( 3000te
EENG223: CIRCUIT THEORY I
Ex. 6.4: Determine the current through a 200- F capacitor whose voltage is shown in Fig 6.9.
Capacitors and Inductors
Solution
The voltage waveform can be described mathematically as:
otherwise0
43V 50200
31V 50100
10V 50
)(tt
tt
tt
tv
EENG223: CIRCUIT THEORY I
Ex. 6.4: Determine the current through a 200- F capacitor whose voltage is shown in Fig 6.9.
Capacitors and Inductors
Solution
The Since i = C dv/dt and C = 200 F, we take the derivative of v(t) to obtain:
otherwise0
43mA10
31mA10
10mA10
otherwise0
4350
3150
1050
10200)( 6
t
t
t
t
t
t
ti
Thus the current waveform is :
EENG223: CIRCUIT THEORY I
Ex. 6.5: Obtain the energy stored in each capacitor in circuit below under dc condition.
Capacitors and Inductors
Solution
Under dc condition, we replace each capacitor with an open circuit. By current division,
mA2)mA6(423
3
i
,V420001 iv
V840002 iv
mJ16)4)(102(2
1
2
1 232
111 vCw
mJ128)8)(104(2
1
2
1 232
222 vCw
EENG223: CIRCUIT THEORY I
Series and Parallel Capacitors
Capacitors and Inductors
Parallel-connected N capacitors
equivalent circuit for the parallel capacitors.
Niiiii ...321
EENG223: CIRCUIT THEORY I
Series and Parallel Capacitors
Capacitors and Inductors
• The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances.
Niiiii ...321
dt
dvC
dt
dvC
dt
dvC
dt
dvCi N ...321
dt
dvC
dt
dvC eq
N
kK
1
Neq CCCCC ....321
EENG223: CIRCUIT THEORY I
Series and Parallel Capacitors
Capacitors and Inductors
Series-connected N capacitors
equivalent circuit for the series capacitors.
)(...)()()( 21 tvtvtvtv N
EENG223: CIRCUIT THEORY I
Series and Parallel Capacitors
Capacitors and Inductors
• The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.
For N=2:
EENG223: CIRCUIT THEORY I
Series and Parallel Capacitors
Capacitors and Inductors
• Summary:
These results enable us to look the capacitor in this way: • 1/C has the equivalent effect as the resistance. The equivalent capacitor of capacitors connected in parallel or series can be obtained via this point of view, so is the Y-D connection and its transformation.
EENG223: CIRCUIT THEORY I
Ex. 6.6: Find the equivalent capacitance seen between terminals a and b of the circuit in the figure below.
Capacitors and Inductors
EENG223: CIRCUIT THEORY I
Ex. 6.6: Find the equivalent capacitance seen between terminals a and b of the circuit in the figure below.
Capacitors and Inductors
F4520
520
F302064
:seriesin are capacitors F5 and F20
capacitors F20 andF6 with the parallel in iscapacitor F4
capacitor. F60 the withseriesin iscapacitor F30
Solution
F20F6030
6030
eqC
EENG223: CIRCUIT THEORY I
Ex. 6.7: For the circuit given below, find the voltage across each capacitor.
Capacitors and Inductors
Solution
mF10mF1
20
1
30
1
)2040(
1
eqC
EENG223: CIRCUIT THEORY I
Ex. 6.7: For the circuit given below, find the voltage across each capacitor.
Capacitors and Inductors
Total charge
C3.0301010 3 vCq eq
This is the charge on the 20 mF and 30 mF capacitors, because they are in series with the 30 V source. ( A crude way to see this is to imagine that charge acts like current, since i = dq/dt)
V101030
3.03
2
2
C
qv
,V151020
3.03
1
1
C
qvTherefore,
EENG223: CIRCUIT THEORY I
Ex. 6.7: For the circuit given below, find the voltage across each capacitor.
Capacitors and Inductors
Having determined v1 and v2, we can use KVL to determine v3 by
V530 213 vvv
Alternatively, since the 40-mF and 20-mF capacitors are in parallel, they have the same voltage v3 and their combined capacitance is 40+20=60mF.
V51060
3.0
mF60 33
qv
EENG223: CIRCUIT THEORY I
Capacitors and Inductors JOSEPH HENRY (1797-1878)
American physicist, discovered inductance and constructed an electric motor.
EENG223: CIRCUIT THEORY I
• An inductor is made of a coil of conducting wire.
Capacitors and Inductors
l
ANL
2
Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H).
EENG223: CIRCUIT THEORY I
Capacitors and Inductors
(H/m)104 7
0
0
2
r
l
ANL
turns.ofnumber:N
length.:l
area. sectionalcross: A
coretheoftypermeabili:
(a) Solenoidal inductor, (b) toroidal inductor, (c) chip inductor.
EENG223: CIRCUIT THEORY I
Capacitors and Inductors
(a) air-core, (b) iron-core, (c) variable iron-core.
• Circuit symbols for inductors:
EENG223: CIRCUIT THEORY I
• Voltage(v) -current(i) relationship of an Inductor:
Capacitors and Inductors
integrating both sides gives:
voltage-current relationship of an inductor.
+
-
v
i
L
EENG223: CIRCUIT THEORY I
• Physical Meaning:
Capacitors and Inductors
• When the current through an inductor is a constant, then the voltage across the inductor is zero, same as a short circuit.
• No abrupt change of the current through an inductor is possible except an infinite voltage across the inductor is applied.
• The inductor can be used to generate a high voltage, for example, used as an igniting element.
+
-
v
i
L
EENG223: CIRCUIT THEORY I
• Important Properties of inductors:
Capacitors and Inductors
1) An inductor is a shaort to dc.
2) The current through an inductor cannot change abruptly.
Current through an inductor: (a) allowed, (b) not allowable; an abrupt change is not possible.
4) A practical, nonideal inductor has a significant resistive component. This resistance is called the winding resistance. The nonideal inductor also has a winding capacitance due to the capacitive coupling between the conducting coils. We will assume ideal inductors.
3) Ideal inductor does not dissipate energy. It takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit.
EENG223: CIRCUIT THEORY I
• Important Properties of inductors:
Capacitors and Inductors
• An inductor has memory.
• Inductor current depends on the past history (-∞ →to) of the inductor current. Hence, the inductor has memory.
+
-
v
i
L
EENG223: CIRCUIT THEORY I
• Important Properties of inductors:
Capacitors and Inductors
• Energy Storing in Inductor
power delivered to the inductor is
energy stored in the inductor:
note that
+
-
v
i
L
EENG223: CIRCUIT THEORY I
Ex. 6.8: The current through a 0.1-H inductor is i(t) = 10te-5t A. Find the voltage across the inductor and the energy stored in it.
Capacitors and Inductors
Solution
,H1.0andSince Ldt
diLv
V)51()5()10(1.0 5555 teetetedt
dv tttt
isstoredenergyThe
J5100)1.0(2
1
2
1 1021022 tt etetLiw
EENG223: CIRCUIT THEORY I
Ex. 6.9: Find the current through a 5-H inductor if the voltage across it is Also find the energy stored within 0 < t < 5s. Assume i(0)=0.
Capacitors and Inductors
Solution
0,0
0,30)(
2
t
tttv
.H5and L)()(1
Since0
0 t
ttidttv
Li
A23
6 33
tt
tdtti
0
2 0305
1
EENG223: CIRCUIT THEORY I
Ex. 6.9: Find the current through a 5-H inductor if the voltage across it is Also find the energy stored within 0 < t < 5s. Assume i(0)=0.
Capacitors and Inductors
Solution
0,0
0,30)(
2
t
tttv
5
0
65 kJ25.156
0
5
66060
tdttpdtw
thenisstoredenergytheand,60powerThe 5tvip
before.obtainedas
)0(2
1)5(
2
1)0()5( 2 LiLiww
kJ25.1560)52)(5(2
1 23
Alternatively
EENG223: CIRCUIT THEORY I
Ex. 6.10: Consider the circuit below in (a). Under dc conditions, find: (a) i, vC, and iL.
(b) the energy stored in the capacitor and inductor.
Capacitors and Inductors
before.obtainedas
Solution
EENG223: CIRCUIT THEORY I
Ex. 6.10: Consider the circuit below in (a). Under dc conditions, find: (a) i, vC, and iL.
(b) the energy stored in the capacitor and inductor.
Capacitors and Inductors
before.obtainedas
Solution:
,251
12Aii L
,J50)10)(1(2
1
2
1 22
cc Cvw
J4)2)(2(2
1
2
1 22 iL Lw
)(a :conditiondcUnder capacitor
inductor
circuitopen
circuitshort
)(b
V105 ivc
EENG223: CIRCUIT THEORY I
Series and Parallel Inductors
Capacitors and Inductors
Series-connected N inductors
equivalent circuit for the sieries inductors.
Neq LLLLL ...321
EENG223: CIRCUIT THEORY I
Series and Parallel Inductors
Capacitors and Inductors
Neq LLLLL ...321
• Applying KVL to the loop,
• Substituting vk = Lk di/dt results in
Nvvvvv ...321
dt
diL
dt
diL
dt
diL
dt
diLv N ...321
dt
diLLLL N )...( 321
dt
diL
dt
diL eq
N
KK
1
EENG223: CIRCUIT THEORY I
Series and Parallel Inductors
Capacitors and Inductors
Paralel-connected N inductors
equivalent circuit for the parallel inductors.
Neq LLLL
1111
21
EENG223: CIRCUIT THEORY I
Series and Parallel Inductors
Capacitors and Inductors
Neq LLLL
1111
21
• Using KCL,
• But
Niiiii ...321
t
t k
k
k otivdt
Li )(
10
t
t
t
t s
k
tivdtL
tivdtL
i0 0
)(1
)(1
0
2
01 t
tN
N
tivdtL 0
)(1
... 0
)(...)()(1
...11
00201
210
tititivdtLLL
N
t
tN
t
teq
N
kk
t
t
N
k k
tivdtL
tivdtL 00
)(1
)(1
01
01
EENG223: CIRCUIT THEORY I
Series and Parallel Inductors
Capacitors and Inductors
• The inductor in various connection has the same effect as the resistor. Hence, the Y-Δ transformation of inductors can be similarly derived.
EENG223: CIRCUIT THEORY I
Capacitors and Inductors
EENG223: CIRCUIT THEORY I
Ex. 6.11: Find the equivalent inductance of the circuit given below.
Capacitors and Inductors
before.obtainedas
EENG223: CIRCUIT THEORY I
Ex. 6.11: Find the equivalent inductance of the circuit given below.
Capacitors and Inductors
Solution:
10H12H,,H20:Series H42
H6427
427
H18864 eqL
: Parallel
EENG223: CIRCUIT THEORY I
Ex: Calculate the equivalent inductance for the inductive ladder network given in the circuit below.
Capacitors and Inductors
EENG223: CIRCUIT THEORY I
Ex 6.12: For the circuit given below, If, Find:
Capacitors and Inductors .mA)2(4)( 10teti
,mA 1)0(2 i)0( (a)
1i
);(and),(),((b) 21 tvtvtv
)(and)((c) 21 titi
EENG223: CIRCUIT THEORY I
Ex 6.12: For the circuit given below, If, Find:
Capacitors and Inductors .mA)2(4)( 10teti
,mA 1)0(2 i)0( (a)
1i
);(and),(),((b) 21 tvtvtv
)(and)((c) 21 titi
.mA4)12(4)0(mA)2(4)()(a 10 ieti t
mA5)1(4)0()0()0( 21 iii
H53212||42 eqL
mV200mV)10)(1)(4(5)( 1010 tt
eq eedt
diLtv
mV120)()()( 1012
tetvtvtv
mV80mV)10)(4(22)( 1010
1
tt eedt
ditv
isinductanceequivalentThe)(b
Solution:
EENG223: CIRCUIT THEORY I
Ex 6.12: For the circuit given below, If, Find:
Capacitors and Inductors .mA)2(4)( 10teti
,mA 1)0(2 i)0( (a)
1i
);(and),(),((b) 21 tvtvtv
)(and)((c) 21 titi
Solution:
t t t dteidtvti0 0
10
121 mA54
120)0(
4
1)(
mA38533mA50
3 101010 ttt eet
e
t ttdteidtvti
0
10
20 22 mA112
120)0(
12
1)(
mA11mA10
101010 ttt eet
e
)()()(thatNote 21 tititi
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