Fundamentals of Electric Circuit - Chapter 6: Capacitors and Inductors

rge:
eq
q C v C
310.10 .30 0,3   (charge on the C1 and C2 )
 Another solution:
q q
v V v V
C C
1 23 3
1 2
0,3 0,3
15 ; 10
20.10 30.10
 
       v v v V
3 1 2
30 5    
C C C mF
3 4 34
/ / 20 40 60   
C is in series with C and C
34 1 2

q
v V
C
3 3
34
0,3
5
60.10

   
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
C 1
C 2
C 3
C 4
12
Chapter 6: Capacitors and Inductors
III. Series and parallel capacitors
Ex 6.4: Find the voltage across each capacitor.
60V
+ -
+
-
v1
v4
40μF
30μF
+ -
v2
60μF
20μF
+
-
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
13
Chapter 6: Capacitors and Inductors
IV. Inductors
 An inductor is a passive element designed to store energy in its magnetic field.
 An inductor consists of a coil of conducting wire.
 If current is allowed to pass through an inductor, the
voltage across the inductor is directly proportional to
the time rate of change of the current.
  
t
t
di
v L i v t dt i t
dt L
0
0
1
; ( ) ( ) L: inductance of the inductor [H]
Typical form of an inductor
Circuit symbols for inductors: (a) air-
core, (b) iron-core, (c) variable iron-core
L
L L+
-
v
+
-
v
+
-
v
iii
(c)(b)(a)
Voltage-current relationship
v
Slope = L
0
di/dt
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
14
Chapter 6: Capacitors and Inductors
IV. Inductors
 Inductor classification:
 Fixed - variable
 Linear - in-linear
 Core: iron, steel, plastic, air, 
 Physical dimension
 Construction
N A
L
l
2

N: number of turns
l: length of coil
A: cross-sectional area
μ: permeability of the core
Toroidal inductor Solenoidal 
wound inductor
Inductor
 Inductance is the property whereby an inductor exhibits opposition to the charge
of current flowing through it, measured in [H]. It depends on:
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
15
Chapter 6: Capacitors and Inductors
IV. Inductors
 Note that:
 An inductor acts like a short circuit to DC
 The current cannot change instantaneously.
 The ideal inductor does not dissipate energy.
 Non-ideal inductor has a significant resistive component:
 winding resistance (very small)
 Winding capacitance (very small, except at high frequencies)
 
   
 
di
p v i L i
dt
. Power:
 Energy:
  
 
   
 
  
t t t
di
w pdt L idt L idi
dt
  w Li2
1
2
Current through an inductor: (a) 
allowed, (b) not allowable, an abrupt 
change is not possible.
R
L
C
Circuit model for 
a practical inductor
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
16
Chapter 6: Capacitors and Inductors
IV. Inductors
Ex 6.5: Find the current through a 5H inductor and the energy stored within 0<t< 5s,
if the its voltage is:
     
t t
t
i v t dt i t t dt t A
L
0
2 3
0
0
1 1
( ) ( ) 30 0 2 ( )
5
 The current through the inductor:
 
 

t t
v t
t
230 , 0
( )
0, 0
 The power:  p v i t
5. 60
 The energy:     
t
w pdt t dt kJ
55 6
5
0 0
60 60 156,25
6
 The energy could be calculated by applying the equation:
  w kJ
2
31 .5. 2.5 156,25
2
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
17
Chapter 6: Capacitors and Inductors
IV. Inductors
Ex 6.6: Find i, vC, iL, energy stored in the capacitor and inductor under DC condition.
 Under DC condition, replacing:
 capacitor open circuit
 Inductor short circuit
   

L
V
i i A
R R
1 3
12
2
 The energy in the capacitor:   C Cw Cv J
2 21 1 .1.10 50
2 2
 The energy in the inductor:   L Lw L i J
2 21 1. . .2.2 4
2 2
R 3
R 2
R 1
L
C 2H
4Ω12V
5Ω1Ω
1F
iL
i
   
C
v R i V
3
. 5.2 10
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
L 1 L 2 L n
18
Chapter 6: Capacitors and Inductors
V. Series and parallel inductors
 Applying KVL to the loop:
    
N
v v v v v
1 2 3
...
v
i

 
       
 

N
N k eq
k
di di di di di
v L L L L L
dt dt dt dt dt
1 2
1
...
vN+ -
    
eq N
L L L L L
1 2 3
...
+ - + -v2v1+
-
Leq
v
+
-
i
 The equivalent inductance of series-connected inductors is the sum of the
individual inductances.
 Consider a series connection of N inductors
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
L 1 L 2 L n
19
Chapter 6: Capacitors and Inductors
V. Series and parallel inductors
 Applying KCL:
    
N
i i i i i
1 2 3
...
v
i
         
t t t
N
Nt t t
i vdt i t vdt i t vdt i t
L L L
0 0 0
1 0 2 0 0
1 2
1 1 1
( ) ( ) ... ( )
   
eq
N
L
L L L
1 2
1 1 1
...
+
-
Leq
v
-
 The equivalent inductance of parallel inductors is reciprocal of the sum of the
reciprocals of the individual inductances.
 Consider a parallel connection of N inductors
i1 i2 i1
+
 
 
     
 
  
t tN N
k
k kk eqt t
i vdt i t vdt i t
L L
0 0
0 0
1 1
1 1
( ) ( )
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
20
Chapter 6: Capacitors and Inductors
V. Series and parallel inductors
        
n n
i i i i i i mA
1 2 2 1
(0) (0) (0) 4 ( 1) 5
 From i1(t) = 4.(2 - e
-10t) mA i1(0) = 4.(2 - 1) = 4mA
Ex 6.7: Find i2(0), v(t), v1(t), v2(t), i2(t), in(t) if i1(t) = 4.(2 - e
-10t) mA and in(0) = -1mA.
L 1
L 2 L n
i1
-
+ v1
2H
v 4H 12H
-
+
v2
+ -
 The equivalent inductance is
     

eq N
L L L L H
2 1
4.12
/ / 2 5
4 12
 Thus:
     t t
eq
di
v t L e e mV
dt
10 101( ) 5.( 4).( 10) 200
     t t
di
v t e e mV
dt
10 101
1
( ) 4 4.( 4).( 10) 80
 Thus:  v t v v1 2( )
    tv t v t v t e mV10
2 1
( ) ( ) ( ) 120
i2 in
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
21
Chapter 6: Capacitors and Inductors
V. Series and parallel inductors
       
t t
t t
i t v dt i e dt e mA
10 10
2 2 2
0 0
1 120
( ) (0) 5 8 3 ( )
4 4
 Current i1
 Current i2
       
t t
t t
n n
i t v dt i e dt e mA
10 10
2
0 0
1 120
( ) (0) 1 ( )
12 12
L 1
L 2 L n
i1
-
+ v1
4H
v 4H 12H
-
+
v2
+ -
i2 in
Ex 6.7: Find i2(0), v(t), v1(t), v2(t), i2(t), in(t) if i1(t) = 4.(2 - e
-10t) mA and in(0) = -1mA.
 For validation:  
n
i t i t i t
1 2
( ) ( ) ( )
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
Ex 6.8: Find i2(0), i2(t), i(t), v(t), v1(t), v2(t) nếu i1(t) = 0,6.e
-2t
và i(0) = 1,4A
L 1
L 2
L n
22
Chapter 6: Capacitors and Inductors
V. Series and parallel inductors
i1
-+
v
v1
-
+
v2
+
-
i2
i
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
 Circuit elements (R, C) are available in either discrete form or integrated circuit
(IC) form, but inductance are difficult to produce on IC substrates.
23
Chapter 6: Capacitors and Inductors
VI. Applications
 The inductor are used in some applications:
 Relays, delays, sensing devices, pick-up head
 Telephone circuits, radio, TV receivers
 Power supplies, electric motors, microphones, loudspeakers
 Capacitors and inductors possess 03 special properties:
 Useful for generating a current or voltage in short period of time (DC circuit).
 Useful for suppression and converting pulsating DC voltage into relatively
smooth DC voltage (DC circuit).
 Useful for frequency discrimination (AC circuit).
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
VI.1. Integrator
24
Chapter 6: Capacitors and Inductors
VI. Applications
 An integrator is an op amp circuit whose output is
proportional to the integral of the input signal.
 In practice, note that:
 The op amp integrator requires a feedback resistor to reduce DC gain and
prevent saturation.
 The op amp operates within the linear range so that it does not saturate.
4
3
7
G N D
R 1
C
vi
-
+
v0
+
-
iR
iC

R C
i i
 At node 3:
    i
R C
v dv
i i C
R dt
0     
t
i
v t v v t dt
RC
0 0
0
1
( ) (0) ( )
  
t
i
v t v t dt
RC
0
0
1
( ) ( )
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
4
3
7
G N D
R 1
C
R 2
VI.1. Integrator
25
Chapter 6: Capacitors and Inductors
VI. Applications
Ex 6.9: Find vO in the op amp circuit if v1 = 10.cos2t (mV) and v2 = 0,5t (mV)
(assume that the voltage across the capacitor is initially zero)
3MΩ
 This is a summing integrator
   v v dt v dtR C R C0 1 2
1 2
1 1 100kΩ
2μF
VO
V2
V1
 
   
t t
v tdt tdt
0 6 6 3 6
0 0
1 1
10cos2 0,5
3.10 .2.10 100.10 .2.10
     
t
v t t t mV
2
2
0
1 10 1 0,5
sin2 0,833sin2 1,25 ( )
6 2 0,2 2
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
4
3
7
G N D
R
C
VI.2. Differentiator
26
Chapter 6: Capacitors and Inductors
VI. Applications
 An differentiator is an op amp circuit whose output is proportional to the rate of
change of the inputs signal.
 Note that:
 Differentiator circuits are electronically unstable because any electrical
noise within the circuit is exaggerated by the differentiator.
 Differentiator circuit is not as useful and popular as the integrator.
vi
-
+
v0
+
-
iC
iR

R C
i i
 Applying KCL at node a
    i
R C
v dv
i i C
R dt
0
  i
dv
v t RC
dt
0
( )
a
Fundamentals of Electric Circuits – Viet Son Nguyen - 2011
4
3
7
G N D
R 1
C
R 2
VI.2. Differentiator
27
Chapter 6: Capacitors and Inductors
VI. Applications
Ex 6.10: Find the output voltage with the given input. Take
vO = 0 at t = 0.
-
+
v0
  RC 3 6 35.10 .0,2.10 10 This is a differentiator with:
vi
5kΩ0,2μF
t (ms)
4
0 2
vi
64 8
 For 0 < t < 4ms or 4 < t < 8ms, the input voltage is:
   
 
    
i
t t ms t ms
v 
t t ms t ms
2 0 2 ,4 6
8 2 2 4 ,6 8
    
    
   
i
O
mV t ms t msdv
v RC 
dt mV t ms t ms
2 0 2 ,4 6
2 2 4 ,6 8
t (ms)
-2
0
2
vi
64 8
2