Download Review: Lecture 3 • Linear circuit • Superposition • Equivalent Circuits

Review: Lecture 3
• Linear circuit
• Superposition
• Equivalent Circuits
—Simple Circuits
—Y- Transforms
—Source transformation
—Thevenin's theorem
—Norton's theorem
• Maximum power transfer
Lecture 4 DC Circuits
Capacitors and Inductors
Objectives
• Review the basics of capacitors and inductors
• Series and parallel capacitors/inductors
• Applications
—Integrator
—Differentiator
—Analog Computer
Capacitors
Capacitors: fundamental passive components.
— Basic form: two conductive plates separated by an insulating dielectric
— Capacitance: the ability to store charge (electric potential energy)
The charging process…
Vs
The external source cannot move charge
any more: fully charged
The physical dimensions of
the capacitor determine the
charge ‘storage ability’
Conductive plate
dielectric
VS
+
+
+
+
+
+
+
+
+
+
A +
E










 B
A capacitor with stored charge can act as a temporary battery.
q  Cv
Parallel-plate Capacitors
The permittivity of
the dielectric
q A
C 
v d
The surface area
of the plates
The spacing
between plates
Circuit behaviour of capacitors
• Current and voltage relation
t
1
v   idt  v(t0 )
C t0
— q  Cv
— Linear element
dq
dv
i
C
dt
dt
•
v  kv0  i  C
Capacitor voltage depends on
the past history of the current
Circuit theorems valid
Superposition, Thevenin…
d kv0 
dv
 kC 0  ki0
dt
dt
—Passive sign convention:
• Being charged: v  0, i  0; v  0, i  0
Passive elements p=vi >0,
‘consume’ energy:
charging
• A capacitor is an open circuit to dc
—
v  const,
dv
 0, i  0
dt
Current is undefined.
• The voltage on a capacitor cannot change abruptly
—
iC
dv

dt
• Power and energy stored:
—
—
dv
Power: p  vi  Cv dt
Energy stored: w 
t
t
Ideal capacitors do not
dissipate energy
v (t )
dv
1 2
p
dt

Cv
dt


- dt 2 Cv v (  )
Series and Parallel Capacitors
• Parallel-plate capacitors
A
— C d
A
1
d d
1
1
d d d , C 
,


 
— Series:
d d C
A A C C
 (A  A )
— Parallel: A  A  A , C  d  C  C
1
T
1
2
1
2
T
1
T
1
2
T
2
1
2
T
1
2
• Behaviour consideration
t
— Series: KVL v  v1  v2  ...  vN
1 t
 1 t

 1 t

v    idt  v1 (t0 )    idt  v2 (t0 )  ...  
idt

v
(
t
)
N 0 

 C1 t0
  C t0

 C N t0

1
v   idt  v(t0 )
C t0
t
t
1 t

1
1
   idt   idt  ... 
idt
  v1 (t0 )  v2 (t0 )  ...  v N (t0 )
C t0
C N t0 
 C1 t0
t
1 1
1 
    ... 
 idt  v(t0 )
C N  t0
 C1 C
1
1 1
1
   ... 
CT C1 C
CN
— Parallel: KCL i  i1  i2  ...  iN
dv
dv
dv
 C2
 ...  C N
dt
dt
dt
dv
 C1  C2  ...  C N 
dt
i  C1
CT  C1  C2  ...  C N
2
iC
dv
dt
Inductors
Inductors: fundamental passive components.
— Basic form: a length of wire is formed into a coil
— Store energy in its magnetic field
Inductance is greatly magnified by
adding turns and winding them on a
magnetic core material.
Current
change
opposite
Air core
Iron core
number of turns
of wire
inductance
in henries
N 2 A
L
l
Ferrite core
permeability in H/m
(same as Wb/At-m)
Variable
Magnetic
field change
Induce
voltage
Circuit behaviour of inductors
• Current and voltage relation
di
vL
dt
t
1
i   v(t )dt  i (t0 )
L t0
—
— Linear element
•
i  i1  i2  v  L
d (i1  i2 )
di
di
 L 1  L 2  v1  v2
dt
dt
dt
—Passive sign convention:
• Energy stored: v  0, i  0; v  0, i  0
• An inductor is an short circuit to dc
—
i  const,
di
 0, v  0
dt
• The current on an inductor cannot change abruptly
—
vL
di
dt
• Power and energy stored:
di
p  vi  Li
dt
— Power:
— Energy stored:
t
t
Ideal inductors do not
dissipate energy
i (t )
di
1
w   pdt   Li dt  Li 2
dt
2
i (  )

-
Series and Parallel Inductors
• Series: KVL v  v
1
 v2  ...  v N
di
di
di
 L2  ...  LN
dt
dt
dt
di
 L1  L2  ...  LN 
dt
v  L1
Leq  L1  L2  ...  LN
• Parallel: KCL
i  i1  i2  ...  iN
1 t
 1 t

 1 t

i    vdt  i1 (t0 )    vdt  i2 (t0 )  ...    vdt  iN (t0 )
 L1 t0
  L2 t0

 LN t0

t
t
1 t

1
1
   vdt   vdt  ... 
vdt   i1 (t0 )  i2 (t0 )  ...  iN (t0 )

L2 t0
LN t0 
 L1 t0
t
1 1
1 
    ... 
 vdt  i (t0 )
LN  t0
 L1 L2
N
1
1

Leq n 1 Ln
Typical passive elements
As long as all the elements are of the same type, the
delta-wye transformation can be extended to C and L
Integrator
iR  iC
vi
iR 
R
iC  C
vi
dvo
 C
R
dt
dvo
dt
vi
dvo  
dt
RC
t
1
vo (t )  vo (0)  
vi dt

RC o
t
1
vo (t )  
vi dt

RC o
Assume vo(0)=0
Differentiator
iR  iC
vo
iR  
R
dvi
vo   RC
dt
iC  C
dvi
dt
Analog Computer
Analog Computer
t
vo (t )  
1
vi dt
RC o
0.5
Analog Computer
t
vo (t )  
1
vi dt
RC o
Analog Computer – after class practice
Evaluate the circuit:
Appendix
Practical issues of capacitors
• Leakage resistance
• Dielectric breakdown
— insulating materials forced to conduct
— sudden surge of current
—Burn, melt, vaporize: permanent damage
• Capacitor types
— paper, air, polycarbonate, polyester, polypropylene,
polystyrene, mixed, silvered mica,
electrolytic(aluminium, tantalum), ceramic…
• Capacitor labelling
—Small capacitor: 0.01.. Microfarads
—103: 10x103 pF: Third digit is the multiplier
Practical issues of inductors
• Winding resistance and
capacitance
• Inductor types
The winding resistance is in series
with the coil; the winding
capacitance is in parallel with both.
Encapsulated
Torroid coil
• Applications:
—Power supplies, transformers,
radios, TVs, radars, and electric
motors.
CW
RW
Variable
L
Applications
• Capacitors
— temporary voltage source
—frequency discrimination
• Inductors
— temporary current source
— spark/arc suppression
—Converting pulsating dc voltage into relatively
smooth dc
— frequency discrimination
Lecture 5 DC Circuits
Transient Circuits