Transcript No Slide Title
CAPACITANCE AND INDUCTANCE
Introduces two passive, energy storing devices: Capacitors and Inductors CAPACITORS Store energy in their electric field (electrostatic energy) Model as circuit element INDUCTORS Store energy in their magnetic field Model as circuit element CAPACITOR AND INDUCTOR COMBINATIONS Series/parallel combinations of elements
CAPACITORS
First of the energy storage devices to be discussed
Typical Capacitors Basic parallel-plates capacitor CIRCUIT REPRESENTATION NOTICE USE OF PASSIVE SIGN CONVENTION
C
d A
Dielectric constant of material in gap PLATE SIZE FOR EQUIVALENT AIR-GAP CAPACITOR 55
F
8 .
85 1 .
10 12 016 10 4
A
A
6 .
3141 10 8
m
2 Normal values of capacitance are small.
Microfarads is common.
For integrated circuits nano or pico farads are not unusual
Basic capacitance law
Q
f
(
V C
)
Linear capacitors obey Coulomb’s law
Q
CV C
C is called the CAPACITANCE of the device and has units of
charge voltage
One Farad(F)is the capacitance of a device that can store one Coulomb of charge at one Volt.
Farad
Coulomb Volt
Linear capacitor circuit representation EXAMPLE Voltage across a capacitor of 2 micro Farads holding 10mC of charge
V C
1
C Q
1 2 * 10 6 10 * 10 3 5000
V Capacitance in Farads, charge in Coulombs result in voltage in Volts Capacitors can be dangerous!!!
Capacitors only store and release ELECTROSTATIC energy. They do not “create” The capacitor is a passive element and follows the passive sign convention Linear capacitor circuit representation
i
(
t
)
C dv
(
t
)
dt
Q C
CV C
Capacitance Law If the voltage varies the charge varies and there is a displacement current One can also express the voltage across in terms of the current … Or one can express the current through in terms of the voltage across
V C
(
t
) 1
C Q
1
C
t
i
C
(
x
)
dx
Integral form of Capacitance law
i
C
dQ
C dV
C
dt dt
Differential form of Capacitance law The mathematical implication of the integral form is ...
V C
(
t
)
V C
(
t
);
t
Voltage across a capacitor MUST be continuous Implications of differential form??
V C
Const
i C
0 DC or steady state behavior A capacitor in steady state acts as an OPEN CIRCUIT
CAPACITOR AS CIRCUIT ELEMENT
i C
v C
i C v C
(
t
) (
t
)
C dv c
(
t
)
i R v R
1
v R Ri R
LEARNING EXAMPLE
C
5
F
DETERMINE THE CURRENT
i
(
t
)
C dv
(
t
)
dt
60
mA i
5 10 6 [
F
] 24 6 10 3
V s i
(
t
) 0 20
mA elsewhere
CAPACITOR AS ENERGY STORAGE DEVICE
i C
Instantaneous power
p C
(
t
)
v C
(
t
)
i C
(
t
)
W
i
C
(
t
)
C dv
c v C
(
t
)
dt
v C
(
t
) 1
C
t
i C
(
x
)
dx
q C
(
t
)
C
p
C
(
t
)
Cv
C
(
t
)
dv
c
dt
p C
(
t
)
C d dt
1 2
v C
2 (
t
)
p C
(
t
) 1
C q C
(
t
)
dq C dt
Energy is the integral of power
w C
(
t
2 ,
t
1 )
t
1
t
2
p C
(
x
)
dx p C
(
t
) 1
C d dt
(
t
) 1 2
q c
2 (
t
) If t1 is minus infinity we talk about “energy stored at time t2.” If both limits are infinity then we talk about the “total energy stored.”
w C
(
t
2 ,
t
1 ) 1 2
Cv C
2 (
t
2 ) 1 2
Cv C
2 (
t
1 )
w C
(
t
2 ,
t
1 ) 1
C q
2
C
(
t
2 ) 1
C q
2
C
(
t
1 )
EXAMPLE
C
5
F
Energy stored in 0 - 6 msec
w C
( 0 , 6 ) 1 2
Cv C
2 ( 6 ) 1 2
Cv C
2 ( 0 )
w C
( 0 , 6 ) 1 2 5 * 10 6 [
F
] * ( 6 ) 2 [
V
2 ] Charge stored at 3msec
q C
( 3 )
Cv C
( 3 )
q C
( 3 ) 5 * 10 6 [
F
] * 12 [
V
] 60
C
C
4
F
.
FIND THE VOLTAGE
v
( 0 ) 0
v
(
t
)
v
( 0 ) 1
C
0
t i
(
x
)
dx
;
t
0
v
(
t
)
v
( 2 ) 1
C
2
t i
(
x
)
dx
;
t
2 0
t
2
v
(
t
) 2
t
8 10 3 [
V
] 2
t
4
ms
Flux lines may extend beyond inductor creating stray inductance effects
INDUCTORS NOTICE USE OF PASSIVE SIGN CONVENTION
Circuit representation for an inductor A TIME VARYING FLUX CREATES A COUNTER EMF AND CAUSES A VOLTAGE TO APPEAR AT THE TERMINALS OF THE DEVICE
A TIME VARYING MAGNETIC FLUX INDUCES A VOLTAGE
v
L
d
Induction law
dt
FOR A LINEAR INDUCTOR THE FLUX IS PROPORTIONAL TO THE CURRENT
Li
L
v L
L di L dt
DIFFERENTIAL FORM OF INDUCTION LAW THE PROPORTIONALITY CONSTANT, L, IS CALLED THE INDUCTANCE OF THE COMPONENT INDUCTANCE IS MEASURED IN UNITS OF henry (H). DIMENSIONALLY HENRY Volt Amp sec INDUCTORS STORE ELECTROMAGNETIC ENERGY.
THEY MAY SUPPLY STORED ENERGY BACK TO THE CIRCUIT BUT THEY CANNOT CREATE ENERGY.
THEY MUST ABIDE BY THE PASSIVE SIGN CONVENTION Follow passive sign convention
v
L
L di
L
dt
Differential form of induction law
i L
(
t
) 1
L t
v L
(
x
)
dx
Integral form of induction law
i L
(
t
)
i L
(
t
0 ) 1
L
0
t t
v L
(
x
)
dx
; A direct consequence of integral form
i L
(
t
)
i L
(
t
);
t t
t
0 Current MUST be continuous A direct consequence of differential form
i
L
Const
.
v
L
0
Power and Energy stored DC (steady state) behavior
p L
(
t
)
v L
(
t
)
i L
(
t
)
w L
(
t
2 ,
t
1 )
t
1
t
2
d dt
W 1 2
Li L
2 (
x
)
dx
p L
(
t
)
L di L dt
(
t
)
i L
(
t
)
d dt
J 1 2
Li L
2 (
t
) Current in Amps, Inductance in Henrys yield energy in Joules
w
(
t
2 ,
t
1 ) 1 2 2
Li L
(
t
2 ) 1 2 2
Li L
(
t
1 )
w L
(
t
) 1 2
Li L
2 (
t
) Energy stored on the interval Can be positive or negative
EXAMPLE FIND THE TOTAL ENERGY STORED IN THE CIRCUIT In steady state inductors act as short circuits and capacitors act as open circuits
W C
1 2
CV C
2
W L
1 2
LI L
2 @
A A
V A
9
V A
6
9
0
V
A
81 [
V
5 ]
I L
1
3
A
I L
2
I L
1
1.2
A V C
1
I L
1
V C
1
16.2
V V C
2
6 6
3
V A
10.8
V I L
2
V A
9
1.8
A
EXAMPLE L=10mH. FIND THE VOLTAGE
v
(
t
)
L di dt
(
t
) ENERGY STORED BETWEEN 2 AND 4 ms
m
20 10 3 2 10 3
s A
10
A s
m
10
A s
THE DERIVATIVE OF A STRAIGHT LINE IS ITS SLOPE
di dt
10 ( 10 (
A A
0 / /
s
)
s
) 0 2
t t
elsewhere
2
ms
4
ms di dt L
(
t
) 10 10 (
A
10 3 /
s H
)
v
(
t
) 100 10 3
V
100
mV w
( 4 , 2 ) 1 2
Li L
2 ( 4 ) 1 2
Li L
2 ( 2 )
w
( 4 , 2 ) 0 0 .
5 * 10 * 10 3 ( 20 * 10 3 ) 2 J THE VALUE IS NEGATIVE BECAUSE THE INDUCTOR IS SUPPLYING ENERGY PREVIOUSLY STORED
CAPACITOR SPECIFICATIONS CAPACITANC E RANGE
p F
IN STANDARD VALUES
C
50
mF
STANDARD CAPACITOR RATINGS 6 .
3
V
500
V
STANDARD TOLERANCE 5 %, 10 %, 20 % INDUCTOR SPECIFICATIONS INDUCTANCE RANGES 1
nH
IN STANDARD VALUES
L
100
mH
STANDARD INDUCTOR RATINGS
mA
1
A
STANDARD TOLERANCE 5 %, 10 %
C
v
i L i
v
IDEAL AND PRACTICAL ELEMENTS
i
(
t
)
i
(
t
)
i
(
t
)
i
(
t
)
v
(
t
) IDEAL ELEMENTS
v
(
t
)
i
(
t
)
C dv
(
t
)
dt v
(
t
)
L di dt
(
t
)
v
(
t
)
v
(
t
)
i
( CAPACITOR/INDUCTOR MODELS INCLUDING LEAKAGE RESISTANCE
t
)
v
(
t
)
R leak
C dv
(
t
)
dt
MODEL FOR “LEAKY” CAPACITOR
v
(
t
)
R leak i
(
t
)
L di dt
(
t
) MODEL FOR “LEAKY” INDUCTORS
SERIES CAPACITORS NOTICE SIMILARITY WITH RESITORS IN PARALLEL
C s
C C
1
C
2 1
C
2 Series Combination of two capacitors 6
F
3
F C S
2
F
i
(
t
) PARALLEL CAPACITORS
i k
(
t
)
C k dv
(
t
)
dt C P
F
SERIES INDUCTORS
v k
(
t
)
L k di dt
(
t
)
L eq
7
H v
(
t
)
L S di dt
(
t
)
PARALLEL INDUCTORS
i
(
t
)
i
(
t
0 )
j N
1
i j
(
t
0 ) 4
mH i
(
t
0 ) 3
A
6
A
2
A
1
A
INDUCTORS COMBINE LIKE RESISTORS CAPACITORS COMBINE LIKE CONDUCTANCES 2
mH
LEARNING EXAMPLE FLIP CHIP MOUNTING IC WITH WIREBONDS TO THE OUTSIDE GOAL: REDUCE INDUCTANCE IN THE WIRING AND REDUCE THE “GROUND BOUNCE” EFFECT A SIMPLE MODEL CAN BE USED TO DESCRIBE GROUND BOUNCE
MODELING THE GROUND BOUNCE EFFECT
L ball
0 .
1
nH V GB
(
t
)
L ball di G dt
(
t
)
m
40 40 10 3 10 9
A s
IF ALL GATES IN A CHIP ARE CONNECTED TO A SINGLE GROUND THE CURRENT CAN BE QUITE HIGH AND THE BOUNCE MAY BECOME UNACCEPTABLE USE SEVERAL GROUND CONNECTIONS (BALLS) AND ALLOCATE A FRACTION OF THE GATES TO EACH BALL