Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Capacitance and Inductance J Irwin
Typology: Lecture notes
1 / 58
This page cannot be seen from the preview
Don't miss anything!
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
RC OP-AMP CIRCUITS
Integration and differentiation circuits
Basic parallel-plates capacitor
CIRCUIT REPRESENTATION
NOTICE USE OF PASSIVE SIGN CONVENTION
Typical Capacitors
Basic capacitance law ( ) C
Q f V
Linear capacitors obey Coulomb’s law C
Q CV
C is called the CAPACITANCE of the device and has
units of
voltage
charge
One Farad(F)is the capacitance of a device that can
store one Coulomb of charge at one Volt.
Volt
Coulomb
Farad
EXAMPLE Voltage across a capacitor of 2 micro
Farads holding 10mC of charge
6
C V
Capacitance in Farads, charge in Coulombs
result in voltage in Volts
Capacitors can be dangerous!!!
Linear capacitor circuit representat
The capacitor is a passive element
and follows the passive sign convention
Capacitors only store and release
ELECTROSTATIC energy. They do not “create”
Linear capacitor circuit representation
( ) ( t )
dt
dv i t C
C
C
c
C
t
C C
t
t
t^ t
0
0
0
0
t (^) t
t
C C C
t
t
C C C
0
0
The fact that the voltage is defined through
an integral has important implications...
R R
R R
Ohm’s Law
c O v t
i ( t ) 0 elsewhere
( ) ( t )
dt
dv i t C
mA
s
i F 20
6 10
3
6
60 mA
p ( t ) v ( t ) i ( t ) C C C
Instantaneous power
( ) ( t )
dt
dv
i t C
c
C
dt
dv
p t Cv t
c
C C
( ) ( )
C
t
C C
C C C
Energy is the integral of power
2
1
( , ) ( ) 2 1
t
t
C C
w t t p x dx
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.”
( )
2
1
( )
2 v t
dt
d
p t C C C
( )
2
1
( )
2
1
( , ) 1
2
2
2
2 1
w t t Cv t Cv t C C C
( )
2
1 1
( )
2 q t
dt
d
C
p t C c
( )
1
( )
1
( , ) 1
2
2
2
2 1
q t
C
q t
C
w t t C C C
W
C
v
C
i
0 ) 0 ( v
0 t 2
2 t 4 ms
3 v t t V
0
i x dx t
C
v t v
t
2
i x dx t
C
v t v
t
0 ) 0 ( v
i t t
3 ( ) 8 10
p ( t ) 8 t , 0 t 2 ms
3
2 t 4 ms
p ( t ) 0 , elsewhere
LEARNING EXTENSION
( ) ( t )
dt
dv i t C
s
i F 3
6
s
i F 3
6
2
C
2 * 10 [ ]* 130 sin
2
Charge stored at a given time
C C
C
6
C V C
C
C
C
6
A
Electric power supplied to capacitor at a given time p ( t ) v ( t ) i ( t ) C C C
Energy stored over a given time interval
W
( )
2
1
( )
2
1
( , ) 1
2
2
2
2 1
w t t Cv t Cv t C C
J
C 2 F
v ( t )
WHAT VARIABLES CAN BE
COMPUTED?
SAMPLE PROBLEM
5 10 Compute voltage as a function of time At minus infinity everything is zero. Since current is zero for t<0 we have
s A t ms A i t
C
10 10 3 5 15 ( )
3 3 6
[ ] 4 * 10 3 * 10 ( 0 ) 0 ( ) 0 6 3 V V t xdx V t C C [ ]; 0 5 * 10 [ ] 8
3 t V t s In particular [ ] 8 75 [ ] 8 3 * 10 *( 5 * 10 ) ( 5 ) 3 3 2 V ms V mV C
C
t C C V ms mV V t A sdx 3 5 * 10 6 6 3 ( 10 * 10 )[ / ] 4 * 10 1 8 75 * 10 [ ] ( ) 8 75 ( 5 )
3 3 3 3
C
Charge stored at 5ms
C C
[ ] 8 75 * 10 ( 5 ) 4 * 10 [ ] * 3 6 q ms F V q ( 5 ms )( 75 / 2 )[ nC ] Total energy stored 2
C
Total means at infinity. Hence [ ] 8 25 * 10
C
2
c
[ mV ]
Formal description of a piecewise analytical signal
A TIME VARYING MAGNETIC FLUX
INDUCES A VOLTAGE
dt
d
v L
Induction law
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
FOR A LINEAR INDUCTOR THE FLUX IS
PROPORTIONAL TO THE CURRENT
L
Li
dt
di
v L
L
L
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
sec
Amp
Volt HENRY
LEARNING by Doing
Follow passive sign convention
dt
di
v L
L
L
Differential form of induction law
t
L L
Integral form of induction law
0 0
0
t
t
L L L
L L
A direct consequence of differential form (^). 0 L L
i Const v DC (steady state) behavior
Power and Energy stored
p ( t ) v ( t ) i ( t ) L L L
W^
( ) ( t ) i ( t )
dt
di p t L L
L L
Li t
dt
d
L
1
2
2
2
2 1
L L
Energy stored on the interval
Can be positive or negative
( )
2
1
( )
2
w t Li t L L
“Energy stored at time t”
Must be non-negative.
Passive element!!!
2
1
2
2 1
t
t
L L
Current in Amps, Inductance in Henrys
yield energy in Joules
