ESE 271 / Spring 2013 / Lecture 9
Last time : energy storage elements ‐ capacitor.
Charge on plates
Energy stored in the form of electric field
Passive sign Passive
sign
convention
Voltage drop across real capacitor can not change abruptly V
lt
d
l
it
t h
b tl
because instant voltage change means instant change of accumulated charge and this, in turn, requires infinite current.
1
ESE 271 / Spring 2013 / Lecture 9
Last time : capacitor charge/discharge.
Charging capacitor by practical DC source
Energy gets stored in the capacitor
the capacitor
Discharging capacitor
Energy previously stored in the capacitor gets dissipated in resistor
2
ESE 271 / Spring 2013 / Lecture 9
Energy can be stored in circuit element in the form of magnetic field.
Ampere’s law – magnetic field created by electric current in vacuum
magnetic flux density
Coil of wire can be used to store energy in the form of magnetic field
i h f
f
i fi ld
Biot‐Savart law:
3
ESE 271 / Spring 2013 / Lecture 9
Magnetic flux density generated by current in the coil of wire.
Magnetic field in center generated by
Magnetic field in center generated by full coil of wire
Biot‐Savart law:
FFor circuit element containing N i it l
t
t i i N
coils and in media with magnetic permeability
In every case depends on geometry but always:
4
ESE 271 / Spring 2013 / Lecture 9
Inductance.
Number of turns inside the closed path
Closed path for Ampere’s law calculation
Total number of turns in solenoid
f
Length of solenoid
Magnetic flux density inside solenoid core:
Magnetic flux inside solenoid core:
Inductance (self‐inductance) relates the magnetic flux to the current that created it.
Depends on geometry of the circuit element.
5
ESE 271 / Spring 2013 / Lecture 9
Linear inductor.
Energy stored in inductor
Recall that in parallel plate capacitor with plate area A and spacing d:
Volume energy density of magnetic field:
, where volume energy density l
Energy stored in solenoid of length and cross‐section of the core A:
In solenoid we just considered:
6
ESE 271 / Spring 2013 / Lecture 9
Voltage drop across ideal linear inductor.
When DC current is flowing through ideal inductor the voltage drop across it is zero.
What happens if the current is changing with time?
Magnetic flux:
Magnetic flux:
Magnetic flux change due to change of current:
F d ’ l
Faraday’s law –
electromotive force:
l t
ti f
Recall EMF in battery:
In inductor:
Change of current in inductor generates
inductor generates the voltage drop.
7
ESE 271 / Spring 2013 / Lecture 9
Voltage and current in inductor.
Passive sign convention
Voltage drop appears because the induced EMF force opposes the change of current
Assume:
moved charges in the direction opposite to the direction of current change
moved charges in the direction opposite to the direction of current change.
8
ESE 271 / Spring 2013 / Lecture 9
Inductor power and energy.
Initial current
Current in inductor implies presence of magnetic field, hence, current is associated with energy and energy can not be changed abruptly without infinite power.
Hence, current through practical inductor can not be changed abruptly since it would imply infinite voltage
would imply infinite voltage.
9
ESE 271 / Spring 2013 / Lecture 9
Let’s put the energy into inductor.
1. Simple ideal case:
2. More realistic case with practical power supply:
‐ Time constant of the RL‐circuit
10
ESE 271 / Spring 2013 / Lecture 9
Increase of current through inductor.
11
ESE 271 / Spring 2013 / Lecture 9
Increase of current through inductor by practical voltage source.
Obviously, everything is just the same since we used Thevenin form of the practical
Thevenin form of the practical source and it is equivalent to Norton one used on slide 11.
12
ESE 271 / Spring 2013 / Lecture 9
Decrease of current through inductor – removal of energy.
1. Simple ideal case:
See ignition coil in cars
coil in cars
2. Realistic case:
13
ESE 271 / Spring 2013 / Lecture 9
Series connection of inductors
Physical sense:
14
ESE 271 / Spring 2013 / Lecture 9
Parallel connection of inductors
Physical sense – current divider.
15
ESE 271 / Spring 2013 / Lecture 9
Example 1.
Assume DC A
DC
steady‐state
16
ESE 271 / Spring 2013 / Lecture 9
Example 2.
‐ Current trough inductor in DC steady‐state
‐ Energy stored in inductor
17
ESE 271 / Spring 2013 / Lecture 9
Example 2 – cont.
Energy stored in inductor:
Energy dissipated in the circuit during Energy
dissipated in the circuit during
transition from initial DC steady‐state to final DC steady‐state condition:
18
ESE 271 / Spring 2013 / Lecture 9
Example 2 – cont.
KVL:
19
ESE 271 / Spring 2013 / Lecture 9
Example 2 – cont.
KVL:
20
ESE 271 / Spring 2013 / Lecture 9
Example 2 – cont.
Find B from initial conditions:
21
ESE 271 / Spring 2013 / Lecture 9
Example 2 – cont.
22
ESE 271 / Spring 2013 / Lecture 9
Superposition.
23
ESE 271 / Spring 2013 / Lecture 9
Superposition ‐ cont.
24
ESE 271 / Spring 2013 / Lecture 9
Superposition ‐ cont.
25
ESE 271 / Spring 2013 / Lecture 9
Superposition ‐ cont.
26
ESE 271 / Spring 2013 / Lecture 9
Superposition ‐ cont.
27
ESE 271 / Spring 2013 / Lecture 9
Example.
DC steady‐state
y
Energy stored in inductor will get dissipated in resistor.
Eventually all voltages and currents will become zero since circuit will contain no sources and there is resistor that does not store but dissipates energy. To find the transient values of voltage and currents we will need to solve integro‐
differential equation – not an easy task, in general.
28