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Electrical Overview
Ref: Woud 2.3
N.B. this is a long note and
repeats much of what is is the text
Q
Q
=
charge
C
=
1
C
C
=
1 coul
I
=
t
(2.50)
t
=
time
min
=
60 s
s
=
1
s
I
=
current
A
=
1
A
work done per unit charge = potential difference two points
U
=
volts
V
A1
W
1V 1
⋅
=
=
1
V
aka electromotive force (EMF)
source, resistance, inductance, capacitance
resistance
⋅
Power
U t
()
It
()
⋅
A
1 watt
=
=
(2.51)
friction in mechanical system
resistance
=
R
=
Ω
1
Ω
ohm
=
1
Ω
Ohm's law
(2.52)
Ut
() It
()R
⋅
=
2
2
power in a resistor .
..
(2.53)
Power
U t
()It
⋅
⋅
R
1
Ω
(1A)
=
1W
=
=
⋅
⋅
⌠
⎮
⎮
⌡
⋅
⋅
inductance
mass of inertia in mechanical system
Vs
H
⋅
⋅
0
⋅
U
I
LI
HA
⋅
inductance
=
L
H
1
henry
1 H
=
=
t
A
d
dt
(2.54)
() L It
1V
or .
..
d
1A
t
=
⋅
=
=
=
L
H
s
⎛
⎝
⎞
⎠
(2.55)
A
d
I
P
=
=
=
dt
s
t
I
I
t
⋅
⌠
⎮
⌡
⋅
⌠
⎮
⌡
⋅
⋅
⌠
⎮
⎮
⌡
⌠
⎮
⌡
0
0
0
0
capacitance
spring in mechanical system
⎛
⎝
⎞
⎠
1
2
I
d
I
(2.56)
inductive_energy_stored
E
ind
Pt
() d
d
dI
dI
⋅
L
t
t
=
=
=
=
⋅
→
2
dt
2
A
⋅
H
1J
=
capacitance
=
C
F
=
F
farad
1 F
=
t
⋅
⋅
⋅
⌠
⎮
⎮
⌡
As
0
d
FV
V
⋅
V
U I
⋅
d
dt
P
⋅
F
=
(2.57)
() C
⋅
⋅
1A
or .
..
CU
d
1V
t
=
=
=
=
C
F
s
=
=
dt
s
t
U
U
t
1
2
U
(2.58), (2.59)
⋅
⌠
⎮
⌡
⋅
⌠
⎮
⌡
⋅
⋅
⌠
⎮
⎮
⌡
⌠
⎮
⌡
d
0
0
0
0
1
11/13/2006
capacitive_energy_stored
E
cap
dU
dU
⋅
C
t
t
=
=
=
=
⋅
→
2
dt
2
V
⋅
F
=
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View Full Document Kirchhoff's laws
first .
..
number_of_currents
sum_of_currents_towards_node
=
0
∑
⎣
I
i
()
t
⎦
=
0
(2.60)
second .
..
i
=
1
sum_of_voltages_around_closed_path
=
0
direction specified
number_of_voltages
∑
⎣
U
i
t
⎦
=
0
(2.61)
i
=
1
series connection of resistance and inductance .
..
imposed .
.. external
Ut
()
:=
U
m
⋅
cos
(
ω⋅
t
)
U
m
=
amplitude_of_voltage
V
=
1 V
(2.62)
1
ω
=
frequency
Hz
=
1
s
t
=
time
min
=
60 s
resulting current assumed also harmonic
It
:=
I
m
⋅
cos
(
t
−
φ
)
I
m
=
amplitude_of_current
A
=
1 amp
(2.63)
φ
=
phase_lag_angle
it is useful to represent this parameters as vectors using complex notation, where the values are represented by the real
parts
Uz t
:=
U
m
⋅
cos
(
t
)
+
U
m
⋅
sin
(
t
)
⋅
i
Iz t
:=
I
m
⋅
cos
(
t
−
φ
)
+
I
m
⋅
cos
(
t
−
φ
)
⋅
i
plotting set up
0
0.5
1
Uz(t)
Iz(t)
Imaginary parts of Uz(t), Iz(t)
0
0.5
1
Real parts of Uz(t), Iz(t) = U(t), I(t)
over R voltage drop will be .
..
U
R
t
:=
⋅
→
⋅
⋅
(
−ω
)
⋅
t
=
⋅
⋅
(
)
(
−α
)
R I t
RI
m
cos
⎣
+ φ
⎦
U
R
t
R I
m
cos
t
− φ
cos
α
=
cos
over L voltage drop will be .
..
U
L
t
:=
L
⋅
d
→
LI
⋅
m
⋅
sin
⎣
(
−ω
)
⋅
t
+
φ
⎦
⋅ω
dt
L
⋅
d
=
−
I
m
⋅ω⋅
L
⋅
sin
(
t
−
φ
)
=
I
m
L
⋅
cos
⎛
π
+ ω⋅
t
−
φ
⎞
dt
⎝
2
⎠
2
11/13/2006
⎛
π
⎞
→ −
()
⎛
π
⎞
⎛
π
⎞
⎛
π
⎞
cos
+ α
sin
α
cos
+ α
=
cos
⋅
cos
α
−
sin
⋅
sin
α
=
0 cos
α
−
⋅
⎝
2
⎠
⎝
2
⎠
⎝
2
⎠
⎝
2
⎠
or .
..
⋅
1 sin
α
in complex (vector) notation .
..
Uz
R
t
:=
R I
⋅
m
⋅
cos
(
ω⋅
t
− φ
)
+
RI
⋅
m
⋅
sin
(
t
− φ
)
⋅
i
⎛
π
⎞
⎛
π
⎞
Uz
L
t
:=
I
m
⋅ω⋅
L
⋅
cos
+ ω⋅
t
− φ
+
I
m
L
⋅
sin
+ ω⋅
t
− φ ⋅
i
⎝
2
⎠
⎝
2
⎠
plotting set up
0.2
0
0.2
0.4
0.6
0.8
1
Real parts of Uz(t), UzR(t), UzL(t) = U(t), UR(t), UL(t)
at this point these vectors are shown with two unknowns included I
m
and
φ
i.e. directions are correct relatively given
φ
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.611 taught by Professor Davidburke during the Fall '06 term at MIT.
 Fall '06
 DavidBurke

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