1
00:00:00,920 --> 00:00:05,595
>> In this video, we're going to
demonstrate how phasor diagrams

2
00:00:05,595 --> 00:00:09,360
or showing the phasors
in the complex plane can

3
00:00:09,360 --> 00:00:12,765
demonstrate the relative phases

4
00:00:12,765 --> 00:00:17,370
and magnitudes of voltages and
currents within a circuit.

5
00:00:17,370 --> 00:00:20,650
To do this, we're going
to remind ourselves that

6
00:00:20,650 --> 00:00:27,125
phasor V is equal to the
phasor I times the impedance.

7
00:00:27,125 --> 00:00:29,360
Now for example, in a resistor,

8
00:00:29,360 --> 00:00:33,605
the impedance is R and V then is simply

9
00:00:33,605 --> 00:00:37,980
equal to I times R. R is a real number.

10
00:00:37,980 --> 00:00:41,975
So there is no phase term associated
with the impedance of a resistor,

11
00:00:41,975 --> 00:00:45,680
and we say that the voltage is in

12
00:00:45,680 --> 00:00:55,845
phase with the current in a resistor.

13
00:00:55,845 --> 00:01:05,220
So if we have just arbitrarily choose
the current to have a zero phase angle,

14
00:01:06,860 --> 00:01:10,520
then the phasor voltage

15
00:01:10,520 --> 00:01:12,440
associated with the resistor or the voltage

16
00:01:12,440 --> 00:01:14,885
across the resistor call it V sub r,

17
00:01:14,885 --> 00:01:18,069
will have the same angle as I,

18
00:01:18,069 --> 00:01:19,790
be in the same direction,

19
00:01:19,790 --> 00:01:24,875
but just have a different magnitude than I.

20
00:01:24,875 --> 00:01:30,990
In fact the magnitude of this
will be R times I. All right.

21
00:01:30,990 --> 00:01:34,415
Unlike in a resistor, a capacitor
introduces a phase difference,

22
00:01:34,415 --> 00:01:40,790
we know that V across the
capacitor is equal to I times Z.

23
00:01:40,790 --> 00:01:44,620
Well, the impedance of
a capacitor is one over

24
00:01:44,620 --> 00:01:49,400
j Omega c. We bring that
j up in the numerator

25
00:01:49,400 --> 00:01:52,115
with a minus sign and we'll
have then this is equal to

26
00:01:52,115 --> 00:01:58,320
I times a negative j over Omega c. Well,

27
00:01:58,320 --> 00:02:06,230
that negative j in rectangular
coordinates is the same thing as

28
00:02:06,230 --> 00:02:10,430
a minus 90 degree phase term
in polar coordinates

29
00:02:10,430 --> 00:02:14,570
or we have then that V is equal

30
00:02:14,570 --> 00:02:24,660
to I times one over Omega c
times e to the minus j90.

31
00:02:24,970 --> 00:02:28,960
So whatever the phase of the current is,

32
00:02:28,960 --> 00:02:31,460
when we multiply the current
times the impedance there is

33
00:02:31,460 --> 00:02:34,790
a negative 90 degree phase term
that gets added to the current.

34
00:02:34,790 --> 00:02:41,750
So once again if we let the current
have a zero reference angle,

35
00:02:41,750 --> 00:02:47,480
then the voltage in a
capacitor is going to be

36
00:02:47,480 --> 00:02:51,620
90 degrees less than

37
00:02:51,620 --> 00:02:54,320
the phase of the current or we

38
00:02:54,320 --> 00:02:59,300
say the voltage is 90 degrees
behind the current,

39
00:02:59,300 --> 00:03:03,985
or we can say the current is
90 degrees ahead of the voltage.

40
00:03:03,985 --> 00:03:07,625
Finally, we look at the
voltage across the inductor,

41
00:03:07,625 --> 00:03:11,000
and we have phasor V. So that one,

42
00:03:11,000 --> 00:03:13,280
of course these are all
phasor terms over here.

43
00:03:13,280 --> 00:03:19,250
Phasor V sub l is equal to I times Z.

44
00:03:19,250 --> 00:03:24,350
Well the Z the impedance of
an inductor is j Omega L,

45
00:03:24,350 --> 00:03:31,620
which is equal to I times Omega L times j.

46
00:03:31,620 --> 00:03:38,970
Well, j is the same thing
as e to the positive j90.

47
00:03:39,160 --> 00:03:41,300
So in an inductor

48
00:03:41,300 --> 00:03:45,430
the voltage will be 90 degrees
ahead of the phase of the current.

49
00:03:45,430 --> 00:03:46,700
Whatever the phase of the current

50
00:03:46,700 --> 00:03:49,910
is and it gets multiplied by
the impedance of the inductor,

51
00:03:49,910 --> 00:03:52,510
which has this positive
90 phase term with it,

52
00:03:52,510 --> 00:03:54,740
that positive 90 gets added to

53
00:03:54,740 --> 00:03:57,020
the phase of the current
and the voltage will be

54
00:03:57,020 --> 00:04:01,790
90 degrees ahead of the current.

55
00:04:01,790 --> 00:04:06,320
So over here we had that being
I once again we'll reference I,

56
00:04:06,320 --> 00:04:11,605
just arbitrarily choosing
it to have a zero phase.

57
00:04:11,605 --> 00:04:18,709
I then V sub l will

58
00:04:18,709 --> 00:04:25,220
be 90 degrees ahead of phasor I.

59
00:04:25,220 --> 00:04:28,745
So we say that the current lags the voltage

60
00:04:28,745 --> 00:04:33,590
in an inductor or the voltage
leads the current in an inductor.

61
00:04:33,590 --> 00:04:37,610
Notice the voltages of the capacitor and

62
00:04:37,610 --> 00:04:42,080
the inductor are 180 degrees
out of phase with each other.

63
00:04:42,080 --> 00:04:44,915
The voltage and the current in the voltage

64
00:04:44,915 --> 00:04:48,170
of the capacitor lags by 90 degrees,

65
00:04:48,170 --> 00:04:52,645
the voltage in the inductor
leads the current by 90 degrees.

66
00:04:52,645 --> 00:04:55,410
There's a mnemonic that
goes along with this,

67
00:04:55,410 --> 00:04:59,870
today's clear back to the late 1800s
when much of electrical work that was

68
00:04:59,870 --> 00:05:05,050
being done was associated with
refrigeration and creating ice,

69
00:05:05,050 --> 00:05:12,730
and the mnemonic is ELI, the ICE man.

70
00:05:12,970 --> 00:05:16,895
What this is suggesting is that E

71
00:05:16,895 --> 00:05:20,620
represents voltage as in
physics we talk about the EMF.

72
00:05:20,620 --> 00:05:22,105
So E is voltage,

73
00:05:22,105 --> 00:05:25,015
I is current, L is an inductor.

74
00:05:25,015 --> 00:05:27,145
So in an inductor,

75
00:05:27,145 --> 00:05:33,905
the voltage leads
the current by 90 degrees.

76
00:05:33,905 --> 00:05:40,010
In an inductor, the voltage
leads the current by 90 degrees.

77
00:05:40,010 --> 00:05:47,720
In a capacitor, the current
leads the voltage by 90 degrees.

78
00:05:47,720 --> 00:05:49,755
So in the capacitor c,

79
00:05:49,755 --> 00:05:51,750
the current is 90 degrees ahead.

80
00:05:51,750 --> 00:05:53,550
So as in the word ICE,

81
00:05:53,550 --> 00:05:59,370
I comes ahead of E. So I is
leading the voltage over here.

82
00:05:59,370 --> 00:06:03,840
In ELI, the voltage is leading the current.

83
00:06:05,860 --> 00:06:09,860
Now let's demonstrate how
these phasor diagrams can be

84
00:06:09,860 --> 00:06:13,100
used to demonstrate the various or

85
00:06:13,100 --> 00:06:19,850
the relative phasors of different voltages
and currents within a circuit.

86
00:06:19,850 --> 00:06:24,770
Here we have a single loop circuit
and writing one KVL,

87
00:06:24,770 --> 00:06:32,130
we know then that phasor
V sub s is equal to.

88
00:06:33,470 --> 00:06:39,650
Well, first of all V sub s sum of
the voltage is equal to V sub r,

89
00:06:39,650 --> 00:06:41,480
plus V sub c,

90
00:06:41,480 --> 00:06:48,830
plus V sub l and of course
these are all phasor voltages.

91
00:06:48,830 --> 00:06:53,150
Now let's go ahead and put
in that V sub r is equal

92
00:06:53,150 --> 00:06:58,295
to I times R. In this case,

93
00:06:58,295 --> 00:07:00,900
R is eight ohms

94
00:07:02,030 --> 00:07:07,340
plus the voltage across the capacitor
is just going to be I times Z sub c,

95
00:07:07,340 --> 00:07:14,160
which is I times a negative j8 plus
the voltage across the inductor.

96
00:07:14,160 --> 00:07:16,070
That's a J2 ohm inductor.

97
00:07:16,070 --> 00:07:22,185
So the voltage across
that will be I times J2.

98
00:07:22,185 --> 00:07:25,010
So we have three terms here.

99
00:07:25,010 --> 00:07:27,800
Three different phasors.

100
00:07:27,800 --> 00:07:30,830
These are phasor voltage is equal to 8I,

101
00:07:30,830 --> 00:07:35,580
minus J8I, and positive J2I.

102
00:07:36,670 --> 00:07:40,180
Showing this explicitly with the phases,

103
00:07:40,180 --> 00:07:42,860
this will be I times eight.

104
00:07:42,860 --> 00:07:45,140
There is no phase difference
between the voltage

105
00:07:45,140 --> 00:07:48,260
across the resistor and the current.

106
00:07:48,260 --> 00:07:53,600
On the other hand we have I times eight

107
00:07:53,600 --> 00:07:58,770
times e to the minus j90 in the capacitor,

108
00:07:58,770 --> 00:08:09,730
plus I times two times e to
the positive j90 for the inductor.

109
00:08:09,730 --> 00:08:12,555
So again, what this is saying is that

110
00:08:12,555 --> 00:08:20,540
the phase of the voltage

111
00:08:20,540 --> 00:08:25,640
across the inductor is 90 degrees
greater than the phase of the current.

112
00:08:25,640 --> 00:08:30,590
On the other hand, the phase
of the voltage across

113
00:08:30,590 --> 00:08:36,049
the capacitor is 90 degrees
behind the phase of the current.

114
00:08:36,049 --> 00:08:42,110
So let's just start out and
say that this is the current.

115
00:08:42,110 --> 00:08:45,620
I will say that it has
a magnitude I naught.

116
00:08:45,620 --> 00:08:51,574
The voltage across the resistor
is eight times that

117
00:08:51,574 --> 00:08:58,670
or V sub r is equal to eight
times I naught in length,

118
00:08:58,670 --> 00:09:03,110
and it has the same angle as I.

119
00:09:03,110 --> 00:09:09,650
The capacitor on the other hand
is 90 degrees behind the current.

120
00:09:09,650 --> 00:09:14,059
So the voltage V sub c is 90 degrees

121
00:09:14,059 --> 00:09:20,775
behind the current V sub c,

122
00:09:20,775 --> 00:09:29,570
and it's down here at
negative 8I naught, and the voltage.

123
00:09:29,570 --> 00:09:31,835
Let's see that's the
voltage of the capacitor,

124
00:09:31,835 --> 00:09:40,560
and the voltage of the inductor
is 90 degrees ahead of V sub l,

125
00:09:42,080 --> 00:09:48,230
the length of it is equal to two times

126
00:09:48,230 --> 00:09:53,930
I naught and it is at an angle of
90 degrees ahead of the current.

127
00:09:53,930 --> 00:09:56,035
So our three voltages.

128
00:09:56,035 --> 00:09:59,645
The voltage across the resistor
is in phase with the current,

129
00:09:59,645 --> 00:10:02,990
capacitor is 90 degrees
behind the current and

130
00:10:02,990 --> 00:10:06,380
the inductor voltage is 90 degrees ahead.

131
00:10:06,380 --> 00:10:08,785
Now we have here that V sub s,

132
00:10:08,785 --> 00:10:12,980
the source voltage is equal to
the sum of those three terms.

133
00:10:12,980 --> 00:10:14,870
So when adding phasors,

134
00:10:14,870 --> 00:10:17,660
you do it just like you do vectors
and that is tip to tell them.

135
00:10:17,660 --> 00:10:22,280
Here's V sub r, add in V sub c,

136
00:10:22,280 --> 00:10:29,590
that comes down here like that
and then add in V sub l. That

137
00:10:29,590 --> 00:10:37,909
brings us back two units back that way
to this point here at negative 6I0,

138
00:10:38,080 --> 00:10:43,555
and so we end up at that point
and that is V sub s then.

139
00:10:43,555 --> 00:10:46,360
Let see how good of a job I
can do drawing a vector down,

140
00:10:46,360 --> 00:10:48,160
they're not very good.

141
00:10:48,160 --> 00:10:51,200
That's a straight line vector.

142
00:10:51,870 --> 00:10:55,165
That's better I guess.

143
00:10:55,165 --> 00:11:01,905
So V sub s then is equal to V sub r,

144
00:11:01,905 --> 00:11:03,480
plus V sub c, plus

145
00:11:03,480 --> 00:11:09,795
V sub l. This results in phasor
is V sub s. What does it tell us?

146
00:11:09,795 --> 00:11:13,910
Well, it tells us that V sub s is something

147
00:11:13,910 --> 00:11:18,920
less than 90 degrees
behind the current or that

148
00:11:18,920 --> 00:11:22,310
the current in this circuit I is leading

149
00:11:22,310 --> 00:11:28,790
the voltage source by some angle
less than 90 degrees.

150
00:11:28,790 --> 00:11:37,050
It also tells us the relative length of
this voltage is longer than V sub r,

151
00:11:37,060 --> 00:11:47,230
it's shorter than V sub c because
V sub c and V sub l are opposites.

152
00:11:49,610 --> 00:11:54,570
I guess to be more accurate if
you to say that V sub c and

153
00:11:54,570 --> 00:11:59,465
V sub l are in opposite directions
and they tend to cancel each other.

154
00:11:59,465 --> 00:12:01,910
Let me say that one more
time just a little bit

155
00:12:01,910 --> 00:12:04,295
differently and see if I can
make it a little more clear.

156
00:12:04,295 --> 00:12:07,790
V sub s the source voltage is equal to

157
00:12:07,790 --> 00:12:11,400
the phasor sum of V sub l, plus V sub r,

158
00:12:11,400 --> 00:12:16,500
plus V sub c. By diagramming
or drawing in V sub l,

159
00:12:16,500 --> 00:12:19,665
V sub r and V sub c and then
adding them tip to tail,

160
00:12:19,665 --> 00:12:23,600
we can see the relative length and phase

161
00:12:23,600 --> 00:12:27,740
of the source to the voltage
across the resistor,

162
00:12:27,740 --> 00:12:29,705
which is in phase with the current.

163
00:12:29,705 --> 00:12:33,340
The relative length and

164
00:12:33,340 --> 00:12:36,980
the relative angle of the source
relative to the voltage across

165
00:12:36,980 --> 00:12:40,370
the inductor and the relative length and

166
00:12:40,370 --> 00:12:46,720
relative phase of the source relative
to the voltage across the capacitor.

167
00:12:46,720 --> 00:12:48,980
Again, this doesn't give
us a real accurate,

168
00:12:48,980 --> 00:12:52,640
but it does give us some feel
for the relative phases,

169
00:12:52,640 --> 00:12:55,520
the phases of one of each of

170
00:12:55,520 --> 00:12:59,880
these voltages relative to the current
and relative to the source.