WEBVTT

00:00:00.200 --> 00:00:03.690
>> In the last few videos, we've been
developing the tools that we need

00:00:03.690 --> 00:00:07.105
to analyze these circuits that are
being driven by sinusoidal sources.

00:00:07.105 --> 00:00:10.675
Specifically, we've developed the construct

00:00:10.675 --> 00:00:12.660
of the mathematical tool of a phasor,

00:00:12.660 --> 00:00:15.330
we've introduced the concepts of impedance,

00:00:15.330 --> 00:00:17.580
and now we're ready to move on and show how

00:00:17.580 --> 00:00:21.750
Kirchhoff's and Ohm's Law is can be
used in terms of phasors impedances,

00:00:21.750 --> 00:00:23.550
to analyze these types of circuits.

00:00:23.550 --> 00:00:24.570
My name is Lee Brinton.

00:00:24.570 --> 00:00:25.830
I'm an Electrical Engineering Instructor

00:00:25.830 --> 00:00:28.090
in Salt Lake Community College.

00:00:29.780 --> 00:00:36.410
Specifically, we're going to derive or
demonstrate how Kirchhoff's voltage and

00:00:36.410 --> 00:00:39.500
current laws apply in these
circuits that are being

00:00:39.500 --> 00:00:43.225
driven by sinusoidal sources,

00:00:43.225 --> 00:00:46.070
and we'll be analyzing it
in this phasor domain,

00:00:46.070 --> 00:00:50.570
and then we'll give
an example of how these laws

00:00:50.570 --> 00:00:51.770
and these tools that we've developed at

00:00:51.770 --> 00:00:55.330
this point are used to analyze circuits.

00:00:55.330 --> 00:00:57.955
First of all, Kirchhoff's Voltage Law.

00:00:57.955 --> 00:01:00.380
We have here a circuit defined involving

00:01:00.380 --> 00:01:06.385
three different devices with a voltage
reference here plus to minus v_1 of t,

00:01:06.385 --> 00:01:08.310
plus to minus v_2 of t,

00:01:08.310 --> 00:01:11.040
and plus to minus v_3 of t. We know from

00:01:11.040 --> 00:01:12.860
Kirchhoff's Voltage Law
that the sum of the voltage

00:01:12.860 --> 00:01:15.095
drops around that loop must equal 0,

00:01:15.095 --> 00:01:21.195
or v_1 of t plus v_2

00:01:21.195 --> 00:01:27.380
of t plus v_3 of t must all add to be 0.

00:01:27.380 --> 00:01:32.405
Now, if we assume that we're operating
in the sinusoidal steady-state,

00:01:32.405 --> 00:01:36.745
and we've asserted that in
that type of a circuit,

00:01:36.745 --> 00:01:40.010
all of the voltages and
currents associated with

00:01:40.010 --> 00:01:43.685
that circuit will be oscillating
at the same frequency.

00:01:43.685 --> 00:01:46.580
Let's just say that
the source is oscillating at

00:01:46.580 --> 00:01:49.995
some Omega radians per second,

00:01:49.995 --> 00:01:52.670
that means that v_1, v_2
and v_3 will also be

00:01:52.670 --> 00:01:55.130
oscillating at that same frequency,

00:01:55.130 --> 00:01:57.605
but they'll have different amplitudes
and different phases.

00:01:57.605 --> 00:01:59.405
Or more specifically then,

00:01:59.405 --> 00:02:05.530
let's write it as v sub m1,

00:02:05.530 --> 00:02:10.620
cosine of Omega t plus Theta 1,

00:02:10.620 --> 00:02:13.790
so the amplitude of v_1 is v sub m1,

00:02:13.790 --> 00:02:16.085
and it has a phase of Theta 1,

00:02:16.085 --> 00:02:23.825
then plus v sub m2 cosine
Omega t, it's the same Omega.

00:02:23.825 --> 00:02:29.100
Omega t plus Theta 2 plus finally

00:02:29.100 --> 00:02:36.840
v sub m3 cosine Omega t plus Theta sub 3,

00:02:36.840 --> 00:02:40.115
and the sum of those three
terms has to equal 0.

00:02:40.115 --> 00:02:44.450
Now, let's represent these in
terms of phasors, and by that,

00:02:44.450 --> 00:02:49.830
we mean then that the real part
of whether this v sub

00:02:49.830 --> 00:02:57.345
m cosine Omega t plus Theta 1 can be
thought of as the real part of v sub m1,

00:02:57.345 --> 00:03:00.135
e to the j Theta 1,

00:03:00.135 --> 00:03:03.120
e to the j Omega t,

00:03:03.120 --> 00:03:06.200
plus the second term here
can be thought of as being

00:03:06.200 --> 00:03:09.790
the real part of v sub m2,

00:03:09.790 --> 00:03:13.200
e to the j Theta 2, that's a j,

00:03:13.200 --> 00:03:18.070
Theta 2, e to the j Omega t,

00:03:20.060 --> 00:03:25.695
plus the real part of v sub m3,

00:03:25.695 --> 00:03:28.829
e to the j Theta 3,

00:03:28.829 --> 00:03:34.770
e to the j Omega t, must equal 0.

00:03:34.770 --> 00:03:38.060
Now, realizing that we're talking
about the real part of all of

00:03:38.060 --> 00:03:41.770
these and that this e to the j Omega t
is common to all of them,

00:03:41.770 --> 00:03:46.435
we can rewrite this then
as the real part of,

00:03:46.435 --> 00:03:52.790
and also pointing out that that term
right there is just phasor v_1,

00:03:52.790 --> 00:03:57.380
and this right here is phasor v_2,

00:03:57.380 --> 00:04:03.330
and then of course right
there is phasor v_3.

00:04:03.330 --> 00:04:06.850
We can now rewrite this as the real part of

00:04:06.850 --> 00:04:16.290
phasor v_1 plus phasor v_2 plus phasor v_3,

00:04:16.290 --> 00:04:20.380
e to the j Omega t,

00:04:21.620 --> 00:04:24.270
and that must equal 0.

00:04:24.270 --> 00:04:26.890
Well, we know that e to
the j Omega t doesn't equal 0,

00:04:26.890 --> 00:04:29.110
therefore the sum of the phasors v_1,

00:04:29.110 --> 00:04:31.645
v_2, and v_3, must equal 0.

00:04:31.645 --> 00:04:36.399
Or specifically, phasor v_1 plus

00:04:36.399 --> 00:04:43.205
phasor v_2 plus phasor v_3 must equal 0.

00:04:43.205 --> 00:04:44.915
This is incredibly important.

00:04:44.915 --> 00:04:46.820
What that's saying is that up here,

00:04:46.820 --> 00:04:48.570
we have this circuit
operating in the time domain,

00:04:48.570 --> 00:04:51.800
we know that the sum of
those three voltages has to equal zero.

00:04:51.800 --> 00:04:55.160
What this demonstrates is
that not only must the sum

00:04:55.160 --> 00:04:58.240
of the time signals around
that closed loop equals zero,

00:04:58.240 --> 00:05:01.550
but the sum of the phasor
representations of

00:05:01.550 --> 00:05:05.495
each of these voltages
must also equal zero.

00:05:05.495 --> 00:05:08.045
In other words, and this
is the final thing,

00:05:08.045 --> 00:05:12.780
Kirchhoff's Voltage Laws apply or Law,

00:05:12.780 --> 00:05:15.620
singular, the Kirchhoff's
Voltage Law applies in

00:05:15.620 --> 00:05:21.470
this phasor domain in exactly the same way
that it does in the time domain.

00:05:21.470 --> 00:05:24.745
Now, let's take a look at
Kirchhoff's Current Law.

00:05:24.745 --> 00:05:27.730
We have a similar set of circumstances here

00:05:27.730 --> 00:05:31.360
where the sum of the currents i_1 of t

00:05:31.360 --> 00:05:37.690
plus i_2 of t plus i_3 of t, must equal 0.

00:05:37.690 --> 00:05:39.590
I'm not going to take
the time right now to do it,

00:05:39.590 --> 00:05:41.830
suffice it to say that
the same type of analysis

00:05:41.830 --> 00:05:44.350
that we just did for
Kirchhoff's Voltage Law,

00:05:44.350 --> 00:05:47.230
results in the sum of

00:05:47.230 --> 00:05:52.120
the phasor currents associated with

00:05:52.120 --> 00:05:57.810
the node must also equal zero.

00:05:57.810 --> 00:06:01.484
So once again, in terms of
impedances and phasors,

00:06:01.484 --> 00:06:05.120
the sum of the phasors or the sum of

00:06:05.120 --> 00:06:10.430
the currents represented in
the phasor form must equal zero also.