>> In the last few videos, we've been developing the tools that we need
to analyze these circuits that are being driven by sinusoidal sources.
Specifically, we've developed the construct
of the mathematical tool of a phasor,
we've introduced the concepts of impedance,
and now we're ready to move on and show how
Kirchhoff's and Ohm's Law is can be used in terms of phasors impedances,
to analyze these types of circuits.
My name is Lee Brinton.
I'm an Electrical Engineering Instructor
in Salt Lake Community College.
Specifically, we're going to derive or demonstrate how Kirchhoff's voltage and
current laws apply in these circuits that are being
driven by sinusoidal sources,
and we'll be analyzing it in this phasor domain,
and then we'll give an example of how these laws
and these tools that we've developed at
this point are used to analyze circuits.
First of all, Kirchhoff's Voltage Law.
We have here a circuit defined involving
three different devices with a voltage reference here plus to minus v_1 of t,
plus to minus v_2 of t,
and plus to minus v_3 of t. We know from
Kirchhoff's Voltage Law that the sum of the voltage
drops around that loop must equal 0,
or v_1 of t plus v_2
of t plus v_3 of t must all add to be 0.
Now, if we assume that we're operating in the sinusoidal steady-state,
and we've asserted that in that type of a circuit,
all of the voltages and currents associated with
that circuit will be oscillating at the same frequency.
Let's just say that the source is oscillating at
some Omega radians per second,
that means that v_1, v_2 and v_3 will also be
oscillating at that same frequency,
but they'll have different amplitudes and different phases.
Or more specifically then,
let's write it as v sub m1,
cosine of Omega t plus Theta 1,
so the amplitude of v_1 is v sub m1,
and it has a phase of Theta 1,
then plus v sub m2 cosine Omega t, it's the same Omega.
Omega t plus Theta 2 plus finally
v sub m3 cosine Omega t plus Theta sub 3,
and the sum of those three terms has to equal 0.
Now, let's represent these in terms of phasors, and by that,
we mean then that the real part of whether this v sub
m cosine Omega t plus Theta 1 can be thought of as the real part of v sub m1,
e to the j Theta 1,
e to the j Omega t,
plus the second term here can be thought of as being
the real part of v sub m2,
e to the j Theta 2, that's a j,
Theta 2, e to the j Omega t,
plus the real part of v sub m3,
e to the j Theta 3,
e to the j Omega t, must equal 0.
Now, realizing that we're talking about the real part of all of
these and that this e to the j Omega t is common to all of them,
we can rewrite this then as the real part of,
and also pointing out that that term right there is just phasor v_1,
and this right here is phasor v_2,
and then of course right there is phasor v_3.
We can now rewrite this as the real part of
phasor v_1 plus phasor v_2 plus phasor v_3,
e to the j Omega t,
and that must equal 0.
Well, we know that e to the j Omega t doesn't equal 0,
therefore the sum of the phasors v_1,
v_2, and v_3, must equal 0.
Or specifically, phasor v_1 plus
phasor v_2 plus phasor v_3 must equal 0.
This is incredibly important.
What that's saying is that up here,
we have this circuit operating in the time domain,
we know that the sum of those three voltages has to equal zero.
What this demonstrates is that not only must the sum
of the time signals around that closed loop equals zero,
but the sum of the phasor representations of
each of these voltages must also equal zero.
In other words, and this is the final thing,
Kirchhoff's Voltage Laws apply or Law,
singular, the Kirchhoff's Voltage Law applies in
this phasor domain in exactly the same way that it does in the time domain.
Now, let's take a look at Kirchhoff's Current Law.
We have a similar set of circumstances here
where the sum of the currents i_1 of t
plus i_2 of t plus i_3 of t, must equal 0.
I'm not going to take the time right now to do it,
suffice it to say that the same type of analysis
that we just did for Kirchhoff's Voltage Law,
results in the sum of
the phasor currents associated with
the node must also equal zero.
So once again, in terms of impedances and phasors,
the sum of the phasors or the sum of
the currents represented in the phasor form must equal zero also.