L7 3 1 Kirchhoffs Laws With Phasors

>> 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.