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>> In the last few videos, we've been[br]developing the tools that we need

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to analyze these circuits that are[br]being driven by sinusoidal sources.

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Specifically, we've developed the construct

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of the mathematical tool of a phasor,

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we've introduced the concepts of impedance,

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and now we're ready to move on and show how

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Kirchhoff's and Ohm's Law is can be[br]used in terms of phasors impedances,

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to analyze these types of circuits.

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My name is Lee Brinton.

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I'm an Electrical Engineering Instructor

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in Salt Lake Community College.

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Specifically, we're going to derive or[br]demonstrate how Kirchhoff's voltage and

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current laws apply in these[br]circuits that are being

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driven by sinusoidal sources,

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and we'll be analyzing it[br]in this phasor domain,

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and then we'll give[br]an example of how these laws

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and these tools that we've developed at

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this point are used to analyze circuits.

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First of all, Kirchhoff's Voltage Law.

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We have here a circuit defined involving

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three different devices with a voltage[br]reference here plus to minus v_1 of t,

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plus to minus v_2 of t,

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and plus to minus v_3 of t. We know from

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Kirchhoff's Voltage Law[br]that the sum of the voltage

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drops around that loop must equal 0,

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or v_1 of t plus v_2

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of t plus v_3 of t must all add to be 0.

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Now, if we assume that we're operating[br]in the sinusoidal steady-state,

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and we've asserted that in[br]that type of a circuit,

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all of the voltages and[br]currents associated with

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that circuit will be oscillating[br]at the same frequency.

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Let's just say that[br]the source is oscillating at

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some Omega radians per second,

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that means that v_1, v_2[br]and v_3 will also be

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oscillating at that same frequency,

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but they'll have different amplitudes[br]and different phases.

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Or more specifically then,

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let's write it as v sub m1,

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cosine of Omega t plus Theta 1,

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so the amplitude of v_1 is v sub m1,

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and it has a phase of Theta 1,

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then plus v sub m2 cosine[br]Omega t, it's the same Omega.

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Omega t plus Theta 2 plus finally

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v sub m3 cosine Omega t plus Theta sub 3,

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and the sum of those three[br]terms has to equal 0.

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Now, let's represent these in[br]terms of phasors, and by that,

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we mean then that the real part[br]of whether this v sub

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m cosine Omega t plus Theta 1 can be[br]thought of as the real part of v sub m1,

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e to the j Theta 1,

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e to the j Omega t,

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plus the second term here[br]can be thought of as being

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the real part of v sub m2,

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e to the j Theta 2, that's a j,

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Theta 2, e to the j Omega t,

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plus the real part of v sub m3,

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e to the j Theta 3,

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e to the j Omega t, must equal 0.

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Now, realizing that we're talking[br]about the real part of all of

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these and that this e to the j Omega t[br]is common to all of them,

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we can rewrite this then[br]as the real part of,

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and also pointing out that that term[br]right there is just phasor v_1,

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and this right here is phasor v_2,

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and then of course right[br]there is phasor v_3.

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We can now rewrite this as the real part of

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phasor v_1 plus phasor v_2 plus phasor v_3,

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e to the j Omega t,

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and that must equal 0.

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Well, we know that e to[br]the j Omega t doesn't equal 0,

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therefore the sum of the phasors v_1,

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v_2, and v_3, must equal 0.

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Or specifically, phasor v_1 plus

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phasor v_2 plus phasor v_3 must equal 0.

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This is incredibly important.

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What that's saying is that up here,

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we have this circuit[br]operating in the time domain,

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we know that the sum of[br]those three voltages has to equal zero.

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What this demonstrates is[br]that not only must the sum

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of the time signals around[br]that closed loop equals zero,

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but the sum of the phasor[br]representations of

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each of these voltages[br]must also equal zero.

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In other words, and this[br]is the final thing,

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Kirchhoff's Voltage Laws apply or Law,

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singular, the Kirchhoff's[br]Voltage Law applies in

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this phasor domain in exactly the same way[br]that it does in the time domain.

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Now, let's take a look at[br]Kirchhoff's Current Law.

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We have a similar set of circumstances here

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where the sum of the currents i_1 of t

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plus i_2 of t plus i_3 of t, must equal 0.

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I'm not going to take[br]the time right now to do it,

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suffice it to say that[br]the same type of analysis

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that we just did for[br]Kirchhoff's Voltage Law,

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results in the sum of

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the phasor currents associated with

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the node must also equal zero.

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So once again, in terms of[br]impedances and phasors,

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the sum of the phasors or the sum of

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the currents represented in[br]the phasor form must equal zero also.